Experimental Fluid Mechanics R. J. Adrian . M. Gharib . w. Merzkirch D. Rockwell· J. H. Whitelaw Springer-Verlag Berlin Heidelberg GmbH Engineering springeronline.com ONLINE LlBRARY H.-E. Albrecht M. Borys N. Damaschke c. Tropea Laser Doppler and Phase Doppler Measurement Techniques Springer Prof. H.-E. Albrecht Bräsigweg 18 18069 Rostock Dr.-Ing. M. Borys Physikalisch-Techn. Bundesanstalt Fachlabor 1.41 Bundesallee 100 38116 Braunschweig ISBN 978-3-642-08739-4 Dipl.-Ing. N. Damaschke Technische Universität Darmstadt Strömungslehre und Aerodynamik Petersenstraße 30 64287 Darmstadt Prof. Dr. -lng. C. Tropea Technische Universität Darmstadt Strömungslehre und Aerodynamik Petersenstraße 30 64287 Darmstadt ISBN 978-3-662-05165-8 (eBook) DOI 10.1007/978-3-662-05165-8 Library of Congress Cataloging -in -Publication-Data Laser doppler and phase doppler measurement techniques / H.-E. Albrecht... [et al.l. p. cm.-- (Experimental fluid mechanics) Includes bibliographical references and index. 1. Fluid dynamic measurements. 2. Laser Doppler velocimeter. I. Albrecht, Heinz-Eberhard. H. Series. TA357.5.M43 L374 2002 620.1 '064--dc21 2002032404 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitations, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003. Softcover reprint ofthe hardcover Ist edition 2003 The use of general descriptive names, registered names trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: data delived by authors Cover design: design & production, Heidelberg Printed on acid free paper 6113020/M - 5 4 3 2 1 Series Editors PROF. R. J. ADRIAN University of Illinois at Urbana-Champaign Dept. of Theoretical and Applied Mechanics 216 Talbot Laboratory 104 South Wright Street Urbana, IL 61801 USA PROF. M. GHARIB California Institute of Technology Graduate Aeronautical Laboratories 1200 E. California Blvd. MC 205-45 Pasadena, CA 91125 USA PROF. DR. W. MERZKIRCH Universität Essen Lehrstuhl für Strömungslehre Schützenbahn 70 45141 Essen Germany PROF. DR. D. ROCKWELL Lehigh University Dept. of Mechanical Engineering and Mechanics Packard Lab. 19 Memorial Drive West Bethlehem, PA 18015-3085 USA PROF. J. H. WHITELAW Imperial College Dept. of Mechanical Engineering Exhibition Road London SW7 2BX UK Integrated Solutions in Laser Doppler Anemometry and Phase Doppler Anemometry • High accuracy LDA and PDA measurements • State-of-the-art software package and high quality electronics/optics • Ideal for 1D, 2D and 3D point measurement of velocity and turbulence distribution in both free flows and interna I flows • On-line measurement of the size, velocity and concentration of spherical particles, droplets and bubbles suspended in gaseous or liquid flows Read more about Dantee Dynamies' complete solutions for Laser Doppler Anemometry and Phase Doppler Anemometry on www.dantecdynamics.com Preface The laser Doppler and phase Doppler measuring techniques are both relatively young. The laser Doppler technique was first proposed in 1964 but came into widespread use only in the 1970s. The phase Doppler technique exhibited a similar development about 20 years later. Both techniques share a number of commonalties, not only in the hardware but also in the fact that both are most widely used in the fluid mechanics community. Therefore the technical overlap of the two techniques also extends to a strong 'user' overlap and this was one of the prime motivations for addressing both techniques in one volume. This book has arisen out of need. A comprehensive book about the phase Doppler measurement technique does not exist. Neither are the more recent developments of the laser Doppler technique weIl documented in a single volume. The student or user of these techniques presently relies on a combination of contributions from conference proceedings, journal publications and manufacturers' documentation. Furthermore, the fundamentals involved come from a wide variety of disciplines, e.g. electromagnetic theory, signal processing, etc., fields which are generally not so familiar within the fluid mechanies community. This book is an attempt to consolidate some of this information for the reader. The authors have intended this book to be both a reference book and an instructional book. This expresses itself in quite a varied degree of complexity in the different chapters. A reasonable attempt has been made to be thorough in the citation of literature to direct the reader to many details wh ich cannot be included within the scope of this book. At the same time, the reader will find some novel results in this book, especially on the subject of particle characterization. In preparing this book, the authors have drawn on the experience and advice of a large number of colleagues within their respective institutes who deserve special mention and thanks. In Rostock this includes Dr. H. Bech, Dr. W. Fuchs, Dr. W. Kröger and Prof. Dr. E. Müller. Prof. Dr. K. Bauckhage from the University of Bremen at the Institute for Material Science initiated a joint project from the Deutsche Forschungsgemeinschaft with Rostock, which stimulated new ideas about the computation oflight scattering on small particles in homogeneous and inhomogeneous fields. At the Physikalisch-Technische Bundesanstalt in Braunschweig, where M.B. worked at the Department of Fluid Mechanics until 2000, the collaboration with Prof. Dr. D. Dopheide, Dr. R. Kramer, Dr. H. Müller and Dr. V. Strunck was much appreciated. At the Lehrstuhl für Strömungsmechanik in Erlangen, where C.T. worked until 1997, interaction with Prof. Dr. G. Brenn, Dr. J. Domnick, Prof. Dr. F. Durst, Dr. A. Naqwi and T.-H. Xu is gratefullyacknowledged. In Darmstadt the authors had the pleasure of working Preface VIII closely with Dr. L Araneo, Dipl.-Ing. K. Heukelbach, Dr. H. Nobach and Dr. I.V. Roisman on various application aspects. The authors first came into contact with each other through a joint project from the Volkswagen Foundation (Contract 1/66 487) and then through subsequent grants from the Deutsche Forschungsgemeinschaft (Mu 1117/1, Tr 194/9). The authors gratefully acknowledge the finaneial support of these funding ageneies for enabling this initial collaboration and its continuation over the past years. Unavoidably there exist errors and omissions in this book and the authors take full responsibility for these. Readers who have suggestions for improvements are welcome to contact the authors (tropea@springer.de). Rostock / Darmstadt / Braunschweig 2002 H.-E. Albrecht M. Borys N. Damaschke C. Tropea Contents 1 Introduction ........................................................................................................... 1 1.1 Historical Perspective .................................................................................. 1 1.2 Use ofthe Book ............................................................................................ 3 PART I: FUNDAMENTALS 2 Basic Measurement Principles ............................................................................. 9 2.1 Laser Doppler Technique .......................................................................... 12 2.2 Phase Doppler Technique ......................................................................... 23 2.3 Time-Shift Technique ................................................................................ 25 3 Fundamentals ofLight Propagation and Optics .............................................. 27 3.1 Electromagnetic W aves ............................................................................. 27 3.1.1. Description of Electromagnetic Waves ...................................... 27 3.1.2. Polarization ................................................................................... 33 3.1.3. Boundary Conditions and Fresnel Coefficients ........................ .35 3.1.4. Laser Beams ................................................................................. .37 3.1.5. Optical Mixing of Electromagnetic Waves................................ .44 3.1.6. The Doppler Effect ...................................................................... .45 3.2 Optical Components ................................................................................. .47 Matrix Transformation for Imaging .......................................... .47 3.2.1. 3.2.2. Propagation ofLaser Beams Through Lenses and Apertures .. 53 3.2.3. Optical Gratings and Bragg Cells ................................................ 56 3.2.4. Optical Fibers ................................................................................ 65 3.2.5. Photodetectors .............................................................................. 70 4 Light Scattering from Small Particles ................................................................ 79 4.1 Scattering of a Plane Wave ........................................................................ 81 4.1.1. Description using Geometrical Optics (GO) .............................. 85 4.1.2. Description using Lorenz-Mie Theory and Debye Series ......... 96 4.1.3. Scattering Characteristics for a Plane Wave ............................ 100 4.2 Scattering of an Inhomogeneous Field .................................................. 127 4.2.1. Extension to the Method of Geometrical Optics (EGO) ......... 128 4.2.2. Description using Fourier Lorenz-Mie Theory (FLMT) ......... 134 4.2.3. Scattering Characteristics of an Inhomogeneous Field .......... 146 4.3 Characteristic Quantities ofLight Scattered by Sm all Particles .......... 162 X Contents PART 11: MEASUREMENT PRINCIPLES 5 Signal Generation in Laser Doppler and Phase Doppler Systems ................ 169 5.1 The Signal From an Arbitrarily Positioned Detector ........................... 169 5.1.1. Fundamental Relations .............................................................. 172 5.1.2. Signals from very Sm all Particles ............................................. 177 5.1.3. Signals from Large Particles ...................................................... 199 5.1.4. Visibilityofthe Signal ................................................................ 214 5.1.5. Shift Frequency Influence ......................................................... 219 5.1.6. Measurement and Detection Volumes..................................... 221 5.1.7. Statistical Time Series ofParticle Signals ................................. 227 5.2 Laser Doppler Technique ........................................................................ 231 5.2.1. Dual-Beam Configuration ......................................................... 232 5.2.2. Reference-Beam Configuration ................................................ 233 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique .......... 244 5.3.1. Determination ofIncident and Glare Point Positions ............ 247 5.3.2. Phase Doppler Technique ......................................................... 250 5.3.3. Reference Phase Doppler Technique ....................................... 254 5.3.4. Time-Shift Technique ................................................................ 259 5.4 Refractive Index Determination ............................................................. 266 5.5 Moire Models ........................................................................................... 267 6 Signal Detection, Processing and Validation ................................................. 273 6.1 ReviewofSome Fundamentals .............................................................. 275 6.1.1. Discrete Fourier Transform (DFT) ........................................... 276 6.1.2. Correlation Function ................................................................. 281 6.1.3. Hilbert Transform ...................................................................... 283 6.1.4. Signal Noise ................................................................................ 287 6.1.5. Cramer-Rao Lower Bound (CRLB) .......................................... 290 6.2 Signal Detection ....................................................................................... 300 6.3 Estimation ofthe Doppler Frequency ................................................... 305 6.3.1. Spectral Analysis ........................................................................ 307 6.3.2. Correlation Techniques ............................................................. 311 6.3.3. Period Timing Devices .............................................................. 313 Quadrature Demodulation ........................................................ 315 6.3.4. 6.4 Determination of Signal Phase ............................................................... 317 6.4.1. Cross-Spectral Density .............................................................. 317 6.4.2. Covariance Methods .................................................................. 321 6.4.3. Quadrature Methods .................................................................. 322 6.5 Model-Based Signal Processing .............................................................. 323 6.5.1. Fundamentals ............................................................................. 323 6.5.2. Example Applications ................................................................ 324 Contents XI 7 Laser Doppler Systems ..................................................................................... .33 7 7.1 Input Parameters from the Flow and Test Rig ..................................... .338 7.1.1. Description of the Flow Field ................................................... .338 7.1.2. Necessary Spatial and Temporal Resolution .......................... .351 7.1.3. Flow and Flow-Rig Parameters ................................................ .358 7.2 Components and Layout of the Transmitting Optics ......................... .363 7.2.1. Collimators ................................................................................ .363 7.2.2. Beamsplitters and Polarizers ..................................................... 369 7.2.3. Methods for Achieving Directional Sensitivity ....................... 371 7.2.4. Generation ofthe Measurement Volume ............................... .377 7.3 Layout ofReceiving Optics .................................................................... .383 7.4 System Description .................................................................................. 389 7.4.1. One-Velo city Component Systems .......................................... .389 7.4.2. Two-Velocity Component Systems ......................................... .392 7.4.3. Three-Velo city Component Systems ....................................... .396 7.4.4. Multi-Point Systems .................................................................. .401 7.5 Laser Transit Velocimetry ..................................................................... .405 8 Phase Doppler Systems .................................................................................... .409 8.1 Selection of the Optical Configuration ................................................. .411 8.2 Single-Point Phase Doppler Systems ................................................... .417 8.2.1. Three-detector, Standard Phase Doppler System .................. .417 8.2.2. Planar Phase Doppler System .................................................. .425 8.2.3. Dual-Mode Phase Doppler ....................................................... .430 8.2.4. Dual-Burst Technique ............................................................... .436 8.2.5. Extended Phase Doppler Technique ....................................... .446 8.2.6. Reference Phase Doppler Technique ....................................... .449 8.3 Further Design Considerations for Phase Doppler Systems ............. ..454 8.3.1. Influence ofthe Gaussian Beam ................................................ 454 8.3.2. Slit Effect ..................................................................................... 466 8.3.3. Non-Spherical and Inhomogeneous Particles ........................ .467 8.4 Multi-Dimensional Sizing Techniques ................................................. .470 8.4.1. Interferometric Particle Imaging (IP!) ..................................... 470 8.4.2. Global Phase Doppler (GPD) Technique ................................ .478 8.4.3. Concentration Limits ................................................................ .481 9 Further Partic1e Sizing Methods Based on the Laser Doppler Technique .. .491 9.1 Techniques Based on Signal Amplitude ................................................ 491 9.1.1. Cross-sectional Area Difference Technique ........................... .491 9.1.2. Combined Laser Doppler and White Light Sizer .................... 500 9.2 Time-Shift Technique .............................................................................. 501 9.2.1. Time-Shift Technique in Forward Scatter ............................... 504 9.2.2. Time-Shift Technique in Backscatter ...................................... .506 9.3 Rainbow Refractometry .......................................................................... 517 9.4 Shadow Doppler Technique .................................................................... 523 XII Contents PART III: DATA PROCESSING 10 Fundamentals ofData Processing ................................................................... 529 10.1 Statistical Principles ................................................................................ 529 10.2 Stationary Random Processes ................................................................ 533 10.3 Estimator Expectation and Variance ..................................................... 535 10.3.1. Estimators for the Mean ............................................................ 535 10.3.2. Estimators for Higher Order Correlations ............................... 539 10.3.3. Estimators for Transient Processes .......................................... 542 10.4 Propagation ofErrors .............................................................................. 543 11 Processing of Laser Doppler Data .................................................................... 545 11.1 Estimation of Moments ........................................................................... 547 11.2 Estimation of Turbulent Velo city Spectra............................................. 552 11.2.1. The Slotting Technique ............................................................. 554 11.2.2. Reconstruction with FFT ........................................................... 558 11.2.3. Post-Processing Steps ................................................................ 561 11.3 Correlation Estimates from Multi-Point Systems ................................ 563 11.4 Measurements in Transient Processes .................................................. 566 11.4.1. Effect ofWindow Size on Phase and Ensemble Statistics ...... 567 11.4.2. Energy Partitioning in Transient Flows ................................... 568 11.5 Data Simulation ....................................................................................... 569 12 Processing ofPhase Doppler Data ................................................................... 573 12.1 Validation Procedures ............................................................................. 573 12.1.1. SNR Validation ........................................................................... 573 12.1.2. Phase Difference Validation ...................................................... 574 12.1.3. SphericityValidation ................................................................. 574 12.1.4. Amplitude Validation ................................................................ 574 12.1.5. Transit Time Validation ............................................................ 575 12.2 Particle Statistics ...................................................................................... 576 12.2.1. Flux Density Vectors and Concentration ................................ 576 12.2.2. Distribution ofParticles ............................................................ 579 12.2.3. Geometry of the Detection Volume ......................................... 582 12.2.4. Estimation ofthe Number ofParticles ..................................... 590 12.2.5. Summary and Examples ............................................................ 591 l2.3 Post-Processing of Phase Doppler Data ................................................ 595 12.3.1. Particle Size Distributions ......................................................... 595 12.3.2. Mean Diameters ......................................................................... 598 l2.3.3. Non-Spherical and Inhomogeneous Particles ......................... 599 Contents XIII PART IV: ApPLICATION ISSUES 13 Choice ofParticles and Partide Generation ................................................... 605 13.1 Particle Motion in Flows ......................................................................... 606 13.2 Particle Generation .................................................................................. 613 13.2.1. Droplet Generation .................................................................... 614 13.2.2. Solid Particle Generation ........................................................... 619 13.3 Introducing Particles into the Flow ....................................................... 621 13.3.1. Liquid Flows ................................................................................ 622 13.3.2. Gas Flows ..................................................................................... 622 13.3.3. Two-Phase Flows ........................................................................ 623 13.3.4. Natural Seeding .......................................................................... 624 14 System Design Considerations ........................................................................ 627 14.1 System Design Guidelines ....................................................................... 627 14.1.1. Laser Doppler Systems ............................................................... 628 14.1.2. Phase Doppler Systems .............................................................. 635 14.1.3. Alignment and Adjustment... .................................................... 638 14.2 System Design Examples ......................................................................... 642 14.2.1. Velo city Measurements in a Narrow Channel Flow ............... 642 14.2.2. Drop Size Measurements in a Diesel Injector Spray ............... 647 14.3 Refractive Index Matching ...................................................................... 655 14.3.1. Matching with Flow Containment... ......................................... 655 14.3.2. Matching for Variable Density.................................................. 660 Appendix ................................................................................................................... 661 List of Symbols and Acronyms ........................................................................ 662 Derivation of Equations Describing a Laser Beam ........................................ 681 Internal and Near Field Solution ...................................................................... 686 Bibliography ............................................................................................................. 689 References .......................................................................................................... 690 Books (or parts thereof) on the Laser or Phase Doppler Techniques .......... 718 Periodicals Dealing with the Laser or Phase Doppler Techniques ............... 719 Conference Series devoted to Laser or Phase Doppler Techniques ............. 720 Index ......................................................................................................................... 723 2 lIntrodl,Iction very early stage several suggestions were made about how to obtain more information about the scattering centers themselves, especially their size. Initially the amplitude (or the modulation depth) of the scattered intensity was considered. However, amplitude-based techniques have a number of drawbacks, not the least ofwhich is the need for calibration, which, even today, have hindered their widespread use. In 1975 Durst and Zare (1975) first published the idea of measuring partiele size using phase measurements. They wrote: "Double element photo detectors with fixed spacing detect different signal amplitude differences for different fringe spacing and, hence, can be used to record a signal sensitive to the size of spherical particles. The authors used a double element photodiode with elements spaced 2 mm apart to obtain information on the sphere diameter through phase measurements between the two detected signals." They related the fringe spacing in space to the radius of curvature of the partiele; however, they proposed measuring the fringe spacing through the amplitude difference. Although they recognized the phase difference between the two signals, apparently they did not realize it could be measured reliably. They came to the conelusion: "However, it is apparent that size measurements of this kind require the distance between the photodetectors to be matched to the fringe distance and, hence, to the particle size to be measured. This requirement is a disadvantage for practical measurements of size distribution." In the final-year thesis of Flögel (1981) entitled "Investigation of partiele velocity and partiele size using a laser Doppler anemometer", equations relating particle diameter to the phase difference between signals detected by two photodetectors were given and a system was tested by measuring drop size distributions in a spray. In 1984 three groups presented phase Doppler systems, Bauckhage and Flögel (1984) (also documented in the Ph.D. thesis of Flögel 1987), Saffmann, Buchhave and Tanger (1984) and Bachalo and Houser (1984). The basic physical ideas were thus available and a rapid period of development of the phase Doppler technique followed. An account of this initial phase of the instrument development was assembled by Hirleman (1996). The phase Doppler technique uses a single scattering mode, usually reflection or first-order refraction, to determine partiele size. Whereas the signals in the reflective mode are only sensitive to size and detector position, in the refractive mode the index of refraction is also an influencing parameter. In recent years, several instruments have been demonstrated which, through a combination of reflected and refracted light, are capable of determining also the refractive index of the particle. These developments are very much on-going. A second path of development is the measurement of non-spherical partieles, whereby many suggestions to date can no longer be strictly called phase Doppler instruments. Some of these topics will be addressed in chapters 8 and 9. A third measurement technique has been ineluded in this book, the 'timeshift' or 'volume-displacement' technique, which was first introduced by AIbrecht et al. (1993) and is also used for partiele sizing. This technique is still in its infancy and has yet to be realized as a commercial instrument. On the other hand, in combination with a phase Doppler system, the time-shift technique of- 1.2 Use of the Book 3 fers the potential for particle characterization beyond just the size. This technique is possible to implement only when shaped beams are used for illuminating the measurement volume; however, this is virtually always the case with laser Doppler and phase Doppler systems. The basic principles of this technique are described in detail in this book and corresponding guidelines for system design are given. In Fig. 1.1 the three techniques discussed in this book and their various implementations are compared with other laser measurement techniques for single and multi-phase flows. The techniques have been arranged according to the number of velo city components they measure (u, v, w) and the dimensions in which the flow field is sampled (x,y,z,t). The possibility of measuring size is also no ted. Time { I DGV - Doppler global velocimetry FRS - Filtered Rayleigh scattering GPD - Global phase Doppler IPI - Interferometric particle imaging LDV - Laser Doppler velocimetry LFT - Laser Flow Tagging LTV - Laser transit velocimetry PD - Phase Doppler PIV - Planar Doppler velocimetry PIV - Particle image velocimetry PTV - Particle tracking velocimetry Fig. 1.1. Overview oflaser measurement techniques for single and multi-phase flows 1.2 Use of the Book There are many common elements between the laser Doppler and phase Doppler techniques, not only in the optical system but also in the signal processing and data processing. This is reflected in the organization of this book, as illustrated in Fig. 1.2. Part I covers the fundamentals of light propagation and light scattering in detail and is essential for those readers concerned with the design and layout of laser Doppler and phase Doppler instruments. 4 lIntroduction Phase Doppler Technique (5.3) Fig. 1.2. Organization ofbook chapters 1.2 Use ofthe Book 5 Part II deals with specific measurement principles, more fundamentally in chapter 5 and more application orientated in chapters 7 and 8 for the laser Doppler and phase Doppler techniques respectively. The overlapping topic of signal processing is covered in chapter 6, with an introductory section on fundamentals. A number of novel techniques for particle sizing are introduced in both chapters 8 and 9. Part III deals with data processing, with some fundamentals covered in chapter 10 and more specific issues in chapters 11 and 12. Part IV discusses tracer particles in chapter 13 and specific design considerations in chapter 14, albeit only a small selection of possible applications can be considered. More tedious derivations, primarily from chapters 4 and 5, have been relegated to the Appendices. The Bibliography has been broken down into books, periodicals, and archival papers, arranged in alphabetical order. Already from this brief overview it is apparent that this book draws on many different disciplines: physics, electromagnetic theory, optics, electronics, signal processing and data processing theory, fluid mechanics and two-phase flows. Each discipline and community has developed its own nomenclature and conventions and it is not surprising that if these were all retained, a great deal of repetition of symbols would occur. Nevertheless, we have chosen to do exactly this, so that each reader will hopefully recognize quantities in their accustomed form. As an aid, we have added a comprehensive list of symbols in Appendix I. PART I FUNDAMENTALS 10 2 Basic Measurement Principles a Signal in time domain: v·r \ Parlide Lighl source AmplilUde ;(/) / i(t) Receiver b ßs , Signal in time domain Amplilude Spalial graling Signal in frequency domain i(t) '---+T----''--T-_--.I(fl c LL AmplilUde f, f Pha e .u) lL..-_Il-'_ a f d Signal in time domain Amplitude e i(t) Signal in correlation domain Amplitude ;(')~ Ilt Fig. 2.1a-e. Flow measurement techniques using an optically fixed measurement volume In Fig. 2.1b,c a spatial grating has been introduced either on the transmitting side or on the receiving side of the system. The former case is designated as a real substructure, the latter as a virtual substructure, since it is only present from the point of view of the receiver and normally changes with particle diameter or with the position of the detector. In principle, any type of grating can 2 Basic Measurement Principles 11 be used, for instance, a multiple-line grating, as shown in Fig. 2.1b,c, or a twoline grating, which will result in a 'time-of-flight' measurement, as shown in Fig. 2.1d,e. However, the latter is disadvantageous at higher flow turbulence levels, since the particle trajectory may be such that only one of the two grating lines is crossed, resulting in a missed signal and thus lower data rates and biased averages. For exactly this reason, the spatial extent of the grating is often kept to a minimum in practical systems. A full grating on the other hand resolves the velo city over the entire measurement volume and indeed, using uniformly spaced lines, the frequency f of the resulting signal pulses is directly proportional to the velocity normal to the grating lines LIx Vx =y= f LIx (2.2) where LIx is the line spacing and T is the period between pulses. The frequency can be determined from the signal either in time or frequency domain. Information relating to the particle radius is contained in the amplitude, in the modulation (visibility), in the phase and in the arrival time of the particle signals. Amplitude and visibility techniques (Umhauer 1996, Gebhardt 1989) require one detector and must be calibrated. Phase differences (phase Doppler technique) or arrival time differences (time-shift technique) are measured using at least two detectors and require no calibration. Using a CCD-line array, a CCD matrix (Christophori and Michel 1997, Michel et al. 1997) or a matrix of optical fibers (Petrak and Hoffmann 1985, Morikawa et al. 1986) as a receiver, the grating becomes essentially apart of the receiver and a virtual measurement volume is obtained. In this case, incoherent light is adequate. However, the formation of real measurement volumes of sufficient precision is not possible using incoherent light. For this reason monochromatic, coherent laser light is used. This leads to the well-known laser Doppler and phase Doppler optical configurations. In certain laser Doppler configurations (reference-beam mode), an interference pattern is formed on the detector surface through the superposition of a scattered light field and a reference wave. This interference pattern can be interpreted as a virtual measurement volume (Yeh and Cummins 1964). For the 'time-of-flight' arrangements shown in Fig. 2.1d,e, use of either incoherent or coherent light is possible in principle; however, practically the necessary spatial resolution is only possible with laser light. These systems are thus known as 'laser two focus' (L2F) or 'laser transit velocimeter' (LTV) systems (SchodI1975, 1977, Schodl and Förster 1988). This case is actually a special case of the continuous line grating, where the distance Llsx is now defined by only two limiting bounds. Although the data validation rate with LTV systems decreases dramatically with increasing flow turbulence level, by rotating the optical system about its optical axis to several different orientations, some statistical information regarding the turbulence field is obtainable. Nevertheless, the use of 12 2 Basic Measurement Principles such systems is generally restricted to weH directed flows, e.g. as found in turbomachinery blading. The foHowing discussion concentrates on the most commonly used methods from those listed above for defining the measurement volume, Fig. 2.1b, the laser Doppler and phase Doppler technique as weH as Fig. 2.1d, the laser transit velocimeter and the pulse delayvelocimeter. 2.1 Laser Doppler Technique The laser Doppler technique (Vasilenko et al. 1975, Watrasiewicz and Rudd 1976, Durst et al. 1976, Durrani and Greated 1977, Rinkevicjus 1978, Drain 1980, Dubiscev and Rinkevicjus 1982, Wiedemann 1984, Albrecht 1986) uses monochromatic laser light as a light source. The interference of two beams crossing in the measurement volume or the interference of two scattering waves on the detector creates a fringe pattern. The velo city information for moving scattering centers is contained in the scattered field due to the Doppler effect. Strictly speaking, the laser Doppler technique is an indirect measuring technique, since it measures the velo city of inhomogeneities in the flow, typically tracer particles. This represents the flow velo city only if no appreciable slip velocity is present. Otherwise the slip velocity must also be determined. The basic principle of the laser Doppler technique is illustrated in Fig. 2.2. The Doppler effect (section 3.1.6) is invoked twice, once when the incident laser light of the transmitter system, characterized by the wavelength Ab and frequency Ib (subscript b for beam), impinges on the moving target, and once when light with a frequency I p (subscript p for particle) is scattered from the moving target particle and received by a stationary detector with the frequency Ir (subscript r for receiver) (Goldstein and Kreid 1967). eb·v p 1 Ir=Ip 1--- e ·v 1-~ =Ib c "" Ib + Ib e \ 1-~ (2.3) c vp.(epr-e b) C vp.(epr-e b) Ib +----'-----------'-Ab where cis the speed oflight in the medium surrounding the particle. The second term in the second line of Eq. (2.3) contains the Doppler shift of the incident wave frequency. The difference of the normal vectors appears when the direction of propagation of the incident and scattered wave differs. The Doppler shift is directly proportional to this difference and to the velocity of the particle. For typical flow systems the Doppler shift is of the order 1. .. 100 MHz, which compared to the frequency oflaser light of approximately 10 14 Hz is very small and thus virtually impossible to resolve directly. One exception is a direct detection with the help of an interferometer (Paul and Jackson 1971, Jackson 2.1 Laser Doppler Technique 13 Fig. 2.2. Defining geometry for a pplying the Doppler effect in the laser Doppler technique and Paul 1971, Smeets and George 1981) or through the use of frequency dependent absorption cells, the latter leading to the Doppler global velocimeter (DGV) (Komine 1990, Komine et al.1991, Meyers 1995), sometimes called planar Doppler velocimetry (PDV) (Mosedale et al. 2000). However, conventional optical arrangements work with two scattered waves, each exhibiting a different Doppler shift. Alternatively one laser beam can act as a reference beam and be mixed with a scattered wave. The two waves are mixed on the detector surface in a process known as optical heterodyning, yielding the beat frequency, which typically lies in a much more manageable frequency range for signal processing. There are several alternatives to practically realize such systems using one ineident beam, two of which are shown in Fig. 2.3. In Fig. 2.3a a dual-scatteredwave system is shown and in Fig. 2.3b a one-reference-beam, one-scattered-wave system, both of which have been successfully demonstrated (Yeh and Cummins 1964, Forman et al. 1965, Goldstein and Kreid 1967). In both cases the difference (beat) frequency fD is obtained through the optical mixing of waves with frequencies fl and f2 on the detector. For the onebeam configurations these frequencies are given as a b ..I. x I. x Bearn spli tter ,. ~ Lascr bearn A.,f. Z e~ , / " emirellecting mirror '" e, e/" 2 Lascrbcarn A.,!. , /~ . Recc"'cr /' l\lirror ", / " Receh'cr Semirellecting mirror Fig. 2.3a,b. Optical configuration of single incident beam system. a Dual-beam scattering configuration, b Reference-beam configuration 14 2 Basic Measurement Principles • Dual-scattered-wave configuration (Fig. 2.3a): (2.4) (2.5) • Reference-beam configuration (Fig. 2.3b): (2.6) (2.7) The measurement volume is defined in both cases using an aperture on the detector, thus a virtual measurement volume is realized. These systems are not commonly used, mainly because the small aperture required to limit the measurement volume also leads to a highly reduced intensity level of the detected light and the difference frequency is dependent on the receiver position. The more widely used optical configuration is based on two incident waves, as illustrated in Fig. 2.4. Figure 2.4a shows the so-called dual-beam configuration, in which areal measurement volume is formed at the intersection of the two incident waves and the scattered waves are detected with a single detector (vom Stein and Pfeifer 1969, Rudd 1969). • Dual-beam configuration (Fig. 2.4a): (2.8) a La er beam A"f, b x Laserbeam }..,,f, ~ ----------.~ ~~--~~--I~Z x Receiver - z \ Receiver Laser bea m I.., ,f, Laser bea m A, ,f, Fig. 2.4a,b. Optical configuration for dual-incident-beam systems. a Dual-beam configuration, b Reference-beam configuration 2.1 Laser Doppler Technique 15 Figure 2Ab illustrates the reference-beam configuration, in which case the detector is positioned directly in the path of one of the beams (e pr = e 2 ). Typically the incident reference beam is much lower in intensity than the incident scattering beam (5:95). This configuration is seldom used; however, it does show some advantages for measurements in highly absorbing media. • Reference-beam configuration (Fig. 2.4b): (2.10) (2.11) Noteworthy is the fact that the difference frequency is independent of the receiver position for the dual-beam configurations in Fig. 2.4. If the intersection angle of the two beams is denoted by 8, then the difference frequency on the detector is given by I I _ 2 sin <:% - 2sin<:% fD- - - v p cosa----vp.L Ab (2.12) Ab as clarified also in Fig. 2.5. The flow direction a is measured with respect to the perpendicular of the beam bisector. Thus the frequency difference is linearly proportional to the velocity component in the x direction, denoted by v p.L or v px' For very small tracer particles, the very illustrative fringe model can be used to explain the measurement principle of the laser Doppler technique. This model is based on the spatial energy density in the measurement volume, as de- n = e, - Cl = 2sin o/, e, Fig. 2.5. Vector relations relevant to determining the Doppler frequency 16 2 Basic Measurement Principles scribed by the following. A linearly polarized homogeneous electromagnetic wave can be described with the electric field vector (see section 3.1.1.1) (2.13) or in complex number notation (2.14) where Wb is the angular frequency, k b is the wave vector in the wave propagation direction of the laser light having a wavenumber of kb = 2n I Ab' e E is a unit vector (orientation of polarization) and Bo the amplitude of the electric field, r is a vector defining an arbitrary point in space, where the electric field strength is to be determined, and rpb is the phase of the electromagnetic wave at the origin and for time t = o. If the two incident beams are of equal intensity, with a polarization perpendicular to the x-z plane in which the beams symmetrically lie, then the fields can be described by (see Fig. 2.6a,b) Ql = B o exp(j [wbt- kb(xsin %+ zcos%)+ rplJ) (2.15) Q2 = B o exp(j [wbt- kb(-xsin %+ zcos%)+ rp2]) (2.16) The electric field in the intersection volume of the laser beams crossing with intersection angle of is given by the superposition (see Fig. 2.6c) e (2.17) The energy density, see Eq. (3.27), of the electromagnetic field in the measurement volume is given by w= t:B 2 = 4t:B o2COS 2(k bxsin 19/2/ - 2(w t-k zcos19/+ rpj +rp2) rpl-rp2)COS 2 b b /2 2 (2.18) This energy density in the measurement volume can be interpreted as a wave propagating in the z direction with an amplitude modulated in the x direction of 4t: B 2cos 2 (k o b x sin 19/ _ rpj -2 rp2 ) /2 (2.19) The intensityl of the electromagnetic wave is obtained by time averaging over one period I Hecht (1998) p.49: "In the past physicists generally used the word intensity to mean the flow of energy per unit area per unit time. By international, if not universal, agreement, that term is slowly being replaced in optics by the word irradiance". Both terms come into use in this book but theyalways refer to the same quantity. 2.1 Laser Doppler Technique 17 1T with (JU») =- f!(t) dt (2.20) Ta In complex form, the averaging reduces to a multiplication with the conjugate complex value ~*), yielding for the intensity when both phases are equal m· rpj = rp2 = rp 1= cC(E 2) = ccgg* = c(w) = 2ccE~ cos2(kbxsin~) (2.21) Electric field strengths of incident waves E, =E. cos(ro.t-k.(-xsin 0/, + 2 cos o/,») b ,------~~x Electric field strength in the intersection area ofthe incident waves Intensity proportional to the temporal mean of the electric field strength 1= EC E~ cos' (k b x sin 0/,) E=E, + E, - p. .~ - ,. .~~ 2 Fig. 2.6a-d. Generation of the interference structure of two homogeneous waves. a,b Electric field strength of incident waves, c Superposition of electric fields, d Intensity 18 2 Basic Measurement Principles I=t:cE~ [ 1+eos( 2sin'7i J] 21tT,X (2.22) The spatial dependenee of the intensity in the interseetion volume ean be interpreted as an interferenee field with fringes parallel to the y-z plane (Fig. 2.6d, see also seetion 5.1, Fig. 5.14). The fringe spacing is given by the argument of the eosine function in Eq. (2.22) above: L1x=~ (2.23) 2 sin '7i If the position variable x is now replaeed by x = v pJt, Eq. (2.22) beeomes (2.24) whieh offers a very physieal interpretation of Eq. (2.12). A small particle d p «L1x passing through the interferenee pattern effeetively sampies the loeal intensity, whieh is eonstant over its diameter. A particle of diameter d p pereeives a mean power of 1t 2 d p «Ab P"",IAp ""'I-d, 4 p (2.25) and seatters this power in all spaee. The scattered wave is modulated in its amplitude and has the carrier frequency of the laser beam. Therefore an eleetrieal signal i(t) is obtained from the photodeteetor, whose amplitude is modulated with the differenee frequeney fD. The frequeney fD is ealled the Doppler frequeney (Eq. (2.12», but refers to the differenee between the two Doppler shifted waves. (2.26) (2.27) The velocity eomponent perpendieular to interferenee fringes is then inversely proportional to the period of the fringe erossing TD , L1x VP.L=y (2.28) D For different phases, (jJj and (jJ2 in Eqs. (2.15) and (2.16), the interferenee pattern is only shifted in the x direetion or in time for the signal obtained from a moving particle (2.29) 2.1 Laser Doppler Technique 19 Note that this 'interference' or fringe model of the laser Doppler technique is strictly only valid for very small particles fulfilling the condition d p «Ab' since only then can the amplitude and phase, or the intensity of the field be considered constant over the partide diameter. The partide interacts with the field and generates a scattered field of strength proportional to the sum of the individual field strengths, as expressed by Eq. (2.17). The energy flux c(w) is equal to the intensity at the photodetector. The photodetector averages the power density tempo rally due to its finite response time (section 3.2.5) and integrates the intensity spatially over its photosensitive surface. The electric signal obtained from the photodetector is directly proportional to the spatial energy density in the intersection volume. The small partide effectively sam pIes the local intensity of the interference pattern in the intersection volume. For partides larger than the wavelength oflight this model fails. Both the amplitude and the phase of the incident waves vary across the diameter of the partide. Effectively the partide images certain parts of the incident waves onto the photodetector, as interpreted in terms of geometrical optics in Fig. 2.7. Thus only certain areas of the partide surface are involved in defining signal properties. The position and size of the receiving aperture define the position and size of these interaction areas. The area of the first interaction with the field is known as the "incident point" and the source area of the scattered wave is called "glare point" (Fig. 2.7)1. The size of the incident areas I "points" and glare areas I "points" is proportional to the size of the detection aperture. Figure 2.8 pictures the reflective and refractive glare points on the surface of a water droplet for detection at 30 deg. The scattered waves detected by the receiver each have an amplitude, which depends on the position of their glare points. Each is proportional, through the scattering functions ~" ~2 (chapter 4), to the field strength at the incident points. ~o exp(j Cj)" + 1jI,,) lio :r cxp(j Cj)"+ IjI ,, +1jI'2') ~ Fig. 2.7. Signalorigin for large particles 1 Correctly speaking the names incident point, glare point and interaction point can only be used for a point-like receiver. For receivers with a finite size aperture, these points become areas. However, according to convention, the term point will be used for both situations throughout this book. 20 a 2 Basic Measurement Principles With background illumination b Fi rst-order refraction [rom background ill umination c Without background illumination Particle outline added for clarity d I neident laser Glare I ineident point retlection Background illumination (neident point refraction \ Glare point refraction Fig. 2.8a-d. Glare points on a water droplet in air IJ, = 30 deg. a With background illumination, b Without background illumination and with the shape of the partide indicated, c Schematic configuration of camera and light sources, d Light paths and generation of incident points of reflection and refraction Depending on the shape of the particle and the different propagation directions of the incident beams, phases at the incident points are different for each beam (lPlr' lP2,)' and according to the particle material and different locations of the glare points, an additional phase shift for each wave can result, (VIIr' Vl2r). The field strength on the detector arising from each incident beam is then (2.30) 2.1 Laser Doppler Technique 21 (2.31) For very small particles the glare points merge, and shape or material induced phase shifts vanish «(jJ2r - (jJlr = (jJ2 - (jJI' 1f/1r -If/ 2r = 0), yielding the fringe model. In contrast, for larger particles the scattered waves interfere on the surface of the photodetector as shown in Fig. 2.7. Thus, the measurement volume is virtual and only exists for the photodetector. In a manner similar to Eq. (2.29), the signal from the detector is given by i(t) - ce E~ [ 1+ cos( 2rcfvt- (jJlr + (jJ2r -If/lr + If/2r)] (2.32) which is still modulated by the Doppler frequency. The main difference to the small particle result is the added phase shift differences 1f/2r -If/lr and (jJ2r - (jJlr =j:. (jJ2 - (jJI· In comparison with Eq. (2.29) the fringe pattern is shifted in phase. However, this phase shift is of no consequence because the exact phases of the incident beams at the time when the particle is at position x p = 0 are not known anyway. When the particle traverses across the measurement volume the interference pattern moves across the detector surface. Equations (2.26) and (2.29) remain valid. The optical arrangement discussed above yields the velo city component normal to the interference fringes; however, its sense is no longer contained in the received signal. The two particles shown in Fig. 2.9a, moving with equal but opposite velocities through the measurement volume, will generate the same electrical signal on the detector. Directional information is recovered when incident laser beams of different wavelengths are used. A wavelength shift of one or both of the laser beams can be achieved using acousto-optic modulators (Bass 1995, Vol. 11 Chapt.12), for example Bragg cells (Chang 1976) (see section 3.2.3.2). If an acousto-optic modulator is mounted in the path of beam 1, the frequency of the beam can be shifted by an amount f'h' yielding fl = fb + f'h or fl = fb - f'h (2.33) Since the frequency fis the derivative ofthe phase with time f=_1 d(jJ 2rc dt (2.34) for a stationary wave the frequency shift can be expressed as a linear change of phase with time (2.35) In the fringe model this corresponds to a movement of the fringes in the -x or +x direction with a constant velo city. After the optical mixing of the two scattered waves on the detector surface, the modulation for the configuration in Fig. 2.4a becomes (2.36) 22 2 Basic Measurement Principles Thesignal frequency exhibits an offset equal to the shift frequency. A stationary particle will result in a signal with a modulation of f'h. A particle moving with the fringes yields a lower frequency and movement against the fringes, a higher frequency (Fig. 2.9). Strictly speaking the shift frequency changes the wavelength of the light and thus, the light scattering properties of the particle. However this change relative to the frequency oflight is so small (1:10 12 ) that it can be neglected. A conventionallaser Doppler optical arrangement is summarized in Fig. 2.10. The laser beam is split into two beams of equal intensity and polarization using a beam splitter and brought to intersection with a lens. A collimator is used for adjusting the beam properties in the measurement volume and the Bragg cell provides a frequency shift used for the directional sensitivity. The Doppler frequency is determined using a signal processor and the data analysis for computing flow properties is performed in a computer. The actual realization of these components in a measurement system can be extremely varied, involving for instance optical fiber transmission between the laser and the focussing optical components. Several such systems operating at different wavelengths can be integrated into a single optical arrangement to yield several flow velo city components simultaneously. Further illustrations of practical systems are given in chapter 7. a Fig.2.9a,b. Explanation of the frequency shift technique for directional sensitivity. a Without frequency shift, b With frequency shift Laser Trdn mi l1ing len Collima lor Rccci ver probe C mpcn 31ion gJass or bragg cell Fig. 2.10. Dual-beam laser Doppler anemometer 1\ volume 2.2 Phase Doppler Technique 23 The laser Doppler technique sampies the flow velo city at discrete times corresponding to the passage of a partide through the interseetion volume. The velocity sampled at these times can be considered as a primary measurement quantity. The derivation of flow parameters or secondary measurement quantities, such as mean flow velo city, turbulence level or turbulence spectra, requires further data processing. The details of this data processing and in particular the means to achieve a given accuracy of the secondary quantities are the subject matter of chapters 10 and 11. 2.2 Phase Doppler Technique The identification of spatial structures within the measurement volume will rely either on time delays or phase differences and this necessitates detectors at two or more positions in space. For homogeneous spherical particles only one parameter must be deterrnined, the diameter of the partide. For this the minimum of two detectors is already sufficient, using either time delays or phase differences. The standard arrangement for the phase Doppler technique is shown in Fig. 2.11 (Durst and Zare 1976, Flöge11981, Baudmage and FlögeI1984, Bachalo and Houser 1984). The incident beams correspond to the same optical arrangement used in the laser Doppler technique. The two detectors are positioned out of the plane of the incident beams at an angle rfJr' usually known as the off-axis angle. The detectors are also placed symmetrie out of the y-z plane by the angles ±If/ r' the elevation angles. The analysis begins with the signal given in Eq. (2.32). For very small partides, which effectively sampie the interference pattern in the intersection vol- Receiver probes Receiver front lens y x 1tan mil1er probe direClion Fig. 2.11. Optical arrangement for the phase Doppler technique 24 2 Basic Measurement Principles ume, both detectors yield the same signal phase. This corresponds to the glare points on the surface of the particle merging with one another. In this case, no useful measure of particle size is possible using the conventional phase Doppler optical arrangement. For larger particles, the situation as depicted in Fig. 2.7 is valid and the phase difference Ll4'12 between signals received on detectors 1 and 2 will depend on the respective path lengths of the two beams to the two detectors (4 paths involved) hence, on the particle diameter. A further phase difference will arise due to composition (refractive index) of the particle. Since the positions of the four incident points and the four glare points are determined by the positions of the detectors, a further index must be fore seen for each detector being considered. The signals at the detectors are given by (index br; b beam 1 or 2; r receiver 1 or 2) i l (t) - ecEH1+ co, 21tfDt - (<<P11 -«P21 + '1'"11 -'1'"21)]) (2.37) (1+ cos[ 21tfDt - (<<P12 -«Pn + '1'"12 -'1'"22)]) (2.38) i2(t) - ec E~ for detectors 1 and 2 respectively. For the particle structure identification only the alternating (modulated) component (AC) of the signals is of relevance iIAC(t) - ecE~ cos[ 21tfDt-(<<P11 -«P21 + '1'"11 -'1'"21)] = ecE~ cos( 4'1) (2.39) i2AC (t) - ecE~ cos[ 21t fDt -( «P12 -«P22 + '1'"12 -'1'"22)] = ecE~ cos( 4'2) (2.40) The signals again exhibit a Doppler frequency, whereas the particle characteristics and the detector position influence the relative phase. The phase Doppler technique employs the phase difference of two signals ,14'12' received at the same time for both detectors (2.41) which arises for all particles, dependent on shape and composition. The first term in Eq. (2.41) is influenced by the shape of the particle. The second term is dependent on both shape and composition (refractive index). The phase difference is, therefore, influenced in the case of reflection only by the shape of the particle and in the case of refraction by the shape and composition and is independent of time or particle position. Equation (2.41), although derived using plane waves, is valid not only for all particle shapes and composition, but also for both homogeneous and inhomogeneous incident waves. The optical arrangement given in Fig. 2.11 allows the measurement of only one free parameter, thus it is suitable only for the measurement of homogeneous, isotropic, spherical particles. The remaining task is to determine a unique relationship between the phase difference given in Eq. (2.41) and the shape and composition of the particle, as well as specifying the necessary size and position of the detector apertures to fulfIl this relation. These questions are addressed in sections 5.3 and 8.2. Cleady the advantage of the phase Doppler technique lies in the fact that size (and ve- 2.3 Time-Shift Technique 25 locity) can be measured for each individual particle and furthermore, that no calibration is required. 2.3 Time-Shift Technique The time-shift technique is a further measurement principle which has application to particle sizing. This technique was first introduced by Albrecht et al. (1993) and is possible only when the particle has a well-defined curvature, e.g. spherical or elliptic, and when the particle is larger than about one third of the illuminated measurement volume. These conditions are often met when measuring with a phase Doppler system and indeed, the time-shift technique can be realized with exactly the same hardware. The measurement principle will be briefly introduced in this section and further details about its application in measurement systems will be given in the seetions 5.3.4 and 9.2. The time shift is an effect which arises solely from an inhomogeneous illumination of the particle. No time shift between signals arises if the particle is illuminated with homogeneous waves. The basic principle of this technique can be illustrated by considering a single laser beam illuminating a moving particle. The situation is illustrated in Fig. 2.12, which shows the particle at several positions within the illuminating beam and the detected light intensity due to light reflected by the particle. The intensity is shown for two different receiver collection angles. In time, receiver 1 attains a maximum signal intensity before receiver 2. Otherwise, the signals are expected to be identical and they represent a simple imaging of the incident wave by the particle onto the detector. Thus, the two signals will be shifted a time Llt with respect to one another. Because the positions of the incident points on the particle surface are a I v Moving parlide "/ I I " ~\.Receiver I I -. Dclcclcd signals " ~ r-r, I ~I I,( I\. I , I" "'" Intcnsily profile of incident wavcs -- I Receiver 2 ~ Fig. 2.12. Origin of the time shift for reflection caused by an inhomogeneous illumination field 26 2 Basic Measurement Principles function of the particle size, it is clear that the magnitude of the time shift will also be a function of the particle size. This function is monotonic and calculable for the case of spherical particles. Thus, the time-shift technique requires two receivers and provides particle size information if also the velo city of the particle is known. The velocity allows the measured time shift of two signals to be expressed as an effective measurement volume displacement. For this reason the time-shift technique is best combined with a conventionallaser Doppler system for velocity measurement and can use exactly the same hardware as the phase Doppler technique. However, a more detailed analysis in section 9.2 will show that appropriate receiver positions for the time-shift technique may be different than those for the phase Doppler technique. The measurement volume displacement as basis of the time shift between signals has been previously exploited in other optical configurations. Pavlovski and Semidetnov (1991) and Lin et al. (2000) use the time shift between singlebeam signals from two detectors and measured the velo city with a laser transit velocimeter. The technique was called the pulse displacement technique. Hess and Wood (1993) used alaser Doppler configuration and the time shift between different scattering orders on one detector and Onofri et al. (1996) recognized the time shift between signals as an additional source of size information in the dual-burst phase Doppler technique. Nevertheless, the technique has not yet been realized in a commercial instrument. The above explanation of the time-shift technique is based on the light reflected from the particle. In fact, a similar effect arises from other components of scattered light and this leads to a number of possible enhancements to conventional phase Doppler systems. These possibilities are discussed in sections 5.3.4 and 8.2. Finally it is noteworthy that the magnitude of the time shift is independent of the incident beam intensity profIle. For small particles the time shift is no longer weIl defined and the resolution is insufficient to perform size measurements. 28 3 Fundamen tals of Light Propagation and Opties For a harmonie oscillator, Maxwell's equations can be simplified by using a complex form for the field strengths, ~and H, where the underline denotes a complex quantity and m is the angular frequency curl!!=~+jm.Q, divQ = P , ~=K~, curl~=-jm~ divB = 0 .Q=C~, ~=JL!! (3.8) (3.9) (3.10) For charge-free space (p=O), Eqs. (3.8) - (3.10) lead to two identical partial differential equations for the two field parameters AE+eE=O (3.11) AH+eH=O - -- (3.12) more commonly known as the wave equations of the electromagnetic fieId. Any solution of these equations can be interpreted as a wave. The wavenumber k is determined by the frequency of the wave and by the material properties of the medium in which the wave is propagating. The relation between these parameters is known as the dispersion relation 1s. = ~ cJLm2 - jmKJL = mJiii (3.13) The propagation speed of an electromagnetic wave (speed of light) is determined by the material properties, which can be combined in the index of refraction n 1 Co Co Jiji ~crJLr n C=--=---=- Co = 1 ~ =299,792,458 m s~ (3.14) (3.15) VcoJLo From Eq. (3.13) a complex dielectric constant for a conducting medium (K > 0 ) can be defined .K ~=c-J­ m (3.16) Furthermore, the complex index of refraction is also deterrnined from these material properties (3.17) The refractive index for several typical tracer particles used in flow studies is given in chapter 13, Tab. 13.3. Generally the refractive index is also frequency (wavelength) dependent. Dielectric constants are used when describing static electric fieIds, whereas the refractive index is used for optical applications. Furthermore, the refractive index 3.1 Electromagnetic Waves 29 can be dependent on the pressure and temperature. The temperature and frequency dependence of the refractive index of water is illustrated in Table 3.1 (Thormählen et al. 1985, Schiebener et al. 1990, Lide 1997 p. 10-257 and LanboltBörnstein 11/8 28512). Generally only the relative refractive index is considered when studying the scattering characteristics of small particles, for instance the ratio of a particle refractive index!!:.p to that of the surrounding medium nm (3.18) To analyse spherical wave propagation, as is appropriate for the light scattering from spherical particles, it is advantageous to consider the solution of the wave equations in spherical coordinates (r, rp, 0). The solution of Eqs. (3.11) and (3.12) can be found in spherical coordinates using two scalar potentials. For a scalar potential ll, Eq. (3.11) becomes 1 d2 --2 r dr (rll) + d (. dll) 1 d2 II 2 smo- + 2 • 2 2 +k ll=O r sm 0 dO dO r sm 0 drp 2 1 • (3.19) Indeed, in analyzing the laser Doppler and phase Doppler techniques, a number of different solutions of the wave equation are necessary and therefore, several of the important solutions are discussed below. Tab1e 3.1. Temperature and wavelength dependence oi the reiractive index oi water (pressure 1 bar, ) Temperature [Oe] Wavelength [nm] 0 20 40 60 80 100 257 Ar+ 476.5 Ar+ 488 Ar+ 514.5 Ar+ 632.8 532 Nd:YAG He-Ne 1.37454 1.37357 1.37067 1.36644 1.36117 1.35506 1.33824 1.33760 1.33522 1.33166 1.32718 1.32193 1.33759 1.33696 1.33460 1.33106 1.32660 1.32l37 1.33626 1.33564 1.33331 1.32980 1.32538 1.32021 1.33549 1.33488 1.33255 1.32907 1.32468 1.31953 1.33229 1.33162 1.32930 1.32585 1.32153 1.31649 1064 Nd:YAG 1.32782 1.32548 1.32176 1.31720 1.31205 1.30646 3.7.7.7 Homogeneous Plane Waves The simplest solution of the wave equation results for a non-conducting medium (K = 0) in cartesian coordinates. It is easy to show by substitution that f(t- ek·r / c) is a solution to the wave equation, where e k is an arbitrary unit vector, r is a vector to an arbitrary point in the field and f 0 is an arbitrary function. This solution describes a wave propagating in the direction of +e k with a speed c. The argument of the function f, t - e k • r / c, is the phase of the wave. All points lying on a arbitrary plane perpendicular to e k will have the same 30 3 Fundamentals of Light Propagation and Optics phase for a given time and, therefore, this solution fulfills the conditions of a plane wave. A homogeneous plane wave also requires that the solution value is constant in every plane of a constant phase. The solution does not specify changes of amplitude in the direction of e k for a constant time or time dependence of the solution value for a point in space. Assuming a sinusoidal wave behavior in time and space, e.g. cos[w(t -e k . r I c) I, Eqs. (3.11) and (3.12) lead to the following solutions for the field strengths in complex form (3.20) Since the amplitudes ~o and Ho do not vary for constant phases, this solution describes a homogeneous plane wave. Often only the phrase "plane wave" is used for this specific time and spatial dependence. It is convenient to introduce the wavevector (3.21) The magnitude k is known as the wavenumber and remains real for (Eq. (3.13)). The wave in Eq. (3.20) can then be written as l( =0 (3.22) The orientation of the electric field strength vector gives the polarization of the wave, as discussed further in the next section. In Fig.3.1 a (homogeneous) plane wave polarized in the x direction and propagating in the z direction is illustrated. Substituting the solutions for the electric and magnetic field strengths (Eq. (3.22)) into Maxwell's equations (Eqs. (3.8)-(3.10)), demonstrates that for a x Phase hunts z E(",t) = E., sin(w/ - k z)c, Fig. 3.1. A homogeneous plane wave 3.1 ElectromagneticWaves 31 loss-free medium, the propagation direction, the electric and the magnetic field are all perpendicular to one another (3.23 ) Thus, the wave is designated as a transverse electromagnetic wave (TEM). As an example, the equations for aplane wave polarized in the x direction and propagating in the z direction are g = Eox exp[j (OJt- kz)] e x H = {fE (3.24) (3.25) ox exp[j (OJt- kz)] ey Since the electric and magnetic field strengths are coupled, it is generally sufficient to work onlywith the electric field strength. The energy density Eq. (3.6) of this plane wave is given by 1 w=-(c:E 2+,uH2), 2 E=ml, H=I!!I (3.26) Equation (3.26) allows the energy density to be expressed as a function of the electric field only (3.27) and shows that the electric and the magnetic wave contain the same amount of energy. The Poynting vector (Eq. (3.7)) is related to the energy density through (3.28) The energy present in space is transported spatially with the speed oflight. Fluctuations of the energy density cannot be directly measured, since the inertia of electron emission in an optoelectronic detector prohibits such high frequencies to be resolved. Thus, a temporal averaging occurs, leading to the concept of intensity, being the temporal average of the Poynting vector 1 ,+T J (J(x))=- f(x)dx T, (3.29) The complex notation for the electric field strength has the advantage that the integration leads to a simple conjugate multiplication 1= Ce E.E' 2-- (3.30) Note that in the literature, the argument in the exponential function in Eq. (3.22) may take different signs. In the following discussion, a wave propagating in the positive z direction will be denoted by exp[j (OJt- kz)). The results are briefly summarized. The wave 32 3 Fundamentals of Light Propagation and Optics (3.31) (3.32) pro pagates in the z direction and is polarized in the x direction. The phase of the wave at time t = 0 and location z = 0 is cp. For a lossy medium (K:t 0), the wavenumber is complex according to Eq. (3.13) (3.33) and the argument of the exponential function describing the electromagnetic plane wave is also complex (3.34) The real part of the exponential function describes the wave damping and the complex part describes the harmonic oscillation. If aplane wave, propagating in a loss-free medium 1 is perpendicularly incident on an interface to a lossy medium 2 (K:t 0), the wave will be exponentially damped in medium 2, depending on the conductivity. If the medium 2 is not too thick, then the wave emanates from the medium 2 with a finite measurable amplitude. This essentially describes transparency or opaqueness of the material. Highly conductive materials, such as metals, are opaque for modest thickness, whereas purely dielectric media are transparent. Since the damping factor ßk is related to the frequency, the transparency of the medium is also related to the wavelength. 3.1.1.2 Spherical Waves In many laser Doppler and phase Doppler applications, the sphericity of the particles is assumed and thus, solutions of the wave equation in spherical coordinates are desirable. The particles act as a source for spherical waves. A general solution of the wave equation in a spherical co ordinate system (Eq. (3.19» is given by II = -C {~n(kr)} Pnl( cos iJ ){sin(lCP)} kr Sn (kr) cos(lcp) (3.35) where the curly brackets indicate that either function can be chosen. Both 1 and n are integers, originating from the separation ofvariables in Eq. (3.19) and Cis an arbitrary amplitude. The functions ~n(kr) and Sn(kr) are the Riccati-Bessel functions of the 3rd kind and of order n and P~ (cos iJ) are the associated Legendre functions of order n (see section 4.1.2 and Appendix). In general, the electric and magnetic field strengths can be computed from two scalar potentials IIj and II 2 given by Eq. (3.35). For each potential the radial component of either the magnetic or electric field strength vanishes (van de Hulst 1957,1981, Kerker 1969, Bohren and Huffman 1983). Each pair ofintegers 3.1 Electromagnetic Waves 33 n and 1 eorresponds to a spherical partial wave. The Legendre functions and the sine and eosine funetions eontain the angular dependenee of the spherieal wave, whereas the radial dependence is expressed by the Rieeati-Bessel functions. Arbitrary eleetromagnetie waves ean be specified by an appropriate superposition of partial spherieal waves and choice of their amplitude. It is important to note the asymptotie behavior of the Rieatti-Bessel functions. For large values, these functions ean be approximated with an exponential funetion having a eomplex argument r exp( -j kr) for kr ~ oe (3.36) (n(kr) "" j"+l exp( +j kr) for kr ~ (3.37) ';n(kr) "" 1 00 This argument already appeared in a similar form in Eq. (3.22) far homogeneous waves. A wave with exp(-jkr) propagates in the positive radial direetion, whereas exp( +jkr) indieates a wave propagating in the negative radial direction. For the simplest ease of n = 0 and 1= 0, two solutions for a potential are obtained, one expanding in the positive radial direction I l1 = ~r exp[-j(kr+;7i)] (3.38) and one eontraeting in the negative radial direetion Il2 = ~ exp[+j(kr+;7i)] (3.39) For spherieal waves, the amplitude decreases with 1/ (kr) in the positive radial direetion. Furthermore, the simple waves of Eqs. (3.38) and (3.39) have a eonstant phase of +1t /2. The amplitude of these simple waves goes to infinity for r=O. In the laser Doppler and phase Doppler teehniques the detector is generally positioned in the far field and thus the wave amplitude decreases inversely with radius from the measurement volume. This means the intensity and the signal amplitude, deereases with the square ofthe inverse radius. A solution of the wave equation for a homogeneous plane wave incident on a homogeneous, spherieal partide will be diseussed in detail when presenting the Lorenz-Mie theory (seetion 4.1.2). 3.1.2 Polarization A homogeneous plane wave propagating in z direction ean have two field components perpendieular to the propagation direetion k = ezk (3.40) This equation can be interpreted as the sum of two independent partial waves with orthogonal field eomponents and a relative phase shift cp x - cp y 34 3 Fundamentals of Light Propagation and Optics gx = Eox exJ{j( wt - kz + qJ J] (3.41) gy =Eoyexp[j(mt-kz+qJy)] (3.42) The polarization of the waves is given by the orientation of the vector is expressed more conveniently by rearranging Eqs. (3.40) - (3.42) as ~. This (3.43 ) (3.44) In the Ex -E y plane the trace of the vector tip 1980, Born 1981, Bass 1995, Vol. I Chapt.5) ~ is given by (Born and Wolf +[~)2 -2~~COS(qJy -qJx) = sin 2 (qJy - qJx) ( ~)2 Eox Eoy Eox Eoy (3.45) which is the equation of an ellipse with the major axis orientated at an angle _ 1 (2EoXEOY cos(qJy -qJx)] 2 2 Eox -Eoy a - -arctan 2 (3.46) Depending on the phase difference LlqJ = qJ y - qJ x and the amplitudes of the partial waves Eox and Eoy ' a right or left circular polarization, elliptical polarization or a linear polarization can be obtained. Light with various states of polarization is shown in Fig.3.2. Elliptically polarized light arises when the phase difference of the two waves is LlqJ"* mt /2 (Fig.3.2c) or in the case of different amplitudes when LlqJ = n1t (Fig.3.2d). If both of the waves are completely out of phase LlqJ = (n + l'i)1t, the major and minor axes lie on the Ex and Ey axes. Equal amplitudes result in circularly polarized light when LlqJ = (±l'i + n)1t (Fig. 3.2b). Linearly polarized light arises when LlqJ = n1t (Fig. 3.2a). Every linearly polarized homogeneous plane wave created by two partial homogeneous waves can be transformed into only one partial homogeneous wave by rotation of the coordinate system around the z axis. The polarization of scattered light from the particles is affected by reflection and refraction. The polarization state of the incident light and the received light can also be altered using polarizers or polarization filters in the transmitting and receiving optics. A half-wave plate preferentially retards one of the waves by 1t and thus rotates linearly polarized light by 1t /2 or changes the rotation direction of circularly polarized light. A quarter-wave plate retards one wave by 1t /2 and thus transforms linearly polarized light into circular polarized light or circular polarized light into linearly polarized light (see section 7.2.2). Note that only light oflike polarization can exhibit interference. 3.1 Electromagnetic Waves 35 Fig. 3.2a-d. Polarized light. a 45 deg linear, b Right hand circular, c Left hand 45 deg elliptical, d Right hand 0 deg elliptical 3.1.3 Boundary Conditions and Fresnel Coefficients According to Maxwell's equations, the tangential component of the field strength of both the electric and magnetic fields must be continuous at an interface between a medium 1 and a medium 2. This is a fundamental boundary condition which must be invoked when the interaction of electromagnetic fields with matter is considered and is paramount for the computation of light scattering from small particles. Generally, the incident wave E(;)(r,t) in medium 1 is known. After interaction with some optical inhomogeniety (medium 2), the reflected or scattered wave E(r)(r,t) in medium 1 and the transmitted wave E(t)(r,t) in medium 2 are sought. These three waves are related at the interface through the boundary conditions (3.47) For a homogeneous plane wave impinging on an interface with the normal vector n I the electric field strength can be given in complex notation as n I x~~) exr[j (mt-kU) .r)]+n I x~~) exp[j(liJt-k(r) or)] (3.48) 36 3 Fundamentals of Light Propagation and Optics The boundary condition also specifies that the three wave vectors k U), k(r) and k(t) alllie in the incident plane. If the angles between these vectors and the normal are designated by (Ji (incident angle), (Jr (reflection angle) and (J, (refraction angle), the laws of reflection (J , = (J r (3.49) and refraction (3.50) follow. Furthermore, the ratios between the complex incident amplitude of the incident wave m~) and the reflected m~l) and refracted m~l) waves can be derived. These relations are known as the Fresnel equations and are dependent also on the polarization. For a wave polarized perpendicular to the incident plane the reflection and refraction amplitude coefficients are given by Hecht (1998) and Born and Wolf (1999) n _ I COS(Ji J11 n _=.Lcos(J, J12 (3.51) n 1 COS(Ji 2- J11 n _ I COS(Ji J11 (3.52) n +=.Lcos(J, J12 For a wave polarized parallel to the incident plane these coefficients are r = -11 (E(rl] tJ:. 2 -cos = J12 _0 E Ul _0 11 (J i n =.LCOS(Ji J1 2 _0 (J n +_1 cos(J, J1 1 (3.53) n 1 COS(Ji 2_ (E('l] t ll = E~i) nl --cos , J11 J11 11 (3.54) tJ:. 2 cos(J , +!'lLcos(Jt J12 J11 The conductivity of the medium can be accounted for in the complex refractive index. The Fresnel (amplitude) coefficients are complex, the magnitude pertaining to the amplitude and the phase giving the phase shift of the electric field. The intensity of the reflected and transmitted wave can be computed from the reflectance Rand transmittance T. Splitting the wave into the two components parallel and perpendicularly polarized to the incident plane the factors become RII.-L--P -Ir 12 (3.55) 3.1 Electromagnetic Waves T: -.6. 11 II,.L - n1 2 cosB, 112 cosB; It 1 2 37 (3.56) -II,.L when the case of total reflection is excluded. The sum of reflectance and transmittance must adhere to energy conservation, i.e. (3.57) thus, it is sufficient to specify either one to determine the intensity. For the special case of total reflection (B; > arcsin(n 2 I n 1), n2 < n1 ) Eqs. (3.55) to (3.57) are not valid. For this case the reflectance becomes unity, R = 1, and the transmittance vanishes, T = 0 (Born and Wolf 1999). Example reflectance and transmittance are given in Fig. 3.3 for an air/glass interface as a function of incident and refraction angle. For incident waves perpendicular to a surface (B; = 0 deg), 4% of the intensity is reflected on the surface. Therefore, each uncoated surface in an optical system results in an energy loss of 4%. 30 0 -" u " 8, [deg] Rerfraction angle 1.0 t=--=---=----=----~--~~-:;=:--~-~-_::_::_--:.-=:.--=----=---=='----=---------....;;,:;--------------------------- -- ..... - --- .8 .~ -- - - - -- - - -- "...'" I:-< "u " .8 u 0.5 -- Reflectance parallel polarization Reflectance perpendicular polarization Transmittance parallel polarization Transmittance perpendicular polarization ,, - - - - -- - - - - - -- - - - - - -- - - - - - -- - -- - - - - - -- - - - - - -- - - - - - -- - -- - - - - --- - - - - - -- - - - - - -- - - -- - ~ , " " 0:= ~ , .... -- - -- ,,' ,."" --- \ \ R.L " .' ............ #' ................ ... ........ - .............. - .. 0.0 ~--------------=--....- -~-- =~=--~---~~=~=:~----~-=~~---~--~~~~~~--=---=--=--~----~--~--------- -------- -- ---:-=,= o 45 Brewster angle Incident angle Fig. 3.3. Reflectance and transmittance for an air/glass interface (n, 8, [deg] 90 = 1, n, = 1.5, f.l, = f.l,) 3.1.4 Laser Beams Certain characteristics of laser beams make them indispensable for the laser Doppler and phase Doppler techniques. The choice of wavelength and power depends on the application, although for practicality, visible wavelengths are generally preferred. The necessary laser power for a desired signal power may depend on many factors, including flow velo city, scattering efficiency of the particles, position and focallength of the receiving optics and also the flow me- 38 3 Fundamentals ofLight Propagation and Optics dium. Due to the central importance of the laser beam, which is also a solution of the wave equation (3.11), a more precise description of its properties will be given. 3.7.4.7 Physical Charaderistics of a Laser Beam In a laser (Light Amplification by Stimulated Emission of Radiation) two basic physical processes are exploited: stimulated emission, which leads to a high degree of monochromatic light and the amplification of the radiation in an optical resonator, which leads to high power densities and a long coherence length with small divergence angle. Four lasers are in general use for measurement systems: the gas laser, the semiconductor laser, the solid-state laser and the fiber laser (Table 3.2). More details about each of these laser types can be found in the literature (Milonni and Everlyn 1988; Koechner 1992; Siegman 1986; Kleen and Müller 1976, Svelto 1976, Bass 1995 Vol. I Chap. 11, Eichler and Eichler 1998, Czarske et al. 1997, Czarske 2002). For laser measuring techniques, the important parameters include wavelength, line width, power and coherence length. In some cases the physical dimensions of the laser may be of importance. In recent years a trend from gas lasers to semiconductor lasers can be observed, as the power density of the latter increase. Similar small dimensions and low power consumption are also afforded by the Nd-YAG laser. The bandwidth given in Tab. 3.2 arises mainly due Table 3.2. Typical parameters of lasers suitable for laser Doppler and phase Doppler applications (linewidth/coherence length given for single mode emission) Wavelength Color Bandwidth of gain [GHz] Linewidth 1 Power Coherence length [MHz] 1 [m] [W] red 1.6 300/1# 0.5xlO- 3 ••• O.051.0 0.5 ultra violet violet 4 blue green 400010.07 5x10 3 •• .20 (allIines) 1.5 1 red 100* 13# 0.2XlO-3 ••• 1 ~1.5 0.001...0.3 ~1.1 <1 [nm] Gas lasers: He-Ne Ar+ 632.8 257 476.5 488 514.5 Solid -sta te lasers: Laser diode 640 ... 690 Nd:YAG 1064 532 1 infrared green Fiber lasers: 1030 ... 1130 infrared ~p-conver- 635 slOn red 100 120 <1/300# ~3000 ~744 0.0001 3 ••• 100 5x10 0.001...10 0.1...110 0.1 Beam quality factor M 2 [1] Amplitude Noise [% ] ~1 <1.1 ~1 1 Frequency doubled; # Note: single-mode emission (at low power); * 10 MHz; 3m (for DBR resonator); 2600GHZ; 200llm (for multi-mode emission) 3.1 Electromagnetic Waves 39 to spontaneous emission and is therefore a function of the laser medium used. Depending on the quality of the resonator and the stabilization, small bandwidths ofthe emitted light and coherence lengths of some millimeters up to several kilometers are available. 3.7.4.2 Mathematica/ Description ofthe Gaussian Laser Beam A mathematical description of the laser beam is essential for the layout of laser Doppler and phase Doppler systems. It arises from the solution of the wave equation (3.11) for the field strength, given the boundary conditions as dictated by the physical construction of the laser. Gas lasers have axissymmetric resonators. The laser light is emitted at one end through a half-silvered mirror and the polarization is determined by the orientation of a Brewster window at the other end. Typical apertures placed within the resonator lead to a monochromatic beam of narrow wavelength line width and with a Gaussian distribution of the intensity, as is illustrated in Fig. 3.4. The beam waist typically lies at the output mirror. Describing a laser beam as an electromagnetic wave, the electric field strength for the wave propagating in the Zb direction and linearly polarized in the Yb direction is given by (Kogelnik and Li 1966; Davis 1979; see also Appendix) (3.58) where the components are (3.59) (3.60) and 2 1 == nrwb Rb A (3.61) b is denoted as the Rayleigh length on the Zb axis. The electric field amplitude of the wave decreases by .J2 relative to the maximum amplitude at Zb == O. EOb is the amplitude of the electric field in the center of the beam at the waist (origin). The amplitude isafunction of the input laser power and the beam waist radius, given as (3.62) 40 3 Fundamentals ofLight Propagation and Optics P(x,.y,.-,,) Gaussian ficld 1.. E, E, E e E _l EOl I I 2" I e.fi. e 1 decay of field slrenglh (e · dccay of inlensily) relative 10 -' axis Fig. 3.4. The Gaussian laser beam In Eq. (3.59) the amplitude is complex because the laser beam can have a phase unequal to zero at the origin (3.63 ) The longitudinal Zb component of the field is now necessary because the wave no longer has a homogeneous field distribution perpendicular to the propagation direction. The Zb component generally has no relevance for light scattering. The beam propagation can be described by its radius (3.64) defined by the width at which the intensity falls to e-2 of the maximum value on the axis (e- 1 of the wave amplitude). At Zb = 0 the beam exhibits a minimum width or beam waist of radius rwb ' Constant phases are obtained on circles with radius Rb around a virtual origin ZR on the Zb axis, as illustrated in Fig. 3.5. For positions Zb > 0 the origin is located at ZR <0 and for negative coordinates, Zb <0, the origin will be ZR >0. The wavefront curvature Rb depends on the Zb position (3.65) where is the virtual origin of the wave on the Zb axis. In the near field a Gaussian intensity distribution with plane wavefronts is obtained (ZR --7 ±oo). The phase in the far field can be described by a spherical wave. The arctangents in Eq. (3.59) converges for large Zb to 1t /2 and gives a phase offset ZR (Zb «lRb) 3.1 Electromagnetic Waves 41 Fig. 3.5. The beam and wave coordinate system between the far and the near field. The phase offset ean be interpreted as a phase jump in the foeus point (waist) of the beam, known from geometrieal opties (van de Hulst 1981 p 207). While Rb expresses the wavefront eurvature on the Zb axis, Rw gives the radius at an arbitrary point in spaee R w =Jx; + Y; +(Zb -zS (3.66) and should be used instead of Rb(Zb) in Eq. (3.59). The orientation eXbxb +eybYb +eZb(Zb -ZR) e w = sgn (Zb ) -----;=========~ ~ x; + Y; + (Zb - ZR)2 (3.67) gives the loeal propagation direetion at this point. These relations are useful for defining an ineident homogeneous plane wave for a small particle, whose amplitude is the loeal amplitude of the laser beam at the particle position .J E~b + E;b and whose propagation direetion is such that there is no longitudinal eomponent of the wave. A loeal wave co ordinate system arises using a rotation through the angle rfJ w about the x b axis and a rotation through the angle lfI w about the Yb axis. m =aretan~ 'l'w R ' b x b lfI = aretan- w Rw (3.68) Further relations are (3.69) ab is the half divergenee angle in the far field. At the Rayleigh length lRb' the beam radius beeomes .J2 rwb and the intensity is half of the maximum value in 42 3 Fundamentals of Light Propagation and Optics the waist center (see Fig. 3.4). The geometry for the Gaussian laser beam can always be described by two parameters, for instance wavelength and beam waist radius, half divergence angle and Rayleigh length or a combination of these four parameters. Figures 3.4 and 3.5 illustrate the various parameters of a laser beam. So me simplifications of Eq. (3.59) are possible using the amplitude factor FG and the phase value ([JG in comparison to homogeneous plane waves (3.70) with (3.71) ([JG r2 Zb 2R b lRb b =-kb--+arctan- (3.72) For convenience, the complex beam parameter (Kogelnik and Li 1966) is introduced (3.73) The wave can now be described in terms of magnitude and phase as E =E ~exP[J·(OJt- q kb _yb _Ob rmb (,) Zb 2 ( _b Zb ) (x 2+y2))] b b (3.74) All these relations for a Gaussian beam are valid for the first approximation of the wave equation, see Appendix. In reality, the beam profile does not have an exact Gaussian shape (Kogelnik and Li 1966, Davis 1979, Barton and Alexander 1989). However, the higher order terms in the field description can be neglected for most applications. When the waist diameter of the beam is of the order of the wavelength, then this approximation also becomes increasingly inaccurate. 3.1.4.3 Non-Gaussian Beams The optical properties of a laser beam from a semiconductor laser are somewhat more complicated and of considerable importance, especially when small focal diameters are desired to obtain small measurement volumes in laser Doppler and phase Doppler systems. The physical dimensions of the resonator in a semiconductor laser lead to an elliptical beam profile, where two different divergence angles and beam waist positions (virtual origins) can be identified. The separation between the virtual origins along the Z axis is known as astigmatism (Lls in Fig. 3.6) and can amount to as much as 40 /lm. Thus, thefocussing of the beam can be more difficult. The elliptic beam profIle and (he astigmatism can be corrected using an appropriate collimation, as discussed in section 3.2.2 and 7.2.1. 3.1 Electromagnetic Waves 43 The electric field of a laser diode can be approximated by the product of a Gaussian and a Lorentzian distribution (Naqwi and Durst 1990, Zeng 1992). (3.75) Here rmb (Zb,) is the radius for an e- 1 decrease of the field amplitude of the Gaussian profile in the x b direction and mb(Zb) is the half-width of the Lorentzian distribution in the Yb direction. R bx and R by are the radii of the wavefronts for the Gaussian distribution in the x b direction and for the Lorentzian distribution in the Yb direction. 8 x and 8 y are the FWHM angles (Full Width Half Maximum) in the two orthogonal directions. They are related to the half divergence angles a x and a y in the two directions (Fig. 3.6) through r tanax = (ln 2) -~ 2 8 x tan-·_ for the Gaussian profile (3.76) for the Lorentzian profile (3.77) 2 8 Y tanay = .je-l tan2 Typical values for 8 x and 8 y are 10 deg and 30 deg, corresponding to full divergence angles of 12 deg and 39 deg, which according to Eq. (3.69), lead to mean waist diameters of 3.9 11m and 1.2/lffi (Ab = 650 nm ). Fig. 3.6. The laser beam from a semiconductor laser 44 3 Fundamentals ofLight Propagation and Optics 3.1.5 Optical Mixing of Electromagnetic Waves The electric current of a photodetector is directly proportional to the intensity given by Eq. (3.30), whereby the sum of all field strengths impinging on the detector surface, independent of polarization, must be considered. This process is called "optical mixing ofwaves". In a laser Doppler or phase Doppler system, small particles (d p « ,1) interact directly with the Poynting vector of the superposition of the two waves. For larger particles this superposition takes place on the detector surface. Additional scattering contributions may also be present, for example scattering from side walls in the test section. As an example, the mixing of two elliptically polarized spherical waves will be examined, having the wavenumbers k l , k 2, the angular frequencies 0)1' 0)2 and the phase shifts rp xl' rp x2' rp y1 and rp y2 in the two tangential polarization directions. The phase shifts may depend on the refractive index and the size of the scattering particle being considered as a source for the two waves. If the detector is aligned such that the normal to the detection surface (x-y plane) is directed towards the scattering particle, then to a first approximation, only tangential field components of the waves must be considered. The distances between the source points of the spherical waves and the detector are 1j and r2 • The field strength at the detector becomes (3.78) (3.79) (3.80) (3.81) (3.82) This expresses only the component of the field strength lying parallel to the detector surface so that the Poynting vector and the energy flux direction points into the detector surface. The intensity falling onto the detector using Eq. (3.30) is described by I=CE 2 -E .E - ,=CE(E 2 _xl +E _x2 )(E* _xl +E* _x2 )+(E _y1 +E _y2 )(E* -yl +E*) _y2 CE( 2 2 2 2 1=2 EXl+Ex2+EYl+EY2 (3.83 ) + 2E xl Ex2 cos[ (0)2 - 0)1 )t- (k 2r2 - kl'i) + (rpX2 - rpXl)] + 2E yl EY2 COs[ (0)2 - 0)1 )t- (k 2r2- kl'i) + (rp y2 - rp Yl) J) 3.1 Eleetromagnetie Waves 45 The signal received by the detector corresponds to the integration of Eq. (3.83) over the surface of the detector. The first sum in Eq. (3.83) corresponds to the DC part of the signal 2 +E 2 +E 2 +E 2 ) I vc =Ce - (E xl x2 Yl Y2 2 (3.84) The next two sums involve modulated amplitudes, which will depend on the coherence of the light. These can be combined as l (3.85) with (3.86) 1 ExlEx2 sin( lPx2 - lPXl)+ Eyl EY2 sin( lP y2 - lPYl) lP = arctan[ ----------'---=-----';--'-----'---'cExl E x2 cos( lPx2 - lPXl)+ Eyl EY2 cos( lP y2 - lP Yl) (3.87) Equations (3.84) and (3.85) allow expression (3.83) for the intensity to be simplified (3.88) where the DC part and modulated AC part have been combined in the visibility (3.89) which is a common measure of signal quality in the laser Doppler and phase Doppler techniques. If the amplitude and phase of each primary wave are equal, the visibility is equal to 1 (r = 1). Furthermore, if the wavenumbers kl and k 2 are equal and the source points of the spherical waves are the same, Eq. (3.88) can be written as (3.90) 3.1.6 The Doppler Effect The Doppler effect accompanies any movement of either the transmitter or receiver of electromagnetic radiation (Vogel and Gerthsen 1995). The principle of the Doppler effect is illustrated in Fig. 3.7. An electromagnetic wave emanating from a moving transmitter (generally a particle) with velo city v p and with 1 The sign of the eosine argument ean be positive or negative, beeause the eosine funetion is even. The ehoiee is made sueh that a particle moving in the positive x direetion results in a positive frequeney differenee, aeeording to Fig. 2.5 and Eqs. (2.4) to (2.12) 46 3 Fundamentals ofLight Propagation and Optics b a A, Fig. 3.7a,b. The Doppler effect. a Moving transmitter and stationary receiver, b Stationary transmitter and moving receiver transmitting frequency jp will be compressed in the direction of movement and expanded in the opposite direction (Fig. 3.7a). This results in a change of wavelength and frequency as given by A = _c_-_v---,p~·_e-,-p_r I - _c _ _ Ip r r - Ar - -,--I-,-p_ vp·e pr (3.91) 1--C The perceived wavelength Ap and frequency jp of a moving receiver (generally a moving particle) with a relative velo city v p with respect to the stationary transmitter (in general a laser) (fl' AI) is given by (Fig. 3. 7b) A P = __A-'.I__ V P ·e lp (3.92) 1--C Ifboth the transmitter and the receiver are moving, the Doppler effect can be invoked twice and the perceived frequency at a stationary receiver for a stationary laser and forlight scattered from a moving particle becomes (Iv pi« c) 1- elp'v p Ir =II e \ l--~ c 3.2 Optical Components 47 (3.93) 3.2 Optical Components Fundamental to the realization of optical measurement systems is the generation and the detection oflight and the propagation oflight through optical components. In the following sections these fundamentals are discussed with respect to components typically employed in laser Doppler and phase Doppler systems. 3.2.1 Matrix Transformation for Imaging Generally the paraxial approximation can be made when analyzing laser Doppler or phase Doppler optical systems. Paraxial means that all rays subtend only small angles with respect to the optical axis and thus, the sine and tangent of the angle can be replaced by the angle itself. In this way the equations of paraxial optics become linear and can be conveniently computed using matrices (Kleen and Müller 1986, Hecht 1989). The influence of optical components on the light path can be expressed in terms of a matrix, characteristic for that element, e.g. for thin lenses. Through the multiplication of matrices, the accumulated effect of many components can be evaluated. Imaging by a lens can be considered as the sum of two elementary processes, the refraction of light from one medium into another one and the propagation of light in the lens medium. Examining first the propagation of light through a medium of refractive index n and thickness d yields (Fig. 3.8a) (3.94) (3.95) a y b Ligh I beam Medium I Medium 2 O~~~------------------,-~+ z Fig. 3.8a,b. Propagation of light. a Through a medium of thickness d, b Refraction at a plane surface 48 3 Fundamentals of Light Propagation and Optics which can be expressed as (3.96) where At is know as the transfer matrix. This matrix takes the transmitted ray at the entrance to the medium and transforms it into the incident ray at the exit from the medium Refraction on a plane surface is described by (Fig. 3.8b) n2 sin a 2 = n1 sin a 1 (3.97) (Snell's law) or for paraxial rays (3.98) which for Y2 = Yl yields (3.99) Now the case of refraction at a spherical surface is examined, as illustrated in Fig. 3.9. For paraxial rays Snell's law can be written as (3.100) where the second index designates the surface number in the optical configuration. In relation to the optical axis the angles become ßl1 = a+a l1 and ß21 = a+a 21 (3.101) so that the ray position at the interface can be described by sina=~ and R1 ' Y21 = (3.102) Yll Y Surfacc 1 a ßII 21 a R, Yl1 O~--~~--~L-----~r---------~--------~---' l\ledi um J ", A Fig. 3.9. Refraction at a spherical in terface Medium 2 B z 3.2 Optical Components 49 and using Eqs. (3.100) and (3.101) n 2-nI n2a 21 =nla ll ----Yll RI (3.103) Equation (3.103) can be written in matrix form (3.104) where D is the refractive strength of the interface (3.105) and Ar is known as the refraction matrix. It transforms the entrance ray into the exit ray after refraction. If B, the origin of the curvature, lies to the right of A, the radius is positive (R >0) and for B left of A, R <0. For a plane interface (R» YI) Eq. (3.103) just reduces to Eq. (3.98). Using these fundamental matrix relations, more complicated elements such as lenses or lens systems can be examined. Light propagation through a simple lens with a refractive index of n/, an input surface curvature of RI , a thickness d and an output surface curvature R2 in a medium with refractive index nm can thus be described as (3.106) The system matrix for a thick lens is thus (3.107) For a thin lens with d ~ 0, Eq. (3.107) becomes (3.108) which contains the negative of the inverse focallength. To discuss the imaging properties of a lens, several planes are of special interest (Fig.3.1O). The plane tangential to the lens surface on the optical axis is 50 3 Fundamentals ofLight Propagation and Optics known as the vertex plane. The front focallength fl (f.f.1.) and the back focal length f2 (bJ.1.) are measured from these planes. The front and back focal points are the points where light, propagating parallel to the optical axis, is focused. If the parallel light rays entering the lens are extrapolated to the interseetion point of extrapolated focused rays, as shown in Fig. 3.11, a curved surface is defined. For the paraxial approximation this becomes a plane and is known as either the front (H1 ) or back (H 2 ) principal plane. Iflenses are analyzed using principal planes, the incident and exiting rays can be extended to the principal planes and between the planes the rays are parallel to the optical axis, as shown in Fig. 3.11. For a thin lens, the two principal planes merge at the center of the lens. Then there is only one common focallength. More complex optical systems are often reduced to an equivalent thin lens, with known matrix and two translation operators. o I z, I I ) I l~ ", ~ ZII Fronl loeal plane • Fronl prindpal plane ", I 1\ J ", ", Fronl ver lex plane I F, ~ d /, Object plane v, H, Zn 11 m /, Back verlex plane Back principal plane Back focal plane Image plane Fig. 3.10. Important planes of a lens. 0: object plane, H",: front and back principal planes, I\,,: front and back focal planes, V;,2: vertex planes, I: image plane From prinei al plane I' ront principal plane ßack principal plane z Fig. 3.11. Explanation ofthe principal planes of a thick lens (Hecht 1998) 3.2 Optical Components 51 The principal planes of the equivalent system are given by a translation h1,2 from the vertex planes, as shown in Fig. 3.10. Note that for H, to the left of V" h, >0, and for H 2 to the right of V2, h2>0.' The equivalent optical system is given as a matrix with unknown translation coefficients. h 1 0)1 [ 1 -' _[1-~ nmf ~_h'h2] nm n~f n 1 h m o - 1 -- f (3.109) 1 _ _'_ nmf Comparing this matrix with the matrix for a complex optical system ( A Bi=[I- D) c n~f h'n: _~ h2 - :t;] 1-~ f (3.110) nmf the equivalent focallength for a thin lens f=-~ (3.111) C and the principal planes can be determined A-l h2 =n - m C (3.112) As an example, the thick lens from Eq. (3.107) can be transformed in this way. The equivalent focallength is then given by f = R,R 2 (n/ - nm { R, - R2 - d( 1- :~ )1 (3.113) and the principal planes are located at h' = -f!i... R nm 2 (1 - nn/ J m ' (3.114) With known principal and focal planes, the system can be analyzed similar to a thin lens. Using the system matrix, the ray paths (angles and distance to the optical axis) after the lens are related to those before the lens, thus the image plane can be determined from a known object plane. Applying Eqs. (3.106) and (3.108) to the situation pictured in Fig. 3.12 leads to the so-called thin-Iens equation (h1,2 = 0). , Some literature uses a notation in which for H 2 to the right of V;, h1 > 0, and H, to the left of V;, h, > 0 . 3 Fundamentals of Light Propagation and Optics 52 Thin lens y ParuUellight nay Light nay through the focal point OL-~~~ ____________- '______ ~~~-L _ _ _ _ _ _~~_ _~~~ z ]' f z ZI , Fig. 3.12. Focal relations for a thin lens 1 1 1 Z2 ZI -=--+- f (3.115) Parallel light entering the lens (al = 0) will foeus at the foeal point f. Light emanating from the foeal point of the lens (ZI = f) will be collimated, with a foeal point at Z2 = (a 2 = 0 ). The transverse image magnifieation at the image plane (Z2) of a thin lens is given by 00 ß = Image size _.2 Objeet size (3.116) ZI whieh leads to (3.117) The numerieal aperture of a lens is defined by NA=sinB =!!L A 2f (3.118) where dl is the free aperture diameter of the lens. In photography the f-number is often used, which is just the half inverse of the numerieal aperture 1 f/#= 2NA The intensity of the image will inerease with the square of NA. (3.119) 3.2 Optical Components 53 3.2.2 Propagation of Laser Beams Through Lenses and Apertures In the far field a laser beam can be treated as a spherical wave. A thin lens will transform an incident spherical wave with wavefront curvature Rw1 into a spherical wave with wavefront curvature R w2 (Fig. 3.13) according to (3.120) or 1 1 (3.121) according to Eqs. (3.106) and (3.108). A similar relation can be written for the complex beam parameter Eq. (3.73) (Kogelnik and Li 1966) 1 1 1 ~2 ~1 f (3.122) which allows the imaging of a laser beam through a lens to be calculated. For laser Doppler systems the imaging ofbeam waist rw1 at point ZI in front of the lens to a beam waist rw2 at point Z2 behind the lens is of importance (see Fig. 3.14). The distances ZI and Z2 are measured relative to the front and back principal planes respectively. The arrangement in Fig.3.14 is then analyzed using Eqs. (3.96) and (3.108) and assuming nm = 1 (air), yielding (Bernabeu 1989) (A B)=(1 Z2)( C D 0 1 1 0)(1 1 0 -r 1 ZI)=[I-~ _..!.. 1 f ZI+ Z2- Z 1-~ ;2J (3.123) f The complex beam parameter after the thin lens can be calculated by q _2 Aq +B =-=,,-1_ Cq +D (3.124) _1 At the beam waist (Rb ~ 00 ) the real part of the beam parameter vanishes (3.125) Since the beam waist before the lens lies at the position ZI and after the lens at the real part of the beam parameter is zero. Thus, two equations to determine the beam waist radius and length are obtained by examining separately the real and imaginary parts ofEq. (3.124) Z2' 54 3 Fundamentals of Light Propagation and Optics y Sphcrical wavcfronl z f f Fig. 3.13. Imaging of a spherical wave using a thin lens l! , Fig. 3.14. Imaging of a laser beam and its waist by a lens (3.126) The position of the imaged laser beam waist is given by the real part (3.127) and the beam width by the imaginary part rwJ (3.128) The quantities are visualized in Fig. 3.14 and the dependencies expressed by Eqs. (3.127) and (3.128) are shown graphically in Fig. 3.15a and b. These figures 3.2 Optical Components 55 b ~ f 2 o -4 -2 o 2 4 6 z.ff -4 -2 o 2 4 6 z.ff Fig. 3.15a,b. Imaging characteristics of a Gaussian beam for movement of the incident waist about the front focal point (f = 40mm, d w• = 100~, A. b = 488nm). a Beam waist radius, b Position ofbeam waist illustrate the beam waist radius and its position for movement of the incident waist about the focallength ofthe lens. A laser beam waist at the front focal point will be imaged to the back focal point with the diameter (3.129) This relation follows immediately from the imaging of a spherical wave. Dickson (1970) derived this result by applying Kirchhoff's diffraction integral to a centered circular pupil with a Gaussian pupil function. Rempel and Fischer (1982) showed that the result is a good approximation also for asymmetrie beams. Otherwise, Fig. 3.15 indicates that even minute movement of the waist from the front focal point will result in large movements ofthe waist on the focussing side. In the example given in Fig. 3.15, a 15 mm shift from the front focal point results in a 50 mm shift in the back focal plane. For larger shifts away from the front focal plane, the output beam waist moves over large distances with virtually constant diameter, i.e. the beam is collimated. This behavior is exploited in the collimators described in seetion 7.2.1. Any apertures used in the system must be chosen large enough to avoid diffraction effects, either within the laser cavity or in the following optical system. To estimate allowable aperture sizes, a circular aperture of diameter 2ra is examined and the Kirchoff diffraction integral is solved for a Gaussian beam, the result being shown in Fig. 3.16. The maximum intensity transmitted through the aperture I2' related to the intensity before the aperture II' is plotted against the radius of the aperture ra relative to the local beam radius rmb at the position of the aperture. According to this result, apertures should be a minimum of 4rmb in diameter to avoid any influence of the aperture on the beam profIle (Dickson 56 3 Fundamentals ofLight Propagation and Optics ..... N ~ 1.0 .i? '1;J c:: .5" " .~ '" Ol 0.5 ~ 0.0 ----~ o 2 Relative size of aperture ',/r,"b Fig. 3.16. Influence of the aperture radius r. on the centerline intensity of a Gaussian beam 1970, Hofman 1980). Therefore in laser Doppler and phase Doppler systems, small apertures which truncate the beam should be avoided. Laser beams from semiconductors have more complex imaging properties since they exhibit different wavefront curvatures in directions orthogonal to one another (Naqwi and Durst 1990). When sm all beam divergence or a localized circular beam waist is sought, the elliptically shaped beam must be specially treated with shaping collimators. One preferred collimator system begins by adjusting one of the divergence angles of the beam to infinity, usually in the plane exhibiting the largest divergence. Plane waves are obtained in this direction. The waist in this direction then lies in the focal plane of the collimator. Following this a cylindricallens is used to correct for the astigmatism in the other direction. In this way a Gaussian beam profile can be approximated and further analysis can be based on the established relations given above. 3.2.3 Optical Gratings and Bragg Cells Diffraction gratings and Bragg cells can be used effectively as beam splitters and if the grating is moved (rotated), both devices also result in a frequency shift, a necessary function for making laser Doppler systems sensitive to the direction of the particle velo city. More recently, diffraction gratings have been integrated into fiber optic interfaces (Czarske 1999) and holographie techniques now allow very precise gratings with very specific characteristics to be manufactured. Nevertheless, the Bragg cell is more commonly used in laser Doppler systems, both due to its high stability of frequency shift and because there are no moving parts. Kerr cells and Pockels cells have also been demonstrated as frequency shift devices. However the Kerr cell exhibits a low limiting frequency while requiring 3.2 Optical Components 57 a high driving voltage (Bass 1955). Both the Kerr cell and the Pockels cell generate a phase variation with time, which cannot be maintained indefinitely without a sweep-back. This leads to a discontinuity in the resulting Doppler signals. Thus, neither ofthe devices have become common in laser Doppler systems. 3.2.3.1 Diffraction Grating The diffraction grating offers a simple and inexpensive means for beam splitting. Either amplitude or phase splitting using a step or sinusoidal grating can be used. The sinusoidal grating allows higher power to be directed into the diffracted orders. A frequency shift is achieved by rotating a radial grating at a constant frequency. If the assumption can be made that the grating length L is much larger than the grating width b, the standard expressions for diffraction through a slit can be used to express the field strength a distance r from the grating (see Fig. 3.17) (Hecht 1989, Klein et al. 1986). Qp==Cfrg'Texp(-jkr)dA A (3.130) r Here the assumption is also made that both the source and the receiver are far away from the grating in terms of wavelength I/, and that b > 21/,. The field strength in front of the grating g' in Fig. 3.17 can therefore be assumed constant. T is the transmission function for the grating and A is the illuminated area of the grating. For amplitude splitting, the transmission function T can be expressed in terms of the transmission factor Tg x' z y Fig. 3.17. Simple line diffraction grating Diffracl ion grating (area A) 58 3 Fundamentals ofLight Propagation and Optics I for b nb ~ x ~(2n+ 1)- for b ' (2n+l)-<x«n+l)b T(x)=Tg { o 2 n=0,±I,±2,... (3.131) 2 Phase gratings can be even more effective than amplitude gratings. A phase grating can be realized by simply varying the thickness d of a transparent medium with relative refractive index m. The transmission factor is then given by exP(-j 27t (m -1)d) for T(x)=Tg { A o for b nb ~x~(2n+ 1)2 b (3.132) (2n+ 1)- < x < (n+ l)b 2 Further improvements are achieved if the amplitude or phase change is varied sinusoidally across the grating, for instance for an amplitude grating the transmission function becomes 27tX) T(x)=Tg ( l-COSb- (3.133 ) For a phase grating realized through variation of the grating thickness, the transmission function takes the form (3.134) The result of diffraction through a grating can be expressed in terms of the intensity at a point P . 2(2X 7tb (sm . ß -sma . )) sm (;~ (sinß-sina)J x=br' (3.135) The positions of intensity maxima are given by the coherence condition .ß . pA sm p -slna=±-, b p= 0,1,2, ... (3.136) The incident angle a simply shifts the position of the maxima. If the grating is rotated, like in Fig. 3.18, the diffraction occurs on the moving edges of the grating lines. Thus, the grating is a moving receiver for the incident beam and a moving source for the radiated beam. The individual maxima exhibit a Doppler frequency shift, whose magnitude is dependent on the diffraction angle. For a = 0 deg this shift is f sh -- vsinß A (3.137) 3.2 Optical Components 59 Inten ity I r-----~ Laser beam with frcqucm;y I. I. + f ...", ß Fig. 3.18. Rotating diffraction grating where V is the velo city of the grating. For a radial grating with N lines, a grating constant of band a rotational speed of OJ disc the frequency shift becomes v = P OJdisbJdisc fSh(ß p ) = Pb PJ.f"disc N = PJ.Egrid (3.138) The frequency shift is therefore a function ofboth rotational speed and ofthe grating line density. Lines of 5 flm width and 10,000 to 20,000 on one disc are achievable. At rotational speeds of 10,000 rpm, shift frequencies of 10 .. .20 MHz in the first diffraction order are achievable. Using the two first diffraction orders as beams for a laser Doppler velocimeter, the shift frequency doubles. A refined analysis of the rotating grating treats the transmission function as a wave packet or as a wave with wavelength b, which intercepts the incident wave at an angle TC /2 - a and with frequency fg. As a wave packet, the transmission function can be expressed using a Fourier series (Menzel et al. 1974). Tg Tg [ ( OJ . t 2TCX) 1 [( 2TCX)] T(x , t)=-+2cos - --cos 3 OJ gTid. t 2 TC grld b 3 b (3.139) By solving the diffraction integral (Eq. (3.130)) for a Gaussian beam with its waist at the grating leads to the following intensity distribution p=2n+l n=0,1,2, ... (3.140) 60 3 Fundamentals ofLight Propagation and Optics The diffraction integral can be interpreted as a Fourier transform in space, thus the result is simply the convolution of the transformed illuminating field !i with the transform of the transmission function T. For an amplitude grating, all uneven diffraction orders appear, together with the zeroth order. Furthermore, for a Gaussian input beam the intensity distribution in each order is also Gaussian. The zeroth order is unshifted in frequency. The high er orders are frequency shifted according to Eq. (3.138). Their angular positions are given by Eq. (3.136). With proper layout of a sinusoidal phase grating, the zeroth-order beam can be suppressed and a large percentage of the energy concentrated into the ±1 sI order. This requires an optimization of the local grating separation b with the grating wavelength A = v I f grid. This optimization is very difficult for steplike phase gratings. A very elegant method of realizing a phase grating is afforded by Bragg ceHs. 3.2.3.2 Bragg Ce/ls Whereas a sinusoidal phase grating can be achieved mechanically by varying the thickness of the grating material, a Bragg ceH achieves this using acoustically generated pressure waves in a crystal. The basic principles of a Bragg ceH can be described with the help of the sketch in Fig. 3.19. An optimal interaction between the acoustic and electromagnetic waves is given when the vectors of the incident light wave koc' the detlected (refracted) wave k!c and the acoustic wave kac satisfy the relation (3.141) The individual components are resolved as (3.142) (3.143) Fig. 3.19. Bragg ceH 3.2 Optical Components 61 and with Aoc "" Alc "" Ac' Eq. (3.141) yields the diffraction condition ~-A(· m/l,c - ac smac _·P) Sin c (3.144) where A ac is the acoustic wavelength. The extent of the acoustic wave is significantly larger than that of the incident light wave (L» Tm) and thus it can be treated effectivelyas a one-dimensional grating. Therefore, only beam intensity variations in the x direction must be accounted for. The transmission function takes the form (3.145) The amplitude of the density fluctuations or refractive index fluctuations in the crystal Lln are determined by the choiee of material and the applied acoustic power. The acoustic wavelength, A ac ' can be controlled through the acoustic excitation frequency. The solution of the diffraction integral requires integration along the x and z axes; however, even for plane waves, no closed solution exists. Nevertheless, a useful description of acoustic and electromagnetic wave interaction can be obtained using the wave equation, as presented by Chang (1976) and Quate et al. (1965) (Bass Vol. II, part 2, 1995). This begins with the twodimensional wave equation o2~Ax,z,t) + o 21L(x,z,t) ox 2 OZ2 f1 o 2(c(X,t)Qc(x,z,t)) ot2 (3.146) where the subscript c denotes the medium of propagation. The acoustic pressure wave is described by (3.147) whieh leads to spatial and temporal variations of the dielectrie constant B. For an isotropie medium the polarization does not change. The change in B can be expressed in terms of the relative dielectric constant B r and the optical elastic coefficient p e(x,t) = B[ I-BrpSa(X,t)] (3.148) A solution of the wave equation for plane waves is assumed as the sum of a zeroth-order and first-order wave (3.149) with Qoc(x, z,t) = Qo (z)exp[j(f1J oJ- kocx sin a c- kocz cos Pm)] (3.150) Qlc(X,z,t) = Ql (z) exp[j(f1JIJ + klcxsin a c-klczcosPm)] (3.151) 62 3 Fundamentals ofLight Propagation and Optics In general amismatch between the acoustic wave and the light wave must be allowed for, such that (3.152) This arises due to an improper choice of incident angle a or amismatch of the acoustic wavelength. This mismatch has been given by Chang (1976) to be equal to (3.153) The optimal angle of incidence is found under the condition Llk aem . mAo sIna e = - - 2nA ae = 0 to be (3.154) and the Bragg angle at which a maximum intensity can be found in the first order is then (3.155) Using Eqs. (3.147)-(3.151) the relation between the field strength and the acoustic wave can be expressed as dgoc • 1 k 4 *E - - = J - on On1 p Sa le dz 4 (3.156) (3.157) Higher orders can be neglected, assuming only a weak coupling between the acoustic and optical fields (Lle« e). Using the boundary conditions gOe (0) = go and gle (0) = 0, the field strength and the intensity can be obtained. The ratio of first -order intensity at the exit [le (L) to the zeroth -order incident intensity [Oe(O) is a figure of merit for Bragg cells and is given by [ (L) le [oc(O) '" 17 . Llk ae L]2 [ SIn-- 2 &;eL (3.158) where 17 is the efficiency factor of the Bragg cello Assuming equal refractive indexes for each order, the efficiency factor can be expressed as 17 - 3 SL J2 -~MP 2 (~) ( nnP 2A a - 2A20 2 a H o (3.159) 3.2 Optical Components 63 The acoustic power Pa is related to the wave amplitude through 1 3 2 (3.160) Pa =-pv SaLH 2 AU material properties in Eq. (3.159) are found in the factor n6 p 2 M 2 =--3 (3.161) pv Land H are the dimensions of the excitation transducer, p is the mass density of the medium and v is the acoustic velocity therein. For matched conditions (Eq. (3.155» the intensity ratio ofEq. (3.158) reduces to [lc (L) (3.162) --""17 [oc(O) Thus, all quantities influencing the acousto-optical interaction are known. Table 3.3 summarizes the relevant material properties and characteristic quantities for various Bragg ceU media, including allowable wavelength ranges and light powers. Some distortion of the incident beam will occur; however, this distortion can be minimized using a suitably small beam diameter with respect toH. TypicaUy, the center frequency for operation lies between 40 MHz and 500 MHz. The efficiency 17 decreases with increasing acoustic wavelength A ac ' which in part can be compensated by using a high er acoustic power. However this leads to increased power in higher order beams and is therefore not practical over large ranges of driving frequencies. Amismatch of the transducer to the acoustic field away from the center frequency also leads to a decrease in coupled acoustic power, hence also to a decrease in efficiency. Such a dependence is pictured in Fig. 3.20, which also indicates that efficiencies of up to 90 % can be reached under optimal conditions. The phase mismatch of the Bragg ceU can be expressed in terms of the acoustic wavelength A ac as Table 3.3. Materials and their properties used in Bragg cells (NEOS 1997) Polarization Laser power n density lkW cm 2 ] [ -] Material Optical range [nm] Flint Glass SF6 AMTIR Fused Quartz Tellurium Oxide Germanium Gallium Phosphide Lithium Niobate 450-2000 Random 0.12 1060-5000 Random 5.0 185-4500 Rand./Lin. >50 400-5000 Rand./Circ. 35 2000-15,000 Linear 0.5 630-10,000 Rand./Lin. 0.5 600-4500 Linear 0.05 1.8 2.6 1.46 2.25 4.0 3.3 2.2 Acous- Acoustic M 2 XlO- l5 tic velo city mode [kms l ] [m2 W- l ] L L L/S L/S L L/S L/S 3.15 2.6 5.96/3.76 4.26/0.62 5.5 6.3/4.13 6.6/3.6 8 140 1.5/0.46 34/750 180 44/17 7/15 64 ~ 3 Fundamentals ofLight Propagation and Optics 100 ------- !:::- ... ..... , .8 u eS >.. u t:: " TJ (.::i '+-< ~ 50 0.0 0.5 1.0 Acoustic power Pa [W] Fig. 3.20. Diffraction efficiency of a Bragg ceil as a function of acoustic power (AcoustoOptic Modulator A-lOO, A-150 Hoya Optics 1988) (3.163 ) using the characteristic length (3.164) and (3.165) A aeB is the acoustic frequency for the Bragg condition, Eq. (3.155). In this case the phase mismatch becomes L LI<p=-m(m-1)1t 2Lo (3.166) The phase mismatch for the first mode solution of the intensity function, Eq. (3.158), is Lltp=0.451t. The acoustic bandwidth of the first-order beam is then Llfac = 2 fae - facB facB facB (3.167) 3.2 Optical Components 65 3.2.4 Optical Fibers A very high percentage of commerciallaser Doppler and phase Doppler systems today are delivered with fiber optie probes on both the transmitting and receiving side of the system. On the transmitting side, fiber optics link the laser and transmitting optics to the front lens elements mounted in a probe. On the receiving side, the lenses and apertures in a probe are linked to the photomultipliers. The main advantage is the flexibility of probe placement and simplicity in traversing. The bulkier components of the system: laser, Bragg eeH, photomultipliers, ete., can be mounted stationaryand remote from the measurement loeation. This also opens the way for miniaturization of probe heads for special applications. In some particularly novel systems, optical fibers have been used as delay lines (Czarske and Müller 1995) or as pre-amplifiers (Többen et al. 1999). Three types of optical fibers are available with standardized dimensions: step index, graded index and single mode fibers. As their names suggest, they differ in the refractive index profile and in the field distribution of transmitted light through the fiber, as illustrated in Fig. 3.21. The step index fiber has a core diameter in the range 40 .. .400 11m with a cladding index of refraction (n 2 ) about 1 % lower than the core (n j ). Light is transmitted through total intern al reflection, thus there will exist a limiting launeh angle BA above which the light will not be transmitted and will exit to the cladding. This angle ean easily be deterrnined by applying SneH's law for the ease of Y taking the limiting value, Y = YR: (3.168) and assuming the refractive index outside the fiber to be unity. For total intern al reflection COSYR . n2 nj =Slna=- (3.169) and thus NA = sinBA =~n~ -n~ =nJiJJ (3.170) where Li is the relative index of refraetion difference. (3.171) The quantity sinB A is known as the numerieal aperture (NA) and expresses the range oflaunch angles whieh will result in transmission. Some numerieal examples of the numerieal aperture for various fibers are given in Table 3.4. The large NA for step index fibers facilitates the coupling of light into the fiber. On the other hand the maximum path-Iength inerease between an axial ray and one entering at BA amounts to a factor 11 eosy and this contributes to dispersion. Thus, the step index fiber is not suitable for single mode transmission. 66 3 Fundamentals ofLight Propagation and Optics a r 11 medium 11 m b 11 "J "'(" -- \ ~ - -- _._-- - c r IJ a -, ,, ,, , ~1I2 , _.J. _____ " fit 1/ 2 11 - - - _.- - -- _._._.- - - -~ _.- - - - I'J - _.- - - - - - - ", Fig. 3.21a-c. Types of optical fibers. a Step index fiber with ray path, b Graded index fiber with ray path, c Single mode fiber with field distribution Its main use in the present context is on the receiving side of the optical system, where the ease of coupling light into the fiber can be exploited. The graded index fiber exhibits a parabolic profile of refractive index over the core, usually expressed as apower law for p~1 for p>1 (3.172) where p = r lais the non-dimensional radius and p = 2 for graded index fibers. The limiting launch angle is now dependent on radius (3.173) 3.2 Optical Components 67 Tab1e 3.4. Typical fiber specifications for A. = 850 nm Fiber Core di- Numerical Limiting ameter Aperture launch angle NA [-] a [flm] BA [deg] relative index of re- Attenuation fraction difference [dBkm 1] LI [%] Step index Graded index Single mode 40 25 5 1.45 0.91 0.23 0.25 0.2 0.1 14.5 11.5 5.7 4.5 3.0 3.0 and the numerical aperture is understood as being the extreme value taken over the core radius. The graded index fiber is not normally employed in laser Doppler systems. As the core radius of a fiber decreases, the number of possible propagation modes of light also decreases. The single mode fiber has core radii of 3...6 !J,m and, dependent on wavelength, allows only one mode to pro pagate. This condition is met when the normalized frequency 21ta v=;:NA max (3.174) lies below certain limiting values. For a step index fiber this value is V c = 2.405, for a graded index fiber (p = 2) V c = 3.6 (Kokuben and Iga 1980), and is known as the mode cut-off condition. Manufacturing tolerances prohibit single mode, graded index fibers in large quantities. The transmitting optics of laser Doppler and phase Doppler systems use exclusively single mode, step index fibers, with the associated stringent requirements for low-Ioss in-coupling. From Eq. (3.174) it is clear that single mode fibers should be wavelength matched, hence, color separation in a two or three-velocity component measurement system is recommended before coupling into the fibers. However on the transmitting side, also the polarization of the light must be preserved and this is generally not the case in fibers, even if the light is linearly polarized at the launch end. Small imperfections in the core shape or intern al stresses through bending or twisting lead to intrinsic or extrinsic birefringence. These effects are avoided by the use of polarization preserving, single mode fibers, which are slightly ellipsoidal or pre-stressed in the cladding to create a very high linear birefringence. This leads to a preference for polarization in one direction. Typical pre-stressing elements are illustrated in Fig. 3.22 for the PANDA fiber (polarization and absorption optimized, Fujikura) and the Bow-tie fiber (sometimes referred to the Hi-Bi fiber, York 1986). These are often elements with different coefficients of thermal expansion, which then induce a stress when the fiber is cooled after the drawing process of manufacturing. Of course the polarization preserving (PP) fiber must be properly aligned with the polarization direction of the input beam. All fibers exhibit a transmission loss due to absorption, scattering and leakage. These losses are expressed as an experimental power attenuation or damping over a length L 68 3 Fundamentals of Light Propagation and Optics a b Fig. 3.22a,b. Fibers showing pre-stressing elements in cladding to preserve polarization. a PAND A fiber, b Bow-tie fiber (3.175) where the coefficient ß is usually given in dB km-I and is wavelength dependent. Losses in optical fibers used in the telecommunications industry are very low; however, these operate at wavelength windows far above visible light. The theoreticalloss limits are dictated by Rayleigh scattering from molecular density fluctuations, which are independent of material concentrations and defects. Since Rayleigh scattering goes with A,-4, this loss increases strongly towards visible wavelengths. At 488 nm the absorption is about 33 dB km-I, while at 1.06!JlIl it is about 2 dB km-I. The small core diameters of single mode fibers lead to high power densities, which eventually can damage the fiber. Special fibers are available which use no core doping but rather a cladding dopant to achieve index of refraction variations. Power levels of 5 W at 488 nm are achievable. The damage threshold for pulses of short duration, for example as used with particle image velocimetry, is 1.4 kJ cm-2 at 532 nm for a pulse of duration 15 ns, in a fused silica fiber. The largest losses when using fiber links are usually associated with the coupling losses when entering the fiber. These can easily be as large as 50%; however, careful layout and alignment of the coupling can reduce this value to 20% .. .30%. For a single mode fiber the fundamental propagation mode (LPoI : linear polarization) can be approximated weH by a Gaussian beam profile and is thus weH suited to the transmission oflaser light originating from a laser, lasing in the TEM oo mode. Conditions for coupling a laser beam into a single mode fiber include achieving an appropriate spot size at the fiber end face, with a numerical aperture smaller than that given by the fiber and also that the beam axis is properly aligned (translation and two tilt angles) with the fiber axis. The spot size is generally matched to the core diameter and can be achieved using the collimation guidelines relations given in section 7.2.l. The beam waist (e-2 ), after focussing through a lens of focallength f, is given by Eq. (3.129) as (3.176) 3.2 Optical Components 69 and should be approximately 1.1 times the core diameter. The far-field divergen ce ofthe beam is given by Eq. (3.69) A. (3.177) ab=_b1trwb and should be less than BA' generally not a difficult condition to meet. The remaining alignment is however critical and can afford up to six different degrees of freedom to achieve optimal results. A very useful element for coupling a laser beam into a fiber is the graded index lens, marketed by Nippon Sheet Glass under the trade name SELFOC (selffocussing). The lens is essentially a graded index fiber of larger diameter and shorter length. The index of refraction is given by (3.178) where A is the gradient constant. The ABCD matrix (see section 3.2.1 and Fig. 3.23) is given by (3.179) For a parallel input beam (nB1 =Odeg) at z =0, Eq. (3.179) yields (3.180) Y2 = YI COS.,fAZ2 thus the beam propagates in a sinusoidal manner in the lens with period p = 21t / -JA. For a Gaussian beam the propagation in terms of the parameter q (section 3.1.4.2) can be described using - r 11 COS.,fAZ2 + (.,fAn o sin .,fAz q = _2 -q_I .fAno sin.fAz 2 + COS.,fAZ2 r (3.181) y I/(r) Fig. 3.23. Ray path in a SELFOC lens 70 3 Fundamentals of Light Propagation and Optics Ifthe lens is exactly P /4 long (X - pitch), an input beam with its waist on the front lens surface will be focused to a waist at the lens back surface, with a radius (3.182) As a numerical example, a X - pitch SELFOC lens with -JA = 0.0966 mrn -1, Z = 16.26 mm and a wavelength of 632.8 nm is considered. An input beam of diameter 1 mm exists with a spot diameter of 2.7 /-Lm, quite sufficient for coupling into single mode fibers. A further advantage of the SELFOC lens is its relative insensitivity to translation. This is seen directly from the lens matrix, in which for a X - pitch lens the Y2 co ordinate no longer depends on Yl. A coupling efficiency of 70 ... 80% can be achieved using SELFOC lenses. Due to aberrations, an improvement can also be expected for lenses with a slightly shorter length (10 ... 20 /-Lm) than X- pitch (Nicias 1981). Some examples of laser Doppler systems employing SELFOC lenses for fibers coupling can be found in Jones et al. (1985), Hironaga et al. (1985) or Stieglmeier and Tropea (1992). no = 1.547, 3.2.5 Photodetectors Detection of scattered light is achieved using various photodetectors, including photomultipliers (PM), PIN diodes or avalanche photodiodes (APD). The choice depends on the wavelength, the desired response time and the intensity. The detector determines to a large extent the overall sensitivity of the system. An overview of photomultiplier and avalanche photodiode characteristics, together with further references can be found in Dopheide (1995) and Bude (1980). The photodetector influences the system in different ways: • The absolute spectral sensitivityl Sc ().,) (generated current / incident power) or the quantum efficiency l}q().,) (generated electrons / incident photons) determines how effective the light is transformed into a current. • The sensitivity of a photodetector is limited by the measurable current arising when the detector is covered - the dark current i dc • • Due to the response time of the detector, most fluctuations in the light intensity above a frequency, je, of about > 10 8 Hz will be averaged, thus a modulated light beam will result in a DC signal at the detector. Any fluctuations of intensity below the limiting frequency, fg, will be followed virtually exactly. • Photodetectors are often combined with amplifiers. In the case of photomultipliers the amplifier is realized directly in the dynode chain, in APDs, amplifiers are often integrated onto the same chip. In this way a strong amplification can be achieved before noise enters the electronic system, hence achieving low noise amplification. 1 In the literature sometimes the spectral sensitivity is also called spectral response or radiant sensitivity. 3.2 Optical Components 71 The power deteeted is ealeulated by integrating the Poynting veetor (Eq. (3.7» over the surfaee of the deteetor, whereby an averaging oeeurs for all eontributions above the limiting frequency Je (Eq. (3.29». This averaging oeeurs due to the inertia of the electron emissions. Pr =~ fJ JC~(t) x!!<t») dt·dA r ; A, T = c; fJ ~II .~I; dAr (3.183) JJ = I dAr A, A, In this equation the parallel symbol refers to veetor eomponents of the seattered field perpendieular to the normal surfaee veetor Ar. Only the component perpendieular to the deteeting surfaee contributes to electron emissions. Radial components of the eleetrie field veetor result in a Poynting veetor component parallel to the deteetor and represent losses. The produet with a eomplex eonjugate is equivalent to optieal mixing (heterodyning see seetion 3.1.5). This power eorresponds to aquanturn eurrent equal to dN q =_r P i = __ q dt hf (3.184) where f is the frequeney of the light and h is Planek's eonstant. This quantum eurrent generates an eleetron eurrent. The ratio of generated eleetrons to incident photons is ealled quantum efficieney 17 q (/L,} and depends on the wavelength A of the ineident light. The quantum effieieney is related to the speetral sensitivity through the relation (3.185) where q is the elemental eharge and c the speed of light. Figure 3.24 illustrates the sensitivity for the three most eommon photoeathodes. Curves for quantum efficieneies 17 q have been included. Figure 3.24 also illustrates typieal speetral sensitivity and quantum efficieney for semiconduetor deteetors. Semieonduetor deteetors exhibit mueh higher quantum efficieney than photoeathodes. For a quantum efficieney of 17 q , the quantum eurrent i q is transformed into an eleetrie eurrent equal to dN q - q17 q ceJJE ·E* dA e-q17 q dt - hf" 2 -11 -11 r i - J (3.186) A, whieh shows that the eleetrie eurrent density S (see Eq. (3.3» generated from the deteetor area is direetly proportional to the incident intensity (3.187) 72 3 Fundamentals ofLight Propagation and Optics "i \., ,, , 300 400 500 600 700 800 900 1000 Wavelength 2 [nm] Photocathode S-20 (Na-K-Sb-Cs) Photocathode S-ll (Sb-Cs) Photoeathode S-l (Ag-O-Sc) ---0-- Semineonductor Si- Diode Fig. 3.24. Spectral sensitivity 5,(,1) and quantum effideney 17iA) for photoeathodes and semieonductor detectors and the output electric current is just the integral of this intensity over the detector surface i e =ffSdA Ar = qTJ q r hf ffldA = qTJ q p r hf (3.188) r Ar In PMs and APDs the generated signal is amplified by the gain factor G. For PMs it is the ratio of the anode current to the generated photoelectric current from the photocathode. A PM with n stages (dynodes) and a secondary emission Because the secondary ratio of per stage has a current amplification of G = emission rate can be directly changed with the supply voltage, also the gain factor is very sensitive to any change of voltage. The output current of the detector is given by an . a ir =G qTJq hf ffldA r (3.189) A. In most cases an intern al electronic amplification M is integrated into the detector. Furthermore, the dark current from the cathode i dc , generated in the case of no illumination, is amplified by G and results in a detector signal of idca from the anode (index a) and this limits the sensitivity of the detector. The source of dark current is the thermal emission of electrons. Cooling the detector, which for some practical measurement tasks is necessary, can reduce the dark current. 3.2 Optical Components 73 Since all detectors involve a time averaging above a limiting frequency, the output current of a detector in a laser Doppler or phase Doppler system will contain both a DC and AC (modulated) part. (3.190) where m is termed the visibility or modulation depth. The visibility expresses the ratio of the AC to the DC part and is given by m = i AC iDC = imax -imin with imax + im;n imax = max(i r) (3.191) im;n = min(i r) Noise limits the accuracy with which any parameter can be determined from the signal. One method for quantifying the noise level is the signal-to-noise ratio (SNR), which expresses the ratio of the power in the signal Ps to the power in any added noise PN , given in decibels SNR/ =1010g /'dB 10 ~=20Ig iefis P 10. N (3.192) lefi N Equation (3.192) indicates that the signal and noise currents in the terminating resistor R of the receiving electronics can be replaced by their respective powers. The amplitude of the AC part can be computed with the help of the visibility m and the amplification factor G, i AC = m Gi DC • The total noise in an amplification chain is determined to a large extent by the noise in the first element of the chain, in this case the photodetector. For this reason two noise sources at the detector will be examined in more detail, shot noise and thermal noise. 3.2.5.1 Shot Noise Shot noise is unavoidable and integrally related to the electron emission process. It is a white noise, whose magnitude is related to the available system bandwidth and the signal amplitude i'hot =~2q,dfi (3.193) Often shot noise is referenced to the bandwidth (3.194) or expressed from Eq. (3.188) as an optical noise power (3.195) The sensitivity of a photodetector is limited by the measurable current arising when the detector is covered, the dark current idc • The dark current is also a 74 3 Fundamentals ofLight Propagation and Optics white noise source related to the electron emissions and can be directly added to the shot noise. The cathode dark current of a PM with gain factor G can be changed into an equivalent noise input power. For a signal power of (3.196) the SNR is unity. The internal amplification of the detector, either the dynode chain in a PM or the amplifier of an APD, adds noise to the signal and is accounted for by a noise increase factor F. This factor is simply the ratio of the signal power output including noise to the signal power output when a noise free amplification is assumed. For example a PM with n stages in the dynode chain, each with the same multiplication factor of 0 and an internal amplification G, exhibits an output current noise amplitude of i'hot = G 2qLi! {1+ ~ + ~2 +... + ~ J (3.197) Comparison with Eq. (3.193) for a noise-free amplified signal i'hot = G~2q Li! i (3.198) and squaring for the power ratio gives 1 1 1 FpM =1+8"+Y+···+ 0 on = 0-1 (3.199) For an APD with an intern al amplification in the p-n junction, the process is more noisy, as shown by McIntyre (1966), and leads to a current noise amplitude (3.200) and a noise increase factor of (3.201) FAPD =G 3.2.5.2 Thermal Noise Thermal noise (index t) arises from spontaneous fluctuations in electron current in any resistance. It is a white noise and limited only by the system bandwidth Li! . The working resistance R is a parallel circuit consisting of the terminating resistor of the photodetector R d and the input resistance of the subsequent amplifier R; 1 1 1 R Rd R; -=-+- (3.202) 3.2 Optical Components 75 The equivalent voltage due to this noise is given as J (3.203) u, = 4k TRL1f where T is the temperature in degrees Kelvin and k is the Boltzmann constant. The thermal noise voltage can be converted into a thermal noise current (3.204) 3.2.5.3 Photodetector Model An equivalent circuit of a photodetector and a first amplification stage is pictured in Fig. 3.25. The quantum current iq (Eq. (3.184)) in the cathode gives rise to an electric current ie with a shot noise current ie sho, given by Eq. (3.194). Thermal emissions at the cathode lead to a dark current idc and an associated shot noise current Idc shot' The internal amplification with a factor G, amplifies both signal and noise. The noise sources of the intern al amplifier are taken into account through the factor .JF. The anode current of the photodetector contains therefore (3.205) and the associated noise of these parts (3.206) The detector is terminated with a resistance R d connected in parallel to the input resistance Ri of the first amplifier stage. Each resistance leads to a source Genera I ion Genera Iion of signa l of noise ~~ '-----y-------''' ' - - y - - " Elcclron emission olthe calhodc Internal amplitlcr ~~----y-----~" Amplilicr Fig. 3.25. Equivalent circuit of a photodetector and a first amplification stage 76 3 Fundamentals ofLight Propagation and Optics of thermal noise, shown as a voltage source. The input current of the external amplifier is given by (3.207) The detector has a capacitance Cd' which together with the resistances R d and R; in parallel, represent a low-pass filter. Assuming that the signal does not vary more than 1 % about the maximum value, the cut-off frequency can be approximated by (Durst and Heiber 1977) f = c 0.14 (_1 21tCd Rd +~J R; " 0.14 ~ (3.208) 21tC d R; Use has been made ofthe fact that usually R d » R;. The time constant CdR; has typical values of 10-9 s (R; = 100 n, Cd"" 10 pF) and thus, the cut-off frequency lies around 20 MHz, although much higher values are also achievable. The signal and the noise is amplified with M a and Ma-JF,: respectively and generates the output signal U s and output noise uN • 3.2.5.4 Signa/-to-Noise Ratio (SNR) The SNR of the photodetector can be derived using the definition given in Eq. (3.192) and considering the alternating part of the signal and the noise component (3.209) Usually data is available on the internal amplification factor G, the anode dark current i dcAn and the absolute spectral sensitivity Sc (.,1,). Using these parameters, together with the incident power Pr' Eq. (3.209) can be reformulated as (3.210) The system bandwidth Af may be limited by use of a low-pass filter with a cut-offfrequency corresponding to the highest expected flow velocity. However, if no such filter is employed, the cut-off frequency is determined by the capacitance and resistance ofthe photodetector, see Eq. (3.208). Figure 3.26 shows typical curves of SNR for a PM and for a semiconductor detector. Ideal conditions for M ~ 00 and m = 1 at a bandwidth of 1 MHz are shown. Curves are also included for m = 0.5 . The curves exhibit two distinct regions. At higher power levels shot noise dominates and a JP: dependency is observed. At low power levels the dark current dominates and the SNR is proportional to Pr. 3.2 Optical Components 77 --Ideal - - - Photomultiplier, m=l --Photomultiplier, m=O.5 -- .... --Photodiode, m=l ---- -- Photodiode, m=O.5 -50 -100 L....l...J....l.WJll--1....L.JJJWL--'-J...L.IJLWL.-L..L.ll.LWL...J....l..1.WIlL-.L..l..JLl.UJIL....l...J....l.WIIl--L.J..llW1l.--L.JLJ..llWL..J...I.J.U1IIJ 10. 14 10~ 10' Signal Power P, [W] Fig. 3.26. Dependence of SNR and signal power level (for ,1j = 1 MHz, R = 100 n, T=300K) The SNR of semiconductor detectors at high power levels deviates only marginally from its maximum attainable value. The disadvantage of semiconductor detectors lies in the relatively high values of dark current, which lead to an earli er bend in the curves of Fig. 3.26. The dark current of semiconductor detectors lies in the range I d "'1...10nA, whereas for photomultipliers the value is I d ::::lnA. Semiconductor detectors are therefore wen suited to medium and high scattering light intensity (Dopheide et al. 1990). Their compact form and low power consumption are particularly advantageous. For low light levels the PM is preferable, in extreme cases with auxiliary cooling, e.g. photon detection. Further analysis of the SNR must include subsequent amplification stages. For N stages the noise power becomes N-l Pn = 4kTLlf F,I1 Mn = 4kTLlf F,M, (3.211) n=l where M s is the amplification factor and F , =F, + F2 -1 + F3 -1 + ...... +~ 1 M M M 1 1 I1M N-l 2 (3.212) n n=l is the noise factor of the amplification chain. The first stage is the photodetector and is clearly instrumental in determining the final noise power and therefore deserves particular attention. The rise time of photodetectors is approximately r '" 1...5 ns so that waves above a frequency of f '" 10 8 Hz in Eq. (3.183) are averaged. Note that the dura- 78 3 Fundamentals of Light Propagation and Optics tion of a photon is about r phot"" 2 x10 6 To , thus for A = 632.8 nm, r phot = 4 ns (Orear 1987). However the photodetector is also a low-pass fllter due to its circuitry as represented in Fig. 3.25, with a cut-off frequency of 1 fc= (3.213) eR a a lying in the same order of magnitude. Thus, the low-pass filter characteristic also leads to an integration effect (Eq. (3.208». Above this cut-off frequency any signal modulation will be attenuated accordingly. The lowest cut-off frequency in the signal processing chain will dominate the low-pass filter effect and the amplitude attenuation will take the form I AC 1 = r I DC ----r===== 1+(2n j (3.214) J The SNR then becomes (3.215) 2/2 M 2 Pq(IDC 2kT(R +R)ßi + I dc )+ a, LI!f RR. a , This effect is particularly important for high signal frequencies, as shown in Fig.3.27. 'i:Ci' ~ -==------..... _-::..:.-:-.~, - - ..... _..._ ..... ~ 30 I - - -...... ,, 20 10 o ,, ,, ,, Cut-off frequency J; 1 MHz 10 MHz 100 MHz ... 1 GHz 10' .... ,, ,, ,, ,, , 10' .•. ,, ,, ,, 10' ,, ,, ,, , ,, ... ,, , ,, ,, ,, , 10' 108 Signal frequency Fig. 3.27. Influence of the cut-offfrequency on SNR ,, f [Hz] 80 4 Light Scattering irom Small Particles lutions of the scattering problem are required which can also account for such phase and amplitude distributions over the surface of the scattering particle. To begin, a plane wave will be assumed and GO and LMT solutions for small spherical particles will be discussed. Subsequently, these solutions will be modified to account for an inhomogeneous incident field. One method of achieving this is to decompose the inhomogeneous incident wave into a spectrum of plane waves and to apply the principle of superposition to a large number of plane wave solutions. This leads to the Fourier Lorenz-Mie Theory (FLMT). The principles of geometrical optics can also be extended to account for inhomogeneous fields and since a comprehensive description of this approach is not yet available in the literature, a detailed discussion of this method will be given. The method will be denoted Extended Geometrical Optics (EGO). It provides the basis for the derivation of the phase differences and time shifts of the detector signals used in the phase Doppler and time-shift techniques presented in section 5.3. It is beyond the scope of this book to discuss all possible approaches to treat the light scattering in homogeneous and inhomogeneous fields for different types of particles. For instance Gouesbet et al. (1985, 1989) and Grehan et al. (1986) have generalized the LMT to be applied to a particle arbitrarily placed in a Gaussian beam (Generalized Lorenz-Mie Theory, GLMT). Extensions for cylindrical and layered particles are derived within the framework of more recent GLMT developments (Gouesbet and Grehan 2000). Besides purely analytical solutions, there exist a number of numerical methods, in general for smaller particles but applicable for inhomogeneous incident waves and non-spherical particles. Using field theory, there are two basic approaches to the light scattering problem: solving the wave equation with respect to the boundary conditions or to relate the solution to known source problems. Examples for the former are the Extended Boundary Condition Method (EBCM) (Barber and Yeh 1975, Iskander et al. 1983) or the T-matrix method (Waterman 1965, Varadan and Varadan 1980). The Coupled Dipole Method (CDM) also known as Discrete Dipole Approximation (DDA) or digitized Green's function method (Purcell and Pennypacker 1973, Singham and Salzman 1986, Lakhtakia 1990, Hoekstra 1994) is an example far the latter type of approach, using known dipole solutions and replacing the particle with an appropriate combination of coupled dipoles. An overview of some methods for the computation of light scattering from small particles, limited to those methods often used for laser Doppler and phase Doppler system calculations, is given in Table 4.1. For further details about other methods to compute the scattering properties of small particles the reader is referred to the extensive literature in this field (e.g. Yeh et al. 1982, Barber and Hill1990, Harrington 1987, Hafner 1990, Hafner and Bomholt 1993, Doicu et al. (2000». A review of elastic light scattering theories is presented by Wriedt et al. (1998). 4.1 Scattering of a Plane Wave 81 Tah1e 4.1. Approaches often used in laser Doppler and phase Doppler for the computation oflight scattered by small spherical particles in a plane wave and their extensions to inhomogeneous waves Computations oflight scattering from spherical particles Geometrical optics approach dp »Ab Wave solutions Incident plane wave Geometrical Optics (GO) Parallel rays oflight incident on the particle surface are considered which contribute to the detector signal. The scattering of these rays is computed under the assumptions and using the rules of geometrical optics. Lorenz-Mie Theory(LMT) The incident plane wave is replaced by a sum over spherical waves in space. The wave equations are then solved for each spherical wave and a superposition follows. Incident inhomogeneous wave Extended Geometrical Optics (EGO) The inhomogeneous wave is approximated over small surface segments of the particle as locally homogeneous and plane. For each surface segment the GO rules are valid and can be applied to the collection of rays which reach the detector. Fourier-Lorenz-Mie Theory (FLMT) The inhomogeneous wave is expanded into a number of plane waves by a Fourier transform. For every spectral coefficient the LMT is applied. The superposition of the partial waves on the detector surface represents the inverse transform. Generalized Lorenz-Mie Theory (GLMT) The incident inhomogeneous wave is replaced by a sum over spherical waves in space. The wave equations are then solved for each spherical wave and a superposition follows. 4.1 Scattering of a Plane Wave The light scattering according to geometrical optics (GO) and Lorenz-Mie Theory (LMT) is wen documented in the literature (Kerker 1969, Born and Wolf 1980, Born 1981, van de Hulst 1981, Bohren and Huffman 1983) and therefore only results will be discussed below. An extension for arbitrary polarization will be presented (Arst et al. 1990), as weH as the use of Debye series (Hovenac and Lock 1992), which provides a correspondence of the wave solution to the scattering orders of GO. These weH-known solutions oflight scattering of a plane wave from a spherical particle make several assumptions. • The incident wave is homogeneous and plane, meaning that the incident wave amplitude is constant in all space and that the phase is constant in every plane perpendicular to the direction of propagation, see section 3.1.1.1. The wavelength Aw and the wavenumber k w of the incident wave is connected 82 4 Light Scattering from Small Particles with the vaeuum wavelength ,1.0 and wavenumber in vaeuum ko through the refraetive index of the surrounding medium nm : ,1. w = ,1.0 I nm , k w = k onm • • The particle is spherical, homogeneous and isotropie, exhibits no fields itself (p p = 0), has a relative magnetic permeabilityI of Ji p = Ji o' a complex refraetive index of I1 p and a diameter of d p • • The surrounding medium is infinite, homogeneous, isotropie, exhibits no damping of electromagnetic waves (Km = 0), exhibits no field itself (Pm = 0), has a relative magnetie permeability of Ji m = Ji o and areal refractive index of nm · In reality nearly none of these assumptions are strictly fulfilled; however, the complexity of the scattering process would otherwise not allow a closed form solution for any configuration. The nomenclature in the following discussion adheres to that given in Fig. 4.1. The incident plane wave is defined in a wave eoordinate system (WeS) and propagates in the Zw direetion with an arbitrary polarization in the X w - Yw plane of ~w. The origin of the wes is loeated at the center of the particle. The referenee system for the polarization direction is the scattering plane defined by the wave vector of the incident plane wave k w and the seattering veetor aligned between the particle center and the detector r pr • The angle between these two vectors is known as the scattering angle tJ,. The scattering plane lies at an angle qJ, to the xw-z w plane. The polarization in the e<ps direction, perpendicular to the seattering plane, is associated with the §.l scattering function and the parallel polarization in the el1s direction is associated with the §.2 seattering funetion. Different seattering orders will be specified by a superscript (N) • As a first step, the eleetric field veetor of the incident wave must be projected pherical system coordin~ t e ~~~-L Y. Inciden t lalle wave Fig. 4.1. Definition of the wave co ordinate system (WeS) and the seattering coordinate system (SeS) 1 The condition for the magnetic permeability is easy to avoid (Kerker et al. 1983). 4.1 Scattering of a Plane Wave 83 onto the two polarization directions perpendicular (lPs) and parallel (zJ,) to the scattering plane. To determine the scattered field strength, the scattering functions must be calculated and applied to this incident field. (4.1) The zJ s component and the fP s component of the scattered field ~s' as weH as the scattering vector rpr ' are all orthogonal to one another. They define the scattering coordinate system (SCS). The X s axis of this system is aligned with the scattering angle zJ" (ezJs>' the Ys axis is aligned with the angle lPs> (eq>,), and the z s axis is along the scattering vector rpr ' (e pr)' The origin of the SCS is located at the point where the properties of the scattered field are sought, generally where the receiver optics are to be placed. Two dimensionless parameters, caHed Mie-Parameters, are important for the scattering nd p Aw x =-M (4.2) n nd YM=mxM=.:=l'...~ nm (4.3) /("w where m is the relative refractive index of particle and medium: e.g. m > 1 for particles and m < 1 for bubbles. The particle properties enter through the scattering functions §.! and §.2' This chapter is mostly related to the methods for determining these scattering functions and their properties. For laser Doppler and phase Doppler simulations more than one incident wave is used. For this reason it is helpful to define for every incident wave its own co ordinate system (WCS) and relate these systems to a global reference, main coordinate system (MCS). Furthermore, the position of the receiver(s), the particle and the scattering co ordinate system can also be related to this main coordinate system. In Fig. 4.2 the four co ordinate systems and their relations are illustrated. The superposition of several scattered waves, for instance from different laser beams and/or scattering orders, is performed in the receiver coordinate system (R CS), also shown in Fig. 4.2. The x r - Y r plane lies parallel to the detector surface and zr is aligned perpendicular to it. The origin of the RCS can be, for instance, the center point of the receiver surface. The angles between the directions of the scattered field components e zJs ' eq>s and e pr in the scattering co ordinate system and the unit vectors of the RCS, e rx , ery and erz> are respectively ß(zJ,q>,r)(x,y,z)' Normally for the superposition of scattered fields only the zJ s and fPs components are important, since the detector surface is aligned perpendicular to the 84 4 Light Sca ttering from Small Particles Deleclor surface Plane wave y z Fig. 4.2. Coordinate systems for the computation of light scattering by a particle in a plane wave scattering direction e pr and, in the far field, the component along the scattering radius can be neglected. A transform from the SCS into the RCS is achieved using an appropriate directional matrix !irxJ (cosßax COSß<f'XJ E =M E ß-s ~r =( !iry = cosßtJy cosßrpy -., (4.4) The directional cosines are given by the scalar product of the unit vectors eos and e~s of the ses with the unit vectors e rx and e ry of the RCS, all components expressed in the MCS. For real detectors, the components of the scattered field will vary across the detector surface. Also the orientation of the SCS over the detection surface must be considered. To deal with this the detector surface is sub-divided into small segments, such that the scattered field can be considered constant over each segment. For each segment the components of the scattered field are computed and then projected with Eq. (4.4) onto the RCS. Note that in special applications with curved detector surfaces the RCS can be also vary for each detector segment. On every segment n of the detector the received ligh t intensity is (4.5) and thus the DC and modulated part of the current in magnitude and phase can be computed. The sum in Eq. (4.5) is performed over all incident waves N w on the particular detector segment. The total signal power Pr received by the detector is obtained by summing over the power on each of the N detector surface segments. The power of each 4.1 Scattering of a Plane Wave 85 segment is the intensity, Eq. (3.83), times the segment size An' Eq. (3.183). For N segments and N w incident electromagnetic waves this leads to (4.6) which can be expressed as a DC part (see section 3.1.5) (4.7) and a modulated part The AC part can be expressed with an amplitude, related to the DC part with the visibility m , and a phase of a eosine function. PAC = P AC eos( rp AC) , (4.9) For a known geometrical configuration the task is now to determine the scattering functions in Eq. (4.1). This is the most eomputationally expensive part in light scattering calculations and will be presented in the following sections. 4.1.1 Description using Geometrical Optics (GO) Light scattered from an air bubble in water was eomputed by Davis (1955) using geometrical optics. Glantsching and Chen (1981) examined scattering from a water droplet in air. The principles the GO approach have been summarized by van de Hulst (1981). The validity of GO is only given for particles fulfilling dp » /Lw. To eompare results of the GO method to solutions given by the LMT, diffraction and surface waves must also be accounted for. Surface waves will not be eonsidered in the following analysis, however Hovenac and Lock (1992) give a detailed treatment. The effect of diffraction will be treated below. 4.7.7.7 Diffraction (oth order) The diffraction of an electromagnetic wave by a spherical particle can be weIl approximated using Fraunhofer diffraction from a circular disCo Solutions for diffraction of a plane wave through a circular hole can be found in standard textbooks (Born and Wolf 1980, Klein and Furtak 1986, Hecht 1989). Considering the geometry of Fig. 4.3, the field strength due to diffraction at a point Pis given by 86 4 Light Scattering from Small Particles A. Plane wave x •. k •. catleri ng pla ne Fig. 4.3. Diffraction geometry (4.10) Equation (4.10) can be approximated in the far field (rpr » 1989) as d~ I Aw , Hecht (4.11) with the Bessel function of the first kind and first order Jl (x). Equation (4.11) already takes the Babinet principle into account, i.e. the fact that diffraction from a circular disc exhibits a phase shift of n; compared to diffraction through a circular hole (change of sign, van de Hulst 1981). The diffraction by the particle is independent of polarization direction and can be summarized by the function S(O) =S(O) =x 2 JJxMsintJ,) ex (_.~) _ 2 M '.Q P J2 _1 xM (4.12) slnus In the following treatment the diffraction will be denoted as the scattering of order N=O. In the discussion ofthe Debye series (section 4.1.2), the combined contribution of diffraction and reflection are considered in the Debye order p=l. 4.7.7.2 Reflecfion and Refraction Whenconsidering the scattering of an electromagnetic wave from a spherical particle using geometrical optics, the intensity and the phase of the reflective 4.1 Scattering of a Plane Wave 87 and refractive components can be considered independently. These two properties of the scattered wave will therefore be discussed separately. Intensity. For a homogeneous spherical particle the intensity of the different scattering orders is influenced by two fundamental factors: • Amplification or attenuation due to the geometry of the ray paths; i.e. focused or divergent rays. • Separation of the primary field into reflected and refracted rays (absorption will be neglected) at every interface according to the Fresnel equations (see section 3.1.3). Influence oi the Geometry oi the Ray Paths on the Intensity. Following the derivation ofDavis (1955), the amplification or attenuation of the scattered field due to the geometry of the scattering process is to be expressed in terms of gain factors G(N). Considering a differential circular surface element dA of the incident plane wave centered around the Zw axis (see Fig. 4.4) dA = nr;[ sin 2 (B; + dB;) - sin 2 B;] = (4.13) nr; sin(2B; +dB; )sin(dBJ The area ratio to the projected scattered field dA(N) can be analyzed for each scattering order. The light power dPdA distributed over the area dA is given by the incident intensity ofthe plane wave I w ' Considering a detector placed at an arbitrary point in space at a distance rpr from the middle of the scattering particle, the sum of all possible detector positions will be simply a sphere centered around the particle. The illumination area dP dA Fig. 4.4. Reflective and refractive components of a cylindrical light beam of vanishing thickness impinging on a bubble (Davis 1955) 88 4 Light Scattering from Small Particles on this sphere due to reflection (N = 1) can be given for the far-field condition rpr » d~ I /Lw as dA(1) = 4n:r:r sin(2e; +de; )sin(de;) (4.14) The area ratio leads to an intensity being received on the detector surface due to reflection of 2 dA rp (1) Ir = I --(1-) = I w - -2 dA 4rpr W (4.15) As a comparison, a point source at the center of the particle with an intensity I PO at the particle radius, would generate an intensity I p at radius rpr of (4.16) The gain factor for reflection is simply the ratio of these intensities 1(1) 1 Ip 4 G(1) =_r_ _ (4.17) For higher scattering orders (N > 1) the gain factor is a function of the angle of deviation D(N) (at the N th surface) of the N th order. From the scattering geometry, the area dA (N) is given as (4.18) With reference to the power falling onto the circular ring area dA, the intensity oflight over the area dA (N) is given by I (N) r dP dA =--=1 - - = 1 dA(N) W dA(N) W r; sin(e;) co s(e;)d e; · (D(N))dD(N) rpr2Sin (4.19) Elementary geometrical considerations lead to a relation between the angle of deviation D(N) and the incident angle e; D(N) =2(N -1)arcsin( si:e; )-(N -2)n:-2e; (4.20) The gain factor for N>1 is then given by a combination ofEqs. (4.16), (4.19) and (4.20) G(N) = sine; cose; ------'-----'''-'---r=====1 sinD(N) cose. 2(N -1)--' -2 m (4.21) 4.1 Scattering of a Plane Wave G(N) = sinBicosBi sinD(N) (2 (N -l)cosBj mcosB, 89 -2r) For Bi = 0, Eq. (4.21) is not defined. A limiting value for Bi ---70 yields for this special case: limG(N) = 8,-70 1 N 1 4( --;;---:1 (4.22) )2 Although the scattering angle tJ s is restricted to the range O:S: tJ s :s: n, the angle of deviation D(N) has no such limitation. Since usually the scattering angle tJ s is known and not the incident angle Bi' the incident angle must be determined through Eq. (4.20), which is thus reduced to the problem of finding zeros of the function f(B;) =_D(N) +2(N -l)arcsin( Si:Bi )-(N -2)n-2B =0 j (4.23) within the range -n/2:S:Bi :S:n/2 for all possible D(N)=±tJs -2kn and with k:S: (N -1) /2. Due to symmetry, tJ s must be considered in both the negative and positive directions. Every solution for D(N) in Eq. (4.23) can be interpreted as a possible ray path or partial ray through the particle. For instance in Fig. 4.5, the following values of D(N) can be found for the 6th order: 10 deg, -10 deg, -350 deg, -70deg and -710deg (Davis 1955, tJ s =lOdeg, m=0.75, air in water). Using Eq. (4.20), the corresponding values of Bj , and, with Eq. (4.21), the gain factors can be calculated. The scattered intensity of a given scattering order at a receiver r" is the sum of all possible D(N) solutions superimposed, while also preserving the phase. Wafer " m =1.333 D(6\ =-350 deg D(') =-7l0deg Air ", = I - ,. D(6) =-370deg Fig. 4.5. Possible partial rays of 6th scattering order and scattering angle of 10 deg for an air bubble in water m = 0.75 90 4 Light Scattering [rom Small Particles Separation of the Intensity at Interfaces. The separation of the primary field at every interface and for every ray path into a reflected and a refracted ray is governed by the fact that the tangential components of the electromagnetic field must be continuous across the interface. The resulting Fresnel equations give a relation between the field strengths of the reflected and refracted light as a ratio to the incident field strength. The field strength of the reflected and refracted beam in ratio to the incident beam is dependent on the polarization direction, either parallel (z3, component) or the perpendicular (fP, component) with respect to the scattering plane and is given in section 3.1.3 by Eqs. (3.51) to (3.54) (Hecht 1989, Born 1981). For dielectric media (f.Lm = f.L p = f.Lo) the relative magnetic permeability cancels out and only the relative refractive index must be used in the Fresnel equations. ErtJ mcos(}; -cos(}, rtJ=--= E;tJ mcos(}; +cos(}, cos(}; -mcos(}, cos(}; +mcos(}, E,tJ 2cos(}. ttJ=--= " E;,J mcos(};+cos(}, E,lp t ---- 2cos(}. , lp - E;lp - cos(}; +mcos(}, (4.24) (4.25) with . (sin(}) (), = arCSln ~ (4.26) The intensity oithe reflected and refracted waves can be computed from the reflectance RtJ.lp (Eq. (3.55» and transmittance TtJ,lp (Eq. (3.56» T tJ,lp =~=mcos(};lt I ; () tJ,lp COS, 12 (4,27) When applying the Eqs. (4.24) - (4.27) to separate reflective and refractive components at interfaces of a spherical particle, it is important to differentiate between internal and external interfaces for the high er scattering orders N > 1. For external interfaces the relative refractive index m pm = m = np / nm must be used; however, for internal interfaces its inverse value mmp = m-I = nm / np and the respective angles are required. Since the incident and reflected angles for a given ray path are always the same within a spherical particle, the Fresnel reflection coefficients rtJ and rlp only change sign for the two types of interfaces. The reflectance RtJ,lp does not change at all. The Fresnel coefficients for the exemplary cases of air/water and water/air interfaces as functions ofthe incident angles are illustrated in Fig. 4.6. The increase or decrease of intensity due to the ray path through the particle can be expressed in terms of an intensity coefficient for each scattering order. i(N)(x,m,(}) tJ.lp , =(n:d A, p w )2 a(N)G(N) = x M2a(N)G(N) tJ.lp tJ,lp (4.28) 4.1 Scattering of a Plane Wave 91 b a I I I-- n, = 1.333 n, =1 n2 = 1.333 112 Ir.I=lr~1 ,/ ,, ,, , , 0.5 .. ' .' =1 ,, ,,, ,, - ., Ir.I=lr~1 -.- , ....... " Ir"I=I~I~ 0.0 o 30 60 Incident angle 90 0 ~ 30 [deg] .. - ,, , l/ I 60 Incident angle 90 ~ [deg] Fig. 4.6a,b. Dependence of the reflectance and transmittance on the incident angle for two different polarization directions. a Water/air interface m = 1.334, b Air/water interface m = 0.75 with a(N) - 7J,rp - R { 7J,rp RN-2(I-R 7J,rp for N=1 2 7J,rp ) for N> 1 (4.29) In analogy to the Lorenz-Mie theory (section 4.1.2), scattering functions for each scattering order and for each polarization component can be defined for the light scattered according to geometrical optics ~;N) =.p: exp(j </J~N)) and ~;N) = p,f exp(j </J~N)) ( 4.30) By considering also the phase of each scattering order, </J<:'~, it is possible to properly superimpose all scattering contributions at each point on the surface of the detector. Phase. Three physical effects enter the computation of the phase for each scattering order!: • Phase jumps due to reflection at interfaces: </J~N) • Differences in the optical path length: </J~~) • Phase jumps due to ray bundle focussing: </Jj~) The conventional phase functions are described in detail in van de Hulst (1981). ! The sign of the phase jumps depends on the form chosen for the propaga ting wave, i.e. exp[j(mt-kr)] or exp[-j(mt-kr)]. Subsequent derivations will use exp[j(mt-kr)], as introduced already in Eq. (3.22). 92 4 Light Scattering from Small Particles Phase Jumps due to Reflection. At each interface a phase change of the field strength can occur in the reflected order. This is indicated in Fig. 4.7 for an water/air and air/water interface. Incident angles for phase jumps are the Brewster or polarization angle Bj = Bp = arctan(m) (4.31) and the angle of total reflection (critical angle) Bj = Be = arcsin(m) (4.32) Phase jumps of 1t can be seen directly from the sign of the amplitude coefficient using the relation for N=1 for N>1 (4.33 ) The case of total reflection must be considered separately. This case only occurs in homogeneous spherical particles for reflection (N = 1) and for a relative refractive index m < 1, because for incident angles of Bj > arcsin(m) the reflection amplitude coefficient r/J,rp is complex. Rather than taking the discrete values o and 1t, as suggested by Eq. (4.33), the phase change for total reflection is given by (4.34) a 0.0 180 r-c---------~~------<lJ arg(r.) = arg(r~) ~ ~ E:'" - 90 f- arg(r_l = arg('!I) arg(r.) = arg(r~) o ,, - -90 fn1 =1, -180 !- o n, = 1.333 n2 =1 ,, n2 =1.333 I 30 I 60 ,, , ... ,, ,900 Incident angle (); [deg] 30 60 90 Incident angle (); [deg] Fig. 4.7a,b. Dependence of the phase of the reflected rayon the incident angle for the two different polarization directions. a Water/air interface m = 1.334, b Air/water interface m = 0.75 4.1 Scattering of a Plane Wave 93 and rjJ~) = 2arctan [ ~sin2B.-m2l ' 'P cosB; (4.35) Path-Length Differences. Path-length differences through the particle are usually derived with respect to a reference path connecting the source and detector through the center of the particle, as pictured in Fig. 4.8. Examining the ray path for reflection yields ",(I) = 'l'pl 2k w rp cos(}l) , (4.36) and for refraction ",(2) k (,,2) ,(2)) 'l'pl =2 wrp cosui -mcosu; (4.37) Each additional scattering order leads to a further phase change of -2kwm rp cosBi N ). Thus, for the N 'h scattering order the phase change becomes rjJ~~) = 2k w a Rct1cclcd rA COSd;N) - (N -l)mcosB;N)] (4.38) b r~y Fig. 4.8a,b. Phase changes due to path-length differences through the particle. a Reflection, b Refraction Phase Changes Through Ray Bundle Focussing. Considering each ray as a "pencil of light" of finite width, this ray will be focused by the particle as illustrated in Fig. 4.9 for an air bubble in water (m < 1) or for a water droplet in air (m > 1). According to van de Hulst (1981), an astigmatic beam experiences a phase change of 1t /2 when passing the focalline. ~if' ~n[(N -1)-~[l+~n( d~;) )l) (4.39) 94 4 Light Sca ttering from Small Partic1es 0, Fig. 4.9a,b. Focussing of a finite width "penci1 of light". a Spherica1 bubb1e, b Spherica1 particle The sum of all influences on the phase of the scattered light leads to the following expression A.(N) 'I'';,rp == A.(N) 'l'p ';,rp + A.(N) + A.(N) 'l'pl 'l'fl (4.40) These procedures for computation of the amplitude and phase of the sc attered field strength from individual rays of incident light must be applied to each partial ray of each scattering order, i.e. for each solution D(N) according to Eq. (4.23). The sum over all scattering orders with all ray paths yields the scattering functions (4.41) and the field strength observed at a given point on the detector surface within the scope of geometrical optics is given, together with Eqs. (4.1) and (4.4) as (4.42) 4.1 Scattering of a Plane Wave E= _r 95 exp(-jk r ) wprMMME k ß-S rp_w wrpr The decisive advantage of geometrical optics is that the scattered light received by the detector can be considered in terms of scattering orders and each contribution can be analyzed separately E(N) _r = exp(-jkwrpr)M M(N)M E k ß-S rp_w wrpr (4.43 ) 4.1.1.3 Rainbow and Airy Theory A number of additional scattering phenomena must now be considered, which cannot be explained using geometrical optics and the concept of scattering orders. These include the rainbow, the Airy theory and surface waves. The properties of the rainbow will be discussed further in section 4.1.3.3 and its exploitation in a measuring instrument in section 9.3. Further details can be found in Hovenac and Lock (1992) and van Beeck (1997). A rainbow exists when two rays of the same scattering order pass through the particle on different paths but exit with the same scattering angle. Their interference with one another is called the rainbow. The scattering angle, or more precisely, the angle of deviation, exhibits an extreme value at the rainbow angle. dD(N) --=0 dB; (4.44) Any small variation of the incident angle results in a directional change in the scattering angle variation. At this incident angle, given by cos 2LJ{N) u· , RB m 2 -1 =----::-- (N _1)2 -1 (4.45) the gain function, Eq. (4.21), exhibits an infinite value, which prohibits a computation of scattered light intensity in the vicinity of the rainbow angle. The rainbow angle is given by geometrical optics as (4.46) This situation can be resolved by introducing diffraction effects near the rainbow. According to the Huygens-Fresnel principle, the propagation of a wave can be described with the help of Eq. (4.10). Every unobstructed point on a wavefront at time t serves as a source of spherical secondary wavelets. The amplitude of the optical field at any point beyond is the superposition of all of these wavelets at time t+ L1t. 96 4 Light Scattering [rom Small Particles The wavefront in the vicinity of the rainbow for the N = 3 scattering order can be approximated according to geometrical optics using a cubic function j(v). Inserting this function into the diffraction integral Eq. (4.10), and for Qw = 1, the electric field strength in the rainbow region can be approximated as with and the integral can be transformed into the Airy function. 4.1.2 Description using Lorenz-Mie Theory and Debye Series 4.1.2.1 Lorenz-Mie Theory (LMT) A closed solution for the scattering of a plane wave from a spherical, homogeneous, isotropie particle was first presented by Lorenz (1890) and by Mie (1908). This solution is weIl documented in the literature (Born and Wolf 1980, Born 1981, Kerker 1969, Bohren and Huffman 1983), thus only final results will be presented, although an extension using the Debye series will be considered in more detail (Debye 1908, Hovenac and Lock 1992). This extension allows the computed scattered field to be interpreted in terms of scattering orders. The solution presented by Mie was obtained und er the assumptions and geometrical considerations presented at the beginning of section 4.1. For most applications, the approximation of the far field is sufficient and the following discussion is therefore related to this case. For the interested reader, the exact solution and the near and intern al field is given in Appendix. All properties of the particle enter through the scattering functions ~1 and ~2' which can be interpreted as a transform between the incident and scattered fields. The approach used to determine the scattering functions is to solve the wave equations in a spherical co ordinate system (Eq. (3.19) and section 3.1.1.2). The incident plane wave is first decomposed into spherical partial waves, with the origin of the co ordinate system placed at the center of the particle. Thus, the surface of the particle also coincides with a constant value of the radial position co ordinate f = f p ' Two further waves are introduced before solving the wave equation, a scattered wave emanating outwards from the particle and an intern al standing wave propagating inside the particle. The amplitudes of these waves, designated the partial wave amplitude of the scattered wave and internal wave, follow the constraint that the tangential field strength at the particle surface must be continuous in the radial direction. It is essential to consider two independent solutions for each of the incident and scattered waves, one having no radial electric component (transversal electric wave), the other one having no magnetic radial component (transversal magnetic wave). The total scattered field is obtained by summing over a sufficient number of scattered spherical partial waves. In the far field some simplifications can be made and the scattering functions can be written as 4.1 Scattering of a Plane Wave 97 = ~1 (z9,) = LQnJZ"n( z9,) + Qn 1"n (z9,) (4.48) n=l = ~2 ( z9,) = LQn 1"n (z9,) + QnJZ"n (z9,) (4.49) n=l where JZ"n(z9,) and 1"n(z9,) are related to the associated Legendre functions (Born and Wolf 1980) and give the angular dependence of the scattered light in the direction of the scattering angle z9" The scattering functions in the far field are dependent only on the scattering angle z9, (van de Hulst 1981 pg. 121). The complex values Qn and Qn are the partial wave amplitudes. The partial wave amplitudes depend on the Mie parameters (Eqs. (4.2), (4.3» and are given by _ 2n+l m~n(Y )~~(XM)- ~n(XM)~~(Y) Qn I I n(n+l) m~n(Y _M )qn(XM)-qn(XM)~n(Y _M ) b _n = 2n+ 1 ~n(Y) ~~ (x M)-1:11. ~n(XM)~~(Y ) n(n+l) ~n(Y )q~(xM)-mqn(xM)~~(Y ) _M _M (4.50) (4.51) Thefunctions ~n(xM)' IjIn~l.M) and qn(XM) are theRiccati-Besselfunctionsofthe 1st and yd kind, order n (Abramowitz and Stegun 1965), and the superscript I denotes the derivative of the functions. The necessary number of partial waves for convergence of the sums in Egs. (4.48) and (4.49) depends on the Mie parameter xM" The maximum summation index is given by n max =x M +4.05x73' +2 M (4.52) for the relative error of 10-7 • For relative errors smaller than 10-14 the summation limit is n max =x M +8.1x73' M +2 (4.53) More complete details of the solution can be found in the literature (Born and Wolf 1980, Born 1981, Kerker 1969, van de Hulst 1981, Bohren and Huffman 1983). For light polarized in the x direction, the solution Eq. (4.1) turns into the solution given by van de Hulst (1981, pg. 124). It yields results comparable to those obtained by the geometrical optics approach, Eq. (4.42), however without the possibility to split the result into scattering orders. 4.1.2.2 Debye Series Decomposition On the basis of the work of Debye (1908) Lock (1988) and Hovenac and Lock (1992) demonstrated that the partial wave amplitudes Qn and Qn could be expanded into series, whose terms can be interpreted as individual scattering orders as described by geometrical optics. When a partial spherical wave is incident on a particle, a part will be reflected and a part will be refracted into the 98 4 Light Scattering from Small Particles particle. In the same mann er, when the intern al wave propagates to the outer particle radius, apart of the wave will be refracted out of the particle and apart will be reflected and remain in the particle. By summing over all these interactions, the partial wave amplitudes from Eqs. (4.50) and (4.51) can be written in the form (4.54) where pis the scattering order, R is the reflection coefficient and r the transmission coefficient for a spherical wave. The superscript (MM) denotes the reflection from the outer particle surface (medium - medium), (PP) is the inner refleetion (particle - particle), (MP) is the transmission from the outer medium into the particle (medium - particle) and (PM) is the transmission from the particle into the surrounding medium (particle - medium), (see Fig. 4.10)1. It is important to note that the reflection and transmission coefficients are different for a and b . The reflecti~n an<r"transmission coefficients can be expressed in terms of the Riccati-Bessel functions ofthe third kind, Sn and ';n, and their derivatives R(MMl = _"",b, R(PPl = -a",b, r(MPl _""b" = g';n(Y )';~(XM)- !!';n(XM)';~(Y M) -g';n (2:)S~(XM)+ !!Sn(x M),;~ (2:) (4.55) gSn(2:)S~(XM)- !!Sn(XM)(~(Y) -g';n (2:)S~ (x M)+ !!Sn(x M),;~ (2:M) (4.56) -2jm -g';n (2:M )S~ (x M)+ !!Sn (x M),;~ (2:) (4.57) (4.58) a={!!!.1 - for f!n for Tzn and !!={~ (4.59) The coefficients g and !l correspond to the two waves, transverse electric (TE) and traverse magnetic (TM) respectively. Each term in the sum (4.54) can be interpreted as an individual scattering order and the contributions of a spherical partial wave for a given scattering order p can be obtained by for p=l for p> 1 1 (4.60) To avoid confusion between the scattering order p and the index p for particle the indexes for particle and medium are changed to the capitalletters P and M 4.1 Scattering of a Plane Wave Rclractcd spherical wavc T"lPl 99 Internal rellcctcd sphcrical wavc T(AfPI (R(PP))' .... ••• @:g rartial wave Outside rclleclcd sphcrical wave Scatlercd wave = r(M"r(PA!) R (M"') 2 d rclraclcd sphcrical wave I" re(rdctcd spherical wavc + r (MPI R (PPl r (PMI Fig. 4.10. Reflection and transmission coefficients used in the Debye series Note that for p = 1 only a difference is calculated. It can be interpreted as the sum of diffraction, eliminating the undisturbed incident field in the shadow of the particle, and reflection contributions to the scattered field. In contrast to geometrical optics with N = 0 for diffraction and N = 1 for reflection, the scattering order p = 1 yields the diffraction peak in near forward scatter and the reflection part at all angles. Summing over all partial waves n for a given scattering order p, yields a scattering function equivalent to that of geometrical optics (Eq. (4.30». ~~P) (ß s) = = Lgn (p )Jl"n( ß,) +~n (p )1"n( ß s) (4.61) n=l (4.62) n=l Summing over all scattering orders yields the total scattering function as given by the LMT (Eqs. (4.48) and (4.49» = ~l(ßS)= L~~P)(ßs)' p~l ~2(ßs)= L~;P)(ßs) (4.63) p~l A further difference between the Debye series solution and geometrical optics, disregarding the combination of diffraction and reflection, is that the Debye series solution includes also surface waves (Hovenac and Lock 1992), since they are already included in Eqs. (4.48) and (4.49). Thus, with the Debye series it is possible to interpret the Lorenz-Mie theory in terms of scattering orders, as in geometrical optics (Eq. (4.43». 100 4 Light Scattering from Small Particles 4.1.3 Scattering Characteristics tor a Plane Wave The methods described in the previous sections can now be used to compute the scattering characteristics of small particles in an incident plane wave. Specifically, Eq. (4.6) together with Eqs. (4.1) and (4.63) are used. The results lead to insight about the scattering dependencies on particle size, refractive index and polarization, which are essential to properly understand the laser Doppler and phase Doppler techniques. Many of the discussions in this section pertain to the suitable choice of detector positions for phase Doppler systems. In general the aim is to choose a position in which a single scattering order dominates the total received intensity. This has been briefly mentioned in section 2.2, but will be discussed in fuH detail only in sections 5.3.2 and in section 8.1. In the foHowing section most example calculations will be presented for the case of a water droplet in air (m "" 1.333), since the resulting parameter dependencies are quite representative of a large number ofliquid droplets. 4.1.3.1 Intensityas a Fundion of Particle Diameter Both the laser Doppler and the phase Doppler techniques exhibit a particle size bandwidth, limited on the lower side by the minimum intensity detection level and on the upper side by signal saturation in the photodetectors or in subsequent electronics. A typical curve of scattered intensity as a function of the Mie parameter is shown in Fig. 4.11 for a conductive material with a real part of the refractive index of a water droplet in air, m = 1.333 - j 0.1, and for a defined scattering angle. For small particles, the intensity increases with the 6th power of particle diameter and this scattering behavior is known as Rayleigh scattering. The steepness of this curve leads to a rapid approach to the detectability limit for small particles. For particles with a Mie parameter x M > 10, geometrical optics can be applied and the scattered intensity increases with the second power of the particle diameter. Between the regions of Rayleigh scattering and geometrical optics (1 ~ x M ~ 10) the scattered intensity exhibits strong oscillations. This region is referred to as the Lorenz-Mie region. However, it should be emphasized that the Lorenz-Mie theory is in fact valid throughout all size ranges. In the case of less absorbing materials, the oscillations are stronger in the range of geometrical optics, as shown in Fig. 4.12. For medium conductivity the scattered intensity follows the non-conductive curve for small particles. Light that enters the particle is only weakly attenuated before exiting the particle. With increasing size however, the damping of internal waves increases and for large particles very little light exits the particle and the scattering is dominated by reflection. The influence of the refractive index on scattering properties will be discussed in more detail in section 4.1.3.5. The scattering behavior shown in Figs. 4.11 and 4.12 is typical for all scattering angles when an appropriate scaling factor is used. Only one exception must be mentioned. As seen in Fig. 4.11, at the scattering angles 1'), = 90 deg and for 4.1 Scattering of a Plane Wave 101 Partide diameter d p [11m] 0.01 0.1 10 100 --:~ 10 ' '" >- '"c " C 10 ' .. ___ ~ ___ ,, __ I!!\..i'. _._._ ..•..••...• .;.. .•..............•..... '\ / .\' ..t' J/;' .... ...... .:.................... .. .. /. \ ....................... ~ .................. .. .... / :/ ..[ .... .. ..................................... _.................... .. ............ · r ;.. · ............ ·........ ·· ...... ·............··· -....... t .... .............................. ........... . - ........:..................... . ~._ ~ . ./ . .. ·i.. .... .. . .............. T··.................. ........... / ........... :................................... .. i / J .- .................. --_ .. __ .. ..... . " 10: - x.. = 90 deg) . '/' ... (tJ s .............. ........... - _..... - .. --..- ...;. -.... -.... ..... ,~ ... . ... .... ~ 0.1 100 Mie parameter x" [-I 10 Fig. 4.11. Scattered intensity as a function of the Mie parameter XM (!!!. = 1.333 - j 0.316, parallel polariza tion, solid line 13, = 30 deg, dashed line 13, = 90 deg) Partide diameter dp [!-Im l 10' 10 0. 1 /11 ::i ,; = 1.333 /11 - 1.33.\ 111 = 1..\3.\ 111 1..\33 100 j 1000 (Rcnection) j 0.1 j 0.01 10 • 10 I 10 100 Mie parameter x .. [-I Fig. 4.12. Scattering intensity as a function ofthe Mie parameter X M , computed for an insulator, m=1.333, and three conductive materials !!!.=1.333-jO.Ol, !!!.=L333-jO.l, !!!. = 1.333 - j 1000 (13, = 30 deg, A = 514.5 nm, perpendicular polariza tion) 102 4 Light Scattering [rom Small Particles parallel polarization, the scattered power of very small particles drops with the 10th power of particle diameter and becomes small extremely rapidly. 4.1.3.2 Intensity Distribution os 0 Function of Portic/e Size The polar intensity distribution is shown in Fig. 4.13 for four values of the Mie parameter in both linear and logarithmic scales. For small values of x M this distribution resembles a dipole (Fig. 4. 13a,b ). The polarization component perpendicular to the observation plane shows no dependence on scattering angle, whereas the parallel component exhibits two symmetrie scattering lobes in the forward and backscatter directions. At the value of x M = 1 an asymmetry in the distribution is evident (Fig.4.13c,d) and at values between 2 and 3 the first scattering lobes appear (Fig. 4. 13e,f). Furthermore, the qualitative differences between the two polarization components at tJ s = 90 deg are no longer so distinct. A very strong scattering lobe is present in the forward direction, corresponding to the contribution from diffraction, which is independent of polarization direction. Beginning at Mie parameter values of 10, a larger number of scattering lobes appear, as is typical in the geometrical optics region (Fig. 4. 13g,h). The intensity distributions shown in Fig. 4.13 lend insight into the intensity curve shown in Figs. 4.11 and 4.12. For small particles, which have no polar dependency of the intensity, only the amplitude changes with size. As the first scattering lobes appear, they move across the detector surface with changing particle size, thus leading to the observed oscillations in the Lorenz-Mie region. As the particle becomes even larger, the scattering lobes become smaller in angular extent and due to the finite size of the detector surface, an averaging takes place. This again leads to a smooth curve, as observed in the region of geometrical optics (Fig. 4.14). This behavior has some direct consequences for the phase Doppler technique. The optical system of the phase Doppler technique is chosen to insure a monotonie relation between particle size and the phase difference between the two detectors of the receiving unit. Realizing that not only the intensity but also the phase varies throughout the scattering lobes, it is evident that for particles in the Lorenz-Mie region, the monotonie behavior may be difficult to maintain. Thus, the size of the detector aperture must be chosen to integrate over at least one scattering lobe. In fact this requirement often represents the lower measurable limit of the phase Doppler technique, where the relation between phase difference and particle size begins to oscillate. For even smaller particles, the scattering lobes disappear (Fig. 4.13a-d) and the phase Doppler technique can again be applied. For this case, planar phase Doppler optical configurations (section 8.2.2) are used because of the strong dependence of phase difference on particle diameter. Number of Scattering Lobes as a Function of Partide Diameter. The number of scattering lobes between 0 deg:s; tJ s :s; 180 deg is shown as a function of the Mie parameter in Fig. 4.15. In Fig. 4.16 the mean angular extent of the lobes and the minimum and maximum angular distances between lobes are shown. The mean 4,1 Scattering of a Plane Wave Li nea r scale Logarithmic seale X/of = 0.\ b --.- - .... , ~/" ". odeg o 10' " 10 10' " 3x \0 Ix lO \0·" 11 :' "'. ::;\;;:: ," .. ' .. ' ......... . Odeg I' ,:\ 12 6X \0' " . \ :// '. '1i : .. ' : I '. ' .. . I . : J . 10 " 11 . - '" . ...... . .... 10' " 2X\0,1I 4X\0 103 I . \ ....... ~:..._.,,"" ' ' ... _';'~.;' 10 1,'1,.,1 1,'1" ,1 0 \0 10 - octcg 10 • 5x \0 ' 10'· 10 ) IX IO' e 1,'1,.,1 0 X., .... =3 f l,'i..,l Odeg 2xlO ' 10·' 10·' 4x 10 ' 6x 10 ' 10·' 8x I0" 10 I Ix lO ' X., = 10 h 1,'1,.,1 o Odcg 10·' 10 ' 10 ' 6 Fig. 4.13a-h. Scattering function at four different Mie parameter values computed for water droplets in air (m = 1.334), solid: perpendicular polarization, dashed: parallel polarization 104 4 Light Scattering from Small Particles Particle diameter dp [flID] 10 10 100 100 Mie parameter xM [-] Fig. 4.14. Scattering intensity as a function of the Mie parameter XM. Computed for a water droplet in air (m = 1.333 , Aw = 514.5 nm ) at an scattering angle of 7J, = 30 deg and a circular receiver aperture of radius R, = 10 mm number of scattering lobes is linearly proportional to the Mie parameter, with a proportionality factor of 11m. At scattering angles where the lobe amplitude is large, i.e. where mixing of different scattering orders can be expected, the dis- ~o :;: J Particle diameter dp [flm] 50 100 • Calclulated number oflobes per 180 deg 400 A",rn rn 200 ." ." o ...-.- .....'.' ........ ... .... .., 200 .......'." ......' 400 Mie parameter XM [_]600 Fig. 4.15. Counted number of intensity maxima representing the number of scattering lobes between 0 deg and 180 deg as a function of the Mie parameter x M and particle diameter (,1=514.5 nm), caIculated with the Lorenz-Mie theory. The solid line corresponds to x M Im, the points correspond to the number oflobes counted (m = 1.334 ). 4.1 Scattering of a Plane Wave 1 Particle diameter dp [flml 10 105 100 b0100~------.---~r--.--~,-.-~-.-------.----.---.-~-.-.,-~ '" ~ x Minimal distance Maximal distance Mean distance • o • • ...• •...... .". . ..~ .._ . e• •e x 10 100 Mie Parameter xM [-] Fig. 4.16. Angular distance representing the width of scattering lobes between 0 deg and 180 deg as a function of Mie parameter x M and particle diameter d p (A = 514.5 nm). The solid line corresponds to 180 degxm I x M ' the points correspond to calculated minimal (x), maximal (.) and mean (0) distances oflobes (m = 1.334) tance between the lobes becomes an important design parameter for the receiving aperture. The receiving aperture should be large enough to average over severallobes and to smooth out strong amplitude fluctuations with size. The minimum distance between lobes is a quantity, which must be considered when the detector is placed at scattering angles where the mixing of several different scattering orders can be expected. Origin of Scattering Lobes. The appearance of scattering lobes (maxima and minima) in the polar intensity distribution (Fig. 4.13h) corresponds to interference of at least two waves emanating from glare points on the surface of the particle. These waves could be waves of different scattering orders or waves of the same scattering order but different partial rays. This is clarified in Figs. 4.17 and 4.18, which show the dependence of the scattered intensity of a lOOllm water droplet in air as a function of scattering angle for both polarization components. The Debye series decomposition, Fig. 4.17 and the geometrical optics 4.18 has been used to show also the contributions of each individual scattering order. The modulation depth of the scattering lobes increases sharply at scattering angles where two or more scattering orders are of similar amplitude and begin to interfere with one another. The angular frequency of the lobe structure (intensity oscillations) depends on the distance between the glare points and therefore on the scattering angle and on the scattering order. The glare points can be seen as point sources of scattered spherical waves -larger glare point separations produce higher angular frequencies. 106 4 Light Scattering from Small Particles 4.1.3.3 Scattering Orders, Scattering Modes Diffraction (N=O,p=l). The field strength of diffraction in the far field, given by Eq. (4.11), is pictured in Fig. 4.19 in a plane perpendicular to the direction of propagation of the incident wave and for small scattering angles. As expected, a ~ 10' ..'! Alcxander's dark band >. -;;; "" 2"d rainbow \ I" rainbow ,-'-., ,-'-,r-""-.., ..<:: 10 ' 10 ' .u-L-L~~LW~-L~~~~-:~~~~~auu-~~~-L~~~~~~ - -SlIlll - - 2 nd rcfraction ~Re tlcction Parallel polarization (rclatcd 10 5'h refraction - - - - 61h refraction 7'h rcfraction 4'h refTaction --0--1 " refr.lclion b ----v-- 3'd refraction -.- - -.-. 8"' refraclion ------- 9'h refraclion scattcring plane) Diffraction peak (p= I) c: '" "<: 10 ' 10 ' WU~-L~~~~-L-Lm-L-~~-L~~mw~-L~~~~~-L~~~~~U o 45 90 Brewster angle 13~SCatlering angle 180 Rainbowangle tJ, ldegl Fig. 4.17a,b. Scattered light intensity as a function of scattering angle, decomposed für the first ten scattering orders calculated with Lorenz-Mie theoryand Debye series decomposition für an incident plane wave (d p = 100!1.ill, A. = 488 nm, XM = 643.8, m = 1.333, pointlike receiver). a Perpendicular polarization, b Parallel polarization 4.1 Scattering of a Plane Wave 107 the diffraetive effeets deerease rapidly with seattering angle and they are rotationally symmetrie about the Zw axis. The diffraetion peak in the forward direetion is identieal for both polarization states of the incident plane wave. In Fig. 4.20 the dependenee of the field strength amplitude on seattering angle f}, and particle diameter d p is shown. Clearly the region affeeted by diffraetion and the a --; 10' ~ PcrpcndicuJar polarizalion (rclalcd .!!. Diffraction peak (N=O) 10 scallcring plane) Alcxander's dark band 2"" rainbow \ Refleclion (N== I) Sum I" rainbow ,.......J---..r..---'--, 10 ' 10·' a.u.......u...-'--'--J.......JL.......L....u...-'--'--.......L.......L...J......L.I....L.-'O:::;"'L.......L-L....L....J......L.....L..J1I-l..J.L...L..i...L.....L...I---'.....L....J....I - - 2nd refraction ---<>-- Sill refract ion gO' refraction - -$ u 111 ----<>--Renect ion 3m refraction b --; 10' - - - - 6'h refraction 7'h rerract ion 9'h refraction ,,-.-..-.-,,-.-..-.-,,-.-..-.-,,-.-..-,-~,,-.-.,-.-.,-.-.,-; Parallel polarization "'" Diffraclion peak (N", O) RcflcClion (N= I) 10' 10 ' 10 ' L.I-L....L...JIIL~L......-~h='=:h::r::::::i::Litt::::::::::;±I$:=:j:=::::;::::::::~ ° ßrewster angle Scallering angle 180 Rainbow a nglc tJ, [deg] Fig. 4.18a,b. Scattered light intensity as a function of scattering angle, decomposed for the first ten scattering orders calculated with geometrical optics for an incident plane wave (dp=lOOllm, A=488nm, x M =643.8, m=1.333, point-like receiver). aPerpendicular polarization, b Parallel polarization 108 4 Light Sca ttering from Small Particles EOIIT 400 40 40 -40 Fig_ 4.19. Diffraction by a spherical par tide with the diameter of d p (Bob = 2x 10 5 Vm- I , rp' = 80 mm, A = 488 nm) = 10 ~m Particlc d iam eter d, [J.lm) Fig. 4.20. Field strength amplitude for diffraction as a function of the partide diameter and sca ttering angle (Ba = 2 X 10 5 V rn-I, rp, = 80 mm, A = 488 nm ) angular extent of the diffraetion lobes deerease rapidly with inereasing particle diameter. The degree to whieh diffraetion must be eonsidered in eomputing the seattered field depends largely on whieh seattering orders are being eonsidered and their respeetive amplitudes. However, to eorreetly aeeount for diffraetion in the deteetor signal at small seattering angles, the field strength of the incident laser beam must be added to the sum with the eorreet phase (eompare with near field in section 4.1.3.6 and referenee-beam method in seetion 5.2.2). This will depend on the deteetor distanee from the beam axis and also on the beam radius. 4.1 Scattering of a Plane Wave 109 Retlection (N=l, p=l). Retlection can be found at all scattering angles. This is easily confirmed by looking at a metal sphere and observing that the entire surroundings are imaged on the surface. For transparent spheres the Fresnel condition dictates that retlection will vanish at the Brewster angle for parallel polarized light, Fig. 4.21. At this angle all light is refracted into the sphere. Thus, at the Brewster angle the high er scattering orders dominate over retlection and water droplets, for instance, exhibit an intensity ratio of nearly 1000 between first-order refraction and higher scattering orders (Fig. 4.17b). Such conditions are particularly interesting for the phase Doppler technique, since when one scattering order dominates, the lobe structure vanishes and the relation between particle diameter and phase remains linear. Since the diffraction and reflection are not separated in the Debye series expansion, the curve for the scattering order 1 represents only reflection at scattering angles above about 10 deg. The distinction between diffraction and reflection is clarified in Fig. 4.21. The sum of the diffraction and reflection, as obtained by applying the Debye series expansion, is compared to the individual contributions (and their sum) as computed using geometrical optics. ~ 10' t-ro-''-''~-''-~~-''-''~-r.-''~-''-''ro-.,-"~-.,-,, ..::!. - - Diffra<::lion and rcllcclion (Debye) .... .~ -- ---- DiffraClion and rcllcction (GO ) - - Pure rellection (GO) '-------,> '"c: 10' \Ot=::::;::~=:;::::=t=::::;:=:z::::::;::~~±::~ o 10' 2 4 ............... Pcrpend icular polarizalion (90 deg) /' Diffraclion and reOedion (Debye serics) • • Diffraclio n and relleclion (GO) :.:~_::.: Pu re reOcction (GO) Parallel po larizali n (0 d g) 1O ' L.....J.--'-....L....J.....JL.....J.....L.....L...L-L-L--'-....L.....L...1L-L....L....L..L-L....I.....L.....L...L-L-L....J.......L....J........L...J.....L.-'-L-W o 45 90 135 180 Scaltcring angle ". Ideg l Fig. 4.21. Contribution of diffraction and reflection to scattering intensity as computed by geometrical optics and LMT with a Debye series expansion (dp=lOO/lm, A=488nm, x M = 643.8, m = 1.333) First-Order Refraction (N=p=2). In the forward scatter region (10 deg < tJ s < 50 deg ... 100 deg), first-order refraction usually dominates. Geometrical optics predicts a limiting angle for first-order refraction, for example 82.79 deg for water droplets in air (m = 1.333). This behavior can be seen in Fig. 4.22. The Debye series expansion yields a less sharp decrease in intensity of firstorder refracted light with scattering angle. The discrepancy can be attributed to 110 4 Light Scattering from Small Particles the existence of surface waves, which are not accounted for in geometrical optics. For more accurate results with geometrical optics, surface waves can be calculated and added to the solution (Hovenac and Lock 1992). The difference between the scattering orders of Debye series and geometrical optics yields also an estimation of the surface wave strength. This demonstrates one advantage of using Debye series computations together with results from geometrical optics, especially in overlapping regions of different scattering orders. One further example is seen in Fig. 4.17b, where in the region 80deg<t9, <90deg the firstorder refraction is the strongest scattering order, in disagreement with the result obtained using geometrical optics in Fig. 4.18b. o • l't refraction p=2 (Debye series) 1,t refraction N=2 (GO) Parallel polarization (0 deg) Perpendicular polarization (90 deg) 10-9 L-I.---'--'--'--L-I.---'--'--'--L-I.---'--'--'--L-I...J....L..-'--'--.L......IL......L-'--'--'--L-I.l.....L...:.L--'--.L......IL......L-'---'---' 180 o 45 90 135 Scattering angle iJ, [deg] Fig. 4.22. Contribution of refraction to scattering intensity as computed by geometrical optics and LMT with a Debye series expansion (dp=lOOllm, A=488nm, x M =643.8, m = 1.333) Higher Order Refraction (N)2, p>2) and Rainbows. According to geometrical optics, second-order refraction has two scattering modes above scattering angles of about t9, = 137.2 deg for water in air (m = 1.333). This is illustrated in Fig. 4.23, in which two modes exist with the same scattering angle. Interference between these two modes leads to strong scattering lobes within a single scattering order for perpendicular polarization. For parallel polarization the intensity of the second-order refraction is weaker. This is because for water in air, the Brewster condition is met for first-order refraction. In comparison to the perpendicular polarization, most of the light that is refracted inside the partiele leaves the partiele immediately. Only a small part is reflected inside the partiele and creates higher scattering orders. This is also evident for the much higher intensity seen for first-order refraction with parallel as opposed to perpendicular polarization in Fig. 4.22. Because the Brewster condition can be met only for parallel polari- 4.1 Scattering of a Plane Wave 111 Fig. 4.23. Illustration of the two scattering modes for second and third-order refraction and further rays (reflection, surface waves) influencing the intensity in the rainbow region zation, this also leads to weaker mode oscillations for higher scattering orders for this polarization state (Fig. 4.17). This behavior of rainbow suppression for one polarization direction cannot be generalized too widely, as illustrated later in Fig. 4.35 for other refractive indexes. The term rainbow is used to designate angular regions in which more than one solution exists for a single scattering order, in which case the individual solutions are known as modes or partial rays. One such example has been illustrated in Fig. 4.23. The second-order refraction in water droplets forms the wellknown rainbow observed in nature at around zJ, = 137 deg. A second rainbow formed by third-order refraction is also seen in nature und er favorable conditions. The region between the two rainbows, where reflection dominates, is known as Alexander's dark band since the intensity drops sharply (Figs. 4.17 and 4.18). Both of these rainbows can be seen in the picture shown in Fig. 4.24. Higher order rainbows are generally not seen in nature due to their low intensity level. In Fig. 4.25 the intensity distribution over the scattering angle range of the rainbow (132 deg ~ zJ s ~ 152 deg) is shown. The interference between the different partial rays of the second-order refraction generates the dashed curve, known as the main rainbow maximum and supernumerary bows sometimes seen also in natural rainbows. The rainbow angle is the limiting angle of secondorder refraction according to geometrical optics. At this angle both partial rays are of similar intensity and the rainbow attains its highest amplitude. For larger scattering angles one ray always dominates and the interference intensity decreases. The interference pattern from second-order refraction is further modulated by interference between second-order refraction and reflection. This is known as ripple structure and is shown as asolid curve in Fig. 4.25. From the above description it is evident that the particle size can be obtained by measuring the scattering lobe separation. Especially in the rainbow region 112 4 Light Scattering from Small Particles 2n~ rainbow (N ,I) l"rainbow(N 3) A1exander's dark band Fig. 4.24. Natural rainbows observed under favorable weather conditions :-. .;;; 600 c: u c: ---Sum or all scatlcring orders (LMT soluti on) ------- 2 nd refraction (p - 3) on ly (Debye solution) 400 135 140 145 150 callcring angle I', Idegl Fig. 4.25. Detail of scattered intensity near the rainbow angle for (d p = 100 11m, m = 1.333, A = 488 nm, x M = 643.77) this possibility has been exploited in instruments and the Airy theory of rainbows (seetion 4.1.1.3) gives a very good approximation of the relation between particle size and the supernumerary bow distribution (section 9.3). In Fig. 4.26 the intensity distribution is shown as a function of the particle diameter for m = 1.32. In Fig. 4.26a alI scattering contributions from LMT are pictured and the supernumerary bows are disturbed by the ripple structure. In Fig. 4.26b only the contributions of second-order refraction have been used, thus eliminating the ripple structure. 4.1 Scattering of a Plane Wave b a [ 113 500 ",,"" . " 400 <l E .~ "" '" :Q 300 . " 0.. 200 100 134 136 138 140 112 144 134 136 110 138 142 144 Scallering angle /J, (deg) Scallering angle /J, (deg ) Fig. 4.26. Intensity distribution in the rainbow region as a function of particle size (A. = 632.8 nm, m = 1.32, perpendicular polarization). a Complete LMT solution, b Second-order refraction only, from Debye series (p = 3 ) The position of the maximum intensity of the supernumerary bows is not a strong function of particle diameter, however the ripple structure is more sensitive. In Fig. 4.27 the complete Debye series has been used to generate the scattered intensity over a particle size range of 400 nm. Already for size changes of 260 nm, the ripple structure is shifted in scattering angle by one period. Thus, exact size measurements for particle diameter variations significantly smaller -200.4 E :0. .." <:; E " '6 " :Q 200.2 i: " 0.. 200.0 134 136 138 140 142 144 Scallering angle /J, (deg) Fig. 4.27. Scattered intensity distribution over the rainbow region as a function of very small particle size changes (m = 1.32, A. = 632.8 nm, perpendicular polarization) 114 4 Light Scattering from Small Particles than the wavelength are possible in prineiple by observing the rainbow strueture closely (Han et al. 1998). Another problem for particle size and refraetive index determination from the rainbow pattern is evident from Fig. 4.27. Due to the ripple strueture, it is diffieult to determine the exaet loeation of the rainbow angle or the angles of maximum intensity in the supernumerary bows. This is the limiting faetor of aeeuraey in rainbow refraetometry and will be diseussed further in seetion 9.3. The angular positions of the seattering orders are funetions of refraetive index, henee also of the position of the rainbow. In Fig. 4.28 the intensity distribution is shown as a funetion of refraetive index for a particle of 200 /-lm (,1 =632.8 nm). Together with information from Fig. 4.26, both the refraetive index and the particle diameter ean be determined. The refraetive index is related to the absolute angle of the rainbow maxima and the particle diameter to the period of the supernumerary bows. This teehnique is diseussed in more detail in seetion 9.3. b a -:: 1.35 " ., ü .,... <.!: 1.34 '- 0 >< " .5 "C 1.33 \.32 134 136 138 114 140 142 Scatlcring angle iI, Ideg] 134 136 138 140 142 144 c3ltering angle tJ, ldeg] Fig. 4.28. Intensity distribution in the rainbow region as a function of refractive index (A = 632.8 nm, d p = 200 ~m). a Complete LMT solution, b Second-order refraction only from Debye series (p = 3 ) 4.1.3.4 Scattering Order Dependence on Partic/e Diameter In Figs. 4.29-4.32 the intensity distributions of the first 10 seattering orders are given for three particle sizes, 10 11m, lO0/-lm and 1000/-lffi (m = 1.333 water in air, ,1 = 488 nm) and for both polarization eomponents, ealeulated with geometrieal opties and Lorenz-Mie theory. Aeeording to geometrieal opties, the ray paths through a spherieal particle will be independent of size and the intensity distribution of the individual partial rays as a funetion of seattering angle should 4.1 Scattering of a Plane Wave 115 not change with size. This behavior is seen in Figs. 4.29 and 4.31 for reflection and first-order refraction (N <3), both of which exhibit only one partial ray. The curves for these two scattering orders are only scaled in intensity for different particle sizes. The influence of particle size becomes evident only after a superposition of all partial rays (preserving the phase, Eq. (4.41) and the multiplication of their individual intensities by the particle diameter squared, Eqs. (4.28) and (4.30). For this reason, the intensity of scattering orders with two or more possible partial rays (N > 2) changes qualitatively with particle size. Nevertheless, the sharp decrease of the scattering orders in the region of the limiting angles is always present. Figures 4.30 and 4.32 show the same scattering orders as in Figs. 4.29 and 4.31, calculated using the Lorenz-Mie theory and Debye series decomposition. As shown in Figs. 4.30, and 4.32, the limiting angles for refraction and high er scattering orders are clearly not independent of particle size, again confirming the lower size limitations of geometrical optics. Besides the larger number of scattering lobes for larger particles, the limiting angles of different scattering orders also become much sharper and approach the geometrical optics solution. For smaller particles these limits become less distinct due to surface waves. As an illustration, two limiting angles predicted by geometrical optics are marked in Figs. 4.29 to 4.32 with vertical dashed lines: the furthest extent of first-order refraction (l9 s = 82.79 deg) and the rainbow angle (l9 s = 137.9 deg). Regions of strong oscillations also become smaller for larger particles, since the oscillations arise partly from interfering orders. Important to note is that the reflection at the Brewster angle no longer disappears for parLicles sizes near the limit for geometrical optics, Fig. 4.32a. For large particles and perpendicular polarization, reflection dominates in the regions 75 deg < 19 s < 110 deg and 130 deg < 19 s < 140 deg, in some cases by a factor exceeding 200 (Fig.4.30c). This suggests favorable conditions for phase Doppler systems; however, for small particles this is no longer valid, since high er scattering orders become much stronger. In fact, it is unlikely that a phase Doppler system can size water droplets less than 10 IJ,m using reflected light (Fig. 4.30a). Clearlya correct layout of a phase Doppler system is not possible only on the basis of the intensity distribution of reflection (N = P = 1), first-order (N = P = 2) or second-order (N = P = 3) refraction. For water in air for instance, the third-order refraction (N = P = 4) even exceeds reflection at some scattering angles. For larger droplets, the fourth-order refraction (N = P = 5) achieves high values near 19 s "" 40 deg (Figs.4.29c, 4.30c) and the 6th order refraction (N = P = 7) at 19 s "" 130 deg (Figs. 4.29c, 4.30c, 4.31c, 4.31c). Also, the seemingly large dominance of first-order refraction over reflection at the Brewster condition is diminished by rather strong contributions from third, fifth and eighthorder refraction (N = 4,6,9) (Fig. 4.32a). 116 a 4 Light Scattering from Small Particles 10' ::l ..:i. .,q <J<U :5 10 ' b c 45 90 --Surn - - 2nd refraction ----<- Reflection ~ 3 rd refraction -----1 st refraction -----+---4 th refraction 135 Scattering angle 13, [deg] -~-~--~- 8th refraction ~ 5th refraction - - - - 6th refraction ~ ~~~~ ~~~~ ~ i h refraction - - - -- - ~ 9th refraction Fig.4.29a-c. Intensity distribution of the first 10 scattering orders for different diameters of water droplets in air calculated with geometrical optics (m = 1.333, Aw = 488 nm, perpendicular polarization, point-like receiver). The dominating scattering order is marked with the correspondingvalue ofN. a d p =lOJ.tm, b d p =100 J.tm, c d p =1000 J.tm 4.1 Scattering of a Plane Wave a 117 10' ::! ~ .€ .:§"= 10·' b c - - - Sum -=--1 ,t refraction --'-- Reflection and - - - 2nd refraction ~ 3'd refraction Diffraction Scattering angle 1'J, [deg] ..... 7'n refraction ----T---4 tn refraction ~ stn refraction stn refraction - - - - 6tn refra ction 9tn refraction Fig. 4.30a-c. Intensity distribution of the fIrst 10 scattering orders for different diameters of water droplets in air calculated with LMT and Debye series decomposition (m = 1.333, /Lw = 488 nm, perpendicular polarization, point-like receiver). The dominating scattering order is marked wi th the corresponding value of p. a d p = 10 f-lm, b d p = 100 f-lm, c d p = 1000 f-lm 118 a 4 Light Scattering from Small Particles 10' ::! ~ ~ .-0;:: C3 0) .5 10 ' 90 --Surn _ _ 2nd refraction - - - Retlection - - 1 " refraction ~4'h refraction ~ 3'd retraction 135 180 Scattering angle 0, [deg] _._-~-~- 8 th refraction --<>-- s'h refraction - __ - 6'h refraction - - - - - - ~ 9'h refraction ~~ ~.~~~~~ 7'h refraction Fig. 4.31a-c. Intensity distribution of the first 10 scattering orders for different diameters of water droplets in air calculated with geometrical optics (m = 1.333, Aw = 488 nm, parallel polarization, point-like receiver). The dominating scattering order is marked with the correspondingvalue of N. a d p =10 /-Im, b d p = 100 /.Im, c d p = 1000 /.Im 4.1 Scattering of a Plane Wave a 119 (0) - - 1 ,t refraction --Sum --'-- Reflection and - - - 2nd refraction Diffraction ________ 3'd refraction Scatteringangle tJ, [deg] .... 7'h refraction refraction _._-_._. 8th refraction ---v-- 5111 refraction - - - - 6111 refraction ------- 9th refraction ~111 Fig. 4.32a-c. Intensity distribution of the first 10 scattering orders for different diameters of water droplets in air calculated with LMT and Debye series decomposition (m = 1.333, Aw = 488 nm, parallel polarization, point-like receiver). The dominating scattering order is marked with the corresponding value of p. a d p = 10 11m , b d p = 100 11m, c d p = 1000 11m 120 4 Light Scattering from Small Particles 4.1.3.5 Intensity Dependence on Refractive Index The real part of the relative refraetive index of the particle affeets primarily the angular position of the seattering orders and the loeation of speeifie limiting angles. The diffraetion in forward seatter remains unaffeeted. The eurves shown in Fig. 4.33 ean be computed from Eq. (4.20), whieh reduees to the following expressions: • for total refleetion I')TR = 1t - 2aresin(m) = 2areeos(m) (4.64) • for the Brewster angle I') p = 1t - 2 aretan(m) (4.65) • for the limiting angle of first-order refraetion (N = 2) _{1t- 2aresin(m)1) m < 1 I')c- 1t - • (m 2 aresm (4.66) m >1 • and for the rainbow angle in second-order refraetion (N = 3) I') RB f+-m2 f+-m2 = -4 aresin -1 - - - + 1t + 2 aresin - - m 3 The Brewster or polarization angle, \ (4.67) 3 I') p , Brewster \ angle inereases with deereasing relative re- Limiting angle of 1,I refraction \ \ .. " .... .... .... ,; ,, , ; ~ ___ - - .... Rainbow angle of 2nd refraction o o 45 90 135 180 Scattering angle iJ, [deg] Fig. 4.33. Dependence oflimiting angles on the relative refractive index 4.1 Scattering of a Plane Wave 121 fractive index, merging with the angle for total reflection (f) TR) at low values of m. For m> 1 the angular range offirst-order refraction increases rapidlywith m. The second-order rainbow exists in the range 1< m::; 2. These dependencies can be identified also in the results presented in Figs. 4.34 and 4.35, showing the scattering intensities of all scattering orders for various values of m smaller (Fig.4.34) and larger (Fig.4.35) than 1. These results were obtained using the Debye series expansion. For relative refractive indexes smaller than unity (m < 1) and parallel polarization, the coincidence of total reflection and Brewster angle at small m leads to strong intensity oscillations in the reflection curve, since the two effects directly oppose one another. A phase Doppler photodetector placed at the Brewster angle to detect only second-order refraction (N =3) is only appropriate down to a refractive index value of m = 0.75 and also demands good angular alignment. It is much easier to chose a position at which reflection dominates, since also the scattered amplitude is much larger, especially near angles of total reflection. For relative refractive indexes less than unity (m < 1) the total reflection angle f)TR is a good choice for a phase Doppler detector, since this is also the limiting angle for first-order refraction (N = 2). It is evident from Fig. 4.34 that near the angle of total reflection, the scattered intensity is almost independent of scattering angle, confirming the angular independence of the gain factor given in Eq. (4.17). Indeed for m < 1 only reflection and the first two refractive orders are of any significance for the total scattered intensity. For perpendicular polarized light, not shown here, even the second-order refraction is no longer significant. Thus, for applications of phase Doppler when m< 1, the detectors are best positioned where reflection dominates. However, a backscatter arrangement is not advisable since the dominance of reflected light is not strong enough. A further three scattering effects for the case of m < 1 are noteworthy. Secondorder refraction (N = 3) exhibits very strong oscillations in forward scatter and virtually no oscillations in backscatter. This is because two modes of second- order refraction exist in the forward direction and lead to interference, while in backscatter only one mode is present. A second effect is the dependence of the scattering lobe modulation depth on the dominance of any single scattering mode. Whenever two scattering modes or partial rays, e.g. reflection and refraction in Fig. 4.34 in backscatter, are approximately equal in amplitude, the modulation depth increases dramatically. Finally, all higher order scattering modes (N > 3) exhibit a behavior similar to the third-order refraction (N = 4), except with diminishing amplitude. Therefore, for clarity, only the third-order refraction is shown in Fig. 4.34. For a refractive index larger than unity (m > 1), the dynamics of the scattering orders with changing refractive index are much more dramatic. A generalization over many scattering orders is no longer possible since once the scattering angle for the highest intensity of one scattering order exceeds 0 deg or 180 deg with increasing m, the sign of its dependency on m changes. One example is the thirdorder refraction (N = 4). The position of the highest intensity increases in Fig. 4.35a,b. After reaching 180 deg, the position decreases with increasing relative refractive index, Fig. 4.35c-f. The number of such sign changes with m in- 4 Light Scattering from Small Particles 122 a 10' =! ~ ;... "' " ;:" 10 "' b c 10' 1 0 ) .r-L~~~~~~~~~~ 10' <}'Wl'5il'lii\~,WiI",,, d e f 135 - - Sum (LMT) - - Diffrac t ion and Reßecl io n (p - - - I " refraction (p = 2) Sca tt cring angle = I) -b-- 2"" refract ion (r - .') --0--:\ 'd refract ion (r - 4) Fig. 4.34a-f. Intensity distributions for parallel polarized light and varying refractive indexes calculated by LMT with Debye series (d p =100f.lm, A=488nm, x M =643.77). a m = 0.9, b m = 0.85, c m = 0.8, d m = 0.75, e m = 0.7 , fm = 0.65 4.1 Scattering of a Plane Wave a 123 10J :::I ~ >- "'"'-' E 10 J b 10 3 IO.J t:;:=.;::::t.L"==t:::=d'n:=hd9:....l'L~...L..........L.1.....::................-.....J<a:::...:..:.. I 0 3 .~or:-.......;t!.... c d IO ) ~~~~ ~~-L~~~~~~~~~~~~~~~~-L~~~~ IOJ ~l"-~-~ __ e IO J __ ~J.....J........1~..4....L.......1..-.JL~.~::::r:::::l::::;.::::;~s21~=:;:=::;::::~~ o 135 IBC um (LMT) ,d rcfra cli on (p _ 4) ---<>---- Diffracli on and reOeclion (p - I) --.-<1'" rcfracliol1 (p - 5) --0--1 " refraclio n (p 2) _ _5'" refraclion (p - 6) _ _ 2"" refraclion (/) 3) - - - _6'h refraclion (p - 7) Scaltering angle tJ, Ideg! ............7'" refraclion (p 8) _._._._._lI 'h rcfraclion (p - 9) --- --.9'" refraclion (p - 10) Fig. 4.35a-f. Intensity distributions far parallel polarized light and varying refractive indexes calculated by LMT with Debye series (d p =100/lill, A=488nm, x M =643.77). a m = 1.1 , b m = 1.2, C m = 1.3, d m = 1.4, e m = 1.5, fm = 1.6 124 4 Light Scattering from Small Particles creases with the scattering order and is given by N -3. This can be seen directly from the solutions for the angular deviation D(N) in Eq. (4.20). The use of detectors for phase Doppler measurements at the Brewster angle is only possible starting at a refractive index of m = 1.27, as seen in Fig. 4.33. However, at larger values of refractive index the signal may be disturbed by higher order scattering contributions. A further possibility for positioning a phase Doppler detector for m > 1.35 is in the range of dominant reflection near Os = 135 deg, although the linearity of the signal phase difference with particle diameter can also be disturbed here by higher order scattering for very small changes of m. This behavior has consequences for the operation of phase Doppler systems since the refractive index is generally a function of temperature and may change during a measurement. As an example, the refractive index of 1.4 is considered. According to Fig. 4.35d measurements using reflected light in backscatter (120 deg < Os < 140 deg) appear to be possible, since reflection dominates. However with changes of the refractive index to m =1.5, the 5th and 6th order refraction quickly gain in importance and will detrimentally affect the linearity between particle diameter and phase difference. The imaginary part of the refractive index, determined by the conductivity of the particle medium, is the second influencing factor of the scattering function. Figure 4.36 illustrates the total angular scattering function for varying values of the imaginary part of the refractive index. For a conductive medium, the influence of the imaginary part dominates the Fresnel coefficients. A large imaginary refractive index part strengthens the reflection, in fact only reflection and dif- -----+-- In = 1.333 - j 100 -----'-- m = 1.333 - j 0.891 -----<>-- =1.333 - j 0.02 (Rcileaion and diffraction only, p = I, for 111 = 1.333) In . . . . . .. m = 1.333 - j 0.00224 _ _ ",=1.333 45 90 135 180 SC3llcring angle t?, (°1 Fig. 4.36. Dependence of the scattering function on the imaginary part of the refractive index fOr(A = 488 nm, d p = 50 J.tm, parallel polarization) 4.1 Scattering of a Plane Wave 125 fraction exist and virtually all of the incident wave is reflected. Due to the constant gain factor in Eq. (4.17), the reflection amplitude is essentially independent of scattering angle. As the imaginary part decreases, a portion of the ineident wave is refracted into the particle and the scattering function corresponds to that of a transparent particle. In this case the reflective portion of the incident wave is determined by the real part of the refractive index. The light which is refracted into the particle is strongly attenuated and cannot leave the particle as higher scattering orders. The real part of the refractive index is significantly larger than the imaginary part and essentially determines the Fresnel coefficients. However, the imaginary part is important in determining the attenuation factor in the particle. If the imaginary part is further decreased, the attenuation in the particle is no longer so strong and some light willleave the particle. This happens first with the first-order refraction, since the path length within the particle is the shortest in this case. The higher orders follow with decreasing magnitude of the imaginary part. The point at which the imaginary part can be fully neglected depends on the particle size. As already shown in Fig. 4.12, this occurs sooner for sm all particles than for large particles. This underlines the necessity to compute calibration curves for phase Doppler systems which cover the actual, expected ranges of particle size and refractive index. It is also important to consider all scattering orders, not just the dominant ones. Using only a few scattering orders can give an approximate indication of a suitable phase Doppler layout but cannot replace a full computation. 4.1.3.6 Internal Field and Near Field To gain further insight into the parameter dependencies of the scattered light it is instructive to examine the internal field and near-field scattering of the particle as illustrated in Fig. 4.37. In Fig. 4.37a the full scattering process is illustrated. Outside the particle the ineident plane wave and the scattered wave superimpose. The ineident plane wave has been blanked out in 4.37b for better visualization of the scattered wave. The shadows behind the particle in Fig. 4.3 7a arise due to interference between the incident wave and the diffractive part of the scattered light. The same effect can be seen in Fig. 4.37c and d, where only the contribution of reflection and diffraction is illustrated, with and without the ineident wave. The light area on the right in Fig. 4.37d is attributed to the diffraction part of the Debye decomposition p = 1, which is phase shifted by 180 deg relative to the incident wave and has the same amplitude as the ineident wave. Superimposing the ineident wave with the diffraction part results in the shadow behind the particle, Fig. 4.37c. Furthermore, the disappearance of reflection at the Brewster angle can be seen in Fig. 4.37d. In Fig. 4.37e and f the intensity of the intern al field and the near field arising from first-order refraction (p = 2) and second-order refraction (p = 3) is plotted using Debye series decomposition. The focussing behavior of the particle in first-order refraction and the rainbow in the second-order refraction can be 126 4 Light Scattering from Small Particles a Scattered tield superimposed with incident wave b Scattered and interna I field only Focal poinl FirSI -order refraclion (p = 2) rcfraclion (f> = 3 , rainbow) c Diffracted and reflected wave (p =1) superimposed with incident wave d Diffracted and reflected wave only =1) f Second-order refraction (p = 3) only Fig. 4.37a-f. Distribution of intensity in the internal field and in the near field of a scattering particle, calculated by LMT and Debye series decomposition (..1 = 488 nm, d p = 20 flm, m = 1.3 , parallel polariza tion) 4.2 Scattering of an Inhomogeneous Field 127 elearly recognized. The intensity distribution of the rainbow is already apparent in the near field. The superposition of all scattering orders results in Fig. 4.37b. In the right part of Fig. 4.37b, additional fringes in forward scatter are visible, compared to Fig. 4.37c,e. From Fig. 4.32 it is reasonable to assurne that these fringes come from the interference between first-order refraction and reflection. In Fig. 4.37b, the interference phenomena in backscatter can be identified as interference between different higher scattering orders, creating local maxima and minima of intensity. In the edge zone of the partiele, a elear structure can be identified in the zoomed portion. This structure corresponds to so-called resonances of the partiele. Rays in this zone fulfIl conditions for total intern al reflection on every surface and thus, never leave the particle. The light is coupled into this zone by wave effects but the propagation can be described by geometrical optics (Roll et al. 1999 and Roll and Schweiger 2000). The maxima and minima of intensity arise due to resonance conditions being met around the circumference of the partiele. The angular structure is due to the fact that both right and left running waves exist, thus leading to standing waves. Optical resonances similar to a laser cavity induce further effects in the scattering functions. The condition of resonance is fulfilled for specific partiele diameters and at these values the scattering function changes very rapidly with small partiele diameter changes. This is also a limiting factor for the partiele diameter resolution of the phase Doppler technique, which is of the same order as the wavelength of the incident light. 4.2Scattering of an Inhomogeneous Field There are a number of approaches to compute light scattering for inhomogeneous incident fields; however, even a brief review of all of these techniques would be too lengthy for the present purposes. The following discussion restricts itself to two possible approaches, both ofwhich build on the solutions given above for an incident plane wave. The geometrical optics solution will be extended to yield the so-called Extended Geometrical Optics (EGO) approach. The Lorenz-Mie solution will be modified to superimpose many incident plane waves, the sum of which yields the incident inhomogeneous electromagnetic field. Fourier decomposition is used for this transformation, thus the technique is known as Fourier Lorenz-Mie theory (FLMT). Further details on other approaches for inhomogeneous fields can be found in the literature (e.g. Hoekstra 1994, Gouesbet and Grehan 2000, Yeh et al. 1982, Wriedt 1998, Doicu et al. 2000). For this extension of geometrical optics and Lorenz-Mie theory to arbitrary fields, all assumptions and coordinate systems introduced in section 4.1 can be retained. One additional coordinate system, the beam coordinate system (BCS) must be introduced. Thus, the wave coordinate system (WCS) is now coupled to the beam coordinate system rather than to the main co ordinate system (MCS). 128 4 Light Scattering from Small Particles To extend the geometrical optics approach, a plane wave will be postulated at the incident point on the particle surface, with characteristics corresponding to the actual field strength in the wes at that point. Similarly, the FLMT reduces the computations to that of many plane waves, thus justifying all preliminary assumptions made in section 4.1. The five coordinate systems to be used are pictured in Fig. 4.38. 4.2.1 Extension to the Method of Geometrical Optics (EGO) The theoretical basis to compute the light scattering using geometrical optics has been outlined in section 4.1.1 for the case of an incident homogeneous plane wave. The idea behind EGO to account for inhomogeneous fields is to identify the position of all interaction points for all scattering orders reaching the detector surface, (see Fig. 2.7 in section 2.1 for the definition of incident, glare and interadion points). The task is to compute the local amplitude, phase and propagation direction of the incident wave at each incident point (Borys et al. 1998). The superposition of sufficiently many such waves, paying attention to their correct phase, leads to particularly intuitive results compared to the computationally more complex Generalized Lorenz-Mie theory (Gouesbet et al. 1989) or Fourier Lorenz-Mie theory (Albrecht et al. 1995). Dclcclor Surfa..:c y /'lIes z Fig. 4.38. Coordinate systems for light scattering computations of incident inhomogeneous fields 4.2.1.1 Ray Tracing for EGO The field distribution across the surface of the particle depends on the particle position in the measurement volume. However, since the area which corre- 4.2 Scattering of an Inhomogeneous Field 129 sponds to that portion oflight eventually ending up at the detector is not known beforehand, neither is the amplitude, phase or direction of propagation known. The individual solution(s) for each scattering order must be determined iteratively, employing the rules of geometrical optics. The orientation of the partial plane wave at the incident point differs generally from the bearn axis. Nevertheless, the iteration can begin by assuming that the propagation direction at the incident point e w is coincident with the bearn axis e b , regardless of which interaction point is being considered. Figure 4.39 illustrates the necessary vectors to be considered in determining the incident point on the surface of the particle. For scattering of a homogeneous plane wave, these vectors are immediately defined once the incident wave vector, the detection direction and the refractive index is known (see section 4.1.1). Because in the following description only one incident laser beam and one scattering direction (one receiver) is analyzed, no further indexes, as used in chapters 2 and 5 are necessary. The basic rules of geometrical optics in vector form, as applied already in section 4.1.1 (Hecht 1989), can be used to formulate the problem, i.e. the law of reflection (N = 1) er!) -e pr W = 2e(!) cos(). I (4.68) reflection inside the particle (N > 2) (4.69) refraction into the particle (N = 2) Fig. 4.39. Decomposition of an incident ray into reflection and refraction bya spherical particle 130 4 Light Scattering from Small Particles me(l) -e =(cos8. -mcos8t )e(l) t W (4.70) 1 and refraction out of the particle meiN-I) _eiN) =(mcos8t -cos8.)e(N) t pr I (4.71) Eliminating e~N) using e(N+I) _ eiN) e(N)= _ _ __ , 2cos8, (4.72) yields the ray path for the N'h scattering order when N 2':: 2 (Albrecht et al. 1996, Borys et al. 1998) -e w e(l) 0 e(2) 0 =M GO e(3) eiN) (4.73) eiN) pr with A B 0 0 0 0 C D C 0 0 0 MGO = 0 C D C 0 0 0 0 0 0 B A m A=cos8;-mcos8,+--- , 2cos8, 1 C=----, 2 cos 8, B= (4.74) m 2cos8, 1 D=---2cos8 cos8, ' (4.75) (4.76) The rank of the matrix corresponds to the scattering order N. The superseripts (1) to (N) identify the various segments of the ray inside the particle. The vectors e~~) and ew define the scattering angle iJ s cosiJ s =e(N)·e pr w (4.77) of the respective scattering order N, after which the incident angle 8; and the refractive angle B, = arcsin(sin 8; Im) for each ray and scattering order N can be found using Eq. (4.20). Inverting the system of equations in Eq. (4.73) leads to a solution for the coordinates of the incident points and glare points on the particle surface for each scattering order N (N 2':: 2; for N = 1 Eq. (4.68) is sufficient). The incident electromagnetic field is sampled around each of the incident points 4.2 Scattering of an Inhomogeneous Field 131 and projected onto the detector using the governing relations described in section 4.1.1. Knowing the field distribution within the laser beam, the amplitude E b , the phase ({Jb and the propagation direction of the wavefront e w at each incident point are also known (Eq. (3.59), (3.67». The divergence of an inhomogeneous laser beam leads to small deviations in the orientation of the wavefront vector e w at the incident point, which in turn yields a slightly different angle of incidence B; and scattering angle iJ, for a given scattering order. Thus, the correct coordinates for the vectors e(1) ... e(N) must be found by iteration. The iteration procedure used for all possible partial rays and scattering orders which reach the detector can be expressed in terms of the following steps: 1. Definition of e w = eb (as for a plane wave), the scattering vector (see Fig. 4.38) e pr = (rOr -rop)/lror -ropl and the scattering angle 0, = arccos(e pr ·e w ); 2. The scattering angle iJ, leads to the incident angle B;, Eqs. (4.20) and (4.23); 3. Deterrnine the vectors e(1) ... e(N) and thus the interaction points rop + r pe(1) and rop + rpe(N) on the particle surface using an inversion of Eq. (4.73) (Eq. (4.68)for N = 1); 4. Compute the new wave vector e w at the incident point rop + rp e(1); 5. Correct the vector from the glare point to the receiver e~~) = [ror - (rop + rpe(N» I/lror - (rop + rpe(N» I; 6. Compute the new scattering angle iJ, = arccos(e~~) 'e w ); 7. Repeat steps 2 to 6 until the coordinates of the vectors rpe(1) ... rpe(N) no longer change more than some prescribed amount t:« A (t: = 10-3 X A is generally sufficient). 4.2.7.2 Intensity and Phase In addition to determining the interaction points on the surface of the particle it is also necessary with inhomogeneous fields to evaluate the modified phase rfJ~~J for computation of the amplitude functions .S.\~), according to Eq. (4.30). Considering the ray path from the incident wave to the detector, the resulting phase can be considered as the sum of various contributions, as in the case of a plane wave (see Fig. 4.40). The starting point is the phase of the laser beam at the incident point ({J~,N) of the N th scattering order on the particle surface rop + r pe U,N).1 This phase can be taken directly from the analytic description of the input beam. According to Eq. (3.59), this would be for the case of a Gaussian beam mU,N)=m(rU,N»)=_k [ Z(i,N)+ 'rb 'rb Op b 0 ( XU,N»)2 +(yU,N»)2J 0 0 2R ( (i,N») b 1 Zo U,N) +arctan~ 1 (4.78) Rb To avoid confusion with the numbering of the scattering orders, the superscript (i,N) will be used for the incident point and (g,N) for the glare point of the N ili scattering order. 132 4 Light Scattering from Small Particles x ,-' " Fig. 4.40. Refraction (N = 2) for a sphere positioned peripherally in a laser beam where the position of the incident point is defined by XU,N)j rU,N) Op = r Op + rPe(l,N) = [ y~i'N) 0 (4.79) Z(i,N) o The phase change of the incident beam between the center of the particle and the incident point (4.80) is the first term, which contributes to the phase of the scattering function. Note that this phase is not equal to rp k w • eU,N) for inhomogeneous waves, because the wavefronts in general are not parallel to each other. The phase change within the particle amounts to 1 t/J~N) = -2(N -l)kw m rp cosB, ' k w =kb (4.81) The phase change between the glare point and the detector is A,(g,N) 'I' pr =-k w IrOr -(rOp +rp e(g,N))1 (4.82) Similar to the case of a plane wave, the phase jumps due to reflection t/J~NJ,'P' Eqs. (4.33) to (4.35), and due to the beam focussing t/J(P, Eq. (4.39), can be used without change. The phase at the detector surface for each scattering order is obtained by summing over all phase terms 2 m(N) 'rr 1 2 == mU,N) + A,(N) + A,(N) + A,(N) + A,(g,N) 'rb '1', 'l'PO,ep 'l'fl 'l'pr (4.83) The sign depends on the definition of the wave, Eq. (3.22). These relations are expressed in terms of a single scattering order N; however, a final computation must include all partial rays of each scattering order. 4.2 Scattering of an Inhomogeneous Field 133 Because in Eq. (4.1) the phase change from the particle center to the detector surface f/Jpr = -kwrpr is already considered and the initial phase of the incident wave has no intluence on the scattering function, the phase of the scattering function ~.\~) is given by n,(N) 'l'zJ.'P = n,U.N) 'l'w + n,(N) + AN) + n,(N) + n,(g.N) _ 'l't 'l'PzJ.'P 'l'fl 'l'pr (4.84) n, 'l'pr The intensity coefficients i<:'J can be found using the same expressions as with a plane wave, Eq. (4.28). The scattering functions relate the electric field strength at the center of the particle to the scattered electric field strength. Therefore, the electric field strength at the incident point Ei;·N) has to be related to the electric field strength at the particle center Eb (rop ), and this intluences the magnitude of the scattering function. 4.2.7.3 Summary ofthe Results All quantities to compute the scattered field strength from Eq. (4.1) are therefore available. S(N) _1 _ - E~;,N) 0N) (. n,(N») ( ) VI 'P exp J'I' 'P Eb r op and S(N) _2 _ E~;·N) 0N) --(--)VI zJ E r b op (. n,(N») exp J'I'zJ (4.85) The electric field at the receiver can be computed by summing over all scattering orders and using the relation from Eq. (4.1) (4.86) (4.87) In addition to the various scattering orders, the diffraction can be included for the case of an inhomogeneous incident field. A simple approximation can be obtained by assuming that the amplitude and phase of the incident wave does not vary considerably over the diameter of the particle. Thus, only the amplitude and phase of the incident field ~b at the position of the particle must be considered, as in Eq. (4.11). For larger particles, in which the amplitude and phase of the incident wave cannot be considered constant over the particle diameter, other methods must be employed. Possibilities include a numerical solution of Eq. (4.10) or application of Eq. (4.11) to all plane partial waves of the Fourier transformed incident wave. This latter method is outlined in the next section as part of the Fourier Lorenz-Mie theory (section 4.2.2). Also a comparison of results using EGO and FLMT will be presented in section 4.2.3. All scattering functions for all scattering orders are then available. For areal detector the computation must be repeated for all the detector segments. The results are then transformed into the detector co ordinate system, Eq. (4.4), before integrating over the active area of the detector, Eqs. (4.5) to (4.9). 134 4 Light Scattering [rom Small Particles 4.2.2 Description using Fourier Lorenz-Mie Theory (FLMT) In signal theory, the analysis of instationary signals is often reduced to that of stationary signals using a Fourier decomposition of the time series. The Fourier analysis yields amplitude and phase distributions in frequency domain, typically in discrete form. The response of the system to an instationary input can be found by summing over all of the discrete frequency contributions. In a very similar mann er, an inhomogeneous wave can be constructed as the superposition over many plane waves, since light propagation is a linear phenomenon. In time series analysis the Fourier transform is between time and frequency domain. For an inhomogeneous wave in space, the two-dimensional Fourier transform yields an amplitude and phase spectrum in wavenumber domain. Each individual spectral coefficient represents a homogeneous plane wave. Thus, the inhomogeneous wave is decomposed into many homogeneous plane waves. The contribution of each homogeneous plane wave to the scattered field can be determined using geometrical optics or the Lorenz-Mie theory. Summing over all contributions yields the total scattered field. This technique falls into the category of Fourier optics (Goodman 1968, Menzel et al. 1974) and is often called the plane wave approximation (Clemmow 1966). In the following section the decomposition of an arbitrary inhomogeneous wave into plane waves and, in particular, the constraints on the discretisation will be discussed. As a special example, the transformation of a Gaussian laser beam will be given. The scattered field will be obtained using the Lorenz-Mie theory (section 4.1.2) and thus this approach is called the Fourier Lorenz-Mie theory (FLMT). 4.2.2.1 The Spectrum of Plane Waves According to Goodman (1968), an arbitrary two-dimensional field strength distribution extending in a plane z = zr can be transformed onto the image plane using a two-dimensional Fourier transform 1 +00+- f(l,m) = A? LLQAx,y)exp(jk(lx+my)]dxdy (4.88) (4.89) 4 is an arbitrary normalization factor, although it is useful to use the wavelength of light in the medium of propagation. The inverse Fourier transform gives then the components ofthe initial input field (Clemmow 1966) +00+- Qx(x,y) = JJE(l,m)exp[ -jk (lx + my)]dldm (4.90) QAx,y) = -J (4.91) Jg(l,m)exp(-jk(lx+my)]dldm 4.2 Scattering of an Inhomogeneous Field In the image plane (l,m) the transformed field an electric field, namely ~(l,m)also 135 takes the form of ~2D(I,m) = exp"(l,m )+eyg(l,m) (4.92) Adding a third component B.(l,m) in the direction of e z ~(Z,m) = exp"(l,m) + ey g(l,m)+ ezB.(I,m) (4.93) leaves the initial field exQx(x,y)+eyQy(x,y) and the two-dimensional transformed field exP"(l,m) + eyQ(l,m) unaltered for z = zr. The inverse transformation --J ~(x,y,z)= J[exp"(I,m)+eyg(l,m)+ezB.(I,m)] (4.94) x exp( -jk[lx +my+ n(z -Zr )])dldm now leads to a three-dimensional electric field distribution in (x,y,z) space. Due to linearity, the term under the integral must still satisfy Maxwell's Eqs. (3.1)(3.3). This leads to the wave equation ~~(l,m)+k2~(l,m)=O (4.95) for which Eq. (4.94) represents a solution. The field must also be divergence free div(E~)=O (4.96) therefore the wavenumber indices I and m determine n through n=±.Jl-1 2-m 2 , f+m 2 :::;;1 (4.97) The boundary condition Eq. (4.96) leads in the transformed plane to the relation R (l,m ) = - 1p"(I,m)+mQ(I,m) n (4.98) The z component of the field is therefore determined by the x and y components. The essential point is that the spectrum is expressed as a field strength spectrum and the integrand in Eq. (4.94) can be interpreted as a homogeneous plane wave with the wave vector (4.99) and the vector of the electric field strength p(l,m)j ~(l,m)= [g(l,m) B.(l,m) (4.100) 136 4 Light Scattering from Small Particles k is the magnitude of the wave veetor, i.e. the wavenumber, and the spatial frequencies (l,m,n) are the direetional eosines of the wave veetor. The inhomogeneous field in the original domain has thus been deeomposed into a spectrum of homogeneous plane waves in the transform domain. The field strength and the wave veetor are orthogonal for eaeh homogeneous plane wave, i.e. ~(l,m) .k(l,m) = 0 (4.101) The amplitude speetrum E(l,m) and the phase speetrum rp(l,m) are given by Eq. (4.100). The eleetrie field strength in the original domain is obtained by integrating over all homogeneous plane waves in the transformed plane, as expressed by Eq. (4.94). These relations are pietured in Fig. 4.41. After transformation, the three-dimensional eleetrie field strength is represented by a speetrum of homogeneous plane waves with their wave veetors all ending on a sphere of radius k. The amplitudes and the phases of the two eomponents of the electrie field strength perpendieular to k are related to E. and Q as follows: E e = wx _wx (p .J ~1-m2 2 1-1 -m 2 + _Q m1 ~ nvl-m- 1 (4.102) e wx (4.103) This implies that the Zw eomponent of the eleetrie field of eaeh homogeneous k lD (I,III) = -2lt ( I ) A. 111 Transforma tion plane Wave veclors 0 f plane wave k waves Plane of reconslruclion BCS ...~----____~~~~~~~__________-:z. (MCS) Fig. 4.41. Interpretation of the spectrum in image domain as plane waves in original domain 4.2 Scattering of an Inhomogeneous Field 137 plane wave is zero, as expected for a medium with no damping (4.104) !iwz =0 The field vector in coordinates of the partial plane wave is (4.105) The direction of the two electric field components and the propagation direction ofthe homogeneous planes are given by ml n ~1-m2 e wx = 0 1 ~1-m2 e wy = ~1-m2 -~1-m2 , mn ~1-m2 e_ =[:J (4.106) The wave vector in the coordinate system of the partial waves is therefore (4.107) The reconstruction of the three-dimensional field vector at an arbitrary point ZI =Z-ZT gives the propagation distance of the homogeneous plane waves in the direction perpendicular to the transformation plane. For the reconstruction of the two-dimensional field strength from Eqs. (4.89) and (4.90), ZI is equal zero, ZI = Z - ZT = 0, and the situation simplifies such that all three-dimensional wave vectors are now projected onto the Z = ZT plane. Note that these projeeted vectors have different wavenumbers. Even if only the x and/or y component of the inhomogeneous field strength in the transformation plane is known, the conditions in Eqs. (4.95) and (4.96) lead through Eq. (4.98) to the Z component for a wave which propagates in the Z direction. The coordinates in the spectral domain can be used as directional eosines of the plane wave vectors in the original domain. Equation (4.97) states that for n > 0, the energy transport of an inhomogeneous wave is in the positive Z direction and that the incident wave also propagates in the positive Z direction. For n < 0 the energy transport is in the negative Z direction. To keep the integration limits for 1and m symmetrie about zero, the transformation is chosen perpendicular to the Poynting vector of the inhomogeneous incident wave. An inhomogeneous wave is constructed as the sum of a number of plane waves with various amplitudes, phases and propagation directions but with the same wavenumber. (x,y,z) is given by the Eq. (4.94). The value Transformation of a Gaussian Beam. The transformation of a Gaussian beam (seetion 3.1.4.2) into aplane wave spectrum will be given as an example. 138 4 Light Scattering [rom Small Particles Equation (3.59) gives the Yb component of a polarized Gaussian beam depending on Zb' The transformation can be placed at an arbitrary Zb = ZT position in the beam by 1 +=+= g(l,m)=y f fgYb(Xb'Yb,Zb =zT)exp[jkb(lxb+mYb)]dxbdYb b (4.108) -== If polarization in the x b direction exists, the second component can be written as 1 +=+= r.(l,m) =yb __ f f= gXb(Xb'Yb,Zb =ZT )exp[jkb(lxb+ mYb)]dxbdYb (4.109) These integrals can be solved analytically and the field strengths of the plane waves in the spectrum are (4.110) (4.111) Figure 4.42 illustrates the amplitude and phase distributions of the field strength for a laser beam in each of the original and image domains respectively. Note that the Fourier transform of a Gaussian distribution is also a Gaussian distribution, as exhibited by the function Q in this figure. However, it is also noteworthy that the Q distribution would remain unchanged, regardless of the ZT value at which the original x b - Yb plane was chosen. Nevertheless, it can still be recommended to choose the original plane at a ZT position where the desired inverse transform is sought, since extrapolation errors in the phase increase for z\ = Zb - ZT values farther removed from the original plane. 4.2.2.2 Numericallmplementation The implementation treats the integrals in Eqs. (4.88) and (4.89)as summations for any position of the transformation plane ZT: (4.112) 4.2 Scattering of an Inhomogeneous Field a b W.), I 0.8 139 lliob)' I 0.6 0.1 0.2 100 o c 3xlO' o x. IQI W••),I o Y. [).ImJ - 100 l).I mJ 100 100 y.l).ImJ 100 d 2x 10' 17.68 17.7 0.01 0,01 0,0 I 11-1 0.0 1 0.0 1 o 1111-1 1 [- J 0.0 1 f e .>xlO' 2XIO' 11-1 0.01 0.0 1 Fig. 4.42a-f. Amplitude and phase of a laser beam in the original domain and the image plane (Ab = 488 nm, rwb = 50 /lII1). a,b Relative amplitude and phase of electric field strength ofthe laser beam at Zb = 1 ~m, c,d Amplitude and phase ofthe spectrum ofplane waves transformed at ZT = 1 ~m , e,f Amplitude and phase of the spectrum of plane waves transformed at ZT = 1000 ~m (4.113) as also those in Eqs. (4.90) and (4.91) 140 4 Light Scattering from Small Particles (4.114) (4.115) with Xi = -xmax + Ax(i -I) , Iv =-lmax +AI(v-I), Yj = - Ymax + Ay(j -I) m w =-mmax +Am(w-I) (4.116) Al and Am are the sampie intervals in the image (transformed) domain and Ax and Ay are the corresponding intervals in the original domain. The inverse transform of Eq. (4.94) for a three-dimensional field distribution in terms of summations is given as N,. N" ~(Xi'Yj'Z) = II[ex!!Jlv,mw)+eyg(Zv,mw)+ ezR(Zv,mw)] v=l w=l (4.117) x exp( - jk[lvxi + mwYj +n(Zv,mw)(z - ZT )])AIAm together with Eqs. (4.97) and (4.98). Sampie Interval. The original wave can be reconstructed to any degree of accuracy by specifying an appropriate number of component plane waves N 2 = N/N m • The task is to choose an appropriate number N and the sampie intervals Al and Am. This issue is weIl known for the discrete Fourier transform and a comparison with other application fields is instructive in choosing the number N and the sam pie intervals. The Fourier transform for equally spaced sampies ai , i = 1,2, ... ,N is given by! Av _ - 1 ~ exp[.J 21t(V-I)(i-I)) r;::; L... ai "I/N i~l (4.118) N For the two-dimensional Fourier transform Fv,w of the function f,j, i = 1,2, .. . ,N and j=I,2, ... ,N V,w IN _ 1 LN F exp(.J21t--'----------'---'---------'-------'----'--'---'(V-I)(i-I)+(W-I)U-I)] N i~l j~l J ',J N F -- (4.119) Comparison ofthe two Eqs. (4.112) and (4.119) leads to AxAy E(Zv,mw)=~NFv,w (4.120) with 1 Several different normalization factors and signs of the phase are in use in defining the Fourier transformations. 4.2 Scattering of an Inhomogeneous Field lvx; (v-nU -1) A N 141 (4.121) The transform between P(l ,m ) and F can exploit symmetry conditions in the periodic transformation: leading to v,w A v-I 1 =--v Llx N 1 = v far A N -(v-I) Llx N N l:S;v:S;- 2 (4.122) for N -+I:S;v:S;N (4.123) for N l:S;w:S;- (4.124) N -+I:S;w:S;N (4.125) 2 and A w-l m =--W Lly N m = W A N-(w-l) Lly for N 2 2 These relations lead then to the interval spacing in the image plane (Llv=I, Llw=l) A Lll = lV+l -lv =- - , (4.126) LlxN Thus, relations between the sampIe intervals in the original and in the image domain have been established. Although these intervals can be chosen arbitrarily, the choice in the original domain should be made such that the beam is not truncated significantly (see section 3.2.2) 2 rmax -_~ x max + Ymax 2 >2 rmb - (4.127) For large particles or particle positions at a large distance from the beam axis, the particle surface can exceed the limit of rmb • In this case the periodicity of the discrete Fourier transform can distort the intensity profile. For such cases a larger distance r has to be inserted into Eq. (4.127) for rmb • 2 rmax =~ x max + Y max 2 ->2 r (4.128) The sampIe interval in the image domain determines the accuracy with which the laser beam can be reconstructed from the sum over all partial plane waves. Experience has shown that in the image domain, amplitudes down to about a e-8 amplitude decay should be considered. Using Eq. (4.110) for a Gaussian beam, this threshold becomes (4.129) and the amplitude bandwidth is then 142 4 Light Scattering from Small Particles (4.130) Thus, the number of partial waves N and their extent in wavenumber can be determined as follows: • The sampled region in the original domain of the laser beam is chosen as (4.131) For particle positions X5 p + Y5 p > '~b or for particle diameters sampled region is computed as 'p > 'mb' the (4.132) This situation may occur with large particles, where one side of the particle lies far beyond the 2 'mb limit of the laser beam. • The sampIe interval in the image domain is calculated according to Eq. (4.126) A Al = Am = _ b _ 2'max (4.133) • The sampled region in the image domain is chosen as the e- 8 amplitude decay, giving the bandwidth according to Eq. (4.130) as I max =m max = Abmax 2 =2-J2~ 1t 'wb (4.134) • The number of sampIe points is then given as the ratio of the bandwidth to the sampIe interval Ab Al Ab Am I Al m Am 8-J2 r N=-=--=2-'!:!!J2:..=2~=--~ 1t 'wb (4.135) • The total number of partial plane waves is N 2 • As an example, laser beams with a wavelength of Ab = 488 nm and beam waists of 'wb = 10 J.!m ; 50 11m; 100 11m are considered. The amplitude and phase at four points are computed via the inverse transform. The results are compared with those using Eq. (3.59) (Kogelnik and Li 1966) in Table 4.2. The normalized deviations lie typically weH below 1%, validating this method of representing inhomogeneous waves. In Fig. 4.43 the accuracy of the reconstructed beam as a function of the number of sampIes and the sampled region is plotted. If the sampled region is too small, the spectrum is truncated and the reconstructed beam is incorrect, regardless of the number of sam pIe points chosen. If the number of sampIe points is chosen too small, the sampled region in the spectrum becomes too large and an error also occurs. The sampIe interval depends on the sampled region and on the number of sampIe points. Therefore the gradient in the right part of Fig. 4.43 4.2 Scattering of an Inhomogeneous Field 143 Table 4.2. Comparison of amplitude and phase of laser beams according to Eq. (3.59) and using the inverse transform of a discretely sampled Fourier transform according to Eqs. (4.110) and (4.114) for the points A(rwb/2,rwb/2,0.1I.lm), B(2rwb ,2rwb ,O.ll.lm), C(rmb /2, Y,nb /2,1 mm) and D(2rmb , 2rmb , 1 mm) (Ab = 488 nm). The transformation plane is always located at ZT = o. # The error ofthe phase is normalized by 360 deg Point Waist Amplitude field strength rwb Original [flm] [Vm A 10 B Error Original Reconst. Error# Imax N2 [Vm [% ] [deg] [ -] [ -] l] [deg] 0.6065307 0.6065303 6.12xlO- s -73.7660 -73.7656 6.24x10 5 0.324 4.71xlO-4 0.302 [% ] 1.1lxlO- 6 0.0439 49 -73.8328 -73.7837 l.36xlO-4 0.0439 400 -52.1886 -52.1918 8.89X10- 6 0.0439 169 0.790916 0.333229 1.27xlO-3 0.0439 1369 0.6065307 0.0003355 0.6053632 0.0003348 100 0.6065307 0.6065303 6.24X10 5 -73.7704 -73.77 1.11X10 6 0.0043949 0.0003355 0.0003344 0.324 -73.7711 -73.7216 l.38X10 4 0.00439400 5 0.6064575 0.6064579 6.02xlO- -64.4731 -64.4724 1.94X10 6 0.0043949 0.0003354 0.0003343 0.335 -71.1481 -71.1363 3.28X10 s 0.00439400 C 0 0.6065303 0.0003344 0.6053661 0.0003338 -73.7703 -73.773 -63.1426 -89.8427 l.39XlO- 6 0.0087949 l.38X10 4 0.00879400 -63.1415 3.06x10 6 0.0087949 -89.9402 2.71X10 4 0.00879400 50 B A B C 0 Reconst. 0.0003355 0.0003344 0.324 0.3283150 0.3283219 0.00208 0.0001816 0.0001822 0.358 C 0 A l] lE:byl Phase field strength arg(E:b) -73.7698 -73.7235 Poinl B in Table 4.2 Samp led region 1",,,,, .111,,,,,,, I-J 0,15 40 Fig. 4.43. Dependence of the normalized error of the reconstructed electric field strength on the number of sampie points and sampled region. The maximum error is estimated by comparing the original and the reconstructed field strength at 441 equally spaced points in the original domain -2rwb < x < 2rwb and -2rwb < Y < 2rwb (Ab = 488 nm, rwb = 10 I.lm) 144 4 Light Scattering from Small Particles is not parallel to an axis. The plateau for a large number of sampie points and a large sampled region is caused by round-off errors of the computer. 4.2.2.3 The Fourier Lorenz-Mie Theory (FML T) Using the spectral decomposition of an inhomogeneous electromagnetic wave into partial plane waves allows existing solutions of light scattering for a plane wave to be generalized to inhomogeneous waves. The example of the LorenzMie theory generalized to the Fourier Lorenz-Mie theory will be discussed as an example. For each ofthe indices (l,m) the Lorenz-Mie theory (Eqs. (4.1), (4.48) - (4.51)) can be applied to compute the contributions of each partial plane wave to the scattered intensity (Colak 1978, Albrecht et al. 1995). The final result is obtained by a summation over all scattered contributions from the plane waves, preserving the correct phase. This corresponds in principIe to the inverse transformation back to the original domain. Generally, the geometrie relation between the incident wave vector, the partiele and the receiver position will be different for each partial plane wave. First, the field strengths ~(l,m) of the partial waves from Eq. (4.100) are considered. The phase is with reference to the origin of the transformation plane rT • If the center of the partiele is not coincident with the origin, than the separation must be accounted for as a propagation distance of the partial wave. ~p(l,m) = ~(l,m )exp[ -j k(l,m). (rop - rT)] (4.136) The field strength gp(l,m) ineludes the polarization of each incident plane wave. It is this field strength which is used as an incident wave for the LorenzMie computation. The two polarization components of each partial plane wave lying perpendicular to the propagation direction can be computed according to Eq. (4.106). _ ~wp(l,m)- [ I 2 -Jl-m mn n 2 O -Jl-m ml -Jl-m 2 -Jl-m 2 ~ J ~p(l,m) -Jl-m 2 : n;;t;][pJ~ ~M.!L"eXJ{-jk(l,m)(r", [~ ml (4.137) -r, II The required components perpendicular and parallel to the scattering plane, according to Eq. (4.1), are given by ~LMT(l,m)= ( -sinIP,(l,m) COSIP,(l,m)J (1). coslP"m (I smIP"m ) ~wp(l,m) =M9'~wp(l,m) (4.138) 4.2 Sca ttering of an Inhomogeneous Field 145 Using the scattering angle 1'J" which is now dependent on 1 and m, the scattering functions are given by . 21 (1'J,(Z,m)) = Lf!nll"n{ 1'J,(Z,m ))+ Qn T n{1'J,(I,m)) (4.139) n=l ..22 (1'J s(Z,m)) = Lf!n T n{1'J,(I,m))+ Qnll"n{ 1'J,(I,m)) (4.140) n=l The scattered field in the far field for each partial plane wave is given by (I ) = exp( -jkwrpr ) -E scw ,m kwrpr (..21 (1'J,(I,m)) 0 ) (I) 0..22 {1'J, (I,rn )) -E LMT ,m (4.141) The scattering from the inhomogeneous wave is then given by integration or summation over all scattered partial waves. This corresponds to the inverse Fourier transform from the image plane to the physical plane. The exponential function required for the inverse transformation enters through Eq. (4.136). (4.142) It is important to remember that the field strength vector given by Eq. (4.141) depends on the coordinate system chosen, since the scattering angles depend on 1 and m. Before integration ofEq. (4.142), all field strengths must be expressed in the same coordinate system and that of the receiver is the most convenient. The transformation matrix is denoted as M ß , as introduced in Eq. (4.4). To summarize, the computation of the scattered field using FLMT can be expressed in matrix notation as ~r = exp( -jkwrpr ) J= =J k r Mß(I,m)Ms(I,m)Mq>(I,m)Mw(I,m) w pr (4.143) -00--00 X~2D(I,m)exp( -j[ k(I,m ).(rop - rT )]) dldm Similar to this solution for the far field, allother solutions based on incident plane waves can be extended to the case of inhomogeneous waves, in particular the solution for the near field or intern al field. For the near field, Eqs. (4.139) and (4.140) are replaced by Eqs. (A.35) to (A.37) and for the intern al field, by Eqs. (A.44) to (A.46). Equation (4.141) then yields a three-dimensional field strength vector, which also contains a radial component. It is important to note that for the near field, the radial phase dependence, expressed in the exponential function exp(-jkwrpr ) in Eq. (4.143), is already included in the scattering function ..21,2. Also the Debye series decomposition of the LMT solution, as described in section 4.1.2, can be extended to the case of inhomogeneous waves in this manner. In this case, the scattering function will depend on which scattering order is 146 4 Light Scattering from Small Particles being considered and the total contribution from scattering order p can be computed according to Eq. (4.143) Decomposition into scattering orders of the near field and internal field proceed in a similar manner. 4.2.3 Scattering Characteristics of an Inhomogeneous Field The most common inhomogeneous incident wave is the laser beam, which exhibits in general a Gaussian beam intensity profile (Davis 1979). Because the intensity of the incident beam is spatially shaped and not constant in space like a plane wave, the scattering properties are now dependent on particle position. The intensity in the near field of a spherical particle with diameter of 20 J.!m moving through a laser beam of 10 J.!m beam waist diameter is shown in Fig. 4.44. The intensity is calculated with the FLMT and the beam axis is marked with an arrow on the left of each diagram. In Fig. 4.45 the scattered intensity in the far field is given as a function of scattering angle iJ s for the same configuration as in Fig. 4.44. In comparison to the plane wave case, Fig. 4.13, the scattering functions are asymmetrical. The particle surface is partially illuminated and only these areas are relevant for the scattered intensity. For the plane wave the complete scattered field is translated in space with any motion of the particle, whereas with an inhomogeneous incident wave the scattering function changes during the passage of the particle. For any point in space, the amplitude and phase of the scattered field will also depend on the position of the particle in the incident beam. In Fig. 4.46 several scattered rays as computed using EGO are shown for comparison with the FLMT results. The use of Debye series yields a graphical representation which immediately lets the result be interpreted in terms of geometrical optics. In Fig. 4.47 the internal field and the near field are separated using Debye series decomposition for the case of Fig. 4.44d. In comparison to Fig. 4.37 it is reasonable that only a part of the scattering function is selected with a shaped beam. The particle regions which are not illuminated suppress regions in the scattering function. A good example for this is the Brewster angle for reflection. On the illuminated side of the particle, the Brewster angle is clearly visible, whereas on the lower part of the particle the scattering function of reflection is completely suppressed and the characteristic of the Brewster angle has no influence on the scattering function. The scattered intensity of a given scattering order at a specific angle is proportional to the intensity at the incident point for the selected ray path. For the case of a plane wave, the incident intensity is equal for all incident points. In the 4.2 Sca ttering of an Inhomogeneous Field 147 case of an inhomogeneous incident wave, the intensity at the incident point is dependent on the partide position. The intensity at the incident point of each scattering order is weighted with respect to a homogeneous incident wave. The results shown in Fig. 4.45 are repeated in Fig. 4.48 but with the decomposition into various scattering orders. It can be seen that for any given scattering angle, the relative magnitude of the various scattering orders can change with partide position in the incident beam. The contribution of any one scattering order to the total intensity distribution decreases with the distance of the incident point from the axis of the laser beam. In Fig. 4.48 the scattering function of first-order refraction for aplane wave with the same intensity as that on the laser beam axis is additionally plotted. The curve for the inhomogeneous case touches this symmetric curve at different positions. The scattering orders appear to rotate as the partide moves through the beam. Furthermore, interference effects can be recognized, which lead to intensity oscillations and local intensity maxima and minima. There are two types of interference phenomena present. One is between different scattering orders and the other is with light of the same scattering order but different scattering mo des. The latter is known as the rainbow for each scattering order (see section 4.1.3.3). The rainbow of the second-order refracted light is only present for partide positions from x op = 3 11m to 9 11m (Figs. 4.48b - 4.48d). In the other cases, the incident field illuminates only one of the two incident points significantly. Therefore, scattering orders or partial modes can be suppressed with a shaped beam depending on partide position. When a partide moves through a beam, the dominating scattering mode for a given receiver position can change with partide position. This can lead to a temporal separation of the various scattering orders on a fixed detector, which is not observed for a homogeneous wave. Clearly this dynamic behavior of the scattered field with partide motion has direct consequences for the laser Doppler and phase Doppler measurement techniques. For a given detector position the amplitude, the phase and the composition of the scattered field in terms of scattering orders will depend on the partide position. This is the physical origin of the "Gaussian beam effect" or "trajectory effect" (section 8.3.1) as known from the literature about the phase Doppler technique. For partides of diameter much sm aller than the beam waist, the scattering from an inhomogeneous field tends towards the plane wave case. If the amplitude deviation of the incident light is limited to 5% ... 10% across the partide, the maximum allowable partide-diameter-to-beam-waist ratio is about d p < (0.2 .. .0.3)dwb (dwb-diameter of the beam waist). Nevertheless, the influence of the shaped beam is basically determined by the distance of the incident points to the axis of the beam, which varies with partide size, partide position, refractive index and receiver position. For larger partides the inhomogeneous nature of the incident wave must be accounted for. Together with the signal generation analysis presented in chapter 5, an approximation for evaluating this influence in laser Doppler and phase Doppler systems is given in section 8.3.1. 148 4 Light Scattering from Small Particles -;::- 40 c ::1. >< c .9 20 .;;; o Q.. o - 20 40 40 E ::1. >< c .9 20 '"o Q.. o -20 40 _ 40 WI:l........ . - E ::1. >< C .g 20 "§ Q.. o -20 -20 o 20 40-40 Position y [flm] -20 o 20 10 Position y [flm] Fig. 4.44a-f. Scattered intensity of the near field and internal field of a 20/lll1 particle as a function of particle position in a Gaussian beam of radius rwb = 5 J.Lm calculated with FLMT. The incident beam is indicated with white lines (x op = Yop = zOp = 0 J.Lm, d p = 20 /lll1, m =1.333, Ab = 488 nm, rwb = 5 J.Lm, parallel polarization). a Xb = 0 /lll1, b Xb = 3 J.Lm, c Xb =6 /lll1 , d X b = 9 /lll1 ' e Xb = 12 J.Lm, f Xb = 15 J.Lm 4.2 Scattering of an Inhomogeneous Field a 90 c 90 b 90 d 270 90 270 Xp 149 =9 11m o 270 e 90 270 90 f Xp =151l1ll o 270 - - - - Scattering function laser beam 270 Scattering function plane wave Fig. 4.45a-f. Total scattered intensity in the far field (logarithmic scale) for cases shown in Fig. 4.44 calculated with FLMT. The diffraction peak at 0 deg is excluded for clarity (dl' = 20 Ilm, m = 1.333, Ab = 488 nm, rwb = Sllm, parallel polarization) 150 4 Light Scattering from Small Particles E 40 ~rnnTwr~TTrT__ :::1. c .2 20 20 o 20 Positi on 10 Z [jl m) Fig. 4.46. Visualization of the light paths for different scattering orders calculated with extended geometrical optics in comparison with the near and internal field calculated with FLMT (x op = YoP = z op = 0 11m, d p = 20 /lffi, m = 1.333, Ab = 488 nm, rwb = 5 /lffi, Xb = 10 11m, parallel polarization) 4.2.3.7 Comparison between Extended Geometrical Optics and FLMT As in the case of a plane wave, both calculation methods, Fourier Lorenz-Mie theory and extended geometrical optics complement one another for the calculation and interpretation ofthe scattering diagrams. Figure 4.49a shows the total scattered intensity for a 300 Ilm water droplet in a 100 Ilm laser beam waist as a function of the scattering angle for the three methods (LMT, FLMT, EGO). The FLMT and EGO solutions agree very weIl with one another but deviate considerably from the Lorenz-Mie theory result, as expected. In Fig. 4.49b the gopd agreement between results of the two methods is emphasized by also plotting their difference as a function of scattering angle. Particularly good agreement is found in regions where one scattering order dominates, for instance in backscatter with second-order refraction. The largest deviations occur in regions in which scattering orders are mixed, for instance near 30 deg with first and second-order refraction or at 80 deg with reflection and third-order refraction. The largest deviations occur coincident with large negative dips in the scattering function, corresponding to destructive interference between scattering orders. The smallest variations of computed amplitude, for example through neglect of surface waves in EGO, can therefore lead to rela- 4.2 Scattering of an Inhomogeneous Field S 151 40 ~ >< t:l .Si 20 .~ 0 "'" 0 -20 S -40 40 ~ >< t:l 0 :.s 20 '"0 "'" 0 -20 S -40 40 ~ >< t:l :~0 "'" 20 0 -20 -40 -40 -20 o 20 40-40 Position y [[lm] -20 o 20 40 Position y [[lm] Fig. 4.47a-f. Scattered intensity of the near field and internal field of a 20/lID particle decomposed in scattering orders with Debye series. The incident beam is indicated with lines (x op = YoP = zop = 0!lm, d p = 20 !lm, m = 1.333, Ab = 488 nm, rwb = 5!lm, Xb = 10 !lm, parallel polarization). a Incident beam and reflection, b Reflection only, c First-order refraction, d Second-order refraction, e Third-order refraction, f Fourthorder refraction 152 4 Light Scattering from Small Particles a 90 .. b . : )f. ~-'" ,.,.•.~... . . . ........ _ .). l' ~~ ~ '~'... ~.~m::: ," :.:.. .. o ':'''' - - 90 o 270 d x , = ~I-\ nl 90 x , = 91-\ nl 270 90 c '" . .... 270 c 90 270 90 .... 270 - - - - - , Diffraction and retlection (p = 1) p = 2 .................. P = 3 Refraction: -0-- 270 1 st order refraction plane wave, p = 2 P=4 '" P= 5 • P=6 Fig. 4.48a-f. Intensity distribution in the far field (logarithmic scale) of various scattering orders from ca ses Fig. 4.44 calculated with FLMT and Debye series decomposition (d p =20 /lIIl, m =1.333, Ab = 488nm, rwb =5 f.lm, parallel polarization). 4.2 Scattering of an Inhomogeneous Field 153 ~ 10 ' Q .: ~ 10' :::'-'0 c:.. 10 ' "0 0) >.<; u er" 10 • 10'" 3: 10 • L- "::: 0 Q. "uc: " L- ~ es 45 90 Fig. 4.49a-c. Comparison of scattered power as a function of scattering angle computed using LMT, FLMT and EGO (Ab = 488 nm, rwb = 50 /ll11, d p = 300 /ll11, ro, = 200 mm , A, = 3.1415 mm' , point-like receiver, PI =1 W, ro p =0, EGO: sum over flrst 9 scattering orders). a Comparison between received power calculated with LMT, FLMT and EGO, b Absolute difference between FLMT and EGO, c Relative deviations between FLMT and EGO tively large normalized errors. However, for finite detector sizes, these effects become negligible due to their very low absolute amplitude and very restricted spatial extent. Figure 4.50 shows the decomposed scattered field for each of FLMT with Debye series and EGO separately. Also here the good qualitative and quantitative agreement of the results is evident. Figure 4.51 provides even more details in comparing the two methods by examining individual scattering orders and the difference between results com- 154 4 Light Scattering from Small Particles Scallering angle iJ, Idegl --A---- FLMT: Diffraction and reflection (p =1), EGO: reflection (N =1) --{)-1 ,I refraction (p, N =2) - - 2nd refraction (p, N =3) ~ refraction (p, N =5) 1h -<J--- 5 refraction (p, N =6) -----o--7 1h refraction (p, N =8) ----t>- 81h refraction (p, N =9) 41h ___ 3'd refraction (p, N =4) ---D--- 6th refraction (p, N =7) Fig. 4.50. Comparison of the first 9 scattering orders as a function of scattering angle computed using FLMT and EGO from case shown in Fig. 4.49 puted using the two methods. At very small scattering angles (Fig.4.51a) the differences are large since FLMT provides reflection and diffraction together. For this calculation the EGO computation does not account for diffraction. For first-order refraction (Fig. 4.51b) good agreement between the two methods is found up to about 83 deg, beyond which the surface waves (Hovenac and Lock 1992) become dominant in the FLMT solution. These are not accounted for in EGO, however the deviations shown in this diagram provides a first estimate of their magnitude. Similar estimates can be made for the high er scattering orders, second-order refraction (Fig. 4.51c) or third-order refraction (Fig. 4.51d). Generally the deviations between methods increase near the limiting and rainbow angles, Eqs. (4.44), (4.66)-(4.67); however, since the absolute amplitudes here are small, little effect on the final result integrated over a detector surface is observed, as seen from Fig. 4.49. 4.2 Sca ttering of an Inhomogeneous Field ~ JO"' Diffrat ion and rcO ection FLMT ... "?:0 155 10 ' 10 ' "'" 10 " 10 ' 101\ 10 11 10 '" 10"" 10'" JO ' - 10 ' 3: nd 2 order Rcfracti on: ... 6" JO ' 10 ' •••••••• EGO Q., - - IFLMT.EGOI JO ' 10 " JO B - - FLMT 10" -------- EGO - - IFLMT.EG lOB 10 " 0 90 Scattering angle 180 ~ [deg] 0 90 180 Scattering angle tI; [deg] Fig. 4.51. Comparison of the dominant scattering orders, reflection, first, se co nd and third-order refraction, computed using FLMT and EGO plotted together with the absolute deviations between both theories from the case shown in Figs. 4.49 and 4.50 In conclusion, either method provides results which can be considered adequate to analyze optical measurement systems involving light scattering from particles in inhomogeneous ineident waves. 4.2.3.2 Imaging Properties of a Particle Depending on where the particle is situated in the incident beam and where the detector is placed, the incident beam is effectively sampled at the incident points of each scattering order. For a one-dimensional case this is illustrated in Fig. 4.52. Each scattering order or each incident point creates a virtual image of the ineident intensity distribution, only visible from the chosen receiver location (Fig. 4.52a). Because of the shaped structure of the beam, the virtual images are separated in space. The larger the distance between the ineident points, the 156 4 Light Scattering from Small Particles Incidenl Gaussian beam a (B) Receiver signals: 1= 0 Timel \ Cenler posilion Itime of Ihe parlicle Fig. 4.52a-c. Imaging properties of a particle in a one-dimensional shaped beam. a Dependence on receiver location, b Dependence on scattering order, c Dependence on particle size larger the separation between the virtual images. This displacement of virtual images becomes more distinct as the ratio of partide size to beam diameter increases, Fig. 4.52. In Fig. 4.53 the three-dimensional imaging behavior is visualized for a partide, which creates three virtual images of the incident beam. Fig. 4.53. Virtual images of a laser beam centered on the z axis bya large particle. The images correspond to three different scattering modes 4.2 Scattering of an Inhomogeneous Field 157 Note that the absolute intensity of the images could be different because of the different scattering amplitudes of each scattering order. In the following section the scattered light on the surface of adetector fIxed in space is computed for particles of various sizes placed with the particle center in the YoP = 0 plane. In fact, the particle may produce several virtual images of the laser beam on the detector, depending on its position, the position of the detector and which scattering orders are involved. Computations have been carried out using FLMT for three different particle sizes. The optical confIgurations and the results are shown in Fig. 4.54. In Fig. 4.54a a particle of diameter 1 f..lm is traversed through a beam of 6 f..lm diameter at its 10, Real laser beamand partide motion 10, / Plane 01' parlide motion c Real laser beam c dccay 100 10, 50 / Di placcd virtual image ofthc beam ~~~o -20 d p = 16lim 20 xo" IjlI111Double pC'dk from reflection and refraction :.L'I particle motion Real laser beam Fig.4.54a-c.Virtual images of laser beam (A b =512nm, d wb =6Jlm) on adetector ( rjJ, = 10 deg, lfI, = 11.3 deg) using an oil drop let of different sizes in wa ter (m = 1.08). a d p =lJlm, b dp =8Jlm, c d p = 16Jlm 158 4 Light Scattering from Small Particles waist and the resulting intensity distribution, created by the dominant scattering order, resembles dosely the intensity distribution existing in the real incident beam. In this case the scattering could have been computed equaHy weH using a homogeneous wave approximation, where the amplitude of the homogeneous wave depends on the position of the partide center. For a larger partide, dp = 8 )..Lm, the virtual image of the laser beam is displaced towards negative x values and also exhibits a double peak behavior in space. This becomes even more exaggerated for the largest partide, d p = 16)..Lm, shown in Fig. 4.54c. For large partides the virtual images created by each of the scattering orders are separated significantly in space. Indeed, for some geometries, even a single scattering order can produce more than one virtual image, for instance second-order refraction in backscatter. Notice in Fig. 4.54c that a large partide positioned at the origin is not even visible to the detector. Figure 4.55 provides an interpretation of this effect in terms of geometrical optics. An explanation can be given in terms of the incident points. Each glare point seen by the detector images the laser beam with a displacement in space, depending on the partide size. This is in fact equivalent to the time-shift technique (sections 2.3, 5.3.4 and 9.2) applied to each scattering order individually. For a partide at position 1 in Fig. 4.55, the incident point for the dominant scattering order lies outside a region of significant incident amplitude, whereas for position 2 the incident point is weH within the bounds of the beam. The double peak in x direction corresponds to the two dominant scattering orders for this detector position: reflection and first-order refraction. The exact position of the incident point depends on the scattering order considered, the partide diameter, the relative refractive index and the detector position (Albrecht et al. 1994, 1996). Partide position 2 Particle p ition 1 Incidcnt point x r... (z) e. z Laser bcam y Glare point 0-. ep, elNI'II: I" Receiver Fig. 4.55. Sampling of a laser beam through the ineident interaction point on a particle surface 4.2 Scattering of an Inhomogeneous Field 159 The position of the interaction points can be determined using methods of geometrical optics. The amplitude of the scattered field for the particular scattering order N considered is proportional to the field strength amplitude at the incident point position (see Fig. 4.55) XOpJ = [Yop rop dp , r =p 2 (4.145) zop Since the position of the incident point may be very different for different scattering orders, different amplitudes of the incident beam proflle will be imaged onto the detector. For a Gaussian laser beam, whose axis is aligned with the z co ordinate, the intensity at the detector is proportional to (Fig. 3.4, Eqs. (3.30), (3.59)) (4.146) r [2(XOP+x(;))2+(YOP+y'nfj 1 r (N) 1+ z-2( ZOP+Z (;))2 exp - 2(-2( I+Z z op +Z (;))2) r wb Rb (4.147) Rb with the beam radius Eq. (3.64). The coordinates of the particle position at which a maximum intensity is obtained at the detector are designated as (4.148) At this location the incident point is situated at the origin of the incident beam, i.e. at its intensity maximum. For reflection, the location of the incident and glare points with respect to the center ofthe particle is given by Eq. (4.68) (I) (1) r (,,1. ) = r e (.',1 ) = r e-e pr w p PI e(1) pr -e (1)1 w (4.149) where rp is the particle radius. For first-order refraction (N = 2), the position of the incident point is given by Eqs. (4.70) and (4.71) ( ;2) r' =r P (;2) e' =r (2) ( 2 cos 8 ,cos ( 8;-mcos, 8)) e(2) mepr-m+ w ----'---'----,-----------'----------:p2(cos8; -mcos8,)(m+cos8, (cos8; -mcos8,)) cos8; = 1- (4.150) m 2(I-e(2) .e(2)) pr w -27(-I-+-m-2-_---'---m-r2=(=I=+=e=~=~=.e=~=2)=)~) (4.151) 160 4 Light Scattering irom Small Particles (4.152) In these expressions e<J') is the local unit vector of the incident wave at the incident point of the scattering order N, e~~) is the unit vector of the scattered wave from the glare point to the receiver, Bi is the incident angle and B, is the refr ac ted angle. For the calculation of the amplitude of the scattered field, the vector from the glare point to the receiver e~~) and the wave vector at the incident point e~N) can be replaced by the detector orientation vector e pr and the incident beam vector eb respectively, since the local deviations are extremely small. For the example of dominant first-order refraction and using the previous example of an oil droplet in water (m = 1.08) the partide position for a maximum intensity can be obtained from Eqs. (4.148) to (4.152) and for a dp = 16 /ll1l partide as X max = 5.4 11m, Ymax = 4.7 11m , Z max = 3.6 11m (4.153) This is the location of the center of the partide when the incident point for the first -order refraction, N = 2, is exactly at the middle of the beam waist. If the partide is now moved in the plane YoP = 4.7 11m, the virtual image of the laser beam will appear as though a very small partide is sampling the incident beam in the y = 0 plane, as illustrated in Fig. 4.56. A direct comparison can be drawn between Figs. 4.54a and 4.56. For the condition of Fig. 4.54c, i.e. a partide traversed in the y = 0 plane, the maximum intensity is obtained for the partide center location Yop = 0: x max = (4.154) _X(N) , Yo, ,/ parlide mOIion Y = ~.7 ~un Real lase r bca m Fig. 4.56. Image oi a laser beam on a detector through an oil droplet in water (d p =161-1m) with its center in the plane YOP = Ymax = 4.71-lm (Ab =512nm, d wb =6I-lm, (P, = 10 deg, f/lr = 11.3 deg, ro, = 200 mm, m = 1.08) 4.2 Scattering of an Inhomogeneous Field 161 whieh eorresponds direetly to the maximum of an x-z eross-seetion through the laser beam. For a 16 11m diameter partide in the y = 0 plane, intensity maxima for first-order refraetion (N = 2) are obtained for x max = 5.4 11m, Ymax = 0 11m, zmaxl = -105.4 11m, zmax2 =112.6 11m (4.155) These values correspond to the position of the peaks shown in Fig. 4.54e. In Fig. 4.57 the eontributions to the total deteeted signals are shown for a partide traversing along a zap = -200 11m trajeetory. The shift ealeulated with the FLMT of the refraeted part in x direetion agree weH with the ealeulated X max value of 5.4 11m from the EGO. Furthermore, the loeal maxima are seen to be interferenee between first-order refraction and refleetion/diffraetion. The results presented above demonstrate that either the FLMT or EGO methods are eapable of eomputing the seattered light field of an inhomogeneous ineident wave on a spherieal partiele. Compared with seattering from a plane wave, the non-uniform intensity and the wavefront eurvature of an ineident wave lead not only to a displaeement of the virtual beam image away from the geometrie laser beam loeation (volume displaeement), but also to multiple images, depending on the deteetor position, the seattering order, the size and the refraetive index of the partide and its trajeetory through the ineident beam. Obviously these imaging properties have consequenees for laser Doppler and phase Doppler teehniques. These imaging properties are also the foundation of several other measurement teehniques. The dual-burst teehnique (seetion 8.2.4) arranges the optieal eonfiguration of a phase Doppler instrument sueh that partides pass on trajeetories whieh exhibit two virtual images of the incident beam pair, one for first-order refraetion and one for refleetion. Thus, two phase Dop- ~10 0..' Total received power Diffraction and reflection First -order refraction Pa rtiele diameter " ) Beam diameter " ) -20 -10 o 10 Particle center position 20 x op [flml Fig. 4.57. Scattered intensity for a oll droplet moving through a laser beam (d p = 16 11m , m = 1.08, zop = -200 11m, YoP =0 11m ) 162 4 Light Scattering from Small Particles pIer estimates of size can be obtained in quick sequence as the particle traverses the volume. This redundancy can be exploited as a size verification or to estimate refractive index. Also the shift in the intensity maximum can be used as the basis of a sizing technique known as the volume displacement or time-shift technique (section 9.2). With this method the distance between the intensity maxima of one scattering order and two different receiver locations is size dependent. This distance is measured by knowing the particle velo city and the time between signal maxima when a particle traverses a phase Doppler measurement volume. Single beam/sheet sizing techniques have also exploited this shift of image maxima (Hess and Wood 1993, Albrecht et al. 1993), as discussed further in section 9.2. Finally, these imaging properties are also the cause of the "Gaussian beam effect" in classical phase Doppler systems. The FLMT and the EGO computational methods therefore offer a means to analyze such systems and estimate potential errors (see section 8.3.1). 4.3 Characteristic Quantities of Light Scattered by Particles The scattered electric field strength from small particles can be computed using GO (EGO) or LMT (FLMT) according to Eq. (4.1). For an incident wave linearly polarized in the x direction and a scattered wave propagating in the radial direction to the receiver, the field strength on a point detector is given by (4.156) where Eo is the incident field strength and 11 kr is the decrease of the field strength due to of the propagating, scattered, spherical wave. The scattering characteristics of the particle are contained in the scattering function (4.157) The scattered intensity at a point detector is then (van de Hulst 1957) [Sc E; )2 [0 ( )2 = EC--2 5sc (.Q u,,{jJs = - - 2 5sc tJs,{jJ, (kr) (4.158) (kr) and through integration over the detector surface (see Eqs. (4.5)-(4.9» the total received power is PSc = Jf!sc dA = ~~ Jf 5l c( = ~~ Q)dQ il, A, J J5 .JzJ s L1tp~ sc ( tJ" {jJs)2 (4.159) sin tJs d tJs d{jJs 4.3 Characteristic Quantities ofLight Scattered by Particles 163 (4.160) where Qr is the solid angle of the receiver. The integral scattering function G( 19" qJ,) = f f5 sc ( 19" (4.161) qJ,)2 sin 19, d 19, dqJs .1tp s L1t9 s contains therefore specific scattering characteristics (XM' m) of a particle with respect to the position (19" qJ" r) and size (A19 s, Aips) of the receiving optics (see Fig. 4.58). Since for large particles the scattering characteristics can vary strongly with trajectory, such an integral scattering function must be used with caution. Only in the case of d p «d wb can the scattering function be considered independent of the incident wave characteristics. In this case the integral function computed for a particle in the middle of the measurement volume will yield directly the maximum signal amplitude and can be used for system design. In Fig. 4.59 the integral scattering function for a detector aperture with a receiver angle of rr = 8 deg (see Fig. 4.58) and for a water droplet in air is shown for the case of various forward and backscatter detector positions. The scattering function in backscatter is typically 100 times weaker than for forward scatter. For larger particles, the maximum signal amplitude will not necessarily occur for a particle positioned in the center of the measurement volume. To obtain an indication of maximum signal amplitude, the integral scattering function must be evaluated for the particle position of maximum scattering intensity. If absorption is neglected, the total scattered power can be obtained by integrating Eq. (4.160) (4.162) z Fig. 4.58. Receiver position and receiving angle t!, 164 \.:J 4 Light Scattering from Small Particles 1000 - -,.--:..:-:..-..::.. ..... - ... ---:: .. ............ . .- . . ,::.-:.:=-_.--_ . -'- ~ 0 ..;j u ~ c2 100 /" ;"""' ... ;' ~ .~ ~ :-.. --- Ir~,/..... , ' CI) = '" - -'"' ~ .,.. 10 I' /",/ .. )\..,,1 \ I • ... \ I : '" :s .... CI) '\I" ~, <U " Receiver loeation tJ, '" 6 deg tJ, '" 12 deg tJ, '" 18 deg tJ, '" 174 deg 0.1 0.01 0 20 10 30 ParticIe diameter dp [flml Fig. 4.59. Scattering characteristics for water droplets in air for different scattering directions (Ab = 488 nm, d wb = 400 11m, = 8 deg) r, and in analogy to gas discharge (van de Hulst 1981), a scattering cross-section can be defined Ppse ASe (4.163) =-- Io as weH as a normalized scattering efficiency Ppsc (4.164) 17sc =-I--2 o 1trp given with respect to the particle projected area. The surn of absorption and the scattering cross-section is the extinction cross-section (4.165) which for negligible absorption is just equal to the scattering cross-section. By substituting the actual scattered power at the detector (Eq. (4.160» into Eq. (4.164), the coHection efficiency can be obtained. PSc PSc 17 Seeff = - I - - 2 = 17SC -P o 1t rp (4.166) pSc Figure 4.60 shows the coHection efficiency for the detector defined in Fig. 4.59. The coHection efficiency is in fact larger for srnaller particles, reaching 10% for forward scatter configurations. 4.3 Characteristic Quantities ofLight Scattered by Particles 165 Receiver location: Forward scattering 0, '" 6 deg 0, '" 12 deg Backward scattering 0, '" 174 deg I I, ' " ..... \. " - " '" ... _ _________________ _____ ________ _ , . ,'"' ... " , .. ' , f ''',' .. \." ',01' _....... .......... "... ....... "'" "~~"" .. ..' .......... ___ .... ', ................... _... ,r, .. " .. ,# 105L-i-~-L-L~~~L-~~-L-L~~ o 10 __L-L-~~-L-L-L~~~L-i-~-L-L-L~~ 20 30 Particle diameter dp [11m] Fig. 4.60. Collection efficiency as a function of particle size for different scattering directions (,1 = 488 nm, d wb = 400 /-Lm, y, =8 deg, m = 1.334) PART 11 MEASUREMENT PRINCIPLES 170 5 Signal Generation in Laser Doppler and Phase Doppler Systems to the spatially invariant axis of this main coordinate system. The analysis is presented in a general form, applicable for a Gaussian beam, section 3.1.4.2. In the case of plane waves, several terms can be omitted. The illuminated volume is formed by the intersection of two beams, with an orientation of the wave vectors along the laser beam propagation axis. = 2rc k b Ab ' (5.1) b=1,2 The two laser beam axes are mirrored about the y-z plane, in the x-z plane and the center of the illuminated volume is placed at the origin of the main coordinate system. The position of the laser beam waists can be arbitrarily placed along the axes of the beams, as seen in Fig. 5.2. The two waists do not coincide when the system is not weIl aligned. The laser Doppler or phase Doppler detectors are placed at the position r Or from the origin of the main coordinate system, at an off-axis angle of f/Jr (x rotation l ) and at one elevation angle, IfI r , for laser Doppler systems or at two elevation angles, ±1fI r' for phase Doppler systems (y rotation) (5.2) r=1,2 z Fig. 5.2. Geometry of the laser bcam intersection volume (for the case of poor alignment) 1 The direction of rotation around the x axis for the off-axis is defined mathematical positive. For illustration, a negative off-axis angle is used in Fig. 5.1. 5.1 The Signal From an Arbitrarily Positioned Detector 171 A signal arising from a particle moving with the velocity v p is to be examined. Initially, the detectors will be considered as point detectors, meaning that the scattered light intensity remains constant over a small detector surface M r • The particle is spherical and has the diameter d p = 2rp and an instantaneous position rap- The trajectory of the particle is defined by the initial position r pa at time t = 0 and the velo city vector v p as shown in Fig. 5.3. (5.3) The analysis of how the particle images the illuminated volume onto the detector can follow either as a spatial or temporal analysis. The spatial analysis examines the imaging properties of the particle over all possible particle positions, rap' at a fixed time t. The temporal analysis transforms a three-dimensional trajectory of the particle into a one-dimensional time signal of scattered light intensity. In either case, the analysis can be characterized by the particle diameter. • For very small particles (d p «Ab)' real interference fringes are present in the illuminated volume and this interference structure of the laser beams is sampled through the velo city and the trajectory of the particle (Eq. (5.3». The scattering properties of the particle reduce to an intensity factor. • Medium sized particles (d p '" Ab) can be analyzed by scattering theories based on the Maxwell's equations. In most cases, the incident amplitude distribution over the particle surface can be approximated by a homogeneous wave, but the phase cannot be assumed constant. • For large particles (d p »Ab)' the laser Doppler or phase Doppler system can be analyzed using geometrical optics or Maxwell's equations. Normally, the intensity and phase changes over the particle surface must be taken into account. Fig. 5.3. Vector notation for incident and glare points 172 5 Signal Genera tion in Laser Doppler and Phase Doppler Systems This chapter examines the signal generation according to this particle size classification (very small or large) in space and time domains. The analysis begins with the scattered electromagnetic vector on the surface of one detector, placed at arbitrary angles rp, and If/" Eq. (5.2), with respect to one laser beam with an arbitrary rotation angle of ~ < 45 deg, Eq. (5.1). For a laser Doppler or phase Doppler detector, which receives the scattered waves from two beams, the superposition of two scattered waves from two laser beams on the detector surface is examined. However first some remarks about the scattering order must be made. For a given detector position rp, and If/ r' the incident points and the glare points on the surface of the particle are determined by the scattering order, N. The scattering order can therefore always be considered as one additional independent variable. The following analysis is valid for any scattering order, therefore, the superscript (N) is dispensed with. If one particular scattering order is dominant (its amplitude is always 10 times larger than all others), then the signal structure can be derived directly from the equations presented below. If more than one scattering order makes significant contributions to the detected signal, then a superposition of the electric fields from all contributing scattering orders on the detector surface must be performed. In such cases, the signal can then become more complex, e.g. dual or multiple bursts (section 8.2.4). 5.1.1 Fundamental Relations The notation ofthe vectors related to the particle is given in Fig. 5.3. The vector r6;) denotes the incident point, (i), on the particle surface of the incident wave b with respect to receiver r, measured from the center of the particle. The position of the incident point in the main co ordinate system is given by (i) U) rO,br = rop + r br J [ x op + x br(i)J (;) = [ YO,b, (;) = Yo p + Ybr(i) XO,b, U) zO,br zop (5.4) + Zbr The radius r~!) refers to the glare point (g) of the incident wave b with respect to the receiver r, after a number of interactions with the particle surface and measured from the center of the particle. The position of the glare point in the main coordinate system is given by X (g) o br r(g) O,b, = r Op J [x + r(g) = [ y(g) = br O,br + x br(g)J YOp + y(g) br op (g) zO,br (5.5) (g) zop + zbr The incident wave at the position of the incident point is defined by the vector k~: and the complex amplitude vector ~~:, which can be split into an amplitude factor and a phase factor. 5.1 The Signal Fram an Arbitrarily Positioned Detector 173 (5.6) For larger particles in inhomogeneous waves, the wavefront curvature may influence the phase of the primary wave at the incident point, as illustrated in Fig. 5.4. In such cases, the local orientation of the incident wave k~: and the field strength ~~: may deviate from the values at the waist center k b and ~Ob The amplitude and phase of the beam at the incident point can be computed according to Eq. (3.59). However Eq. (3.59) is cast in a beam coordinate system and therefore a transformation from the main co ordinate system (x, Y, z) to the coordinates ofthe beam (x b ' Yb' Zb) is necessary x b =xcosiji'+zsiniji' Yb =Y Zb = (5.7) b=1,2 ±x siniji'+ zcosiji' + Ztb If the system is not perfectly aligned, Ztb expresses the displacement of the beam waist relative to the intersection point of the two beams, as illustrated in Fig. 5.2. At the incident point, a Gaussian beam polarized in the Y direction, and neglecting the Z component, has the following magnitude and phase U) _ Ebr - EOb "i;) r br =r + (i))2 YO,br J = e y EU)br exp(.J If/U)) br ' EU) _br r wb exp[ U) U) el - (i) • e/)2 ( XO,br COS /2 +ZO ,br Sin /2 C) 2 rm,br (5.9) ' Glare point . E Id •• --' \ y,br (r;,br) Rel:eivcro~ r•• (5.8) IIr(;} _ -----............ . - ...... - --. .. ..--"" ............ oint Laserbeam Beam waist Fig. 5.4. Vector notation with respect to the laser beam Phase front at ineiden. point 174 5 Signal Generation in Laser Doppler and Phase Doppler Systems t kb (i) _ lf'br - Wb - [+ " BI Sin /2 (i) -XO,br U) ( XO,br + (i) ZO,br COS BI /2 + Ztb U) • BI)2 (i) COS BI /2 + ZO,br sm 12 + ( YO,br + )2] (5.10) 2R(;) br ±X~~r sin o/z + Z~~r COSo/z + Ztb +uctm' +w 1 , r Ob' lf'Ob = lf'Y ,Ob Rb The beun radius (i) (i) • BI U) BI 1+ ( + _XO,br Sin 12 + ZO,br COS /2 + Ztb _ rm,br - rwb J2 (5.1l) 1Rb md the radius of the wavefront U) • BI U) BI )2 + 12Rb (+ -xO,br sin 12 + ZO,br COS 12 + Ztb ±X~,~r sin o/z + Z~,~r COS o/z + Ztb (5.12) refer to the local values at the incident point (see Fig. 5.4); however, in many cases the values for the center of the particle may be adequate. The above phase expression, Eq. (5.10), may be rearranged using the unit vector of the beam axes, Eq. (5.l), and the position vector of the incident point rci:L Eq. (5.4), to yield (;) _ lf'br - wbt - k b ,rop '- - kbz tb (;) (;) + lf'ob - k b 'rbr + lf'G,br /~ "-y---J "V Planewave Particle Gaussian beam (5.13) with U) If/G,br k b • rop = kb(±xOP sin o/z + zop coso/z) (5.14) U) sin BI + ZU) COS BI) k b ,rU) = kb (+x br br /2 br /2 (5.15) ( XU) COSBI =+= zU) O,br /2 O,br = -k sin BI)2 + (y U) /2 O,br )2 2R(;) b br + U) • BI (;) BI ) _X O br Sin 12 + Zo br COS 12 + Ztb [ +arctan' (5.16) , 1Rb The first four terms in Eq. (5.13) apply to the case of plme waves propagating in the direction of the beam axis, describing their time and spatial dependence and the phase at the origin, lf' Ob' Equation (5.14) corresponds to the phase change of a plme wave between the origin of the main co ordinate system and the center of the particle, and kbz tb corresponds to the phase change between the waist of the beun and the main coordinate system. 5.1 The Signal Fram an Arbitrarily Positioned Detector 175 The fifth term in Eq. (5.13) corresponds to the phase change of a plane wave between the particle center and the incident point (Eq. (5.15». For very sm all particles, d p « Ab, this term vanishes. The last term, expressed as \f/~~b" arises due to wavefront curvature beyond the incident beam waist (Eq. (5.16». For particles passing remote from the beam waist, this difference arises from the difference of the wave vector at the incident points and on the beam axis. The different direction of k~) compared with k b influences only the scattering function (different scattering angle), the phase change due to the wavefront curvature is already included in the term \f/~~b" In a similar manner, the incident amplitude from Eq. (5.9) (5.17) can be split into a spatially constant part from a plane wave EOb ' and a factor that is due to the Gaussian proflle of the beam p(i) G,br = r wb r(i) m,br ex [ p ( (i) (i) • 8/)2 (i) )2] + ZO,br Sin /2 + (YO,br 81 - XO,br COS 12 ( (i) rm,br )2 (5.18) The particle effectively images the incident field at the ineident point onto the photodetector. If the scattering plane is not parallel to the polarization of the ineident field (in this case the Y direction), two polarization components on the detector surface are obtained, corresponding to the two scattering functions ~l and ~2 in Eq. (4.1). If the receiver only sees one polarization, for instance by using a polarization filter in front of it, one polarization component vanishes and the vector equation for the scattered field reduces to a scalar equation for one polarization component. On receiver r, the scattered field strength from beam b is then E (r) -_br E(r) br (r») exp(.JIf/br - E(i) Sbr [.( _br ~exp J If/S,br b rp,br kbrp,br (g) nl)] + 12 (5.19) The scattered spherical wave propagates with an amplitude decrease of (kbr;$~tl and a phase of kbr;$~ + ~. Additionally, an amplitude Sbr and a phase change If/ S,br due to the scattering is included. The scattering factor Sbr and the phase If/S,br can be computed using extended geometrical optics (EGO) or Fourier Lorenz-Mie theory (FLMT), and are related to the scattering functions ~l and ~2 of the particle (see chapter 4). The detector integrates the intensity of this field strength over the photosensitive surface (see Eqs. (4.6)-(4.9» and performs also a time averaging of frequeneies beyond its cut-offfrequency (see section 3.2.5 and Eqs. (3.183) (3.208». The intensity must include all scattered fields reaching the detector, in this case all N ili order contributions, as discussed in section 4.2.1. For the signal generation on the detector of a laser Doppler or phase Doppler system, the intensity resulting from the superposition of two scattered waves (one scattering order) from two different laser beams with different propagation 176 5 Signal Genera don in Laser Doppler and Phase Doppler Systems vectors (b = 1,2 Eq. (5.1» will be considered. Furthermore, only one polarization component is assumed in the following discussion. The intensity at the receiver is then (5.20) + (k I' rl(i)r - k (i») + (k 1rp,l (g)r 2' r 2r - k2 r p(g»)] ,2r and is illustrated in Fig. 5.5. In this case the illuminated volume has been scanned by a very small particle in the plane y = o. The scattered intensity of a very small particle located in the center of the beam waist of one beam is used as the reference intensity in Fig. 5.5 (5.21) The intensity consists of a direct part Ir,DC and a modulated part, with amplitude Ir,AC and phase lfJr' For a well-adjusted laser Doppler or phase Doppler system, some simplifying assumptions can be made. For the case of a symmetrical optical set-up, the two beam waist diameters are the same (5.22) By using a power meter, the light power of the two beams can be matched with high accuracy and together with assumption Eq. (5.22), the amplitude of the electric field in the beam waist is the same for each beam -2 -I o Fig. 5.5. Intensity in the region of the illuminated volume (rwb = 50 f.lID, Ab = 488nm, d p « Ab) 2 e = 4 deg, 5.1 The Signal From an Arbitrarily Positioned Detector BOI = BOl = Bo =mOb I 177 (5.23) The frequency Wb and the wavenumber kb of one beam may differ from the other. This is the case if frequency shifting or if different colors are being used, usually for the purpose of resolving the flow direction. In the time independent terms of Eq. (5.20), the wavenumbers can be assumed to be equal, also for frequency shifted beams. (5.24) The variation of the amplitude of the electric field in Eq. (5.19) by moving the particle throughout the illuminated volume can also be considered negligible. The scattering function is almost constant, which means that the scattered amplitude is not a function of particle position d lJI S,br = d lJI S,br = d lJI S,br = 0 dz op dx op dyo p (5.25) The change of the scattered wave amplitude (not phase) in Eq. (5.19) due to the propagation of the spherical wave between the glare point and the detector is basically independent of particle position and glare point position. This is because the distance between the particle and the receiver is much larger than the illuminated volume and the particle diameter, thus r;1,~ and rpr do not vary significantly with particle position. 1 1 d(kbTpr ) d(kbTpr ) dx op dyop kbT~1~ "" kbTpr ' (5.26) The resulting signal can be interpreted either in time domain or in space. The description in space is basically an analysis of the imaging of the laser beams and their illuminated volume by the particle, the time domain description is an imaging of the spatial description according to the particle trajectory and velocity. The discussion now turns to the signal generation from very small particles, d p «Ab' Very small particles follow the flow with little or no slip, therefore this discussion is particularly relevant for laser Doppler systems. 5.1.2 Signals from Very Small Particles The intensity given in Eq. (5.20) is the intensity seen by the detector for a specified position of the particle in the illuminated volume of the laser beams. It comprises several terms, most of them dependent on the particle diameter. For very small particles (d p «Ab), the following simplifications can be made: • For very small particles, the incident and the glare points coincide with the center of the particle. Equations (5.4) and (5.5) reduce to r(O O,br = r(g) O,br = r 0P , rb(ri ) = 0, r(g) br =0 (5.27) 178 5 Signal Generation in Laser Doppler and Phase Doppler Systems • The beam waist radius and the radius of the wavefront at the incident point are equal to the values in the partiele center, not dependent on the receiver position but different for the two beams. (5.28) • The influence of the partiele diameter on the phases of the incident fields at the incident points is negligible. In the vicinity of the partiele, the amplitude and phase of the laser beams is practically constant. Therefore, the plane wave assumption over the partiele surface can be made, but for different partiele positions the Gaussian characteristic of the beams from Eqs. (5.16) and (5.18) must be considered (5.29) • Furthermore, the distance between the glare points and the receiver, influencing the amplitude and phase of the scattered wave, is the same for both scattered waves r(g) = r(g) = rpr = const p, 1r p,2r (5.30) • The scattering function can be assumed to be independent of scattering angle for very small partieles (see Fig. 4.13a,b) 51r = 52r = 5 = const , lfI S,lr = lfI S,2r = lfI s = const (5.31) 5.1.2.1 Spatial Description ofthe Signal (Very Sm all Partie/es). The intensity Ir and the phase ({Jr given in Eq. (5.20) inelude several terms, all of which depend on the partiele position in the illuminated volume of the laser beams. If these terms and their sum are now plotted as a function of the partiele position (Eq. (5.3», the measurement volume structure will be obtained. Spatial Description ofthe DC Amplitude (Very Small Partic1es). The above conditions lead to the following expression for the spatial distribution of the DC part (5.32) 5.1 The Signal Prom an Arbitrarily Positioned Detector 179 For the condition that the 10cal beam diameter does not vary with particle position, the DC part consists of two Gaussian peaks, with separate maxima when the particle is centered on the axis of either laser beam (Fig. 5.6) XDcmax bJ r DC max.b = [ Y DCmax,b = Z [±tan 0 %] , (b = 1,2) (5.33) 1 Z DCmax.b The superposition of these Gaussian pulses in Eq. (5.32) yields only one local maximum in the region of the illuminated volume and two separated maxima on the beam axis for larger Z coordinates, as illustrated in Fig. 5.7. For very small particles, the positions of the two maxima are given by the transcendental equation ('rn) = 'm2 = 'm) - zop tan . e/ tanh(~ xDCmax - /2 2 XDCmaxZOp 'm . e/ Sin /2 e/) COS /2 , Yo p - 0, I - Zap I~. ~ (5.34) 2sm% · 1 / " ,IX - / .. 0,6 0.4 0,2 0 Izr.ex: / .. 0,6 0.4 0,2 -2 -I o z lmml 2 0 Fig. 5.6. Separated DC parts of the two laser beams and the curves oflocal maximum intensity. The gray levels are relative to the waist intensity of one beam (rw == 50 11m, B=1deg, Ab=488nm) 180 5 Signal Generation in Laser Doppler and Phase Doppler Systems Fig. 5.7. Superimposed DC parts ofboth laser beams and the curve oflocal maximum intensity(rw = 50 Ilm, e= 4deg, Ab = 488nm) For the outer regions, the hyperbolic tangent takes the value of unity and the positions of the maximum intensity are on the axes of the two laser beams, dependent on zop' The absolute maximum is obtained for r op = 0 at the center of the intersection region of the beams. The z coordinate for which the maximum in the intersection region at x op = 0 splits into two separated maxima is given by rm Izsplit I="21 sin% (5.35) As defined in the next seetion, this position corresponds to half of the measurement volume length for the modulated part of the signal. The determination of the z location of the particle trajectory in the region of illuminated volume is therefore only possible by separating the two scattered waves from the two beams. Note that a small dislocation Ztb of one beam waist from the other along the beam axes results in no displacement of the DC part intensity. Spatial Description of the AC Amplitude (Very Small Particles). In the laser Doppler and phase Doppler techniques, the AC, or modulated part of the signal is used for velo city and size measurement. Thus, the AC part effectively defines the illuminated, measurement and detection volumes of the system. The illuminated volume is defined without any particle in the interseetion area of the laser beams and corresponds to the e-J intensity decay of the interference structure. The spatial structure of the AC signal amplitude can be obtained from Eq. (5.20) (5.36) by assuming that the beam diameter is constant in the region of the illuminated volume rm ) = rm2 = rm = const . 5.1 Thc Signal From an Arbitrarily Positioned Detector 181 The spatial intensity maximum lies at rop = 0, i.e. in the center of the intersecting laser beams. For a very small partiele in the center of the illuminated volume, the maximum intensity at the detector becomes 1r,Acmax = C{E o_ S _)2 = 10Kpr (5.37) kbrpr with the assumption that the system is perfectly aligned (Ztb = 0). 10 is the maximum incident intensity at the center of the measurement volume; K pr contains all parameters dependent on the partiele and on the position of the detector. If the boundaries of the measurement volume are arbitrarily set by the condition (5.38) then the volume is defined by in ( Xop crwos'li)2 +(YrwOp)2 +(Zop srw 'li)2 --1 (Ztb =0, rm =const=rw ) (5.39) Equation (5.39) is independent of partiele properties or detector position. It describes an ellipsoid with the axes a0-~ - cos'li b = rw '0 , c=~ o sin 'li (5.40) Thus the amplitude of the AC part of the signal effectively defines the measurement volume. For small partieles, this is areal measurement volume because the volume coincides with the illuminated volume of the laser Doppler system and is independent of the receiver position. In contrast, for large partieles, the measurement volume is virtual and the position of the volume will depend on the receiver position and on the imaging properties of the partiele, i.e. its size and refractive index, as weIl as on which scattering order is being detected (see also section 5.2). The actual volume of the measurement volume is given by 8 3 V=~~ o 3 sine (5.41) In Fig. 5.8, the dimensions of the measurement volume are visualized for the configuration from Figs. 5.5 to 5.7. The dimensions of the measurement volume remain independent of the partiele diameter, even though large partieles scatter higher intensities, because its dimensions are all defined relative to the maximum scattered intensity. However, only signals exceeding some minimum detection intensity 1d will be registered at the photodetector, thus, the detection volume may not coincide precisely with the measurement volume. The minimum detectable intensity will set lower limits for the size of detectable particles and also establishes a relation 182 5 Signal Generation in Laser Doppler and Phase Doppler Systems 2 o -I z lmml 2 Fig. 5.8. Intensity of the modulated part relative to the intensity of one beam. The extent ofthe measurement volume (a o =50.04 11m , bo = 50 11m, Co = 1.43 mm, Va = 0.015 mm 3 ) is indica ted with a line (rw = 50 11m, e = 4 deg, Ab = 488 nm ) between the detection volume and the measurement volume. The minimum intensity is determined by the sensitivity of the photodetector, which in turn can be influenced by the intern al electronics. Furthermore, the detection volume will be dependent on the particle properties and on the position and size of the detector surface (Eq. (5.37». The dimensions of the detection volume can be written as 1 Ir,Acmax -1 n---, 2 .!.ln Ir,Acmax bd = b0 Id 2 , Id C d =c 0 1 Ir ACmax - 1n - ' - 2 I (5.42) d Sm aller particles lead to smaller detection volumes. Particles resulting in signal intensities less than some acceptance threshold I d are not registered by the system. It is the detection volume, which governs the signal generation in laser Doppler and phase Doppler systems. The particle concentration may be dependent on size, thus the lower limit imposed by the I d threshold effectively establishes which sub set of particles will contribute signals. The effective particle concentration can be regulated by adjusting the acceptance threshold of amplitude. For particle sizing, the detection volume must always be referred back to some reference volume, for instance the measurement volume. Otherwise computed fluxes and concentrations will depend on the scattering properties of the particles. If the detection threshold is chosen too high, some sm all particles may not enter the statistics at all. Figure 5.9 illustrates the detection volume dimensions relative to the measurement volume (Fd = ad I ao = bd Ibo = Cd I co) as a function of particle size for water droplets in air. For a detection threshold of Pm;n = Pd = ArId = lO-BW all particles above 0.4 /J.m are registered, while for Pm;n = Pd = 10-6 W, this lowerlimit rises to 2.2/J.m. The actual volume of the detection volume is given by V d 1 I r,ACmax )~ = V ( -ln 0 2 I d (5.43) 5.1 The Signal From an Arbitrarily Positioned Detector - -'". ,=0 183 2.5 11 ~' 2.0 ~ .~ c .5'"' "E" 1.5 Ö > c 1.0 ::l 0 'ü '-' <) ., "0 0.5 > '" <l 0:.: 0.0 8 10 Particle diameter h.lm] 6 2 0 Fig. 5.9. Dimensions of the detection volume relative to the measurement volume as a function of particle size (Ab = 488 nm, e = 4 deg, m = 1.333, ro, = 300 mm, 1fI, = 0 deg, q), = 90 deg, 7;. =25 mm) The deteetion volume may also exhibit additional limits for large particles. For high scattering intensities, the photodeteetor may go into saturation. If the ratio of the AC to DC part (modulation depth or visibility) is low, then the signal-to-noise ratio may also be greatly redueed. Thus, the deteetor may no longer reeognize large particles in the center of the deteetion volume. This situation arises in laser Doppler systems when large deteetion apertures are usedat angular positions ofhigh loeal scattering intensity, e.g. near the diffraetion peak in forward seatter or near the rainbow angle (see also seetion 5.1.4). Spatial Deseription of the Phase and Frequency of the AC Part (Very Small Partides). The diseussion eontinues with the phase of the AC part of the scattered light in Eq. (5.20). Assuming the eonditions given in Eqs. (5.22)-(5.31) are still valid, the phase of the eosine function in Eq. (5.20) ean be written as {(Jr =(0)2 -O)l)t+(k l -k 2 ) ·r Op + (k1z'l -k2 z,J (5.44) + ('1/ 02 -'l/oJ + ('l/G,2 -'1/G,l) and deseribes, together with the amplitude of the AC part, the interferenee of the two waves within the interseetion volume of the laser beams. The spatial frequeney of the interferenee fringes in the x direetion, used for the velocity measurement, ean be ealeulated from Eq. (5.44) using 0) r =2nf = dd{(Jr r x op and the interferenee fringe distanee is the inverse of this equation (5.45) 184 5 Signal Generation in Laser Doppler and Phase Doppler Systems ÖX = 21t = O)r 21t[ dcp, )-1 dx op (5.46) The individual terms ofEq. (5.44) can be interpreted as follows: For non-equal frequencies, 0)1 and 0)2' the phase changes linearly with time and a stationary particle will create a cosine change of intensity. In space, this can be interpreted as a moving fringe pattern passing over the particle. The frequency of the received oscillations from a stationary particle is identical to the frequency difference 0)2 - 0)1. This frequency shift can be used for directional sensitivity or for detection of very small velocities in laser Doppler systems. The different starting phases, If/ 01 and If/ 02' create a constant offset. If allother phases are zero (small particle in the center of the measurement volume, no shift frequency and plane waves), the detector will probably not detect the highest possible intensily because the interference field is not symmetrically arranged about the y-z plane. In laser Doppler systems, this difference is caused by the different optical path lengths of the laser beams from the beam splitter to the measurement volume. Near the beam waist, the incident waves are plane and the orientation of the waves al the incident points on the surface of the particle corresponds to that of the beam axes (Eq. (5.1). The fourth term in Eq. (5.44) contains the properties of plane incident waves (5.47) The sum and difference of the inverse wavelength appearing in Eq. (5.47), indicates that the symmetry about the y-z plane is broken when the two beams are of different wavelengths. The interference planes no longer lie parallel to the y-z plane, but are slightly rotated about the yaxis. However, these effects are usually so small that a single wavelength can be assumed; Al = A2 = Ab or kl = k 2 = k b • For instance, the rotation of the interference planes for a system with a crossing angle of 1 deg, a wavelength of 488 nm and a shift frequency on one beam of 40 MHz, amounts to one angular second. The second term in Eq. (5.47) can be set to zero. The remaining term (5.48) can be related to the interference fringe spacing in the real intersection volume. The interference fringe spacing becomes Llx=~ 2sin% (5.49) Equations (5.40) and (5.49) show that within this measurement volume, the number of interference fringes is 5.1 The Signal From an Arbitrarily Positioned Detector _ _ 2a o _ 4rw e/ 185 (5.50) N fr -No - - - - t a n /2 Llx Ab independent of the particle size. Small particles about the size of the wavelength sampie the interferenee pattern as they pass through the interseetion volume (eompare to Eq. (2.22» and generate a real image as illustrated in Fig. 5.10. The number of interferenee fringes in this ease is No"" 14. The last term in Eq. (5.44) deseribes the effeet of eurved wavefronts of a Gaussian beam on the phase of the AC part of the signal. The magnitude of this term ean be computed using Eq. (5.16), applied to eaeh laser beam at the ineident points on the particle surfaee. It is important to note that the radius of curvature is not neeessarily equal for eaeh beam. This term quantifies deviations from the ideal ease of plane wavefronts in the laser Doppler and phase Doppler teehniques. To evaluate loeal variations of the spatial frequeney or spaeing of the interferenee fringes, Eqs. (5.45) and (5.46) must be applied to Eq. (5.44). The spatial frequeney in the x direetion is given by OJ x t =2n!x = ddtpr =kb[2sin%-eOS%( :lzl\ - : 2Z 2 x op Zl + Rl Z2 + R2 ) (5.51) -(Xl + X 2 + Yl + Y2 )-(Gl +G2 )] The fringe spacing ean be expressed as (Miles et al. 1996) ÖX=2n[ dtpr dx op )-1 ~ 2sin% (5.52) Fig. 5.10. Interference pattern in the measurement volume. The size of the measurement volume is indicated with a line (I,,~ = I"AC cos(tp), Llx = 7 ~m, No = 14.3, rw = 50 ~m, e = 4 deg, Ab = 488 nm ) 186 5 Signal Generation in Laser Doppler and Phase Doppler Systems The coordinates of each laser beam, as given by Eq. (5.7), including any waist translation along the beam axis, must be substituted into Eq. (5.52). In Eq. (5.51) the wavefront curvatures (5.53 ) and contributions due to the ~ phase shift in the far field Gb =_I_-_d_(arctan~J= 1 1Rb 2kb dx op 1Rb 2kb1~b+zi (5.54) have been abbreviated. However, to first order these terms can be neglected and according to Miles et al. (1996) these terms are limited by 1 max( Gb ) = max(Xb ) = max(Yb ) = 21 Rb (5.55) and must only be considered for small beam waists. Fig. 5.11a visualizes interference fringes in the x-z plane, centered on the intersection volume of two laser beams. The spatial frequency of the fringes in the x direction is shown graphically in Fig. 5.11b. This variation of spatial frequency can also be illustrated using Moire fringes, as demonstrated in section 5.5. The relative deviation of the spatial frequency, illustrated in Fig. 5.11 c, and the fringe separation from the plane wave case can be expressed as (5.56) bx Err "'[2tane/(~12 + /2 2 ZI Rl X 2Z 2 12 Z2 + R2 2 J-1-1]-1 (5.57) The effect becomes more significant for highly focused beams, in which case the beam exhibits strong divergence in the vicinity of the beam waist (measurement volume), hence, a strong wavefront curvature. The effect also arises if the laser beams do not intersect exactly at their beam waists; This effect is often the limiting accuracy factor for velocity and size measurements using the laser Doppler or phase Doppler techniques. The curvature of the wavefronts within the measurement volume directly influences the accuracy of the velo city measurement through the spatial frequency deviations. These deviations are shown as relative frequency error in Figs. 5.12 and 5.13, dependent on the beam waist radius, the intersection half-angle and dislocation of the beam waist from the center of the intersection volume. Computations have been performed for various detection volumes, ranging from 0.1 to 3 times the measurement volume dimensions (Eq. (5.42». For a given beam waist radius, the error decreases as the size of the detection volume decreases, as seen in Fig. 5.12. 5.1 The Signal Prom an Arbitrarily Positioned Detector 187 - 1.0 50 a cos (c)I , ) 0.5 o o -0.5 ·50 - 1.0 IXI O& 50 15 % c / ,,' E:1. >< - 9% o 6% 3% 0% z hun) · 500 500 Fig. 5.11a-c. Influence of wavefront curvature of Gaussian beams to the frequency of interference fringes. The size of the measurement volume is indicated with a line ( rw = 15 /JlIl, = 4 deg, Ab = 488 nm). a Interference field scattered bya very small particle in the x-z plane, b Spatial frequency in the x direction, c Relative frequency error related to plane wave case e A decrease of detection volume size can be achieved using slits or pinhole apertures in front of the detector (sections 7.3, 12.2.3 and 14.2.1). For small beam waists, the wavefront curvature increases, hence, also the frequency error (Fig. 5.12a). By increasing the intersection half-angle, this error can be partially compensated because the measurement volume is decreased in length (z direction). In Fig. 5.13a,c, both beam waists have been dislocated from the intersection volume center in the same direction, Ztl = Z'2 resulting in longitudinal distortion of the fringes. For such a misalignment, as for a perfectly aligned system (Fig. 5.11), the frequency deviation depends primarily on the z coordinate. In Fig. 188 5 Signal Generation in Laser Doppler and Phase Doppler Systems b 10 -3 L---L...LJL.l..l.Lll1_-'-.L..LJ....U..llL~~--L.JU!U..U 100 10 1 10' 10' 0.1 Beam waist radius rw [fim] 1 Full intersection angle 10 e [deg] Fig. 5.12a,b. Relative frequency error of a very small particle in the detection volume of a perfectly aligned system. The dimensions of the detection volume are 0.1, 0.2, 0.5, 1, 2, 3 times the measurement volume dimensions The verticallines mark the value, which was held constant in the other diagram. a Dependence on be am waist radius (e = 4 deg, Ab = 488 nm, Zn = Z'2 = 0), b Dependence on fuU in tersection angle (rw = 50 J.tm, Ab = 488 nm, Zn = Zt2 = 0 ) 5.13b,d, the waists have been dislocated in opposite directions, Ztl = -Z'2' resulting in transverse distortion of the fringes. In such cases the frequency deviations are dependent primarily on the x coordinate (see section 5.5). The relative error increases with increasing bearn waist dislocation. If the detection volume size is assumed to be constant for all dislocations, the error decreases above the Rayleigh length, Fig. 5.13a,b. With increasing Zb' the radius of curvature of the wavefront increases and for smalilocal regions in the beam, the wavefronts can be assumed planar. Therefore, the fringe spacing converges for very large dislocations, Z'b' to the ideal plane wave case, given in Eq. (5.49). This interpretation is only valid for constant detection volume sizes, e.g. when using a pinhole in front of the detector. Without such spatial filters, the beam radius changes with the beam dislocation, Eq. (5.11), and the measurement and detection volume size increases. If this influence is also considered, then the relative frequency error increases monotonicallywith bearn waist dislocation (Fig. S.13c,d). For all dependencies shown in Figs. 5.12 and 5.13, the maximum intensity in the measurement volume has been assumed the same for all cases, which is not strictly correct when using a laser with constant power. A further analysis of the fringe distortion in the measurement volume in terms of parameters of the optical transmitting system is presented in section 7.2.4. 5.1 The Signal From an Arbitrarily Positioned Detector ~ 189 15 b a c...:; ~ ~ 0 30 c...:; t c...::; ,.., ,..,,..,0 20 >u Q) C Q) '" <l:: V' Q) Q) .,; 10 v'" ~ Rb 25 50 75 100 0 Bearn waist dislocation zn = z" [rnrn] Rb 25 50 75 100 Opposite waist dislocation zn = -zt2 [rnrn] Fig. 5.13a-d. Relative frequency error of a very small particle (d p -7 0) in the detection volume for longitudinal and transverse distortion. The dimensions of the detection volurne are Fd =0.2,0.5, 1,2,3 times the measurement volume dimensions (e=4deg, Ab = 488 nm, rw = 50 ~m). a Dependence on beam waist dislocation for constant detection volume size, b Dependence on opposite dislocation of beam waist for constant deteetion volume size, c Dependence on beam waist dislocation for beam diameter corrected detection volume size, d Dependence on opposite dislocation of beam waist for beam diameter corrected detection volume size 5.1.2.2 Temporal Description of Signals From Very Sm all Partie/es For the photodetector, only a description of the signal in time domain is relevant. The spatial and temporal descriptions of the signal arising from sm all partides are related through the trajectory and the velo city of the partide in the measurement or detection volume, as pictured in Fig. 5.14. To further analyze the signal generation in time domain, the following assumptions about the partide motion will be made: 190 5 Signal Generation in Laser Doppler and Phase Doppler Systems y Fig. 5.14. Signal generation in the detection volume • The particle velo city is given by v p. The laser Doppler instrument measures only the velo city component perpendicular to the interference fringes (Fig. 5.14) and the system is usually orientated such that this corresponds to the main flow direction. This will be assumed to be aligned with the x axis. • For flows with low levels of turbulence, a cartesian representation of the velocityvector is useful m=~ z (5.58) whereas for highly turbulent flows, a vector in spherical coordinates will be used (Fig. 5.15) y Fig. 5.15. Velocityvector in spherical coordinates 5.1 The Signal From an Arbitrarily Positioned Detector vx ] v p = [ vy Vz =vp [s~ntJvc~stpv] sm tJ sm tp v v 191 (5.59) costJ v For tp v = 0 and tJ v = 1t /2, the velo city vector has only an x component. • The particle moves through the volume with a constant velo city, meaning no changes of direction or variations in speed. The influence of velo city fluctuations during the particle transit through the measurement volume and their influence on the signal parameter estimation will be addressed in sections 6.5.2 and 8.2.3. • The reference position used to characterize the particle trajectory is the position within the measurement volume (for small particles) or detection volurne (for larger particles) at which the trajectory of the particle center crosses the plane x = 0 (5.60) • The reference time is set to zero (t = 0) at this position. The cartesian description of the particle position rap leads to (Fig. 5.3) (5.61) and the description in spherical coordinates yields (5.62) Both coordinate systems will be employed, where first the DC part and then the AC part of the detector signal will be analyzed. Amplitude of DC Part in Time Domain (Very Small Particles). For very small particles, the spatial intensity distribution within the measurement volume will be directly imaged by the motion of the particles onto thephotodetector in time domain. The DC part consists of two pulses, corresponding to the crossing of the two laser beams. For a further analysis, it is convenient to assurne that the beams have identical parameters (Eqs. (5.22) to (5.24» and that the detector is far removed from the scattering center (Eqs. (5.25), (5.26». Furthermore, the assumptions given in Eqs. (5.27) to (5.31) are used. For the DC part the signal is described by 192 5 Signal Generation in Laser Doppler and Phase Doppler Systems with xo p = vxt. It is also interesting to note the time between the two DC pulses. If the y and z components of the velo city are non-zero, then both the time at which the signal maximum is observed and the time between the two pulse-like signals will be dependent on the particle trajectory. In general, the position of maximum amplitude will occur at ±z tano/z(1+m tano/z)pO XpDCmaxb = VJDCmaxb = 2 z 2 2 (m myypo cos2o/z )2 (b = 1,2) (5.64) (1+ mz tan o/z) + --y- coso/z assuming a constant beam radius for the complete trajectory (rm *- !(xoP,YoP,zop»). For particle trajectories parallel to the main flow direction x (my = m z = 0), Eq. (5.64) reduces to XpDCmaxb = V x tDCmaxb = ±zpo tan O/Z, (b = 1,2) (5.65) in agreement with Eq. (5.33). Interestingly, the time between the DC pulses L1x DC12 = xpDCmaxl - xpDCmax2 = vxLltDC12 = 2z pO tan o/z (5.66) yields the Z position of the particle trajectory interseetion with the plane x =0, if the velo city is known. For particle motion parallel to the laser beam plane ( vy = 0), the duration between pulse signals becomes 5.1 The Signal Prom an Arbitrarily Positioned Detector A.x DC12 =vxL1tDC12 =2z po tan% 1 I-m z tan% 193 (5.67) For particle motion without a z velocity component, the pulse separation is given by A.x DC12 =VxL1tDc12 =2z po tan% 1 2 (5.68) 1+ ( my ) cos~ The last two expressions are independent of the intersection co ordinate y pO' The additional term in Eqs. (5.67) and (5.68), compared with Eq. (5.66), represents a first-order error approximation for determining the particle trajectory. For small deviations from the main flow direction, it can be neglected. The term in Eq. (5.68) can be measured when using a two-velocity component laser Doppler system aligned to measure directly the x and y velo city components. In this manner, the estimation of the particle trajectory can be improved. The error in Eq. (5.67) from the V z component can be reduced by decreasing the intersection angle. This error vanishes for the limiting case of e = 0 deg. In comparison, the error in Eq. (5.68) reduces, but is still present for small intersection half angles, as in a laser transit velocimeter. Within the beam intersection area, the two DC pulses begin to overlap towards the center of the intersection volume until only one maximum is obtained, as illustrated in Fig. 5.7. An analytic expression can no longer be given for the maxima in this region. The DC part of the signal can be utilized for a z position estimate of the particle if the two pulses are separated by using different optical properties, such as different polarization for each of the two laser beams. Amplitude of AC Part in Time Domain (Very Small Particles). The amplitude of the modulated part of the signal can be obtained from Eq. (5.36), substituting the particle trajectory given by Eqs. (5.61) or (5.62) Ir,Ac = t:C[Eo_S_~)2 exp[--;-([xop cos%f + [myxOp + Ypof kbr r r pr m m (5.69) The maximum modulation amplitude is obtained for the position X pACmax = • 2 BI myypo+mzzposm ;/2 2+ 2 · 2 BI + 2 BI m m sm /2 cos /2 y (5.70) z Thus, the position at which this amplitude is obtained depends on the particle trajectory. For trajectories passing through the origin or parallel to the main flow direction (x), this position will always lie on the x = 0 plane. Any displace- 194 5 Signal Generation in Laser Doppler and Phase Doppler Systems ment from this plane will depend linearly on the intersection point at which the trajectories pierce the measurement volume. This displacement will be smaller for trajectories eloser to the intersection point of the two beams. These relations are better understood for turbulent flows using a trajectory description in spherical coordinates, Eq. (5.62). The trajectory parameters iJ v and qJv are related to the velo city components through (Fig. 5.15) iJ v = arctan ~V2 +v 2 x y (5.7l) Vz The velo city vector is perpendicular to the projected reference plane, which itself is rotated in the spherical coordinate system according to the parliele trajectory. This projected reference plane lies in the plane'; = 0 of a rotated cartesian coordinate system (,;, 1],(). The spherical co ordinate system for defining the partiele trajectory and the reference co ordinate system have the same origin. The partiele trajectory pierces the projected reference plane at the point P(1]po,(po) and has at this position the shortest distance to the intersection point ofthe laser beams (rpo·v p = 0). In the (x,y,z) system, this corresponds to (5.72) = 0 and iJ v = rr; /2 (v p = v x), both coordinate systems coincide and 1] pO ----t ypo, (po ----t z pO. By using the coordinates of the intersection point For qJv (Eq. (5.72» and the spherical representation of the trajectory direction (Eq. (5.59», the cartesian coordinates of the trajectory from Eq. (5.3) are given by (5.73 ) with r=vpt. The components of Eq. (5.73) will be substituted into Eq. (5.39) to yield the intersection points of the trajectory with the measurement volume bounds as a function of the spherical coordinates (r, iJ v' rp v) and the parameters 1] pO and (po. (5.74) (5.75) 5.1 The Signal From an Arbitrarily Positioned Detector 195 Cl = -;..( -21] pO sin tp vcostp vsin 1'J v - 2S pO cos 2 tp v sin 1'J v cos 1'J v) ao + :2 (21] pO sintp v costpv sin 1'Jv - 2Spo sin 2tpv sin 1'J v cos 1'J y) (5.76) o + -;"(Spo sin 1'J vcos 1'J v) Co (5.77) 1 72 • +2~ pO Sill 2.0 U y -1 Co The path length of the trajectory in the measurement volume is given by Llr = 1i - r2 = _1_~ C; C2 4C 2 CO (5.78) For trajectories parallel to the main tlow direction (x direction, 1'J v = 1t /2, tp v = 0 ) the path length in the measurement volume reduces to 72 Llr =2a .L 0 2 1-~- 1]po b2 (5.79) 2 o Co The number of periods in the signal results by projecting the trajectory onto the x-y plane and dividing by the interference spacing N =2costpvsin1'Jv v LIx..jC; ~ C; 4C 2 ~ C; -C =N 0 OV 4C 2 -C (5.80) 0 N Ov is the maximal number of periods for all trajectories in the respective direction. The square root term gives the dependence on the location of the intersection point. For Llr = 0 in Eq. (5.78) an equation for the projected reference area will be obtained (5.81) with (5.82) 196 5 Signal Generation in Laser Doppler and Phase Doppler Systems (5.83) (5.84) Equation (5.81) describes an ellipse in the 1]- ( plane. Because the semi-axes of this ellipse are not parallel to the 1] and ( axes ( Cry( -:f- 0), a rotated co ordinate system (~, 1]',(') parallel to the semi-axes is defined. This coordinate system arises by rotating the (~, 1], 0 system around the ~ axis by the angle ß 2 tan ß = C -C ( Cry( ry + [ C -C ( ry ) +1 (5.85) Cry( and the projected reference area in (1]',(') becomes elliptical with the semi-axes bovand cOv (5.86) (5.87) Thus, the projected reference area A ov can be obtained as a function of the partide trajectory direction (Buchhave 1975, Fuchs et al. 1983) (5.88) This area is the projected area of the measurement volume seen from a flow in the direction given by f)v and ({Jv' ao' bo' Co are the dimensions of the measurement volume (Eq. (5.40» and A o is the reference area for a flow in the main flow direction v p = e x v x and is required in the data processing step for estimating fluxes. The volume of the measurement volume is independent of the partide trajectory. By replacing the measurement volume dimensions a o' bo and Co with the detection volume dimensions ad , bd and Cd' the same relations are valid for the detection volume Vd , the detection area A d = rrbdcd and the projected detection area Adv = rrbdvcdv = AdFAv ' because only the size of the detection volume scales with partide diameter. The dependence of the projected reference and detection area dimensions and the path length on the angles f)v and ({Jv is shown in Fig. 5.16a,b. The influence of ({J v on the projected reference area is by no means negligible. The dependence of the number of periods in the signal on f) v is shown in Fig. 5.17a. Flows with a large V z component yield both a small reference area and 5.1 The Signal From an Arbitrarily Positioned Detector a ::!: 197 b 10 ~~~~~~~~~~~~~~ e o Llr 1.0 J, = 90 deg 1-------=---------.::~--__1 .;;; e " E '6 --- ----------- - -~~~-~~~~~--- - ---­ --- CJ, COP " .~ =45 deg co '" ~ 5 ---0-- CJ, =30deg 0. 5 <1>---0---0-0---0-...:...0--<""::'-0--0--=4> Ao• =F Ao A, CJ, = ISdcg --- ..---... --+---..... _- ..---.. ---.. ---.. - - ----c>--.. -O - CJ, = Odeg .. -O---- <>- .... O()..... - '<) .... -.c> ..... - -Q - - 0.0 L....I......L.....L--'-.L.......L.....L.....L....J1 L.......I.....J......I....J....JL......L...J......I....J o 45 90 0 45 Trajeclory angle tJ.(deg] 90 Trajcctoryangle 'P, (deg ] Fig. 5.16. Relative dimensions of the path length and the projected reference area (19=13deg). aDependence on 13 v (IPv=Odeg), bDependence of FAv on IPv with parameter 13 v a b ~90~~~~~~~~~~~-r~ " ~ s. -;..,,0 .2 0 <::-"- .4 "2 ·0 0- .6 e .g 0 v "~ .8 " "2 No, ' No - 0.9 1.0 JO 60 90 TrajeclOry angle 11, (degl -I o ln lerseclion poinl I Spo' (0 I-I Fig. 5.17. Relative number of periods in the signal. a Maximum number of periods as a function of 13 v and IPv «(po = lJpo = 0), b Dependence on the interseetion coordinates (po and lJpo (13 v =90deg, IPv=Odeg) a low number of signal periods. For rp v =0 and iJ v = 1t /2, the velo city vector has only the component v x • In this case Eq. (5.80) reduces to (5.89) 198 5 Signal Generation in Laser Doppler and Phase Doppler Systems Thus, the number of periods in the signal is determined by the coordinates at which the particle pierces the reference area ypo, zpo or 1J po, (po' Figure 5.17b shows this dependency for various trajectories. It should be emphasized that the above analysis refers always to the trajectory-dependent measurement volume. Due to the detector amplitude threshold, there will exist a relation between the measurement volume and the detection volume as given by Eq. (5.43). Thus, the number of signal periods will also depend on the detection threshold, detector position and the light scattering properties ofthe particle. These dependencies are examined further in section 5.1.6. Phase of the AC Part in Time Domain (Very Small Particles). The signal phase in spatial domain is given by Eq. (5.44). By replacing the coordinates of the partide by the time-dependent position of the partide, Eq. (5.61) or (5.62), and with the approximation, Eq. (5.48), the phase can be expressed as 2sin~ (jJr = ,dwt+ 21t-,i-vxt+(k 1z'l -k 2 z,J+( Vl"02 - Vl"0l) +( Vl"G,2 - Vl"G,I) (5.90) b The frequency ofthe signal can be obtained by the change of phase in time 1 1 d{jJr Ir = 21t W r= 21t dt = (5.91) 2 sin ~ d ( 21t +~vx + dt Vl"G,2 ßW - Vl"G,1 ) The term ,dwt = (w 2 -wl)t indudes the effect of a frequency shift, arising from the use of acousto-optic modulators in one or both of the laser beams or because two lasers with different wavelengths are employed to generate the two beams. The second term in Eq. (5.91) gives the influence of the interference field on the AC part of the signal; the Doppler frequency 2sin~ 2sin ~ ID = - , i - vx =-,i-V-L b (5.92) b which is independent of the partide trajectory and linearly dependent on the velocity component perpendicular to the interference fringes. This equation embodies the basic measurement principle of the laser Doppler technique. The partide trajectory is only reflected in the term (VI" G2 - VI" GI)' which expresses a spatial dependence of the signal phase (see Fig. 5.1I) anel results in an uncertainty of the frequency estimate. For ,dW = 0, the detector sees a stationary interference pattern set in the intersection of the two laser beams. The signal with ,dW"# 0 can be interpreted as the signal obtained from a moving set of interference fringes. In actual measurement systems, the shift frequency is chosen to satisfy I,dWI> WD = 21tID; ,dW> 0 leads to fringe movement in the negative x direction and ,dw< 0 leads to movement in the positive x direction. The measured frequency then corresponds to the relative velo city of the partide with respect to the interference fringes. Thus, 5.1 The Signal Fram an Arbitrarily Positioned Detector 199 the particle velocity is determined with reference to the velocity L1v =!shAb I (2sin o/z). Stationary particles result in a signal with frequency !sh' These relations are illustrated in Fig. 5.18. For v x > 0, the measured frequency is greater than the shift frequency and for Vx > 0, the measured frequency lies below !sh' The maximum measurable velocity is therefore determined by the maximum measurable frequency, while allowing for any shift frequency. The minimum frequency which can be processed determines the minimum measurable velocity or, if frequency shift is being used, the maximum measurable velo city in the direction of fringe movement. A stationary particle in the detection volume generates a signal with exactly the shift frequency, but no DC amplitude modulation. Some processing electronics may not validate such signals that are not burst-like in character. a f ~~-r-~------------~~ 0.0 c ~ b J J.... +---1---;>(' Y, Fig. 5.18. Measured frequencies for particles crassing the measurement volume in positive and negative x directions. a Without frequency shift, b With frequency shift 5.1.3 Signals from Large Particles In section 5.1.2, the signals from very small particles d p «Ab were examined as a special case of Eq. (5.20) for the laser Doppler measurement technique. For determining particle size with the phase Doppler technique, it is necessary to also consider the phase and amplitude changes of the incident light over the particle surface. The incident and glare point positions are no longer assumed to be coincident with the center of the particle. Larger particles can be grouped into two different classes: 200 5 Signal Generation in Laser Doppler and Phase Doppler Systems • For medium-sized partides, Ab « d p «rwb, the phase change of the incident wave over the partide surface is not negligible, but the intensity distribution can be assumed constant. In this case the plane wave assumptions can be used for calculating the scattered light (see section 4.1). • For partides large compared to the spatial structure of the incident wave, rwb ::; d p , the intensity change over the partide surface generates a more complex scattering behavior (see section 4.2). The following discussion is related to the most general case of large partides. The plane wave case of medium partides can be easily derived from the resulting equations. The imaging of the incident beams through glare points onto the detector surface, and the resulting properties of the signal are central issues to be introduced and discussed in this section. 5.7.3.7 Spatial Description ofSignals from Large Partie/es The illuminated volume of the intersecting laser beams is the same for small and large partides. The dimensions and location of the illuminated volume, which coincide with the measurement volume for small partides, are given by Eqs. (5.40) and (5.39). The description ofthe measurement volume for large partides in space is basically an analysis of the imaging of the illuminated volume of the laser beams by the particle. This imaging process results in a virtual measurement volume which will be displaced from the illuminated volume. The displacement depends on the partide size, partide refractive index, the scattering order and the detector position. The basic difference in treating very small partides and large partides is that for large partides, the distance between the incident and glare points to the center of the partide can no longer be neglected. The position of the incident points and the glare points on the partide surface is now related to the partide position through Eqs. (5.4) and (5.5) (i) _ ra,lr - r(g) O,lr rap (i) + r lr (i) , = rOp + r(g) Ir' r a,2r = rap r(g) 0,2r (i) + r 2r = rOp + r(g) 2r (5.93) (5.94) The analysis continues with a spatial discussion of the DC and AC parts of the measurement volume for large partides. Again the simplifications given by Eqs. (5.22) to (5.24) can be used. Furthermore, using the condition r~~:» 2rp , the change ofthe amplitude (not the phase in Eq. (5.19» on r~~~ is the same for each scattered wave (Fig. 5.3) and does not vary significantly with partide position (Eq. (5.26». The assumption given by Eq. (5(5.25) that the scattering amplitude is not a function of the partide position, can be used in most cases, but the scattering function is different for each of the two laser beams because of its complex behavior and large angular variations for large partides, as seen in section 4.2.3. 5.1 The Signal From Arbitrarily Positioned Detector 201 Spatial Description of the Amplitude of the DC Part (Large Particles). The DC part is simply the addition of the two individual squared field strengths of the laser beams, each exhibiting a Gaussian distribution in all planes parallel to the X b - Yb plane of the beams. For large particles this condition leads to the following expression for the DC part. I r,De ( )sm<% .)2 = Ce Bo rw 2[[ ~ )2 ex ( -2 (() X;,rl cos<% - Z;,d [ ) 2 k brpr (,j rm,lr P ( )2 rm,!r + (())2 Y;,rl U s)2 ex [ -2 (X U0,r2) COS<%2+ ZU)0,r2 sin <%)2 + [~ 2 +(y0,r2) )2 r(n rn,Ir p ( J (i) (i) rm ,2r )2 J] (5.95) It is important to note that the local radii r~1r (b = 1, 2; r = 1, 2) not only depend on the partide position, as for very small partides, but also on partide properties and receiver location and can be different for each beam. For calculation of the intensity maxima, the influence of the local radii variance can be neglected and the position of the two individual maximum intensiti es can be obtained through _ rVCmax,br - Xpmax,br [ Ypmax,br Zpmax,br + tell an (+ tel] an [ +Z J _ -Xbr (i) -=-Zb!i) U) - Ybr 0 0 1 /2 - /2 , b=1,2 (5.96) For very small partides, the distance between the incident points and the center of the partide is negligible; the second term remains and this simply describes the axes ofthe laser beams (Eq. (5.33». In Fig, 5.19 the line for the position of the maximum DC amplitude is shown for a very small partide and for a large partide (d p = 70 ~m). The small partide Fig. 5.19. Generation of a virtual image of one incident beam for a large particle and position of the DC amplitude maximum (bold line) (rw =50flm, 8= 40deg, Ab = 488nm, 1fI, = -30 deg, d p = 70 flm, pure reflection, xl:) = -31.7 flill, z::) = -14.8 flm ) 202 5 Signal Generation in Laser Doppler and Phase Doppler Sytems simply traces out the position of the incident beam (Fig. 5.6), whereas the large particle sees a line of maximum amplitude displaced by an amount just equal to the distance between the incident point and the center of the particle xl~). For large particles, the incident point sampies the laser beam and generates a virtual image of the beam. Note that also the waist positions of the beams are imaged differently. If now the DC amplitude contributions from each beam are combined according to Eq. (5.95), the position of the DC maximum results in a transcendental equation of the form J c2 --c Cl +C2 exp( _2_ 2 1- =0 rmb (5.97) with cose/=tz(i) sin e/ Cb =X(i) O,br /2 O,br /2 , 2 Cb=C b (i) )2 + ( YO,br , b = 1,2 (5.98) where the coordinates of the incident points are again related to the particle position through Eq. (5.4). Two amplitude maxima are obtained far from the intersection of the laser beams. Only one maximum is found near the intersection area. The lines for the position of the overall maximum amplitude are drawn in Fig. 5.20 for the case of a very small particle and a large particle. Additionally, the imaged waist positions are indicated in Fig. 5.20. Even though the system is perfectly aligned, the imaged waist positions are not coincident. This means that a perfectly aligned system can be effectively misaligned by large particles (d p > lOd w ) and velocity measurement errors like in Fig. 5.13 can occur. -2 o z lmml Fig. 5.20. Position of the total DC part for a large partiele (bold line) and a small partiele (thin line). The images of the beam waist centers are indicated with crosses. The position of the overall maximum intensity is indicated with a drele. (rw = 50 11m, e = 4 deg, Ab = 488 nm, 1fI, = 170 deg, 9, = 0 deg, d p = 500 11m, pure reflection) 5.1 The Signal From Arbitrarily Positioned Detector 203 Spatial Description of the Amplitude of the AC Part (Large Particles). The amplitude of the AC part is obtained through multiplication of the scattered field strengths from the two laser beams and can be expressed by ( XU) Q,lr cosel _z(n /2 0,1r ( ( sin el)2 + (yU) )2 /2 O,lr (i) rm,lr )2 U) Sin • el)2 X O,2, COS el /2 + ZO,2, /2 + (U) YO,2r )2] )2 r (5.99) (i) ( (i) m ,2r The posItIOn of the single spatial maximum, obtained by neglecting the change ofbeam radius with particle position and for the same beam dimensions, is given by U) U) U) _X-,-I!.."_+_x-,2"-.r 2 U) - Z 2r 2 el tan i2 YIrU)+y U2r) XAcmaxJ r ACmax = [ YACmax = Z ACmax + Z I, zU) + Z(n Ir 2r 2 (5.100) 2 + X(n - X(n Ir 2, 2 cot,:% 2 For very small particles, all terms tend to zero and the maximum amplitude is found at the center of the beam intersection volume. For larger particles, the position of maximum amplitude moves as illustrated in Fig. 5.21; however, the movement is linear with the distance of the incident points from the particle center. Because the light path inside the particle does not change significantly with particle motion in the illuminated volume for a given receiver position, the position of the maximum amplitude is linearly dependent on particle diameter 0.5 o Fig. 5.21. Scattered intensity of lhe modulated (AC) part for a large particle (d p = 500 jlm). The measurement volume is indicated with a bald line for the large particle and for comparison with a narrowline for a small particle (rw =50jlm, e=4deg, Ab = 488 nm, lfI, = 170 deg, 11,. = 0 dcg, pure reflection) 204 5 Signal Generation in Laser Doppler and Phase Doppler Sytems in all three directions. Especially this feature can be exploited for measurement purposes, using the time-shift technique (section 9.2). The terms (5.101) in Eq. (5.100) describe the center point on the line connecting the incident points and the values X = r U) U) x lr -x 2r 2 = Z r U) U) zlr -z2r (5.102) 2 are the half-distances or half-separation in each of the x and z directions between the incident points. For small intersection angles the separation of the two incident points is in most cases small compared with the distance from the center of the particle. Furthermore, the position of the overall maximum amplitude for the AC part corresponds to that of the DC part, as for small particles. The spatial extent of the measurement volume for a particle seen by the detector can be defined using an intensity threshold given by Eq. (5.38) for the AC part of the signal. From Eqs. (5.99) and (5.38), the measurement volume can be defined as This equation describes an ellipsoid displaced by the amount rAC,max in space. The major and minor axes are given in Eq. (5.40). The total measurement volurne is given by Eq. (5.41). If the beam waists are near the origin of the co ordinate system, as is normally the case for laser Doppler and phase Doppler systems, the local diameter of the laser beams is nearly the beam waist diameter and rm can be replaced in Eq. (5.103) by rw • The displacement of the measurement volume is, in general, a complex function of receiver position (scattering angle), refractive index and order of scattered light, and depends nearly linearlyon the particle diameter. Figure 5.22 shows pictorially the measurement volume displacement for a given detector position and for two light scattering orders, reflection and first-order refraction. This displacement is for a specific particle diameter. Because the displacement depends on scattering order, each scattering order creates its own measurement volume. Thus, the amplitude of the AC part of the scattered light effectively defines a virtual measurement volume for each scattering order. For very small particles it will be areal measurement volume because the volurne coincides with the non-displaced illuminated volume (Eq. (5.39» of the laser beams (rAC,max = 0) and is independent of the receiver position, scattering or- 5.1 The Signal Fram Arbitrarily Positioned Detector 205 y Virtual mcasurcmcnt volume (e.g. first -order rcfract ion) ILluminated volum<' from Virtual mcasuremcnt volume (e.g. rell <:Iion) Fig. 5.22. Position of the displaced virtual measurement volumes der, refraetive index and particle diameter. It ean be understood as an identieal overlapping of the measurement volumes from all seattering orders due to a vanishing measurement volume displaeement. Aecording to the definition ·of the measurement volume size (Eq. (5.38», it is independent ofwhether the volume is real or virtual and is the same as the illuminated volume of the system. The size is also independent of the particle diameter, even though large particles seatter high er intensities, beeause the dimension is defined relative to the maximum seattered intensity. As with small particles, a deteetion volume ean be defined as the volume within whieh the signal from a particle exeeeds a given threshold and, whieh ean be related to an arbitrarily defined volume given by Eqs. (5.42) and (5.43). This also provides the relation between deteetion volumes for different particles. Figure 5.23 shows the deteetion volume displaeement for different deteetor positions and for two light seattering orders, refleetion and first-order refraetion. Although the displaeed measurement volumes for eaeh seattering order are always the same size, their deteetion volumes ean vary signifieantly, as indieated in the example shown in Fig. 5.23 Spatial Deseription of the Phase of the AC Part (Large Particles). The phase of the AC part of the signal from large particles will now be eonsidered. The phase of the eosine funetion ean be written from Eq. (5.20) as and deseribes, together with the amplitude of the AC part, the interferenee of the two waves on the deteetor surfaee. The first to fourth terms in Eq. (5.104) are the same as for very small particles (Eq. (5.44». The other terms have the following meanings: 206 5 Signal Generation in Laser Doppler and Phase Doppler Sytems Time Position Receive r ignal • x Virtual delection volumes :;::;.~j~jl~!:;li~~~ ßeam 2 e.g. fusl -order refrac lion e .g . reil eC'~\' I ______~~~~~~________~~__________ ßeam I -+__ ____ ~ 4-~ = Fig. 5.23. Position of the displaced detection volumes The flfth term in Eq. (5.104) represents the optical path-length difference of the two light beams through the particle from the incident points to the glare points. This phase will vanish for the reflection mode or for very small particles, because the incident and glare points coincide. This phase is strongly influenced by particle diameter and refractive index. The difference (5.105) in Eq. (5.104) includes a phase difference arising from the curvature ofthe particle. For small particles, this phase difference vanishes (rl(;) = ri~) = 0). For large particles, this phase difference is related to particle size and also represents the fundamental principle of the phase Doppler technique. The phase shift is proportional to the sum of the x distances of the incident points from the particle center. Note that for symmetrie incident points (receiver in the y-z plane), both values have the same amount but different signs and thus, the phase is zero. In the case of spherical particles, the geometrie relations of the light path through the particle do not change, only the scaling is linear with the particle size. Thus, the use of this effect in the phase Doppler technique for the sizing of spherical particles is easily applied. Because this phase is related to the incident points, it is different for each scattering order. The difference kI rp,l(g)r - k2 r p(g),2r -- kb ((g) rp,l r - (g) ) r p ,2r (5.106) in Eq. (5.104) gives the phase difference arising due to the path-Iength difference between the source points of the scattered wave on the particle surface (glare points) and the detector. This difference also vanishes for small particles (r~1~ = r~~)r)' For larger particles, the surface curvature leads to an effective path- 5.1 The Signal From Arbitrarily Positioned Detector 207 length difference, so that this term is also important in the phase Doppler technique. If the receiver is far away from the partide, the value can be calculated from the glare point positions and the directional unit vector from the partide center to the receiver (g) (g) )-k ((g) (g») -kb ((g) (g») k b ( rp,lr - rp ,2r f 2r - flr ·e pr b rp,lr - rp ,2r ·e pr - (5.107) The two scattered waves from each pair of glare points create an interference field, similar to the case of two point sources. This interference field is the origin of the laser Doppler signals for large partides. The fringe distance decreases linearly with the glare point separation distance and this distance is also linearly dependent on the partide diameter. A large partide produces narrow fringes in space, a small partide produces wide fringes. The phase Doppler technique measures the angular spatial fringe distance and the partide diameter can be easily calculated from this quantity. Note that the glare points move and the phases of the scattered waves change when the position of the receiver is changed. All phase terms in Eq. (5.104), except (5.108) are, to a first approximation, independent of the partide positlon f op ' Any x movement of the partide causes the interference fringes from large partides to move over the detector surface with the same dependence as for very small partides. Thus, the velo city in the x direction can be determined in the same way. Using two detectors or two measurements, it is possible to eliminate the partide position x op and to measure the partide-dependent phase of the signal, or the spatial fringe separation. This is the basic principle of the phase Doppler technique. Generally, this can be done by forming the difference of two measured phases from two detectors r = 1,2 LI<P12 = <PI - <P 2 = (~S,21 - ~S,l1) - (~S,22 - ~S,12) U») + (k l ' fl1U) -k 2 ' f21U») - (k(l 'if) 12 -k 2 ' f22 (5.109) U) + ( ~G,21 U») - (U) ~G,22 - ~G,l1 U») ~G,12 The determination of these position-independent phasesl for spherical partides, hence the calculation of the phase difference I diameter relation for phase Doppler systems, will be examined in section 5.3. I The symbol cIJ, will be used for the position and time-independent phase of the detected signal, whereas the symbol 'P,. includes temporal and/or spatial dependence of the phase. Strictly speaking, the influence of the wavefront is not included in cIJ, because it is position dependent but it arises as a measurable phase difference error. 208 5 Signal Generation in Laser Doppler and Phase Doppler Sytems The last term in Eq. (5.104), appearing also as the last line in Eq. (5.109), is produced by the phase deviation of the Gaussian bearn from the plane wave case. This difference carries the spatial frequency error, as discussed in section 5.l.2 for very small particles. Furthermore, an influence on the measured fringe spacing by the phase Doppler technique is included in this term. The exact determination of this phase error is dependent on the incident point, thus, on the receiver angle. As an example, the phase difference error _ (U) (U) ) If/G.21 -If/U») G.ll If/G.22 -If/U) G.12 (5.110) ,;jr[) err - for a 30 ~m particle in a laser beam of 30 ~m diameter is plotted in Fig. 5.24 as a function of the particle position in the illuminated volume. The optical transmitting configuration is the same as that used in Fig. 5.11 and the receivers are symmetrically positioned at If/ r = ±2 deg and f/Jr = 30 deg. The measurement volume for reflection has been considered in Fig. 5.24. The maximum phase difference error in the reflective measurement volume is in this case 9.2 deg (8.9%) but changes with the size of the detection volume and therefore with the detection threshold. The situation gets worse if the transmitting system is not weH aligned, as shown in Fig. 5.2. In Fig. 5.25, the maximum phase difference error is plotted as function ofbeam waist diameter and particle diameter for different detection volumes. In Fig. 5.25a the relative phase difference error or relative diameter measurement error is shown for a particle with the same diameter as the measurement volume or bearn waist. For the same ratio of particle diameter to beam waist the error decreases for larger diameters and as expected, the error increases for larger detection volume sizes. In Fig. 5.25b the relative error is shown for two different measurement volume diameters as a function of particle size. The absolute phase difference error increases linearly with particle diameter and, therefore, the relative error is independent of particle size for a given configuration and a constant detection volume. Again the error increases for smaller beam waist diameters and larger detection vol- - 15 dcg LJ(/Jm 10 dcg 5 deg o deg 00 Fig. 5.24. Phase difference error in the in tersection area of the laser beams as a function of particle position for a perfectly aligned system. The displaced measurement volumes of the two receivers (1jI,=±2deg) are indicated with white lines (rw=15~m, e=4deg, Ab = 488 nm, d p = 30 ~m, 1jJ, = 30 deg, LllP12 = 103.7 deg, reflection) 5.1 The Signal From Arbitrarily Positioned Detector a ~102~~~~~~~"Tn~--"rr~ 209 b - - - 20 flm ~" --- -- 200 flm '1 -t 10 ~'1 _~o __ a~~D __ D __ D __ D __ O_-~_~b , I lO-\L-...l......J....J...LJ.lll.llO--..l...-L.l..w.llllO.LO--......,....l.J-'LJ..llJIOOO I Waist and particle diameter dp ' d", [flml 10 100 1000 Particle diameter dp [flml Fig. 5.25a,b. Dependence of relative phase difference error due to Gaussian beam phase divergence as a function of beam size and particle diameter for different detection volurne sizes (1f/,.=±2deg, 9,=30deg, reflection, Ab =488nm, e=4deg). aDependence on beam waist diameter for a constant ratio of particle diameter and beam waist diameter (2rw = d p ), b Dependence on particle diameter for two different beam waist diameters urne sizes. For a laser or phase Doppler system, the detection threshold is a constant value adjusted for detection of the smallest partide sizes. For larger partides the detection volume increases and for this reason the relative error increases. This effect is indicated in Fig. 5.25b with the thin dashed lines. As an example, a system with a detection volume of ad = 100 I.,tm (0.5 x 200 !Lm = rw ) capable of detecting l!Lm partides cannot measure partides with 250!Lm with an error less than 10 % by reflection dominated scattering. By reducing the size of the detection volume, e.g. by reducing the laser power, the accuracy of the measurement can be increased for larger partides but the number of small partides detected and, therefore, the signal rate is reduced. Each of the two receivers generates their own detection volume. The displacement of each of these two detection volumes increases with partide size. A phase difference measurement is possible only in the overlapping region. The sharp decrease of the curves in Fig. 5.25b for larger partides indicates that the detection volumes no longer overlap and a phase difference measurement becomes impossible. Furthermore, the error of the velocity measurement is influenced by the partide size. As already mentioned in this section and illustrated in Fig. 5.20, the virtual images of the beam waists are not coincident for a perfectly aligned (Z'1 = Z'2 = 0) system. Additionally, the measurement volume is displaced. This results in a misalignment of the system for partides much larger than the beam waist (d p > lOd w ). In normal laser Doppler or phase Doppler systems this effect can be neglected but for the time-shift technique, very small beam waists have advantages. 210 5 Signal Generation in Laser Doppler and Phase Doppler Sytems Note that all errors depend on scattering order, refractive index and receiver position and increase rapidly for misaligned systems, as already shown for small particles in Fig. 5.l3. The phase Doppler user must be aware of this error and should make a pre-estimation for the system to determine the maximum error for each diameter. 5.1.3.2 Time Domain Description of Signals from Large Particles The particle trajectory is given by Eqs. (5.61) and (5.62), together with the related assumptions. The position coordinates of the particle rop can be replaced by the time-dependent description of the trajectory. The trajectory of the particle deterrnines how the particle-generated, three-dimensional virtual measurement volume is sampled, as discussed in the previous sections. Time Domain Description of the DC Amplitude (Large Partides). As for very small particles, the DC signal consists of two pulses. The assumption is made that the incident beams are identical (Eqs. (5.22) to (5.24» and that the detector is far from the scattering center (Eq. (5.26». For large particles, the DC signal in cartesian coordinates is given by Eq. (5.99), replacing the main coordinates of the incident points by the position of the particle center in terms of trajectory parameters (Eq. (5.61» and the coordinates of the incident points relative to the particle center (Eq. (5.4». (5.111) The x component of the particle position can be replaced by v J. Of interest is the time interval between the two separated pulses. For particle trajectories parallel to the main flow direction x, the pulse maxima occur at the particle positions/times - t -- X br(;)+( (;))t an/ e/ ZpO+Zbr 2 xpDCmax,b-vxDCmax,b- , b= 1,2 (5.112) For very small particles, Eq. (5.112) reduces to (5.65). If the interval between pulses, the velocity V x and the particle diameter are known, the Z position of the trajectory intersection with the reference plane can be computed. If now the y and Z velocity components are non-zero, then the time of the pulse maxima and the duration between them will be a function of the particle trajectory. In general, the maxima occur at the positions b= 1,2 (5.113) 5.1 The Signal Fram Arbitrarily Positioned Detector 211 The difference of these two values is equal to the separation of the pulse maxima. In the region of the beam interseetion, the DC pulses are additive and only one amplitude maximum occurs. Time Domain Description of the AC Amplitude (Large Partides). The amplitude of the AC part of the detector signal can be derived from Eq. (5.99) using the particle trajectory and the coordinates of the incident points according to Eq. (5.111). The particle x position yielding the maximum AC amplitude, under the assumption of a constant beam diameter, is given by XpACmax = For very small particles, the incident points and the glare points converge to the position of the particle center, thus, Eq. (5.114) reduces to Eq. (5.70). The time at which the maximum AC signal is obtained depends on the particle trajectory, as for small particles, and on the particle size. By comparing Eq. (5.114) with Eq. (5.113) it is obvious that the x particle position yielding the maximum AC amplitude is the mean value of the x particle positions yielding the maximum DC amplitudes for particle trajectories parallel to the x-y plane. Therefore the particle dependent time shift can be extracted from the AC part alone, the DC part alone or from the complete signal containing AC and DC part. For particle motion parallel to the x axis, Eq. (5.114) reduces to XpACmax = V x tACmax = -x r + Zr tan %= x(n Ir + x(n 2 2r + ZU) - Z(;) Ir 2r 2 tan BI 12 (5.115) whereby the maximum pulse position of the particle signal is now only a function of the position of the incident points. Since these points are linearly related to the size of the particle for spherical particles, this affords a further method of particle sizing. This technique is known as the time-shift technique and will be discussed further in seetion 5.3.4. The position of maximum amplitude in Eq. (5.115) is related to the x = 0 plane. In practice this reference position can be determined by positioning a detector in this plane x = o. Such adetector will receive the maximum AC amplitude when the particle is at position x op = O. Generally, any other receiver 10cation outside of the x = 0 plane can be used, so the time difference between the two signals from the two detectors can be used for particle diameter determination Llx 12 = -vx LltI2 = xpACmax,I =.!.[-x(n _ 2 11 x U) + x(n 21 12 xpACmax,2 = -Xl + X2 +(ZI - zJtan % + x(n + (Z(i) _ 22 11 ZU) - z(n 21 12 + ZU)) tan BI] 22 12 (5.116) 212 5 Signal Generation in Laser Doppler and Phase Doppler Sytems Important to note is that the velo city of the particle1 must be known for determining the particle diameter using the time-shift technique. By comparing Eq. (5.109) and (5.116), the difference ofthe phase Doppler and time-shift technique is seen clearly. • The phase Doppler technique measures the spatial interference fringe separation or the distance between glare points on the particle surface through phase difference measurement. These fringes are generated by scattered waves originating from the glare points and influenced by the path-Iength differences. • The time-shift technique measures only the incident point separation by using the inhomogeneous amplitude of the incident wave. The positions of the incident points are related to the receiver position and depend on the ray paths. For plane waves, the time shift cannot be defined, because the incident waves have no local maxima. Knowing the velo city V x of the particle and the time shift of the bursts, the spatial displacements of the signal maxima can be derived. If the second and third velo city components, v py and v pz ' are measured with a two or threevelo city component system, the other terms in Eq. (5.114) can be determined. Therefore, the time-shift technique can be used to determine the particle size but the system must be adapted to the respective measurement conditions (see sections 5.3.4 and 9.2). For more general trajectories in a turbulent flow, the position of the shifted signal maximum is given by Eq. (5.99), together with Eqs. (5.4) and (5.73). Using the equation for the displaced measurement volume ellipsoid (Eq. (5.103)), the points at which the particle pierces the reference area of the displaced detection volume for arbitrary trajectories is given with Eq. (5.72) direct1y by -"po sinlPv - (po coslPv sin tJ v + XPAcmaxj [ rpACmax = v pt = "po coslPv - (po sin lPv sin tJ v + YpACmax (po sin tJ v (5.117) + Z pACmax The difference to the analysis for very small particles in section 5.1.2.2 is, that all volumes, planes and areas are displaced in absolute coordinates by the measurement volume displacement and thus, can be spatially separated for different scattering orders and receiver locations. The relative relations and sizes, e.g. reference and detection area size and number of periods, do not change. Therefore all relations are still valid for large particles and can be used for system configuration. I The factor for converting the spatial signal displacement to the temporal time shift is the negative particle velo city. The negative sign is introduced so that the phase difference in the phase Doppler method and the time shift in the time-shift technique have the same slope with particle diameter for the same scattering order. The definition is given later in seetions 5.3. 5.1 The Signal Fram Arbitrarily Positioned Detector 213 Time Domain Description of the AC Part Phase (Large Partides). The phase of the AC signal part follows directly from Eq. (5.104), by substituting for the partide position coordinates Eq. (5.61) or Eq. (5.62). Again, two identicallaser beams and a rather distant position of the detector has been assumed. The first four contributions to the detector phase for small particles have already been discussed in detail in section 5.1.2. The second term contains the Doppler frequency, Eq. (5.92). The 5th to Sth terms are dependent, to a first approximation, only on the detector position, particle medium and particle diameter and can be assumed constant over the entire intersection area of the laserbeams. The fifth term, IJI S,2r -IJI S,lr' contains the phase shift within the particle for different orders of refracted light. This term is dependent on the particle size, particle medium and also on the scattering order being detected. This term is the basis for particle size measurements and for the determination of the refractive index. The sixth term, k l • ri;) - k 2 . ri~), contains the phase difference between the ineident waves at the ineident points on the particle surface. This contribution is directly related to the surface curvature near the incident points. The seventh term, k b (r~~: - r~:,),), contains the phase difference of the two scattered waves between the particle surface and the detector, and is dependent on the local surface curvature near the glare points and on the scattering order being detected. This term can be computed knowing the detector position and the position of the glare points on the particle, kb (ri~: - r~§),) "" kb (rif) - r\\g»· ep,' The scattered waves from the glare points create the spatial interference field. As analyzed in the spatial description of the phase for large particles, the measurement of the fringe distance is the basis of phase Doppler technique. The last term in Eq. (5.104) contains the effect of the phase curvature of the ineident beams for the case of Gaussian beams. The complete contribution of the Gaussian beam cannot be reduced only to this term, but it gives a good approximation. In the case of curved wavefronts, the local wave vector and the 10cal propagation direction of the ineident wave is changed. This influences the ineident angle of the wave, hence, the ray path inside the particle. Based on geometrical optics, this means that the ineident and glare points move slightly when the particle moves. These more complex relations can be analyzed by extended geometrical optics (section 4.2.1) or Fourier Lorenz-Mie theory (section 4.2.2). The total contribution dependent on the particle properties amounts to (5.llS) This is expressed independent of particle shape, but is of course the basis of the phase Doppler technique for spherical particles. The connection to particle medium and shape is discussed in sections 5.3, 5.4 and chapter s. 214 5 Signal Generation in Laser Doppler and Phase Doppler Sytems 5.1.4 VisibHity of the Signal After examining the behavior of the DC and AC parts of the Doppler signal for small and large particles, it is now possible to turn to their relative magnitudes, known as the signal visibility or modulation depth (Eq. (3.89». The modulation depth is a function of the trajectory and the position of the particle trajectory, the particle scattering properties, and the aperture size and position of the receiver. Therefore, a correct analysis requires that the signal be obtained by integrating the two interfering scattered fields (Eq. (5.20» over the surface of the detector. For the two beams b = 1, 2, the scattered intensities at a point on the detector surface, I lr and I 2r , can be found from Eq. (5.20). r Ir = I lr + I 2r + 2~ I l,[2r cos( OJ,t+ lPr) (5.119) If these intensities are now replaced by the local intensity of each of the incident points Il~) and Ii;; times a scattering function 51r and 52r (see section 4.3), the dependence on the trajectory and the particle properties can be separated. (5.120) The modulation depth r, after integration over the receiver aperture and with the assumption of an angular independent scattering function, can then be approximated as (Adrian and Orloff 1977) ('); I ()'r 5 5 2 11 2 1r 2r r - rr 1(°5 2 + 1(°5 2 _ Ir Ir Ir 2r U) (52 + 52 ) I IrU)I 2r Ir 2r 52) (IU)52 + IU)52 ) 251r 52r = rr (52Ir + 2r Ir Ir 2r 2r (5.121) = rr (52 +52 ) Ir r 2r The first factor r in Eq. (5.121) contains the integration over the receiver aperture and the position of the receiver; the second factor p contains the dependence on particle properties; whereas the third factor v is trajectory and position dependent and embodies the capability of the incident waves to interfere. r r (5.122) 5.1 The Signal From Arbitrarily Positioned Detector 215 The influence of the partide trajectory on the modulation depth, given by is now examined. Equal intensities at the incident points, e.g. at the center of the measurement volume 1\;) = n;/ =10 , yields a modulation depth dependent only on the seattering and the receiver properties (5.123) Yv =1, If the two incident points are dose together (ri;) "" ri;), e.g. for sm all intersection angles, the ratio of the intensities is given by (5.124) and for very small partides (5.125) r The dependence of v on partide position (x op Ja o) is shown in Fig. 5.26a für very small partides. The parameters used are zop JCo and the scattering funetions assumed equal S2 r = SIr. There is no dependence of the trajectory-dependent modulation depth on Y op for very small partides, since the intensity ratio is constant for a plane perpendicular to the intersection plane, YoP = o. In the plane x op = 0 and zop = 0, the b a """ 1.0 ................t+:~~~!'e:jl:+"-............ .<: C. "c; "0 .2 '" :; "0 o ~ 0.5 -0.5 1.0 0.0 0.5 Parlicle pos ilion x. p I a. I-1 - 180 · 90 0 90 18C TrajeclOry direclio n !?, Idegl Fig. 5.26a,b. Trajectory dependent modulation depth y v for very small partides (5],. =5" .). aAs function of partide position xo p for different trajectories z OP (YoP =0), b Expected modulation depths inside the measurement volume as a function of the trajectory direction (qJv = 0 deg) 216 5 Signal Generation in Laser Doppler and Phase Doppler Sytems modulation depth is also not dependent on particle position. Allother trajectories lead to a decrease of modulation depth towards the outer edges of the measurement volume. The influence of particle trajectory on modulation depth is shown in Fig. 5.26b for the case of Yop = 0 (qJv = 0 deg). At the signal maximum, all modulation depths inside the shaded region can be expected for the corresponding trajectory directions piercing the measurement volume. Figure 5.27 shows the detector signals for a very small particle and two particle trajectories v p = e x vpx ' near and far from the center of the measurement volurne. Figs. 5.27a-c and 5.27d-f each show the DC part, the AC part and the total signal. The modulation depth corresponds closely with the signal-to-noise ratio (SNR). A low modulation depth leads to a low SNR. A significant variation of the signal quality (SNR) is observed, dependent on trajectory. Because of this it is favourable for the signal processing to identify and operate on the central portions of the signal. - ;;: 1.0 a " ",,'" <lJ ] i. 0.5 S < 0.0 - ",,0 1 b e c f " ",,"' <lJ :9 0 ~ S < -1 - 2 ",,0 "" :9 " <lJ :&S 1 < -2 -1 0 1 2 -2 Particle position x op I a o [-] -1 0 1 2 Particle position x Op I ao [-] Fig. 5.27a-f. Detector signals and their corresponding DC and AC parts for two different particle trajectories (v p = ex VPX> 5h = 52'.)' a-c rpo = (0, O.lbo, O.lcoY, d-f rpo = (0, 0.2bo, 0.7 coY 5.1 The Signal From Arbitrarily Positioned Detector 217 Also the scattering characteristics of the particle influence the trajectory dependent modulation depth of the signal. If the scattering functions SIr and S2r are not equal, the modulation decreases in the center of the measurement volurne and increases at the outer regions. Therefore, the maximum signal amplitude does not necessarily correspond to the maximum modulation depth. A further decrease of modulation depth is due to the factor (5.126) In scattering regions where the incident points are not spatially separated, e.g. diffraction in forward scatter or for very rough particles, and for large apertures and large particles, an early analysis by Farmer (1972) can be used to approximate P' According to this approximation the size dependence of the modulation depth is given by the Bessel function of first order r (5.127) In this analysis the entire particle cross-section contributes to the signal generation, hence it is only valid for the near forward direction with separated incident and glare points, large detector apertures and small intersection angles. Bachalo (1980) gave a more rigorous analysis of the visibility by numerical integration ofthe scattering function over the aperture ofthe receiver and Semidetnov (1983) derived an analytic approximation for dominant reflected and refracted light. He assumed the dominance of one scattering order and a constant interference fringe distance L\X(N) over the aperture. For circular apertures his integration of the fringe system over the aperture results in a Bessel function. (5.128) r where R is the full receiver aperture angle. Z(N) is the normalized mean angular fringe spacing of the scattered light, relative to the particle diameter d p and the interference fringe spacing in the measurement volume Llx and is given by Z(N) =~ fJ Llx .0, 2n roIgrad( cP;N) (lf/r' ~r' ro»)1 dil (5.129) r where il r is the solid angle of the receiver aperture. The normalized mean fringe spacing of the scattered light Z(N) can be calculated by using the relation of geometrical optics and is independent of the interference fringe spacing of the illuminated volume and particle diameter. A Lorenz-Mie computation of the 218 5 Signal Generation in Laser Doppler and Phase Doppler Sytems modulation depth and a comparison with Eqs. (5.127) and (5.128) are shown in Fig. 5.28a for water droplets in air and a receiver position of t/Jr = 30 deg. Indeed, certain diameters yield a modulation depth of zero. This relation is only of practical use for particle sizing when the SNR is quite high or when a single scattering mode dominates. In Fig. 5.28b the modulation depth is shown for circular and rectangular apertures located on the optical axis. The modulation depth decreases rapidly with increasing particle diameter for the circular aperture. On the outer edges of the aperture the diffraction from the respective beams dominates. In these regions the scattered intensity from one beam is very high (diffraction) and from the other much smaller (reflection and refraction). Therefore, nearly no interference at the outer parts of the aperture occurs and the modulation is very low. In the central part of the aperture the scattered intensities of the two beams are equal and the modulation is high with low intensity. Integrating the intensity over the aperture results in a low modulation of the detected signal. This effect is important for aperture positions at which the scattered intensity exhibits strong spatial variations, e.g. diffraction in forward scattering or at the rainbow angle. The modulation can be increased when the scattered intensities of the two beams are comparable over the complete aperture. By using a small rectangular aperture in the y direction in forward scatter, the modulation increases dramatically, as illustrated in Fig. 5.28b. The overall signal amplitude decreases, but the SNR can be improved significantly. For larger apertures, receiver locations displaced from the optical axis can be chosen, where a constant intensity appears (see section 4.1.3) b a , e:: 1.0 , .<:: P.. ~ c:: -------- Pure reflection , , , , ·ß ..,, ···, ·. First-order refraction: - - Lorenz-Mie - - Approximation by Semidetnov , :g'" o ~ 0.5 Lorenz-Mie Pure reflection and diffraction Approximation by Farmer Rectangular aperture .. . 0.0 ,, ,, I-L...l....1...J...J--'-L..J...:'''-'-.L.J...J...JI>4...:,,",,-.J....l...J.....l...L.;.>...t.:.I o 100 200 Particle diameter dp [11m] o 50 100 150 Particle diameter dp [11m] Fig. 5.28a,b. Particle size dependence of modulation depth for water droplets in air (A b =488nm, e=4deg, r,=300mm, Vlr=Odeg). aReeeiver loeation 9r=30deg and circular aperture (R,. = 10 mm), b Receiver loeation 9, = 0 deg and cireular (R,. = 10 mm) and rectangular (R x = 0.125 mm, Ry = 10 mm) aperture 5.1 The Signal From Arbitrarily Positioned Detector 219 5.1.5 Shift Frequency Influence In section 5.1.2.2 it has been shown that the number of periods in a Doppler signal depends on the particle trajectory through the measurement volume. Indeed, in turbulent flows, in which the particle may have a very low velocity in the x direction, even the maximum possible number of periods will depend on the particle trajectory (Eq. (5.80». To facilitate measurements of particles with low velocities in the x direction, or particles passing through the peripheral region of the measurement volume, frequency shifting has been introduced, i.e. the use of laser beams with two different wavelengths. In principle two lasers with different wavelengths can be employed, however, the use of one or two Bragg cells is more common (Durst et al. 1976, 1981). Introducing a frequency difference between the two beams leads to a moving interference pauern. For instance, if beam 1 is shifted by the frequency !sh' the resulting signal modulation frequency is given by Eq. (5.91) as (5.130) For Vx > 0 (!D > 0), the generated signal has a modulation frequency lying above !sh' whereas for V x < 0 UD< 0), the signal frequency lies below !sh. A stationary particle yields a signal frequency of exactly !sh. For some signal processing algorithms, the estimation of the signal frequency requires a minimum number of signal periods, N m;n. Thus, the choice of frequency shift can influence the size of the detection volume. For a stationary interference field, the detection area is a function of the flow direction (rp v' 13 v> as given by Eq. (5.88) for an arbitrary flow direction. Using the minimum number of required periods N m;n' the Doppler frequency !D and the frequency shift !sh given by Eq. (5.130), the trajectory and period dependent projected reference area is A OvN (N m;n'!D'lPv' 19) Ao (5.131) The projected detection area can, therefore, be approximated as a function of the Doppler frequency (Fuchs et al. 1983) (5.132) where C2 and FAV are defined in Eqs. (5.75) and (5.88). The projected detection area depends on both the particle trajectory through the measurement volume and on the chosen shift frequency. This is also true for the maximum possible number of signal periods. For arbitrary trajectories, the number of signal periods is given by Eq. (5.80). If N ov is replaced by the maximum number of signal 220 5 Signal Generation in Laser Doppler and Phase Doppler Sytems periods using a shift frequency, then an expression is obtained for the maximum number of periods for an arbitrary flow direction and with frequency shifting N oV'h(f'h' rp v' 1'J J (5.133) No Both the detection area and the maximum number of signal periods are related to the shift frequency through the factor (5.134) Figure 5.29 illustrates the magnitude of this factor as a function of the ratio of Doppler frequency to shift frequency. From Fig. 5.29 it is apparent that for a finite shift frequency and for JD = 0, the number of signal periods becomes infinite N ---7 00 , i.e. for a stationary particle, the modulation amplitude remains constant in time. For an increasing Doppler frequency, the number of signal periods decreases, reaching an asymptotic limit of N ov for JD» J'h. This also corresponds to the case of a stationary interference field, i.e. no shift. If the particle and the interference pattern move in the same direction, then the case of zero signal periods arises when the velo city of each are the same. For JD «J'h' the N ov asymptote is again achieved, albeit no longer with directional -4 -2 o 2 4 Doppler frequency in relation to shift frequency Fig. 5.29. Shift frequency influence factor as a function of the ra tio JD I J,,, Iv I !,h [-] 5.1 The Signal Prom Arbitrarily Positioned Detector 221 sensitivity. Fig. 5.29 is a helpful aid for recognizing the allowable measurement range for a given shift frequency or choosing suitable f:tlter limits for the signal (Tropea 1985). Ifthe minimum number of signal periods is also to be maintained for a partide passing the measurement volume off-center, then the maximum number of periods must be chosen 20% .. .30% high er. Furthermore, for trajectories defined by the angle f) v and rp v' a further increase is necessary. For instance, for f)v = 30 deg and rpv = 30 deg, a factor 2.3 must be applied to the maximum number of periods in the center of the volume. Altogether therefore, the measurement volume must be designed for (2.8 .. .3.0)N O. According to Fig. 5.29, a usable range of -O.3::;!D I !sh ::; 1.35 results. As an example, if the shift frequency is 10 MHz, then the measurement range of Doppler frequencies is -300 kHz::;!D ::; 13.5 MHz. This gives directly the measurement range in velocity. The appropriate range for high and low-pass filters is given by 9.7 MHz::;! ::;23.5 MHz. 5.1.6 Measurement and Detection Volumes Signal detection is accomplished at the signal processing stage and the method used for signal detection will influence the dimensions of the detection volume. The signal detection can be based, for instance, on an amplitude level chosen above the background noise level and/or on a test of periodicity, either in time domain or frequency domain. In either case, this influences the dimensions of the detection volume. The e- 2 decay of the modulation amplitude was chosen rather arbitrarily to define the dimensions of the rneasurernent volurne. For equal flow conditions therefore, all partides exhibit the same measurement volume. For very small partides, the measurement volume is identical to the illuminated volume at the intersection of the beams, Eq. (5.39)-(5.41). For large partides, the measurement volume becomes virtual, equal in size to the illuminated volume but displaced in position by an amount dependent on the scattering properties of the partide and on the detector position (Eqs. (5.103), (5.100), Fig. 5.22). The volume from which signals are received is designated the detection volurne, which can differ from the measurement volume. The detection volume is defined largely by the requirements placed on the signal and can be either smaller or larger than the measurement volume. The signal amplitude at the detector is influenced directly by the scattering properties of the partide: partide size, partide material, and by the properties of the intersection volume. With a detection using signal amplitude for instance, only partides exceeding a certain minimum diameter will be detected. The detection volume may vary significantly for different partide populations depending on any further validation requirements placed on the signal. For a given flow direction, the partides effectively see the projection of the measurement and detection volume perpendicular to the trajectory direction, as 222 5 Signal Generation in Laser Doppler and Phase Doppler Sytems given in Eqs. (5.78)-(5.84). These areas are called projected reference area and projected detection area respectively. For particle trajectories in the main flow direction (v y = V z = 0) the dimensions and position of the projected reference and projected detection area are identical to the detection area (Eq. (5.42)) and the reference area (Eq. (5.40)). For concentration and flux measurements the reference area is used to relate all samples to the same value. The various requirements and influences are discussed in the following sections 5.7.6.7 Influence of an Amplitude Threshold For some arbitrary threshold voltage Ud' or threshold current id through a given resistance, the detection volume is defined by the minimum intensity I d required to reach the threshold. For Eq. (5.103), this condition can be expressed as ! 2 ln( uACmax (dp)) =! ln(iAcma::(dp )) =! ln( I ACm"" (d p)) ~ 2 =( ~ xop -a:Acmax J 2 ~ + (Yop - ~AC max J -2 J (5.l35) + ( Zop cmax UAcmax(dp)' iAcmax(dp) or IACmax(dp) are the maximum voltage, current or intensity achieved for a particle with a given diameter d p positioned at the center of the measurement volume. 1 The following analysis is based on the intensity of the signal. All amplitudes are therefore related to intensities. Alternatively, signal current or signal voltage can be used by exchanging the corresponding variables. The dimensions of the detection volume can be related to ao' bo and co' Eq. (5.40), as follows (5.l36) where bd and Cd are the dimensions of the detection area A d for particle trajectories parallel to the x axis. The detection area is the cross-sectional area within the detection ellipsoid, on which the particle generates the maximum signal for a velo city vector in the x direction (see Fig. 5.30) and is given by A d =Ad(dp)=nbacd = nboco ln(IAcmax(dp))= Ao ln(IAcmax(dp)) 2 Ia 2 Id (5.l37) Ao = nboco is the corresponding area, the reference area, within the measurement volume. 1 All quantities connected with the detection volume, e.g. A d , ad , bd , Cd' UACmax' ete., are particle size dependent. This ean be indicated explicitly as f(d p ). Nevertheless, the subscript d indicates the particle size dependenee implicitly (I d is one exeeption). 5.1 The Signal Fram Arbitrarily Positioned Detector 223 Since the scattered intensity I ACmax (d p ) is directly proportional to the integral scattering function G(d p ) (Eq. (4.161), the threshold intensity I d also determines the smallest detectable particle in an ensemble of polydispersed particles. 5.1.6.2 Influence of a Minimum Number of Periods If a minimum number of signal periods N min is chosen to validate signals, then the measurement volume is reduced in the x direction according to aON N. = ao----12:!!!1.. (5.138) No The other dimensions are obtained using Eqs. (5.39) or (5.103) and (5.138) as (5.139) No is the maximum possible number of periods in the measurement volume (Eq. (5.50», without including the influence of a shift frequency, i.e. for stationary fringes. The same relations are valid for the detection volume by replacing a o ' bo and Co by a d , bd and Cd. The detection area is then defined by (5.140) which is a special case of Eq. (5.132) for particle trajectories in the main flow direction. 5.1.6.3 Influence of Partic/e Density Distribution Already in section 5.1.2.2 it was shown that the projected detection area A dv and the projected reference area A ov are trajectory dependent (Eq. (5.88». A relation to the reference area of the measurement volume Ao is now given. In practice, several of the above influences may be present and should be considered for the system layout, especially with respect to the signal rate. The main measurement result of a phase Doppler system is a particle size distribution N pd = Np (d p ), collected over a time interval Llt. Together with the frequency measurement, both the size and velo city of each particle is available, enabling the volume (mass) flux and concentration to be computed. Since the dimensions of the detection volume are dependent on detector requirements (minimum amplitude, minimum number of periods) and on the scattering properties of the particles, a computation of fluxes and concentrations with respect to a reference volume is necessary. The measurement volume can be used as this reference volume. Consider first the flux of monodispersed particles Qpd (index d for monodispersed particles with one diameter and therefore one detection area) at the ve- 224 5 Signal Generation in Laser Doppler and Phase Doppler Sytems locity v p = exvx . The measured number concentration npd of particles with diameter dp during the time T and for the N pd particles collected is given by (5.141) bd and Cd are the dimensions of the detection area for particles with diameter d p •1 Knowing the individual particle size allows also the mass flux density qmd to becomputed (5.142) where Pp is the mass density of the particle material. One main difficulty is that the quantities bd and Cd are themselves dependent on the particle size. Thus, for a given detection amplitude threshold Ud' the detection prob ability is larger for large particles than for small particles. Therefore, large particles are weighted stronger in the size distribution. Especially for wide size distributions, this error can be very significant. Some correction is necessary when either the size distribution or the mass flux is to be computed from the statistics of Npd(d p )' In the following, a correct estimator will be derived based on the ellipsoidal shape of the measurement volurne. In many cases, this volume is further restricted using a slit aperture on the detector. Such systems will be analyzed later in chapters 8 and 12. Given an intensity detection threshold I d , the dimensions of the ellipsoidal volume are given by Eq. (5.136). The detection area is given by Eq. (5.137). Related to I d , IAcmax(dp) is the maximum scattering signal amplitude possible for a particle of diameter d p in the detection volume. For inclined trajectories, the modification given in Eq. (5.88) must be used. The validated signal rate N pd (number of signals per time) for monodisperse particles with d p is then (5.143) For a trajectory parallel to the x axis passing an arbitrary point P(y po, z po) on the plane x = 0 (Fig. 5.30), a Doppler burst [(xo,) ~ [m~(d,)e+( :z )'] (5.144) with the maximum signal amplitude 1 The detection area A d = 1tbd cd in Eq. (5.143) is constant for all monodisperse particles and could be placed in front of the summation. It is kept in the summation here because in later analyses (section 12) the detection area is particle size dependent. 5.1 The Signal Fram Arbitrarily Positioned Detector Refcren.:e plane (x == 0) / 225 j'\'Ieasuremenl volumc: u. ,v. ,co lI ~e -' " dem., y Deleclionvolume: ud,li. ,c d II ~ II " Rcferencc are'd: A. == n/)."o ' ". ~e ' II A em. , Dele.:lion arca: Aß = nl>dcd ' " 0 ~ II J Measuremenl volumc x ~ ~ Parlide Uo I vo]umc Fig. 5.30. Detection plane dimensions for correction of partiele statistics (5.145) will be obtained. Dependent on the referenee posItIOn in the plane x = 0, P(Ypo,z po}, amplitudes I d ~ I max (d p) ~ I Acmax (d p) are possible. The dimensions bd and Cd and the size ofthe deteetion area A d ofthe partiele group with d p are given in Eqs. (5.136) and (5.137). The burst amplitude Imax(d p } ean be related to the amplitude eorresponding to the threshold I d • (5.146) Sinee the exaet trajeetory of a partiele is seldom known, these relations ean only be applied in a statistieal sense. On the other hand, partiele diameters and the distribution of maximal amplitudes for eaeh elass of diameters ean be determined by measurements. This is aehieved by averaging over all maximum amplitudes Imax(d p } aeross the deteetion area. This average beeomes (Borys 1996) (5.147) 226 5 Signal Generation in Laser Doppler and Phase Doppler Sytems Using Eq. (5.147), the detection area (Eq. (5.137» can be computed from (5.148) Using the measured maximum signal amplitudes Imax(d p )' the partide size dependent detection area can be computed. Normalizing the detection area to the reference area Ao' yields the correction factor (5.149) with which each partide size dass must be multiplied. This factor insures that all size dasses are referenced to the same reference area Ao of the measurement volume. This correction is apre-requisite for computing the partide size distribution (Eq. (5.141) or mass flux (Eq. (5.142». Similar expressions accounting for the partide size dependent detection volume has been given by Qui and Sommerfeld (1992) for a slit aperture in the receiving optics. One disadvantage of this approach is that the signal amplitude is not measured by all signal processors and that signal amplitude saturation can falsify the results, especially for large partides. For this reason, a more practical approach based on burst length will be introduced in section 12.2.3. Furthermore, it is possible to deterrnine experimentally the integral scattering function G. The relation to the scattering function is given by Eq. (4.160). PACmax rrrpIo = AJACmax (d) p = 2 kb G (5.150) which yields for the integral scattering function (5.151) where 10 is the maximum intensity in the center of the detection volume, Ar is the aperture size of the receiving optics, rand p are the modulation depths for receiver integration and partide properties as defined in section 5.1.4, and Pd and I d are the power and intensity threshold for detection. The modulation depths are important for larger aperture sizes and larger intersection angles. The maximum amplitude uAcmax(dp) or maximum scattered intensities IAcmax(dp) can be determined from the statistics of the amplitudes for each partide size dass using Eq. (5.147). For the determination of the integral scattering function, the transfer function of the system must be known. For the inverse case, when the integral scatteringfunction is known (see chapter 4), the transfer function of the measurement system can be calculated on-line (Borys 1996). r r 5.1 The Signal From Arbitrarily Positioned Detector 227 5.1.7 Statistical Time Series of Particle Signals The purpose of this section is to derive some preliminary guidelines according to which a laser Doppler system can be matched to a given flow field. Each particle (subscript i) generates a signal, as given in Eq. (5.20), and using Eqs. (3.189) and (3.190), the generated current is equal to (5.152) The time t;, is the time of the signal maximum with respect to so me reference position ofthe particle, in this case rpo • ioc,;(t-t;) is the DC part, ri(t-t;) is the modulation depth (visibility, Eq. (5.121)) and f/J; is the signal phase at the signal maximum of the signal generated by the particle i. All amplitudes are understood to be values integrated over the detector surface. The high-pass filtered signal, as shown in Fig. 5.31, is used for further processing (5.153) where io,; depend on the particle trajectory and are, therefore, statistically distributed. OJ D ,; depends on the flow velocity fluctuations and will also exhibit a distribution. The width or e- 2 decay of the bursts tb are the same for all signals generated by one particle, whereas the burst length, given by the threshold of the detector, is different. For system layout, the number of detected or processed signals per unit time is of importance. For this computation, a random and homogeneous spatial distribution of particles in the flow is assumed. Only in cases where seeding is injected in the immediate vicinity of the volume would a non-uniform distribution be expected. The mean number of particles in the measurement volume is I, I, I" Fig. 5.31. Irregular arrival ofburst signals I. Time I 228 5 Signal Generation in Laser Doppler and Phase Doppler Sytems 4 _ N pa = -TC aoboconp (5.154) 3 and the mean signal rate (number of signals per unit time) for particle trajectories through the measurement volume parallel to the x axis is (5.155) In these expressions the overbar indicates a statistical time average and the index 0 that the values are related to the measurement volume. For a validation rate of 100% the particle flux QPd through the reference area is identical to the given signal rate. If now arbitrary particle trajectories are allowed, the influence of the trajectory on the signal rate can be investigated. The signal rate will depend on the size of the projected reference area perpendicular to the instantaneous velo city vector (Eq. (5.88» (5.156) The dependence of the factor FAv = coC~/2 on the angles ({Jv and iJ v is shown in Fig. 5.16 and although only a weak dependence on ({Jv is observed, the effect of iJ v is very strong. Clearly, the zvelocity component will be strongly underweighted. This suggests that the volume should always be orientated such that the main velo city component is aligned either with the x or yaxis of the measurement volume. The same relations can be given for the mean number of particles N dp' the signal rate Nd and the trajectory dependent signal rate Ndv for a given detection volurne size by replacing the measurement volume dimensions with those of the detection volume. For practical system layout, it is of interest to investigate the change of signal rate when only signals above a certain user changeable amplitude threshold I d are used for analysis. The threshold I d defines the detection volume, within which a signal will be detected and processed. First, all particles are assumed similar in their scattering properties, thus, they would have the same overall maximum signal amplitude IAcmax(dp) when moving through the center of the detection volume. The spatial distribution of the maximum signal amplitudes Imax(d p) for arbitrary trajectories is derived by using Eqs. (5.86), (5.87) and by replacing the measurement volume dimensions by detection volume dimensions. ( ll~o J2 + ( S~o J2 bdv cdv = k, 0 :-s; k :-s; 1 (5.157) Bursts of equal amplitude will be obtained for each value of k, where k = 0 when a particle passes through the center of the detection volume and k = 1 at the periphery of the detection volume. Using a variable amplitude intensity, the contribution of different signal amplitudes to the total signal can be derived. Within a projected detection area of 5.1 The Signal Fram Arbitrarily Positioned Detector dA dv 1 2 =--A Ov dI I 229 (5.158) - signals of constant amplitude contribute to the total particle flux as d QPd - - =npdv pd d dI I 1_ - (5.159) A dv =--npdvpdAov- 2 where Vpd is the mean velocity of particles with the size d p . In case of no correlation between particle size and velo city, the mean velocity of all particles vp can be used instead of Vpd (see section 12.2.1). Integrating over all groups of the same amplitude yields the particle flux of monodisperse particles of size d p Qpd 1 - - A Ov 2 =--npdv pd f I d IAcmax(dp) 1 dI =-npdVpdAo 1- ln I 2 v - II- - - -(d' - -») ACmax P Id (5.160) which is the rate of particle-producing signals exceeding the specified threshold. One assumption of this analysis is that the particle flux density of a given particle size qpd (particles per area and per time) must be uniform across the projected detection area as illustrated in Fig. 5.32 and given by (Albrecht and Fuchs 1987, Albrecht et al. 1990 and 1993, Hintze 1993, Borys et al. 1993) dQPd __ qpd =--=npdvpd dA dv (5.161) The increase of signal rate with a lower threshold is illustrated in Fig. 5.33a. The particle flux through the projected reference plane Qpo = npdvpAov /2 is used as a normalization factor. The amplitude range associated with a higher signal quality contributes more to the total signal rate than the outer regions of the measurement volume, which produce signals oflower quality. This is confirmed by examining the distribution of the signal rate as function of signal amplitude I 4---~~----+1 k e-2 I Acm""(d p)+----~~---+-----=-........- qp qpd+-------,,....----I- - - - - - P~ticl~ flu; d-;;nsity Detected particle flux density A d" Ao" Fig. 5.32. Uniform distribution of particle fiux acrass the detection plane 230 5 Signal Generation in Laser Doppler and Phase Doppler Sytems (5.162) as illustrated in Fig. 5.33b. Furthermore, if the particles are polydisperse in size, the integral must also be performed over the size distribution function. The maximum amplitude for particles can be expressed in terms of the maximum modulation depth rand the integral scattering function Gp (see section 4.3) I ACmax (d p) = Cr(dp)G p(d p) (5.163) All detection parameters are combined into the constant C. Equation (5.160) then becomes (5.164) (5.165) For all parameters, an integration over the detector surface has been assumed. Equation (5.164) can be inverted as a fundamental means to measure particle size distributions. For this, the scattering function must be "continuous". In a similar mann er, the size distribution can be derived from the distribution ofthe number of periods in the detected signals as discussed in section 9.1.1 for the cross-sectional area difference method (Albrecht and Fuchs 1987, Albrecht et al. 1990 and 1993, Hintze 1993, Borys et al. 1993). ~ i3 (Ö 0,6 .[n '" .~ 0,4 '" ~ I=e'I (d) d AClIlax p 0,2 ( A d" = A o" ) 0, ° ° L..J....u.J...-'-'--'-'--'--'L.....L.l...-1--'--'---'---"---L-JWLJ...J....J.J 2 4 Relative threshold 6 IAcm~ 8 ( dp ) lId [-] 0, ° L...L--1..L.J.......J.---'-..J......JL.......L..-'-'--'--'-..J......J----'----'-""" e 0,5 LO Relative signal amplitude Im~ I IAC mox [-] 0,0 Fig. 5.33a,b. Influence of the specified amplitude detection threshold on signal rate 5.2 Laser Doppler Technique 231 The laser Doppler technique is an indirect measurement method, which sampIes the flow velocity at discrete times. The sampling function is dictated by the detection and validation of the signals generated by scattering particles. Most flow properties of interest will in some way rely on a minimum signal rate being achieved, which is now the topic of closer scrutiny. Given a homogeneous and random distribution of particles in the flow, the signal rate can be derived using Poisson statistics. The probability function p(N p) that Np particles are simultaneously in the detection volume is given by (5.166) where Np is the expectation (average particle concentration times volume of detection volume). For Np < I the most likely probability occurs for Np = 0, i.e. no particles in the volume. Generally it is desirable to avoid multiple particle signals, at least from the point of view of the signal processing. This condition is often referred to as the single realization mode. It is now interesting to specify the optimum particle concentration to maintain the single realization mode, while maximizing the signal rate. Accepting multiple signals only less than 0.5 % of the time p(N p > I) < 0.5 0/0 results in Np = 0.1 (p(N p > 1,0.1) = 1- p(O,O.I)- p(I,O.I) = 0.0047; p(O,O.I) = 0.9048; p(I,O.I) = 0.0905). The allowable particle concentration is then given directly by Eq. (5.154) for the detection volume dimensions. For Np> 0.1, the signal overlap from multiple scatters is no longer negligible and the validated signal rate may begin to decrease. Three concentration ranges can be distinguished: • Np <0.1: single realization with a probabilityof99.5 0/0 • N p :2: 0.1 : multiple particle signals, in which the superposition of random amplitudes deteriorate signal quality and may lead to a lower signal rate • Np> 10: a quasi-continuous signal. As an example, the condition Np< 0.1 leads to an average particle concentration of np < 6.6 mm- 3 for the system specifications used in Fig. 5.5. If the concentration is too high, more stringent validation, such as a higher amplitude threshold, can be used to decrease the detection volume and thus insure single realization operation. 5.2 Laser Doppler Technique The laser Doppler technique is most widely used as a technique for flow velo city measurements, although industrial applications involving solid surface velocities are also not uncommon. For flow studies, the two optical configurations shown in Fig. 2.4 are used almost exclusively: the dual-beam technique (Fig. 2.4a) 232 5 Signal Generation in Laser Doppler and Phase Doppler Systems or the reference-beam technique (Fig. 2.4b). The latter is more of historical significance, being the first realization of the laser Doppler principal, but has been used also recently in some configurations for long range (Dopheide et al. 1990) and boundary layer measurements, in which it is combined with a time-shift to yield also particle position (Strunck et al. 1994, 1998). In both configurations the illuminated volume is formed by the intersection of two beams, onto which the detector is focused. The flow velocity is proportional to the modulation frequency in signals generated by tracer particles moving through the illuminated volume. The proportionality constant is given uniquely by the wavelength and intersection angle (Eq. (5.92», hence no special calibration of the technique is necessary. As with all tracer-based methods of flow measurement, the light scattering particles must follow all flow fluctuations slip-free. The requirements placed on the particle to insure that this is the case are discussed fully in section 13.1. and will be assumed to be fulfilled in the following discussion. 5.2.1 Dual-Beam Configuration Most laser Doppler systems correspond to the configuration shown in Figs. 2.4a and 5.1 , the so-called dual-beam technique. With this optiCal arrangement, there are two illuminating beams and the detector is positioned either in the forward scatter direction,CPr = 0 deg and IfI r = 0 deg or in backscatter direction, CPr =Odeg and IfI r =180 deg (Fig. 5.34). The advantage ofthis optical configuration is that the measurement volume is uniquely defined by intersection of the beams. Since sm all particles are preferred with the laser Doppler technique, the corresponding relations derived in section 5.1.2 define the measurement and the detection volumes. y Rc~ciycr in forward scattcr dircction Rc~civcr in ba..:kward scattcr dircct io n Fig. 5.34. Dual-beam laser Doppler arrangement with forward-scatter and backscatter deteetion 5.2 Laser Doppler Technique 233 Due to symmetry, the scattering functions for each beam can be considered equal, Eq. (5.31). The signal comprises a DC part, arising from the scattering of each beam (Eq. (5.32)) (5.166) X h( r; xopzop sin BI cos BI] cos 4 . 12 12 and a modulation part, arising from the superposition of the two scattered waves, Eq. (5.36) (5.167) Indeed, the particles used with the laser Doppler technique are usually small enough that the interference model (Eqs. (2.15) to (2.29)) is applicable. The measurement volume therefore corresponds exactly to the illuminated volume at the beam intersection. The dimensions of the measurement volume are given by Eq. (5.40) and the volume itself by Eq. (5.41). The dimensions of a detection volume prescribed by a given intensity threshold can be computed according to Eqs. (5.42) and (5.43) and considering the scattering properties of the particle. The path length and the number of periods for a particle moving through the detection volume vary with flow direction. The Doppler frequency, the maximum number of periods in the burst signal and the projected detection area seen by particles can be computed using Eqs. (5.92), (5.80) and (5.88) respectively. Laser Doppler systems are generally equipped with acousto-optic modulators (Bragg cells), which allow the flow direction to be resolved (section 5.1.5). This also permits very small velocities to be correctly measured. Because focused laser beams, e.g. Gaussian beams, with curved wavefronts are used for the laser Doppler techniques, the most important uncertainty is produced by the diverging fringe system. A description of the influence is given in sections 5.1.2.1 and can be reduced to Eq. (5.56). A more complete description ofhow a dual-beam system can be designed to best match a given flow study is given in chapter 7 and in section 14.1.1. 5.2.2 Reference-Beam Configuration Historically, the first laser Doppler device was a reference-beam arrangement (Yeh and Cummins 1964). In fact, the reference-beam method requires only one illuminating beam in the flow, although the reference beam is generally also passed through the flow to localize the measurement position when adjusting. A superposition of the reference beam with scattered light from tracer particles 234 5 Signal Generation in Laser Doppler and Phase Doppler Systems generates the Doppler signal. Recently, the reference-bearn technique has been used more frequently, mainly due to lower noise levels in lasers and semiconductor detectors. Semiconductor detectors can be placed directly in the path of the reference bearn, as opposed to photomultipliers, which saturate or suffer damage. Figure 5.35 (Fig. 2Ab) illustrates a typical reference-beam arrangement. The detector is placed in the path of the reference beam, i.e. at fjJ r = 0 deg and lfI r = ± %. The second bearn is used for illumination of the particles and generates the scattered wave. Both beams have a waist radius of rw1 = rw2 = rw and the same wavelength Ab. Generally an unequal amplitude ratio is used, favoring the illuminating bearn, e.g. 5 % /95 %. In this way the amplitude of the scattered wave compared to the reference wave can be improved, which yields a high er modulation depth in the signal. A system will now be analyzed in which the detector (r = 1) is placed directly in the line with one bearn (b = 1) e 01 = e 1 (IfIr = +%), as shown in Fig. 5.35. The position vector of the detector (Eq. (5.2» is given by (5.168) For simplicity, the assumption is made that all waves on the detector surface have the same polarization er A small surface area Mn on the detector will be considered. In the analysis no consideration is given to which scattering order falls onto the detector, since for the laser Doppler technique only the frequency of the signal is used far velocity measurement and the phase of the detector signal is not relevant. Three waves contribute to the signal on the elemental area Mn: the reference X ;",,,.,,":~:Virtual fnnges L- 11 "" Beam 2 k, Main flow direclion ßeam I Fig. 5.35. Reference-beam laser Doppler arrangement 5.2 Laser Doppler Technique 235 beam (gbl ) and the scattered waves from each of the two beams (gll' g2l ) (5.169) Since all components exist on the same elemental area, subscript n will dispensed with. §.bl is the wave amplitude of the reference beam (b = 1) on the detector surface (r=I), as given by Eqs. (3.59) and (5.8)-(5.10) (Fig. 5.36) (rwl =rw2 =rw ' ZRI = ZR2 = ZR) (5.170) rml (rOl ) is the radius of the reference beam at the detector, 'l,n is the position of the surface element Mn in the laser beam and R1 (rm ) is the wavefront curvature at the detector. §.ll is the scattered wave from beam 1 on receiver 1, Eq. (5.19). +arctan( X OP siniJ2' +zop cosiJ2'] + If/ll ZR II ßeam2 Beam I Fig. 5.36. Relation between scattered wave and detector surface 236 5 Signal Generation in Laser Doppler and Phase Doppler Systems 511 is the scattering function, rm1 (rüp ) is the beam radius and R 1 (rüp ) the wavefront curvature at the position of the particle r üp ' The relations between the scattered wave, the reference beam and the detector surface are clarified in Fig. 5.36. The phase 1f/ 11 includes phase differences caused by the particle, which for small particles vanish. The scattered wave from beam 2 is similar in form +arctan( -xüp sin,% + züp cos'% ZR J+ ]J If/ 21 The total power for the detector is therefore given by Eq. (3.183) PI = c2e JJ (gI + Qll + Q21 )(Q~ + Q;'1 + Q;I) dAr (5.173) A, The assumption of a point detector is made, meaning that the field strength does not vary significantly in the immediate neighborhood of the chosen point. Exact computations can be carried out using EGO or FLMT. Figure 5.37 illustrates the individual contributions to the total detector signal for the parameters given in the figure caption. Two particle trajectories on different sides of the illuminated volume were considered: rüp = (x üp ' 0, 100 11m) and rüp = (x üp ' 0, -10011m). The detector power is normalized with the input laserpower. The detected signal comprises four components: 1. The three waves generate a DC part of the signal equal to (5.174) Ce , I DC ,bl =2Qblgbl' I DC,21 = Ce E E* 2 _21_21 (5.175) The first term I DC bl dominates and arises directly from the reference beam (Fig. 5.37a,b). The'next two terms I DC,l1 and I DC,21 are DC parts due to the two scattered waves. These are the same parts as the DC parts in the dual-beam technique. They do not contribute substantially to the total DC part of the signal (Fig. 5.37c,d). 2. A negative DC part arises from the superposition of the reference beam with the scattered wave from the reference beam, i.e. with a Gaussian attenuation. 5.2 Laser Doppler Technique Trajectory A: a 237 Trajectory B: b '9 8 500 'C . 'E' '" '"" 0 "" ] :aEi < -500 c '":=; 0.4 ~~ '" 0.2 :&Ei 0.0 < '":=; 'C ". :' ,/1' . :9" lvI! :~··~t " ---_.....,..,.'.,. " I, " .. I, " "' ~, ': ,-~"" -0.2 e 500 0 "" '"" -500 :9 :& -1000 "," Ei < -1500 -20 o 20 -20 Particle position xOp [11m] o 20 Particle position x Op [11m] Fig. 5.37a-f. Signal contributions in a reference-beam arrangement for two different partide trajectories (left side: Zpo =Zop =100llm, right side zpo =zop =-lOOllm) according to Eqs. (5.170)-(5.181) (rw=lOllm, e=7deg, 511 =5.5, 512 =4, ro1=100mm, Yop=O). a,b Signal relevant contributions, c,d Signal parts with minar significance, e,fFull detector signal [De2 CC( QblQll* + QllQbl* ) =2 (5.176) Essentially this corresponds to the energy removed from the reference beam through scattering by the particle. Thus, the total DC part of the signal experiences a negative dip in amplitude as the particle traverses the volume, as shown in Fig. 5.37a,b. Exact computations for this signal can be obtained using FLMT or EGO. Since the negative dip results from interference of the reference beam with the scattered wave from the reference beam, the negative dip of the DC part is shifted with respect to x = 0, such that the minimum amplitude is obtained for the particle position 238 5 Signal Generation in Laser Doppler and Phase Doppler Systems XDCmin =zpo tan~ (5.177) This property of the DC signal part has been exploited to determine also the position of the trajectory. For this technique, two reference-beam systems were operated simultaneously with one detector on each beam (Strunck et al. 1993). The time delay between the signal minimum corresponds to Llt DC = 2x DCmin I v x and the z position of the particle trajectory can be measured by Z z 0 p = Vx Llt 2tan~ (5.178) 3. The essential part of the signal for the velo city measurement is the alternating part arising from interference between the reference beam and the scattered light from beam 2. I AC1 = 2CC( Qb1 Q21* + Qb1* Q21 ) (5.179) This part of the signal is illustrated in Fig. 5.37a,b. It reaches a maximum when the scattered wave from beam 2 reaches its maximum, i.e. when a small particle is at the center of beam 2. The position of the burst maximum is given for small particles (d p < 3Ab) by X ACmax =-zpo tan~ (5.180) Therefore, also this modulated part of the signal is shifted in time compared with the time when the particle is at the position x op = O. The distance on the trajectory between the burst maximum and the DC part minimum is 2z po tan ~, which can be used directly to determine the z component of the particle trajectory (Strunck et al. 1993). If two, symmetrie reference-beam systems are used, i.e. adetector is placed directly in the path of each beam of a laser Doppler system, then the z coordinate of the particle trajectory can be determined using the distance between the two burst maxima, 2XpACmax (Strunck et al. 1993). This is analogous to the distance between the two burst minima, given in Eq. (5.177). 4. A further modulation part of the signal arises from the interference of the two scattered waves (Fig. 5.37c,d). This corresponds to the AC part in the dualbeam technique I Ac2 CC( QllQ21* +.Qll.Q21 * ) =2 (5.181) However, this contribution is significantly smaller in amplitude than the portion given above in (Eq. (5.179». The principle contributions to the signal are summarized as: • DC part ofthe reference beam (Eq. (5.174» • Superposition of reference beam with the reference-beam scattered part (negative DC part, Eq. (5.176» 5.2 Laser Doppler Technique 239 • Superposition of the reference beam and the scattered light from beam 2 (modulated part 1, Eq. (5.179» • Superposition of the reference-beam scattering and the scattered light from bearn 2 (modulated part 2, Eq. (5.181» The total sum of all contributions is shown in Fig. 5.37e,f. Since the reference bearn contributes directly to the modulated signal part, relatively good signal quality can be obtained using small apertures in the x direction and wide apertures in the y direction (e.g. slits or cylindricallenses) in front of the detector. Typically, semiconductor detectors (e.g. PIN diodes) are employed, due to the high light levels involved. A further approach is to use large stand-off distances between the transmitting optics and the measurement volume and between the measurement volume and the detector. This has been exploited byDopheide et al. (1990) for a system used in wind tunnels. The measurement volume of a reference-beam anemometer is a virtual measurement volume, existing only for the detector. It is positioned along the axis of the illuminating bearn. For a detector aligned on beam 1, a measurement volume along beam 2 is perceived. Such a virtual volume is pictured schematically in Fig. 5.38 and computed using EGO in Fig. 5.39. The extent of the measurement volume in a reference-beam system is determined by two factors: first by the centerline amplitude of the illuminating beam and second by the modulation depth of the interference between the reference beam and the scattered wave from beam 2, integrated over the detector surface. Since the field strength of the reference bearn over the detector surface acts as a constant factor, the amplitude-limiting extent of the measurement volume is determined mainly by the amplitude of beam 2, the illuminating beam. The measurement volume dimensions computed for a point-like receiver are illustrated as a function of the intensity ratio within a laser beam in Fig. 5.40. The volume boundaries can be approximated in the x and y coordinate directions using the e- 2 amplitude decay, with ao '" bo '" rw and in the z coordinate direction using co'" 2.53 xl R cos~. For the example pictured in Fig. 5.40, this results in amplitude boundaries of (l R = 13.27 mm) ao '" bo '" 60 ~m and Co '" 33.6 mm. The second limitation of the virtual measurement volume is dictated by the modulation depth of the interference between the reference beam and the scat- ---- Virtual mcasurcmcnt volumc x Rcfcr ncc bcam Photodiode ~~~~z .- - - - - - - -= ~ - --:-----/ e Partide Illuminating bcam Fig. 5.38. Arrangement oflaser beams, receiver and measurement volume 240 5 Signal Generation in Laser Doppler and Phase Doppler Systems ~... IOO "" ~ I I I - I I ~ ...0 Power p la.u·1 49.0 . ... ~ . .... " o Q.. -50 1- ...... .'.-._.• ,- -48.7 Parlicle position zOr II'I11J Fig. 5.39. Virtual measurement volume for a reference-beam arrangement, detector in beam 1, computed using EGO (Borys et al. 1998), (Ab = 852 nm, e = 9 deg, rw = 60 ~m, d p =8~m, m = 1.33, rOl = 700mm, 1j =0.5 mm) - 1 1.5 .... ~o '"::I 'i5 l.': '"E 1.0 ::I Ö > C ., § ::I .,"'" E 0.5 <J .~ '" C2 2 MC'dsurmcnt volume lcnglh relalcd 10 Raylcigh lcnglh 3 <"01',1-1 Fig. 5.40. Measurement volume dimensions as a function of intensity ratio in the illuminatingbeam tered wave from the illuminating beam, integrated over the detector surface (Drain 1980). Since the detector surface is small compared to the distance between the measurement volume and the detector, the amplitude and phase of the beam can be considered constant over the detector surface. However, the 5.2 Laser Doppler Technique 241 scattered wave is a spherical wave, for which the amplitude, the phase and the propagation in the receiving direction will depend on the particle position in beam 2 (Fig. 5.38). The angle a between the reference beam and the scattered light from the illuminated beam depends directly on the position of the particle in the illuminating beam, as shown in Fig. 5.38. Fig. 5.41 illustrates the interference pattern at the detector plane for two different particle positions along the axis of the illuminating beam Z2' These results have been obtained using FLMT computations and for the parameters given in the figure caption. The fringe spacing on the detector surface is given by L1s(a)=~ (5.182) 2sin a 2 and from Fig. 5.42, this can be expressed as a = arctan(_Z-=2~S_in_e_-_J r01 - (5.183) Z2 cose For small angles a this can be simplified to Z2 sine (5.184) a~-"--- and b a E 1.0 E ..'" ~ 0.5 'S 1: c o § 0.0 ~ ,..,o · 0.5 · 1.0 - 1.0 · 0.5 0.0 0.5 1.0 - 1.0 Location on receiver x, [mm l -0.5 0.0 0.5 1.0 Location on receiver x,lmml Fig. 5.41a,b. Interference pattern on the detector plane of a reference-beam system arising from the reference beam and the scattered light from the illuminating beam for two positions of the scattering partide along the axis of the illuminating beam z, (FLMT calculations, A. b = 852 nm, 6J= 9 deg, rw = 65 ~m, r01 = 700 mm). a z, = 1 mm, b z, = 4.5 mm 242 5 Signal Generation in Laser Doppler and Phase Doppler Systems z Partidc Fig. 5.42. Geometry for the interference between the reference and scattered wave on the receiver (5.185) Therefore, not only can the z co ordinate of the particle be determined from the time shift of the signal maximum (Strunck et al. 1993), but also the spatial frequency of the interference pattern at the detector plane can provide this information (Borys et al. 2000a, b). The spatial extent of the virtual measurement volume in a reference-beam system is given by the geometry of the system and the wavelength. For adetector diameter of 21i, constructive interference occurs on the surface of the detector for 21i I Lls::; 0.5, i.e. over an angle A a <_b c - (5.186) 41i This angle (coherence cone, Drain 1980) limits the measurement volume to z < 2 - "01 sina c sin(EhaJ (5.187) For angles a> a c the amplitude decreases (destructive interference) up to complete elimination at 21i I Lls = 1. This corresponds to an angle Ab a d =2a c = 21i (5.188) The amplitude of the modulated signal at the detector is given by an integration of the interference field over the active surface of the detector. Analogous to computing visibility for a spherical particle in an interference field (Drain 1980), the modulation amplitude for a circular aperture is given by (5.189) with 5.2 Laser Doppler Technique r Ij r1z 2 sine As Ab rOl 243 (5.190) ':> =2n-"'21t....!.....c~- Since the function 2/1 (I;) I;; has its first zero at 21j I Lls =1.22, the angles a c and a d according to Eqs. (5.186) and (5.188) are actually larger for a circular aperture. They can be computed as 1.22 Ab a d =2ac = -21j (5.191 ) The Eqs. (5.186) and (5.188) are really only valid for a rectangular receiving aperture. An integration of a rectangular aperture with the area Ar = drxd ry yields an expression similar to Eq. (5.189), namely . 1tdrx Sln- PAC '" 10 drxd ry _--=A~s,­ nd rx As (5.192) These properties of the virtual measurement volume of a reference-beam system can be verified either numerically, using a light scattering computation, or experimentally. In either case, it becomes evident that the influence of the modulation depth dominates the generated signal and that the intensity decrease along the axis of the illurninating beam can be neglected in comparison. Fig. 5.43 shows the measured and computed (EGO) variation of modulation depth as a function of particle position for the measurement volume already pictured in Fig. 5.39 (Borys et al. 1998). The measurements and computations agree weH in terms of the main lobe and side-Iobe positions. According to Eq. b a PAC [mW] 0.08 o ·5 o -10 5 Position z [mm] Fig. 5.43a,b. Position and extent of measurement volume for the reference-beam technique (Borys et al. 1998) (A = 852 nm, rw = 60 ~m, B=9deg, rOl = 700 mm, Ii = 0.5 mm). a Measured values, b Computation with EGO 244 5 Signal Generation in Laser Doppler and Phase Doppler Systems (5.191) the modulation depth vanishes for a d = 0.0596 deg, which, using Eq. (5.187), yields a z position of z = Z 2 cos~ = 4.6 mm. This agrees well with the results presented in Fig. 5.43. 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique For particle sizing, the laser Doppler technique uses a minimum of two detectors, placed at an off-axis angle f/Jr and at the elevation angles of ±VJr (Fig. 5.44). This allows determination of the diameter of spherical, homogeneous particles of dimension significantly larger than the wavelength (d p »Ab)' The diameter can be inferred from both the amplitude and the phase of the resultant signals. The glare points on the surface of the particle effectively sam pie the amplitude and phase of the incident laser beams, thus, both amplitude and phase of the signal contains information about the surface curvature. Scattering orders above reflection depend not only on the size and shape of the particle, but also on the medium. The quantitative relation between amplitude, phase and particle size depends on the relative position of the incident and glare points on the surface of the particle. It is now necessary to derive this expression as a function of the scattering order. The analysis will be carried out for detector 1 (r = 1) in Fig. 5.44. The position for the incident points on the particle surface relative to the origin are given by Eq. (5.4) and for the glare points on the particle surface relative to the origin are given by Eq. (5.5). The signal at the detector is given by Eq. (5.20). For size measurements only the modulated part of the signal is of concern. In principle, laser beams of dif- E, e, Fig. 5.44. Optical arrangement for particle sizing using the laser Doppler technique 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique 245 ferent wavelengths could be used for the size measurement, although a single wavelength will be assumed below. The particle size can be determined on the basis of • either the instantaneous phase shift between modulated signal parts of the two detectors (phase Doppler technique) • or the time shift between the signal maxima on the two detectors (time-shift technique) If the intensity at the two glare points is approximately equal, then only the phase of the modulated signal contains information about particle size. This condition is met when the particle diameter is small with respect to the measurement volume dimensions. A statie analysis of the Doppler signal yields the local phase of the detector signal. This is the basis of the phase Doppler technique. A dynamic signal analysis must consider also the Gaussian intensity distribution of the incident beams (inhomogeneous field). The influence of the beam intensity distribution on the signal amplitude is investigated, which leads direct1y to a time shift of the signals at the detector. This aspect leads to a technique known as the time-shift technique for particle sizing. The time-shift technique does not preclude the simultaneous use of the phase shift. In this respect, it is useful to recall that the phase shift remains when the incident beams are plane waves, while the time shift no longer exists. In Fig. 5.45 the relative amplitude at the incident points of the two beams and for one exemplary positioned receiver is given as a function of the normalized particle position. The ratio of the particle diameter to the beam waist diameter Intensity at incident point of: - - first beam -1 second beam dPIldI ' =0.1 o 2 Normalized particle postion x op I dp [-] Fig. 5.45. Relative intensity at the two incident points as a function of normalized particle position. The ratio of particle diameter to beam waist diameter has been varied (if/,.=üdeg, ~,.=3üdeg, e=2üdeg, N=l) 246 5 Signal Generation in Laser Doppler and Phase Doppler Systems has been varied in the diagram. For very small partieles the amplitude is constant for all partide positions and over the whole partiele diameter. For larger partieles this is only true over smaller ranges. This demarcates the limits between the phase Doppler technique (static analysis of signals) and the time-shift technique (dynamic analysis of the signals). In the static signal analysis the dependencies on partiele diameter are investigated only for partieles positioned at the origin. This analysis fails for larger partieles, because the amplitude decreases. Since for shaped beams the inhomogeneous amplitude is inherently associated with non-plane wavefronts, also the phase cannot be weIl described by the static analysis for large partieles. Therefore the dynamic analysis has to be used for larger partieles. Often a quasi-static analysis can be used for the phase. For the case shown in Fig. 5.45, this could be performed for the normalized partiele position at xO p I dp = -0.48, where the static conditions are always fulfilled. For such a quasi-static analysis the dependencies on system and partiele parameters e.g. receiver location, intersection angle and refractive index have to be known very weIl beforehand and the analysis has to be dynamically adapted for every individual case. In the following analysis the assumptions given in Eqs. (5.23) to (5.26) are made. Furthermore, the wavefront curvatures are neglected and therefore the Doppler frequency does not vary in the signals. The signal amplitude changes are considered. The AC part of the signals is given by the intensities at receiver 1 (see section 5.1.3) 11(t) = 1AC,I (t- tl )cos[ OJ D(t- tl ) + cPI ] +" t~t,)'] oo~ =Im.' exr[ w, (t -t, )+<J>,] , tl = t ACmax,1 (5.193) t2 = t ACmax,2 (5.194) and receiver 2 12(t) = 1AC,2(t- tJcos[ OJ D(t - tJ + cPJ =Im., .• "'{-( v, t~t, )}o.J: w, (1-1,)+ <J>,] , The temporal shift of the bursts Atl2 = -(tl - t 2 ) is given by the Eqs. (5.116) and (5.114) and the phase shift AcP12 = cPI - cP 2 from Eq. (5.109). Both the time shift and the phase shift are functions of the incident points and glare points 0n the surface of the partiele. As such they are also functions of partiele size and shape, relative refractive index m, intersection angle e, detector position rfJr' IfI r and scattering order. The phase shift results from the sampling of the wavefronts, the time shift results from the sampling of the intensity profile of the beam by the incident points. The incident and glare point positions on the partiele surface depend also on the partiele position in the volume, thus with partiele movement the phase difference and time shift will also change (Borys 1996). However, in practice this effect can be almost neglected (see sections 5.1.2.1, 5.1.3.1 and 7.2.4). 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique 247 5.3.1 Determination of Incident and Glare Point Positions Reflection and first-order refraction are the most commonly used scattering orders for the phase Doppler and time-shift technique. The relation between the incident point position, the glare point position and the diameter of a spherical homogenous particle will now be derived for these orders. 5.3.1.1 Reflection (N = 1) For reflection the incident point and the glare points are coincident (5.195) For a spherical particle of radius rp ' the glare point at position (Fig. 5.4) (5.196) is determined by the unit vector of the scattered wave e~~b' using the law of reflection (Fig. 5.46a) ') r'( =r br p (i) Ie (i) p,br - and the incident wave (i) e p br - e w br ' e~~b, , (i) ew,br (5.197) I With a large receiver distance and a particle position near the intersection point of the laser beams e(i) - e - e p,br ~ pr'-'- Or) e(i) "" w,br eb (5.198) Eq. (5.197) yields (5.199) Thus the position of the reflection points on the particle surface are given by the detector position and beam orientation. a Fig. 5.46. Vector relations for scattering from a spherical particle. a Reflection, b Refraction 248 5 Signal Generation in Laser Doppler and Phase Doppler Systems Using the beam vectors of Eq. (5.1) and the detector positions according to Eq. (5.2), the glare point positions for reflection(index br) can be computed as: (;) _ rll J sin lfI r - sin !Ji cos IfIr sin9r [ coslfl r cos9r-cos!Ji rp - r::: ~ ,,2 1- sin IfIr sin!Ji - cos IfIr J - sin lfI r - sin ~ cos IfIr sin 9 r [ (5.200) cos9r cos!Ji coslflrcos9r-cos~ (;) r I2 = rp r::: ,,2~I+sinlflr sin!Ji-coslflr (5.201) cos9r cos!Ji (5.202) [ -sin lfI r + sin ~ coslflr sin9r J (;) coslflr cosr -cos!Ji r 22 = rp r::: ,,2 ~1- sin IfIr sin ~ - cos IfIr cos9r cos~ (5.203 ) 5.3.1.2 First-Order Refraction (N = 2) Using the law of refraction (see section 4.1.1.2) and knowing the unit vectors of the incident beam and the detector position, the primary glare point position can be derived with the assumption e~~b' "" eb and with the help of Fig. 5.46b e(;) br = (g) [ ., (. ., mep,br- m+2slnT br slnTbr-mslnT br )] eb 2 (sin Tbr -msin T' br )[ m +sin T' br (sin T br - m sin T'br )] (5.204) (5.205) The position of the secondary glare points is given as [ • I (. • ,)] m+2Sln T br Sln Tbr -mSln T br (g) ep,br -m e b 2 (sin T br - m sin T' br ) [m + sin T'br (sin T-m sin T' br )] r(g) br Using the law of refraction = r e(g) p br (5.206) (5.207) 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique cos'lbr np nm (5.208) --=-=m cos'l~r 249 (5.209) and the relations U)) e"br' (-e br = .-, SIn" br (5.210) the terms (5.211) sin 'l br = (5.212) can be given explicitly. For plane waves and (f = 1, 2, für» f p (e~~b, "" ep' "" e o,.) Eq. (5.204) now simplifies to b = 1, 2 ) [ m (±sin If/r) - [m + 2 sin 'l'br (sin.'l rb - m sin T'br)] (±sin 1Ji)] m coslf/r SlllyJr m cos If/r cosyJr - [m + 2 sin T'br (sin Tbr - m sin 'l'br)] coslJi . e(d - --'-----,----------=----:-;0----,--------"---,,--"-- 2 (sin Tbr -m sin T'br)[ m + sin T'br (sin Tbr -m sin T'br)] br - (5.213) and (5.206) to (±Sin If/r) [m + 2 sin 'l'br (sin Tbr - m sin T'br)] - m (±sin 1Ji)1 [ [m + 2 sin T'br (sin Tbr - m sin T'br)] coslf/r sin yJr [m + 2 sin T'br (sin 'l br -m sin T'br)] COS If/r cosyJr - m coslJi e(g) br =- " " - - , - - - - - - - - - - - : - ; 0 - - - " - - , - - - - - - - - - - , , - - - " 2 (sin Tbr -m sin 'l'br)[m+sin T'br(sin'lbr -m sin T'br)] (5.214) with sin 'lbr = sin 'l'br = (5.215) (5.216) 250 5 Signal Generation in Laser Doppler and Phase Doppler Systems These relations allow the position of the incident and glare points for firstorder refraction to be computed. Note that with the assumptions given in Eq. (5.198), the Eqs. (5.213) to (5.216) are independent of partide position and therefore the position and time-independent phase r[Jr can be used instead of the instantaneous phase f/J r (see section 5.1.3.1). For symmetrie receivers like in Fig. 5.44, and for large distances to the partide (rpr »dp )' the plane wave case yields additionally e1 ·ri;) =e 2 ·ri2 sin r'll = sin r'22 e1 ·ri~) =e 2 ·ri? (g) - sin 7'12 = sin 7'21 (g) e p,ll • rl! - e p,22 . r 22 ep,12 . r12(g) -- e p,21 . r 21(g) = sin 7 22 sin 712 = sin 7 21 sin 7l! (5.217) Similar relations for higher order incident and glare points can be derived in an analog manner. This is in fact the basis of the EGO light scattering approach. 5.3.2 Phase Doppler Technique Equal amplitudes at both glare points occur only for an approximately homogeneous incident field around the scattering partide or when the glare points are very dose together. This defines the condition for which the static signal analysis is valid. The phase of the detector signal is an instantaneous quantity which changes with time and must therefore be referenced to a particular position of the particle. The principle of the phase Doppler technique lies in positioning the detectors such that the instantaneous phase difference is directly proportional to the diameter of a spherical homogeneous particle. The detectors must be positioned such that light of only one scattering order dominates the signal. For this condition the relation between phase difference of the detector signals Eq. (5.109) + kbeOl' (r 21(g) -rl!(g)) - kb e 02 ' ((g) r 22 -r12(g)) (5.218) + (VFS,21 -VFs,l!)-(VFs,22 - VFS,12) and the partide diameter must be found. This is be carried out for each scattering order, e.g. N = 1 (reflection) and N = 2 (first-order refraction) in the next two sections. For N > 1 the refractive index of the partide medium must also enter the computation. 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique 251 5.3.2.1 Reflection (N = 1) For reflection (N = 1) the primary and secondary glare points are coincident, thus Eq. (5.218) reduces to (Albrecht et al. 1993) (5.219) +k b [e ol ' (r 21(g) (g)) - e - r l1 (g) 02 ' r 22 (g))] - r l2 The phase difference between the glare point and the receiver can be replaced by the phase difference between the particle center and the receiver and the particle center and the incident point. Therefore Eq. (5.219) reduces to ·r(i)-k .r(;j)-(k ·r(i)-k .r(i))] LllP(J)=-LllP(J)=lP(J)-lP(J)=2[(k 12 21 I 2 I 11 2 21 I 12 2 22 (5.220) With these relations for the glare points, the relation between the particle diameter and the phase difference is given by LllP~V = 2rr.,[i dp(~l-coslf/r costPr cos~+sinlf/r sin~ /L b (5.221) -~l-coslf/r costPr cos~-sinlf/r sin~) for symmetrie receiver locations as shown in Fig. 5.44 (tPl = tP2' IJI I = -1JI2)' This relation was given first by Flögel (1981) for the condition of a homogeneous field around the particle. Figure 5.47 shows the detector signal for the reflective mode. The dependence of the phase difference on the particle diameter for reflection is shown in Fig. 5.48. 5.3.2.2 First-Order Refraction (N =2) Most phase Doppler systems use first-order refraction (N = 2) scattering. The contribution to the phase difference arising from the light passage through the particles (Fig. 5.46b) is given as 4rrm. If/S,br = - kb m r"br =--/L-fp SIll T br I (5.222) b so that Eq. (5.218) for first-order refraction becomes LllP~;) = - LllP~~) = lP~2) _ lP~2) = k (rl(:) -rl(~)) - k 2 • (ri:) j • ri;)) + k b [ e OI • (rif) -rl(p)- e 02 . (rit) - + k b m(r,.11 - r,,21 - f,,12 + f,,22) rin] (5.223 ) 252 5 Signal Generation in Laser Doppler and Phase Doppler Systems LldJ(2)=dJ(2)_dJ(2)=k [e .(r(O-r(O)-e 12 1 2 bIll 12 2.(r(i)-r(O)] 21 22 +kb [e 01 ' (r 21(g) -r11(g)) -e 02 ' ((g) r 22 -r12(g))] (5.224) m p (' , . -' . , . ') +41t-r sln'Z'1l-sm'21-s1n'Z'12+s1n'Z'22 Ab With these relations, together with Eqs. (5.213)-(5.217), some algebraic manipulation leads to a relation between phase shift and particle diameter - l+m -m.[i ~1-sinlf/r sin~+coslf/r cos9r cos~ ) 2 (5.225) This relation was also first given by Flögel (1981). A similar procedure can be used to derive the relations for higher order scattering modes. For the refractive mode, simulated signals are shown in Fig. 5.47 and the phase diameter relation in Fig. 5.48. 5.3.2.3 Phase Conversion Factors In summary, the approximations used to deriveEqs. (5.221) and (5.225) are as follows: • The incident waves can be considered plane around the particle (e~~br = eb ). • The distance to the detector is much larger than the particle diameter (ror » d p , e~~l, '" ep,)' • The detectors are positioned symmetrically about the y-z plane (lf/l = -If/ 2' 91 = 92)' • The particle is symmetrically positioned with respect to the detectors, i.e. near the coordinate origin (e pr =e or )' Und er these conditions the position and time-independent phases on each receiver are half the phase difference A,ffi(N) ,ffi( N) _ _LJ_'#_1_2_ ':PI , 2 A,ffi(N) ,ffi(N) _ _ _ LJ_'#_12_ '#2 - (5.226) 2 For a configuration with 19= 7deg, 9r = 30deg, If/r = 3deg, signals for reflection (N = 1) and first-order refraction (N = 2) have been simulated and shown in Fig. 5.47. For reflection and for a positive velocity (v x > 0), the phase difference LldJ\Y is positive (LldJW > 0, LldJW < 0), meaning the signal at detector 1 leads the signal at detector 2 (Fig. 5.47a). For refraction, the phase difference LldJW is negative (LldJg) <0, LldJi~) >0), i.e. the signal at detector llags the signal at detector 2 (Fig. 5.47b). The relation between particle diameter and phase difference (Eqs. (5.221), (5.225» is shown in Fig. 5.48 for reflection and for firstorder refraction. 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique b a - 253 1.0 ~ Reflection -;::/ ; (N= 1) 0.5 .. ,," ': ,' , (0 t:: b.O Vi ' .., First-order Receiver 1 Receiver 2 ,, ,, ,, ,, ", 0.0 , refraction (N= 2) " ,," ,, I, I, ' , ,, , ,' , : ' ,, .,, ,, ,, , ,, , ,, -0.5 ,' ,, ,, '' , ,' " " :',, """, . -1.0 -20 -10 0 -10 10 0 10 20 Time t [flS] Time t [flS] Fig. 5.47a,b. Example of the modulated part of phase Doppler signals (e= 7 deg, Ab =488nm, d w =20J..Lm, dp =6J..Lm, m=1.33, v p =lms- 1 , fjJ,=30deg, 1f/,=3deg). a Reflection, b First-order refraction 0:0 " ~ 270 SOl '1 " "... ~ 180 u t:: '-'-' - - Reflection (N =1) 90 '6 ~ ..c: '" A., 0 ............ -90 ---'- .. .................. - ------ First-order refraction (N= 2) ... - ...... ............ --- --- --- ........... --- ............ -180 "-- ... ........... .......... -- -........... -270 0 10 20 30 Particle diameter dp [flm] Fig. 5.48. Example of phase difference as a function of particlc diameter computed using geometrical optics from Eqs. (5.221) and (5.225) (e=7dcg, A b =488nm, m=1.33, fjJ, = 30 deg, If/, = ±3 deg) The particle diameter using the phase Doppler technique follows from Eqs. (5.221) and (5.225) for reflection and first-order refraction respectively. The constant giving the slope between phase difference and particle diameter and including the system dependent parameters is normally combined in the phase conversion factor 254 5 Signal Generation in Laser Doppler and Phase Doppler Systems (5.227) with F~l) = :~ [J2( ~1- cos V/r cosf/J rCos'J'i + sin V/r sin 'J'i (5.228) -~l-cosV/r cosf/Jr cos'J'i-sinV/r sin'J'i)r for reflection and F~2) = :~ [2( 1+m 2- m J2 ~1 + sin V/r sin'J'i + cos V/r cosf/Jr cos'J'i - l+m 2 -mJ2 ~l-sinV/r sin'J'i+cosV/r cosf/Jr cos'J'i )r (5.229) for first-order refraction. The particle size can be calculated by multiplying the measured phase difference with the phase conversion factor. For illustration, the phase conversion factor is always positive, therefore the magnitude of the phase difference can been used to obtain the particle diameter. (5.230) 5.3.3 Reference Phase Doppler Technique As in the reference-beam laser Doppler technique, the detectors of a phase Doppler system can also be used as reference-beam detectors in the laser beams. For such a reference phase Doppler system, illustrated in Fig. 5.49, the phase difference I diameter relations from Eqs. (5.221) and (5.225) are not valid because the e< E., Beam I k, Fig. 5.49. Reference-beam phase Doppler arrangement 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique 255 signal is generated by the interference of only one scattering wave and the reference beam .. By using the reference detectors in the illuminating beams, the dominant scattering order is first-order refraction for transparent particles. For opaque particles reflection dominates in forward scatter. Phase difference I diameter relations for reflection and first-order refraction will now be derived. 5.3.3.1 Reflection (N =1) The AC part of the Doppler signal on receiver r arises from the superposition of the reference-beam (b = r) field strength g~r) = Eb (ror> cos[ (Ob t + \Vb (ro,)J, with the scattered wave from the other beam (r (r) E _br - E(i) br exp[.J ( (Ob t + \V(i) br - k b =r (5.231) * b) (g))] b rp,br (5.232) , The term \Vb(rOr ) in Eq. (5.231) gives the phase ofthe reference beam b at the receiver r and \Vb~) in Eq. (5.232) is the phase of the illuminating (object) beam at the incident or glare point (r6:~, = r6,1~ for reflection), as illustrated in Fig. 5.50. Both phases can be computed using the equation for a laser beam, Eq. (3.59) and a transformation between beam and main coordinate system as given in Eq. (5.7). The AC part of the intensity at receiver r =1, which is relevant for velocity and diameter measurements, is then given by (Eq. (5.179» (i) I 1 - E1 (r m )E 21(i) cos( -\VI (r m) + \V21 k b rp,21 (g)) (5.233) (g)) + k b rp.12 (5.234) - and at receiver r = 2 by I2 - E2 ( r 02 )E 12(i) cos ( \V 2 (r 02) - (i) \V12 x BC81ll2 objcCI bca 111 k,~ z Bcam 1 rcfcren.:e beam Fig. 5.50. Signal generation with the reference phase Doppler for reflection 256 5 Signal Generation in Laser Doppler and Phase Doppler Systems The phase difference between the two signals is given as LldJ(1) 12 = dJ(1)1 - dJ(1) 2 (5.235) The phase lf/b: l of the object beam b at the incident and glare points rci:l, can be approximated by = rci,i: (5.236) with lf/ Ob being the phase ofbeam at the origin. The change of phase between the glare point from beam band receiver fis given by ( )J2 l-~f(g).e + ( fo,tr Q,br Or fOr fOr , (5.237) which for fOr »f~,t; reduces to (g) k b fp,br '" 2n ( T fOr - fO,br . eOr (g) ) (5.238) b The absolute glare point position can be replaced by the particle position and the glare point position relative to the particle center. Because the receivers are located in the laser beams, the wavenumber and the unit vectors to the receivers can be substituted by the wave vector of the beams, which yields for the two receivers locations (5.239) For the condition fOr »lRb = nf; lAb (Rayleigh length), the phase of the reference beam at the receiver becomes lf/b(fOr )'" lf/ob n -k b ' fOr +-, 2 b= f (5.240) where lf/ Ob is the phase of the reference beam at the origin. Substituting Eqs. (5.236), (5.239) and (5.240) into Eq. (5.235) and setting r~:l = rH l leads to the phase difference between the signals of reflected light from a particle centered in the intersection volume (rop = 0, r6:11 = ri;!, r6:12 = rl(p) (5.241) Due to symmetry (9r = 0, lf/ r = ±%) , the relations . e/ kbeOI . r21U) -- kbe02 . f12(;) -- -k 1. fl2(;) -- -k 2 . f 21(;) -- k bfp sIn /2 (5.242) hold, thus (5.243) 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique 257 where L1x is the usual fringe spacing in the virtual interference field. This signal generation is physically pictured in 5.51. The interaction points for reflection sampie the virtual interference fields of the two detectors at different positions. Due to symmetry arguments, the separation of the interaction points corresponds direct1y to the diameter of the particle. The ratio of the partide diameter to the fringe spacing gives the phase difference between Doppler signals in multiples of aperiod. In fact, for the case considered here (f/Jr = 0, IfI r =±%), the path-Iength difference between the reference beam and the scattered wave from the object beam is equal to the difference between the two waves scattered by the partide in the object beam. Thus, the expression for phase difference in a standard phase Doppler arrangement (Eq. (5.221)) will give the same diameter dependent term in Eq. (5.243). The additional shift of 1t accounts for the fact that a 1t /2 phase shift between the scattered wave and the reference beam arises in the far field of a focused laser beam (ror »l Rb)' For a homogeneous reference beam, this term would vanish. Examples of reference phase Doppler signals in reflection mode are given in Fig. 5.54a,c. x Virlual callered wavc from bcam 2 10 receiver I m~dsuremenl Re.:eiver I z callcred wavc from beam I 10 receiver 2 volum of re.:eivcr 2 Receiver 2 Fig. 5.51. Interpretation of the phase difference genera ted in reflection mode in the reference phase Doppler system 5.3.3.2 First-Order Refraction (N = 2) The computation of the phase difference between signals on the two detectors arising from first-order refractive scattering differs from the reflective case only in the additional phase term I (ill_ 21t - -21t (g) k brnr"br r n r br -rbr Ab --rn Ab d p Sin . T, br , bt:r (5.244) which accounts for the phase change when the scattered wave traverses through the partide as pictured in Fig. 5.52. The additional phase change can also be considered in Eq. (5.235) and for first-order refraction the phase difference can then be written as 258 5 Signal Generation in Laser Doppler and Phase Doppler Systems Receiver I x Beam 2 objecl bearn z Beam I rderencc bcam Fig. 5.52. Signal generation with the reference phase Doppler for first-order refraction L1IP~;) = IP~2) = - IPi2) -VI (rOI ) - V2 (rQ2) + v~i + v~l- kb(r;~il + r;~~) - kbm(r',21 + r"IJ (5.245) For a particle centered in the illuminated volume (rop = 0) the simplification used in Eqs. (5.236) and (5.239) can also be applied to Eq. (5.245) L1IPg) '" kbe OI ' rit + kbe 02 • rl~) - k l . ri~) - kbmdp(sin r;l + sin r;J-n - k ri;) 2 . (5.246) U sing the symmetry properties of the optical arrangement, this can be simplified to (5.247) For small e, Eq. (5.247) just corresponds to I I (2) 2n L1IP12 ",--d2m-I-n A. p (5.248) b Also this result can be interpreted in a very physical manner, as illustrated in Fig. 5.53. As the interseetion angle becomes small, the paths of the two refracted waves and the phase of the reference beams all become very similar. The particle-size dependent term in Eq. (5.247) just corresponds to the optical pathlength difference between the reference beam and the scaUered beam in traversing through the particle. The additional phase shift of n again arises due to the nl2 phase difference in theJar field ofa focused beam (rOr »lRb). This term vanishes for a homogeneous reference beam. 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique x 259 ScaHered wave (rom beam 2 10 receiver I z cat tcred "'ave from bcam I to receiver 2 Fig. 5.53. Physical interpretation of the phase difference between reference phase Doppler signals arising from fIrst-order refraction 5.3.3.3 Phase Conversion Factors Example signals of a reference phase Doppler system for reflection and firstorder refraction and two different particle trajectories are illustrated in Fig. 5.54. Figure 5.55 shows an example of the phase difference/diameter relation of a reference phase Doppler system. Comparing Eq. (5.247) with Eq. (5.225) reveals a significantly higher size resolution than a conventional phase Doppler system (Strunck et al. 1994). A further investigation of reference phase Doppler characteristics is given in section 8.2.6. The phase conversion factor for a reference phase Doppler system operating in reflection mode can therefore be calculated from the phase conversion factor of a standard phase Doppler system operating also in reflection mode but with the reference system receiver locations. An additional phase difference offset of 180 deg is necessary. For the refraction mode the phase conversion factor differs from that of a standard phase Doppler configuration and is unique for the reference configuration. dp -O) P<PR _ - p(N)(LJ,Q>(N) <PR 12 LJ,x 2n , + 180 deg) (5.249) (5.250) 5.3.4 Time-Shift Technique The optical configuration of a system measuring the particle diameter with the time-shift technique is similar to the standard phase Doppler configuration as shown in Fig. 5.44. For the dynamic analysis of the time-shift technique, plane wavefronts and a Gaussian beam intensity profile, as used in section 5.1.3, will be assumed. In contrast to the phase difference, the time shift of the signal is dependent on the direction of the particle trajectory through the measurement volume also for 5 Signal Generation in Laser Doppler and Phase Doppler Systems 260 b a .,=! Rcllcction ::;, '" "0 :l "i:O:. E "' "0:c.o Vi ~ ~-r.-,,~-..-,,-r.-,,~~ ., Rcllcct io n ;cop - -60 f.1m - - Receiver 1 ------- Receiver 2 o -40 Time I o 40-40 [f.1sJ 40 Time I If.1s I Fig. 5.54a,b. Example of the modulated part of reference phase Doppler signals (e=18.9deg, Ab =852nm, dwb=40l..tm, d p=20J..lm, m=1.328, vp=lms· I ). a Reflection Zop = 0 J..lm, b First-order refraction zop = 0 J..lm, c Reflection zop = -60 J..lm, b First-order refraction zOp = 30 J..lm the case of plane wavefronts (see section 5.1.3.2). If all three velocity components were measured, v p' rp v and 13 v' the trajectory could be determined (Eqs. (5.58) and (5.59». The time shift of the signals is based on the displacement of the virtual detection volumes. It can be determined by the times t ACmaxl and tACmax2 for the signal maxima on the receivers 1 and 2 (Eqs. (5.114)-(5.116». L1 (N) _ (N) t 12 - - tACmaxl L1 (N) (N»)_ x 12 _ -tACmax2 - - - - vx (N) (N) X pACmax,1 - X pACmax,2 (5.251) Vx The negative sign in the definition of the time-shift is used to guarantee that the time-shift and the phase-difference are both positive for reflection and both negative for first-order refraction. A conversion of the phase and time conversion factors can then be performed without respect to the sign. 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique 261 ~ 3600"".-"".-"".-"""""".-""".-"".-"""-.",,, :S >e.~ 1800 '1 ~ ... <lJ o ~ ................... ~ -1800 '" - - Retlection (N = 1) --- -- --- ................. 1l ~ -3600 -- -- ------ First-order refraction (N= 2) ----------- -- ...................... -5400 ............................. -7200 --------------- 5 15 10 20 25 Particle diameter dp [firn] Fig. 5.55. Phase difference as a function of particle diameter computed using geometrical optics fromEqs. (5.243) and (5.247) (19=18.9deg, Ab =852nm, m=1.328) For arbitrary trajectories the time shift is given with Eqs. (5.101) and (5.102) by L1t 12 =-tACmax,l +tACmax ,2 (5.252) +[ vp ~1+m2y +m z2 [1+m o/z+4)] o/z 2 z tan 2 cos For small intersection angles, i.e. for a slight change of intensity along the z dimension of the detection volume, the influence of the V z component of the particle velo city can be neglected. The time shift is then given by 12 = -tACmax,l + t ACmax,2 = L1t Xl -X 2 +(Yl - yJm (2) Vx y (5.253) l+m y For particle trajectories in the x direction (v p = v xex) Eq. (5.252) reduces to (5.254) 262 5 Signal Generation in Laser Doppler and Phase Doppler Systems If the position of the incident points for reflection, Eqs. (5.200)-(5.203) and first-order refraction, Eqs. (5.214)-(5.216) are inserted into Eqs. (5.252) to (5.254), relations between particle diameter and time shift can be obtained. 5.3.4.1 Reflection (N = 1) For reflection and particle trajectories in the x direction, the relation between time shift and particle diameter is obtained as (Albrecht et al. 1994, Borys 1996) LU(1) 12 =_ L1x~;) =~[ Vx 2J2vx coslf/r cos9r tanr:%+sinlf/r ~1-COSlf/r cos9r cosr:%+ sinlf/r sinr:% cos If/r cos9 r tan r:% - sin If/r (5.255) ] ~1-coslf/r cos9 r cosr:%-sinlf/r sinr:% 5.3.4.2 First-Order Refraction (N =2) For refraction and particle trajectories in the x direction the relation between particle diameter and time shift becomes (Albrecht et al. 1996, Borys 1996) A (2) d [ L1t(2) __ ~_~ 12 - Vx -2J2vx C,-Sllllf/r . ~l+m2-m~2(l+Cc+C,)~l+Cc+Cs (5.256) C, Hll'!V, J with Ce = cos ljI r cos9 r cosr:% , Cs = sin ljI r sin r:% , C, = cos ljI r cos9r tan r:% (5.257) In contrast, with the phase difference technique, which exhibits a 21t ambiguity with increasing particle size, the time-shift technique displays no similar limitations. Thus, there exist no restrictions on the choice of larger detector elevation angles. 5.3.4.3 Time Conversion Factors As for the phase Doppler technique, time-shift conversion factors can be defined for converting the measured time shift into a particle diameter for particle trajectories in the x direction. d = L1t(N) p(N) p with 12 T (5.258) 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique p,0l =2.J2v ( T x coslf/r cosrfJr 263 tan~+sinlf/r ~l-coslf/r cosrfJr cos~+sinlf/r sin~ ==J- ----r==c=0=s=lf/=r=c=0=s=rfJ=r=t=an==!Ji=2=-=s=in=lf/===r ~l-COSlf/r cosrfJr cos~-sinlVr sin~ (5.259) 1 for reflection and (5.260) C, + sin IV r ~~r=1+=m==:=2-=m=;~2=(~1+=c=c~-~C~,)c~--;:l=+=Cc=-=C=, )-1 for first-order refraction. Note that the conversion factors are independent of the wavelength of the incident beams, which is not the case for the phase Doppler technique. Für symmetric receiver locations (lVI = -IV 2' rfJl = rfJ2) the time shift of each signal relative to the particle position at x op = 0 is half the full time shift between both receivers (Nl Llt;~l t ACmax,2 = 2- (5.261) while still considering the sign convention of the time shift. An example relation between time shift and particle diameter computed using geometrical optics is pictured in Fig. 5.57, For positive particle velocities (v x >0), the time shift Lltlfl =-(t~21 -t::'~2) is positive for reflection (Lltj1l >0, LltW < 0) and is negative for refraction (Lltg l < 0, Llt~il > 0). The signals received on the detectors are givenby Eqs. (5.193) and (5.194). Generally more than one scattering mode is present at the detectors at the same time. The beam amplitude proflle could lead to a time dependent selection of detected scattering order, depending on the trajectory of the particle. Such effects are largely suppressed in the phase Doppler technique by choosing detector positions at which one scattering mode is always dorninating. If more than one scattering order/mode is present, the detected signal may exhibit fractional signals. An example and application of such signals is given in sections 8.2.4 and 9.2. The measurement volume displacement as a basis of the time-shift technique is not only in the x direction, but also in the y and z directions. Computations of the volume displacement using geometrical optics are shown in Fig. 5.58a. These computations have been performed for both reflection and first-order refraction. Note that the volume displacement is opposite to the time-shift, whose sign is defined in relation to the phase difference. For comparison, similar computations have been performed using FLMT and are also displayed in Fig. 5.58b. At 264 - 5 Signal Generation in Laser Doppler and Phase Doppler Systems a ~ 1.0 ';::::,E b Retlection "" I, ';::::, '"c: Receiver 1 Receiver 2 , (N= 1) 0.5 t Oll efraction (N= 2) 11 I, •• 11 I, I, " I I 11 I. I ,. " I, I I'" r;; " I I I I 0.0 '1 " '. I I, I I I I \ I' 11 .'1. I. 111 1 , 1 1 , " I I I I I • I, " 1 1 11,1 " 1 1 11,1 " 11,1 " " .' 11 " I, 'I 11 " • 11 II " " 11 -0.5 11,. 11 I I I I I I 11,1 11 " 1I 11 " " U :: ~ I, -1.0 -40 -20 o ::'1,1:: ~ 11 " I, " 11 " ~ ~ : I -60 :! !: :: :: :,:::: 20 40 60 -60 -40 o -20 20 40 60 Time t [fls] Time t [fls] Fig. 5.56a,b. Example of the modula ted part of time-shift signals ((9 = 7 deg, Ab = 488 nm, d wb = 20 flm, d p = 40 flm, m = 1.33, v p = 1 m S-l, 9, = 30deg, /fI, = 20 deg). a Reflection, b First-order refraction - - Retlection (N = 1) o - .............. - ". " . ... - ..... ............ ". '" ............ First-order refracion (N = 2) - ......... " . ... - ...... '" '" '" '" " . - ......... - ....... " . -'. Partic1e diameter dp [flm] Fig. 5.57. Time shift as a function of particle diameter computed using geometrical optics from Eqs. (5.255)-(5.257) ((9=7deg, 9,. =30deg, /fI, =±20deg, m =1.33, V x =1 ms- 1 ) small diameters, some additional scattering orders introduce small oscillations into the relation; however, these would be smoothed out with a finite detector size.Such oscillations, which disturb the linear relation between phase difference and particle diameter, can also be expected for the phase difference if more than one scattering order contributes to the signal. An extensive discussion of this effect in connection with the phase Doppler system is given in chapter 8. 5.3 Particle Sizing with Phase Doppler and Time-Shift Technique iE a 15,,-r,-,,-r'-rT-r'-rT-r'-rT-r~ 265 b :::1. RC(]CCli o n --01.--- 0---e-- Y"'Cmo (I) Z Aemlln - 150~~~~~~~~~~~1~5~~~~ 5 Parliclc diamCIcr d p IlJml First-ordcr refra ction (I) x ACrmr.1' (\) ~ --~ 10 (I) xACmu (2) YACma (2) 2ACrn4.~ 15 20 Particlc diamcter dp IlJm [ Fig. 5.58a,b. Detection volume shifts (e =18.4 deg, 9, = 30 deg, If/, = ±10 deg, m =1.08). a Reflection and first-order refraction computed using geometrical optics, b Computed using geometrical optics and Fourier Lorenz-Mie theory for dominant refraction Physically the scattered interference field moves across the detector surface according to the particle movement, as pictured in Fig. 5.59 (see also the Moire picture Fig. 5.64). The refractive mode leads to movement of the fringes in the direction of the particle and the reflection mode leads to movement in the opposite direction. This fringe movement can be experimentally visualized (Schöne 1993). Fig. 5.59. Interference fringe movement in the plane 9, = 30 deg for reflection and firstorder refraction for a particle moving in positive x direction. The white lines are added as reference lines at a constant angular direction 266 5 Signal Generation in Laser Doppler and Phase Doppler Systems 5.4 Refractive Index Determination The phase difference and time-shift for reflection depends only on the particle diameter for a given optical configuration. With this known particle diameter and an additional phase difference or time-shift measurement from a higher scattering order, e.g. first-order refraction, Eqs. (5.225) or (5.256) can be solved for the refractive index. Such measurements demand that each phase-difference and time-shift ofthe different scattering orders can be determined. One solution could be to use one phase Doppler system in a scattering region of dominant reflection and the second one in a region of dominant refraction. However, due to the complexity ofthe scattering function (see section 4.1), both scattering orders are seldom dominant in different scattering angle regions. A second solution is to separate the scattering orders inside of one signal. Because the sign ofthe time shifts ofreflection and first-order refraction are opposite, both signal parts can be separated temporally by using measurement volurne diameters smaller than the particle diameter. For the system in Fig. 5.44 and a main flow direction parallel to the x axis, receiver 1 detects first reflection and than first-order refraction. For receiver 2 the sequence of the scattering orders are reversed. Therefore two time-shift measurements can be performed. With the known particle diameter from reflection, the refractive index of the particle can be determined from the first-order refraction measurement ~ ~cos'l'r cosrjJr cos'% + 1 +~w+ cos'l'r cosrjJr cos,%-l m "" v 2~---------'-------'------2-w (5.262) with • 2 w=2d2 sm 'l'r P(VxLltg))\l+ cos'l'r cosrjJr cos'%) (5.263 ) By using a shifted phase Doppler system, the main flow direction of the system in Fig. 5.44 could also be parallel to the y axis. Both receivers are located at the same off-axis angle and therefore both receivers detect first reflection and in the second signal part first-order refraction. The like scattering orders appear at the same time at both receivers and therefore temporal separated phase difference measurements for reflection and first-order refraction can be performed. By again using reflection for particle diameter determination, the refractive index can be extracted from the second phase difference (Onofri et al. 1994) m"" J2 ~cos,%cos'l'r cosrjJr + 1 +~2w+ cos'% cos 'l'r cosrjJr-1 2 1-w (5.264) with (5.265) 5.5 Maire Models 267 Both techniques are based on the temporal separation of at least two different scattering orders and are therefore known as dual-burst techniques. A third possibility to deterrnine the refractive index is the extended phase Doppler technique (Durst and Naqwi 1990). For this, two two-detector phase Doppler systems with different off-axis angles are used. The receivers are positioned such that first-order refraction dominates. The ratio of the two independent phase difference measurements is no longer a function of the partide diameter and therefore only a function of the refractive index .(j(P~) AlP~) "" sin If/A sin If/B The equation can be solved for the refractive index (5.267) A =(AlPA Sin1f/B)2 JA ,p AlP B sinlf/A JB Important to note is that the extended phase Doppler technique can operate with plane waves and only one dominant scattering order (N > 1) in comparison to the dual-burst techniques. However, it requires an additional two-detector phase Doppler receiving optics. A more detailed analysis of the dual-burst techniques and the extended phaseDoppler method is given in sections 8.2.4 and 8.2.5. 5.5 Moire Models Moire fringes can be used quite effectively to illustrate the working principles of both the laser Doppler and the phase Doppler measurement principles (Durst and Stevensen 1975). For this purpose, the description of the laser beam phase fronts as given in Eq. (3.59) and Fig. 3.5 and thescattered wave with Doppler shift as given in the Eqs. (3.91)-(3.93) and in Fig. 3.7 are used. The Moire models allow the phase relationships to be visualized. In Fig. 5.60 two modeled laser beams intersect at their waists. Horizontal interference fringes are observed in the intersection area, corresponding to the energy density fluctuations in space, which are sampled by small scattering partides as they pass through the volume. If the intersection angle is varied then the fringe spacing and the fringe number varies correspondingly. The Moire model of interference is also appropriate to demonstrate systematic errors arising from a poor layout of the transmitting optics when the two waists of the laser beams are not located at the intersection point (Hanson 1978). This is illustrated in the smaller images in Fig. 5.60 and further discussed in sections 5.1.2.1 and 7.2.4. 268 5 Signal Generation in Laser Doppler and Phase Doppler Systems Misalignment: Opposite waist dislocation, Fringe distance variation in x direction Misalignment: waist dislocation, Fringe distance variation in z direction Fig. 5.60. Interference pattern observed in the interseetion volume of two laser beams for a perfectly aligned system and two misaligned systems If a scattering center moves with the velocity v p through the interference field, then the partide acts as a source for scattered waves, whose frequency and wavelength variation is given by Eq. (2.3), depending on whether the detector and source are moving towards each other or away from each other. For small partides, the middle of the partide can be considered the source of the scattered wave, whose amplitude is modulated by the Doppler frequency. In the case of two illuminating beams from different directions, the Doppler frequencies detected by a receiver are slightly different. A superposition of the scattered waves yields a visualization of the Doppler difference method, Fig. 5.61. If the two scattered waves are temporally moved relative to one another, then the signal generation can be demonstrated. For small partides, the interference fringes shown in Fig. 5.60 are the source of the scattered wave and the waves shown in Fig. 5.61 are the result of the scattering. Figure 5.62 illustrates the reference-beam operation, in which the scattered wave is superimposed with one of the laser beams. It is dear from this representation, that there are constraints on the positioning of the detector surface within the region of coherence (see section 5.2.2). Shifting the two patterns over one another again illustrates the signal generation and the wave generated from the two patterns moves towards the detector. For large particles, the interference model is no longer valid. In this case individual interaction points on the partide surface can be distinguished for each la- 5.5 Moire Models 269 In.:idenl h"lInis Wavc1cnglh of Ihc wlIve wilh Ih" frcqucn.:y diITerem;c Fig. 5.61. Superposition of scattered waves from a small particle (Doppler difference method) S.:allcrcd wav Fig. 5.62. Superposition of a scattered wave with one of the laser beams (reference-beam method) ser beam. Their position depends on the position of the detector as weIl as on the dominating scattering order, typically reflection or first-order refraction. The superposition of the two scattered waves from the same scattering order now takes place on the detector surface. Figure 5.63 illustrates two glare points on the surface of a large particle, arising from e.g. first-order refraction. These two points are then treated as the source of two scattered waves, which generate interference fringes in space. 270 5 Signal Generation in Laser Doppler and Phase Doppler Systems Small partidc Largc paniclc Fig. 5.64. Superposition of two scattered waves with different source points (glare points) for illustration of size dependent phase difference between two spatially separated receivers Because the positions of the glare points depend on the scattering angle, the source points of the scattered waves depend on receiver location. The interference fringe system in Fig. 5.63 is therefore only valid in a sm all angular region indicated by the white lines. If the partide moves, the two scattered waves emitted from the glare points have slightly different frequencies. The more complex Moire model, illustrated in Fig. 5.64 is in practice only valid in the direction of observation, again indicated by the two white lines. For a receiver with a different location the glare point positions change and a different Moire model appears. If the two scattered Fig. 5.63. Superposition of two Doppler shifted scattered waves with slightly different frequencies from a large particle (phase Doppler technique) 5.5 Moire Models 271 waves are moved relative to one another, then the fringes move across the surface of the detector and generate the modulated part of the phase Doppler and laser Doppler signal. 274 6 Signal Detection, Processing and Validation Technique/ Device Analog spcctrum analyzer I Domain IRemarks Spectral Very inefficientbut a\'ailablc atthe lime of first laser Dopplcr systems Tracker (Phase Spcclral Opcralcs allow SNR or [requency bul requires high dala ra le (e.g. liquid nows) lockcd 1001') f'eriod liming dcviccs (Counter) Time BurSI/Real time speclru m anal)"lcr Speclral Exploils roDuslness of I'I'T and benefits from spcclral dornain ana lysi AuIO corrcla tion I Au IOcova rian..:c Corrcla- Exh ib ils similer bcncfits 10 bursl ti n spcclral analyzers Quadrature demodu lalion Time Model parameter eSlimalion I Reference I Iigh Da ndwidlh out sen ilive 10 NR. Widcspread use inl o 1990s Signa l phase information is more n."adily availablc but rcqu irc highcr SNR lmplcmcnlcd in and operalc Correla- on digilized burst lion. Time I Spcctra l Deighton and eylcs (1971) Ladi ng (1987), Meyers end lemons (t 987), Ibrahim and Bachalo (1992) Ladin g and Andcr cn (1988), Na kajima and lkcda (1990), Jensen (1992) Agnlwal (1984), Czars kc 1'1 al. (1993), Müller el al. CI 994) Nobach (1999) : SOftwdfC Fig. 6.1. Chronological sequence of major signal processing developments for laser Doppler and phase Doppler systems rameters from the statistical functions, e.g. power spectral density, according to a given algorithm. The iterative procedures optimize the estimated parameters further by minimizing the difference between the model signal and the measured signal. This optimization can be performed in time, correlation or spectral domain. Iterative methods of signal processing are presently not widely used; however, their potential is large and the growing speed and flexibility of digital signal processors will enable these methods to be implemented efficiently. Such methods will be discussed in more detail in section 6.5. 6.1 ReviewofSome Fundamentals 275 ignal Proce sing I\lean, variancc Au tocorrclalion funetion Power peet ral densit y Correlalion domain I'requency domain Frcqucncy Phase Arriva l time Residen e time Amplitude Fig. 6.2. Classification of signal processing methods As seen from this preliminary discussion, most contemporary processors are based either on the power spectral density (PSD) or on the autocorrelation function (ACF) of the Doppler signal. Therefore, a brief review of spectral analysis and so me basic fundamentals of signal processing will be given before continuing with adescription of specific processor concepts. 6.1 Review of Some Fundamentals Spectral analysis in laser anemometry typically refers to analysis using the discrete Fourier transform (DFT), although not exclusively. Other transforms have been employed, including Walsh, Wigner, Hilbert or wavelet analysis; however, to date these are not widespread in commercial instruments. Furthermore, all spectral analysis is now performed digitally, using between 1 and 8 bits for amplitude resolution. There are some principallimitations to processing sampled data with I-bit resolution (H0st-Madson and Andersen, 1995), especially at lower SNR (Domnick et al. 1988, Ibrahim et al. 1991); however, in practice these are not too severe. Indeed the induced errors are usually smaller than from other sources in the system and it is probably more important to concentrate on maximizing the number of sampies in the signal (Ibrahim et al. 1994). Moreover, the speed advantage of using fewer bits is rapidly diminishing with the introduction of faster hardware components. The following sections provide an introduction to signal processing fundamentals for both spectral analysis and for estimation of statistical parameters from signals. 276 6 Signal Detection, Processing and Validation 6.1.1 Oiscrete Fourier Transform (OFT) The DFT of a finite series of eomplex values ~n = x(t = n At,) (n = O,I, ... ,N -1), sampled at equal time intervals and over the time duration 0::::; t < T = NAt s ' is defined as k=O,I, ... ,(N-l) (6.1) and its inverse transform as n=O,I, ... ,(N-l) (6.2) where n is the data sampie index at time intervals of Ats and with the eorresponding sampie frequeney of is. The speetral coefficients are computed for the equally spaeed frequencies given by fk =_k- = k is, k = 0, 1, ... , (N -1) NAt s N (6.3) The frequency spacing of the resulting Fourier eoeffieients is therefore (6.4) This is also the lowest frequeney that ean be resolved. Note that the eapital and smaliletter notation will be used for frequeney and time domain respeetively. The Fourier transform yields eomplex speetral values. The real part is associated with the eosine funetion and the imaginary part with the sine funetion. Thus, the real part represents contributions to the signal whieh are symmetrie about zero and the imaginary part deseribes the asymmetrie contributions. The power speetral density (PSD) is given by the squared magnitude of the speetral coeffieients This funetion is symmetrie about k = N /2. It represents the distribution of the total signal power between the frequeneies 0 and is. An alternative representation is the use of negative and positive frequencies. For this ease, all values of k;:>: N /2 are interpreted as negative frequeney values and the speetrum is symmetrie about k = o. In this ease, the funetion is known as the two-sided speetrum. The one-sided PSD simply eonsiders the speetral distribution up to k = N /2 and is given by 6.1 ReviewofSome Fundamentals 277 Note that the power of the signal is not the same because i = 0 is also doubled (No bach et al. 2000). The term density is used because power per frequency bandwidth Ai, is being considered. The maximum resolvable frequency is half the sampling frequency Imax = I, /2 = IN/2 (Nyquist frequency) and the resolution is determined by the data set duration Ai, = 11 T. Graphically the PSD and the parameters involved in computing it are shown in Fig. 6.3. Two properties of the DFT deserve particular attention for laser Doppler applications. Since the time between the sampie points is not infinitely small, the 120wer in the signal at frequencies above imax will appear in the PSD at lower frequencies, an effect known as aliasing. This falsifies the spectrum at the lower frequencies. An example of aliasing is given in Fig. 6.4. The signal in Fig. 6.4a contains two frequency peaks at 2.4 Hz and 9.5 Hz, as can be seen in the correct spectrum illustrated in Fig. 6.4a. By sampling the signal at 15 Hz, the maximum resolvable frequency is 7.5 Hz and thus, the Nyquist criterion is not fulfilled for the signal power at 9.5 Hz. The spectral portion above 7.5 Hz is mirrored about the Nyquist frequency and results in a additional peak at 5.5 Hz (Fig. 6.4b). Furthermore the signal noise at frequencies above imax also increases the noise level at frequencies below imax. Aliasing errors in estimates of PSD are avoided by applying an analog antialiasing, low-pass filter with a sharp cut-off at half the sampling frequency. This procedure is illustrated in Fig. 6.5, using the same signal as used in Fig. 6.4. Before sampling the signal, a low-pass fllter removes the frequencies high er than the Nyquist frequency imax (Fig. 6.5b). The spectrum of the flltered and sampled signal in Fig. 6.5c contains no additional frequency peak. Furthermore, the noise level is reduced to the same level as in the original signal in Fig. 6.5a. Due to the equally spaced sampling of the input data set and also due to the equally spaced coefficients of the spectrum, a periodicity every N sampies is inherent in Eq. (6.1). This effectively means that the DFT perceives and acts on an infinite juxtaposition of the input data record and the inverse DFT effectively Power spectra l densily (PSD) Nyqu ist frequcncy Ar _ ~"12 - I _ f. 2 1,-2" Fig. 6.3. The power spectral density and the sampling parameters 278 a 6 Signal Detection, Processing and Validation . Original signal Analytical spcctrum 3() o ~ ,j Eq ualI)' !13Ced Sam\l1CS 1111 11 111111 11111111 111111 J J J j j bÄ t +. &. ---'--.'---~~.-r:>-tl'~~ •• ..0 ,0 f 11 lzl PSD X Sampled signal o I o .. I "() DislurlJing frcqucnc)' and noi e c:::> 151 yu~~~~~~~~~~~~-:~ o 10 f IHzl Fig. 6.4a,b. Aliasing error in spectrum due to signal frequencies occurring above the Nyquist frequency. a Original signal and spectrum, b Sampled signal and falsified spectrum transforms an infinite juxtaposition of the spectrum. This is illustrated in Fig. 6.6 for a time series. If the beginning and end of the record do not merge smoothly into one another, sudden amplitude jumps are perceived, which give rise to additional frequency components in the spectrum. These 'end effects' are unimportant for records oflong time duration; however, they deserve attention with short records, as encountered with the laser Doppler technique when measuring high speed tlows or with small measurement volumes. These effects are diminished by applying window functions in the time domain. Window functions scale the input data amplitude and force a tapering to zero at the beginning and end of the signal (Marple 1987). If an entire Doppler burst signal is centered in the digitized data record and has approximately the same duration as the data record, then it forms its own (Gaussian) window, since its amplitude begins and ends near zero. This represents an ideal sampling case. If only short intermediate segments of the burst are acquired and processed, then a windowing in the time domain may be necessary. A further consequence of a finite input record duration is spectral broadening. A spectrum of an infinitely long sine wave is adelta function at the signal frequency. A finite length sine wave yields however a broadened peak, in which the peak width is inversely proportional to the input signal duration. This process is graphicallyillustrated in Fig. 6.7. The spectrum of an infinite sine wave is a delta function at the signal frequency (Fig. 6.7a). A finite duration sine wave can be viewed as the product of an infinite sine wave with a rectangular window of 6.1 ReviewofSome Fundamentals PSDY, Original signal ~() 1 I Isl o Low-pass filtered signal AnalyticaJ spectrum Signal frcqucncy c=:::) PSDY. 279 Disturbing frcquc ncy 10 5 J [llz l Low-pass filtered spcctrum Signal frcq ucncy 0 / -I 111111 Illll Equally spaced sampies 111111 lllll 1111111111 Low- pass cut-off frcqucncy 0 5 10 f IlIzl 0 5 10 f IHzl .• . . . ,. .. w· ~ , 0 00 -I Fig. 6.Sa-c. Elimination of the aliasing error by use of a low-pass, anti-aliasing filter. a Original signal and spectrum, b Low-pass filtered signal and spectrum, c Sampled signal and non-aliased spectrum x Fig. 6.6. Implicit periodicity of acquired signal when processing using the finite length DFT duration T (Fig. 6. 7b). The speetrum of the finite sine wave will therefore be the eonvolution of the delta function with the magnitude of a sine funetion, the 280 a 6 Signal Detection, Processing and Validation Time domain Frequency domain -00 ~ Time o b X i Jo ® Co~volution Multiplication PSD o Time Frequency Frequency c PSD Time o Jo Frequencl' Fig. 6.7a-c. A multiplication of two signals in the time domain is equivalent to a convolution in the frequency domain. This can be used to explain spectral broadening due to finite record lengths. a Infinite sine function and related spectrum, b Rectangular function and related spectrum, c Finite sine function and related spectrum transform of a rectangular window (Fig. 6.7c). This can be easily illustrated using the following relations. If a signal y(t) is given in the time domain as the product of two other signals, x(t) and h(t) y(t) = x(t) h(t) (6.7) then the Fourier transform of y(t) is given by the convolution of the Fourier transforms of x(t) and h(t) (Bendat and PiersoI1986). I(f)=2{(f)®H(f)= J2{(a)H(! -a)da (6.8) The power spectral density of y(t) is then 2 , Gk =--YkY k N!s- - , k=O,1, ... ,N!2 (6.9) 6.1 ReviewofSome Fundamentals 281 An obvious consequence of spectral broadening is that the resolution of distinct signal frequencies in the PSD can be improved by sampling a longer portion of the signal. However, in the laser Doppler and phase Doppler techniques the signal duration is limited to the transit time of the particle through the measurement volume. This transit time, which is inversely proportional to the flow velo city, will ultimately limit the accuracy of the frequency estimation. In fact, this is a manifestation ofHeisenberg's uncertainty principle. The product of signal observation time and frequency resolution will be constant. T.1f =1 (6.10) In practical implementations of the DFT, Eq. (6.1) is not used directly but rather a recursive form known as the fast Fourier transform (FFT) is used. There are many realizations of the FFT, but they share one feature in common, namely, that they normally operate on 2" points l : sampie records are restricted to values such as 16, 32, 64, 128, .... The calculation time of the DFT implemented with Eq. (6.1) increases with N 2 • The FFT algorithm reduces the computation time to the order of NlogN. A commonly used technique with the FFT is that of zero padding. Without changing the spectral content of the signal, zero padding forces the FFT algorithm to estimate the spectrum at additional frequencies between zero and fmax' thus improving the resolution. This is easily seen by exarnining a signal doubled in length by adding zeros. Instead of Eq. (6.1) the transform becomes 2N-l (2nnk) K.k= L!.nexp - j - - , n=O 2N However, since !.n k=O,I, ... ,(2N-l) (6.11) = 0 for n = N, N + 1, ... ,(2N -1), this can be written as (. 2nn(k/2)] ' Kk -_ ~ ~!.n exp -J n=O k=O,I, ... ,(2N-l) (6.12) N which is identical to the N-point transform for every other k value. However now Kk is computed also at intermediate k values. The spectral content of the signal has in no way been altered, but with the intermediate estimates, interpolation of peak locations can be improved. Zero padding can also be used to extend input data records up to a length of 2" values, in preparation for an FFT. 6.1.2 Correlation Fundion Principally, the information available in spectral domain is also available in the correlation domain, since the autocorrelation function R( r) forms a Fourier transform pair with the power spectral density (Wiener-Khinchine relation). In digital form this can be expressed as 1 Algorithms exist for FFTs using other record lengths, especially prime number decompositions; however, these are not in widespread use. 282 6 Signal Detection, Processing and Validation fs L N/2-1 ( • 21tnk) Rn =R(r=n.::1r)=G1k1exp +J-2N k=-N/2 N f ( Go +(-1)"G =_s 2N N /2 +2 L G eos (21tnk)J' -k 2 L N/2-1 R1n1ex s n=-N/2 2( =- Ra +(_1)k R N ' 2 fs (6.13) k =O,1, ... ,N/2 (6.14) N k=1 Gk=G(j=fk)=-j n=O,1, ... ,N/2 N/2-1 ~ .21tkn) -J-N (21tkn)J ' +2 L Rn eos - - N/2-1 n=1 N where Llr = Llts is the time lag interval. The autoeorrelation function is by definition symmetrie about r = o. With the mean removed, the autocorrelation funetion is known as the autoeovarianee funetion; however, these two terms will be used interehangeably, always assuming a mean-free input signal. The eorrelation function ean also be eomputed direetly using the estimator (6.15) A eomputation of Rn using the FFT, first to compute the PSD and then to transform to the eorrelation domain, exhibits a speed advantage that inereases with inereasing data reeord length (NlogN eompared to N 2 for a direet ealeulation of the eorrelation funetion). However, there are some subtle differenees between the estimate of Eq. (6.13) and that of Eq. (6.15). The most important of these is the so-ealled 'wrap-around' error (Bendat and Piersol 1986), whieh has its origins in the finite length DFT, Eq. (6.1). The inherent periodicity in time whieh is implied by Eq. (6.1) and illustrated in Fig. 6.6, essentially means that the eorrelation funetion computed aecording to Eq. (6.13) assurnes an infinite juxtaposition of the input signal in time. The derived autocorrelation function will also be based on this assumption and is, therefore, known as the cireular autoeorrelation. This error is avoided by first padding the input signal with zeros at the beginning and end of the original signal to double its length. The autoeorrelation function will exhibit a periodicity at the same period as the original signal. For instanee, the autoeorrelation of an infinite sine wave will be an infinite eosine wave, as illustrated in Fig. 6.8a. Thus, the signal frequeney ean be estimated by measuring the elapsed time over one or more zero erossings of the autocorrelation funetion (period timing). The autoeorrelation of a Gaussian windowed sine wave eentered around t = 0, as shown in Fig. 6.8b, will yield as a eorrelation function a eosine wave with an amplitude deeay direetly related to the window width. Of partieular interest is the effeet of signal noise on the correlation funetion. As illustrated in Fig. 6.8e, the eontribution of signal noise ean be found entirely in the first coefficient of the autocorrelation funetion, i.e. at r = o. This is beeause the signal noise has no inherent time seale, meaning that it is eompletely random and not eorrelated with itself over any length of time. This last property of the autoeorrelation 6.1 Review ofSome Fundamentals 283 R(T) AR + - - - - - " x(t) SNR=OdB A+-----+A Fig. 6.8a-c. Input signal and autocorrelation function. a Sine wave, b Noise-free Doppler signal, c Noisy Doppler signal function is particularly interesting for laser Doppler and phase Doppler signal processing, because it provides a means of separating the noise effects from the signal, thus, improving the estimation of signal frequency and other signal parameters. 6.1.3 Hilbert Transform The Hilbert transform of a function x(t) is defined by y(t) =N{ x(t)} =.!.. Jx( r) d r Tt_j-r (6.16) and is an integral transform, where the Cauchy principal value is taken in the integral. The function y(t) is produced by passing x(t) through a filter with the transfer function H{J) = -j sgn{J) (6.17) A singularity exists at the value f = 0, which, however, does not cause any computational problems. On the other hand, the infinite integral causes problems for signals that are not mean-free. Thus, when processing laser Doppler and phase Doppler signals with the Hilbert transform, it is necessary to first remove the mean, either optically or electronically. 284 6 Signal Detection, Processing and Validation The magnitude and phase of H(f) are IH(J)1=1 (6.18) arg{H(J)}=-lt sgn(J) (6.19) 2 The inverse of the Hilbert transform is given by (6.20) So me typical examples ofHilbert transform pairs are given in Table 6.l. A sampIe signal and its Hilbert transform are shown in Fig. 6.9. For a given input signal x(t) the Hilbert transform is the signal y(t) which is shifted by -90 deg in phase for all frequencies. An analytical (complex) function for a given input signal x(t) can be defined as z(t)=x(t)+j K{x(t)} (6.21) which has spectral values only for frequencies larger than or equal to zero (j ;::: 0). Its Fourier transform is zero for all negative frequencies, or in the discrete case for all frequencies f ;: : N /2. This analytical signal can be used to derive the signal envelope A(t) and the instantaneous signal phase rp(t). A(t) = Iz(t)1 (6.22) rp(t)=arg{ z(t)} (6.23) The envelope and phase of the Doppler-like signal from Fig. 6.9 are shown in Fig.6.10. Table 6.1: Some sampIe Hilbert transform pairs N{ x(t)} x(t) y(t) = const ax l (t) + bX2(t) defined as 0 aYI (t) +bY2 (t) x(at) y(at) x(t - to) y(t-to) [x(t)x(t- 1") dt [y(t)y(t- 1") dt asinbt acosbt -acosbt asinbt 1 1 1tt-a J(t - a) 6.1 ReviewofSome Fundamentals - - Original signal --- --- Hilbert trans form ,"" :, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, , ,\ ,," ':, , ,, ,, ,, ,, , o ,, ,, 1 1 1 1 1 1 1 t, ," ,, ,, ,,, , , ,, ,, 1 1 1 1 1 1 1 1 1 1 , ,, 1 1 1 1 1 1 11 . 1 1 1 1 I, 1 "" '.' 285 , 't ,' "" Time[s] Fig. 6.9. A sample signal and its Hilbert transform, illustrating the -90 deg phase shift b a - - Original signal -----Envelope,,. -_ ,, .,, o ·1 Time[s] Time[s] Fig. 6.10a,b. a Input signal and computed envelope amplitude, b Instantaneous phase of input signal The calculation of the Hilbert transform for a discrete signal of finite length xn=x(t=tn ), tn=nlfs' n=O,l, ... ,(N-l) can be performed in the frequency domain using the Fourier transform 3 or its fast implementation, the FFT. (6.24) with -j bn ={ j for for Os,n<N/2 N/2s,n<N (6.25) 286 6 Signal Detection, Processing and Validation The analytical signal defined in Eq. (6.21) can be obtained as follows (6.26) with bn ={20 for O.:;,n<N/2 for N/2':;'n<N (6.27) These expressions are illustrated graphically in Fig. 6.11, in which the real and imaginary Fourier coefficients of a real input signal are shown. The modified coefficients used in the inverse transform to obtain the Hilbert transform (Eq. (6.24» are shown in Fig. 6.11a, and the modified coefficients used in the inverse transform to obtain the analytical function (Eq. (6.26» are shown in Fig. 6.11b. From this figure it becomes apparent that the Hilbert transform can be implemented using very simple operations in combination with the Fourier transform. N a N Im(5{ x(t)}) Im(b, S{x(t)}) Iiilbcrt trans!orm. 1\ {x(t)} Re(S{X(/)}) N N b Im(S{ x(t)}) n Rc(S{X(/)}) Re{b,S{x(t)}) Fig. 6.11a,b. a Fourier coefficients modified for the Hilbert transform, b Modified coefficien ts for analytical signal 6.1 Review ofSome Fundamentals 287 6.1.4 Signal Noise There are several features of signals from laser Doppler and phase Doppler systems, which make their processing quite unique. Certainly the fact that the signals come at irregular and unknown time intervals, corresponding to the particle passage through the measurement volume, is important to consider. This creates the need for signal detection to indicate when the processing should take place or if the processed result is to be retained (validation). The task of signal detection, and later of signal processing, is made more challenging by the presence of noise superimposed on the signal. Noise is essentially any amplitude deviation of the signal from the analytic form resulting from the signal generation analysis given in chapter 5. Sources of noise are manifold, including stochastic noise coming from the photodetector and electronics (shot noise, Johnson noise, dark current, see section 3.2.5) as weH as from the scattering process itself. The laser may introduce signal noise, especially laser diodes, which, if improperly stabilized, can exhibit modehopping (Dopheide 1995). Noise can arise from unwanted reflections associated with the flow rig or other particles. Poor grounding of the experimental apparatus or poorly designed electronics may also lead to ground noise. Expect for the last source, noise contributions in the system are usually considered to be spectrally white. This refers to the fact that the total noise power is distributed evenly over all frequencies up to the upper bandwidth of the system. Attempts to estimate this noise power usually concentrate on the noise arising from the photodetector and its associated electronics. The power of signal fluctuations (J"; put into relation with the power of noise fluctuations (J"~ is known as the signal-to-noise ratio (SNR) and is generally expressed in decibels: SNR/ = +10 IOg( (J"; I dB (J"2 ) (6.28) n Interestingly, the SNR can vary significantly, not only with the bandwidth of the electronics, but also with the type of photodetector and with the signal intensity. This has been illustrated already in section3.2.5 (Fig. 3.26). An empirical determination of SNR for various semiconductor detectors can be found in Dopheide et al. (1987) and Dopheide et al. (1990). Generally, however, analytic expressions for noise yield only best case estimates, since any optical misalignment or electronic misadjustment can easily become overwhelming in its increase of the noise level. The SNR can be estimated from a given signal segment using various techniques and such an estimation is often used in the detection/validation step of signal processing to indicate wh ether a result can be expected to be reliable or not. This is discussed in further detail below. In Fig. 6.12, a laser Doppler signal, a noise signal and the summation of the two in time, spectral and correlation domain is illustrated. It becomes obvious from Fig. 6.12 that the power spectral density (PSD) or the autocorrelation function (ACF) offer exceHent means to monitor SNR and to determine whether a particle signal is present or not. 288 6 Signal Detection, Processing and Validation TImc (samplcd wilhj) WhiteNoise (Iow-pa s lillcrcd) ign~l Noisy signal Spc<.1l"dl cu. (/) c.. + Ja Frequency I J Correlalion "- "- "- I. <: <: ~ + - U U cr; 0 Time lag • 0 1: o t Fig. 6.12. Representation of a laser Doppler signal, a noise signal and a combination of the two in time, spectral and correla tion domain An idealized graphical interpretation of SNR is given in Fig. 6.13, which shows schematically the power spectral density (PSD) of a Doppler signallogarithmically scaled. The SNR is given by the ratio of the areas A to B. A more detailed Cl Vl c.. o -I A ~ SNR = 1010g B cr!/f. 0.01 B I ~ __ ~ ~_~~_.~_~~~_~~~~~~~~~~J- o 5 f 10 15 J:", ____- . Frcqucncy Jla.u.1 k~----------~--------------------~ Fig. 6.13. Graphical representation of SNR using the power spectral density (PSD) 6.1 ReviewofSome Fundamentals 289 estimation procedure is given by Tropea (1989). The noise appears as a base line floor of width L1f, the bandwidth of the system, and of amplitude O"~ I fs. Any flltering of the signal, for instance using a low-pass fllter, will directly decrease area Band thus increase the SNR, since more of the noise is removed. The use of a band-pass fllter to increase SNR increases the reliability of the signal detection, since the SNR acceptance threshold can be chosen high er. In contrast, the variance of the frequency estimation remains constant because the peak in the spectrum still has the same width. Indeed, such adjustable input fIlters are usually an integral part of any Doppler signal processor. On the other hand there is a danger in flltering with too narrow a bandwidth, since in general the signal frequency is not known apriori. This can lead to truncation of the velo city distribution and to a bias of the estimated moments. The SNR can also be estimated from the autocorrelation function. Noise, being fully stochastic and having zero correlation duration, appears only in the first autocorrelation coefficient, i.e. R( r = 0)1. Thus, the SNR can be estimated by comparing the amplitude of the autocorrelation function at r = 0 to the maximum peak amplitude of the remaining periodicity, exemplary shown in Fig. 6.8c for a high-pass flltered burst signal with added noise. If the frequency of the periodicity f has already been determined, the amplitude of the signal AR (Fig. 6.8) can be estimated by fitting a eosine wave to the measured correlation function at points removed from r = o. This can be computed using the expression A _ _ R.,...('--n_L1....:.r)----,R - cos(21tfnL1r) (6.29) from which the variance of the noise portion of the signal can be computed (6.30) The index n should ideally be chosen at the first maximum or minimum removed from r = O. The SNR is then given as SN%B = 1010g( ;~ J (6.31) The presence of noise in the signal can have both direct and indirect effects on the measurement quantities. In the worst case, the scattering signal from the particle may not be detected at all or a completely wrong frequency or phase estimate could be made from a noisy signal. However, noise may also effect estimates of signal duration, generally leading to an overestimation, which then influences the derived flow quantities in the data processing. In any case, noise increases the variance of the signal frequency and phase estimates, regardless which processing scheme is employed. Thus, noise at the signal processing stage essentially determines the lowest resolvable level of measurable turbulence in a 1 The statistical scatter (errar) of the autocorrelation coefficients increases with SNR for every Lir, given a finite number of sampies. 290 6 Signal Detection, Processing and Validation flow. This level is known as the Cramer-Rao Lower Bound (CRLB), and is discussed in further detail in the following section. 6.1.5 Cramer-Rao lower Bound (CRlB) (Contributed byH. Nobach) The goal of analyzing an acquired signal is to derive several signal parameters according to a given model representing the physical basis of the signal generating process. In laser Doppler techniques, this is either the Doppler frequency corresponding to the velo city of the particle or the phase difference between two signals, acquired from a phase Doppler system. In practical cases, the recorded signals are equidistantly sampled and time limited, so that the amount of available information is also finite. Furthermore, the signal is influenced by noise and this intro duces uncertainty into any parameter determined from the signal. The calculation of signal parameters is therefore called estimation, since it contains a random component. The true values of the parameters to be estimated are seldom known and different estimation algorithms (estimators) will also yield different results. Therefore it is of interest to quantify the accuracy of each estimator statistically. . To begin with, the expectation ofthe estimator should be equal to the true value, i.e. non-biased. Second, the estimator should be efficient, meaning that it uses all available information to estimate the required parameter as accurately as possible. In chapter 10, features of estimators are discussed in more detail. The efficiency of an estimator is quantified by its variance. While the bias should be zero, the finite amount of information yields a lower bound of achievable accuracyand thus, a finite variance. For unbiased estimators this lower bound of variance is given by the Cramer-Rao lower bound (CRLB) (Kendal and Stuart 1963, Papoulis 1988). No unbiased estimator can obtain estimates with a variance smaller than the CRLB, thus this quantity can be used to evaluate the performance of a specific algorithm. On the other hand, the CRLB gives no information about how an algorithm should process a measured signal to reach this lower bound. However, based on estimation theory, and closely related to the CRLB, the maximum likelihood (ML) estimator can be derived. If any unbiased estimator reaches the CRLB, then the ML estimator will also reach it, at least asymptotically (Kendal and Stuart 1963, Kay 1993). For a signal x{t=tJ=x; =m; +n;, with x= i=O,I, ... ,(N -1) l lj lj Xl . Xo X~_l j , m= mol m. m~_l , n= nol n. n~_l , x=m+n (6.32) (6.33) 6.1 ReviewofSome Fundamentals 291 consisting of the model signal m of known type, the noise n and the unknown (scalar) parameter a, the CRLB is given by 1 (6.34) where p(x,a) is the joint prob ability density function, (section 10.1), of the measured signal x for a given parameter a. Since a is normally a vector, Eq. (6.34) is the inverse of a matrix, the Fisher information matrix J, whose typical element is given by (6.35) with dlnp(x,a) Ha = 1 -:. ua j ' (6.36) The bound of the i-th unknown element of the parameter vector a is given by the i-th diagonal element with index ii of the inverse Fisher information matrix (6.37) where no summation is implied. For uncorrelated and signal independent noise with power O"~, and with a Gaussian distribution, the joint prob ability density function p(x,a) becomes (6.38) and the elements of the Fisher information matrix become (Whalen 1971) (6.39) To derive the lower bounds for a given Doppler burst this can be calculated and inverted, at least numerically. As an example, the lower bounds for the estimation of the frequency OJ and the phase (jJ will be derived. Since the parameter vector a must contain all unknown parameters, including those, which are not estimated (hidden parameters), a constant amplitude of unity during the observation time is assumed for simplification. The time dependent signal x(t) = m(t) + n(t) (6.40) 292 6 Signal Deteetion, Proeessing and Validation is eomposed of the model signal m(t) = cos(mt+ tp) (6.41) and the time dependent noise n(t). The measured signal after sampling is therefore X; =x(t=t)=cos(mt; +tp)+n;, (6.42) i=O,I, ... ,(N -1) and each sampie is a function of the two model parameters x; (m,tp) and the noise. The sampling times are given by t; = i I fs. The noise n is uncorrelated and Gaussian distributed. The parameter vector is (6.43) a=(;J The derivatives of the model parameter dependent sampies m; dm d~ d; dm = cos(ox; + tp) are = -t; sin(mt; + tp) (6.44) . ( ) =-sm mt; +tp (6.45) The Fisher information matrix becomes (6.46) The inverse of the Fisher information matrix is -~t; sin 2(mt; +tp)] N-l (6.47) ~); sin 2(mt; +tp) j=O As an example, a numerical simulation was performed for the (true) parameters m = 2, cp =1.1, fs =10 and N = 256. 1 The noise power was varied logarithmically in 25 steps from e-lO to eID, which corresponds to 25 equal steps of SNR, expressed in dB.2 For each noise level, 1000 independent realizations were generated. The individual signals were processed by aleast mean square estima1 2 For this and subsequent examples in this seetion, frequeneies have been nondimensionalized using 2n. Thus, for OJ = 2, fs = 10 eorresponds to IOn sampies per eyde. The noise power has been normalized with the signal varianee, thus lJ; = I corresponds to SNR = üdB . 6.1 Review of Some Fundamentals 293 tion routine, which for Gaussian distributed noise is equal to the maximum likelihood estimation. The Fisher information matrix and its inverse were calculated to be 1 (130 1653 ) J = O"~ 1653 28356 -1 J = 0"2 ( n 0.029787 -0.001737) -0.001737 0.000137 (6.48) (6.49) In Fig. 6.14a a sample signal with SNR = 15 dB is illustrated. The results presented in Fig. 6.14b show that the maximum likelihood estimator meets the calculated CRLB. Furthermore, a threshold noise power O"~max can be seen for the frequency estimate. Above this limit, the noise dominates the spectrum and the algorithm estimates the frequency randomly from the entire frequency range. The frequency of the threshold depends not only on the signal characteristics, but also on the capability of the estimation procedure to find the correct peak in the spectrum. The phase range is limited by ±1t. Therefore, the variance of the phase estimation is also limited. In the case of phase Doppler signals, the Doppler frequency and the phase difference between two signals Xi = cos( OJti + qJx) + n xi , i = 0,1, ... , (N -1) (6.50) Yi = cos( OJti + qJ y ) + nyi , i = 0, 1, ... , (N -1) (6.51) with independent noise components n x and n y are of interest. To derive the CRLB for the phase difference, it is convenient to re-write these signals as b a ~ 10' ~ <U "0 .0..e 1 C 'e'" > '" Ei c «: ..8 11:' '" 12 '~B 0 10-' -1 o 5 10 15 Time [s1 10-' 'LL..u..u.....L...I..llW-.l..1.LL..J-LJULJ.j-LW...L.L.LI.L...l....LJ.JJL....L..I.JJJ 10-' 10-2 1 O:m~ 10' 10' Noisepower [-1 Fig. 6.14a,b. Single-tone parameter estimation. a Sample input signal, b Comparison of the CRLB with computed variance for frequency and phase estimates 294 6 Signal Detection, Processing and Validation (6.52) Yi =myi +n yi =COS(lüti + I)? + AI)?) + n yi ' i=O,I, ... ,(N -1) (6.53) Since the signals are of the same length with independent noise components, the joint prob ability density function p(x, y,a) now becomes (6.54) with m xi =COS(lüti +I)?) m yi = cos( lüti + I)? + AI)?) (6.55) (6.56) and the elements of the Fisher information matrix become (6.57) The vector of unknown parameters is (6.58) Note that I)? is included in the parameter vector since it is unknown, even though it is not used. The derivatives of m x and m y are om. 0;' = -ti sin(lüti + I)?) om. = -sin(lüt. + I)?) oI)? , om =0 __ x, Xi oAI)? om. o~' = -ti sin( lüti + I)? + AI)?) om. = -sin(lüt + I)? + AI)?) oI)? , om. --y' =-sin(lüt + I)? + AI)?) --Y' oAI)? , (6.59) (6.60) (6.61) (6.62) (6.63) (6.64) 6.1 Review ofSome Fundamentals 295 The Fisher information matrix becomes (6.65) with N-I = I/ sin 2 (mt; +tp) Pk (6.66) i=O N-I Qk =I l sin 2 (mt; +tp+Lltp) (6.67) j=O The inverse of the Fisher information matrix is (6.68) As an example, a numerical simulation was performed for the (true) parameters m = 2, tp =1.1, Lltp = -0.8, fs =10 and N = 256. The noise power varied logarithmically in 25 steps from e- Io to e+IO • For each noise level 1000 independent realizations were generated. The individual signals (Fig. 6.15a) were processed by a maximum likelihood estimation routine. The Fisher information matrix and its inverse were calculated to be J r l 55536 3261 1608] [ 3261 258 128 (5 n 1608 128 128 =.-;- 0.000070 -0.000894 [ = (5~ -0.000894 0.019063 0.000008 -0.007794 0.000008 ) -0.007794 (6.69) (6.70) 0.015536 The results presented in Fig. 6.15b show that the maximum likelihood estimator meets the calculated eRLB. Again, a threshold noise power can be seen for the frequency estimate and the phase range is limited by ±n. Note that the eRLB and the empirically derived estimation variance of the phase tp and the phase difference Lltp are different. The expressions of the eRLB derived above are not convenient for practical use in setting up a signal processor. Explicit expressions of the eRLB are required. To derive these, a set of two orthogonal signals with independent noise components n and are considered. n Xi = mi + ni , i = 0,1, ... , (N -1) (6.7l) 296 6 Signal Detection, Processing and Validation '. "\ I I • • I I 1 'I I t :: o I I I : : : I I t ~ , , ,I 1 .' I t ',' I I I ~ : I I I !:::: :: I I : : : I I \ { .,8o:t5 I I I \ f I, I I 1 I I' I ~ 'e~ : 1 ! o : ~~ , :. f -1 : ~ ..\ : '~ • ./ I I ~ 10 Empirical CRLB variance , e" 4 ? • I i 10 5 ./ ..... I I I I l 10-2 ,5 15 10-' LL.Llll...L.L.LILL.U.lL.L....I.J..lL-L.Llll...L.L.LILL.U.lL.L...u..u 10-4 10' 10' Time [sI Noise power [-I Fig. 6.15a,b. Single-tone parameter estimation from two orthogonal signals. a Sample input signals, b Comparison of the CRLB with computed variance for frequency, phase and phase difference estimates (6.72) with = Acos{wt; + lP), i = 0, 1, ... , (N -1) (6.73) m; =Asin(wt;+lP)' i=0,1, ... ,(N-1) (6.74) m; where additionally the amplitude A is unknown. The joint probability density function becomes (6.75) and the elements of the Fischer information matrix are (6.76) The vector of unknown parameters is (6.77) Using the fact that explicitly as m; + in; = A 2, the Fisher information matrix can be expressed 6.1 Review ofSome Fundamentals N-l N-l A2~)~ A2~> ;=0 0 i=O N-l J=_1 A 2L;t (j2 n 297 N-l j A 2L;1 i=O ;=0 0 0 0 N (6.78) N(N -1)(2N -1) 3N(N -1)f, A2 =-2-2 [ 6(jJ, 3N(N-l)f, 6Nf, o 0 2 The zero elements in this matrix indicate that the amplitude ean be estimated completely independent of the frequency and the phase. Thus, the amplitude ean be presumed to be known without eh anging the lower bounds of the frequeney and phase estimator varianee. The inverse of the Fisher information matrix beeomes r = 1 2 2 2 6Nf,2 -3N(N -1)f [ -3N(N -1)f, N(N -1)(2N -1) 2 (jn 2 AN (N -1) ' o (6.79) 0 leading to the CRLB for the frequeney (j2 > 0)- 12(j2 f2 n , A2N(N2-1) (6.80) This is the CRLB for two signals with independent noise eomponents. For only one signal, the information eontent is approximately one half, leading to (j2> 0) - 24(j2r n, A 2 N(N 2 -1) (6.81) Using SNR=~ 2(j~ (6.82) the CRLB for (J) ean be expressed as (Ibrahim et al. 1990, Rife and Boorstyn 1974, Wriedt et al. 1989) (6.83) or if (J) = 21rf is used, this varianee reduees to 298 6 Signal Detection, Processing and Validation (6.84) This expression was derived assuming that the noise is spectrally white. Any filtering used to reduce the signal noise violates this assumption and Eq. (6.84) no longer strictly holds. Thus, while (bandpass) filtering may improve the SNR, the estimator variance may not be reduced. For the case of phase Doppler signals, a second signal pair is required. = m xj +n xi ' m x; = Acos(wt; + Ij?), i = 0,1, ... , (N -1) (6.85) Xj =mx;+rlx;' m x; =Asin(wt;+Ij?) , i=O,I, ... ,(N-l) (6.86) X; y; = m y; +n yj , myj = Acos(wt; + Ij?+ LlIj?), i = 0,1, ... , (N -1) (6.87) Yj = myj + rly;' m y; = Asin( wtj + Ij?+ LlIj?), i = 0, 1, ... , (N -1) (6.88) The joint probability density function becomes (6.89) and the elements of the Fisher information matrix are r =_I_~[dmxk '} ()~ k=O daj dmxk + dmxk dmxk + dm yk dm yk + dm yk dm yk ) da) daj da) daj da) daj da) (6.90) The vector of unknown parameters is (6.91) The Fisher information matrix becomes N-l 2A2~)1 i=O N-l /=_1 2A2~> (}2 n i=O N-l N~) N-l 2A2~> i=Q N-l 2A2~) i=O N-l N~) N-l A2~); o i=O N-l A2~) o i=O N-l A2~) i=O i=O i=O 0 0 0 o 2N (6.92) 6.1 ReviewofSome Fundamentals J= A2 6er2n f2, 2N(N -1)(2N -1) 6N(N -1)f, 3N(N -1)f, 0 6N(N -1)f, 12Nf/ 6Nf/ 0 3N(N -I)!, 6Nf/ 6N!/ 0 0 0 0 12Nf/ A2 299 (6.93) The inverse of the Fisher information matrix is 12f,2 _ 6!, N 2 -1 N+l 5N-l rl=~ -~ 2 2A 2N N+l N+l 0 0 -2 0 0 -2 4 0 0 0 0 A2 (6.94) leading to the following CRLBs (H0st-Madsen and Andersen 1995) (6.95) (6.96) 2er2 er2Ll<p> A2N" (6.97) Note that these are the lower bounds for the four signals with independent noise components. If only the two phase Doppler signals are given, then the information content is one half and the bounds become 2 er2(5N -1) er > n '1'- A 2N(N+l) 6f} 2 N(N -1)SNR (6.98) 5N-l 2N(N+l)SNR (6.99) =---------- 4er 2 2 er2 > ---" = ----Ll<p- A 2N NSNR (6.100) The CRLB is exactly half of the value for a Doppler signal, since the frequency information content in the two phase Doppler signals is twice as large. The CRLB for the phase difference between the two phase Doppler signals is lower than that for the absolute phase by a factor of 0.8, for large N. This can be seen in Fig. 6.16, which presents the estimator variance results from Fig. 6.15b, normalized by the noise power. All of the above derivations were based on a signal model of constant amplitude. In fact, laser Doppler and phase Doppler signals exhibit an amplitude de- 300 6 Signal Detection, Processing and Validation o o 00 0 0 0 0 D---LT-------------------------------------~---------------------o 0 0 0 • • • • • • • • • • • • • • • ------ CRLB (ip) 0.01 Empirical Variance (ip) 0 - - CRLB (dip) • Empirical Variance (dip) 0.00 10.4 10-3 10-2 10- 1 10 Noise power [-] Fig. 6.16. Variance of the maximum likelihood estimator, normalized by the noise power for phase and phase difference. Comparison to the respective CRLBs (Simulation parameters as used in Fig. 6.15) scribed by a Gaussian envelope and for this case, the derivation of CRLB is somewhat more tedious. Results for the frequency, amplitude, arrival time and residence time are presented by H0st-Madsen and Gjelstrup (1996). 6.2 Signal Detection The most primitive form of signal detection is to apply an amplitude level to distinguish between portions of the signal during which a particle is present and those consisting only of noise. This is illustrated in Fig. 6.17 for a high-pass filtered laser Doppler signal. A gate pulse indicates the beginning and the end of each particle signal. The gate pulses can be used to enable further processing of the signal, e.g. a digitizing and storage or frequency estimation. The gate pulses can also be used to measure the duration ofthe burst, the so-called particle transit time or residence time. This is achieved by counting pulses from a high-speed oscillator during the period of the gate pulse, as illustrated in Fig. 6.18. The time measurement has an uncertainty of ±l interval of the high frequency oscillator. Although this rather simple method of signal detection has been widely used for laser Doppler signal processing in combination with period timing devices (section 6.3.3), it does not fulfil the necessary requirements for distinguishing signal from noise. In fact there is no principal reason why the noise amplitude must be smaller than the signal amplitude, although the opposite would be considered particularly unfavorable measurement conditions. Nevertheless, such measurement conditions 6.2 Signal Detection 301 Fig. 6.17. Simple amplitude detection scheme I' _G_at_e..:.p_ul_s_e_ _...... i ! Oscillator pulses , Llllllllllljlllll~~~,~ ~~ 11 , I1111I111111111 Counter N,r~ - - ----- ---- ! J,,,, ! -----,----- --------------------- - .';...r , - - - - - i i Rcsidence lime o ~--~--=~--------------. Rcsel Slarl Siop Read ou l Fig. 6.18. Measurement of the residence time using a high frequency oscillator. No" is the number of counted oscillator pulses during the gate segment are regularly encountered in complex, two-phase flows exhibiting strong variations of particle concentration and background noise. Thus, detection schemes based on a signal amplitude threshold cannot be considered very reliable. Furthermore, amplitude-based detection would lead invariably to a favored selection of large amplitude signals, which in the phase Doppler technique, would translate directly into a bias favoring large particles. Some improvement can be obtained by monitoring the amplitude of the signal envelope, i.e. after low-pass fI).tering the burst signal. This approach can be refined further with a peak detection scheme, such that only the central part of 302 6 Signal Detection, Processing and Validation each signal, i.e. the part with the maximum SNR will be processed (Qui and Sommerfeld 1992, Bachalo et al. 1989). This technique results in improved frequency and phase estimates from the signals and also leads to a decrease of trajectory-related sizing errors in phase Doppler systems (Sankar and Bachalo 1991, Grehan et al. 1992, Aizu et al. 1993), as will be discussed further in chapter 8. Nevertheless, all approaches based on signal amplitude retain a fundamental dependence on the system gain (e.g. photodetector high voltage), which is set by the user or influenced by changing flow conditions. The part of the signals carrying information is distinguished by its periodicity or a coherency over a finite time interval. Noise on the other hand, is spectrally white and as such, is completely uncorrelated with itself over any time interval. Thus, the signal and the noise are separable in the spectral or correlation domain. This property of the spectrum and autocorrelation functions has been reviewed in section 6.1. The detection is preferably performed in advance of the signal processing or even on-line, but can just as easily be applied as a post-detection, in which case it is generally known as a validation step. Increased capabilities of digital electronics has led to a host of new on-line detection schemes based either on the PSD or the ACF, some of which have been patented and/or commercially realized. Several of these schemes will be briefly reviewed. The burst detection method of Jensen (1992) is based on detecting the first negative peak of the ACF falling below a fixed threshold. The burst detector operates on the I-bit digitized (or clipped) input signal. A positive signal voltage is coded as 1 and negative signal voltages as o. The double-clipped correlation operates on this I-bit data and can be very economical to implement since only logic gates are required. The value of R(O) is always equal to n, the sampie record length, and thus the amplitude of other peaks in the correlation function, for instance the first negative peak, in relation to n gives a direct measure of SNR. Therefore, a negative threshold of the ACF can be used for minimum SNR detection. A rough frequency estimate is obtained by determining the shortest time-delay coefficient that falls below the fixed threshold, since the fundamental frequency of the ACF is the same as the signal frequency in time domain. The transit time is determined by measuring the time during which negative valued coefficients fall below the specified threshold. The center of the burst can be determined by marking the beginning and end of the burst captured in the FIFO (First in first out) of the sampier running parallel to the burst detector. The burst detector, pictured as a block diagram in Fig. 6.19, is implemented using an autocorrelator with 16 logarithmically spaced delay taps operating on the I-bit digitized signal. These taps are placed so that they cover a 15:1 dynamic range in frequency. The logarithmically spaced coefficients of the doubleclipped ACF are computed by multiplying the delayed outputs of the delay line by the incoming signals. A short-term running average for each of these delay products is calculated using a simple RC filter and compared to the preset threshold. The time constant is chosen to correspond to about 4 cycles of the incoming signal. Further details can be found in Jensen (1990) or TSI Inc (IFA 755). 6.2 Signal Detection I-bit digit ized signal 303 Logarilhmic dc\ay line ShOTt time average Reference threshold Threshold comparator Deleclor logi..: 4-bit frequcn..:y Burst gate Fig. 6.19. Block diagram ofrS! 755 burst detector AI-bit correlation approach is also used in the minimum cross-correlator (MMC), introduced as part of a laser Doppler sensor concept by Damp and Sommer (1993) and Damp et al. (1994). In this system, massively parallelized correlators on an ASIC (Application-specific integrated circuit) simultaneously monitor signal coherency over a very broad bandwidth by cross-correlating with the simplest possible signal signature, namely (01010101..., 001100110011...), hence the term minimal cross-correlation. The parallel implementation of the previous detection scheme was used to monitor simultaneously a large frequency bandwidth. A similar strategy stands behind the detection system described by Blancha and Murphy (1990), in which the signal is passed to aseries of narrow bandpass fIlters to detect Doppler signals at low SNR in a given frequency range. Obviously the overlap of the filters must be carefully chosen and the system has bandwidth limitations imposed by the hardware realization. The DFT is essentially a digital realization of a filter bank, the steepness of the filter being related to the total observation time and the spacing related to the number of sampie points. Thus, several signal detection units have employed a DFT with subsequent computation of the power spectral density. Almost all hardware implementations of the DFT utilize quadrature mixing to generate from one input signal, a pair of signals with equal amplitudes and a precise 90 deg-phase relationship. These signals represent the sine and cosine part of the signal and can each be Fourier transformed to yield the real and imaginary contributions to the spectral distribution. Thus, the total record length to be transformed can be halved. Techniques for quadrature mixing are discussed in section 6.3.4. 304 6 Signal Detection, Processing and Validation In terms of signal detection, the question to be asked is; what is the probability of detecting a spectral peak, which actually corresponds to noise, rather than to a coherent Doppler signal (false detection rate)? This question has been investigated empirically by Qui et al. (1994) and theoretically by Ibrahim and Bachalo (1992). Although not directly comparable on quantitative terms, both investigations show clearly that the false detection rate decreases for increasing record length, that it decreases for increasing SNR and, that a rate of less than 1% can be achieved at SNR = -5 dB for arecord length of 32 or larger. Ibrahim and Bachalo (1992) find also a significant advantage of the Fourier-based detector over a correlation-based detector. To perform the DFT at sufficiently high speeds for use as a burst detector or for preliminary frequency estimation, three approaches have been adopted in various devices: • buffered data storage with rather complex (and expensive) hardware-wired processing • recursive algorithms for fixed window, sliding D Fr • I-bit DFT with simplified hardware One example of a processor using buffered data storage is the commercial Burst Spectrum Analyzer, from Dantec Dynamics (Tropea et al. 1988). However this device has in the meantime been replaced with much fast er and more flexible DSP(digital signal processor)-based hardware. Qui et al. (1994) have derived a recursive DFT, which in the notation of Eq. (6.1) becomes X. _k.,+! (.2nk)( 1 x .-x ) =exP J-N- X _k., +n (_N+, -,) (6.101) This recursive approach is particularly attractive if only a few Fourier coefficients have to be computed, i.e. when the frequency of the input signal is known quite weIl apriori. To achieve this, Qiu et al. (1994) employ a novel frequency modulation procedure to transform two bandpass filtered input signals of the same frequency to a signal with a constant known frequency, but with the same amplitude as the first input signal. The constant frequency is chosen such that the Fourier index, k = N 14, i.e. the sampie frequency, is four times larger than the signal frequency. Hence the DFT is only performed in one frequency bin, taking advantage of the recursion formula in Eq. (6.101). The technique can be realized on-line at low cost and also results in the phase difference between the two input signals, which is relevant for phase Doppler applications. For laser Doppler applications, the photodetector is simply connected to both signal inputs of the electronics. The actual signal detection is performed by comparing the spectral coefficient amplitude at frequency index k with apreset threshold. This signal then gates further processing and yields the burst duration (transit time). AI-bit realization of the DFT has been implemented by Ibrahim et al. (1994) (Real-Time Signal Analyzer, Aerometrics/TSI Inc. St, Paul, MN) in which a 32 complex sampie DFT is performed on a sliding window of 75% overlap (8 sam- 6.3 Estimation of the Doppler Frequency 305 pIe shift). The computed SNR is monitored and compared with apreset threshold. Once the threshold has been exceeded, a lower level for detection is used to prevent areset of the gate due to variations of the SNR throughout the signal (US Patent No 5,289,391). In this implementation 20 million DFTs can be performed per second. In closing the discussion on signal detection, it is worthwhile to once again underline the distinction between simply triggering a processing algorithm and the actual signal detection. The detection step can be either a pre- or postvalidation of the result and essentially establishes whether the processed data is to be considered a valid signal or not, and possibly also how reliable the signal may be. By their nature, some signal detection schemes also provide further information, like first frequency estimates, transit time, burst centering information or phase difference information. 6.3 Estimation of the Doppler Frequency The Doppler frequency fD refers to the frequency of the signal periodicity when a particle passes through a laser Doppler or phase Doppler measurement volurne. The frequency is related to the particle velo city through (Eq. (2.27» fD= 2v.L sill <J2' Ab (6.102) Two basic assumptions are usually made in laser Doppler signal processing: the fringe spacing, L1x = Ab sin(<J2') /2, is constant throughout the measurement volurne and the particle velo city is constant during the passage through the measurement volume. In fact neither of these assumptions are strictly true, although for the time being they will be assumed so. Section 5.1.2 examines in detail the necessary conditions for a constant L1x and section 7.2.4 specifies und er what conditions the particle velocity can be assumed constant. Some initial processing strategies for signals with varying frequency or phase are discussed in section 6.5. The Doppler frequency fD is a random variable, meaning even an exact repetition of a particle passage, with precisely the same velocity, will yield a different signal, hence, a different frequency estimate. This is due to the triply stochastic nature of the processes involved: light scattering, light detection and signal amplification (Mayo 1975). The scatter, or variance of fD' given exact repetitions, will also represent the lowest measurable flow turbulence level and as such, is a quantity of considerable interest. Specifically, for the estimation of signal frequency, the Cramer-Rao Lower Bound (CRLB), as discussed in section 6.1.5, presents the lower bound of variance. For the case oflaser Doppler signal processors, digitizing the signal with a sampie frequency of f, and over a set of N sampies extending the length of the Doppler signal, the CRLB is given approximately by Eq. (6.84), whereby the noise is assumed to be Gaussian distributed in amplitude and spectrally white. Although Eq. (6.84) has been derived for a con- 306 6 Signal Detection, Processing and Validation stant signal amplitude, it is also an adequate approximation for Doppler burst signals in the following discussion. The goal of any laser Doppler or phase Doppler signal processor is to determine fD' while achieving the CRLB variance. An estimator wh ich achieves the CRLB is known as an efficient estimator. In fact, this lower bound can be undercut if apriori knowledge of the signal is available. A good example is the use of narrow-band filters before the processor, which essentially increase the SNR, the danger being, that in a turbulent flow the signal frequency may fluctuate outside the chosen filter bandwidth, thus falsifying or biasing the overall flow statistics. In Eq. (6.84), the quantity N / fs represents the duration of the record. If the signal duration is matched to the record duration, this can be replaced by the transit time rand, assuming a system bandwidth Af equal to the Nyquist frequency (f, /2), Eq. (6.84) can be written as ,,;= 21t3 2 (') r 3 Af 1- ~2 SNR (6.103) Assuming that the record length can be continuously weH matched to the signal duration, despite changes due to e.g. partide trajectories, increasing either SNR or the residence time can reduce the variance of frequency estimation. Means for increasing SNR can be inferred from Eq. (3.210), for instance increased laser power, larger collection aperture or larger scattering centers. The residence time can be increased by using a larger laser measurement volume; however, with a given laser power this also decreases the incident intensity and the SNR. Since this decrease is inverse quadratic, a net improvement of the frequency estimation can be achieved with a larger measurement volume. A somewhat more refined analysis, considering also the variation of SNR across the measurement volume, is given by Ibrahim et al. (1994), resulting in the proportionality 2 1 (J - - fJi (6.104) This discussion underlines the importance of not only maximizing the system SNR, but also devising electronics that exploit a maximum length of the available signal, if possible centered about the maximum amplitude. Adaptive burst length processing schemes have been developed for this reason and can be shown to be unbiased (H0st-Madsen and Gjelstrup 1996). Attention is now turned to various estimators of fD' their performance (how dose their variance comes to (J}) and their implementation. One further consideration is the robustness of the estimator, meaning its sensitivity to slightly varied processing parameters, typically understood as the sensitivity to front panel settings or user manipulation. The actual evaluation of signal processor performance is not at all uncontroversial, since a large number of influencing parameters are relevant and not easy to duplicate between laboratories. For more details on testing methods per se, the reader is referred to Tropea et al. (1988), Tropea (1989) or Hepner (1994). 6.3 Estimation of the Doppler Frequency 307 6.3.1 Spectral Analysis The following discussion is restricted to digital signal processing, since analog devices for spectral analysis play virtually no role any longer in Doppler signal processing. The choice of the sampling frequency is dictated by the Nyquist criterion and the number of sampIes is dictated by either the signal duration, or possibly the processing speed, if the signals are all of long duration. This is clearly the first dilemma encountered, since the signal duration depends on particle trajectory and scattering amplitude (among other things) and varies between zero and some maximum value for the ideal measurement volume crossing. Any fixed sampIe size will necessarily be sub-optimal for any given burst. On the other hand, it is not possible to know signal duration beforehand, so that any attempt at variable width spectral analysis necessarily requires buffer memory with post-optimization. Tropea et al. (1988) investigated the influence of the burst length to record length for an FFT processor operating on 16, 32 or 64 sampIes as a function of SNR. The result reproduced in Fig. 6.20, indicates that for ratios above 0.8, no significant improvement is obtained. This also indicates that not N or J, alone, but the total observation time is the important parameter, at least above N = 32. There have been several commercial (Lading 1987, Meyers and Clemons 1987, Ibrahim and Bachalo 1992) and numerous other implementations of FFT signal processors. Indeed, employing a transient recorder, it is possible to realize such a spectral analysis in any laboratory in software, albeit without sophisticated signal detection and often at much lower processing rates. For speed improvement, most commercial processors use a reduced number of bits, typically :::; 4, quadrature mixing, so that only half the number of sampIes are required, and a time domain window, e.g. Hamming, due to the short signal duration. Most b a ~ 4 c 0 .;::: '" .~ 3 'd 'd ... '" '" if) 'd c 2 I -- r\ • - SNR = 25 dB N=8 - 0 - N=64 .\ - \ \"'---. N=8 - - N=64 2 . -........ '. -. - .............. 1 = -7 dB -0- 4 0 ~-o-o-o~o-o-o-o---Olo-o _ 0 SNR - - • 1 6 I 2 Burst length Irecord interval [ -] ~o 0 0 • DDOD-O-D-O_D_D 2 4 Burst length I re cord interval [ -] Fig. 6.20a,b. Variance of an FFT processor (BSA) as a function of the burst length to record length ratio. a SNR = 25 dB, b SNR = -7 dB 308 6 Signal Detection, Processing and Validation processors operate on fixed length transforms, one exception being a high-speed FFT processor from Aerometrics/TSI Ine. (RSA). Irrespective of the presence or absence of the Doppler signal, this processor transforms the incoming signal at a rale of 20 million transforms per second, updated with every 8 neW sampies. Once adetector gate pulse is present, frequency measurements are stored in a circular 4-cell memory buffer. Measurements with N = 64 (32) are used first. Once the gate length exceeds N =128, the frequency values for N =128 are stored and so on until four valid measurements with N = 512 are obtained. If the burst is shorter, the four most re cent frequency values are averaged and a fluctuation tolerance is applied as one validation criterion. The actual frequency estimation from the sampled signal follows from the power spectral density. Consider a signal of the form (6.105) with a as the maximum signal amplitude at the arrival time tm , 17 being related to the inverse of the residence time squared, fD as the Doppler frequency, and rpa as the signal phase associated with the signal. The discrete Fourier transform (Eq. (6.1)) ofthe signal given in Eq. (6.105) is 5( ~t;l )(J,) = A;, J* ex{-.'(f,~- J.)' + j [2.U, - J, )t~ + "'"1J (6.106) and the power spectral density (Eq. (6.6)) is given by G(fk) = AG exp[ -17GUk - fD)2], 2n 2 k = 0,1.. .,N 12 (6.107) 17 = G 17 (6.108) A = nfs a 2 (6.109) G 217N with fk = k fs 1N and assuming that there is no interference between the spectral peaks at fD and fs - fD' This also assurnes that the burst has not been truncated in the time domain, i.e. that the amplitude becomes negligible at the end of the transformed record. The maximum coefficient value is assumed to arise from the Doppler signal. The second maximum is often also identified and compared in magnitude with the first. Only if apreset ratio of the two is not exceeded, will the data be considered valid, thus avoiding validation of very noisy signals. The inherent frequency resolution of the digital PSD is given by Eq. (6.4) and thus, improves with increasing record length. A significant resolution improvement, typically by a factor of 10, is possible by interpolating the peak position around the maximum coefficient. The interpolation curve should account for the shape of spectral broadening, which for a centered burst with Gaussian envelope, is Seen from Eq. (6.107) to also be Gaussian. If the logarithm of the PSD is considered, 6.3 Estimation of the Doppler Frequency 309 the spectral broadening is described by a parabolic curve. The situation is pictured in Fig. 6.21 The interpolation curve is determined using three coefficient values, and the position of the peak normalized with the spectral resolution is given by (6.110) for a parabolic curve. The frequency of the peak value is then (6.111) Variations on this procedure have been suggested, all of which lead to marginal improvements. Hishida et al. (1989) have suggested a five-point regression of the parabola. Matovic and Tropea (1991) have introduced an iterative, variable width interpolation. If the burst is very short the peak broadens more and the two points on either side of the maximum value are also very high. In this case, the points Gk _ 2 ,Gk'Gk+ 2 are used in the interpolation. Also peak detection algorithms using the first moment of the spectral peak have been suggested. To demonstrate the performance of the three-point, parabolic interpolation a numerical simulation has been performed using the signal (6.112) Note that s(t;) corresponds to a noise-free, modeled signal, whereas s(t;) represents the (noisy) measured signal. For the simulated signals, the sampling parameters fs = 10 and N = 256 have been used and the noise is uncorrelated and Gaussian distributed 1• The model has the (true) parameters a = 0.9, 17 = 0.1, fD = 1.3 and ({Jo = 0.8. The noise power2 was varied logarithmically in 15 steps between e-lO and e2 • For each noise level 10,000 independent realizations of the Doppler signal were generated. For each signal the PSD was calculated and the peak detection using the parabola interpolation was performed. In Fig. 6.22 the empirically found variance of the frequency estimates is shown in comparison to the Cramer-Rao lower bound. The variance is seen to be significantly higher than the CRLB and the estimator is therefore not efficient. Furthermore, a threshold noise power can be recognized, above which the noise dominates the spectrum and the algorithm estimates the frequency randomly across the entire frequency range. A second estimation procedure was realized in this simulation, in which fIlters were applied to the signal before calculation of the spectrum. Filtering in1 2 The frequencies in this example have been non-dimensionalized using 2n. The amplitude is given in arbitrary units. The noise power has been normalized with the signal variance, thus a~ = 1 corresponds to SNR = 0 dB. 310 6 Signal Detection, Processing and Validation Frequency Fig. 6.21. Interpolation of the spectral peak position - . - Three-point parabolic interpolation • __________ 0 - - 0 - 0 - Three-point parabolic interpolation (windowed) /:./---------0 - & - Three-point parabolic interpolation ~~~:"W'd ""d fihered) I/'//x~' .---------.---------~*~~~ • __________ ~a ~ .~.--------~*~ .~ ~ 10-6 .~ ~a ~~ 10 -7 ~::...d:::.-l-L.LL.LllJL------'-----'--'-LLJ--'--Ll_----'--L----'---JLLl--'--Ll_----'--L----'---JLLl--'--Ll_----'--L----'---JLLlJ.JJ 10-4 10 3 10' 10- 1 10 Noise power [-1 Fig. 6.22. Empirically determined variance of frequency estimation procedures using parabolic interpolation of the logarithmic PSD compared to the CRLB. Signals have been simulated creases the SNR and thus an improvement is expected in the frequency estimation. The fIltering consists of two steps. First only that part of the signal exceeding e- 2 of the maximum amplitude was filtered. Then a Blackman window (Rabiner and Gold 1975) was used to suppress edge effects. The results in Fig. 6.22 show the advantage of this signal conditioning compared to the original signal processing. Further results relating to Doppler frequency estimation using spectral analysis can be found in Shinpaugh et al. (1992). 6.3 Estimation ofthe Doppler Frequency 311 6.3.2 Correlation Techniques There have been commercial implementations of laser Doppler signal pro cessors based on the autocorrelation function (ACF) or autocovariance function (ACV) (Lading and Andersen 1988, Jenson 1992, Ikeda et al. 1992), as weH as numerous research efforts, which employ the ACF, e.g. Damp and Sommer (1993) or Damp et al. (1994). The digital realization ofJensen (1992) works in combination with the tapped delay line correlator for burst detection, described in section 6.2. The input signal is sampled simultaneously over 256 sampies at 16 overlapping sampie rates, each with a 1:10 bandwidth. According to a rough frequency estimate from the burst detection unit, one of the sampie rates is chosen and the content of that buffer is transferred to the autocorrelator at the completion of burst detection. Burst centering is possible before performing the autocorrelation over 128 delay bins. The frequency is determined by checking the number of delay coefficients that occur between zero crossings of the first and last valid cyde in the ACF. Cydes are validated according to the absolute maximum of each half cyde, which is related to the SNR. The frequency resolution is improved by linear interpolation between the two coefficients that define the first and last zero crossing of the measurement. Further validation criteria indude a minimum number of cydes in the ACF and a minimum number of delay coefficients between the first and last valid cydes. Alternative methods for frequency estimation from the autocorrelation function have been experimented with. Matovic and Tropea (1989) have used a parametric estimator, termed the autocovariance lag ratio method, to match a cosine wave locally to three of the autocorrelation coefficients. For instance, the autocorrelation function ofthe signal given in Eq. (6.112) is given by (6.113) (6.114) (6.115) Very good performance down to SNR = 0 dB was achieved. This work also confirmed that stable results could be achieved using only al-bit digitization (dipped correlation), which was also exploited by Nakajima et al. (1988). One processor based on the covariance function has been implemented in analogue circuitry (FV A, Dantec Dynamics AIS, Skovlunde, Denmark). The principle can be understood from the block diagram in Fig. 6.23, in which the input Doppler signal is labeled set). This signal takes the form set) = a exp[ -1] (t - tm )2] [1 + rcos(21t fDt+ /Po)] +n(t) where r is the modulation depth of the signal. (6.116) 312 6 Signal Detection, Processing and Validation Gatc from Burst dctcctor Fig. 6.23. Block diagram of an analog laser Doppler processor based on the covariance function The input signal is first bandpass filtered, removing its mean value. This signal is then fed to a hybrid coupler and to a delayer, set to the delay time r, after which a multiplication and integration is performed. The 0 deg and 90 deg signals after the hybrid coupler can be expressed together in complex notation as (6.117) If the envelope is expressed as Ac (t) = aexp[ -1](t- t )2] m (6.118) then the real and imaginary parts of the autocovariance can be expressed as CR=Ac(r)cos(21tfDr)+nR(r) (6.119) C[ = Ac(r)sin(21tfD r )+n[(r) where Ac (r) is the autocovariance of the envelope curve Ac(r)= JA(t)A(t+r)dt (6.120) and nR(r)+jn[(r) represents the complex autocovariance of the noise associated with the signal. Choosing r sufficiently away from r = 0, the autocovariance of the noise is zero so that the ratio (6.121) yields an expression independent of signal amplitude and phase. A burst detector gates the correlation integral and the arctangent of the ratio is performed after integration to yield the Doppler frequency. The choice of the delay time, r, has an influence on the resolution of the frequency measurements. The greater this delay, the higher the resolution. However larger delay times limit the measurement range, since then a 21t phase ambiguity can arise. This range limitation is overcome by measuring the autocovariance at two different values of r. Fur- 6.3 Estimation of the Doppler Frequency 313 ther details of this processor can be found in Dantec (1999). Lading and Andersen (1989) have evaluated the performance of the processor. 6.3.3 Period Timing Devices Electronics which deterrnine signal frequency directly in time domain, do this by measuring the elapsed time T over a number of signal periods Np N f v-- - p (6.122) T These are known as period timing devices (or counters). Although widely used in the 1970s and 1980s, these devices are no longer offered commercially and are being replaced by processors operating in spectral domain or in the correlation domain. The main reason for this is that period timing devices are particularly sensitive to noise and improper frequency values, or "outlier" data points are not uncommon at low SNR values. In its simplest form, this sensitivity is illustrated in Fig. 6.24a, in which random noise fluctuations lead to extra zero crossings, which may register as additional periods. For this reason, not zero crossings, but multiple level crossings are used to count periods. Two such schemes are illustrated in Fig. 6.24b,c respectively. In Fig. 6.24b a valid period is indicated by the level-crossing sequence: negative crossing of L), negative crossing of Lo and negative crossing of L_1' with no positive crossing of L) in between. In Fig. 6.24c the period sequence is given as: positive crossing of L), positive crossing of L 2 and negative crossing of L o, where multiple crossings of L 2 are permissible. The same sequence of level crossings can be used to indicate the beginning and the end of a burst signal, thus activating a gate pulse on which to measure T (see Fig. 6.17). In practice, the level crossing amplitudes are held fixed and the signal amplitude is matched to these amplitudes through a variable input amplification factor. This is an extremely sensitive user adjustment, since too small a factor results in low detection rates and too large a factor leads to level crossing domination through background noise. The bandwidth of acceptable amplification factors is generally not large, requiring considerable user experience for a correct adjustment. For any given measurement point, the signal amplitude may vary considerably from one particle to the next, inherently limiting the efficiency of such an amplitude-based detection scheme. Furthermore, the amplification factor may require considerable adjustment from one measurement point to the next, for instance as the measurement volume approaches a solid wall. This inhibits automation of the data collection process. Recognizing that an optimal match between signal amplitude and detection threshold will be rare, additional safeguards are usually foreseen against noiseinduced errors in the frequency estimation. These generally take the form of a redundant frequency measurement, which must agree within certain bounds of the first measurement. Two registers for N and T are started simultaneously and the measured frequency is only validated under the condition that 314 6 Signal Deteetion, Proeessing and Validation a b L, 1---~--.-~~+-~~~H-~~~~~+-~~~~~,-~" ~~~~~~~~~~~H-HH~~~~~Hrl~~~~~~~~--~ L_I +~~~--''-. ·--'-il:-+-i~.J!-t;-++-hl-.r--Ti-H-!t"-1Ji!-.:r-~-='f--P--l-- LI' L,. L.o_ 1 1 ~J . l! I c I ') I! I 1'-1_ Peri od : I 2 3 4 5 6 7 8 9 10 11 12 13 1~ . I Periods: 1 2 :'I 4 5 6 7 8 9 10 11 12 Fig. 6.24a-c. Amplitude threshold deteetion used for eounting signal periods in period timing deviees. a Zero erossing deteetion, b Amplitude levels symmetrie about zero, e Amplitude levels asymmetrie about zero T2 - 11 < t: I.!!.L N2 T; (6.123) where t: is a freely selectable upper bound lying between 10/0 and 10%. Typical choices for N j / N 2 are 4/8, 5/8, 8/16 or 16/32. At the same time, this condition effectively prescribes a minimum number of periods which must be counted before validation is possible, i.e. N 2 • Nevertheless, given the above validation, it may still be desirable to base the frequency determination on all periods in the signal (total burst), since this will result in a high er accuracy. This can be seen by 6.3 Estimation ofthe Doppler Frequency 315 examining the normalized uncertainty of the frequency estimate, t5fD I fD' due to the uncertainty in the measurement of time T t5fD fD 1 Nase fs Npfose --=--""-- (6.124) where Nase is the number ofhigh speed oscillator dock pulses and fase is the frequency of the oscillator (see Fig. 6.17). This uncertainty decreases with increasing NJ" Utilizing the total burst for frequency estimation has the added advantage that T then corresponds to the transit time, which may be required for estimators offlow quantities (chapter 11) Regardless of whether the frequency is computed over a set number of periods or over the total number of periods in the burst signal, it is still necessary to detect the end of the burst. For one, this determines the signal duration (transit/residence time) and furthermore, the electronics must be reset to allow a new measurement to begin. Alternatively, the device can be reset immediately after the validation step, in which case multiple frequency measurements will be made per burst (continuous mode). This may be desirable in certain liquid flows with very high partide density or for some non-fluid dynamic applications of the laser Doppler technique, e.g. measurement of surface velocities. More detailed considerations concerning refinements of the above estimation procedures can be found in Dopheide and Taux (1984), Dopheide et al. (1990) and Ruck and Pavlovski (1995). In all of these investigations, transient recorders with pre-triggering were used for signal sampling, as first introduced by Petersen (1975) and Durst and Tropea (1977). Thus, burst centering could be performed retroactively and, in some cases, even iterative procedures for the frequency estimation have been suggested. 6.3.4 Quadrature demodulation Quadrature demodulation as a means for laser Doppler frequency estimation is relatively young and not yet commercially available. Although it appears to fail at low SNR (SNR < 10 dB) (Czarske 2001 b), the technique exhibits several features which make it attractive for a number of novellaser Doppler systems, induding those employing two frequency-stabilised, monomode diode lasers (Müller and Dopheide 1993) or other tunable high power laser sources. The technique can also yield time-dependent phase difference information between two signals (section 6.4.3), a feature which may be of particular use in specialized phase Doppler systems. The quadrature demodulation method is weIl established in various disciplines of signal processing and can be explained in terms of the Doppler signal of constant amplitude and the same signal shifted -90 deg, written in complex form as ~(t) = a(t)[ cos( 21tfDt + 9'0)+ j sin(21t fD t + 9'0)] = a(t) exp[j ( 21t fD t + 9'0)] (6.125) 316 6 Signal Detection, Processing and Validation This quadrature signal pair of the input laser Doppler signal can be represented as the real and imaginary part of a complex rotating phasor, whose time dependence is given by s (t)) (t)=arctan( _ 1 _ =2n!v t +rpo (6.126) SR(t) as pictured in Fig. 6.25. The Doppler frequency can be determined from the slope of the phasor (t) with time. The absolute phase shift rpo' is not readily available, since for this a reference time t = 0 must be specified. The implementation of the quadrature demodulation technique is achieved by employing the Hilbert transform (section 6.1.3) to generate a second signal shifted exactly -90 deg in phase to the original input signal. These two signals then become the real and imaginary signals SR(t) and s/(t). Note that the sense of the particle velocity can be recognized by the rotation direction of the phasor. Particle motion in the positive x direction leads to d(t) I dt > 0, i.e. a counterclockwise rotation of the phasor, and motion in the negative x direction leads to d(t) I dt < O. This property has been exploited in a number of novellaser Doppler systems to eliminate optical frequency shifting devices. A graphical representation of the quadrature demodulation technique, adapted from Müller et al. (1996), is shown in Fig. 6.26. Note that the phase angle as a function of time also yields the particle position in the measurement volume. rm{s(t)} rm{s(t)} (=0 Wv WD Re{s(t)} 0 t, tJ Iv =ZnID -a -a 0 I, Re{s(l)} acos(znID1+<rol Z tJ Iv Fig. 6.25. Time dependence of the complex rotating phasor after quadrature demodulation 6.4 Determination ofSignal Phase 317 W)-2nn cos(t,; (t» a(t) s(t) A Re{s(t)} Im{s(t)} Fig. 6.26. Quadrature demodulation byevaluating the phase angle time function (t) 6.4 Determination of Signal Phase In phase Doppler systems, not only the signal frequency but also the signal phase, or better the phase difference between two signals, must be determined as a primary quantity. The standard working equations for phase Doppler systems are Eqs. (5.221) and (5.225), linearly relating the measured phase difference to the particle diameter. To a first approximation, the frequency and phase of each of the two signals are considered constant throughout the duration of the signal. As with frequency determination, the phase difference is also a random variable and there exists a lower bound to the variance of its estimation as given by Eq. (6.100) (H0st-Madsen 1995). Several different techniques for signal phase estimation are presented in this section. 6.4.1 Cross-Spectral Density U sing the notation of seetion 6.1.1, the complex cross-spectral density is defined as (6.127) sometimes expressed as 318 6 Signal Detection, Processing and Validation Qxy(Jk)= Cxy(Jk) +j Qxy(Jk) '-----v-----' Coherence (6.128) '-------v-----' Quadrature where Kk and L are the Fourier transforms of the two signals x; and y; respectively. The cross-spectral density is a complex function, the real part known as the coherence function and the imaginary part called the quadrature function. The phase difference between the two signals x; and y; at the frequency fk is given as (6.129) (s Two sinusoidal signals exactly in phase with one another xy (fk) = 0) will lead to all the signal power appearing in the coherence function. Conversely, if the signals are 90 deg out of phase, all signal power will appear in the quadrature function. Standard FFT algorithms can be used to compute G xy and sxy' Two examples are shown in Fig. 6.27 for signals with SNR = 4.4dB and SNR = 20dB. It is dear that the phase difference LlqJ12 = qJl - qJ2 must be evaluated at the Doppler frequency, LlqJxy = qJx - qJy = SxY(!D), which can be determined using the interpolation schemes outlined in the seetion 6.3.1. A linear interpolation of the phase at f D from the coefficients of Sxy (fk) is sufficient. A detailed analysis of this technique using simulated signals provides performance criteria shown in Fig. 6.28 (Domnick et al. 1988). Given a digitizing resolution of 2 or more bits, reliable phase difference estimates can be expected for a SNR as low as 0 dB. A 2n ambiguity of the phase difference exists with this method; however, some additional computations can resolve this. Large partides yield not only a phase shift but also a time shift of the entire signal, related to the detection volurne displacement, as discussed in section 5.1.3. This time shift can be used to determine the correct multiple of 2n to be used for LlqJxy (!D) > 2n. The general form of the two signals e.g. from two receivers (r = 1,2) can be expressed in terms of the time shift for each signal ±tm,r and their phase shifts ±qJo,r (6.130) The cross-spectral density of these signals in discrete form is given by G _12 a 2 nf exp[ 2N17 =_0_ _ ' (6.131) 2 a nf exp[ 2N17 =_0_ _ ' with the amplitude 6.4 Determination of Signal Phase 319 5 ..!. a " = "0 c.. . E -.; c: oJ:) Vi 0 '".. :, ~ -5 ",• o - 10 I 10 10 Time -;- 10 Time t [fis] I [MH z] 1 "- - 0.5 f- - <-::;':> 1.0 o t [fis] I c , .' ,~ ~ >- .;;:; c: " "0 ] Ü "5l'" B Ü 0.0 I . .1 11. I . 'Cti 180 ~ S ..f .." '" ..r: Q.. 0 I~ QS 0.0 Frcq ucnc)' f [111H z] 0.0 QS Frequency I~ Fig. 6.27a-f. Example processing of two time and phase shifted input phase Doppler signals with SNR = 20 dB and SNR = 4.4 dB. a,b Amplitude of the signals, c,d Cross-spectral density function, e,fPhase function 2 a_ nf G (f) = _0 _ ' exp[ 12 k 2N77 (6.132) and the phase function (12(Jk)=2n(JD - fk)L1t 12 -L1tp12 ' (6.133) 320 ~ .... 0 .... .... Q) Q) '"'" 6 Signal Deteetion, Proeessing and Validation 4 5 points I eyde Phase = 73.5 deg 4.4 points I eyde Phase = 54.5 deg 2 - 0 - 64 points -.6.- 128 points -+- 256 points 5: ~ 4-1:> 6 0 -4 2 4 6 8 -10 Resolution [bit] o 10 20 30 40 SNR [dB] Fig. 6.28a,b. Error in the phase differenee determination using the eross-speetral density method. Results obtained using simulated signals. a Influenee of digitizing resolution, b Influenee of SNR (Domnick et al. 1988) Thus the phase function SI2{fk) is a straight line passing through the desired phase difference Lltp12 at the Doppler frequency, 2nfv' A linearization by means of a straight-line fitting is necessary at low SNR (Fig. 6.27e,f). The slope of the line is just Llt12 . This then resolves the 2n ambiguity using the cross-spectral density. The partide size may be determined either from the phase difference or the time shift (see sections 5.3.2 and 5.3.4). Together, the information can be used to extend the measurement range of the technique to large partide sizes. The procedure is as folIo ws: 1. The partide diameter is first estimated using the time shift method, which of course requires an accurate estimation procedure for the time shift, ,1t12 . 2. The phase difference is then chosen with the addition of 2nN, using an N which yields a diameter dosest to that obtained in step 1. Since this phase difference can be estimated more accurately than the time shift, the diameter computed from the phase difference is used as the final value. This extension of the technique to larger partides also avoids erroneous measurements of larger partides, which is particularly important for accurate mass flux measurements. An example of this signal processing is given in Fig. 6.27 for a phase difference of Lltp12 = -1.628 periods and a time shift of Llt12 = -tl + t 2 = -3.12I-1s. The Doppler frequencyofthese simulated bursts is fv = 0.5031 MHz. For a signal-tonoise ratio of 5 dB, the estimated Doppler frequency is fv = 0.5017 MHz, the phase difference is Lltp12 = -1.62 periods and the time shift ,1t12 = -3.67I-1s. At a SNR = 20dB, these values were respectively fv = 0.5031 MHz, Lltp12 = -1.626 periods and Llt12 = -3.064l-1s. 6.4 Determination ofSignal Phase 321 6.4.2 Covariance Methods In a manner similar to the frequency determination, the covariance function can also be used to determine the phase difference between two input signals. An analog implementation of such a scheme is illustrated in Fig. 6.29 and can be described as folIo ws. Using the notation of section 6.4.1, each of the input signals (r = 1,2) takes the form Sr (t) = arex~ -17 (t - tm.r)2] rr COs(2n fDt + rpo.r) (6.134) where the Doppler frequency is assumed to be the same for each signal. Both input signals are first bandpass filtered to remove the mean and to limit the bandwidth of the noise present in the signals. One signal (r = 1) is passed through a hybrid coupler, transforming it from the real form into the complex form (6.135) where Ar (t) is the amplitude function (6.136) Each of the outputs from the hybrid coupler are multiplied by the other bandpass filtered signal and integrated to yield the real and imaginary part of the cross-covariance function (6.137) (6.138) where A12 (r) is the cross-correlation of the amplitude function and Ltrp12 is the phase difference between the signals, Ltrp12 = rpj - rp2' The ratio of these terms at r =0 yields a quantity independent of signal amplitude and frequency, which Gate from Burs t dctcctor Fig. 6.29. Block diagram of an analog processor for phase difference measurement based on the covariance function 322 6 Signal Detection, Processing and Validation can be used to determine the phase difference between the input signals. C (0) LJIP = arctan-I l_2 12 CR12 (0) (6.139) 6.4.3 Quadrature Methods The quadrature method for determining phase difference between two signals follows directly from its use for frequency determination, discussed in section 6.3.4. If two detector signals are both subjected to the quadrature analysis, then two phasor traces in time are obtained, as illustrated in Fig. 6.30. The vertical distance between the lines corresponds to the phase difference between the signals. The validity of the measured phase difference can be made dependent on the burst amplitude exceeding a certain threshold of the maximum amplitude, for example 10%. One advantage of the quadrature method is that the phase difference can be continually estimated throughout the burst and thus, phase difference changes can be registered. This feature can be quite attractive for some applications, e.g. the dual-burst technique, as discussed in section 8.2.4. As with an analysis using the cross-spectral density (section 6.4.1), a 2n ambiguity also remains with the quadrature method. One possibility to resolve this ~ " 180 ~ ."'" o ..c a. -5 -4 Pig. 6.30. Phasor dependence on time within two phase Doppler signals 6.5 Model-Based Signal Processing 323 is to determine which multiple of 2n is correct, by estimating the time shift between burst envelopes. This can be easily implemented using the Hilbert transform and by computing the envelope function ofthe analytical signal, Eq. (6.22). Due to noise, it is advantageous to then fit a Gaussian envelope to the computed envelope function and base the estimated time lag on this fitted curve. Details of this fitting procedure and its performance can be found in Lehmann and Schombacher (1997). Their simulations indicated that with a proper choice of processing parameters, acceptable results can be achieved for SNR values considerably below 10 dB. 6.5 Model-Based Signal Processing (Contributed byH. Nobach) 6.5.1 Fundamentals Recalling the classification of signal processing methods shown in Fig. 6.2, the iterative methods use the results of a direct estimation and improve the model parameters to better fit the signal or its statistical functions. This can be performed in time, correlation or frequency domain. Such a procedure requires an appropriate parametric model of the signal, a value which indicates the accuracy of the fit (figure of merit) and a strategy which automatically improves the model parameters and minimizes the difference between the model signal and the measured signal or of their statistical functions respectively. Assuming a signal model exists, for instance the signal given by Eq. (6.105) in time domain, by Eq. (6.107) in frequency domain or by Eq. (6.113) in correlation domain, the actual measured signal 5(t;), its PSD cU,), or its ACF R( IJ must be compared to the model signal using some figure of merit, for instance an L 2 norm N-l e(a, T/'!D,(jJO) = ~]s(tJ - s(tJt (6.140) ;=0 N/2 _ 2 eG (AG ,1lG,fD) = I[G(!k)-G(!k)] k=O N/2 _ eR (AR' T/R,fD) =I[R(rn)-R(Tn)] (6.141) 2 (6.142) n=O Note that the optimization is performed only in one of these domains, so that only one of the expressions is used. For statistical functions, which are symmetrical and periodic, only one half of the range is used and for the correlation function, the coefficient at r = 0 is not used because of the noise power, which is concentrated there. For any given parameter set, the figure of merit can be now calculated. Starting with the results from direct parameter estimation as an initial parameter set, 324 6 Signal Detection, Processing and Validation the parameters are improved iteratively, yielding a minimum L 2 norm. This can be carried out using several techniques, e.g. random trial and error methods or genetic algorithms. Presuming convexity, the iterative parameter optimization can be performed with an algorithm, which is similar to the tangent algorithm for the calculation of zeros. The convergence rate is very high, so that the accuracy of each parameter is approximately 10-6 after 10 iterations. However, all iterative estimators fail if the L 2 norm dose to the initial parameter set is not convex. Then the iteration procedure is divergent and simple parameter estimation is not possible. The iterative approach to model-based parameter estimation can approach theoreticallimits of accuracy (CRLB) under ideal conditions. Real signals however, pass through several signal-conditioning steps (e.g. analog fIlters and amplifiers) and und er such circumstances robustness is more important than accuracy. Therefore, in many practical situations, a direct estimation may be almost as effective, and the additional computational costs of implementing an iterative approach may no longer be justified. One interesting application area of iterative parametric estimation is when the parameter to be estimated is no longer directly derivable from the PSD or ACF. Several such situations have been selected below to illustrate the power of model-based signal processing. 6.5.2 Example Applications 6.5.2.1 Estimation of Partic/e Acceleration Normally, the partide velo city is assumed to be constant during its passage through the measurement volume of a laser Doppler system. This may not be true in the case of strong spatial or temporal velocity gradients in the flow field and it may be interesting to estimate the partide acceleration from the acquired Doppler signal. For a constant acceleration, the Doppler frequency can be expressed as (6.143) where im is the instantaneous Doppler frequency at the burst arrival time, i.e. at the middle of the burst or at the time of the maximum signal amplitude, t m • The parameter ß expresses the magnitude of the acceleration, through a = ßL1x, where L1x is the fringe spacing. Figure 6.31 illustrates a simulated Doppler burst from an accelerating partide and with added noise. Examples will now be given of three types of estimators to determine im and ß; a non-parametric estimator, a direct parametric estimator and an iterative parametric estimator. Non-Parametric Estimator. The Doppler frequency is a random variable and can only be estimated as an average value from aseries of data points. Thus, to estimate the frequency as a function of time, a sliding time window is required. A rectangular window is not suitable for Doppler bursts, since the signal ampli- 6.5 Model-Based Signal Processing 325 -1 o 5 10 15 20 25 Normalized time [-1 Fig. 6.31. A simulated burst signal from an accelerating particle with added noise tude at the window edges is high and the window truncates the signal, resulting in a biased frequency estimate. Smoothing windows are therefore preferred, e.g. a Hanning, Hamming or Blackman window. A second effect, which also results in a biased frequeney estimate, is the fact that the signal amplitude is not symmetrie within the window. The amplitude asymmetry distorts the frequency peak in the speetrum, resulting in a translation of the peak center. An equalization of the signal amplitude is desirable. Such equalization can be achieved using normalization with the instantaneous amplitude of the signal envelope. The envelope of the burst ean be calculated from the analytical signal, whieh is derived using the Hilbert transform (Eq. (6.21). Given a measured (noisy) signal ofthe form (6.144) (s(t) is the noise-free model signal), the equalized signal s(f;} ean be ealeulated using s(t.) = s(t) , [s(t;l+j t\{s(t;l}[ (6.145) Figure 6.32 shows the result of the equalization proeedure on the burst from Fig. 6.31. A sliding Blackman window of width Tw = 10 has been applied to the sampie signal in Fig. 6.32 1 • For each window position in time, the instantaneous Doppler frequeney is estimated using the position of the speetral peak, based on a threeI In this example, time has been non-dimensionalized using the sample interval, Llt, =11 J,. 326 6 Signal Detection, Processing and Validation o 5 10 15 20 25 Normalized time [-I Fig. 6.32. The equalized Doppler signal from an accelera ting particle point parabolic fit to the logarithmic spectral values. These estimated frequeneies are only statistical values averaged over the window width. Thus, the achievable dynamic range of frequency changes in time is limited. Nonetheless, for a linear frequency change with time, this is suffieient. Only for a non-linear time dependence would this represent a limitation. Figure 6.33 illustrates the derived frequency changes with time for the signal shown in Fig. 6.32. A linear fit to this frequency-time function directly yields es- Normalized time [-I Fig. 6.33. The estimated Doppler frequency as a function of time from an accelerating particle 6.5 Model-Based Signal Processing 327 timates for the middle frequency Jm and the acceleration parameter ß. The procedure can be greatly simplified if a linear dependence is assumed and only two end points are computed, for instance at the times tA and tB == t A + Llt. The estimates could then be made as (6.146) ~ ~ ß== !(tB)- !(tA) (6.147) Llt This estimation procedure was introduced by Lehmann et al. (1990) and Lehmann and Helbig (1999) and has been found to be very robust. The choice of window or window width are parameters, which may be used to optimize the estimates with respect to bias and variance. Figure 6.34 for instance, shows the empirically found variance of the middle frequency and the acceleration parameter as a function of the window width with respect to the burst duration (time between the e-2 points ofthe maximum amplitude). The two windows are placed at the beginning and end of the burst respectively, as indicated in Fig. 6.35. The minimum variance occurs at slightly different window widths for each parameter and a window width-to-burst-length ratio of 0.7 represents a suitable compromise. Direct Parameter Estimator. For a parametric estimation of the particle acceleration, a signal model must be specified. An appropriate model for a linear frequency change with time can be derived directly from Eq. (6.144). The phase gradient becomes (6.148) a ... 10-5 """;-r,--,-r-,...,-r-rr,,;-r,,-r-,,, Non-parametric estimation Non-parametric estimation CRLB CRLB 10~~~~~~~~~~~~~~ 0.2 0.6 1.0 Window width I burst length [-I 0.2 0.6 1.0 Window width I burst length [-I Fig. 6.34a,b. The empirical estimation variance as a function of the window width-toburst-Iength ratio. aMiddie frequency, b Acceleration parameter 328 6 Signal Detection, Processing and Validation -........ Burst signal envelope ../ " Burst signal Burst length ~ :Ei 1 ~-~---7"""_______------------~""'- S' '" ~c ~O~-r~----------~~------4-=-----------~+-Window width o 5 Windowwidth 10 20 15 25 Normalized time [-I Fig. 6.35. The placement of the time windows for estimation of the particle acceleration and integration leads to an expression for the instantaneous phase rp(t)=rrß(t-tm)2 +2rrfm(t-tm)+rpo (6.149) The discrete burst model becomes (i = 0, 1, ... , N -1 ) (6.150) The discrete Fourier transform of this signal, assuming no interference of the spectral peaks at fm and -fm and for the frequency range 0 to fs /2 (Nyquist frequency) is given by g{S(tJ}(fk) = afs 2 ~ exp( V~ rr 2(fk - fm)2 j(2rrfktm -rpo)] (6.151) 1]-jrrß Using the substitution 1]- j rrß= Bexp(jP) = exp(lnB+ jp) (6.152) leads to the amplitude and phase spectrum of the logarithmic discrete Fourier transform In{ g{s(t; )}(fk)} = a (fk - fm)2 + Cl + j [a 2(Jk - fm)2 +bAfk - fm)+ C2] j '-v--------' Amplitude spectrum ' , Phase spectrum (6.153) 6.5 Model-Based Signal Processing 329 with n 2 cos( -P) (6.154) B C =ln[aJ1t)_lnB 2 1 2 n 2 sin(-P) (6.155) (6.156) B (6.157) C 2 P =m ---2n f" t "1'0 2 Jm m (6.158) From Eq. (6.153), it is apparent that the middle frequency fm of an actual signal s(t;) can be estimated from the discrete Fourier transform of the signal 3{s(t;) Hfd by fitting a parabola to the maximum of the amplitude spectrum and its two neighboring coefficients. This is similar to the signal processing without particle acceleration, except that the amplitude spectrum, rather than the power spectral density is used. Transforming the computed coefficients to the argument f - j"" the remaining coefficients in the logarithmic amplitude and phase spectrum, al' cl' a2 , b2 and c3 ' must be found by curve fitting. Using the relations (6.159) the signal parameters can be estimated as follows AB. ß=--smP (6.160) i!=-BcosP (6.161) b2 t =-- (6.162) n A 2n m 21= A fsJ:;exp(c + I~B) 1 P ({Jo = c2 +-+2nfm tm 2 (6.163) (6.164) These model-based parametric estimators perform well if the measured signal resembles the modeled signal closely. It is, however, sensitive to distortions of the envelope and therefore not robust. Such distortions can arise because the burst length will be dependent on the particle velo city and the envelope length will vary compared with the record length. Furthermore, the envelope can be 330 6 Signal Detection, Processing and Validation distorted due to slit apertures used on the receiving optics, non-linear amplification, filters or window functions. In this particular example, the nonparametric estimator yields superior results. Iterative Parametric Estimator. With the iterative parametric estimator, the aim is to stepwise improve an initial parameter set to obtain agreement between the model signal s(t) and the measured signal s(t;}. The L 2 -norm can be used as a figure of merit (6.165) which has to be minimized (least squares method). For Gaussian distributed noise, this leads to the maximum likelihood estimator. For other noise distributions, this is not the best estimator; however, in most cases it is sufficient and robust. The additional weighting factors wi are necessary if the noise power is not constant. They can be used to suppress very noisy parts of the signal. It is unusual to be able to derive the L 2 -norm explicitly and this is the rational behind an iterative optimization of the parameter set. For the case of an accelerating particle, the problems of envelope distortion, as discussed above, remain and therefore, it is advisable to base the signal model on the equalized signal S(ti), having constant amplitude of 1. The appropriate signal model becomes (6.166) The parameter vector to be optimized is (6.167) Since the equalization yields a signal with time-dependent noise power, a Blackman window is used as a weighting function. Wi 21ti ) -0.08cos( 41ti ) =0.58-0.5cos( N-l N-l (6.168) The original signal is presumed to be centered within the data record, with tm =(N-l)/2. The optimum parameter set is given by the system of equations de dß de =0 dim de d(jJo (6.169) 6.5 Model-Based Signal Processing 331 This system is solved using a tangency algorithm, leading to the iteration step from the n tb to the (n + l)th estimate d 2e dß2 d 2e dßdfm d 2e --dßd({Jo d 2e dßdfm d 2e df:' d 2e dfm d({Jo d 2e dßd({Jo d 2e dfm d({Jo d 2e rp·, f~n) - r'j f~n+I) ({J6n) - ((J6n+l) d({J~ de dß + de =0 dfm de d({Jo (6.170) To prevent divergence after each iteration step, the parameter range and the convexity of the figure of merit are checked using O<fm <fs /2 (6.171) -->0 d2e dß2 (6.172) d2e df:' (6.173) ->0 The convexity check is not necessary for the phase because of its periodicity. These estimators are bias-free and almost efficient for the middle frequency fm and the acceleration parameter ß over a wide range of signal parameters. However, the non-parametric estimator is also quite reliable with much fewer computations. Furthermore, the iterative approach requires good pre-estimates of the parameters. The main advantage of the iterative approach is its possibility to expand the signal model to include additional signal parameters, for instance, higher order derivatives of the velo city. 6.5.2.2 Time-Shift Estimation The time-shift technique, as introduced in section 2.3 and discussed further in section 5.3.4, estimates the particle diameter from the shift in time between signals received on two detectors in space. The magnitude of the time shift depends also on the position of the two detectors, on the detected scattering order and possibly on the refractive index of the particle. Figure 6.36 illustrates a simulated pair ofDoppler signals, shifted in time from one another. Time-Shift Estimation Using the Cross-Spectral Density Function. A time shift of At of a signal s(t) in time domain, with the Fourier transform 3{s(t)} = S(f), yields in the frequency domain 3{ s(t- L1t)} = ~(f)exp(-j 2nfLlt) (6.174) For two signals SI (t) and S2 (t) = SI (t - L1tI2 ) with the time shift L1t12 , the gradient ofthe phase in the quadrature spectrum Sl2(f) = arg(§.:(f)~2(f) is 332 6 Signal Detection, Processing and Validation Receiver 1 o -1 Receiver 2 o -1 o 15 10 5 20 25 Normalized time [-1 Fig. 6.36. Time-shifted Doppler signals from two detectors dS df 12 = -2n Llt (6.175) 12 where S;' (f) is the complex conjugate of SI (f). This relation has already been used in section 6.4.1 for resolving the 2n ambiguity in phase difference determination for phase Doppler signals (Eq. (6.134». Therefore, the time shift estimation requires only a simple extension of the normal phase Doppler processing algorithm. First the cross-spectral density function of the two input signals is computed. Close to the Doppler frequency, the phase of the cross-spectral density function is a linear function off A linear interpolation can be used to derive the phase gradient at fD and thus, the time shift Llt12 • Example signals and their cross-spectral density functions have been given in Fig. 6.27. Estimation from Individual Arrival Times. In principle, the time shift between two signals can also be estimated using the difference of their arrival times (times of maximum amplitude). This requires an accurate estimation of the arrival times. Since the power spectral density or the autocorrelation functions of each signal are statistical functions, they cannot retain the phase or absolute time information, hence, they are not suitable to derive the individual arrival times. Although the power spectral density looses the phase or absolute time information, the Fourier transform itself retains this information, since the original signal can always be reconstructed from its Fourier transform. The arrival time t m of a signal s(t - t m ) can be interpreted as a time shift of the signal s(t) relative to the time t =0, leading to a phase gradient, which is proportional to the time 6.5 Model-Based Signal Processing 333 shift or the arrival time respectively. (6.176) For a given discrete signal s(ti ) the discrete Fourier transform g{s(ti )} = S(fk) is calculated. From the phase spectrum ?(fk) = arg(s.(!k»' the gradient of the phase dose to the known Doppler frequency is estimated (after the correction of the 21t ambiguity). The Doppler frequency can be estimated, as described previously, from the cross-spectral density. The arrival time then becomes 1 d? t =--m 21t df (6.177) The time shift between two signals 51 (ti) and 52 (ti) can then be calculated from the difference of their arrival times. Note that this procedure is similar to the estimation from the cross-spectral density, since the phase of the product S~(fk)S2(fk) is the difference of the two individual phases. Accordingly, the phase gradient is proportional to the difference of the arrival times. 6.5.2.3 Dual-Burst Processing When the partide size is large compared with the beam waist diameter in the measurement volume of a phase Doppler system, it is possible to choose detector positions, such that not one but two signals are obtained on each detector for each partide trajectory. These two signals may have varying degrees of overlap, but originate from two different light scattering orders reaching the detectors. Two redundant size measurements can therefore be performed, if the signal processing is able to identify the change of phase within the overlapping signals. Figure 6.37 illustrates a simulated dual-burst signal. Having redundant size measurements provides possibilities for also determining the partide refractive index. More details about the dual-burst phase Doppler technique are given in section 8.2.4. If the two overlapping signals are in phase, it is not difficult to estimate the Doppler frequency, as demonstrated by the PSD shown in Fig. 6.38a. However, if the signals are out of phase, the frequency estimate from the PSD can be erroneous, as illustrated in Fig. 6.38b. Figure 6.39 indicates, for the simulated signal in Fig. 6.37, the expected deviation of the frequency estimate as a function of the phase shift of the interfering signals. A three-point parabolic interpolation has been used as an estimator. Note that the sharp jumps at -1t and +1t become less sharp for noisy signals. The only effective method to correctly estimate the Doppler frequency and the phases of each of the signal portions is an iterative model-based procedure (Nobach 1999). The two overlapping signals are modeled using the signal s(t) = al exq-17 (ti -tmS]cos(21tfDti +lPo,l) (6.178) 334 6 Signal Detection, Processing and Validation o 10 15 20 25 Normalized time [-I Fig. 6.37. A simulated dual-burst signal b a 1.5 0.5 ,----r---,--.---.----,--,----.-----.-,----, I , Q ~ 1.0 r- - 0.5 I- - 0.0 o 2 Normalized frequency [-I o 2 Normalized frequency [-] Fig. 6.38a,b. The power spectral density of a dual-burst signal. a The signals are in phase, b The signals have a phase shift of 180 deg assuming that the Doppler frequency and the burst width are constant in both portions of the signal. The burst width is defined using a fraction of the maximum intensity in each of the burst. Note that this signal model will not be ideal for all trajectories. For a more general model, the burst width of each signal portion must also be modeled separately. An initial parameter set is estimated by splitting the signal in the middle and performing a conventional, non-parametric estimate of the frequency and phase on each signal portion. These initial estimates are then used in an iteration procedure to minimize the L2 -norm 6.5 Model-Based Signal Processing ";;;' 335 2 <U :E -; ,/ :::u <U .§'1 c oS 1;; os;: <U ~o U C <U 6- ... <U ~ -1 .,/' -2 -180 -90 0 90 180 Phase shift [deg] Fig. 6.39. Deviation of the Doppler frequency estimate (in number of spectrallines) over the phase shift of the two in terfering signals N-l e(al'a 2'!D' 1J, fPool' fPool' tmol' tmoJ = I[ s(t;) - s(tj)f (6.179) j=ü In fact, exact solutions to this optimization problem can be found for the amplitudes a 1 and a2 • For the remaining parameters, a tangency algorithm, similar to that described for the estimation of particle acceleration, must be used. Also the convexity of the solution should be checked at each iteration step. 338 7 Laser Doppler Systems 7.1 Input Parameters from the Flow and Test Rig The proper choice and layout of a measurement system can only be made if the required temporal and spatial resolution for the velocity measurement are first determined. For this, an intimate knowledge of the flow field being studied is required and the following section is devoted to reviewing the means of describing flow fields. Quantities which are typically of interest to measure with a laser Doppler system will be identified. 7.1.1 Description ofthe Flow Field 7.1.1.1 Equations of Motion The laser Doppler instrument provides a field description (Eulerian) of the flow, rather than a material description (Lagrangian). The measurement volume of the laser Doppler system is fixed in space and does not foHow an ensemble of material fluid elements l . The flow velocity vector ui is obtained at a particular point in space, x j' at the time t (7.1) The equations governing the flow field are weH established and for a N ewtonian fluid, i.e. a fluid far which the stress on a fluid element is a linear function ofthe rate of strain, they take the foHowing form (Spurk 1997): • Continuity equation Dp dU i -+p-=O Dt dX i (7.2) • Navier-Stokes equations (7.3) • Energyequation (7.4) 1 This is not entirely correct, since the motion of tracer particles measured using the laser Doppler technique corresponds to the motion of a fluid element (Lagrangian). However any changes of particle velocity over the measurement volume are usually neglected - one exception is the direct measurement of acceleration, as described in section 6.5.2.1. 7.1 Input Parameters from the Flow and Test Rig 339 where D / Dt is the material derivative D d dt d ' dx; - = -+ u· Dt d dt d dX 1 d dX 2 = -+ u - + u - + u l 2 d dX 3 3 -- (7.5) P is the fluid density, b; are the components of the vectorial body force acting on an Eulerian fluid element, pis the mechanical pressure, 1] and ..r are the first and second coefficients of viscosity, e is the intern al energy of the fluid element, A is the coefficient of heat conduction T is the temperature and t: is the dissipation rate of mechanical energy into heat (per unit mass), given by (7.6) with the symmetrie strain tensor given by (7.7) Henceforth, only the special case of incompressible and adiabatic flow will be considered, in which case the governing equations can be written as • Continuityequation (7.8) • Navier-Stokes equations (7.9) • Energy equation De p-=pt: Dt (7.10) In non-dimensional form, using a typicallength scale (L) and velocity scale (u), the Navier-Stokes equations take the form (omitting body forces) du; * dU; dP* 1 d2 u; --* +U j - - * =---* +--*--, dt dX j dx; Re dXjdX j (7.11) where the asterisk (*) denotes dimensionless quantities and the Reynolds number is given by (7.12) and represents the ratio of inertial to viscous forces. 340 7 Laser Doppler Systems At high Reynolds numbers the last term in Eq. (7.11) can be neglected and the Euler equation results, which is also valid for inviscid flows, V = o. b dP P Dt -P; Du; _ dx; (7.13) 7.7.7.2 Description of Turbulent Flow Fields Solutions in closed form can seldom be found to the above equations. Exceptions are simple one or two-dimensionallaminar flows or irrotational, inviscid flows, known as potential flows. More common are high Reynolds number, turbulent flow fields, for which no simplifying measures can be taken. Under these circumstances, the velo city field description is often modified by averaging over space or time. The former leads to a Large Eddy ~imulation (LES) description of the flow field, the latter to the Reynolds Averaged Navier ~tokes (RANS) description. The RANS description is now considered further, since the laser Doppler is not a good device for spatial averaging, hence, is seldom used in conjunction with LES descriptions of the flow. For a RANS description of the flow field, the instantaneous velo city is first split into a base flow and a fluctuation component (Reynolds decomposition) (7.14) where the overbar denotes an average over many flow ensembles, u: kl • -u,x"t=lm-L."u, ( ) 1. 1 ~ (kl( x"t ) I ] N -'>= N k~l I (7.15) ] For statistically stationary flow fields, this ensemble average value of the velocity is independent of time and can be replaced by the time average (principle of ergodicity), as discussed in section 10.2. (7.16) The overbar will subsequently be used to denote this time average. If also the pressure is treated in a similar manner, i.e. P= p+ p' (7.17) and the rules for averages are adhered to g=g, g+f=g+f, gf=g f, dg dS - dg dS - the Reynolds equations are obtained from the Navier-Stokes equations (7.18) 7.1 Input Parameters from the Flow and Test Rig _ dU; dP d2 u; d(pU;U;) pu-=pb - - + 1 7 - J dx j , dX; dx jdX j dx j 341 (7.19) The last term in this equation, originating from the substitution U =U + U' into the eonveetive terms on the left-hand side, resembles the gradient of a stress, similar to the gradient of the viseous stress (seeond last term), and is therefore plaeed on the right-hand side. The terms of the symmetrie tensor pu;uj are known as the Reynolds stresses and are the foeus of most modeling efforts in eomputational fluid dynamies (CFD). They physieally represent the mean transport of the i th eomponent of momentum in the j th direetion due to turbulent flow fluetuations. A relative turbulenee intensity ean be defined for eaeh veloeity eomponent. For example the x 1 eomponent is given by Tu 1 = ,,),Y; (U1U 1 _ (7.20) U where again, u is some representative flow veloeity, perhaps a mean inlet flow velo city or an undisturbed flow velo city outside of a boundary layer. The overall turbulenee intensity is given by -,,),Y; 1 (u; u; Tu= 3 u (7.21) If the turbulenee is isotropie then u{u{ = u~u~ = u~u~ and Eq. (7.21) reduees to Eq. (7.20). A further quantity of interest is the turbulent kinetie energy (per unit mass)l 1-,-, Bk = "2 UjUj (7.22) whieh is also an invariant of the stress tensor. Sinee many statistieal turbulenee models are based on the loeal variations of Bk' it is a quantity of great interest to measure. From Eq. (7.22), it is clear that all three eomponents ofvelocity are required for this; however, often eompromises are made. If only two eomponents are available, for example, u1 and u2 , the third eomponent is estimated as the average of u1 and uZ ' leading to (7.23) A general transport equation for the Reynolds stresses ean be derived from the Navier-Stokes equations (Hinze 1975) and, negleeting body forees, ean be written as 1 Turbulent kinetic energy is often designated with k. To avoid confusion with wavenumber, Bk has been used here. 342 7 Laser Doppler Systems dU;U; dt + -dU;U; Ul~ '---v----' '-y-J change with time convective transport =_~(u;u~u~)_~(dU;P + dU;P) dX 1 " \ P dX j v prod~ction , I diffusive transport - du. - dU P ( dU' dU~) dU~dU~ d (dU~U~) -u~u;--' -u~u;-j +- --' +--' -2v--'- ' +v- -'-' 'dX/ 'dX 1 P dX j dX j dX 1dX 1 dX 1 dX 1 , dX j '--v---------' '----y----J ~ pressure I strain coupling dissipation molecular diffusion (7.24) The special case of i = j yields a transport equation for the turbulent kinetic energy Ek (7.25) In turbulence models, many of the terms in the above equations must be approximated with semi-empirical expressions (e.g. Wilcox 1993, Ferziger and Peric 1996) and these must then be validated in se1ected test cases by direct measurements. Besides requiring a direct measurement of the Reynolds stresses, this places demands on the determination of gradients, which, using a point measurement technique, is achieved by an appropriate choice of measurement point grid spacing. Clearly, regions of the flow with higher gradients and second-order gradients require a finer mesh of measurement points. If the velo city fluctuations in the x j and x j directions are completely independent of one another, all products u;uj are zero. A measure of the degree of dependence is the correlation function 1 (7.26) or the normalized correlation function (7.27) Even more general is the space-time correlation function Rjj(x k ,t,Tk' r) = u;(x k ,t) u;(x k + Tk't+ r) (7.28) or its normalized function (7.29) 1 The correlation function, when computed using fluctuating quantities (with the mean removed) is called the covariance function. Further remarks about the correlation function can be found in section 10.1 7.1 Input Parameters from the Flow and Test Rig 343 The temporal correlation function is obtained for rk = 0, the spatial for T = o. The Reynolds stresses, as appearing in Eq. (7.19), are seen to be a special case of the space-time correlation function, when both rk = 0 and T = o. For rk very large, the velo city fluctuations become statistically independent from one another and the correlation function goes to zero. A measure for the spatial correlation extent, the integrallength scale, is given for the u1 velo city component by (7.30) In a similar manner, the integral time scale can be defined as (7.31) This quantity can be graphically depicted as shown in Fig. 7.2. Information about the smallest scales of motion in a fluid can be obtained through dimensional arguments (Tennekes and Lumley 1972). At the smallest scales of motion, turbulent kinetic energy is dissipated into intern al energy (heat) through the action of viscosity. Considering the units of viscosity v[m 2 s- 1 ] and dissipation rate per unit mass c[m 2 s-3 ], a length scale can be obtained as _(~)X 7J K - c (7.32) 7JK is known as the Kolmogorov length scale. A corresponding time scale is given by 1.0 0.5 O.O .J.......--=' r : : ; - - - - - - - - - - - - - - - -~ Tu l 11 Fig. 7.2. Definition of the integral time seale and the Taylar microseale 344 7 Laser Doppler Systems (7.33) and a veloeity seale by (7.34) Note that a Reynolds number built on these seales yields Re= 'hvK =1 (7.35) V If the typieal kinetie energy of large seale motions is given by U /2 and they lose this energy in a typical time seale of L / u, where L is a maero length seale of the flow, the rate of energy loss (per unit mass), whieh must be equal to the dissipation rate (per unit mass) on average, is given by u2 U u 3 t:=--=2 L (7.36) L Thus the ratio of the smallest to largest seales of motion in the flow is equal to (7.37) This relation is of utmost importanee for subsequent eonsiderations, sinee it allows an estimate of the ultimate spatial resolution required from the measurement system in order to eapture all seales of motion, in terms of known quantities, u, L and v. The dissipation rate per unit mass of turbulent kinetie energy, t:, is an elusive quantity at best. It ean be written as !...=2(~)2 +(dU2)2 +(~)2 +(~)2 +2(dU2)2 +(dU3)2 v dX j --- dX j dX j dX 2 dX 2 dX 2 --- (7.38) +- +- +2 - )2 +2--+2--+2-( dU )2 (dU 2)2 dX 3 dX 3 j (dU 3 dX 3 dU dU 2 dX 2 dX j j dU dU 3 dX 3 dX j j dU 2 dU 3 dX 3 dX 2 There have been very few attempts to measure so me or all 12 terms with either hot-wire anemometry (Browne et al. 1978) or with a laser Doppler system (Benediet 1995). Not only are nine different gradients to be measured, but the seal es at whieh these must be evaluated are of the order of 11h to 5 rh. If the turbulenee field is loeally isotropie, Eq. (7.38) ean be approximated by (Hinze 1975) (7.39) 7.1 Input Parameters from the Flow and Test Rig 345 whieh requires information about only one velocity gradient. This gradient ean be aequired in many eases indireetly, using the Taylor hypo thesis. If «~, spatial velocity fluetuations will appear almost undistorted as temporal fluetuations when these are rapidly conveeted through the measurement position by the mean flow. Thus, t= x /uor u; d dx 1 d (7.40) e.g. u dt The temporal gradient is mueh easier to aequire than the spatial gradient, sinee it requires only a single measurement position and ean be computed direetly from the measured time series of velo city. A second method of estimating the velo city gradient term is direetly through the spatial eorrelation function (7.41) Using the notation (7.42) for the spatially averaged mean square of the veloeity fluetuations and a Taylor series expansion, the eorrelation function ean be approximated as (7.43) or ( {0) [1- Pii(xj,r 2 du k )] ) dr: hl (7.44) 2 whieh requires a measurement of Pii (x j ,rk ), i.e. a two-point veloeity measurement with the two measurement positions separated by the veetor rk • If the Taylor expansion is not used, then the spatial derivative is given by (Hinze 1975) d;;;; I. k rk ==0 U~2 (~~; )2 i (7.45) k rJ.. =0 This suggests that several two-point measurements must be performed with varying rk in order to measure the parabolie behavior of the eorrelation function at small values of rk • If a length seale A;;,k is defined as 346 7 Laser Doppler Systems Ir; 1 2 p;; (r) = 1- ----;2 , 2/1. ji,i p.(r )=1- Irk l2 /I k /l.? ii,k (7.46) then (7.47) where /1;;;,k is known as the Taylor microscale (Taylor 1935) and represents the average dimension of eddies responsible for dissipation, Thus, the homogeneous dissipation rate per unit mass may be determined by parabolic fits to the nine different two-point correlation functions, 7.1.1.3 Velocity Spectra All of the flow quantities discussed above can be considered either velocity amplitude related measures or measures related to the spatial structure of turbulent flow fields, Spatial structures, whether quasi-statie (Taylor hypothesis) or dynamically changing, when convected by a velo city measurement point, will result in temporal tluctuations of velo city, Thus, information about the distribution of energy over different length scales will be available through a frequency domain analysis of the velocity time series. The question remains, how much of the spatial information is recoverable from a single-point, or multi-point velocity measurement? The spatial correlation functions R;j(xk,t,rk,O) already carry complete information about the spatial tlow field structure. The Fourier transform of these functions over all possible separations, rk' yields the wavenumber spectrum tensor (or velocity-spectrum tensor.) (Tennekes and Lumley, 1972) (7.48) where the overbar denotes time average and the j in the exponential is ? = -1, not the index j. k; is the wavenumber vector with components k; = 21t I /1;;, /1;; being the wavelength of the disturbance1• The sum of the diagonal components of (JJ;j' i.e. QJIl + 1/J 22 + QJ33 represents the total kinetic energy per unit mass at a given wavenumber. The directional information can be removed by integrating over spherical surfaces of radii Ik; 1= k , yielding the energy spectrum 1 A; is a wavelength, not to be confused with A, the coefficient ofheat conduction in Eq. A;;,k> defined in Eq. (7.46). (7.4) or with the Taylor microseale, 7.1 Input Parameters from the Flowand Test Rig B(k) =.!. 2 JJ lP;;{k) ds 347 (7.49) 5 The integral of this three-dimensional energy spectrum over all k is equal to the kinetic energy per unit mass. (7.50) Thus, B(k)dk is the contribution to Bk from all wavenumbers within a shell of radius k <Ik i I:::; k + dk. The dissipation rate per unit mass can also be related to the wavenumber spectrum through (Pope 2000) JlJ2Ve "21 lP (k = &= ii j ) (7.51) dk j Generally, it is not feasible to measure the three-dimensional, threecomponent wavenumber spectrum, since this involves spatial correlations of all velocity components with separations in all directions. Most measurements involve one-velocity component, one-dimensional spatial correlations or onevelo city component, time correlations, after which Taylor's hypo thesis is used to relate these to spatial correlations in the main flow direction. The most common spatial correlations are Rn (rp r2 = 0,r3 = 0) and R22 (rpO,O), where the main flow direction is along the x j axis. These correlation functions are called the longitudinal and transverse correlations respectively and examples are depicted schematically in Fig. 7.3. The corresponding longitudinal and transverse spectra are JR 2n ~ll (k = j ) 1 += • ll (7.52) (rj ,0,0 )exp( -J k j r j ) drj _= (7.53) Transverse correlation function g (r): Longitudinal correlation functionf(r): Mainflow~ Mainflow ~, directio~ u;. ui direction ~._._._.~~- (u;)' ~rl (u; )' ~ => :--:::~ -;,;U; rj r Fig. 7.3. Longitudinal and transverse correlation functions r 348 7 Laser Doppler Systems The longitudinal integral length scale can be directly related to the onedimensional longitudinal spectrum through r/l11 (0) L _ u - TC (7.54) (U;2) Also, the Taylor microscale ,111,1 can be estimated from the normalized time correlation function. This microseale divided by the mean convective velo city corresponds to the intersection of a parabolic fit to the time correlation at small T values with the x axis, as shown in Fig. 7.2 For isotropie turbulence, the one-dimensional spectrum can be related to the energy spectrum through (Hinze 1975, Batchelor 1953) ~ E(k)=e~(! dr/l11) dk k dk (7.55) If the Kolmogorov similarity hypotheses are applied, even more convenient forms for the one-dimensional spectrum can be obtained. The first hypo thesis states that the statistics of velocity fluctuations in the equilibrium range of wavenumbers are uniquely determined by c and v. The equilibrium range of the spectrum comprises all wavenumbers above the so-called energy containing wavenumbers. The second similarity hypo thesis relates to the inertial subrange, within which a universal form for the spectrum can be expected, dependent only on c. The inertial subrange is that portion of the equilibrium range extending up to wavelengths responsible for dissipation. Within this range, the following functional relations hold (Bradshaw 1971, Pope 2000) E(k)=C c~ k- X C= 1.5 (7.56) r/l 11 (k) = C1 c~ k1 -X 1 Cl ",,0.49 (7.57) r/l 22 (k) = Cl1 c~ k1-X 1 C: "" 0.65 (7.58) where the constants have been empirically deterrnined. These expressions provide a means for estimating the dissipation rate c from measured, onedimensional spectra. The inertial subrange of the spectrum, exhibiting a -5/3 slope with k 1 , can be extrapolated to k 1 = 1 and c can be directly computed from the value of the extrapolation at that point. The energy-containing range of the spectrum below the inertial subrange and the dissipative range above, can also be incorporated into a model spectrum covering all wavelengths. One such model spectrum, discussed by Pope (2000) is given by (7.59) with the energy-containing function 7.1 Input Parameters from the Flow and Test Rig 349 Po is taken as 2 and CL is a positive eonstant. Lk is the length seale Lk = k 3/ 2 I e. IL tends to unity for large k. Furthermore, the dissipative range is deseribed by (7.61) where ß and c'l are positive constants. For small k1J this function also tends to unity, thus reducing Eq. (7.59) to Eq. (7.56) in the inertial subrange. Experiments at high Reynolds numbers have yielded the empirieal values ß = 5.2, c'l = 0.40 and CL = 6.78. Figure 7.4 summarizes the speetral behavior of isotropie turbulenee for a Reynolds number based on the Taylor mieroseale of Re,! = 500. E(k) 105r-""TITIr-""TITIr-~~rr=--~~~-.~~~-.~~~ 1Jv' 10' 10' 10' 10 Fig. 7.4. Model spectrum for Re 4 = 500 (Eq. (7.59» 7.1.1.4 Turbulent Boundary Layers From dimensional arguments and experimental data, the mean velocity profile near the wall seales with the Ioeal wall shear velo city ur' given by (7.62) where r w is the wall shear stress defined by 350 7 Laser Doppler Systems (7.63) The coordinate system for boundary layer investigations is chosen, such that the x axis lies in the main flow direction parallel to the wall and y is normal to the wall. The universallogarithrnic law of the wall describes the velo city profile of a turbulent boundary layer and applies over a wide range of Reynolds numbers. U 1 ur Ur K V - = - 1ny-+C (7.64) often written as 1 u+ =-lny+ +C (7.65) K u+ = Tl / Ur and y+ = YU r / V are known as wall coordinates and K takes the approximate value 0.41 (von Karm1m constant). Cis a constant which can be empirically determined for various flow types and can also vary with pressure gradient. Equation (7.65) is valid in the approximate range y+ > 30, with the upper limit being flow dependent. Graphically, the entire mean velo city profile takes the form shown in Fig. 7.5, indicating also an inner viscous sublayer (0< y+ < 5) and an intermediate (buffer) region (5< f < 30). The viscous sublayer is dominated by viscous forces (over inertial forces) and exhibits a linear velo city profile. .-:: 25 ~ 'ü 0 20 ~ ,/ ~ '" '"..E: / 15 0 Z 10 ~ ~ 5 0 I 10 100 Norma lizcd wall distancc Fig. 7.5. Logarithmic law-of-the-wall for turbulent boundary layers 1000 y· '-1 7.1 Input Parameters [rom the F10w and Test Rig 351 2 - Ur u=-y (7.66) V which also provides a method of measuring Ur (or r w)' provided the spatial resolution is sufficient to resolve several points in this layer, a topic addressed in the following section. A somewhat more useful relation for deterrnining ur from a data fit was proposed by Durst et al. (1996) 2 u=.!:!..!..(y- Yo)+C 2(y- YO)2 +C 4(y- Yo)4 +Cs(Y- YO)5 (7.67) V which is valid up to about y+ < 12. C2 , C4 and C s are constants. For pipe or channel flows C2 = 2R V where R is the pipe radius or the half-channel height. This estimate satisfies the momentum equation and as such provides a more reliable basic for ur' This expression allows both ur and the effective origin of the y translation to be approximated simultaneously -u; / 7.1.2 Necessary Spatial and Temporal Resolution The spatial and temporal resolution of a laser Doppler system depends on the opticallayout, the signal detection electronics, the validation scheme and the particle seeding. Furthermore, there are several subtle interconnections between the spatial resolution and the temporal resolution. Before proceeding to the layout of the laser Doppler system therefore, it is advisable to estimate the necessary spatial and temporal resolution, as dictated by the flow field and by the desired measurement quantities and their corresponding measurement accuracy. 7.1.2.1 Spatial Resolution The flow field dictates the necessary spatial resolution. There are basically three situations to consider, not all of which may be important for a particular application. The three situations are as follows: • Measurements, where a mean velocity gradient exists across the detection volume • Measurement of central moments at a position free of mean gradients • Measurement of spatial correlations (two-point measurements) The last of these situations will be discussed separately in section 7.4.4. Measurements with a Mean Velocity Gradient. Any validated particle passing through the detection volume of the laser Doppler system will contribute to the computed statistics. At first glance, any overall statistic rp will therefore be a volume-time integral ofthe desired quantity. fPDV = lim..!:..fT(_1 frp(x; ,t) d Vd]dt T->=T o Vd Vd (7.68) 352 7 Laser Doppler Systems There are two reasons why this might not be so straightforward. First, the laser Doppler system does not deliver a velo city for all positions along a partide trajectory lying within the detection volume, but only a single velo city. How this value relates to the actual velocity experienced by the partide across the volume will depend on specifics oE the signal processor and possibly the trajectory itself. The second problem with a simple volume integral is that the prob ability of certain trajectories may depend on the flow field. For instance regions of higher mean velocity within the detection volume may experience a higher number of particle occurrences than regions of low velocity. This can lead to a systematic bias of moment estimators and will be discussed briefly at the end of this section. For the present discussion, this effect will be neglected, meaning that the absolute difference of the mean velo city occurring throughout the detection volurne is assumed small compared with the mean velo city, hence the partide arrival rate across the detection volume is constant. Particle concentration gradients across the detection volume will also be neglected. However, the first difficulty remains and some simplifying assumptions will be made before continuing, most of which are quite reasonable for many applications. This treatment paralleIs closely that given by Durst et al. (1993, 1995, 1998). • The seed particles are monodispersed, resulting in the same detection volume for all velo city values • The detection probability is unity for any particle trajectory cutting any section of the detection volume. In some cases, this assumption will require a frequency shift to insure that sufficient signal periods are available for the processor to validate the signal. The situation to be considered is illustrated in Fig. 7.6. The main flow direction is parallel to the Xl axis. A mean velocity gradient exists in the x 2 direction only. Partide trajectories are dose to parallel with the Xl axis; in any case, all 1 X il--__--!~ J\lean vclocity gradient II,(X,) ....... (// ,, ~rl,; ......... ~..._ ..~..- ·······················--·-i - . I 7 Paniclc I rajeClory DC1CCI io n volumc ~---,'----., .... ....... _ _1 1- Deleclion volume dia meier 2b. Registered veloci t y v-aluc Fig. 7.6. Mean flow velocity gradient across a laser Doppler detection volume 7.1 Input Parameters from the Flowand Test Rig 353 particle trajectories are assumed to pass through the center plane of the detection volume XI = xI,c' The registered velo city for each particle corresponds to the velo city value when the particle is in the XI = xI,c plane, thus the XI and x 3 variation of velo city is removed from the problem. The volume integral now becomes an area integral of the quantity of interest over the detection area lying in the XI = xI,c plane (A d ). Generically, the mean value of some quantity rp over the detection volume (DV) becomes (7.69) The overbar denotes temporal averaging, the subscript (DV) denotes averaging over the detection volume, in this case achieved with an area integral. The integration is performed over the x 2 and x 3 directions. A d is the total area of integration. Two integration areas will be considered. In the one case the detection area on the XI = xI,c plane will be considered an ellipse, in the second case a rectangle. These two areas are pictured in Fig. 7.7, the latter corresponding to a measurement volume truncated about its x 3 center, for instance using a slit aperture in a receiving optics placed in side-scatter. The projected slit half-width is denoted by c~. If the time and area integrals in Eq. (7.69) are reversed in order, then the mean value 7j5 will be a function of x 2 only, i.e. 7j5 = 7j5(x 2 ). Then the integration over x 3 can be carried out and this results simply in the x 3 width of the detection volume. For elliptic and a rectangular detection volumes, Eq. (7.69) becomes respectively 2 _ rpDV 2 =;t; X2"J+ bd _ d x",-b d rp 1 (X 2 - X2,c ) d 2 bd X2 (7.70) (7.71) These integrals can now be evaluated for various measurement quantities, 7j5. The mean and fluctuating flow velo city will be examined, as defined by Eqs. (7.14) and (7.16). The mean velocity can be expanded in a Taylor series about its value on the centerline, (7.72) which, when inserted into Eqs. (7.70) and (7.71) for 7j5 and evaluated up to second-order terms, yields for the elliptic detection area 354 7 Laser Doppler Systems b a Rcfcrencc pla ne Deleclion volumc x, Rcctangular dClccl io n arca / DCICClion volumc flow dircclio n Elliplical dctcct ion a rca X~.,( l" Dirccl io n of obscrval ion . . . . . . . . . ._ CJ~d Ao I\. xl .( Fig. 7.7a,b. Detection area on the rectangular detection area XI = XI,c plane. a For an elliptic detection area, b For a (7.73) and for the rectangular detection area. (7.74) These two expressions indicate the first-order difference between the measured mean velocity and the actual mean at the center plane of the detection volume. For a linear mean velocity gradient, no error occurs since the second derivative in Eqs. (7.73) and (7.74) is zero. In narrow channels with strong spatial velo city gradients however, this linear assumption may be invalid and the detection volurne dimension bd must be decreased a,ccordingly. Alternatively, Eqs. (7.73) and (7.74) can be solved for TI; (x 2 .c>, in which case the gradient terms bemme correction terms to the measured me an velo city UI,DV' 7.1 Input Parameters from the Flow and Test Rig 355 The fluctuating velo city registered over the detection volume is the difference between the instantaneous velo city and the mean value over the detection volurne (7.75) whereby Eqs. (7.73) or (7.74) can be used to substitute for U;,DV' ul(xj,t) can be divided into a mean (U; (x;» and fluctuating part (u~(x j,t», and the fluctuating part can be expanded in a Taylor series about the plane x 2 = x 2,c' yielding (7.76) _( - l<t x 2 c , ) 2 - bd d 2U;(x 2 )1 X 2 dX 2 {1/8 forellipsoidaldetectionvolume 1/6 for rectangular detection volume X2=X 2 ,c Ifnow the integrals Eqs. (7.70) and (7.71) are evaluated with 7p=U{2(X2) and only second-order derivatives are retained, the measured mean square velocity fluctuations for an elliptic detection area will be (7.77) and for a rectangular detection (7.78) These equations indicate that the second-order moment must always be corrected in a shear flow and that this correction can be significant, for example in wall boundary layers (Durst et al. 1998). Similar expressions can be derived for high er order statistics. For third and fourth order statistics and for an elliptic detection area, Durst et al. (1998) give the following expressions. (7.79) 356 7 Laser Doppler Systems (7.80) In these expressions, the mean velo city gradient dominates over the error terms involving gradients of velo city correlations. Hence, for purposes of correction, only terms involving the mean velo city are retained. IfEqs. (7.73), (7.74), (7.77)-(7.80) are used for correcting measured data, then both the detection volume dimension bd and the mean velocity gradient du I dX 2 must be known. To determine bd , Durst et al. (1998) suggest performing measurements with the optical system in question in a laminar tlow, in which the velo city gradient is linear and is easier to evaluate. Equation (7.77) becomes -U'-2- = I,DV b~ _d_u_1 (-,--X_2--,--) 2 1 4 dX 2 (7.81) X 2 =X 2 ,c since there are no velocity tluctuations present. This can be re-arranged to relate the 'apparent' turbulence intensity to the mean velo city gradient through the parameter bd (7.82) By plotting the left-hand side of Eq. (7.82) against ~ul~bvdUj(X2)/dx2Ix2=x2' the effective detection volume half-axis can be determined from the slope of the resulting straight line. In a simple laminar tlow, the mean velocity gradient is usually simple to estimate from preliminary measurements. If a laminar tlow is not available for 'calibration' of the system, then the mean velo city gradient must be estimated directly from the measurements, possibly employing an iteration step together with the correction expressed in Eqs. (7.73) or (7.74). In a turbulent boundary layer, it is helpful to express bd in wall coordinates, l.e. d bd+_u,b - (7.83) v from which the correction (7.7l) becomes for y+ < 5 (7.84) 7.1 Input Parameters from the Flow and Test Rig 357 Clearly, a direct measurement of T w requires a measurement volume of the order, 1. This criterion illustrates the fact that the choice of measurement volume size will depend on the Reynolds number of the flow, since Ur is Reynolds number dependent. More general recommendations about the required spatial resolution in turbulent boundary layers have been put forward by Karlsson and Johansson (1988) in terms of the viscous length scale [+ = V I ur. Here a dimension normal to the wall of sI and parallel to the wall of eS 10[+ is recommended (see also Johansson and Barlow 1989). b; ::; b; Measurement of Moments at a Position with no Mean Velo city Gradient. The second situation, for which the spatial resolution of the laser Doppler system must be considered, is when turbulence quantities are to be measured at a position free of mean velocity gradients. In general, all scales of turbulence will be resolved for detection volume dimensions less than the Kolmogorov length scale l]K" This suggests a smaller detection volume for a high er Reynolds number flow, as expressed by Eq. (7.37) However, unlike a hot-wire anemometer, the laser anemometer is not a spatially integrating instrument, at least if the particle concentration is such that only one particle at once is in the measurement volurne (single realization). Therefore, there are no extra considerations with respect to detection volume size demanded for this situation. 7.1.2.2 Temporal Resolution The required temporal resolution depends very much on the flow quantities of interest. As a measure of temporal resolution the data density is used, which refers to the number ofvalidated signals per integral time scale, Tu (Eq. (7.31) expressed as NT", where N is the validation rate of particles. The temporal resolution is adjusted primarily through the seed particle concentration. Moments can be estimated for all data densities, although in chapter 11 it will be shown that the respective estimator must be carefully chosen because so me estimators can exhibit a statistical bias, depending on the data density. Chapter 10 will discuss the fact that for moments, a data density above approximately one-half does not result in a significantly faster convergence of the result. Thus, in terms of optimizing measurement time for obtaining moments, a data density of about NT" = 0.5 is desirable. Anything lower just increases the overall measurement time. Anything higher increases the amount of data to be processed, without improving the overall measurement accuracy. Therefore, some apriori information about the local integral time scale, Tu' is required when selecting the particle concentration. There will also be an inverse relation between the spatial and the temporal resolution, since as the measurement volume is made sm aller, the particle rate will decrease, given the same particle concentration. This relation is modified further by the signal processing, since the detection prob ability will depend also on signal intensity, signal duration and possibly on the number of periods in the signal. More recent signal processing electronics have been successful in minimizing these secondary influences. 358 7 Laser Doppler Systems For estimation of frequency spectra or time correlations, the necessary temporal resolution again depends strongly on the estimator used. The choice of estimator is particularly important because of the random time sampling of the velo city, a unique feature of the laser Doppler and phase Doppler techniques. Conventional estimators result in non-biased results up to frequencies of the order IV /2n (Adrian and Yao, 1987). More re cent estimators, as discussed in sections 11.2 and 11.3, can extend this range to a frequency of IV or even above, if long data records are available. The long data records are necessary to achieve acceptable levels of estimator variance. As a comparison, for data sampled at regular time intervals, an unbiased and unaliased spectral estimate is possible only up to a frequency of IV /2 (Nyquist frequency), provided no energy above the Nyquist frequency is present in the signal. The highest frequencies which occur in the velocity signal are associated with the smallest scales of turbulence being convected through the measurement point with the mean velo city. Tennekes and Lumley (1972) derive an upper limit of the inertial subrange as krh '" 1, as also seen in the generic spectrum shown in Fig. 7.4. Accordingly, Eq. (7.37) yields for the maximum occurring frequency of velo city fluctuations, corresponding to the smallest flow structures, - -U- Re 3/4 f max 2nL (7.85) where L is a macroscale of the flow. To resolve the dissipative range of the spectrum, a frequency resolution up to imax would be necessary, hence with an appropriate spectral estimator, N '" fm=. This limit can now be used to choose an appropriate seeding concentration, as discussed in the following section. 7.1.3 Flow and Flow-Rig Parameters Some very basic information about the flow rig will influence the opticallayout. For instance, the necessary stand-off distance between the planned measurement positions and the transmitting or receiving optics will directly influence the choice of front lens focallength. All traversing clearances, clearances for accessing or cleaning windows, ete. must be considered. üf course the physical dimensions of the flow rig and the optical access available will determine whether forward, backscatter or side-scatter arrangements are possible. Depending on the medium, the penetration depth and the wavelength of the laser, the extinction must be considered when choosing the laser power to achieve a given intensity in the measurement volume. The extinction is expressed as an exponential decrease of intensity. [= [0 exp(-ßx) (7.86) where ß[m-11 is the extinction coefficient. Some typical values for water at various wavelengths are given in Table 7.1. Also the corresponding penetration depth at which the bearn intensity has been attenuated to e-1 times its initial 7.1 Input Parameters from the Flow and Test Rig 359 value is given in Table 7.1. It is clear that the extinction increases dramatically in water at higher wavelengths, thus restricting the use of laser diodes for liquid flows. Table 7.1. Extinction coefficients ß and penetration depths x (e -I intensity loss) for pure water (#Buiteveld et al. 1994, *Pope and Fry 1997) Wavelength Ab [nm] 632.5* 527.5* 515* 487.5* Pure water Extinction coefficient Penetration depth ß x(e- I ) [m] [m- I ] 2.293 0.2995 0.0428 0.0396 0.0144 0.436 3.34 23.4 25.3 69.4 Another very basic quantity to consider is the velocity range to be measured. This has an indirect effect on spatial resolution through the Reynolds number, but influences more directly the required frequency shift and the required bandwidth of the processing electronics. Most commercial laser Doppler systems realize a frequency shift using a single Bragg ceH and electronic downmixing before processing. Thus, the choice of frequency shift becomes a simple user input. However, systems with double Bragg ceHs or with a rotating grating must be designed specifically for a narrow range of possible shift frequencies. The shift must be chosen to insure that all flow velocities encountered will result in a unique (positive) frequency and furthermore, that a reliable estimation of the signal frequency can be performed by the available processor. The first criterion means that the fringe movement against the main flow direction must be faster than any negative velocities that occur. This is most easily understood in terms of frequencies. The signal frequency seen by the receiver, Ir' is the sum of the shift frequency and the Doppler frequency (Eq. (2.36». I I r=Ish+ u-L2sin~ =Ish+D Ab (7.87) and the requirement is that Ish >lID I, typically Ish >21IDI, for the lowest (negative) velo city. The laUer condition considers flow from all directions. This requires some pre-knowledge of the flow properties at the measurement point. If measurements are to be performed automatically at many positions in sequence, then the chosen shift frequency must fulfil this requirement at all the measurement points. The concepts discussed in section 5.1.5 can be used to help choose an appropriate shift frequency for a given application. The expected frequency range of the signal at the detector must also fall within the allowable bandwidth of the processor. In some cases, frequency shifting can be used to better match signal bandwidth to processor frequency range. Note that although the signal frequency is influenced by a frequency shift, 360 7 Laser Doppler Systems the signal duration is not. It is influenced only by the velo city and the trajectory of the particle through the measurement volume. Some processors are dependent on identifying a minimum number of signal periods, typically 8 or 16. Thus, the shift frequency must exceed the condition ish >lid I accordingly. This requirement has been discussed in section 5.1.5. More recent signal processors operate reliably also with fewer signal periods; however, the shift frequency must still be chosen such that the condition ish >lid I is exceeded. The maximum positive velocity can also be an important design parameter if the expected frequencies approach the bandwidth ofthe signal processor. In this case the interference fringe spacing must be increased, L1x=~ 2sin% (7.88) normally by choosing a smaller intersection angle. In exceptional cases such as supersonic flows, a forward frequency shift is employed to lower the signal frequency for a given flow velo city, with a corresponding decrease of signal periods. Far well-directed flows or situations in which two velocity components are measured simultaneously, a higher maximum velocity can be reached by rotating the optics with respect to the main flow direction. An angle of 45 deg results in a ..fi increase in the allowable velo city magnitude. Two important considerations related to the particle seeding are dictated by the flow rig and the flow properties. The first concerns the ability of the particles to follow the expected velocity fluctuations. This is determined by the size and composition of the particles and by the ratio of the particle density to the density of the flow medium. These influences are discussed fully in section 13.1. The second issue is that of the required particle concentration. The concentration is chosen on the one hand high enough to meet the required temporal resolution requirements and on the other hand, low enough to insure single realization operation. Single realization operation is achieved when the probability of more than one particle occurring simultaneously in the detection volume is less than 0.5%. Assuming a random homogeneous distribution of the particles, the prob ability of having Np particles in a volume V will, according to Eq. (5.166), follow a Poisson distribution (7.89) where Np, the Poisson parameter, expresses the mean number of particles simultaneously in the volume, or in terms of the mean concentration np (7.90) The conditions required to insure single realization have been discussed in section 5.1. 7 and lead to the result Np < 0.1, if V corresponds to the volume of the detection volume Vd • In terms of concentration this becomes 7.1 Input Parameters [rom the Flowand Test Rig _ 0.1 np s Vd 361 (7.91) For an elliptic detection volume this becomes (7.92) Here, a d , bd and Cd are the dimensions of an assumed ellipsoidal detection volurne, rw is the beam waist radius, g is the intersection angle and Pd is the scaling factor between the dimensions of the detection and measurement volume (Eq. (5.42) and Fig. 5.9). As an example, the measurement volume used in section 5.1.2 is examined, assuming also that the measurement volume dimensions correspond to the detection volume dimensions (e.g. Fig. 5.9 for d p < 5 11m and Pm;" = 10-6 W): rw=5011m, o/z=2deg, ad=5011m, bd=5011m, cd =l.4mm, Vd =0.015mm 3 • The particle concentration must therefore be below Tfp S 0.1 / Vd = 6.7 mm 3 • While the upper limit for Tfp is determined by the single realization constraint, the lower limit depends on what quantity is to be measured. For the measurement of moments, no lower limit is mandatory; however, a lower particle concentration will lead to a lower particle rate, hence, a longer measurement duration to reach a given accuracy. Exact expressions for the measurement accuracy are presented in section 10.3.2. An optimal rate is NT" = 0.5, as discussed in the previous section. For particle trajectories perpendicular to the interference planes in the measurement volume, the mean particle rate and the mean particle concentration are related through the mean velo city, Tl, N = rrb d Cd np u = A d np u (7.93) leading to a lower bound on Tfp of 0.5 n =--p Tu uA d or (for an ellipsoidal volume) (7.94) Since the integral time scale, Tu' and the mean velo city Tl are not uniquely related, no further simplification is possible. For the measurement of moments therefore, the bounds on Tfp can be summarized as (für an ellipsoidal volume). (7.95) Corresponding expressions for A d and Vd will be necessary if the detection volume is not ellipsoidal, for instance if the illuminated volume has been truncated by the use of slits or pinholes in the focal plane of the receiving probe. From Eq. (7.95) an absolute lower bound on the resolvable integral time scale can be derived, i.e. when the lower and upper bounds on Tfp are equal 362 7 Laser Doppler Systems T > Fdrw u 0.15u (7.96) which simply states that sm aller integral time scales can be resolved if the detection volume dimension is made smaller, either by increasing the detection threshold or reducing the laser beam diameter. Far the measurement of turbulent velo city spectra, the lower limit for the mean concentration np must be chosen according to the highest frequency which is to be resolved and the capabilities of the chosen estimator. Assuming that the estimator can resolve the maximum desired frequency, fmax' under the condition N = fmax (Note that for a velo city series sampled at regular time intervals, double the mean particle rate would be required!), the bounds on np now become (Eq. (7.93» fmax uA d <n < 0.1 - (7.97) Vd p - again assuming particle trajectories perpendicular to the interference planes. Eqs. (5.156) and (5.42) can be invoked instead ofEq. (7.93) for arbitrary trajectories. Unlike Eq. (7.95), some further analysis is possible with Eq. (7.97). The lower limit must always be less than the upper limit. This leads to the following condition for the maximum resolvable frequency f max < 0.1 -Ad uVd or f max < 0.075-u- Fdrw (for an ellipsoidal volume) (7.98) This inequality indicates that high er spectral resolution is possible by generating a smaller measurement volume. Of course, the necessary mean particle concentration increases correspondingly, since bd , Cd and ad are generally coupled with Fdrw • Another general form ofEq. (7.98) for an ellipsoidal volume is (7.99) which is useful for designing the transmitting optical system, responsible for generating the measurement volume. In terms of flow properties, Eq. (7.97) can be re-arranged using Eqs. (7.85) and (7.37) to yield a lower bound for Hp' while resolving the entire inertial sub range of the turbulent motions (krh "" 1) ReX _ ---.::;,np 27tLA d 0.1 '::;'- (7.100) Vd An example using the above relations is instructive. Consider an airflow with a mean velocity of 15 m S-1, a length scale of 1 m and the measurement volume 7.2 Components and Layout ofthe Transmitting Optics 363 dimensions used in the above example. The Reynolds number is 10 6 (v= 15 X 10-6 m 2 s'). Equation (7.100) reads quantitatively 70 mm -3 < _ 6•7 mm -3 _n- p < (7.101) Obviously, if the mean particle concentration is chosen high enough to resolve the small scales of turbulence (high frequencies), the probability of multiple particle signals from the detection volume will become larger than the assumed 0.5 %. Either the detection volume must be reduced in size (a value of rw = 5f.!m or Pd = 0.1 would be required) or the Reynolds number must be achieved using larger models. These relations must be applied locally in the flow and often the local convective velo city is significantly lower than the velo city scale used for the Reynolds number. This relaxes the lower bound on Tfp somewhat. For the presented analysis, the detection volume dimensions or the factor Pd have to be known. One or two detection volume dimensions can be defined by using slit or pinholes in the receiving probe, as presented in seetion 14.2.1. Nevertheless, the third dimension must always be determined by experiments. This can be done by using amplitude or burst-length statistics (sections 5.1.6.3 and 14.2.2.3) or by measurements in a flow with velo city gradients (sections 7.1.2.1 and 14.2.1.2). 7.2 Components and Layout of the Transmitting Optics The transmitting optical system and the optical access to the flow field must be designed with care, since errors introduced at this stage cannot be compensated using subsequent system components. There is an extremely large number of possible transmitting system configurations, although only a few variations are offered commercially. To describe these possibilities and their layout, the following discussion has been divided into the major components of the transmitting system • • • • Collimators Bearn splitters and polarizers Methods of achieving directional sensitivity Generation of the measurement volume Not all of these components and features exist in every system and sometimes components are combined into single elements. Nevertheless, most systems can be described according to this scherne. 7.2.1 Collimators Collimators are used for beam-waist adjustment and/or for beam shaping. Beam shaping is especially necessary when laser diodes are employed, since their beam 364 7 Laser Doppler Systems profile may be elliptical or display astigmatism. An elliptic beam profile arises when the divergence angle is different in two orthogonal planes. This can be corrected using a cylindricallens. Astigmatism refers to different beam waist positions (or virtual origins) for different planes through the beam, as illustrated already in Fig. 3.6. This leads to highly distorted measurement volumes and is typically remedied using a pair of wedge prisms. Figure 7.8 illustrates a full compensation and correction optics for a laser-diode light source. The beam waist must be adjusted either to achieve a certain desired measurement volume size or, more frequently, to achieve a certain beam waist for an optical fiber in-coupling lens. Collimators according to the Keppler (in Fig. 7.8, 12 > 0 ) or Dutch telescope (f2 < 0 ) principle can be used. For beam-waist adjustment an adjustable collimator is used, meaning the distance between the lenses can be varied (Fig. 7.9). Using Eqs. (3.96), (3.108) to (3.110), the system matrix for the arrangement in Fig. 7.8 is given by 1 Z2J( 1 0J(1 A =( 0 1 -12-1 1 0 _ - [ 1-~ I zoJ( 1 0J(1 1 11-1 1 0 h1 +h 2 _hI1h2 1 I ZIJ 1 j (7.102) h1 I-- I Pelt i er cooler Laser diode Collimator Cylindricallens Anamorphic prism pair Fig. 7.8. Beam collimator used with a laser diode to correct for astigmatism and for an elliptical output beam profile z, 1 1 --------- ----------- - -~ --------------- -~-- Fig. 7.9. Collimator using the 'Keppler' telescope principle 7.2 Components and Layout of the Transmitting Optics 365 In this equation the system matrix is equated to that of a single, equivalent thick lens. Introducing the variable distance between the lenses Zo = 11 + 12 + LI, Eq. (7.102) yields, after considerable manipulation, for the beam waist image position (7.103) and for the beam waist radius (7.104) To illustrate these relations an example collimator with 11 = 40 mm and 12 = 80 mm has been selected. The influence of a sm all misalignment of the collimator, LI (ZI =11 + 12 + LI), on the imaged beam waist is studied. The position of the beam waist image is shown as a function of ZI and rw1 in Fig. 7.10. In Fig. 7.10 the radius of the beam waist image is presented for the same parameter variation. These figures indicate that already a very small misalignment of the collimator, LI, can lead to very large displacements of the beam waist. A small positive displacement, LI, places the imaged waist at very large Z2 and can be used to collimate a beam. Placing a further convex lens in the beam's path will create a waist at its focal point. This lens combination can be particularly attractive for beam collimating because of its compactness. A collimator according to the principles of a Dutch telescope can be integrated into the transmitting optics for adjusting beam expansion and the position of the beam waist. For LI = 0 and 12 < 0 Eq. (7.103) can be simplified to (7.105) and Eq. (7.104) becomes (7.106) The beam waist in 11 will be imaged in 12" If two parallel beams are displaced symmetric about the optical axis, the distance between the beams after the collimator, Llb 2 , will vary as a function of the separation distance before the collimator, Llb 1 • 366 7 Laser Doppler Systems '-" + '::5 ,r b a 20 Zl = 100mm Tw1 - 300 firn _._. 400 firn c ----0- 300 mm _._- 400mm d - 5 0 0 firn 3 = 100 firn zl=-100mm ------200mm ----0- . ... ...'l! Tw1 =-100flm ------200 firn -500mm .,8., ........ .~ .,.,S 2 >Q -20 o 20 -20 Displacement LI [mm] o 20 Displacement LI [mm] Fig. 7.lOa-d. Position and size of the beam waist after collimation (fl = 40 mm, i2 =80 mm, Ab =488nm) (7.107) The discussion concerning spatial and temporal resolution in section 7.1 made it dear that the beam waist in the measurement volume must often be reduced in size to achieve the desired system performance. As described in section 3.2.2 (Eq. (3.129», this is most easily achieved by expanding the laser beams before they enter the front focal lens. There are two common approaches to achieving this. The first is employed with fiber optic probes. The beam exits the single-mode fiber with a divergence angle of approximately lX b Ab na "'- (7.108) 7.2 Components and Layout of the Transmitting Optics 367 where a is the core radius of the fiber and Ab is the wavelength of the laser light. The divergence of the beam can be used directly to achieve the desired beam expansion, by correctly choosing and positioning the collimating lens. The situation is pictured in Fig. 7.11. The lens is positioned at its focallength from the fiber and the focal length is chosen according to the desired radius of the collimated beam, rw2 • (7.109) The second approach for beam expansion is to use a combination of a negative lens followed by a positive lens, as pictured on the right side in Fig. 7.12. A front collimator for increasing the beam waist by a factor of 1:2 before the beam splitting is used for the system illustrated in Fig. 7.12. All outgoing beams of the transmitting optics passthrough the same lens, thus simplifying system alignment. f Fig. 7.11. Beam expansion, achieved when collimating after fiber transmission z, =150 260 l ,= -120 l. = 190 l s= 260 I 100 J ll" V" H r.f-+--------t"- -- ---- -- -ci " . , = 0.85 11", = 0.0473 " d " , j-- 0078 • = \.70 'Y-+-------i-----------J, = 161.2 J~ = ,I 1642 Fig. 7.12. One-velocity component transmitting optics with beam expansion and the equivalent optical system. All distances are expressed in millimeters 368 7 Laser Doppler Systems The equivalent system for the right lens combination shown in Fig. 7.12 can be computed as follows. U sing the primary matrix (7.110) the beam matrix can be computed using Z3 = f3 + f4 (7.111) The equivalent system for the optical parameters shown in Fig. 7.12 becomes • Focallength fe = 164.21 mm • Position of the primary principal plane H 1e : hlc = 136.62 mm • Position of the primary principal plane H 2c: h2e = 95.79 mm The positions of the main focal planes are shown in Fig. 7.12. From these data the dimensions of the measurement volume, the number of interference fringes and the highest allowable concentration of seed parameters can be computed as follows: • The beam separation distance is increased by a factor 1:1.58 from Llb l = 60 mm to Llb 2 = 94.8 mm • The front collimator adjusts the waist by the ratio 1:2 and a further beam expansion is achieved in the transmitting optics of 1:1.58. The transmitting optics images the beam waist diameter dwl with the ratio (Eq. (3.129» dw2 = 4AJ d wl (7.112) Jrd: l so that the final waist diameter using dwl = 850 11m (He-Ne laser, Ab =632.8nm), becomes d w3 =2b o =2rw3 =781lm • The intersection angle is e = 20.66 deg. The dimensions of the measurement volume are thus ao = 39.6 11m, bo = 39 11m and Co = 217.5 11m, the fringe distance is Llx = 1.76 11m and the measurement volume contains N fr = 45 fringes. • To insure single realization operation, i.e. only one particle in the detection volume at a time, 4 b 3 Np = 0.1 = -Jrn pao 0 cOFd (7.113) 3 Thus the upper allowable particle concentration is - 2 n < 1.19 x 10- sine 3 =71mmFd rw - 3-3 (Fd =1) (7.114) 7.2 Components and Layout of the Transmitting Optics 369 if the detection volume size is equal the measurement volume size. A wide selection of standard optical components is available from suppliers, enabling systems to be easily designed or modified for specific applications using similar computations. 7.2.2 Beamsplitters and Polarizers In a majority of laser Doppler systems, a single laser beam is split into two or more beams, possibly also with color separation. The intensity distribution among the beams will depend on whether a reference-beam or a dual-beam system is required; however, most beam splitters are designed to yield aSO/50 intensity split. By far the most common beam splitter in use is the Bragg cell, which simultaneously imposes a frequency shift on one beam. An example of how the Bragg cell is used in an integrated transmitting optics will be discussed in section 7.2.3 (see also Fig. 7.15). The angle between the zeroth and first-order beams exiting the Bragg cell is given by (Eq. (3.155» • ,.1,0 a=arCSIn-2A ac (7.115) where A ac is the acoustic wavelength in the Bragg cell and is dependent on the acoustic velo city in the Bragg cell medium, Va' and the driving (shift) frequency, fsh A=~ ac f sh (7.116) A rotating grating can also be used as a beam splitter, which is essentially a mechanical counterpart to the moving acoustic diffraction grating generated in a Bragg cello A laser beam impinging at an angle a (to the normal) on an amplitude line grating with spacing b will result in the characteristic diffraction pattern with maxima positioned according to (Eq. (3.136» . ßp -SIna=. pA , Sin b P=±I, ±2, ... (7.117) For a line grating, the amplitude of the maxima of the diffraction order p follows a Bessel function (Born 1980, Bass 1995, Hecht 1989). Diffraction gratings can also be designed to yield a high proportion oflight power in the ±lst order, for example phase gratings with a sinusoidal change of phase at wavelength b. The exact performance of the grating depends on the amplitude of the sinusoidal change of phase, as discussed in section 3.2.3.1. Furthermore, if the transmitting optics can be designed such that the intersection half-angle fulflls ~ = ß, then the interference fringe spacing in the measurement volume will be independent of the wavelength. 370 7 Laser Doppler Systems L1x=_A_=~ 2sin o/z 2 (7.118) Such a beam splitter is particularly advantageous when used with semiconductor lasers, since mode hopping or wavelength drift due to temperature fluctuations will then not affect the absolute measurement accuracy (Dopheide 1995). With rotating gratings, the diffracted beams are also shifted in frequency byan amount fsh = ±pfgrid , where f grid is the frequency of the grating line passage through the laser beam. Other examples of systems which exploit the achromatic behavior of the diffraction grating can be found in Schmidt et al. (1992), Czarske et al. (1994, 1997) or Czarske (1999). Conventional beam splitters can be used for laser Doppler systems, whereby some designs have been made self-compensating, hence very in sensitive to misalignment. Some possible designs are pictured in Fig. 7.13. With gas lasers, the coherence length of the laser beam is usually sufficient to make path-Iength equalization within the beam splitter unnecessary. This is not necessarily true for laser diodes. Some early designs also allowed continuous adjustment of the separation LIs, hence, the intersection angle of the measurement volume and the fringe spacing. Most beam splitting techniques demand a certain input polarization direction and polarizing elements may be required in the system prior to the beam splitting element to achieve this. Polarizers consist of material with anisotropie properties, exhibiting different refractive indexes in two orthogonal directions (Born 1980, Bass 1995). Transmission in one direction is characterized by the refractive index no (ordinary wave) and in the orthogonal direction by n e (extraordinary wave). The phase difference between the wave components in each of these directions amounts to 211: ,cjb'=(no -n e )dA (7.119) where dis the thickness of the polarizer. If the thickness is chosen such that the ßS Fig. 7.13. Beam splitting prisms 7.2 Components and Layout of the Transmitting Optics 371 phase difference is exactly 1t /2, then A (no -n e )d=4 (7.120) Und er such circumstances, linearly polarized light entering the polarizer will exit as elliptically (or circularly) polarized light The sense of the elliptic polarization will depend on the difference of refractive index in Eq. (7.120) Such a polarizer is known as a quarter-wave plate and transforms linearly polarized light into circularly polarized light or vice versa. If the plate thickness dis chosen such that the phase difference is 1t, A (no -n e )d=2 (7.121 ) r -r, then a wave entering the crystal at an angle will exit at i.e. a half-wave plate will turn a linearly polarized wave through an angle of 2r. Thus, a A /2 waveplate allows the polarization direction to be arbitrarily rotated. Quarter-waveplates are used in conventionallaser Doppler systems between the laser source and the beam splitter. This allows the beam splitter and transmitting optics to be rotated, while insuring correct polarization of light on the beam splitting surface without having to rotate the laser source. This is illustrated in Fig. 7.14. The rotation of the transmitting optics about the optical axis is required when measuring different velocity components. Beam splitter ~-~~~~~======~-------Circ~lar Li;ear Linear polarization polarizalion polarization Fig. 7.14. Use of A /4 plates to ensure correct polarization at the beam splitting surface 7.2.3 Methods for Achieving Directional Sensitivity Conventional methods for introducing directional sensitivity into the laser Doppler system are based on the generation of a moving interference fringe pattern in the measurement volume. This is achieved by imposing a frequency shift of one beam over the other using a continuous, time dependent optical phase variation of at least one of the laser beams. The most common devices used to achieve this is the rotating diffraction grating (Oldengarm et al. 1976) or the 372 7 Laser Doppler Systems acousto-optic modulator (Bragg ceH) (Durao and Whitelaw 1975), both described in detail in section 3.2.3. In Fig. 7.15, two optical arrangements for a laser Doppler system using Bragg ceHs are shown schematically. In the first example, the Bragg ceH also serves as a beam splitter. The net frequency shift, f'h' is just the driving frequency of the cell. In the second example, one Bragg ceH is placed in the path of each beam and the net frequency shift is the difference of the two driving frequencies, fl and f 2' The optimal incident angle of the laser beam into the Bragg ceH depends on the ceH specifications. This incident angle is realized using a wedge prism, as indicated in the diagram. In addition, the output angle of order m depends on the ceH specifications and on the chosen shift frequency, as given in section 3.2.3. . mA mAJ 2A ac n 2en Slna = - -b = - - - ' m (7.122) where e is the acoustic velocity in the Bragg ceH, f is the driving frequency and n is the refractive index of the ceH. The detected signal then consists of the net shift frequency and the Doppler frequency due to the flow velocity. The driving frequency is generated electronicallyand as such can be made extremely stable. The signal processor is generally designed to accept a relatively high driving frequency, typically 40MHz ... 120MHz. For such driving frequencies, the deflection angle through the Bragg ceH for the first-order beam is of the order 0.1 to 0.3 deg. Alternatively, the signal can be electrically down-mixed into a lower frequency range for processing, provided the highest negative flow velocity stiH results in a positive frequency. The electronic down-mixing is typically an elec- a b I, J, Photo- 10+ (/ ,- 1,) Fig. 7.15a,b. Use ofBragg cells in a laser Doppler system to obtain directional sensitivity. a Single Bragg ceIl system, b Double Bragg ceIl system 7.2 Components and Layout of the Transmitting Optics 373 tronic heterodyning, which can be illustrated using two sinusoidal signals of frequency w l and w 2 • (7.123 ) Adding and squaring these two signals yields s(t) = (51 (t) + 52 (t))2 = A 1A 2(sin 2 w lt+ sin 2 w 2t+ 2sinwl tsinw/) = A 1A 2[sin 2 w l t+ sin 2 w 2t+ sin(w l + w 2)t] + A1A 2sin(w l (7.124) - wJt Applying a low-pass fIlter at a frequency below both wl and w2 removes all but the last term in Eq. (7.124), yielding a signal at the difference frequency. For down-mixing in laser Doppler systems, w l is the carrier frequency plus the Doppler frequency and w 2 is the intended frequency of down-mixing. For instance, a Bragg ceIl operated at w l = 40MHz could be followed in the signal processing by a down-mixer operated at w 2 = 35MHz, leaving a net frequency shift on the signal of 5MHz. The down-mix frequency w 2 can be made variable, according to requirements dictated by the flow velo city, while the carrier frequency is held constant. This has the advantage that the Bragg angle and the optical alignment through the transmitting optics remains unchanged, even for different net frequency shifts. In the second diagram of Fig. 7.15, a system employing a Bragg cell in each transmitting beam is shown. In this case, the difference of the two driving frequencies yields directly the net frequency shift, without the need for downmixing. Thus, the necessary detection bandwidth at the photodetector can be substantially lower, resulting ultimately in a beUer signal-to-noise ratio. The disadvantage of this arrangement is that any change of the driving frequencies also changes the exit angle of the beam from the Bragg cell, which must be compensated by using different wedge prisms for adjusting the entrance and exit angles into the Bragg cello Other techniques for generating a moving fringe pattern in the measurement volume have been demonstrated but are not common. Introducing a relative phase modulation between the beams is one such technique. Such modulation can be generated with a saw-tooth-like driving signal, in which case the frequency shift has periodic phase jumps. Alternatively, a single side-band modulation, not unlike that applied directly to the laser diode in so me laser Doppler systems (Schroder 1987) can be used. Phase modulation can be achieved using Pockels cells, fiber expanders (Jones et al. 1985) or integrated optical devices (Pradel et al. 1993). In principle, the moving fringe pattern can also be achieved by using lasers of different wavelengths (frequencies) for each beam, as indicated in Fig. 7.16. Generally, however, the frequency difference is very high (> 100MHz) and exhibits very large fluctuations, thus a constant shift frequency cannot be assumed. One solution is to detect the beat frequency (reference signal) between the laser beams by superimposing a fractional part of each beam onto a photo- 374 7 Laser Doppler Systems detector, as shown in Fig. 7.16. By evaluating the frequency of each the reference signal and the detected signal, the magnitude and sign of the velocity can be determined. Another method of achieving directional sensitivity is through quadrature mixing. Quadrature mixing generates from one input signal a pair of signals with equal amplitudes and a precise 90deg phase relationship. Two techniques are available to do this, the homodyne technique, which is realized optically, and the heterodyne technique, which involves electronic frequency shifting. The homodyne technique is used less frequently, although recently some novel optical systems involving semiconductor or solid-state lasers have revitalized this option. A second set of fringes must be generated in the measurement volume with exactly the same fringe spacing but with a 1t /2 phase shift. The situation is visualized pictorially in Fig. 7.17. Such fringe systems can be created using a single laser wavelength, but exploiting two polarization directions (Dändliker and Hen 1974, Drain 1980). Two photodetectors must then be used with polarization filters for the separation of channels. If two different wavelengths are used, the fringe spacing must be kept constant. The use of a diffraction grating as a beam splitter insures that the lransmill ing lens ..:altcrcd lighl callcring lighl dClcclor From laser 2 Rctcrcncc dctcclor Refcrcncc signal Dctcclcd signal I." 1."'I 10 Fig. 7.16. Generation of a referenee signal with a frequeney equal to the net shift frequeney x Fig. 7.17. Quadrature signal generation (homodyne method) using two fringe systems shifted 1t /2 to one another (adapted from Müller et al. (1996)) 7.2 Components and Layout ofthe Transmitting Optics 375 fringe spacing is independent of wavelength (Müller and Dopheide 1992, Schmidt et al. 1992). In this case, color fIlters must be used with the two photodetectors. A further method is to alternate rapidly between the two fringe patterns with a synchronized multiplexing of the photodetector output, a technique known as time-division multiplexing (Lockey and Tatam 1994). This technique has also been used for two-velocity component systems (see section 7.4.2). The fundamentals of the heterodyne technique of quadrature mixing can be explained using the generic system pictured in Fig. 7.18. The unit accepts two input signals, the measured signal and a reference signal. The reference signal is typically the driver signal to the Bragg cells for frequency shifting (fsh) or the beat signal generated from the laser beam pair, if an arrangement as in Fig. 7.16 is used. In any case, the reference beam is highly correlated with the measured signal in the sense that any frequency (ffl) or phase fluctuation other than the Doppler signal itself should be the same in each signal. The reference signal is then passed through a broad-band, hybrid coupler, which yields two signals of the same amplitude but with a relative phase shift of 90deg. Thus, the reference signal pair after the hybrid coupler can be described hy + ffl,R)t+ <PR (t)] (7.125) SR2 = IR cos[ 21t{JSh + ffl,R)t + <PR (t)] (7.126) SRl = IR sin[21t{Jsh and the measured signal can be written as (7.127) Due to the high correlation between the measured signal and the reference signals, the fluctuating parts of the signal are equal, ffl,M = ffl,R = ffl and the phase variations are equal <PM (t) = <PR (t) = <P(t), These signals are then mixed, amounting to an addition and a squaring of each signal pair, as illustrated with Eqs. (7.123) and (7.124) This leads in each case to two self-products and to the following two mixed products. 2s Rl sn =~IMIR( cos21tfDt-[COS(21t{2fsh +2ffl + fD)t+2<P R(t))]) I\ lcasured signal Rcfcrcncc signa l Fig. 7.18. Block diagram ofheterodyne quadrature mixing (7.128) 376 7 Laser Doppler Systems (7.129) The self-produets and the seeond term of eaeh of the mixed produets are removed by low-pass filtering. This leaves the quadrature signal pair (7.130) (7.131) in whieh neither the shift frequeney f'h nor the frequeney of phase fluetuations appear. The situation is pietured in Fig. 7.19, in whieh it is clear that quadrature mixing ean also be used to determine particle direetion in the measurement volurne, depending on whether the eosine signal leads or lags the sinusoidal signal. Quadrature demodulation teehniques render possible many novel optieal arrangements, especially in whieh separate laser sourees for eaeh beam are used, e.g. two stabilized Nd:YAG ring lasers or diode lasers. Examples of one or multiple velocity eomponent systems ean be found in Kramer and Dopheide (1993), Müller et al. (1993), Müller et al. (1994), Kramer et al. (1994), Czarske and Müller (1995), Müller et al. (1996a), Müller and Dopheide (1997) or Czarske (2001a). A further teehnique for introducing direetional sensitivity, whieh is partieulady well suited to fiber-based laser Doppler systems, exploits stimulated Brillouin seattering in the fibers. This is a non-linear oseillation effeet in the erystal strueture of the glass fibers in whieh a baekseattered eoherent Stokes wave is generated at a frequeney several GHz below the stimulated wave. Further details ean be found in Többen et al. (1994). A last method of resolving the velocity sign to be diseussed is to segment the measurement volume at the imaging plane in the reeeiving opties and then to Negative partide velocity Positive partide velocity Time Ti me cos(2 n (- 10)1) \. si n(2n( - J"o)t) ~--~--~--~-----+ TIme ... li mc ---"" < in(2n I D I) '............ _...... Fig. 7.19. Directional discrimination bya quadrature signal pair 7.2 Components and Layout of the Transmitting Optics 377 employ two photodetectors. Plamann et al. (1998) have realized this using a twinned optical fiber as sketched in Fig. 7.20. The sequence of the signals in time indicates the flow direction. The sum of the two signals can also be used for a more accurate velocity determination. Further segmenting for a two-velocity component system is conceivable. Measurcmcnt t Scallcrcd light rol·\i~ "; F~ Signal I ~ 2 Time 'Iimc Fig. 7.20. Directional sensitivity using a twinned optical fiber in the image plane of the receiving optics 7.2.4 Generation ofthe Measurement Volume In this section, the requirements for achieving uniform fringe spacing within the measurement volume are discussed. The issue is best illustrated by Fig. 7.21, which employs Moire fringes (section 5.4) to illustrate two laser beams, which do not cross at their respective waist positions. In all cases, the fringe spacing is non-uniform and will lead to biased estimates of both the mean velo city and higher moments. Such a situation arises if the transmitting system is improperly designed or due to astigmatism, arising from unequal beam refraction in different planes. Indeed, this source of error is usually the limiting factor in deter- a Longitudinal distortion b Transverse distortion Fig. 7.21a,b. Fringe distortion in the measurement volume when the laser beams do not intersect at their waists. a Longitudinal distortion, b Transverse distortion 378 7 Laser Doppler Systems mining the accuracy with which flow velocities can be measured using the laser Doppler technique. The origin of the non-uniformity is the curvature of the wavefronts away from the waist of a focused Gaussian beam. Hanson (1973,1976) gave approximate expressions for the variation of fringe spacing along the length (longitudinal) and across the width (transverse) of the measurement volume. Durst and Stevensen (1979) extended the longitudinal analysis for large deviations of the waist from the intersection point. A rather rigorous analysis, valid also for nonGaussian beams, for example from laser diodes, was presented by Durst et al. (1990), however this analysis requires a numerical solution. More manageable expressions have been presented by Miles and Witze (1996) and Miles(1996) and have been shown to be equivalent to the more rigorous derivations for most practical situations. Miles and Witze (1994) have also presented a means to visualize the fringe non-uniformity in the laboratory. The analysis of Miles (1996) begins with the description of a Gaussian laser beam as discussed in section 3.1.4.2, in particular using the Rayleigh length lRb (Eq. (3.61), the wavefront curvature Rb (Eq. (3.65», and the local beam radius rmb (Eq. (3.64». The situation to be examined is pictured in Fig. 7.22, showing two Gaussian beams intersecting with the angle e with the coordinates given in Eq. (5.7). from Eq. (5.52) can be approximated by The local spacing between fringes neglecting the terms in Eqs. (5.53) and (5.54) and is given by ox "" ~ --,,>-[ 2sm ~ X 1Z 1 1- X 2Z 2 z; + I;, z; + I;, ( x Z ) 2tan~ + e z;X,2, +l~2 ZI + 2 2 1 1 1 (7.132) Rl z Fig. 7.22. Geometryand coordinates for deriving fringe spacing 7.2 Components and Layout ofthe Transmitting Optics 379 For the special case of a path-compensated laser Doppler transmitting optics with equal beam waists (rwl =rw2 =rw' ZRl =ZR2 =ZR)' each located at equal distances from the beam intersection Ztl = Z'2 = Z" Eq. (7.132) cannot be simplified as in Miles (1996). For this case of equal waist displacements the interference fringe distance varies mainly in the z direction and is nearly constant in the x direction, as pictured in Fig. 7.21a. For an ideally aligned path-compensated system, the transverse variation of fringe spacing is negligible. Therefore Eq. (7.132) can be approximated by the general expression for the fringe spacing along the z axis (x = 0) given by Miles (1996) .-:.( x = 0, z ) =- Ab (z, +zcos'%)zcos'%) - -[ 1 + -'---;-------'---,--..,,-- VA (7.133) z,(z, +zcos,%)+Z; 2sin,% Note that the definition of the beam waist dislocation z" Eq. (5.7), differs from Miles (1996) along the beam axis, and the numbers of the beams (b = 1,2) have been exchanged, therefore also Eq. (7.133) appears different. For the intersection point of the two beams (x = z = 0) the relative fringe spacing change in the z direction is given by ~ d&: &: dz __I_dfD fD dz cos,% (7.134) Rb which is the longitudinal fringe variation as derived by Hanson (1973). Transverse variations of fringe spacing occur when the two beams have different longitudinal positions. The situation in Fig. 7.21b corresponds to equal but opposite distances of the beam waists from the beam intersection (z, = ztl = -Z'2). This occurs for a perfectly aligned system in which the path lengths of the beams are different. The fringe distance varies mainly in the transverse or x direction so that for z = 0 J .-:.( X,Z ) - - .Ab- -[ 1- ----;,2--;2,------XCOS'% - 0 VA 2sm,% 2 z +Z (7.135) ~tan'%+xcos'% Eq. (7.135) can be expressed with the local wavefront curvatures resulting in ~ vx(x z=O)= , Ab Ab Rb 2e/2/ 2 R b sine/+xcos /2 Rb 2 Rb sin,% +x which includes the results from Hanson (1976). By using Rl (7.136) dJx _ dx - (~_ X-X Rl ) Ab 2(X-X Rl ) X-X Rl = X Rl sin,% (7.136) in Eq. (7.137) Rb which expresses the transverse fringe variation at z = 0, as also derived by Durst and Stevensen (1979). 380 7 Laser Doppler Systems Miles (1996) has given an approximation ofEq. (7.133) in terms of system parameters. For equal distances of the beam waist ( ztl = Z'2 = z, ) in a pathcompensated system the longitudinal dependence x = 0 is &(x =O,z) =+[I-Z 2rwo L1z b cos 2 2sm o/z L1b + '% z' co,' ~x('';;'" )'(1- Z 2; .12, 00" n (7.138) % where rwo is the beam waist and L1b is the beam separation in front of the focussing lens, as shown in Fig. 7.23. L1z b is the normalized misalignment, L1z b = (z wb - j) / ZR' also shown in Fig. 7.23 and z is the Z coordinate normalized with half the measurement volume length o/z Z sin Z=-=Z----'--"---Co rm(x=O,z=O) (7.139) Equation (7.139) is valid for rwo / L1b ::; 0.1 and lL1z b l::; 2 or for arbitrary rwo / L1b when the system is perfectly aligned (L1z b '" 0), in which case Eq. (7.138) becomes &(x = O,z) +[1 Z2 =2sm o/z + cos 4 '% ( 2rL1bwo J2] (7.140) The ratio 2rwo I L1b is seen to be instrumental in determining fringe nonuniformity. Indeed, for this ratio to be significant, the separation L1b would typically be small, implying small intersection angles. Thus for modest values of 2rwo I L1b and L1z b ~[1- Z2rwo L1z + Z2( 2rwoJ2 l 2 sin o/z L1b L1b J &(x = O, z ) = (7.141) b or for ideal alignment conditions J Bcam I Z~I Fig. 7.23. Dual-beam laser Doppler transmitting optics f x 7.2 Components and Layout of the Transmitting Optics (rL1bWo )2] &(x =O,Z)=:+[I+Z 2 2sm~ 381 (7.142) This equation represents the minimum possible variation in fringe spacing (o/z -70, L1z b = 0) for a given intersection angle e and waist diameter rwo prior to the transmitting lens. Miles (1996) gives a similar expression for the transverse dependency (ztl = Z, = -Z(2) of the fringe spacing, Eq. (7.135), which, in terms of system parameters, becomes (7.143) where xis normalized with rm / coso/z = ao' The relative deviation of the fringe spacing, given always in the last term in Eqs. (7.133), (7.135), (7.138) and (7.140) - (7.143), increases linearly with the corresponding coordinate. If the detection volume can be decreased in the respective co ordinate, i.e. by using slits or pinholes in the receiving optics, the velo city fluctuations resulting from different particle trajectories can be suppressed. For a path-compensated system, the standard deviation in the velo city can be significantly reduced by using side-scattering. Nevertheless, for a correct absolute velocity measurement the system has to be aligned perfectly (ztl = Z'2 = 0). The analysis of Miles (1996) and its experimental verification in Miles and Witze (1994) give a very general framework for evaluating fringe non-uniformity in laser Doppler systems. The resulting over-estimation of measured velo city moments due to this non-uniformity is addressed, at least for the second moment, in Zhang and Eiseie (1998). Generally, this effect is only significant at very low turbulence intensities. Introducing a fringe distortion number as 1 dJx r- c -Jx dz - (7.144) 0 where Co is the half-Iength of the measurement volume, Eq. (5.40), Zhang and Eiseie (1998) have approximated the relative normalized error in the estimated standard deviation C5u of velocity fluctuations for the full length of the measurement volume to be 2 1 (l+r )+-r 3 2 li 2 --1 O'~ (7.145) which for laminar flow reduces to Tu= ~ =H r 2 (7.146) The apparent turbulence intensity due to fringe non-uniformity is linearly proportional to the fringe spacing gradient in the measurement volume. This analy- 382 7 Laser Doppler Systems sis was carried out for the simple case considered in deriving Eq. (7.137) and must be modified accordingly if more complicated distortions are present. A misalignment of the beam waists away from the intersection origin can be the result of poor system layout, e.g. improper choice of coHimator, or because of distorting components in the path of the laser beams. The laUer is more common in practice and can easily occur if the laser beams must translate through glass or into a liquid flow at an oblique angle. This situation has been studied in aseries of articles by Zhang and Eiseie (1995, 1996, 1998) and is weH summarized in Fig. 7.24, adapted from their work. As seen in this figure, an oblique entrance into a medium of different refractive index can result in several distorting effects: • Two-velocity component measurements may become impossible because the two measurement volumes may no longer be coincident • The data rate of a single component system may be significantly reduced, since the effective aperture of the receiving lens is significantly reduced. • The waist will be translated along the beam axis, resulting in beam distortion and astigmatism. z x z Fig. 7.24. Origins of astigmatism when using a one or two-velocity component laser Doppler system in liquid flows Such difficulties are avoided if the optical axis of the system enters the medium perpendicularly. Where this is not possible, the use of a liquid-filled prism, as shown in Fig. 7.25 can be used as compensation (Booij and Tukker 1994). A general procedure for computing the position and shape of the measurement volumes of a three component system entering the test section through an arbitrarily shaped window has been presented by Doukelis et al. (1996). 7.3 Layout of the Receiving Optics 383 Fig. 7.25. Use of a liquid filled prism to compensate for astigmatism (adapted from Booij and Tukker 1994) 7.3 Layout of the Receiving Optics The detector should ideally receive only light originating from the spatial extent of the measurement volume. Additional contributions from scattering centers in the beams outside of the volume or reflections from walls or optical components lead to an increase in signal noise and to a decrease of signal quality. Therefore, the receiving optics has the task of imaging as closely as possible only the measurement volume over the length 2c o in the z direction. A typical forward or back-scattering receiving system is pictured in Fig. 7.26a. In actual systems, a two-Iens configuration is normally used, primarily to reduce aberrations and to allow arrangements that are more compact. The front lens of focal length f1 can be normally changed by the user to work with different stand-off distances between the measurement volume and the receiver probe. The system consists of two spatial fIlters, an aperture of diameter dr in front of the system and a pinhole of diameter dp;n near the image plane behind the lens. The detector is the sensitive surface behind the pinhole. The optical system in Fig. 7.26b represents the thin lens equivalent system with the focallength fand the equivalent aperture diameter da' as outlined in section 3.2.1. The magnification of the system is deterrnined by the ratio of the two lenses in Fig. 7.26a _ -Z2 _ f _ Z2 - f - -f2- ß---------f1 ZI ZI - f f (7.147) Therefore the observation field in the x direction, L1x, is the pinhole diameter divided by the magnification factor 384 a 7 Laser Doppler Systems X· I, j~ dClc.:l0r b ol, Fig. 7.26a,b. Configuration of the receiving optics. a Optical configuration of a simple two lens receiving system, b Equivalent single lens system (7.148) which depends on the pinhole size, the focallengths and the working distance. Partieles in the focal plane of the system (z p = 0) are imaged as points in the plane of the pinhole. If the partiele is in the observation field of the focal plane (z=O, IXpl<Llx/2), all scattered light collected by the aperture can pass the pinhole. Outside the observation field, the scattered light is blocked. Jf the partiele is in front or back of the focal plane, the image is defocused and becomes the shape of the aperture, i.e. an cirele. For the positions zp = Llz_,Llz+, x p = 0 in Fig. 7.26, the defocused cireles have the same size as the pinhole and all the scattered light of the partiele is still detected. The depth -of-field of the receiving optics can therefore be computed as and depends on the inverse of the aperture diameter. For partieles inside the shaded region (A) in Fig. 7.26, which is defined by the observation field and the depth-of-field, all scattered light reaches the detector. The scattered light from partieles in the region (B) in Fig. 7.26 is only partially detected because apart of the defocused image is blocked by the pinhole. Light from region (C) cannot enter the detector. 7.3 Layout of the Receiving Optics 385 For laser Doppler and phase Doppler receiving systems only the region which is illuminated by the laser beams is interesting. The region Llz A = Llz A+ + Llz A- in which scattered light can be completely detected is computed as (Nakatani et al. 1977) Llz A + = LIx+2c d tan% d +Ax ' Llz A _ 2tan%+~a-- = LIx + 2c d tan % d -LIx 2 tan 'Ji + --".a_ _ (7.150) 2) Using LIx« da this expression becomes Llz A = LIx + 2c d tan'Ji d tan%+_a = LIx + 2a d d tan%+-" 22) (7.151) 22) where a d and Cd are the semi-axis dimensions of the detection volume in the x and the 2 directions respectively, Eq. (5.42). The light scattered from particles with larger distances from the measurement volume is detected only partially and can reduce the detected number concentration. No signal can be received from particles with positions 2 p > 2 B+ or Z p < Z B-. These limits can be computed as Llz B+ = LIx+2c d tan% d _ LIx ' 2 tan % _ a _ _ ----'"c Llz = B- LIx + 2c d tan % d +LIx 2tan%-----"a-- 2) (7.152) 2) and using again LIx« da the region has the overall dimension Llz B = Ax+2c d tan% Ax+2a d d = d tan%-_a tan'Ji-_a 2z) (7.153) 2z) Equations (7.151) and (7.153) can be used to determine the maximum region from which signals, with or without modulation, can be expected. The volume of this region can be used when computing concentration limits to maintain single particle realization. On the other hand, only signals in the ellipsoidal detection volume, Eqs. (5.42) and (5.43), will be detected. Especially for signal processing techniques working with the burst shape, the burst amplitude or the transit time, any disturbance of the signal shape may influence the final results. ExampIes of such techniques include the interpolation of the peak frequency, section 6.3.1, the cross-sectional area difference method, section 9.1.1, or the weighting techniques for unbiased flow parameter estimations, section 11.1. The depth-of-field is determined by the aperture size in Eq. (7.149). When the aperture size is smaller than (7.154) 386 7 Laser Doppler Systems the scattered light from small particles in the ellipsoidal detection volume with semi-axis ad, bd and Cd is detected without loss of information. The observation field LIx should therefore always be J2 times larger than the expected detection volume diameter 2a d • If now also the diffraction at the aperture da is considered ZI ZI A LIx = dp;n - + 2.44-- (7.155) da Z2 and the magnification ß = -Z2 I ZI' the position ZI = (1-11 ß)f and the numerical aperture NA = da I (2n are used, a final expression is obtained as LIz =_ A.B 1 ß dp;n ±1.22~(ß-l)-2adß _ _ _---""N"-'A'---_:::--__ (7.156) ß tan~±NA ­ ß-l The dependence of the depth-of-field on numerical aperture is illustrated in Fig. 7.27 for different pinhole diameters with and without diffraction. Diffraction limits the region of complete scattering light detection for sm all apertures in comparison to the ideal case from Eq. (7.151). By using Eq. (7.155) in Eq. E 6 ,-.--.--,-.--.--,-,--..-,-, Wilhoul diffrac.:lion: E " 1 .~ " 2 .dz : ------A Llz.: - - - - di""- IO~l O ~~~ _ di",- 30 l-lm --- __ L-~~ _ _L-~~_ _~ 6 .--.---,---.-.--,---,,-,...-.---,---,...---, E E O ........................---L_.L--'---'--..I.......................-..I QO QS 1.0 QO Numcrical apcrlurc NA [-J 0.5 1.0 Numerical aperlurc NA [-I Fig. 7.27. Depth-of-field as a function of numerical aperture for different pinhole diameters. All scattered light on the aperture surface is detected for particles with z positions in regions (A). Scattered light from particles in the regions (B) is partially detected. (a d = 76.2 I-lm, e=20deg, Ab =632.8nm, ß=-O.2) 7.3 Layout of the Receiving Optics 387 (7.154) the lower and upper limits of the numerical aperture for an ellipsoidal detection volume can be defined. The lower limit is given by the diffraction of the aperture which is larger than the pinhole diameter. The upper limit is reduced in comparison to Eq. (7.154), because the diffraction limits the region of complete scattered light detection. If the detection volume dimensions are chosen to be the same as the measurement volume dimensions, ad = ao ' Cd = Co' and the pinhole aperture diameter to be twice as large as the measurement volume image diameter, dpin = -4aoß, Eq. (7.156) yields _ 1 L1z AB ' - ß ±1.22~(I-ß)+6~ß NA cos,% ß (7.157) tane/±NA-/2 ß-l as a basic relation for designing receiving optics. Furthermore the modulation depth and therefore the SNR is influenced by the aperture size and shape as described in section 5.1.4, especially for strong intensity variations over the aperture. For such cases rectangular aperture shapes are advantageous, influencing the depth-of-field for different cross sections. The actual implementation of the receiving optics shown generically in Fig. 7.26 may vary, both with respect to the optical components employed and to the angle at which the scattered light is collected. Often a fiber-optic link is required between the pinhole and the photodetector in order to be able to place the receiving probe remote from the larger and bulkier photodetectors. Especially for two-velocity component systems or phase Doppler systems, in which up to four photodetectors are used, this can be advantageous. However, also for systems operating in backscatter, the transmitting and receiving systems can both use fiber-optic links to allow flexibility in positioning and handling of the measurementprobe. Unlike the fibers used in the transmission system, the receiving fibers need not maintain spatial coherence or polarization and therefore, a large diameter, multi-mode fiber is sufficient. Typically, a graded-index or step index fiber of 200/-lm core diameter is employed, which also gready reduces the required positioning accuracy of the fiber input. With proper dimensioning, the fiber input cross-section may also act simultaneously as the pinhole; however, this sometimes leads to distances, Z2' which are no longer tolerable for compact measurement probes. In this case, pinhole foils can be affixed direcdy to the input face of the fiber. For instance, slit type pinholes are regularly used in phase Doppler systems, as discussed in section 8.2. The scattering angle at which the receiving optics is placed is determined by three considerations: • Scattered light intensity • Optical access to the measurement position • Tolerable dimensions of the measurement volume 388 7 Laser Doppler Systems The maximum scattered intensity is at a scattering angle of <Pr '" Odeg, forward scatter, due to high contributions from reflection and diffraction. Light scattering examples given in chapter 4 (e.g. Figs. 4.29 to 4.32) indicate that the scattered light intensity in forward scatter can easily be a factor 500 .. .1 000 larger than in side scatter (<p r '" 90 deg) or backscatter (<Pr'" 180 deg). Nevertheless, the available optical access to the measurement position may not permit collection in forward scatter, for example with internal combustion engines, where only one optical access to the combustion chamber is available. Especially in large installations (e.g. wind tunnels), it may not be feasible to synchronously traverse aseparate transmitting and receiving optics on opposite sides of the tunnel and with the required precision. In such cases the backscatter arrangement is particularly attractive, since then the transmitting and receiving optics can be integrated into a single housing and can use a common focussing lens. Such a fiber-optic based measurement head is shown in Fig. 7.28 in cross-section. Another reason to depart from a forward scatter or backscatter collection angle is to reduce the length of the detection volume. Especially when measuring in flows exhibiting high velo city gradients in the z direction, the length of the detection volume, which is typically 5.. .10 times larger than the diameter, may not afford the necessary spatial resolution (section 7.1.2). Positioning the receiving optics in side scatter (i.e. <Pr = 90 deg) effectively limits the detection volume to a length given by the projected pinhole diameter (7.158) while accepting that the scattered light intensity may be drastically lower than forward scatter and often significantly lower than in backscatter. Normally such measures can be avoided by orientating the system such that the largest flow gradients are aligned in the x or y directions of the measurement volume. In complex, three-dimensional flows or with the phase Doppler technique this may not always be possible. Such side-scatter systems have been discussed in section 7.1.2.1. Righl-angle /llea uremcnl volume Flow Tube lor laser Backsa! Ile rcd lighl \ Excha ngeablc lrom lcns Thin righl -a ngle prism (movdblc) fiber collima lor /IIonomodc fibers Fig. 7.28. Cross-section of a fiber-optic based laser Doppler measurement probe operating in backscatter (adapted fra m DANTEC Dynamics) 7.4 System Description 389 7.4 System Description 7.4.1 One-Velocity Component Systems The basic layout of a one-velocity component laser Doppler system has already been covered in the previous sections. Examples of conventional optics have been given in Figs. 7.14 and 7.15. Integration of the transmitting and receiving optics into a single, fiber-optic probe has been illustrated in Fig. 7.28. The correet choice of measurement volume size, how to achieve this size and a suitable particle concentration to obtain the desired temporal resolution were discussed in sections 7.1 and 7.2. It is nevertheless instructive to review so me ofthe simple trade-offs when selecting an optical system, many ofwhich can be applied without change to two and three-velocity component systems. Most commercial optical systems offer interchangeable front lenses of various focallengths. This is possible because the outgoing beams have been highly collimated before the front lens, meaning ZI in Eq. (3.127) has been made very large and thus Z2 = f , for alliens choices. This insures that the beam waist remains at the interseetion volume. However, a change of focallength changes both the size of the illuminated volume and the intensity. The diameter of the volume will change approximately linearly with the focallength of the transmitting lens (ft) and the intensity will therefore change with r;b I f/ , where r wb is the beam radius at the waist before the front lens. Changing the focallength of the receiving lens (fr) will also change the solid angle of the receiving aperture, with the proportionality d~ I f,2, where da is the diameter of the receiving aperture. Thus an optical 'figure-of-merit' for a laser Doppler system, characterizing the amplitude of received signals could be K =(d J2 a r wb (7.159) fJr or for a backscatter probe, in which ft = fr = f (7.160) This illustrates clearly that, to a first approximation, a doubling of the focal length will lead to a fourfold decrease in signal amplitude and a corresponding decrease in signal-to-noise ratio. The situation can be improved either optically, by employing a beam expander (increase rwb )' by using a larger receiving aperture (increase da) or by using a higher incident laser power. Of course other means of increasing signal amplitude are also available (larger seed particles, higher quantum efficiency at the detector etc.); however, these parameters are often fixed in a given experimental system. In many applications, a one-velocity component system is sufficient. Nonetheless, most single component systems are designed to be rotated about the optical axis. This allows consecutive measurement of different velo city compo- 390 7 Laser Doppler Systems nents at the same position in the flow field. If the flow is stationary, such a procedure can save the investment in optics and signal processing for a second velocity component. Furthermore, it is possible to obtain the Reynolds shear stress with a one-velocity component system if the rotation angles are correctly chosen. To illustrate this, the co ordinate system shown in Fig. 7.29 is considered. A measurement performed with the optical system rotated at the angles a\ and a 2 about the x axis will yield respectively u\ = ucosa +vsina j j =(u+u')cosa +(v+v')sina j (7.161) j u2 = (u + u')cosa2 +(v + v')sina 2 By choosing a j = -a 2 = a for two consecutive measurements, and performing a third measurement U o with a o = 0 deg, the following flow statistics can be readily computed from the measured mean and fluctuating quantities (7.162) (7.163) - U j -U v=---2 (7.164) 2sina (7.165) u'v' = u /2 j 12 - u2 (7.166) 4cosasina Eq. (7.166) yields one component of the Reynolds shear stress under the assumption of stationary flow and requires only two consecutive measurements at Measurement 0 y Measurement 1 Measurement 2 y Fig. 7.29. Rotation angles of a one-velocity component laser Doppler system y 7.4 System Description 391 a single point. Note that the time series of u'v' is not available. Für this a twovelocity component system is necessary. The statistical errors involved in these quantities are now examined using concepts to be introduced in section 10.4. Applying Eq. (10.67) for the propagation of stochastic errors to the measurement of V,2 and u'v' yields (Tropea 1983) ;:- , - ; 2 ( vU v ) ;:~ = vU ( 1 ) 8cos 2 asin 2 a +( -,-; U v cos2a ;: ) va cosasina 2 (7.167) (7.168) where b designates the respective uncertainty interval, evaluated at the same confidence level for all quantities, e.g. 95 %. These equations have been derived assuming that bU[2 '" ÖU~2 '" öuü2 , which means the measurement uncertainty is about equal for each of the consecutive measurements. The last term in each of these equations is small because the uncertainty in the angle adjustment is generally smalL Thus, the stochastic error will scale with the factors shown in Fig. 7.30. From this figure it is seen that the ideal measurement angles for the quantity u'v' are a = ±45 deg, whereas for V,2 the angles a = ±90 deg are preferred. If both quantities are to be measured, then angles closer to a = ±60 deg may be desirable. Angles less than a = ±30 deg should be avoided, otherwise the uncertainty (scatter) of the turbulence quantities will become unacceptably large. ~6 \ \ (ou'v')' ~ ./----;,-------,,-Beos' asin' a \ \ \ \ \ \ \ \ \ ----- (ov")' ~ , \ -._1_,-(1+2COS' a) 2sm a \ \ \ \ \ \ 2 ,, ,, ,, " " '" ", - ... _--- ... _--O~~ o __ J-~~-L __ ~~ __- L_ _ L-~ __ 45 ~~~-L __ L-~ __ ~~~-L~ Tilt angle Fig. 7.30. Factors influencing the uncertaintyintervals for u'v' and V,2 IX [deg] 90 392 7 Laser Doppler Systems 7.4.2 Two-Velocity Component Systems Extending a one-velocity component laser Doppler system to two velo city components requires a second measurement volume with a different orientation about the z axis, usually 90 deg with respect to the first measurement volume. However, it must also be possible for the photodetector to distinguish wh ether the detected scattered light originated from the first or second measurement volume and this is usually achieved by using different colors. The most common two-velocity component system is therefore a two-color, four-beam arrangement, shown generically as a fiber-optic, backscatter system in Fig. 7.31. A variation of this approach combines one beam of each transmitting pair into a mixed color beam, resulting in a two-color, three-beam optical arrangement, as pictured in Fig. 7.32. This system allows the measurement volume to be positioned very dose to a wall without having to tilt the transmitting probe. By far the most common laser source used for two colors is the Argon-Ion gas laser, providing strong lines in the blue (,1 = 514.5 nm) and green (,1 = 488 nm) wavelengths. When two detection volumes are involved in a single system, their alignment, both on the transmitting side and on the receiving side is very important. Poor alignment leads to low data rates, since the coincidence rate drops. In very poody aligned systems, the two channels may even see signals from different partides at the same time, which falsifies correlation values strongly, since then also a spatial correlation is involved. Alignment procedures vary, depending on the optical configuration. For conventional optical systems without fibers and with integrated beam laser bcam co,or~~~~~J~~~~III~ii Ineidenl IWO Ab =488nm,5145 manipulalors ·Ihnsmiller/rceciver probe ---------- U rlow Fig. 7.31. Two-color, four-beam laser Doppler system, suitable for measuring two velocity components. The green (A = 488 nm) and bIue (A =514.5 nm) lines of an Argon-Ion laser have been used 7.4 System Description 393 Green Fronl vicw; Bluc and rcen ~ inlcrfcren 'e fringes ~o nenl ß1ue Fig. 7.32. Two-color, three-beam transmitting probe for measuring two velocity components splitters, alignment of the two illuminated volumes is insured if the beams are parallel before the common front lens and if the front lens has very low chromatic aberrations. Here the main concern is to insure that the waist of each beam is also at the intersection volume. This is most easily achieved by working with highly collimated beams before the front lens, i.e. Zl ~ 00 and Z2 ~ f in Eq. (3.127). Fiber-optic probes may require adjustment of the individual beams. For this purpose, a pinhole is placed at the focallength of the probe and each beam is aligned to focus onto the pinhole in turn. Good alignment is recognized when the diffraction rings behind the pinhole are concentric and centered on the pinhole. Another possibility is to image the measurement volume by a micro-Iens with a very small focallength. By moving the lens back and forth, the intersection region can be imaged onto a screen. If the image of the measurement volurne is not large enough and no space is left, a mirror can be used to project the image in a different direction. The overlap of the beams in the intersection region can be checked by switching on and off one of the laser beams one at a time. In a similar manner the interference fringes can be visualized to check the modulation in the measurement volume (if no shift frequency is used.) Note that the imaging of the measurement volume by a micro-Iens is similar to the scattering of the laser beams by a very large particle. If the system operates in backscatter, the receiving optics must also be aligned with the illuminated volume. Generally, this is achieved by first coupling laser light into the receiving fiber and then positioning the alignment pinhole at the waist of the beam emanating from the receiving fiber. The procedure above with the transmitting beams then follows. Rough alignment of the receiving optics for off-axis or forward scatter is best achieved by positioning a scattering center in the illuminated volume, typically a hot-wire, a glass bead or even a piece of paper. The center of the measurement volume is achieved when the scattering center produces the same diffraction rings for all incident beams. Because a relatively large particle or cylinder causes a detection volume shift (see section 5.1.3) a further adjustment must be made afterwards. This fine adjustment of the receiving aperture is usually possible by 394 7 Laser Doppler Systems examining the amplitude and especially the modulation of the signals from small particles (d p < 0.05 dw ) on an oscilloscope. Further information about the alignment can be found in section 14.1.3. The component recognition at the photodetector does not necessarily have to be achieved by using different colors. By imposing different frequency shifts on the two measurement volumes, a single color can be used for all transmitting beams and detection can follow with a single photodetector. The received signal must then be fed parallel into two band pass filters, adjusted so that they isolate the respective velo city components. The difference in shift frequencies between the two volumes and the bandwidth of the ftlters must be chosen to avoid any truncation of velocity fluctuations in a turbulent flow. A different concept of achieving the necessary frequency differences between components is to use three appropriate monomode laser diodes. Such a system has been realized with DBR (distributed Bragg reflector) laser diodes with almost equal emission frequencies and overlapping tuning ranges without mode hopping. One laser diode has been used for each beam of a three-beam, two component system presented by Müller et al. (1996). Pulsed laser diodes also open several additional avenues to achieve twovelo city component systems. If two lasers are used, the lasers can be alternately pulsed at a high frequency. By synchronizing the laser pulse sequences with the signal digitization, a technique known as »coherent sampling", a single photodetector and data acquisition chain can be used for both velo city components (Wang et al. 1994a). Switch de-multiplexing or time-division multiplexing is used on the detector signal to separate the two velo city components (Wang et al. 1994b). Multiple pulsed laser diodes for multiple velo city component measurements can also be replaced by time-delay techniques, either through additional path lengths or by using fibers of different lengths. The principle of such a system based on path-Iength differences and using a pulse rate of 240 MHz is illustrated in Fig. 7.33, adapted from Wang et al. (1994a). Several two-velocity component systems have been realized using a single laser line and high frequency optical multiplexing between orthogonal measurement volumes (Resagk et al. 1995). The multiplexing has been achieved using an integrated optical switch embedded in a lithium niobate substrate (LiNb0 3 ), manufactured using the proton exchange process. The obtainable power levels through such devices are, however, quite modest. Most two-velocity component systems are arranged with orthogonal components. Karlsson et al. (1993) have investigated errors which arise when the alignment is not exactly orthogonal, but is assumed so. Normally, this error is negligible; however, inthe case of boundary layers, where both the meanvelocityand the magnitude of the fluctuations is much greater in the streamwise direction than in the wall normal direction, even small deviations of the order of 1 deg can have considerable impact on the measurement quantities. To see this effect, the velo city components u and v in the lab-fixed coordinate system (x,y) can be expressed in terms of the measured velocity components um and vm and the angle a, as shown in Fig. 7.34, assuming the x velo city component to be correet, i.e. u = um. 7.4 System Description V=_I_(V m cosa -umsina) 395 (7.169) (7.170) 1 -(um-V" -----;:;.) U" V =m -Um sma (7.171) cosa I\ lirrors Ligh l pulses: -.D " ! =iF-= ! I I --.L.....'~ ~I - ~I ßcam hi fI rcgiSlcr l + [ 0 cilla lor Fig. 7.33. A schematic diagram of a HF-pulsed diode two-velocity component laser Doppler system using coherent sampling (Wang et al. 1994a) Front vicw: y u x z Fig. 7.34. Non-orthogonalityof a two-velocity component laser Doppler system 396 7 Laser Doppler Systems Equations (7.169)-(7.171) can therefore be used to compute corrected values of the flow quantities, if the rnisalignment angle a is known. A second effect pointed out by Karlsson et al. (1993), which applies when a four-beam, two-velocity component system is tilted to traverse the measurement volume closer to the wall, or principally, when using the three-beam, twovelo city component arrangement, is the influence of the z velo city component on the measurement. The geometry of the situation is sketched in Fig. 7.35. The lab-fixed coordinate system (x,y,z), with its corresponding velocity components u, v and w is assumed to be aligned parallel to the wall. The corrected flow quantities in this system orthogonal to the wall are given by 1 V=--Vm cosß u' v' ~ v' 1 1 --u: v: cosß 1-w'2sin 2ß cosß ~V'2 cos 2 ß+ w f2 sin 2 ß (7.172) (7.173) (7.174) These relations have been derived assuming that the quantities W, u'w' and v'w' are zero (in a two-dimensional flow). The influence on 11 and u'v' is negligible, since ß is typically only a few degrees. On the other hand v~ can be substantially influenced for small y, where v' goes to zero but w' remains finite. Beyond y + "" 5 (see section 7.1.1.4) the effect is negligible. Fig. 7.35. Influence of system tilt in the y-z plane 7.4.3 Three-Velocity Component Systems Especially in the field of turbulence research, it is often valuable to acquire three velo city components simultaneously, allowing not only the computation of all six terms of the Reynolds stress tensor but also higher order velocity correlations, e.g. tri pie correlations. The extension of the laser Doppler technique to 7.4 System Description 397 three velo city components is conceptually straightforward, one additional laser source or a third color from an existing two-velocity component system can be used to form a third measurement volume. A receiving optics, a detector with color filter and a third signal processor, complement the system. The layout of a commercial three-velocity component laser Doppler system is shown pictorially in Fig. 7.36. This system uses the blue, green and violet lines of an Ar-Ion gas laser: six beams are focused at the measurement volume. In practice several complicating factors arise in performing measurements with three-velocity component systems. These factors are mostly related to the fact that optical access to the measurement region is often limited. Hence, the three measurement volumes cannot be aligned orthogonal to one another. A further difficulty arises when the measurement volumes are not spatially coincident. Restricted optical access leads to system designs in which all six beams enter the flow channel through a single window. For liquid flows, an externally traversed three-velocity component system will not be feasible, even with refractive index matching, since the measurement volumes will not move coincident with one another (see section 14.3). This holds also for orthogonal systems, if they can be used. Thus, three-velocity component systems for fluid flows are generally designed to be fuHy immersible. The generalized system pictured in Fig. 7.37 measures three velo city components cI' c2 and c3 ' which are not necessarily orthogonal to one another. A transformation to probe-frxed, orthogonal velo city components up u2 and u3 is necessary using a transformation matrix Fig. 7.36. Pictorial of a three-velocity componen t laser Doppler system 398 7 Laser Doppler Systems One-velocity componcnt probc X, Xl Int erseclion planc I'ronc- Ilx.,a orthogonal <.:oordina tc system Fig. 7.37. Three-velocity component laser Doppler system, showing measured velocity components and probe-fixed, orthogonal coordinate system (7.175) A further transformation to a channel-ftxed, orthogonal co ordinate system may also be necessary and can be performed either subsequently or be directly incorporated into the transformation matrix. Since this final transformation is very application specific, the following discussion considers results only in the probe-fixed coordinate system (X p X 2 ,X 3 ). The probe-ftxed coordinate system is aligned with the axis of the two-velocity component transmission optics (x 3 ) and with the normal to the prob es' intersection plane (x 2 ). The transformation matrix is given by S; A= !.L s, s2l s2l 0 = sin(a;) S2. =sin(a 2 -a.) 0; = cos(a;) _!l 0. s2l s2l ° ~ 02S3 - S203 S.03 -0,S30rp 1 S2,t ~0 3 s2l trp03 ° 3Srp 0 srp = sin( 41) (7.176) o~ = cos(4J) trp = tan( 41) where the angles a" a 2 and a 3 and their senses are defined in Fig. 7.37. In many applications, the optical system can be arranged such that a , = 0 deg, a 2 = 90 deg, and a 3 = 0 deg, in which case the transformation matrix takes the simple form 7.4 System Description l] A=[_i ~ tantP 399 (7.177) sintP which in turn reduces to the unity matrix for an orthogonal configuration (tP = 90 deg). A suitable transformation matrix into a cylindrical co ordinate system is given by Carrotte and Britchford (1994). Systematic errors are introduced if the terms a'! in the transformation matrix A are not correct in magnitude. This refers in particular to the system alignment, as discussed in section 7.4.2 for two-velocity component systems. Indeed, the 'angle' error is probably the one main reason why a laser Doppler system may require calibration, for instance using a theodolite, as described by Boutier and Lefevre (1986) or with a rotating wheel (Snyder et al. 1984). At this point, it is important to consider the propagation of errors through the transformation matrix. Errors in the measured velo city components &; will propagate to the probe-fixed system as Ou ; = du; ~ J, uC j = cj (7.178) a;j&j ou: Assuming random errors in measuring C j' then Je) = 0 and = o. This implies that if the individual mean measurements are non-biased, then the transformed mean velocities will also be non-biased. However, the variance of the errors will not be zero, (0e))2 0, thus all the Reynolds stresses will contain errors. * (7.179) assuming ~ = 0, i.e. stochastic errors on different velocity channels are uncorrelated with one another. This equation shows that errors in the normal stresses are always positive, whereas the shear stresses may exhibit errors of either sign. The simplified form of the transformation matrix given by Eq. (7.177) illustrates that the components U~2 and u[u~ are especially prone to errors, since the coefficients a31 and a33 can become very large for small values of the included angle between the probes tP. If the stochastic error is about the same magnitude on the velocity components CI and c 3 ' then oe? = od = &2. The effect of the transformation on the propagation of errors to u? and u{u~ are shown in Fig. 7.38 as a function of the included angle tP. This figure illustrates that errors explode for small angles between the probes. Further examples can be found in Morrison et al. (1990) and in Carrotte and Britshford (1994). This leads to the recommendation that the included angle tP exceeds 45 deg for the measurement of turbulent quantities. 7 Laser Doppler Systems 400 ~ 6r-r-rrr-r-r-r-r-r-T,-.-.-.-.-.-.-.-.r-r-r-r-r-r-r-,-.-,-, , ,, ,, ,, 511;11; &' I I 1 1 tan'ljJ sin 1jJ ~·~-1=--+----1 2 ,, , 511;' 1 &' sinljJ =-1=---1 \ \ , \ \ \ \ 2 30 ,, ,, ,, , ... ........ 60 Included angle ql [deg] 90 Fig. 7.38. Normalized error in Reynolds stress terms due to coordinate transformation Although the expectation of the mean velo city u3 is correct (non-biased), the effect of the error amplification through the co ordinate transformation will be to increase the required measurement time to achieve a given confidence level in the results. This follows from a standard uncertainty analysis, in which the variance of a derived quantity can be estimated from the sum of the variances of the input parameters times their influence coefficients (Kline and McClintock, 1953) (7.180) Both influence coefficients dU 3 I dC! and dU 3 I dC 3 will normally be strong functions ofthe included probe angle 9 for small values of 9. At least one commercial, three-velocity component system has been offered, which transmits all beams through a single front lens, thus greatly simplifying system alignment. However, this necessarily leads to small included angles 9 for the third velo city component and, as analyzed by Chevrin et al. (1993), leads to intolerable errors in the measurement ofturbulence quantities. A further source of error when using three-velocity component laser Doppler systems is the possibility of the signals obtained on each channel originating from different tracer particles. The situation is pictured in Fig. 7.39, in which the measurement volumes of the system from Fig. 7.37 are ShOWll, assuming each probe is operating in backscatter detection mode. In a two-velocity component system, multiple particle signals are avoided by invoking a coincidence time window on each signal processing channel. A coincidence window is also used in three-component systems; however, the measurement volumes are not necessarily spatially coincident, thus, a time coincidence window alone is not sufficient. The resulting error can be significant, since actually the spatial correlation is measured for such particle pairs, rather than the correlation for separation 7.4 System Description 401 Receiver for c, Receiver for c, and c, DeleClion volume for c, and c, Delcction VOllllllC for c, Fig. 7.39. Detail of measurement volumes in a three-velocity component laser Doppler system zero. Thus, the error will depend on the dimensions of the detection volume, in particular with respect to the microscales of the flow, i.e. how fast the correlation falls off with separation. Further estimates of potential errors can be found in Boutier et al. (1985), Eriksson and Karlsson (1995) or Browne (1989). This error is best avoided optically. The receiving optics must be designed to limit every channel to the same spatial volume. If backscatter probes are used, this is achieved by cross-detection, meaning the probe transmitting one channel detects the channel(s) from the other probe. Thus, only partides in the inter section volume are seen by the system. Alignment of a three-velocity component system can be performed similar to that of a two-velocity component system, described in the previous section. Fig. 7.36 pictures a two-velocity component probe and a one-velocity component probe, all focused onto an alignment pinhole. Another alignment aid, especially useful when the angle between the probes is large, is to replace the pinhole with a steel ball bearing. Each beam is aligned until it reflects concentrically back onto itself. The reflected pattern can be visualized by placing a screen between the probe and the ball bearing, with holes punched out for the outgoing beams. 7.4.4 Multi·Point Systems There exists considerable interest in measuring components of the spatial correlation function R i/x k ,t,rk'7:). As outlined in section 7.1.1, this function yields information about the length scales of turbulent motion and is increasingly used for purposes of modeling turbulence. Furthermore, two-point or multi-point laser Doppler systems must not necessarily invoke the Taylor hypo thesis to estimate flow gradients. Thus, the direct measurement of the individual terms in the dissipation equation becomes possible, in principle. Practically, such systems have been realized either by scanning the measurement volume quickly through the flow region of interest (Durst et al. 1981, Chedroudi and Simpson 1984, Shinpaugh and Simpson 1995), by aligning the measurement volumes of separate systems dose to one another (Nakatani et al. 1986) or by generating a larger measurement volume and acquiring additional information about each tracer partide position (Strunck et al. 1998). Also mixed systems, in which three velocity components are acquired at one position while two 402 7 Laser Doppler Systems components are acquired at another position, have been successfully used to investigate the three dimensionality of complex turbulent boundary layers (ÖI<;:men and Simpson 1995). Somewhat related to multi-point systems is the dual cylindrical wave laser Doppler system for the measurement of wall shear stress, first introduced by Naqwi el al. (1984) and analyzed further by Bultynck (1998). In this instrument, tracer particles throughout the entire viscous sublayer of a wall boundary layer are detected and contribute to an estimation of the wall shear stress. In the following discussion, so me important considerations concerning the opticallayout of two-point systems will be addressed. These issues also apply to multi-point systems. Considerations about the processing of data from such systems are given in section 11.3. A prime objective with two-point systems is to make accurate measurements of the turbulent microscales A ii,k (see Eq. (7.46)). For this, the spatial correlation function must be very accurately measured for sm all spatial separations. This is the portion of the function, to which a parabolic curve fit is performed to estimate Aii,k' For large separations, a measurement of the spatial correlation function is far less problematic, but often also of less interest. Despite numerous studies involving two-point systems (Morton and Clark 1971, Bourke et al. 1971, Drain et al. 1975, Fraser et al. 1986), the consequences of the coincidence window and the overlap of measurement volumes at small separations was only first considered by Absil (1988, 1995) and Absil et al. (1990). Applying a coincidence window, within which signal pairs from the two measurement volumes will be accepted, turns a spatial correlation measurement into a space-time correlation. Thus, the coincidence window must be chosen smaller than the microscales of the turbulence, under which the autocorrelation function is still approximately unity R( T cw) "" 1. Indeed, it appears the entire issue of coincidence window choice may be superfluous when appropriate estimators are used, as discussed in section 11.2 (Müller et al. 1998). The finite size of the measurement volumes and their overlap can lead to a distortion of the measured spatial correlation, generally know as the geometric bias. This has been studied extensively by several authors (Tummers et al. 1994, 1996, Belmabrouk et al. 1991) and especially by Benedict (1995). The magnitude of this bias depends also on which component of the correlation function is being measured. The time dependence of the space-time correlation is dropped and only the spatial correlation function of the i th and j th velo city component Ri/x,y,z) is considered. An optical configuration is assumed in which the optical axis is aligned with the z coordinate and x is the main flow direction. The situation for the longitudinal correlation Rll (.dx,O,O) is pictured in Fig. 7.40, adapted from Benedict (1995). If forward scatter detection is used, particles will be detected over the entire length of the measurement volume. Signal pairs arising from two particles will have an effective separation larger than the intended L1x, depending on the exact particle trajectories. A similar bias occurs for the functions Rll (O,L1y,O) and R11 (O,O,L1z). For this reason, microscale measurements can only be successful if a side-scatter detection is employed, which gready reduces the length of the de- 7.4 System Description a Bursl pair from IWO 403 b l3ursl pair from a single panidc panides x · n =J. '+ tu ,." =tu z /l j z Fig. 7.40a,b. Measuremen t volume orienta tion for Ru (Ax,O,O). a Forward sca tter detection volumes, b Side-scatter detection volumes tection volume, as indieated by the projeeted cireular pinhole apertures in Fig. 7.40. Depending on the co in eiden ce time window, single partides may also lead to two signals on eaeh of the ehannels, in whieh ease a preference for faster partides, whieh fall within the time window, will result. This is similar to the bias diseussed by Brown (1989) for non-orthogonal, three-veloeity eomponent systems. In fact, even with side seatter, the finite size of the measurement volume wiU always lead to a geometrie bias and only when the volume dimensions are of the order of one Kolmogorov length seale ean the measurements be eonsidered fuUy unbiased (Benediet 1995). This is a rather stringent requirement and is not easy to verify. One reason is beeause the geometrie bias does not effeet the measured mean and varianee of the tlow velo city and even the maximum eorrelation eoefficient at zero separation may attain values in exeess of 0.999. Nevertheless, the shape of the correlation funetion at sm all separations ean still be biased, thus affeeting microseale estimates. The geometrie bias of the spatial eorrelation funetion eannot be deteeted in the measurement parameters. A good summary of the situation is given in Fig. 7.41, showing the various measurement volume orientations for all Rll eorrelations and assuming a near spherieal measurement volume. Coneeptually, the same applies for the other eorrelation eomponents. If coincidenee in time is demanded, then any volume separation displaying overlap will quiekly become effeetively zero for lateral and spanwise eorrelations, sinee single-partide, dual-signal events will dominate. A typical measured spatial eorrelation funetion, taken from Benediet (1995) is shown in Fig. 7.42. These measurements were performed in the tlow behind a baekward-facing step at a position where the measurement volume size was about two Kolmogorov lengths. Even under these eonditions, the spatial correlation function shows a tlattening for small separations, indieating a residual geometrie bias. Interestingly, the longitudinal eorrelation appears to possess infinite spatial resolution, assuming that this aligns with the dominant velocity eomponent. A logieal eonsequenee of this reasoning is that the spatial resolution for all eorrelation eomponents improves as the turbulenee intensity inereases, i.e. when there is no longer a dominant tlow eomponent. 404 7 Laser Doppler Systems For two-point correlation measurements, estimators must be carefully chosen to avoid such effects, arising from the fact that coincidence of both channels is demanded. One such 'non -coincidence' estimator is presented in section 11.3. Non-overlapping RI1 (Ax,O,O) Xl Longitudinal : & 'ff = Ax Rl1 (O,Ay,O) l Transverse ~' Spanwise • r AX'ff =0=& AX'ff = & 0 0 @ 0 • .. •• • Az Az'ff =O,tAz Az'ff = Az 11 Ay,!! =O=Ay AY'ff = O,tAy AY'ff = Ay RlI (O"O,Az) Totaloverlap Overlap 'ff r" =O=Az Fig. 7.41. Two-point velo city measuremen ts with a dominant velocity component i ••••••••• Xx X·x • • X • c " '0 . 1: ~ 0,98 o u .2 "E'" ::; u 0.96 x SO • SO Ils coincidcncc, 100 Ilm pillholc Ils coillcidence, 200 fll11 pinhole 0.94 0.02 0.01 0.00 0.01 0.02 Position z Ilf I-I Fig. 7.42. Spanwise correlation coefficient measured behind a backward-facing step of height H (Benedict 1995) 7.5 Laser Transit Velocimetry 405 7.5 Laser Transit Velocimetry Particularly at high velocities, the slip velo city between the seed particles and the flow is best minimized by choosing submicron particles; however, these then fall into the Rayleigh scattering range in which the scattered power is proportional to d;. To maintain particle detection probabilities, a drastic increase in incident power is then required. If this is achieved through a stronger focussing of the laser Doppler measurement volume, eventually only two fringes are left and the velo city measurement becomes a time-of-flight method. This, however, is not the most effective way to construct a time-of-flight instrument based on two light barriers. The laser transit velocimeter (LTV) technique is much more effective. In the LTV technique, sometimes referred to as the laser-two-focus (L2F) technique, two laser beams are highly focused next to each other in the flow. Foci of 5 ... 20!-Lm and separations ofAx "" 50 ... 200!-Lm are typical for measuring velocities in the 500 ... 1500 m S-1 range. This measurement principle is sketched in Fig. 7.43, which shows the LTV measurement volume. Two photomultipliers are focused onto each of the beam waists, very often in a backscatter arrangement, since one main application area of LTV is turbomachinery, where only one optical access is available (Schodl 1975, Schodl 1977). A particle passing in the main flow direction through both volumes results in a signal on each of the photomultipliers, with a time delay inversely proportional to the velo city. If, however, the trajectory of the particle has any small deviations, then the particle may only cross one volume and as such, cannot be processed. Thus, the LTV is very weH suited for measuring laminar flows or at least very weH directed turbulent flows. Two methods are commonly used to determine the transit time. The first method is a direct measurement of the time-of-flight, for instance with a multichannel analyzer or a purpose-built electronics. This assurnes that the particles Signal processing 1 M=-(Mt+M L ) 2 VI ~ ~ , I ,( v, /I!casurcmcnl volume Fig. 7.43. Measurement principle of the laser transit technique 1\ •I 406 7 Laser Doppler Systems follow one another at intervals larger than the transit time and furthermore, that there exists negligible background noise from unpaired bursts. Estimates of the allowable particle concentration are given below. A more robust method is to use the cross-correlation between signals from the two photodetectors. Ideally, noise will contribute only to the background level of the correlogram and the peak will correspond to the inverse of the flow velo city distribution. However, many factors can still lead to a distortion of the correlogram. These effects include spatial and temporal velo city gradients (Lading 1979) and transit time broadening (Mayo and Smart 1979). In general these factors restrict any simple transformation of the correlogram to a velo city distribution, to low levels of turbulence, typically less than 15 % (Brown 1983). Erdmann (1983) proposed a model for LTV based on particle statistics. This expresses the shape of the correlogram in terms of characteristic flow parameters, including the velo city moments, the integral time scale and the dimension of the measurement volume. Measurements performed by Schachenmann and Tropea (1987) indicate that using this method, the LTV technique can be used up to turbulence levels of 300/0. For lower levels of turbulence, the procedure is such that aseries of measurements are performed with the optical system rotated about its optical axis at various angles about the main flow direction a. At each angle, a probability histogram of the time-of-flight At is recorded, as shown in Fig. 7.44 for the joint histogram h(At, a). The velo city components u =Iulcosa, (7.181) v =Iulsina are computed for each of k velo city classes and their average values are given by joinl probabi lily "( I,a) [-J 1.0 0.5 0.0 20 Time of[ligh l Flowanglc aldcg ] Ll/llls j o 0.0 Fig. 7.44. Joint probability histogram of time-of-flight and flow angle 7.5 Laser Transit Velocimetry 407 (7.182) (7.183) with (7.184) In these equations, 1 is the number of angular positions, N ij is the number of measured velocities at the angle j in the velo city dass i, which is weighted by the data rate M, since the partide rate is correlated with the instantaneous velocity. The mean absolute velo city is simply (7.185) and the mean flow direction is given by _ u a = arctan-=- (7.186) v Further analysis yielding u/ 2 or v/ 2 is possible. The dimensions of the measurement volume are chosen under several constraints. Schodl und Förster (1991) indicate the effects of a larger or smaller beam focus and separation. A decrease ofbeam separation at constant beam diameters will increase the range of partide trajectories which will be properly registered. The data rate will increase and the number of rotational positions will decrease, also decreasing the measurement duration. However, the measurement error will increase. For a given maximum allowable error, the measurement duration can be minimized by matching the beam separation to the turbulence level. For instance, for a beam waist diameter of 10 /-lm and 1 % turbulence, a 1 % measurement error can be achieved for aseparation of 350/-lm. At 10 % turbulence, a separation of 70 /-lm should be used. Using the 350 /-lm separation in a flow with 10 % turbulence would increase the measurement duration by a factor of five (Schodl and Förster, 1991). To insure single realization operation, the measurement volume can be approximated by an ellipse and the maximum allowable partide concentration is then given by (section 5.1.7 or 7.1.3) _ 0.1 n <-=0.1 Vo 8A b Ab =0.032-3 2.531t do L1x do Llx 2 3 (7.187) For a wavelength of Ab = 488 nm, a beam of dwb = 10 /-1ill and aseparation of Llx = 500 /-lm, this concentration is about 3 x 10 3 cm- 3 • This limit already lies weIl above natural atmospheric seeding, as indicated in section 13.4. 408 7 Laser Doppler Systems Note that the light forming the measurement volume must be neither coherent nor monochromatic. The choice of lasers as a light source is mainly because of their high intensity and good focussing properties. Especially high-powered laser diodes are very attractive for this application, due to their compactness. More re cent systems have extended the technique to also measure two or even three velo city components simultaneously (Schodl 1998, Karpinski et al. 1999). 410 8 Phase Doppler Systems If the particle is assumed spherical, the measured phase tP r from one detector and for one scattering order can be linearly related to the particle diameter. For reflection the relation between particle diameter and detected phase is given by Eqs. (5.104) and (5.200) to (5.203), as in section 5.3.2.1 tP;I) dA~I-COSlf/r cosr/Jr cos~+sinlf/r sin~ = nf b -~I-COSlf/r cosr/Jr cos~-sinlf/r sin~) "" n.fi d Ab sin If/r sin % P ~1-COSlf/rcosr/Jrcos~ 2n % 'f/;{ ",,-d - - Ab P sin~;{ ~n dp ~cos'f/;{ "" sin% sin 1lr "f'r for <t: 1- cos If/r cosr/Jr cos~ for sin8"" 8 /\ sinlf/r "" If/r /\ r/Jr > 28 for r/Jr = 0 deg /\ If/r > ~ (8.2) b and for first-order refraction from Eqs. (5.104) and (5.213) to (5.216), as in section 5.3.2.2 tP(2) = 2n d ( l+m 2-m.fi ~l+sinlf/r sin ~+coslf/r cosr/Jr cos~ r Ab P 2n ""--d p Ab l+m 2-m.fi ~1-sinlf/rSin~+COSlf/rCOSr/JrCOS~) m sin If/ sin ~ r .fi~(I+CJ(l+m2-m.fi~l+Cc) 2n "" --m d Ab ~'f/;{ -:------:--r~~~==:=====c Plcos~;{I~I+m2-2mlcos~;{1 2n d ""--m Ab P Jl % sin 'f/;{ fi or fior r/Jr , Cc=coslf/rcosr/Jrcos~ sin IIr "f' r "" IIr "f' r /\ (8.3) sin 8 "" 8 /\ sinr/Jr > sin28 = 0 deg /\ sin If/r > sin ~ + m 2 - 2mlcos 'f/;{I Using two detectors, the measurable phase difference between the detectors remains linearly related to the particle diameter. The phase difference between two spatially separated detectors is given by These relations are given by Eq. (5.221) for reflective dominated scattering (N = 1) and by Eq. (5.225) for first-order refracted light (N = 2) in a standard phase Doppler configuration. The factors F~N) are known as the phase conversion factors (see section 5.3.2.3) and relate the measured phase difference to the particle size for a given scattering order N. The inverse of the phase conversion 8.1 Selection of the Optical Configura tion 411 factor is the diameter conversion factor, sometimes called the geometrical factor, and is denoted by Pt) = (F~N)tl. Note that these factors are defined for the phase difference and not for the phase of a single receiver. One of the primary considerations when selecting the optical configuration of a phase Doppler system is, therefore, where to place the detectors to insure that only one scattering order dominates. The distribution of the various scattering orders has been discussed extensively in section 4.1.3 for the case ofhomogeneous waves and in section 4.2.3 for the case of inhomogeneous incident waves. It was demonstrated that an ideal detector position will depend on the relative refractive index of the particle, its diameter and the polarization of the incident light. Therefore, the following discussion concerns the selection of detector and aperture positions to match the desired measurement range in terms of these parameters. Following this discussion, single-point phase Doppler systems will be examined in section 8.2 before describing more specialized considerations in section 8.3 and multi-dimensional realizations ofthe phase Doppler principle in section 8.4. 8.1 Selection of the Optical Configuration Already in section 2.2 it became evident that a particle size measurement on the basis of the phase Doppler technique requires a minimum of two detectors focused onto the measurement volume of an otherwise conventionallaser Doppler system. Fig. 8.1 reviews the essential geometric parameters of the optical system: the beam intersection (full) angle B, the off-axis angle (in the y-z plane) rfr' and the elevation angles ±V' r' usually chosen symmetric about the y-z plane for each of the two detectors. Consider first the case ofhomogeneous incident waves on the particle and reexamine for instance the intensity distributions of scattered light given in Fig. 4.32 for a water droplet in air and for parallel polarization of the incident waves. For the range of droplet sizes shown, 10 /Jl11 ... 1000 /-lm, second-order Laser beams Parlide x Receiver aperture Fig. 8.1. Optical arrangement for the standard phase Doppler technique leasured 412 8 Phase Doppler Systems refraction clearly dominates by at least two orders of magnitude in the angular range 30 deg to 80 deg. Therefore an optical configuration could be chosen, such that the scattering angle 13 s lies in the range 30 deg < 13 s < 80 deg for each beam/detector combination. This can be generalized as 13 m;n < 13 s < 13 max. Recalling again the nomenclature of section 2.2 and chapter 5 (index br: b - beam 1 or 2; r - receiver 1 or 2), the following condition should be satisfied for each detector cos 131r = cost/Jr cos( IfIr -~) (8.5) COS 13 2r = cost/Jr cos( IfIr +~) (8.6) Of course there are an infinite number of possibilities to satisfy these relations; however, it is impractical to attempt to independently position two detectors in such a way that they are both focused onto the same measurement volume. Especially when using slits or pinholes for definition and reduction of the detection volume size (section 8.3.2), the alignment is very difficult and the system cannot be moved without re-alignment. For this reason detector positions are usually chosen such that all detectors are focused onto the measurement volume through a common lens or at least using a common probe. One solution, which allows this and can also satisfy Eqs. (8.5) and (8.6) is to choose the same off-axis angle for each detector (t/J1 = t/J2 = t/Jr) and to choose symmetric elevation angles, 1fI 1 = -1fI 2 • Since IfI r is generally very small in such configurations, the condition (8.7) is often sufficient to insure that a single scattering mode dominates. This is the optical arrangement illustrated in Fig. 8.1 and it is known as the standard phase Doppler system. Details ofthis arrangement are given in section 8.2.1 Another configuration satisfying Eqs. (8.5) and (8.6) and still allowing collection through a common lens is to maintain t/J1 = t/J2 = 0, (8.8) 13 max < 13 2r = IfI r + ~ (8.9) and thus the elevation angles of the detectors must lie in the range (8.10) Such an optical configuration is shown in Fig. 8.2 and is known as a planar phase Doppler system, since the detectors lie in the same plane as the beams. This system will be discussed in more detail in section 8.2.2. In general, the two detectors can be placed wherever the two receivers are not detecting the exact same fringe, and thus the phase difference is non-zero for the considered scattering order. In practice, light scattering diagrams as illustrated in Figs. 4.29 to 4.32 are only necessary to calculate for unknown droplet media, since the situation for many common particles has already been analyzed. Table 8.1 summarizes some 8.1 Selection of the Optical Configura tion 413 Fig. 8.2. Optical arrangement for a planar phase Doppler system Tahle 8.1. Relative refractive indices of common fuels, liquids and solids at 273 K Particle and medium Air bubble in water Air bubble in freon Latex in water Glass in water Water in air Ethanol in air Heptane and dodecane in air Diesel in air Benzene Glass in air Latex in air Relative refractive index m 0.75 0.833 1.192 1.132 1.33 1.36 1.39 1.45 1.50 1.51 1.59 common partides and their relative refractive indices. In Figs. 8.3 and 8.4 the regions of dominant (by one order of magnitude in the partide diameter range 10Ilm to 200llm) scattering orders are plotted via scattering angle and refractive index for both polarization components. Some recommendations for detector positions can be made for different partides, as summarized in Table 8.2, adapted from Dantec Dynamics (1999) and from Figs. 8.3 and 8.4. Characteristic (limiting) scattering angles given in Fig. 4.33 as a function of relative refractive index are also plotted in Figs. 8.3 and 8.4. Given that one single scattering order dominates the light received at the detectors, the relation between the phase and the particle diameter should be linear, as expressed by Eq. (8.2) for reflection and Eq. (8.3) for first-order refraction. In fact, this result is obtained only for geometrical optics, because if a Lorenz-Mie computation is performed, deviations from the linear relation occur. These deviations have their origins in the fact that additional scattering orders begin to become influential. Especially for small particles the assumptions of geometrical optics are no longer valid and the scattering characteristics are influenced by surface waves and a smearing of the scattering orders. This is easily seen by inspection of Figs. 4.30 and 4.32 in relation to Figs. 4.29 and 4.31 for the three particle sizes shown. 414 8 Phase Doppler Systems 2.0 D""T"1rTT"1rTT"1"TT"T"TTT"TTTTTTTTTTTT"TTT"I"TT1rTT"1rTT"1"TT"1"TT"T"TTT"TTTTTTTTTTTrTTT"n-nrTT"1n"nn"n"TT"T-r.w. . . ....,.,. >< ~ 1.8 "0 .S <; .:: . . . . . . Ra i nbow angle - - - . Critical angle 1.6 ü -=~" .~ ., IA 1.2 0; .:::: ...... 1.0 ---- Dominant cattcring order >90% Refleclion (I) 0.8 <30% 30% ... 40% ... 50% ... 60% ... 70% ... 80% ... 90% >90% __ _ DD D Sacttering angle Fig. 8.3a,b. Scattering angle regions of dominant scattering order for perpendicular polarization as a function of scattering angle and relative refractive index. The maps are processed from scattering functions of particles in the range X M = 64.4 ... 1290 (10 Ilm ... 200 Ilm). a The regions where 90% of the scattered intensity comes from only one scattering order are indicated with gray levels for the different scattering orders. The contour lines give the 92%, 94%, 96% and 98% regions of dominance, b Minimum percentage of intensity of the strongest scattering order to the overall scattered intensity for all particle diameters X M = 64.4 ... 1290 (10 Ilm ... 200 Ilm) A eomparison between the computed phase differenee using geometrieal opties and Lorenz-Mie is shown in Fig. 8.5a for a typieal standard phase Doppler eonfiguration and for a point deteetor. However, all deteetors exhibit a finite aperture and this has an integrating and smoothing effeet on the eharaeteristie phase differenee/diameter relation, as shown in Fig. 8.5b. For this seeond computation a reetangular aperture of 1.24 degx 6.26 deg has been used. For any new 8.1 Selection of the Optical Configuration -- 2.0 x 1.8 - <J I "a C ... .~ .. / 1.6 ,- / / Ci .!:: ~ 415 1.4 (I) ,,- ... .?; " t:i 1.2 (3) c:.: 1.0 Do minanl callcrin g ord er >90% Rc!lcclion (1) I" rdcr rcfraclion (2) _ 2'''' order rcfraclion (.~) 0.8 -- .= ...x "U .::'" Ci ..::'" ~ '"> '" t:i 0:: 150 <30% 30% ... 40% ... 50% ... 60% ... 70% ... 80% ... 90% >90% _ __ -000 D 180 Sacttering angle 0, [degJ Fig. 8.4a,b. Scattering angle regions of dominant scattering order for parallel polarization as a function of scattering angle and relative refractive index. Thc maps are processed from scattering functions of particles in the range XM = 64.4 ... 1290 (10 11m ... 200 11m) . a The regions where 90% of thc scattered intensity comes from only one scattering order are indicated with gray levels for the different scattering orders. The contoUf lines give the 92%, 94%, 96% and 98% regions of dominance, b Minimum percentage of intensity of the strongest scattering order to the overall scattered intensity for all particle diameters X M = 64.4 .. . 1290 (10 11m ... 200 11m) measurement situation therefore, it is helpful to check the linearity of this curve for the exact detector positions and apertures being employed. It is worth noting that the phase Doppler technique is not very workable in backscatter ( 15 s > 150 deg). This is unfortunate, since backscatter laser Doppler probes have proven extremely convenient in terms of traversing and alignment, while also requiring only one optical access to the measurement position. The 416 8 Phase Doppler Systems Table 8.2: General recommendations for detector positioning and polarization angle in phase Doppler systems, z'J RB - rainbowangle, z'J TR -angle of total reflection Particle type Scattering order Polarization Reflection 11 Scattering angle Totally rejlecting: Im{~}»l or.1 Any angle except forward diffraction region z'J, > arcsin(91 XM) Two Phase jlows: .1 11 or.1 Optimum near total reflection: z'J, = 13 TR z'J, "'(m-1)xllOdeg 11 22.4 degx.Jm -1 < iJ, < 90 degx J1- (m - 2)2 and z'J, >8deg 1.04 <m Rcflection 1sI refraction 1SI refraction 1.04 < m < 1.24 1.38< m < 1.92 2nd refraction .1 2nd refraction 11 Rcflection .1 m<0.9 0.9<m <0.98 Special Examples: Air in Water Reflection Water in Air Refraction z'J, '" z'J RB 13,~z'JRB Separate regions: 105 deg < 13, < 144 deg z'J, =85 deg Near Brewster angle: 65 deg< z'J, < 72 deg b a Cö <> 0 ~ s'" "<:) <.> '.0 c ... <) ~ 'i3 180 Co> ., '" ..t::. C. "0 C '" S ~ ..t::. c.. Particlc dia meier clp IflmJ Parlide diameier clp IflmJ Fig. 8.5a,b. Phase difference/diameter relation calculated with geometrical optics and Lorenz-Mie theory (A b =514.5nm, ljI,=l.72deg, ~,=60deg, m=1.333, e=4.352deg). a A point-like detector, b A finite size detector (Llljl, = 1.24 deg, Ll~, = 6.26 deg, Mask B) phase Doppler arrangements suggested by Figs. 8.3 and 8.4 and in Table 8.2 on the other hand, always require two optical windows. The reason for this restrietion can be seen in the Figs. 8.3 and 8.4. At large scattering angles ( tJ, > 150 deg), which are suitable for integrating both the out- 8.2 Single-Point Phase Doppler Systems 417 going beams and the scattered light into one probe with a single front lens, no one scattering mode dominates over a large size range (Figs. 4.29-4.32), but especially not over a change of refractive index (Figs. 4.34 and 4.35). Bultynck (1998) and Bultynck et al. (1996) have proposed specialized systems in which either reflected light in Alexander's dark band at a high refractive index can be used (Figs. 4.35d-f, 8.3) or second-order refracted light is used at even high er scattering angles. Although a working prototype using this coneept has been demonstrated, both the signal-to-noise ratio and the signal amplitude were low. Further possibilities of realizing a particle-sizing instrument using light seattered at high scattering angles will be discussed in seetion 9.2. 8.2 Single-Point Phase Doppler Systems 8.2.1 Three-Detector, Standard Phase Doppler System The first realization of a phase Doppler system was a standard system, as already illustrated in Fig. 8.1 (Bachalo and Hauser 1984). The relation between particle diameter and phase differenee is given by Eq. (8.4) and can be written for symmetric elevation angles (lf/ r = lf/1 = -lf/ 2) for reflection as LltP~~ = !1J~ d p = J2 2n dA ~1- cos lf/r cosq.)r cos% + sin lf/r sin % Ab - ~1- eos lf/r cosq.)r eos% - sin lf/r sin %) "'" J2 2n d Ab p sin lf/r sin % ~l-coslfFrcosq.)rcos% for <E sin lf/r sin % (8.11) 1-eoslf/rcosq.)rcos% Error 10% E 0.785 1% 0.1 % 0.279 0.089 and for refraction as LltP(2) - /i2)d 12 -}J12 p =22n d ( 1+m 2 -mJ2~l+coslf/rcosq.)rcos%+sinlf/rsin% Ab p - 1+m2-mJ2~1+COSlf/reoSq.)rCOS%-sinlf/rSin%) "'" -2 2n d msin lf/r sin % Ab p v~1+m2-mv' (8.12) v= J2~1+ eos'f/ r cosq.)r cos% It is apparent from Eqs. (8.11) and (8.12), or for example from Fig. 8.5, that a 2n phase differenee ambiguity will oeeur onee the particle size exeeeds a given value. This value ean be approximated for small intersection and elevation angles (see Eq. (8.2» and for regions of dominant reflection as 418 8 Phase Doppler Systems (8.13) or for first-order refraction (see Eq. (8.3» as d(2) p,max = 21tp(2) 4>,12 '" A Icos~;;il ~+m2 -2mlcos~;;i1 b m -'----_--'-V-'---_ _ _ _'---_-'BI If/r 12 (8.14) There are several methods to overcome this 21t ambiguity and to extend the measurement size range beyond the limiting diameter. One method is based completely on software and has been discussed fuHy in section 6.4.1. It exploits the fact that as the particle diameter increases, a time shift of the signals also arises. Therefore, the time shift of the signals must be determined in addition to the phase difference. More commonly however, hardware solutions are used in the form of one or more additional detectors. The most widely used arrangement is the threedetector standard phase Doppler system, pictured in Fig. 8.6. The three detectors are designated U I , U 2 , and U 3 (because they measure the u component of velocity) and they are used in the foHowing manner. Three phase differences are measured for each particle, the phase difference between detectors UI and U 2 , AlP12 , the phase difference between detectors UI and U 3 , AlPI3 and the phase difference between detectors U 2 and U 3 , AlP 23 • Due to the different elevation angles, the phase difference/diameter relations for the three detector pairs are different, the closely spaced pair yielding a relation which is less steep. Two of the three phase differences are linearly independent and can be used for two independent particle diameter estimations. The third measurement can be used as a validation criteria because the sum over all phase differences must vanish. (8.15) This is iHustrated schematically in Fig. 8.7. The measured phase difference AlPI2 could correspond to several different diameters, as shown in Fig. 8.7. However only one of these will be in agreement with the diameter indicated by Particlc Measured curvatur~ z Fig. 8.6. Optical arrangement of a three-detector, standard phase Doppler system 8.2 Single-Point Phase Doppler Systems 419 Phase difference - - - <l> u (<I>, -<1>,> - - - - t.~~ 1l (<I>, - <I) ,) <I> Ideg) w. 360 deg [/, _ U, y-z plane _ U, diameter Fig. 8.7. The phase difference/diameter relations of a three-detector, standard phase Doppler system and their use to extend the size range beyond a phase difference of 2n the phase difference LltP13. The phase difference ofthe outer receiver pair LlcP\~) is ambiguous in diameter and is given, for positive phase conversion factors, e.g. reflection N = 1, as "",(1) LJ'#12 = ß(1)d 12 P - 2nn 21t (8.16) whereas the phase difference of the neighboring receivers LlcPP3) leads to a single diameter (8.17) The number of 2n jumps can be found from the measured phases as n21t = int[_I_(ß)~: LlcP~~) - LltP~~) + 0.5] 2n ß (8.18) 13 In practice not all measurements fulfil exactly Eqs. (8.16) and (8.17). Some tolerance must be accepted in the agreement of the two separately measured diameters and this can be implemented as shown in Fig. 8.8. Here the shaded areas indicate a constant size or constant phase difference tolerance over the entire measurement range. A constant relative size tolerance could also easily implemented in the processing software. The use of three detectors resolves the 2n phase ambiguity and extends the measurement size range of the instrument. Theoretically the size limit can be extended to ab x 360 deg, where a and bare integers which do not have a common divisor and alb = ß 12 I ßl3' Because of measurement uncertainties and the acceptance tolerance, the maximum size range is in practice restricted by the measurement range of the detector pair with the largest phase conversion factor, in general F<P.13' The maximum measurable particle diameter becomes for reflection approximately (8.19) 420 8 Phase Doppler Systems <!:l " [lieg) Non-acccplance band 360 , ,, , , ,, , <!:l ,, [lIeg) Fig. 8.8. Phase difference matrix for a three-detector, standard phase Doppler system. Only measurement points falling within the tolerance areas will be accepted and for refraction (8.20) The size range is increased as the elevation angle difference between detectors U\ and U 3 or the intersection angle becomes smaller. In practice the range can not be extended to the given theoreticallimit in Eqs. (8.19) and (8.20), because the phase differences for small particles exhibit large scatter and can result in "negative" particle diameters, which are interpreted by the system as very large particles. Therefore, a non-acceptance band for phase differences near 360 deg is usually implemented, as illustrated in Fig. 8.8. Phase Doppler systems usually provide a means of changing the position (and size) of each of the receiving apertures to allow the system to be matched to a particular measurement task. One such implementation of a phase Doppler receiving probe is pictured in cross-section in Fig. 8.9, in which an interchangeable aperture plate is shown. Possible aperture plates (masks) are shown in Fig. 8.10. In this implementation, one lens in the probe is segmented, so that the received light is focused onto three different receiving fibers. The apertures of the symmetric mask A correspond to the three lens segments. Mask Band C use only portions of each lens segment. Further variation of the size range can be achieved by changing the focallength of the receiving optics, Ir, or the intersection angle. The achievable size range with each mask can be estimated for reflection by (8.21) and for refraction by d( 2) p,m"",, "'2~lcos~'/1/1+m2 -2mlcos~'/1 IJt m /2 V / 2 (D\ - DJ1b (8.22) 8.2 Single-Point Phase Doppler Systems 421 Measurement volume \----=::.... \ Front lens Aperture plate Multimode fibers Fig. 8.9. Fiber optic receiving probe for a three-detector, standard phase Doppler system (adapted from Dantec Dynamics (1999» Notch corresponding to index pin Fig. 8.10. Interchangeable receiving apertures (masks) for the probe shown in Fig. 8.9. where Ir is the focallength of the transmitting optics, Ir the focallength of the receiving optics, LJb the beam spacing and D] and D3 the receiver mask dimensions (e.g. given in Table 8.3 for the Dantec Dynamics system). From Eqs. (8.19) to (8.22) the maximum measurable size can be seen to be approximately linearly related to the focallengths and inversely proportional to the beam spacing and receiver separation. The sensitivity and size range of a three-detector, standard phase Doppler system can also be varied by selecting a different off-axis angle. Furthermore, the phase difference/diameter relation will depend on the refractive index for Table 8.3. Parameters of the aperture masks of a three-detector, standard phase Doppler system (Dantee Dynamics) Parameter [mm1 MaskA MaskB Maske R Xl 20.05 13.30 20.05 6.70 20.05 2.10 X3 10.00 6.00 Dl D3 12.65 9.35 2.00 7.05 0.00 2.00 4.00 ._= =_ . ... 422 8 Phase Doppler Systems scattering orders involving refraction. These dependencies are illustrated in Figs. 8.11 and 8.12 respectively, in which the inverse phase conversion factor ßlf1 is shown for systems operating in reflection and first-order refraction. Diameter measurements of particles with different refractive indexes, e.g. at different temperatures, can be improved by using off-axis angles in the range of -E n b Retlection (N= I) First·order rcfract ion (N 2) 2r-,-~--.--.--r--.-.--.-~~ 30 n<-..-~-..-~-..-~-..-~~ ::1. OJ) <.I ~ g~ I- 0 Ü 0.75 ...::= 1: .9 ~ 20 w - I) 0.8jj 8 1. 132 ">c: 0 ... 1.192 'J 1.39 1,45 \.50 \.59 10 1333 <.I <J E Regio ns wi thout geometrica l oplics solut ion 6" 0 0 90 0 90 180 Off-axis angle 1/J, ldcgl Off-axis anglc 1/J,ldcgl Fig. 8.11a,b. The change of inverse phase conversion factor with scattering angle as a function of relative refractive index (Ab = 514.5 nm, 8 = 7 deg, 1fI, = 2 deg). a First-order refraction, b Reflection - §. 0 10 dcg OJ) " ~ a~ ...0 100 dcg 5 Ü .ß c: 0 .~ '"c 0 ... > 10 'J " <J E Ci'" 15 0.5 1.0 1.5 ~O ~ Relativc refraclivc index ~o 111 I-I Fig. 8.12. The change of inverse phase conversion factor with relative refractive index at different off-axis angles (Ab = 514.5 nm, 8= 7 deg, 1fI, = 2 deg, first-order refraction) 8.2 Single-Point Phase Doppler Systems 423 ifJ r > 60 deg. The diameter conversion factor becomes more independent of the relative refractive index in this range, as can be seen in Fig. 8.12. The best choice of off-axis angle depends on the expected refractive index range and system parameters such as intersection angle and elevation angle. Larger off-axis angles are generally better for the refractive index independence but the linearity of the phase difference/diameter relation reduces because of the int1uence of other scattering orders. If possible, the ret1ection mode should be used because the ret1ected light is independent of refractive index. As already discussed, the dominance of one scattering order and the resulting linearity between phase difference and particle diameter is the basis of the phase Doppler technique. In Fig. 8.13 this linearity is shown for an example standard system as a function of the receiver position. The mean deviation from the linearity is calculated from the phase difference of 800 different particle diameters between 0.25/lm and 200/lm for water in air by LMT and compared with the linear relation for pure reflection, first-order and second-order refraction. For parallel polarization a large region ofhigh linearity dominance from firstorder refraction can be used, as expected from Fig. 8.4. For perpendicular polarization the linearity in the forward-scatter direction is much less (see also Fig. 8.3). The linearity increases for smaller off-axis angles ifJr '" 0 deg and larger elevation angles, resulting in asymmetrie planar configuration (see section 8.2.2) with parallel polarization. Furthermore, there exist two very narrow regions with high linearity. These can be identified with Fig. 8.3 as regions of dominant re- Mean phase difference deviation [deg]: > 1 0 ... 9 ... 8 ... 7 ... 6 ... 5 ... 4 ... 3> DDDD Fig. 8.13. Mean deviation of the phase difference/diameter relation from an ideal linear over the receiver angles (/fI, = /fit = -/fI2 = 1, 2, ... ,45 deg, relation, plotted (P, = 1, 2, ... ,180 deg) of a standard phase Doppler system and for the two different polarization orientations. (d p = 0.25, 0.5, ... ,200 Jlm, m = 1.333, e = 7.4 deg, Ab = 514.5 nm, circular receiver, receiver collection angle 1 deg) 424 8 Phase Doppler Systems flection and second-order refraction. Phase Doppler measurements with reflection (ifJ r "" 95 deg) can only be performed with small elevation angles, which leads to a low diameter resolution, as can be seen in Fig. 8.11 b. The second-order refraction can be used with larger elevation angles but the aperture shape must be carefully designed because of the sm all usable off-axis region. Nevertheless, a configuration at ifJr "" 140 deg will be very sensitive to parameter and partide shape variations and will be difficult to align. In summary, the measurement range and the sensitivity of a phase Doppler system will depend on and can be adjusted through the following system parameters Intersection angle, e Focallength of receiving optics, Ir off-axis angle, ifJr (also through the dominating scattering order) Elevation angle, lfI r (e.g. choice of aperture mask and focallength of receiving optics) • Relative refractive index, m • Angular region covered by the aperture of the receivers, ,dlfl rand ,difJ r • • • • The selection of optical parameters is usually simplified using commercial software, which displays immediately the achievable measurement range. However, several factors are not considered directly. For instance, changes of focallength may also lead to strong changes in signal amplitude, as with laser Doppler systems. Furthermore, the calculated measurement size range is only a theoretical limit. Two additional factors may come into play before this limit is reached. The first of these factors concerns the sphericity of the particle. Large partides, in particular droplets or bubbles, will generally be exposed to aerodynamic forces and may deform. The following discussion gives a physical interpretation of the sensitivity to non-spherical smooth partides. Further information is presented in section 8.2.3 for oscillating non-spherical droplets and in section 8.3.3 for non-spherical, rough and inhomogeneous partides. Each detector of the standard phase Doppler system collects scattered light from two glare points on the partide surface, one for each laser beam. If, however, all detectors lie at the same off-axis angle, all glare points for all detectors willlie on a meridian line over the partide surface, as shown in Fig. 8.6. If the partide is non-spherical, then the two size measurements performed with the three detectors will only differ insomuch that this meridian line, associated with the meridian line corresponding to the incident points, changes its curvature between the glare points or incident points. If this is the case, the non-sphericity can be detected when the two sizes d p12 = Fp ,12,dtP12 and d p13 =Fp ,13,dtP13 derived from the phase differences find no agreement for any 21t multiple, i.e. when the measurement point lies outside of the tolerance band illustrated in Fig. 8.8. Experience has shown, however, that partides can displaya very large degree of non-sphericity before the local surface curvature is sufficiently distorted to be detected by a three-detector, standard phase Doppler system. This is because the incident and glare points are located dose together on the same median line. The second factor, which limits the measurable size is the displacement of the 8.2 Single-Point Phase Doppler Systems 425 detection volumes in space with increasing size. The three-detector, standard phase Doppler system can be made less sensitive to this so-called Gaussian beam (or trajectory) effect or to the slit effect by choosing appropriate masks. A carefuHy chosen layout of the system can also partly avoid these problems. These effects will be discussed in section 8.3 because they apply to all phase Doppler configurations. The phase Doppler instrument determines the particle size from the measured phase differences. However, the particle velocity is also available from the signal frequency, as with a laser Doppler system. For the system illustrated in Fig. 8.6, the U x velo city component will be measured. If two velocity components are required, two additional beams can be added to the transmitting optics, as in a laser Doppler system. An additional detector must be added on the receiving side, which by means of a color separator can detect light solely from the second measurement volume. This second detector can, however, still use the same optics, receiving probe and fiber coupling as the detectors for the phase Doppler system. 8.2.2 Planar Phase Doppler System The optical configuration of the planar phase Doppler has been illustrated already in Fig. 8.2. In an interpretation similar to that given above, the glare points seen by the two detectors lie on an equatorialline of the particle in the plane of the laser beams. The phase and diameter conversion factors can be calculated from Eq. (8.4) by using an off-axis angle of zero degrees (r/J r = 0) and different elevation angles, If/, ;f. 1f/2. For reflection these factors become ACP;~ =ß;~dp = .{in dpUI-COS(If/, +~)-~l-COS(If/, -~) Ab -~l- cos( 1f/2 +~) +~l- cos( 1f/2 -~)) = 2 2n Ab (8.23) d psin ~ (sign( If/, )cos~ - sign( 1f/2)COS 1f/2) 2 2 2 (If/, > ~,1f/2 >~) and for first-order refraction LlCP~~) = ß~~)dp = 2n dp(~l +m 2 -.{i m~l-cos( If/, Ab -~) -~1+m2-.{im~1-cos(lf/, +~) -~1+m2 -.{im~1-cos(1f/2 -~) +~1+m2 -.{im~1-cos(1f/2 +~)) (8.24) 426 8 Phase Doppler Systems In Fig. 8.14 the diameter conversion factor is plotted as function of the two offaxis receiver angles in the region where one scattering order dominates. Unlike a z z b - 180 r-r-,..,,.,--,--.--.---r--r:......r-:>a t:>Il """ '"'" 135 öb .,c: c: .2 ~ ~ 90 45 c QOI80~~~~~~~~~~~~~~~~~~§§~~~~e=~~~ " ~ ;;;' N " Ob 135 c: '" c: :Rcgic)noT---- --- .~ ~ 90 :dominan l : refra lion '" @ 2 :, , 45 -90 o 90 180 Elc\ration angle I If, [deg l Fig. 8.14a-c. Dependence of diameter conversion factors in rad/11m of a planar system on the elevation angles Ij!, and 1j!2 ofthe receivers (8= 10 deg, Ab = 514.5 nm). aSymmetrie and asymmetrie receiver set-up, b Diameter conversion factor for reflection, c Diameter conversion factor for first-order refraction (m = 1.333). The region of dominant refraction are indicated with a dashed line 8.2 Single-Point Phase Doppler Systems 427 the standard configuration, all four scattering angles tJ11' tJ 12 , tJ 21 and 73 22 are different and therefore the usable angular regions are smaller. For symmetric and relatively small receiver angles, the diameter conversion factor becomes large. Therefore, the planar system is more sensitive to particle changes than the standard system and can be used for measurements of small particles. The 360 deg limit on phase difference for unambiguous diameter measurements leads again to a maximum measurable particle size and can be approximated for reflection and for first-order refraction for symmetrie receivers (Ij!= Ij!l =-1j!2) by ""A~_l_ d(1) p.max d(2) "" p.max A (8.25) L1b cos v;{ It ~l+m2 -2mlcosv;{1 L1b (8.26) m sin v;{ and for side-by-side asymmetric receivers (1j!1.2 = Ij! =+= LlIj!) (8.27) (2) d pmax ""4A . IJr L1bLID mcosv;{ (l+m 2 -2mlcosv;{1) ( l+m 2 ) ]I, 2 -2mlcosv;{1 -(msinv;{) 2 (8.28) where It and Ir are the focallengths of the transmitting and receiving optics, L1b the beam spacing and LID the distance between the apertures of the adjacent planar receivers. In Fig. 8.15 a planar optical arrangement is compared to a standard optical arrangement by examining the diameter conversion factor as a function of relative refractive index and scattering angle for the same elevation angle difference between the detectors. Note that the planar arrangement becomes extremely insensitive to size for large scattering angles. The sensitivity decreases also for a low relative refractive index. The dashed lines indicate the regions where the receiver angle is larger than the critical angle of first-order refraction, according to geometrical optics. Measurements with first-order refraction are not recommended in these regions. However, scattering for angles larger than the critical angle can still be dominated by refraction, because of the contribution due to surface waves surface waves, as seen in Fig. 4.32. Furthermore, surface waves become more important for small particles. When measuring small particles with a planar system this results in higher scatter of the phase difference measurements and even phase differences with opposite sign (negative particle diameters), Fig. 8.16a. This scatter, with opposite sign of the phase differences for small particles, also occurs if the integrating effect of the aperture is considered, Fig. 8.16c. U sing scattering angles far removed from the critical angle can reduce this effect. This is also one reason why 428 8 Phase Doppler Systems S :::1. bfJ <)) 0 ::2. 3~ <:Q. ...0 -2 Ü eS c: 0 .V; ... <)) -- Standard 1, -4 > c: 0 u ... <)) ~ -6 Planar W; --0-- ----*""- ----.0.-- ----'0'- ----<>-- (5'" 20deg 40 deg 60deg 80 deg Regions without geometrical optics solution -8 0.5 1.0 1.5 2.0 3.0 2.5 Relative refractive index m [-] Fig. 8.15. Comparison of the diameter conversion factor of first-order refraction for a standard and a plan ar receiver arrangement. The separation angle of the two receivers is 4deg for both systems (B=7deg, Ab =514.5nm, Standard: lf/,=±2deg, Planar: lf/,=TiJ,±2deg) the planar system is seldom used on its own. More typical is the combination of the planar system with a standard system, known as the dual-mode system. Therefore, the dual-mode system, described in the next section, generally uses smaller off-axis angles than the three-detector, standard system. By using different scattering angles for the two receivers of a planar phase Doppler system angular regions can exist in which different scattering orders dominate on each of the two different receivers. The conversion factors are then a combination of e.g. refraction and reflection. L1<P~;2) = /1;;2) dp = .fi 11: dp (~l- cos( V1 - %) - ~l- cos( V1 + %) Ab -.fi~l +m 2-.fi m~l- COS(V2 +%) +.fi~1+m2 -.fim~1-cos(V2 (8.29) -%)) As with the standard arrangement, the planar arrangement measures the velocity component parallel to the plane of the two beams. Therefore, a planar phase Doppler system is only practical as a two-velocity component device in combination with a standard phase Doppler system. An exclusively planar arrangement, for instance with three detectors at three elevation angles has not been frequently implemented (Damaschke et al.1998), because the dual-mode phase Doppler concept makes such configurations superfluous. 8.2 Single-Point Phase Doppler Systems 429 "'-s'" "1 OJ ~ -180 :.a ~ OJ ..c:'"'" iO-< -360 ,-,-'~',.l........l-...L...-'---'----L----'Uo...-'---..l-...L...-'---'-' o 100 200 Particle diameter dp [flml 0 100 200 Particle diameter dp [flml Fig. 8.16a-d. Phase/diameter and phase difference/diameter relation calculated with gcometrical optics and Lorenz-Mie theory for a planar configura tion (Ab = 488 nm, If/t = 31 deg, 1f/2 = 39 deg, 9, = 0 deg, m = 1.333, 19= 4.352 deg). a Phase difference for a point-like detector, b Phase of detector 1, c Phase difference for a finite size detector, d Phase of detector 2 Although the planar optical arrangement on its own is not wen adapted far partide size and velo city measurements, it is wen suited for measuring the diameter of fibers or cylindrical partides aligned with the yaxis. A fiber aligned with the yaxis scatters light only in the plane of the laser beams. In this case, geometrical optics applied to spheres is also valid far cylinders and can be used to determine the phase conversion factor. If the fiber is tilted, the scattered light forms a cone and the cross-section of the fiber in the plane of the laser becomes elliptical. This application has been studied in detail by Mignon et al. (1996). They demonstrate that the measurement is insensitive to the tilt angle of the cylinder away from the y axis up to angles of 5 deg to 10 deg, depending on the exact configuration. Gouesbet and Grehan (1994) have presented a more rigorous treatment of the scattering characteristics of cylindrical partides in Gaussian beams. If a second pair of laser beams for a second velo city component is added orthogonal to the planar system, then also the axial velocity of the cylindrical partides or fibers can be measured, as discussed by Schöne et al. (1995). For this application, the fiber material must exhibit surface inhomogeneities to insure that ligh t is detected from both beams at the receiver position. 430 8 Phase Doppler Systems 8.2.3 Dual-Mode Phase Doppler The dual-mode phase Doppler is pictured in Fig. 8.17 and is a combination of a standard and a planar phase Doppler optical arrangement, such that all receiving apertures can be housed in a single receiver unit and two velocity components can be measured. A typical phase difference/diameter curve for such a system is presented in Fig. 8.18, indicating that the planar system exhibits a much lower sensitivity to size than the standard system. However, similar to the three-detector arrangeMea ured curvature (standard system) La er g~ r-~~~~~_b:e:a~ms Measured curvalure (planar sy lern) Fig. 8.17. Optical arrangement of a dual-mode phase Doppler system _ Oll e.> 360 ~ncr-r""T""1"",:""""",-.-T,.,.,rr,...,..-.--r.,.-,rr"""'T""T-a--,rr"""''''''''''''''''''''T""I1..,,---r-,...,..,..-,:ii''!Il :::!. "$ ~ 270 c: ~ , e.> s: "0 '" ;:; ..c Q.. 180 90 200 Parlide diamlcr dp IlJml Fig. 8.18. Phase difference/diameter curves for a dual-mode phase Doppler system calculated using Lorenz-Mie theory (symbols) and first-order refraction (lines) (e=7deg, A" = 514.5 nm, Apt = 488 nm, m = 1.333, Ir = 1000 mm, Mask B) 8.2 Single-Point Phase Doppler Systems 431 ment discussed in section 8.2.1, the combination of the planar and the standard arrangements can be used to remove any 2n: ambiguity from the size measurement. So me caution is necessary in the implementation, since the planar system exhibits oscillations of the phase difference/diameter curve for very small partides. Thus, the lower size range, perhaps d p < 10 11m, must rely solelyon the phase difference measurement from the two detectors of the standard phase Doppler system. For the same reason, measured partide diameters with phase differences near 360 deg with the planar system must be rejected or converted, as shown in Fig. 8.19, because they may have arisen from small partides resulting in negative phase differences. Furthermore, the measurement point scatter of the planar system can increase for a slightly misaligned system or poor signal conditions. In such a case the measurements of small partides can reach the next phase difference period of the standard system near LllPV{2 "" 100 deg, 200 deg in Fig. 8.19. This results in improperly validated partide diameter measurements in equally spaced diameter dasses. Therefore the maximum planar phase in Fig. 8.19 has to be carefully chosen for measuring small partides with a dual-mode system. As shown in Fig. 8.17, the dual-mode arrangement effectively sees glare points simultaneously on a meridian line and an equatorialline of the particle. Deformations of the partide, for instance into an oblate or prolate shape, will result in large differences of curvature along the equatorialline and lead to significant differences in measured size between the standard and planar systems (Doicu et al. 1997, Damaschke et al. 1998). Thus, a validation diagram, as illustrated in Fig. 8.20, will be quite sensitive to partide sphericity and can be imAccepta nce band 1I1aximum planar phase 0iJ 360 <J :::. B""S '<l E 270 <J -.; ~ ""2 '"t:: "0 -.;'" 180 ~ t:: .,~ .= "0 ~ ..c " 90 c- O 90 180 270 360 Phase d ifference pla nar sy lem LI<P~~~ Idegl Fig. 8.19. Validation in the phase difference plane of a dual-mode phase Doppler system (e=7deg, A" =514.5nm, Apl =488nm, m=1.333, J, =1000mm, MaskB) 432 8 Phase Doppler Systems phcri..ily linc .~60 dcg TolcrdllCC b~nd RcjCClcd (c.g. non -s pherical mcasurcmcnls) 360 dcg Fig. 8.20. Validation scheme for detecting non-sphericity using a dual-mode phase Doppler system plemented in the processing software. This is one of the main advantages of the dual-mode arrangement compared with a three-detector, standard system. AIthough this generally results in a lower validation percentage than a threedetector, standard system, the non-sphericity detection can be considered more reliable. The main flow direction of a dual-mode system is the y direction, in contrast to the standard and planar configurations. The reason is that the dual-mode system was designed to suppress the Gaussian beam effect (see section 8.3.1). Large particles moving through their detection volume on the side opposite to the receiver Iocation can generate signals from reflection only, although the system is designed for dominant first-order refraction (see chapter 5, Fig. 5.22 and Fig. 8.26). A two-detector standard system or a three-detector standard system with asymmetrie mask cannot recognize this Gaussian beam effect and will measure wrong particle diameters. Sometimes this effect is also called the trajectory effect because it is trajectory dependent. Iflarge particles move mainly in the y direction, then the signal maximum is always dominated by first-order refraction and, therefore, the measurements are more reIiable for the dual-mode configuration. When the Gaussian beam effect is not expected e.g. for smaller particles or when asymmetrie masks for recognition of the Gaussian beam are implemented, the dual-mode system can also be used with a main flow in the x direction. A complete analysis of the Gaussian beam effect is given in section 8.3.1. The dual-mode phase Doppler arrangement can be implemented in a manner similar to the standard phase Doppler system. Fig. 8.21 shows the cross-section of a fiber-optic receiving probe and Fig. 8.22 illustrates the corresponding aperture plates. The detectors corresponding to the planar system are labeled Vj and V2 since they measure the U 2 or v velo city component. The achievable size ranges are determined by the size limit of the planar system and are not unlike 8.2 Single-Point Phase Doppler Systems 433 those for a three-detector, standard arrangement. Typical receiver mask dimensions are given in Table 8.4, e.g. for the Dantec Dynamics system. The choice of scattering angle for the dual-mode receiving probe is more restrictive than for a three-detector, standard system. This is evident from the response curves shown in Fig. 8.15. To achieve sufficient size sensitivity of the planar system, it is necessary to restrict the off-axis angle to smaller values , e.g. 20 deg < ~r < 40 deg (for water in air and first-order refraction mode). The ratio Measuremcnl volume \ I'ronl lens Aperture plate 4 Multimode fibers Fig. 8.21. Fiber-optic receiving probe for a dual-mode phase Doppler system (adapted from Dantee Dynamics (1999)) Notch corresponding to index pin Fig. 8.22. Interchangeable aperture plates for the dual-mode phase Doppler probe shown in Fig. 8.21 Table 8.4. Parameters of the aperture masks of the dual-mode phase Doppler system (Dantee Dynamics) ParameIer Imm] MaskA MaskB Mask e R 20.05 20.05 20.05 x" x pf 7.0 5.5 2 10 10 10 Ypf 16 10 5 D" 13.5 D pf 8.5 12.75 6.5 3 7 . ..: ~ 1 oslIaoo D .. 434 :::!: 8 Phase Doppler Systems 0.6 ~ ~ ~ o U ~ 0.4 First-order refraction: - - - - m = 0.75 - - - - m = 0.833 ------- m = 1.132 - - 0 - - m = 1.192 --Reflection m = 1.39 -1E---- m = 1.45 -----<r-- ~ .9 ~ ~ o u '" 1lp. 0.2 ""'o <I) .9 ~ 45 90 135 180 Off-axis angle of detector unit 1/1, [degJ Fig. 8.23. Ra tio of the diameter conversion factors of the planar and the standard systems of a dual-mode phase Doppler configuration for reflection and first-order refraction and for different refractive indexes (mask B, 8=7deg, Ab=514.5nm, \fIu12=±2.35deg, \fIV12 = (P, ± 1.2 deg) of the diameter conversion factors between the planar and the standard system as a function of the detector unit angle and for mask B is plotted in Fig. 8.23 for different refractive indexes. In commercial systems the ratio is generally larger thanO.2. Not only does the sensitivity of the planar system drop at higher off-axis angles, but the non-linearities of the phase difference/diameter curves increase for small particle sizes, as discussed for the planar system in the previous section. Therefore, the phase difference of the planar system is generally only used to detect the number of 2n jumps in the phase difference of the corresponding standard measurement. The actual diameter estimation is made according to the standard system because of the higher resolution and accuracy. Furthermore, the mask design influences the sensitivity of the dual-mode system to detection volume effects like the Gaussian beam effect and the slit effect, as discussed in section 8.3. The specific aperture configuration of the dual-mode system demands a much more precise adjustment of the receiving optics, often considered by users as a disadvantage of the system. However the resulting scatter of the measurement values in the validation map (e.g. Fig. 8.20) is easily recognized in preliminary "adjustment" measurements and can be eliminated. This greatly improves the detection reliability of non -sphericity. Adjustment of the dual-mode system can be made less sensitive to misalignment by affixing slit apertures only to the detectors of the standard system and not to those of the plan ar system. Since signal detection is usually made using the standard system detectors, there is no detrimental effects with this measure. 8.2 Single-Point Phase Doppler Systems 435 However, the increased detection volume of the planar system (without slit apertures) may lead to coineident signals from different particles at high particle concentration levels. If the sphericity validation discussed above is not implemented, Damaschke et al. (2001) have shown that the dual-mode system can be used to evaluate the volume of non-spherical droplets, oscillating in a prolate/oblate manner aligned with the x axis. When oscillating, the curvatures of the meridian line and equatorialline change with opposite signs, but very systematically. As the photographs of oscillating droplets in Fig. 8.24 illustrate, their shape may change significantly even during passage through the measurement volurne. The signal shown to the right of the photograph in Fig. 8.24 corresponds to the time-resolved phase difference within single burst signals, for a standard and a planar detector pair. Such time-resolved phase difference information can be achieved using the Hilbert transform, as outlined in section 6.4.3. Note that the measured velocities on the two detectors are not uniform throughout the signal but also not the same. Due to the droplet oscillation, the veloeities of each of the incident and glare point pairs are modified from the bulk translation veloeity of the particle. This leads to variations in the measured veloeity. If the droplet oscillation during the burst signal is significant, several experimental points can be used to construct a diagram analog to Fig. 8.25 and an extrapolation through the points to the spherieity line can be used to estimate the volume equivalent diameter of the oscillating droplet. With such a procedure the validation rate of a dual-mode system can be significantly increased (Damaschke et al. 2001). Note that the standard system change is much more sensitive OsLillating dro plcts u .j c fr inges in illuminal cd ~ :~ 180 Pha e diffcrcn.:e of standard phase Doppler system <P,,, Phase differen.:e of planar phase Doppler ystem .d<PVll 'Ö ~ ..r:. '" 0.. -- ................... ,...._.- ....__ . ..... ..... .... .... - . . .,_ ............ Frequcn.:y of standard rc,civer U, Frcquenq of standard rc,civer U, 43. J L...L-'-L...J.....L...L...J......L..JL...L......L..JL...L....L-JL...L....L...I.....L.L..I.....L..L...J.....J....J...J.....J....J...J o 5 10 Time 11).151 15 Fig. 8.24. Measured evolution of the phase difference and signal frequencies for an osciIlating droplet with a volume equivalent diameter of about 113 11m 436 8 Phase Doppler Systems b a 360 $' :11 E ~;... "E" O~O 0 ]"' 180 "' ;;; 0 360 (j ofi\o 0 c ~ ~ '5 110 .," '" ..c ~ o ~~~~~~"-" 180 Phase difference planar system L1lPpl [deg] 0 360 L~~~~~~ 0 360 Phase difference planar system L1lP;~) [deg] Fig. 8.25a,b. Measured phase differences for spheroidal drop lets of various aspect ratios measured with a dual-mode phase Doppler system. a Distribution of measured diameter processed with commercial software, b Phase difference evolution processed with Hilbert trans form for different oscillation states of a droplet for prolate forms than for oblate forms, again underlying the higher sphericity sensitivity of the dual-mode system. In section 8.3.3 further special phase Doppler configurations are presented for detection and measurement of non-spherical, rough and inhomogeneous particles. 8.2.4 Dual-Burst Technique The dual-burst phase Doppler technique was first introduced by Onofri et al. (1994), Onofri (1995) and Onofri et al. (1996). Although this technique has not been developed commercially, it does exhibit some interesting features, which allow additional particle properties other than the size to be estimated, in particular refractive index or absorption coefficient. Some initial remarks about the dual-burst technique have been given in seetion 5.4. The principles are best understood by realizing that each scattering order has its own measurement and detection volumes. The measurement volume is defined by the particle position (center) at which the signal amplitude at the receiver exhibits an e- 2 decay from the point of maximum intensity. Thus, the measurement volumes are the same size and shape as the illuminated volume, despite the fact that they may be displaced and exhibit different absolute intensities. The displacement of each volume depends on particle size and on detector position. Volumes of different scattering order can overlap and the detection volume refers to that volume in which, for a given particle size, the sum of all scattering order intensities results in a received signal amplitude exceeding the detection threshold. The detection volume can, therefore, be larger or smaller 8.2 Single-Point Phase Doppler Systems 437 than the illuminated volume. The detection threshold may be based only on amplitude or, as with several newer processors, it may be sensitive to a given signalto-noise ratio. For larger particles, the distance between the incident points of different scattering orders increases and this corresponds to a larger separation of the respective measurement volumes. Two examples are shown in Fig. 8.26 for two detector off-axis angles of r/J, = 30 deg and r/Jr = 90 deg and for the case that the dominant order is first-order refraction and the less dominant order is reflection with half the intensity of the dominant one. Because no elevation angle is considered, the measurement volumes are centered in the y-z plane and the displacement is only in the y and z directions. As seen from these examples, the signal will depend on the trajectory of the particle and the receiver location. For so me trajectories, two burst signals will be obtained, one from the dominant scattering order and one from the less dominant scattering order. - 0.4 e; e; .2 Cl "-0.0 u ~ ;: '" Q.. 0.4 _ OA E E D.S 0.0 0.5 1.0 Parlic!eposilion z, [mm) Fig. 8.26a,b. Signal intensity distribution summed over reflection (less dominant order) and fIrst-order refraction (dominant order) as a function of particle position for two different receiver positions (e=13.7deg, rw=lOOJ.l.m, lf/,=Odeg, m=1.8, d p =250J.l.m). Two example particle trajectories are indicated with white arrows. a Receiver off-axis angle fjJ, = 30 deg. Trajectory (A) results in a dual-burst signal, trajectory (B) contains only the dominant order, b Receiver off-axis angle fjJ, = 90 deg. Both trajectories result in dualburst signals with the same intensity ratio 438 8 Phase Doppler Systems The relative intensities of the two measurement volumes will depend on the scattering angle and on the imaginary part of the relative refractive index, since this will affect the first-order refractive scattering. The separation of the volumes is dependent on receiver position and particle size (time-shift technique). The size of each volume will depend only on the width of the laser beam. With proper positioning of the receiver and alignment of the main flow direction with the measurement volumes, two bursts may always be expected and this is the working principle of the dual-burst technique. For determination of the absorption coefficient (imaginary part of relative refractive index) the amplitude ratio between reflection and first-order refraction can be used, under the assumption that the ratio is independent of particle trajectory. Therefore, if the main flow direction is assumed to be in the y direction, as shown in Fig. 8.26b, the receivers have to be located at rjJ r = 90 deg. Figure 8.27 illustrates that in this case, all particle trajectories in the y direction produce signals with the same intensity ratio between reflection and first-order refraction because the detection volume displacement in the z direction is the same for reflection and first-order refraction. However, in Fig. 8.26a the two trajectories parallel to the y axis result in different ratios between reflection and first-order refraction and adetermination ofthe absorption coefficient is therefore dependent on particle trajectory. To overcome this problem, laser light sheets in the plane of the laser beam instead of laser beams can used. For such an optical configuration the trajectory dependence vanishes. The two parts of the burst signal will become more distinct if the laser beam is made narrower and the particles are larger, thus the ratio of particle size to __ ~ 5r--r--~-.---r--~~---.--.-~---.--.-~---.--.-~---.--.--, §. §. First order refraction m = 1.1 m= 1.3 ----- m = 1.5 c 0 OB <lJ f;::::::::~~~~~~ '"' ;;3 " <Ei'"' E <lJ 0 ___ -0- - - Reflection m = 1.9 ~~-----=l 'ü ..:: 'H <lJ 0 U 90 Receiver off-axis angle !/J, [degJ 180 Fig. 8.27. Displacement of the center of the measurement volume in the z direction per micrometer particle radius as a function of the receiver off-axis angle for reflection and first-order refraction (e= 13.7 deg, 1jI, = 0 deg) 8.2 Single-Point Phase Doppler Systems 439 beam waist radius should be preferably large to exploit the dual-burst technique. A more rigorous treatment of how to compute the measurement volume positions will be given below in section 8.3. A typical optical arrangement of a dual-burst system is shown in Fig. 8.28, which is a three-detector, standard phase Doppler system, but with perpendicular polarization and with a different main flow direction. Like in the dual-mode configurations, both receivers will observe the same scattering order at the same time and a phase difference calculation for particle diameter measurement is possible. Generally, the dual-burst technique will therefore also require a second velo city component measurement, since the main flow direction is aligned with the y and not the x axis. A typical signal pair obtained from a dual-burst arrangement is shown in Fig. 8.29, in which also the phase difference is shown. The phase difference can be computed either using the Hilbert transform, as discussed in section 6.4.3 or using model-based signal processing, as outlined in section 6.5.2.3. From this example it is evident that a burst signal separation is preferable. If this is not the case, i.e. the bursts overlap strongly, then the phase difference is no longer distinct for each burst and a size determination becomes non-unique. This effect is known as the Gaussian beam effect and will be also discussed in section 8.3. Given that the two burst signals are distinct, size and relative refractive information are contained in the following measurement quantities • Phase difference of signals from the dominant scattering order • Phase difference of signals from the less dominant scattering order • Time shifts of signals of each scattering order from each detector Since there is redundancy in the size measurement, in principle it is also possible to determine the relative refractive index of the particle. The achievable accuracy, although not high, is certainly sufficient to distinguish between different materials, phases or components. Laser beams direCliony Fig. 8.28. Optical arrangement for the dual-burst technique 440 I:l" "... 8 Phase Doppler Systems 0 ~ = L1<P~) -950 deg 'M/'oI~.......-~W"'lHI"IHII- - - - - - '6 -180 ~ = -230 deg- 2 x 360 deg - - - - - - - '" ~-36~~-i-L~~~~-L-L~~~~~-L~~4LO-L~~L-~-L-L-L6~0-L-U Time t [Ils] Fig. 8.29. Typical dual-burst signals and the computed phase difference (Ab = 514.5 nm, d b =50J..lm, j'h=IMHz, e=13.7deg, m=1.32, d p =150J..lm, v p =6ms- 1 , q),=65deg, 1jI, = ±2 deg) 8.2.4.7 Refractive Index Measurements From the detectors U1 ' U2 and U3 the phase differences LllP\Y, LllPg), LllPW, and LllPW can be obtained, which are linearly related to the particle size using the phase conversion factors F8), Fi-}!, FS) and FN) (the diameter conversion factors are ß(]J) = (Fi~)tl), where the factors can be computed using Eqs. (8.11) and (8.12) for reflection and refraction respectively. From the reflected bursts, the particle diameter can be determined using Eqs. (8.16) to (8.18). Knowing the particle diameter, the phase differences of the refracted parts are only functions of the relative refractive index. (2)( dP ß 12 m ) -- A n,(2) LJ'l-'12 - 2nn 27t d ß(2)(m) = LllP(2) p 13 13 (8.30) (8.31) Because the ratio of the diameter conversion factors ßW I ß\j) is, to a first approximation, not a function of refractive index, Eqs. (8.3) and (8.4) (Onofri et al. 1996), the number of 2n jumps can be calculated similar to Eq. (8.18). The inverse of the approximation in Eq. (8.12) yields an expression for the relative refractive index, given by (!fIr = !fIl = -!fI2) m'" -J2 ~cos% cos!flr cos9r + 1 + ~2w+ cos% cos If/r cos9r-1 2 1-w (8.32) 8.2 Single-Point Phase Doppler Systems 441 where W = 2( 2n d p)2 Ab o/z sin 2 If/r (l+COSo/zCOSlf/r COS9r) 2 Sin 2 (L1cJJ~~) -2nn 2n ) (8.33 ) Knowing the procedure to extract the relative refractive index and diameter from the measured phase differences, a compromise must be found for the optical parameters according to the following conditions: • The intensity ratio between reflected and refracted signals should be dose to unity to aid signal processing. • The centers of the measurement volumes should be aligned with the main flow direction. • The phase difference sensitivity to relative refractive index must be maximized. • The size and refractive index dynamic range should be matched to the measurement problem. The difference between reflected and refracted light intensity can be diminished using appropriate off-axis angles. For instance at 65 deg for perpendicular polarized light or 90 deg for parallel polarized light (water in air m =1.333), the intensity of the reflection and first-order refraction in the scattering function is equal, as can be seen in Fig. 4.17. The receiver separation should be in the main flow direction. In this case the signals from all partides will contain reflected and refracted light with the same amplitude ratio. The phase sensitivitywith respect to the relative refractive index 2n 2-d p Ab sin If/ r sin o/z (2 - mv) y, v = .J2~1 +cos If/r cos9r costJi (8.34) v(l+m 2 -mv)' is an increasing function of the product dp If/ r o/z and a decreasing function of the off-axis angle 9r and the relative refractive index. As an example, for 9r = 60 deg, tJi = 5 deg, If/r = 3.69 deg, m =1.33, Ab = 514.5 nm and a partide diameter of 100 11m, L1cJJg) varies by 2.1 deg for a change in relative refractive index from m = 1.33 to m = 1.34. The refractive index and the partide diameter can also be estimated by using the time shift between signals of like scattering order on each of the detectors. For this, the detectors must again be positioned so that both reflective and refractive scattering orders are present in the signal with similar amplitudes and sufficiently separated in time. For the dual-burst arrangement in Fig. 8.28 the partide diameter can be estimated by the time shift between the reflected and refracted signal parts for each detector. Therefore, the number of 2n jumps can determined and the third receiver U 3 is no longer necessary. In contrast to the dual-burst system in Fig. 8.28 the main flow direction could be aligned in the x direction as in a standard phase Doppler system, Fig. 8.6. If the elevation angles are chosen large enough, then the reflection and refraction parts of the signal are separated asymmetrically relative to the center position of 442 8 Phase Doppler Systems First-order refraction First-orde refraction Reflection Reflection Receiver U, o -20 Receiver U, -20 20 Time t [fis] o 20 Time t [fis] Fig. 8.30. Refraction and reflection signal contributions for detectors U, and Uz • Burst displacement Llt12 =16.3 fls (Oll in water m = 1.06, d p = 30 flill, Ab = 512 nm, t9 = 14 deg 1;" = 10 flm, If/, = ±1O deg, ~, = 11 deg) the particle. Typical signals for such a system are shown in Fig. 8.30 using oil in water as an example, where the partide is larger than the beam waist. In this case the signals can be processed using the time-shift technique. For signals in which the reflective and refractive bursts are weH separated, the time shift of the corresponding maxima is suitable for determination of the partide diameter and the relative refractive index. The partide diameter can be determined from the time shift of the reflective signal part Lltgl according to Eqs. (5.258) and (5.259). A second time shift is available for refractive signal portions Lltg l and the refractive index is given by using Eq. (5.258) and an approximations of Eqs. (5.260) (see section 9.2). m= ~ ~cos IfIr cos (jJr COS'% + 1 + ~w+ cos IfIr cos(jJr cos,%-1 V 2-'---'--'----=-'-----------'----:.....:...--'-'---- 2-w (8.35) with • _ d2 w-2 p Sin 2 2 IfIr (vxLlt:~l) (l+COslflrcos~rcos'%) (8.36) The structure of these equations is similar to the Eqs. (8.32) and (8.33), because the time-shift dependencies are, to a first approximation, almost proportional to the phase shift dependencies (see section 9.2). The sensitivity of the refractive index measurement is determined by the sensitivity of the time shift to the refractive index change (8.37) 8.2 Single-Point Phase Doppler Systems 443 The displacement of the measurement volumes, or the time-shift dependency, is relatively small compared with the sensitivity of the phase difference from Eq. (8.34). For tPr =60deg, o/z = 2.5 deg, IfIr =±20deg, m=1.3 and d p =100/lm the displacement varies by less than l/lm for a change in relative refractive index from m = 1.3 to m = 1.4. Therefore, the determination of the relative refractive index by the time-shift technique is practical only for large changes in refractive index, very small beam diameters and special receiver configurations. The time shift is, however, always useful as a method to determine the number of 2n jumps in the phase difference with only two detectors. Both techniques have some limitations in particle size, which have yet to be thoroughly investigated. For smaller particles the two bursts in each signal begin to merge to form an asymmetric signal, which is no longer possible to analyze correctly. Furthermore, the linearity of the phase difference/diameter relation is disturbed by the interaction of the different scattering modes. To overcome this problem a second set of detectors using the same receiving optics can be used together with 45 deg polarized beams laser as illustrated in Fig. 8.31. When the receiver probe is located at an angle where reflection and firstorder refraction have the same intensity for the perpendicular polarization component and first-order refraction dominates for the parallel polarization component, e.g. tPr = 65 ... 75 deg for relative refractive index of m = 1.3 ... 1.5, the dual-burst technique can be used for larger particles with sufficient burst separation, Fig. 8.32. For smaller particles the phase difference from the perpendicularly polarized light is disturbed by the superposition of the different scattering orders. The parallel polarization component generates a pure refracted signal, where the phase difference is linearly related to the particle diameter, Fig. 8.33. For this case the refractive index information cannot be processed but a diameter measurement is still possible. Note that for sm aller particles the diameter measureReOeclion + first order refraction Fir I-order refraction only z polarized light PoJarizalion pre erving fibers Fig. 8.31. Dual-burst configuration for combined diameter measurement of small partides and diameter and refractive index measurements oflarge partides. 444 8 Phase Doppler Systems Para llel polarizalion Pcrpcndicular polarizalion ,;-(F -.i_rs ,- . I. -o,r, d,cr,r_e,fr,a"_·Iy-io.,.n....,--r-i-r"T"'T"'1rT"'T""O (Reßecl ion and firsl-order rcfraci ion) '~~:::=~' ~~l ~E~::::J öö360 ~~~~~~~;=~;=~;=~~ '" Phas e d iffercncc ""$ '" .d (/)~' '<;] '-' '~ <:: '"... ~ =90.8 dcg o .d(/)\:' = -228.6 dcg '.......-oH'l~ll- - - - - - - - - - - - . '6 ~ t'" Phase differencc 36~O~0~~~1~0~O~~~O-L~~~IO~Ou-~~2~OO~~~ILO~ O-L~~0-L~~-IOLO~~-LJ200 Pa rlide posilion y, lflrnl Pa rlide posilion y, lflrnl Fig. 8.32. Single burst signals for parallel polarization and dual-burst signals for perpendicular polariza tion of the same large particle moving through the measurement volume Parallel polariza lion (Firsl·ordcr rcfraCli on o nly) PcrpcndicuJH pola ri za lion (Rcfleclion and firsl-o rdcr rcfraclion) "u<:: ~ .."= 200 Parl ide posilion Y, lflrnl Fig. 8.33. Single burst signals for parallel polarization and burst signals containing firstorder refraction and reflection for perpendicular polariza tion of the same small particle 8.2 Single-Point Phase Doppler Systems 445 ment is slightly influenced by the refractive index. Furthermore, the sensitivity of the refractive index determination, Eq. (8.34) reduces with particle size, which also results in a high er uncertainty for smaller particles. 8.2.4.2 Absorption Coefficient Measurements The signal amplitude of reflected light is virtually unaffected by the absorption coefficient, whereas the signal amplitude of first-order refraction decreases with higher absorption (see section 4.1.3.2). The scattered light intensity from refraction can be approximated by an exponential function of the particle diameter and absorption coefficient ßp (Beer-Lambert law) I~1x = 16~~ax exp(- ßpi) (8.38) where n:~ax is the intensity for ßp = 0 and 1 is the optical path of the refracted rays through the particle, i.e. 1= dp cose, for first-order refraction. The optical path length is a function of the specific optical arrangement and a linear function of particle size. The absorption coefficient is related to the imaginary part of the refractive index through Im{m} = ß~~b (8.39) Eq. (8.38) cannot be used directly since the quantity n:~ax is not known. However the ratio of the maximum reflective signal amplitude to the maximum refractive signal amplitude for ßp = 0, I~~x / Iö~~ax' can be predicted. If (8.40) then ß p =_~[ln(I~~J-Bl 1 I~~ (8.41) which yields the absorption coefficient as a function of themeasured amplitude ratio - refractive to reflective Ii,;,L / I~~x. In Fig. 8.34 the amplitude of the modulated signal is plotted for particles with different absorption coefficients. Not only is the refracted signal influenced by the absorption coefficient, but also the amplitude of the reflected signal, which causes uncertainties in the measurement. This technique clearly assumes that the particle trajectories are weIl aligned with the location of the measurement volumes, see Fig. 8.26. Furthermore, the apertures of the receivers have to chosen carefuIly, such that the modulation depth of the signal is not influenced by the particle diameter (see section 5.1.4). For certain solutions, for example ink in water, the absorption coefficient can be directly related to concentration. For high concentrations, however, the ampli- 446 8 Phase Doppler Systems - - m= 1.32 m = 1.32 + j 10 5 m = 1.32 + j 10-4 m = 1.32 + j 4.6x10-4 m = 1.32 + j 10- 3 ,, " .,' ":'" / / ."," , .. - ......... ..... OL-~~~~~~_~-L_~~~_~-L_L--L~_~~ -200 o __~-L~ Particle position Yp [11 m] 200 Fig. 8.34. Signal amplitude of the modulated part of dual-burst signals for particles with different absorption coefficients. The recalculated absorption coefficients from the ideal signals are: Im{11!.l = 1.95 x 10-5 ,1.65 X 10-4,5.59 X 10-4, L08x 10-3 (parallel polarization, Ab = 514.5 nm, d b = 100 11m, e= 13.7 deg, d p = 150 11m, (Pr = 65 deg, lfI, = ±2 deg) tude of the refractive signals decreases too dramatically and the technique no longer functions. Another limiting factor is the optical resonances in the particle, which change dramatically the intensity ratio between reflection and firstorder refraction, see section 8.4.1. Small deviations from the spherical particle shape or inhomogeneous inclusions disturb these resonances. Generally, the estimation of the absorption coefficient improves for larger particles. Further details of this technique have been developed by Onofri et al. (1998). 8.2.5 Extended Phase Doppler Technique The extended phase Doppler technique was first introduced by Durst and N aqwi (1990), Naqwi et al. (1990) and Pitcher et al. (1990), suggesting its use to measure the refractive index of the particle. The optical arrangement is pictured in Fig. 8.35, in which two phase Doppler receiving units are positioned at two different off-axis angles, l/J A and l/JB" The basic idea of the extended phase Doppler technique is easily understood by expressing the phase difference measured at each of the two receiver units as a function of the geometrical parameters and the relative refractive index and as a linear function of the particle diameter. (8.42) The ratio of the measured phase differences will therefore be a function of relative refractive index alone if the system parameters are known. 8.2 Single-Point Phase Doppler Systems 447 Receiver proue B Transrniller proue x Fig. 8.35. Optical arrangement of an extended phase Doppler system .dt:/J(N) -A--f(m) .dt:/J(N) - (8.43) B As an example, Naqwi et al. (1991) gives the ratio for firs t-order refraction. .dt:/J~) .d t:/J~) sin If/A k B sin If/B k A (I+m -m.jk;) (I+m -m.JC) 2 (8.29) 2 with (8.44) where the detectors are assumed to be symmetric about the y-z plane at angles If/ A = 1f/1A = -If/ 2A and If/ B = 1f/ 1B = -1f/2B for receiver units A and B respectively. Brenn and Durst (1995) solved this equation for the refractive index, yielding (8.45) (8.46) This equations yields real solutions for non-vanishing denominators and positive square root arguments, thus leading to the validation criterion (8.47) Using these equations, the value of the relative refractive index can be determined for every pair of phase shifts detected by the two receiving units. The resolution and accuracy with which the extended phase Doppler system can determine the relative refractive index will depend on the accuracy with which the geometrical parameters of the system are known, on the accuracy of 448 8 Phase Doppler Systems the individual phase difference measurements, LllP A and LllP B , and on the linearity of the phase difference/diameter curves in the diameter range of interest. Volkholz et al. (1998) have considered the first effect on its own and suggest a fitted correlation in the following form erm '" ±[ O.l42(m -1.2)2 + 0.023] (95% confidence) (8.48) For example, at a relative refractive index of m = 1.34, the measurement uncertainty would amount to erm '" ±0.026. The uncertainty in m due to the phase difference measurements depends also on particle diameter and is given byVolkholz et al. (1998) as er~~ erm,M'=+ ddeg (5.4(m-1.2)2+ 0.5) (95%confidence) (8.49) p where er ~<P is the uncertainty of the phase difference measurement itself. Fig. 8.36 illustrates this quantity for example parameter values (er ~<P = 4 deg, m=1.34). Even without considering the non-linearities in the phase difference/diameter relations, Eqs. (8.48) and (8.49) indicate that the expected errors in m with the extended phase Doppler technique will be substantial for typical liquids and furthermore, will become intolerable for small particles. Thus, the technique in this form is more suited to differentiating between particles of large refractive index differences in a multi-phase system. At small particle diameters the additional problem of non-linearities lead to multiple solutions of d p and m. One method of overcoming this problem has , 1.0 .... 0 t: v >< v 0.8 "0 .9 v > 'BcO ~ 0.6 v ~ 0.4 0.2 40 60 80 Particle diameter dp [flm] Fig. 8.36. Uncertainty of relative refractive index due to phase difference errors (a~(/j=4deg, m=1.34, (OA=30deg, (OB=60deg, e=3.38deg, \f/,=3.36deg, a, =5.5 deg) 8.2 Single-Point Phase Doppler Systems 449 been proposed by Volkholz et al. (1998), in which up to 4 detectors at four elevation angles in a planar configuration are used. They used the so-called joint phase match method for determining the correct relative refractive index for a one-component spray. Especially for the optical configuration of the extended phase Doppler the time shift between detector signals at off-axis angles l[J A and l[J B may be quite large and an expression for deducing the refractive index from the time shifts of the signals can be derived. The refractive index is again given by Eq. (8.45) by substituting A(/> with At (8.50) and ga =coslfl a costPa tan~ +sinlfla' gb = COSlflb COStPb tan~+sinlflb (8.51) tan~-sinlfla' hb =coslfl b COStPb tan%-sinlflb (8.52) ha =coslfla costPa Thus, two techniques are available to measure the refractive index of the partide. The time-shift approach is more appropriate for large partides, whereas the phase difference technique is more suited to smaller partides. In any case, the accuracy of the method decreases for smaller partides, since the measurable influence of the partide medium becomes less. One general disadvantage of the extended phase Doppler technique is that two receiving units are required. They must be extremely weIl aligned onto the same detection volume, which can only be achieved using a well-defined test partide, for instance, a stream of monodispersed partides or a glass bead on a rotating wheel. 8.2.6 Reference Phase Doppler Technique All phase Doppler systems considered up to now have been based on receiving scattered light from two illuminating beams and their interference on the detector surface. However, also the reference-beam arrangement, introduced in section 5.3.3, can be used for determining particle size, as shown by Strunck et al. (1994), Strunck and Dopheide (1996), Borys et al. (1999) and Borys et al. (2000b). One of these reference phase Doppler arrangements is pictured in Fig. 8.37. Each detector sees a virtual interference pattern, for instance detector R] sees the pattern lying along beam 2. The length of the interference pattern along the beam is determined by the aperture of the receiving optics in front of each detector. For identical optics, the virtual interference patterns are symmetric. Depending on the z co ordinate of the particle trajectory through the virtual interference pattern, the maxima of the modulated signal component are shifted in time because of the spatial displacement according to Eq. (5.180). The phase difference between the signals will be a function of particle diameter. This refer- 450 8 Phase Doppler Systems x Beam2 Receiver I (J) " el=====---------=~_E_:===_~~' ~ ,"" Fig. 8.37. Beam and receiver arrangement in a reference phase Doppler system (Borys et al. 1999) ence phase Doppler technique is analyzed in section 5.3.3 in terms of geometrical optics, considering each scattering order in turn. The phase difference I diameter relation for reflection is given in Eq. (5.243) as 2n d P Lll/J(I) =2-d 12 A p sin(e/)-n=2n/2 LIx -n (8.53) b The expression for phase difference in a standard phase Doppler arrangement (Eq. (8.2» will give the same result with exception of the additional phase shift ofn. For refraction the phase difference diameter relation can be expressed as (Eq. (5.247» (8.54) The additional phase shift of n again arises due to the nl 2 phase difference in the far field of a focused beam (rw1 »[Rb) and vanishes for a homogeneous reference beam. For a standard phase Doppler system, the phase difference for the conditions (9r = 0, lfI r = ±~) would, according to Eq. (8.3), be given by Lll/J~;L = 2 2n d p [(m -1) -~1 + m 2 - 2m cos~ ] Ab (8.55) Comparing Eqs. (8.55) with (8.54) reveals that the reference phase Doppler exhibits a significantly high er size resolution (Strunck et al. 1994). This difference is illustrated in Fig. 8.38, which shows a comparison of the phase difference/diameter relations for a planar and a reference phase Doppler system. Despite the much larger elevation angle of the planar phase Doppler system, the reference phase Doppler is many times more sensitive. This intro duces a 2 n ambiguity, which must be resolved using additional detectors, as in standard phase Doppler systems. For transparent particles with d p » A, the refractive contribution to the scattered light dominates in the forward direction. However additional disturbances from reflection and diffraction can be expected. This leads to deviations 8.2 Single-Point Phase Doppler Systems bC 451 O"~".-rrTT"~rr"~""rr"~,,no,,rr,,,,,,~rrrrTT" =23~" "1 " i:l -90 ::! ~ :.a ~ -& -180 ·270 21 22 23 24 25 Particle diameter dp [11m] Fig. 8.38. Phase difference in refraction mode for a reference and a planar phase Doppler system as a function of particle diameter (m = 1.328, Ab = 852 nm, e = 18.9 deg, planar system 1fJ, = ±13 deg) from the linear relation between particle diameter and signal phase difference. In Fig. 8.38 computations using FLMT have been compared with the result given by Eq. (8.54) and these confirm the existence of such deviations. These computations have been performed for a point detector and some smoothing of the irregularities can be expected when the scattered light is integrated over a finite detector. 8.2.6.1 Phase Difference for Very Small Particles Phase Doppler measurements in the sub-micron size range exhibit errors arising from the low signal intensity, a lower gradient between phase difference and particle size and oscillations in this relation. Figure 8.39a compares the characteristic curve for a reference phase Doppler system, a planar phase Doppler system and a standard system. The reference phase Doppler exhibits a considerably different behavior in the Rayleigh range d p < 51 Ab in that the phase difference converges to 1t rather than to zero. This is due to the far-field phase effect present with a focused laser beam. Furthermore, for the range Ab 15 ~ d p ~ 3Ab, an almost linear behavior is observed. Since the amplitude of the alternating part of the scattered light power contains the product of the reference-beam field strength with the field strength of the scattered wave, also the decrease of received AC signal amplitude with sm aller particle diameters is significantly lower for the reference phase Doppler system (PAC - d~ instead of d~). This is illustrated in Fig. 8.39b, again comparing the three phase Doppler systems, in this case in terms of scattered intensity. Thus, an extension of the measurement range into the sub-micron range is relatively straightforward with the reference 452 8 Phase Doppler Systems Reference system - - Planar system - - - - - Standard system -180 o 2 3 4 5 Partide diameter dp [!lm] Partide diameter dp [!lm] Fig. 8.39a,b. Diameter and signal power relations for a standard phase Doppler system, a planar phase Doppler system and a reference phase Doppler system (m = 1.328, Ab = 852 nm, 8= 9 deg). a Phase difference/diameter relation (!f/"St = ±15 deg, f/J,.,sr = 30 deg, !f/,.,Pl = ±11.5 deg), b Signal power/diameter relation (!f/,.,St = !f/,'pl = ±20 deg, f/J,.,sr = 30 deg, NA St = NA pl = 0.05, NA Ref = 7.1 X 10-4) phase Doppler, provided a low noise laser source is used, for instance a laser diode or Nd:YAG laser. 8.2.6.2 Example Measuremenfs In Pig. 8.40 particles from a seeding generator (Dantee Dynarnics Type 55118) have been sized simultaneously in coincidence mode by a planar and a reference phase Doppler system. The reference system was triggered by the planar system to insure coincidence. The correlation of the results is high, however the planar system did not allow measurements of smaller sizes. The scatter observed in the measured distribution is partially due to the oscillations in the exact relation between diameter and phase difference, which have not been considered in the data processing. Despite the long wavelength of the laser diode (Ab = 852 nm) and the low power in the measurement volume (30 mW), measurements could be performed also on very small particles, Pig. 8.41. These particles were generated using an aerosol generator based on the Sinclair-La Mer principle as described in section 13.2.1 (Palas, Type MAGE), operated under various conditions (Horton et al. 1991). An improvement could be expected for shorter wavelengths, higher laser power and lower noise in the laser source. If, in addition to the phase difference and signal frequency, also the time delay between the signals at detectors R1 and R2 is measured, then the position of the particle trajectory in the direction can be deterrnined (Strunck et al. 1993, Borys et al. 1999). This is particularly interesting in flow regions where a high velocity gradient exists, for example in a boundary layer. 8.2 Single-Point Phase Doppler Systems 453 2 Parlide diameier by planar syslem dp lJ.1ml Fig. 8.40. Correla tion between measured particle size using a planar and a reference phase Doppler system (m = 1.328, A.b = 852 nm, = 9 deg, Planar system: 1fJ, = ±1l.5 deg) e a - '" 600 C ;:J 0 u ~OO 200 OWU~~~~~~~LU~LU~~ o 2 .l Parl ide diameier d, lJ.1ml 2 ~ Parlidc diamclcr d, lfl ml Fig. 8.41. Measured size distribution of droplets from an aerosol generator using the reference phase Doppler technique (A.b = 852 nm, e= 9 deg, DEHS in air). a Bubbier temperature 180°C, mean diameter l.05Ilm, standard deviation 0.14Ilm, b Bubbler temperature 130 0 C, me an diameter 0.65Ilm, standard deviation 0.161lm 454 8 Phase Doppler Systems 8.3 Further Design Considerations for Phase Doppler Systems 8.3.1 Influence ofthe Gaussian Beam Especially the discussion concerning the dual-burst phase Doppler system revealed the fact that at least one separate measurement volume exists for every scattering order and for every detector position. In most conventional phase Doppler configurations, the scattering angles are chosen such that the intensity of all these volumes except one is insignificant. In the case of the dual-burst system, two measurement volumes were used, so that redundant size information could be obtained, hence also the refractive index. While the size of each measurement volume is the same for all particle sizes and scattering orders, their displacement from the position of the illuminated volume will depend on both parameters. For large particle-to-beam width ratios therefore, the different volumes are well separated in space and if the trajectory of the particle is known, the correct interpretation of the signal e.g. reflection or refraction, is straightforward. This is not generally the case for a three-detector, standard phase Doppler system or for a dual-mode system. With these systems, a particle may pass through a measurement volume that was not intended for use. The situation has been discussed in section 5.1.3.1 and can be illustrated as shown in Fig. 8.42. The example depicted in this figure is for adetector off-axis angle, 1> r = 60 deg, and for m > 1; the first -order refractive scattering is assumed to be dominant over the reflective scattering. For large particles however, the measurement volumes for reflection and first-order refraction are displaced significantly; for reflection in the opposite direction as the receiver location and for first-order refraction in the direction towards the receiver. The detection volume will favor first-order refraction at 1> r = 60 deg (see Fig. 4.17), nevertheless, for y Particle motion is normal to the page Rellection mc~surcment volume Illuminatcd mcasuremcnt volulllc Dircclion to dctcctor Fig. 8.42. Schematic illustration of the origins of the Gaussian beam effect. Particle motion is normal to the page 8.3 Further Design Considerations for Phase Doppler Systems 455 some partide trajectories, here partide trajectory B, only reflective scattered light will be detected. The resulting size measurement, based falsely on firstorder refraction, will be incorrect. Note that such an effect is only possible when the partide size is of the order or larger than the laser beam width and furthermore, the effect will arise for any shaped beam, even if it is non-Gaussian. The example in Fig. 8.42 was prepared assuming m > 1; however, a similar effect occurs for m < 1. For m < 1 the incident points for reflection and refraction are doser together, thus the effect only becomes significant for even larger particles (Grehan et al., 1994). Note further, that similar effects can also occur for higher scattering orders, but reflective and first-order refraction are the most common because of forward scattering advantages. This effect was originally recognized by Saffman (1986) and was later studied by numerous authors (Grehan et al. 1991, Sankar and Bachalo 1991, Grehan et al. 1992, 1993, Gouesbet and Grehan 1994b, Albrecht et al. 1996). It has been called the trajectory effect or the Gaussian beam defect/effect. There have been numerous strategies proposed to avoid or correct for this effect. One suggestion is to monitor also the amplitude of the signal (Sankar et al. 1992). A large partide detected using the wrong scattering mode due to its trajectory will yield low signal amplitude. A size. dependent amplitude threshold could eliminate such mi stakes (Bachalo 1991). Aizu et al. (1993) proposed using a planar phase Doppler system, with the partides moving along the y axis. In this way, contributions from both the desired scattering order and the unwanted scattering order will be received with a time displacement between them. Since the desired scattering order will yield much higher signal amplitude, errors could be avoided by determining the phase difference only around the maximum signal amplitude. A similar suggestion was put forward by Xu and Tropea (1994) using a standard, two-velocity component phase Doppler system. This procedure is implemented in the dualmode phase Doppler system (see section 8.2.3). A third method of recognizing improper size measurements, implemented in all three-or-more detector systems, is the validation criterion inherent in the three-detector, standard or dual-mode arrangement. The phase difference/diameter response curves for different scattering orders exhibit quite different dependencies for the two receiver arrangements. Signals arising from unwanted scattering orders alone would then lie outside of the tolerance bands shown in Fig. 8.20. For the three-detector, standard and the dual-mode systems the phase difference/diameter relation can be chosen by the masks. The phase difference of the dominating order, e.g. refraction, starts from the origin of the phase difference plane and increases for increasing partide size, as illustrated in Fig. 8.43. When the unwanted scattering order has an opposite sign in the phase difference Idiameter relation, e.g. reflection, the phase difference starts also at the origin or at (360 deg, 360 deg) and decreases for increasing partide diameter. If the aperture mask and, therefore, the three detectors are non-symmetric in elevation angle, the ratio of the phase factors of the detector pairs Fl~N) I Fit) will generally be a non-integer. If this is the case (best value: Fl~N) I F1iN ) = n + 0.5, n = I, 2, ... ) the system can identify the Gaussian beam ef- 456 8 Phase Doppler Systems a 360 dcg b '"E '"E ~ '0'" ~ '- 1:lc: 8c: <> <.I '"o ~ ~ ~ 't5 ~ "~ '"'" '" ..c 0.. L-______ Odcg ~~ _______ L_ _ _ _ _ _ _ __ . Phase diffcrcncc 01' system I , c .. ..c:: 0.. 0 deg Pha se difference of system 1 J60dcg d "' ~--------~-----'--?T--~~ <= '" ~ 360 deg '"E a'" '- '"o '-0'" 1:l c: 1:lc: ~ 't5 S ~ <J .... .."'" "C .'" tL-______ <> ..c:: O"L-____- L__ ~ ____~~_L_ _ _ _ _ _~ ~~ ______ _ L _ _ _ _ _ _ _ _~ Odeg Phase difference of system 1 0 dcg Phase dincrcncc 01" syslem I Measurements Sphericity line Increasing particle diameter ~ -,:Dominant order: ___ +-__ "I!o Unwanted order: Fig. 8.43a-d. Identifieation and non-identifieation of the Gaussian beam effeet using different aperture masks resulting in different ratios of the phase eonversion faetors. a The ratio of phase eonversion faetors is 3 and the Gaussian beam effeet eannot be deteeted, b The ratio of the phase eonversion faetors is 2.5 and measurements from unwanted seattering ean be rejeeted. e Measurements with high seatter ean result in wrong validation, d If the sign of the phase eonversion faetors of the dominant and the unwanted seattering order is identieal, the Gaussian beam effeet eannot be deteeted feet, beeause a measurement influeneed by refleetion leaves the toleranee band, as illustrated in Fig. 8.43b. If a symmetrieal arrangement of apertures is used (for instance when small particles are being measured) and the ratio of the phase faetors is an integer value or when the phase differenee/diameter relations have the same sign (e.g. refleetion and second-order refraetion), then the Gaussian beam effect ean no longer be identified using the three-deteetor arrangement (Fig. 8.43a,d). For instanee, the Dantee Dynarnies mask A (see Fig. 8.10) is symmetrie and is therefore not reeommended in situations where the Gaussian beam effeet ean potentially oeeur. However, even for asymmetrie arrangements, an error-free result is not always possible, espeeially when the phase difference exhibits large seatter (Fig. 8.43e). For the dual-mode arrangement the same eonsiderations are valid with the additional remark, that the seatter of the measurement for the planar system is generallyhigher than for three-detector, standard systems (see seetion 8.2.3). 8.3 Further Design Considerations for Phase Doppler Systems 457 Still another proposal for avoiding this effect has been made by Qiu and Hsu (1999), in which they show that for certain detector positions, an additional pair of detectors can be used to eliminate the effect by taking the ratio of the respective phase differences. This technique is based on the fact that the amplitude ratio of the two scattering orders is the main parameter which influences the measurement in case of a Gaussian beam effect. With an additional phase difference measurement this amplitude ratio can be estimated and the particle diameter corrected. Some remarks about this technique are presented in the next section. A final approach to avoid sizing errors due to the Gaussian beam effect is the correct layout of the system to insure that the ratio between the particle size and the beam width does not exceed a certain limit, i.e. by dimensioning the measurement volume large enough. Generally ratios of 5:1 (measurement volume to particle diameter) has been proposed by various authors: however; this does not consider the specific system or the tolerable error. More recently Araneo et al. (2000) have analyzed the situation in detail, both experimentally and theoretically, which will be discussed in the following sections. 8.3.7.7 Signal structure The basis of the analysis can be understood by examining the diagrams in Fig. 8.44, prepared for two arbitrary particle diameters, d pa and d pb = 3d pa • In each diagram, two measurement volumes are pictured, one for the dominant order and one for the unwanted scattering order. The detection volume is that volume within which the sum of all scattering order intensities exceeds some specified threshold, e.g. amplitude or SNR (section 5.1.6). The spatially dependent parameter A (ud) shown in this figure is the ratio between scattered intensity from the unwanted scattering order, subscript (u), to that from the dominant order, subscript (d). The factor A (ud) changes throughout space and must remain below some threshold to insure a proper size measurement. In the example shown in Fig. 8.44, a value of A (ud) < 1 is always exceeded in the detection volume for the small particle (d p = d pa ), whereas for the large particle (d pb = 3d pa), trajectories exist for which A (ud) > 1. In the following analysis, particle trajectories for which measurement errors occur are to be found. The criteria used are the phase error that occurs and the amplitude of the modulating part of the signal, which must exceed some minimum detection level. For this analysis the form of the signal generated at the detector is given by a dual-burst expression whereby A is the signal amplitude of the dominating order, t is time, tb is the burst width, OJ D is the Doppler or burst frequency, t(d) and t(u) are the size de- 458 8 Phase Doppler Systems a :J ,; d l • =d .. ~ Mcasurcmcnl volumc of dominant scallcring o rder >< .§ ~ '" :Q .... ~ Mcasurcmenl vo lume of unwdnlcd sUlllcring order b Pari idc posl ion z , la.u.] ~ .-"'. Mcasurcmcnt vo lumc of dom inant scallcring o rder '" >< <= ~~~~~==:i .2 -;;; 0 Co. <J :g ;:: '" 0.. Measurcmcnl volumc of unw·" nt cd scallering order Partide pos lio n z r la.u.1 Fig. 8.44a,b. Schematic representation of the total scattered intensity (thin lines) the measurement volumes (bold lines), the detection volume (gray regions) and the amplitude ratio A (url) (dashed lines) of the unwanted to dominant scattering order intensity for two different particle diameters. a Small particle diameter d p = d pa , b Larger particle diameter d p =d pb =3dpa pendent time shifts of the two bursts from each scattering order, <p(d) and q;(u) are the size dependent signal phases from the two scattering orders and A (ud) is the maximum signal amplitude ratio of the unwanted to dominant scattering order intensity for the individual signal (A(ud) = A~'::') == const < 1 for plane wave case). In general, dominant scattering means that the amplitude of the dominant order is ten times larger than the amplitudes of allother scattering orders for the plane wave case (A~'::') < 0.1). Angular regions where this is the case are discussed in section 8.1 and illustrated in Figs. 8.3 and 8.4. An exarnple signal is shown in Fig.8.45. A phase Doppler system would determine from such a signal using e.g. a Fourier transform, the signal phase at the frequency (j) == (j) D n,(du) ( 'Y r ) (j) ~ t sin <p( d) + A (ud) sm <p(U) = (j)D = arc.an cos<P(d) + A (ud) cos<P (u ) (8.57) 8.3 Further Design Considerations for Phase Doppler Systems Normalized amplitudeA tb 1 I > e 459 I 1 I \ 1\ ~ n~~"", A o ~ .... ~ nwanted .\1.1J,. I'siina1 jart / Dominant ~,.\{~/ \l )' <I>(u> -1 o Timet Fig. 8.45. A signal arising from a particle of diameter comparable to the measurement volume diameter and passing through the volume such that both the dominant and less dominant measurement volumes are traversed given sufficient periods in the signal (OJ D > 30 / tb ). It is important to note that the detection volume displacement and thus, the time shift of the signal, always exist. The Fourier transform of Eq. (8.56) gives a time-shift dependent spectrum (section 6.5.2.1). Only at the burst frequency OJ = OJ D is this dependency cancelled and this result corresponds to the results from Qui and Hsu (1999). By using the diameter conversion factors between phase and partide diameter of the dominating (ß<t> , e.g. Eq. (8.3) for refraction) and unwanted scattering order (ß';;>, e.g. Eq. (8.2) for reflection) for a two-detector standard phase Doppler system one obtains for the phase at one receiver ßt ) A Sin(d ßt) CO{d/f[)+A(Ud)CO{dp ßt) sin(d l/J(du) (OJ r p + (ud) p = OJ ) == arctan--7----7-------..:,----';D (8.58) only the two parameters d p and A (ud) are unknown. By using a second independent phase Doppler measurement of the same partide it is possible to determine the partide diameter and the maximum amplitude ratio between unwanted to dominant scattering orders from the two equations. This is the technique of Qiu and Hsu (1999) to eliminate the Gaussian beam effect with the third detector of a three-detector system. In this case the 21t ambiguity can be overcome only with a fourth detector (Qiu et al. 2000). Furthermore, this can be used for determination of the amplitude ratio between two scattering orders and for the absorption coefficient measurement with the dual-burst technique, section 8.2.4.2, even though the two bursts cannot be separated for small partides. If the linear relation between partide diameter d p and phase is substituted for each scattering order, l/J~d) = ß<t>d p /2 and l/J~u) = ß);>d p /2, the absolute phase A(ud) 460 8 Phase Doppler Systems error for one receiver of a standard phase Doppler measurement can be expressed as A(ud) Sin( d p ß~) ~ P~») arctan-----'--::----~""7""" l+A(ud) co{ d p P;) ~ P~») (8.59) The phase error O"t/> disappears for certain particle sizes, known as nodal points. 21tn dp = IP;) _P~)I ' n = 0, 1, 2, ... (8.60) Furthermore, the error depends strongly on the amplitude ratio A (ud) . A useful quantity is the critical amplitude ratio A~ud) at wh ich an acceptable error of magnitude (j ([Je (0::; (j ([Je < 1t /2) occurs in the measurement (8.61) The diameter conversion factors ß,;/) and ß';) are generally so large, that this critical amplitude ratio varies strongly with particle diameter. For this reason it is advantageous to define a minimum acceptable critical amplitude ratio, which is particle size independent min( A~ud) (d p ») = sin (j ([Je (8.62) If in any burst signal the maximum signal amplitude ratio between unwanted and dominant reaches this limit of A~ud), the measured phase could deviate by maximally (j ([Je from the expected result (/J~d) • 8.3.1.2 Warst case particle position Equation (8.62) yields, for a given phase error, detector position and particle size, an amplitude ratio of unwanted to dominant scattering order intensity, which must not be exceeded to avoid errors. For plane waves this ratio would never be exceeded (A~ud) <A~~». For inhomogeneous beams however, trajectories will always exist for which A (ud) is larger than A~ud) (see Figs. 8.26 and 8.44). Of particular interest is to find the particle position rpc at which the ratio A(ud) = A~ud) is reached and also at which the intensity exhibits a maximum. It can be shown that this position always lies on the line connecting the centers of the two measurement volumes, as shown in Fig. 8.46. The connecting line between centers g/ in parametric form is given by 8.3 Further Design Considerations for Phase Doppler Systems " " ~":- ;\ (~n) ConncLlinglinc bctwccn measurement volumc centers g, l\laximulll inlcnsily (or 11(0')= 11:"<0 < ;\!~tl) r.,= r,. .2 .--'-- A(lid) > 461 .-- .-- ~ A ~roll"l Partide posi tion top [a.u.1 r;:.. Particle positions which yield the scattered intensity larger than the detection level of 0.4 'CI and a phase error smaller than arcsin( 0.1) l!'!lI Particlc positions which yield thc scattered intcnsity larger than the dctcction level of 0.4 ~ and a phase error larger than arcsin(O.I) I© IParticle position which yield the scattered intensity not reaches the detection level Fig. 8.46. Relation between the measurement volume centers and the position of maximum signal intensityat which a given amplitude ratio is exceeded exI] e I =[e yl ezl = (d) (u) r ACmax - rACmax (d) I r ACmax (u) - rACmax I (8.63) , where the positions of maximum intensity and the centers of the measurement volumes d~~a.x for scattering orders N = u and N = d can be determined by Eq. (5.100). The parameter p in Eq. (8.63) is the position parameter along the line connecting the measurement volume centers. The ratio of unwanted to dominating scattering order intensity along the connection line can be written from Eqs. (5.99) as, A(ud)(g I (p») = A(ud) exp(_l_[_(xU'U) cosel _ pw 2 O,lr 12 ZU,u) rmb _ (XU,U) COS el 0,2r /2 O,lr sin el)2 /2 _ yU,U)2 + ZU,u) sin el)2 O,2r /2 + (XU,d) COSe/ _ O,lr /2 ZU,d) O,lr O,lr _ yU,U)2 0,2r (8.64) sin e/)2 + yU,d)2 /2 O,lr + (XU,d) COSe/ + ZU,d) sin e/)2 + yU,d)2 0,2r /2 0,2r /2 0,2r J) whereby the laser beam radius rmb and the scattering function in the vicinity of the measurement volume can be considered constant and equal for both incident beams. The absolute positions of the incident points rci:b~) are functions of 462 8 Phase Doppler Systems 'p the parameter p, the particle radius and the relative position of the incident points on the particle surfaee r~:,N), Eq, (5.4) and e.g. Eqs. (5.200) and (5.202) for refleetion and Eqs. (5.213), (5.215) and (5.216) for first-order refraction, and therefore on the system configuration O,br p+ eI = rU,N) + Po eU,N) br , p [ ~~~:] =[:::] p+[~t~:::j+[:~i:] 'p ZU,N) e O,br Z(d) zl (8.65) eU,N) AC,max Z,bT This equation allows the posItIon along the connecting line p to be reformulated as a funetion of amplitude ratios, by inverting Eq. (8.64) with respeet to p. The seleeted particle position f op = ec/ P+ Po depends on particle diameter d p , the seleeted amplitude ratio A(ud) along the connecting line, the ratio for plane waves A~~) and the geometrie eonfiguration of the phase Doppler system. p(dp,m,A (ud) 1 dp 2 A(Ud) (ud) _ 1 'mb ,A pw ,lf/r,cfJr,e)-- [ -C 2 --2C 3 +2-ln(;,i) 2C I 2 dp A pw (8.66) with CI = Cßxl eos 2 "; + Cze zl sin 2 <% + (D xezl + D ze xl )sin <% cos<% + Cye yl C2 =E x eos2el+2Fsineleosel+E sin 2el /2 /2 /2 z /2 +E Y C3 = Cx X(d) eos 2 el + (D x Z(d) + Dz X(d) )sin el eoSel C AC max /2 AC max AC max /2 /2 z (d) • 2 el + Z ACmax SIn 12 C (d) (8.67) (8.68) (8.69) + yY ACmax and C = eU,d) x x,Ir C = eU,d) y Z E x y eU,U) _ eU,u) x,Ir x,2r + eU,d) _ eU,U) _ eU,U) + eU,d) _ eU,U) _ eU,U) y,2r 2,2r y,lr y,2r zJr 2,2r = _eU,d) + eU,d) + eU,U) _ x,Ir x,2r x,Ir D = E eU,d) 2,lr Z x x,2r y,lr C = D + eU,d) _ _eU,d) 2,lr + eU,d) + eU,U) 2,2r eU,U) x,2r _ eU,U) 2,1r (8.70) 2,2r =(e U,d»)2 +(e U,d»)2 _(e U,U»)2 _(e U ,U»)2 x,Ir = x,2r x,Ir x,2r (e U'd»)2 +(e U,d»)2 _(e U'U»)2 _(e U'U»)2 y,lr y,2r y,lr y,2r =(e U,d»)2 +(e U,d»)2 _(e U,U»)2 _(e U,U»)2 E 2,lr Z F = eU,U) eU,U) x,Ir 2,lr 2,2r 2,lr 2,2r _ e(;'U) eU,U) _ eU,d) eU,d) x,2r 2,2r x,Ir 2,lr + eU,d) eU,d) x,2r 2,2r For A(ud) in Eq. (8.66), the eritieal amplitude ratio A~ud) from Eqs. (8.61) or (8.62) ean be substituted. Thus, an analytie expression results whieh gives the 8.3 Further Design Considerations for Phase Doppler Systems 463 position r pe = elPe + Po at which a prescribed error occurs while the signal exhibits a maximum intensity (8.71) The constants are found from the geometry of the phase Doppler system. Along the connecting line between detection volume centers, all amplitude ratios between -00 and +00 occur, arising from the superposition of two displaced Gaussian fields. For infinitely sensitive detectors, a particle position can therefore always be found for which an error will occur. Accordingly, some further criterion must be prescribed on which the selection of error free particle sizes can be made. An error can only occur when the scattered light intensity, being the sum of e.g. reflection and refraction, exceeds the detection level, i.e. the particle trajectory passes through the detection volume. Using the critical particle position r pe (d p) in Eq. (8.65), the scattering intensity of the dominating scattering order [ Id)(d r _ ) _ lid) rO p,rop-rpe - d~2 d po exp 1_[( 8/ _ 8/)2 2 XO,lrCOS/2 ZO,lr Slll ( __ (i,d) (i,d) rmb • /2 (i,d)2 +YO,lr + (X(i,d) COS8/ + Z(i,d) sin 8/)2 + y(i,d)2 0,2r /2 0.2r /2 0,2r J) (8.72) and the unwanted order _ ) _ [lu) [ lu) (d r p,rop - r pe - rO d~2 d po exp( 1_[( 2 __ rmb (i,u) 8/ _ (i,u) X O,lr COS /2 ZO,lr • 8/)2 (i,u)2 Sin 12 + YO,lr + (X(i,U) COS81 + Z(i,u) sin 8/)2 + y(i,U)2 0,2r /2 0,2r /2 0,2r J) (8.73) can be computed from Eq. (5.99). The intensities of the individual orders increase with the square of particle size in the geometrical optics range. The maximum intensity in Eqs. (8.72) and (8.73) (8.74) can be expressed with respect to the maxinmm intensity of a reference particle diameter d po for plane waves. To determine the ratio of the scattering order intensities for plane waves A1~) used in Eqs. (8.64) and (8.71) (8.75) 464 8 Phase Doppler Systems for arbitrary particle diameters, the reference particle size d po , which is definitely large enough to be treated using geometrical optics (e.g. d po = 1000 j..lm), can also be considered. Therefore, the scattering intensities I~~) and I~g) in Eqs. (8.72) and (8.73) are determined for the case of a plane wave and are independent of the Gaussian beam effect. If the total scattered intensity exceeds the detection amplitude (8.76) then erroneous measurements can be expected. The detection level I d of the detectors can be chosen such that particles with a diameter d pd will just be detected, e.g. d pd =2 j..lm. The detection level can then be determined with respect to the reference particle diameter. e.g. I d = (2 j..lm)2 (d) 2 (U)) I ro + I ro (8.77) (1000 j..lm) The numerical solution of Eq. (8.76) for d p yields the maximum error-free particle size for the chosen detector positions. This analysis therefore provides the particle size at which first errors due to the Gaussian beam effect can be expected. Note that the error was assumed for one receiver. For the case of a twodetector standard system this error can be expected on both receivers, thus the phase difference error is double the error on one receiver. 8.3.1.3 Example The above analysis was carried through assuming two arbitrary scattering orders interacting in the ratio A(ud) to yield the detected intensity. Indeed, the sum of all scattering orders must be considered for a more exact solution; however in most practical circumstances, off-axis and elevation angles are chosen for the detectors at which higher order scattering is negligibly sm all. Furthermore, the relatively simple expressions given above would no longer be possible. Typical experimental conditions, given in Table 8.5, can be used in the theoretical analysis given above to determine the particle diameter at which the Gaussian beam effect can lead to a phase difference error for a given optical arrangement. One critical input parameter is the minimum detectable particle diameter d pd • The maximum particle diameter in the investigated process is assumed to be 100 j..lm. The fuH dynamic range of the processor is used, without saluration of the electronics at this diameter. If the phase Doppler system has a maximum dynamic range of about 40:1 (1600:1 in intensity) then the smallest measurable particle diameter is about 2 j..lm to 3 j..lm, accounting for some deviation from the value given by the manufacturer. In Fig. 8.47, the particle diameter at which a phase difference error of 10 deg occurs, has been computed as a function of detector position for each of the laser beam waist diameters, 77 flm and 290 flm, using a smaHest detectable particle diameter of 3 flm and the parameters from Table 8.5. Additionally the region of dominant first-order refraction is indicated. 8.3 Further Design Considerations for Phase Doppler Systems 465 Table 8.5. Example configurations of a phase Doppler system for estimating the error-free measurement range. Quantity Symbol 11.1 deg, 5.72 deg 514.5 nm 90 deg Full in tersection angle of the laser beams Wavelength Polarization angle to be am plane Beam waist radius Relative refractive index Reference particle size for ratio determination Smallest detectable particle diameter Maximum allowable phase difference error Maximum phase error per detector Off-axis angle 77 11m, 290 11m 1.334 m d po 100 11m d pd 0<k 0<t>c 9, Eleva tion angle Dominating scattering order Unwanted scattering order 1jI, (d) (u) a Configura tion /2 311 m 10 deg 5 deg 30 deg < 2.5 deg First -order refraction (1) Reflection (0) b 20 o ~~~~~==~~~ o 30 60 90 Off-ald angle 1/I, ldegl 0 30 60 90 Off-alds angle 1/I, ldegl Fig. 8.47a,b. Particle diameter (in 11m) at which a phase difference error of 10 deg is expected (system parameters in Table 8.5). a Laser beam waist diameter of 77 11m, b Laser beam waist diameter of 290 11m In the off-axis range of f/Jr = 70 ... 77 deg and the elevation range of = 0 ... 5 deg, first-order refraction completely dominates reflection (Brewster angle), leading to the maximum measurable particle diameters. For this angle, other scattering orders than reflection influence the particle diameter measurement, as can be seen from Figs. 4.31 and 4.32. For the position of the incident point of the unwanted scattering order, the respective higher scattering order lfI r 466 8 Phase Doppler Systems must be used. Measurements with the detectors in the black shaded areas are not possible (A~ud) < 0.174 = sin(lOdeg», since even for plane waves the critical amplitude ratio is exceeded A(ud) > A~,:!). The predictions in Fig. 8.47 show that the maximum error-free particle diameter is 331lm and 981lm for the beam waist diameters of 771lm and 290llill respectively for a receiver off-axis angle of ifJ r = 30 deg and small elevation angles. The above example demonstrates that this analysis can reduce the rather complicated theoretical aspects of the Gaussian beam effect to a simple limiting diameter which must not be exceeded during the measurements or a minimum beam diameter for a given particle diameter range. Possibilities of increasing this limiting diameter lie in the choice of optical configuration parameters. More details about such computations and comparisons with measurement results for verification of the analysis can be found in Damaschke et al. (2000). 8.3.2 Slit Effect Most commercial phase Doppler systems use a slit aperture in front of the detector or receiving fibers. The slit is orientated, such that the measurement volurne is truncated in the z direction over the length of the projected slit width, L s • The reason for employing slit apertures in the first place is to achieve welldefined detection volume dimensions (see section 12.2.3). Since all further statistics about size and flux distributions must be based on a reference volume or area, the slit helps to define the detection volume dimension in the z direction. Pinholes can also be used instead of slits, then the detection volume shape becomes that of a cylinder (see sections 12.2.3 and 14.2.1). A typical arrangement is pictured in Fig. 8.48, using an off-axis angle of ifJr = 60 deg as an example. In this case, first-order refraction is expected to be dominant. Independent of particle size, trajectories near the boundaries of the projected slit aperture exist in which either reflection is suppressed (particle A) or firstDetectiol1 volume (projection of the slit aper ture) ,, .. , .. . , refraction Fig. 8.48. Illustration of the slit effect. Particle trajectories are into the page Receiving probe 8.3 Further Design Considerations for Phase Doppler Systems 467 order refraction is suppressed (particle B), because the respective scattered light is outside of the projected slit region. Thus, the slit aperture results in a virtual detection volume, which exhibits an inhomogeneous ratio of dominant to unwanted scattering order intensity, similar to the Gaussian beam effect arising from a shaped beam. The slit effect, first identified and quantified by Durst et al. (1994), becomes important when the slit aperture blocks the glare points of dominant order. In contrast to the Gaussian beam effect, which depends on incident point positions, the slit-effect is dependent on the glare point positions on the particle surface. Due to the finite aperture size and diffraction effects, this explanation of the slit effect using geometrical optics is not precise, nevertheless, sizing errors can occur because light from the wrong scattering order may be dominating. Such errors can be avoided using the same techniques used to avoid the Gaussian beam effect, for instance, a three-detector or dual-mode arrangement with a noninteger phase factor ratio. Further remarks about the slit effect can be found in Durst et al. (1994), Xu and Tropea (1994) and Sommerfeld and Tropea (1999). How the slit aperture dimension is incorporated into the data processing is discussed in section 12.2. 8.3.3 Non-Spherical and Inhomogeneous Particles Already in section 8.2.3.1 the special case of oscillating droplets aligned with the x axis of a dual-mode phase Doppler system has been examined. In this section the more general situation of non-spherical or inhomogeneous particles their measurement with the phase Doppler technique will be discussed. maschke et al. (1999) introduced the classification of particles shown in 8.49, distinguishing between spherical and non-spherical and homogeneous and DaFig. and Parlides Rotationally symmetrie particles Iiomogcneous phcri.:al parcidcs A) ß) L:lycrcd [ nhomogencous D) Prolat. E) Cylindcr. disk ol>lal C) Eva poral ing G) uspcnsions F) General Ii) Al mosl I) Agglomera tes dcformations sphcrical Limils -+ Fig. 8.49. Classification of particles (adapted from Damaschke et al. (1998» J)Very irreguJar 468 8 Phase Doppler Systems Fig. 8.50. Schematic representation of steps involved in particle measurement inhomogeneous particles. Indeed, they also recognized that the problem of size measurement is manifold, comprising the four steps shown schematically in Fig. 8.50; mathematical description of the particle, predictability of light scattering from the particle, strategy for solving the inverse problem and a practical implementation in an instrument. For certain idealized particles, some of these steps may be simple; however, in general an exact description or measurement of the particle characteristics will be virtually impossible. The basic principles of the signal generation presented in section 5.1 are valid for both spherical and non-spherical particles; however, any optical technique can only deliver information about the particle at the points of light interaction - incident and glare points. The phase Doppler technique yields local curvature information around these points, and some information about the light path in the particle between incident and glare points may be available. However, a complete determination of shape and structure of a more irregular and inhomogeneous particle becomes increasingly difficult because of the limited number of sampie points on the surface of the particle. A rather extreme example of single particle analysis based on multiple glare points is the 72 detector DA WN-A instrument introduced by Wyatt et al. (1988). Moreover, the inverse problem, i.e. deducing unique particle characteristics from the multiple measurements also becomes increasingly complex with added detectors. The entire problem takes on added degrees of complexity due to the stochastic processes involved: orientation of the particle, surface roughness, inclusions, ete. For these reasons the outlook for employing the phase Doppler technique as a single particle sizing instrument for non-spherical or inhomogeneous particles is bleak. More encouraging is the possibility of deducing statistical statements about an ensemble of measured particles. Small deviations of particle shape from the spherical form, small inhomogeneities or surface roughness lead to a broadening of the measured diameter dis- 8.3 Further Design Considerations for Phase Doppler Systems 469 tribution. Manasse et el. (1994) and Mitschke et al. (1998) experimentally demonstrate this broadening for inhomogeneous particles, e.g. milk suspensions. A deconvolution of the measured size distribution with a known distribution from homogeneous droplets is used as a correction procedure. Further details are given in section 12.3.3. Göbel (1998) found a similar behavior for rough particles and also used a deconvolution algorithm in the data processing. In both instances, information about single particles is no longer available. Another approach for optically absorbent liquid droplets is to use incident beams of high er wavelengths. Manasse et al.(1993), Manasse et al. (1994) and Mitschke et al. (1998) have all demonstrated that by using wavelengths in the near infrared range, the attenuation of refracted beams can be greatly reduced, leading to a significant improvement in measurement results. In particular, the measured size distribution becomes narrower. Rheims et al. (1998) and Rheims et al. (1999) have attempted to preserve the single particle feature of the phase Doppler technique for particles with roughness or light inhomogeneities. The three detectors of a standard phase Doppler instrument were replaced by a CCD line array in the ljI r direction. Each CCD pixel acts as aseparate detector and the far-field interference pattern can be sampled directly. Small perturbations on the particle surface lead to deviations of the interference fringe spacing from the otherwise linear change with pixel number (elevation angle). A linear regression of the phase deviations lead to estimates of local change of curvature on the surface of the particle. The same authors have proposed using a second CCD line array aligned along the off-axis angle at an elevation angle of ljI r = Odeg. The spatial intensity modulation on this array, arising from the scattering lobes, can lead to estimates of further partide characteristics. A se co nd incident laser beam is used only to provide the partide velo city. These techniques are, however, presently limited by the rather low readout rates of available CCD line arrays. J1urther experimental investigations of irregularly shaped particles have been performed by Naqwi (1996), and Naqwi and Fandrey (1997). They demonstrate that the phase Doppler technique used to measure a stream of 'monodispersed' irregular partides yields a peak in the size distribution at the correct mean diameter. The broadening of the distribution could, to some extent, be reconstructed. A further method of measuring irregular partides is through direct imaging. An example of such a technique is the shadow Doppler velocimeter, introduced by Hardalupas et al. (1993, 1994) and discussed in detail in section 9.4. If more that one partide is present at once in the detection volume of a phase Doppler system, multiple scattering can occur. The response of a phase Doppler system to multiple scattering has been studied theoretically and experimentally by Doicu et al. (1998) and Bech and Leder (1999). Both the size of the individual partides and of the agglomerates could be deduced, depending on the exact position of the partides. For small secondary partides, the measured size distribution was broadened in a mann er similar to that for rough partides. Sultan et al. (2000) examined the influence of secondary light scattering outside of the detection volume on the measured size distribution. Secondary scattering was introduced experimentally by positioning an auxiliary spray between 470 8 Phase Doppler Systems the transmitting optics and the measurement volume or between the measurement volume and the receiving optics. Such secondary scatter does influence the measured size distribution, generally leading to an under-representation of small particles. This is because the scattered light intensity dirninishes and falls below the detection level for small particles fits. A deconvolution procedure was used by the authors to correct for this unwanted influence. 8.4 Multi-Dimensional Sizing Techniques 8.4.1 Interferometric Particle Imaging (IPI) A further technique for sizing spherical particles, which is still in its infancy and goes by various names (PMSI - Plan ar Mie scattering interferometry, Planar interferometric imaging (PlI), Mie scattering imaging, PPIA - Planar particle image analysis, ILIDS - Interferometric light imaging for droplet sizing), is based on scattering and imaging from a single laser beam. This technique will be deno ted Interferometric Particle Imaging (IPI). The origins ofIPI technique can be found in König et al. (1986), who focused a single laser beam onto a stream of monodispersed droplets and measured the resulting fringe pattern in the far field. They recognized already the potential for highly accurate size measurements and applied the technique to measure droplet evaporation. Ragucci et al. (1990) also examined the case of scattering from a single droplet using a Lorenz-Mie calculation to find the oscillation behavior of the scattered intensity in the far field. The extension of the technique to a multidimensional method using an illuminating laser light sheet is attributed to Glover et al. (1995), who examined sparse injection sprays in an optical internal combustion engine. Glover et al. (1995) also discuss the possibility of combining the technique with a PIV system for two-component velo city measurements. The principle of the technique can be understood using geometrical optics. Considering a homogeneous spherical particle, a scattering angle can be found for which two scattering orders, e.g. reflective (N = 0) and refractive (N = 1) are of approximately equal intensity, for example f), "" 65 deg for water in air and Foc uscd I Illumi nal ing laser bcam Imaging oplics / ZI Image planes Fig. 8.51. The IPI technique showing both an in -focus image and an out -of-focus image 8.4 Multi-Dimensional Sizing Techniques 471 perpendicular polarization. This situation is depicted in Fig. 8.5l. An in-focus image of the particle (right-most image in Fig. 8.51) will consist of only two glare points, whose projected separation will be directly related to the particle size. In principle this represents one possible mode of recording and processing (Hess 1998); however, the demands on the image resolution are high and indeed no distinct advantages over direct particle imaging, for instance with back-lighting are obvious. Furthermore, the glare point distance does not change linearly with particle diameter, because of optical resonances inside the particle (Schaller 2000). If, however, the particle is imaged out of focus, interference fringes arising from the two scattered rays appear. The shape of the defocused image of each glare point depends only on the shape of the aperture, whereas the size of the defocused images depends on the degree of defocussing. As the degree of defocus increases, the two glare points merge into one single image with interference fringes. The dislocation of the two defocused images can only be detected if the magnification and resolution of the recording media, e.g. CCD camera, is high enough. For particle sizing, the angular spacing of the interference fringes for a detector placed at a scattering angle of iJ r is to be computed as a function of particle size (van de Hulst 1957 and Roth et al. 1991). The situation is pictured in Fig. 8.52 for reflection and first-order refraction. The path-Iength difference between a reflected ray and a refracted ray through the particle and to the detector can be deduced by comparing each of the individual ray paths to a hypothetical ray from the source to the detector passing through the center of the particle, and then taking their difference (see also section 4.1.1.2 and Eqs. (4.36) and (4.37». Ineidcnl ray N=2 Mcdium Rcfcrcncc rdY Refr~Cled [n "idenl ray N I Fig. 8.52. Ray paths for IPI N ~ 2 rdY 472 8 Phase Doppler Systems The complementary angles rare used instead of the incident angle e~N), the reflecting angle (frl) and the angle of refraction e~N) . The relations between these angles are (8.78) The reflected ray differs in path length from the reference ray by Z(1) _z(r) = AZ(l) = -2rp sin r(1) , (8.79) and the refracted ray by Z(2) _z(r) = AZ(2) = 2r p (m sin ,,(2) - sin ,,(2») t , (8.80) and thus AZ(l2) = Z(l) _Z(2) = d (sin r(2) - m sin ,,(2) P t I - sin ,,(1») r (8.81) The receiver angle 7J r is related to the local scattering angles given in Fig. 8.52 through 2r(l) i = 2(r(2) t ,,(2») = iJ r (8.82) I Furthermore, Eq. (8.82) leads to . 7J m S l l l r- sin r(l) = sin ~ , , msin ,,~2) 2 _ sin ,,~2) = 2 cos ,,~2) (8.83) . 7J r mSlnCOST(2) = , 2 (8.84) ~1+m2-2mcos~r The final expression for the path-length difference can be expressed as AZ(12) =dp[sin ~r _~m2 +1-2mcos ~r ) (8.85) Changes of path length equal to Ab correspond to the next interference fringe and can be related to the fringe spacing Atp dAZ(12) Ab ~"" Atp(12) (8.86) or _1 [ 2,1 iJ Atp(12) = __ b cos-r + dp 2 . iJ r m Slll2 ~m2 +1-2mcos ~r J (8.87) 8.4 Multi-Dimensional Sizing Techniques 473 Thus, the angular frequency of the interference fringes is inversely proportional to the particle size d p as already expected from Figs. 4.15 and 4.16. For relative refractive indexes smaller than 1 the sign of the sum in the brackets and the m in front of the eosine change (Maeda et al. 2000). 2..1, .dip(12) =_b dp [ iJ m cos-r _ 2 J- msin-r iJ2 ~m2 +1-2mcos ~r 1 (8.88) Fig. 8.53 illustrates the dependencies expressed in Eqs. (8.87) and (8.88). Physically, the interference fringes correspond to the oscillating intensity lobes apparent in the scattering diagrams computed using the Lorenz-Mie theory, for instance in Figs. 4.29 to 4.32. Anders (1994) has shown that Eq. (8.87) agrees very weH with Lorenz-Mie computations over an scattering angle range iJ r > 10 deg in forward scatter, i.e. beyond regions influenced by diffraction. Massoli et al. (1999) have also presented Lorenz-Mie calculations of the angular frequency of fringes. Schaller (2000) has demonstrated that optical resonances influence the glare point distance as discussed later in this section. An approximate relation for the angular fringe spacing, valid for iJ r '" 30 deg, was given in the original work of König et al. (1986), but Anders (1994) demonstrates that this relation is rather imprecise. The technique is easily extended to a two-dimensional configuration using a laser light sheet and this makes it attractive for a wide variety of two-phase flows. A typical IPI image, taken from a water spray, is shown in Fig. 8.54. Large m= 0.5 m=0.6 m=O.7 m = 0.8 --oQ--- m = 0.9 ---0-- ~~~~::::~-::=--------------jO.61 45 - - m= 1.1 - - - m = 1.2 .. •. m= 1.4 m = 1.5 -----. m= 1.3 -- ... -. m= 1.6 90 Receiver angle tJ, [deg] Fig. 8.53. Angular interference fringe distance as a function of the receiving angle for different refractive indexes and for a particle diameter of d p = 100 11m. The interference results from reflection and first-order refraction (Ab = 532 nm) 474 8 Phase Doppler Systems Fig. 8.54. Defocused image from a water spray (d p = 20 ... 400 11m) partieles exhibit more fringes in the defo cused image and small partieles less. For large partieles, e.g. in the top center of the of Fig. 8.54, two slightly separated cireles can be seen. For this partiele the distance of the glare points can be resolved with the imaging system and each glare point creates its own defocused image, as illustrated in Fig. 8.51. Only in the overlapping region does interference appear. The size of the partiele image on the out-of-focus plane is not related to the partiele size, but will depend on the position of the recording plane relative to the focal plane of the imaging system. The situation is pictured in Fig. 8.55. The image diameter is given by (8.89) using Eq. (3.115) where da is the diameter of the aperture and f is the equivalent focallength of the imaging system (see section 3.2.1). The shape of the partiele image will correspond to the shape of the receiving aperture. Laser shecl lmaging oplics Re.: rding plane d, z, 2, Fig. 8.55. Imaging situa tion fo r IPI 2, l'ocal 8.4 Multi-Dimensional Sizing Techniques 475 The angular window observed in each image, LW r' is approximately equal to L119 r = 2arcsin(NA) = 2arcsin~ (8.90) 2z[ where NA is the numerical aperture of the imaging system defined by Eq. (3.118). Thus, the diameter ofthe particle can be linearly related to the number offringes in the defocused image, N fr = L1iJ r I L1<p(12) = Kd p , using for m > 1 K arCSlnp . 2z d [ = ___----'--1 A 1 msm-r • iJ 19 r 2 cos- + --r========= 2 19 -ym 2 +1-2mcos-t I (8.91) and for m < 1 (Maeda et al. 2000) K arcsmp . 2z d1[ = -------'-A 1 msm-r • 19 iJ r 2 mcos-- --r========= 2 19 -ym 2 +1-2mcos-t I (8.92) The diameter of the particle can be determined by counting the number of fringes in the out-of-focus image, independent of the image size. Alternative to counting fringes within a given angular window, Min and Gomez (1996) suggest the use of a photodiode array as a detector and to use a Fourier transform of the oscillating intensity to determine the spatial frequency. This no doubt avoids the ±1 fringe count uncertainty; however, due to the short record length some windowing will be required, e.g. Hamming. Hess (1998) points out that a true farfield image is required to obtain straight fringes. In the near or middle field the fringes exhibit curvature. The IPI technique performs best when the intensity of the reflected and refracted rays are equal in intensity, which for water in air (m = 1.33) occurs at offaxis angles of about iJ r '" 65 deg for perpendicular polarization (Figs. 4.29 and 4.30) and iJ r ' " 90 deg for parallel polarization (Figs. 4.31 and 4.32). For air bubbl es in water, the reflected and first-order refracted light is about equal in intensity at an off-axis angle of iJ r '" 45 deg, see Fig. 4.34d. In principle, the IPI technique can also be used with combinations of other scattering orders. Generally however, there is a limited number of angles for which only two scattering orders of equal amplitude dominate for a wide particle size range. By using scattering angles with more than two scattering modes, additional fringe systems appear. This is evident in the simulations presented in Fig. 8.56, in which third-order refraction is seen to become influential. The spatial spectrum of the fringes show interference between first-order and third-order refraction as weIl as between reflection and third-order refraction, Fig. 8.56c. The two sets of fringes with a small frequency difference result in a beat frequency with a long modulation, as can be seen in the defocused image in Fig. 8.56b. The 476 8 Phase Doppler Systems c <> '":> ~ '" ~ 2 ::::; <J ReneClion and fir. l-order rcfra cl ion J) Rcneclion a nd \ /Ihird -ordcr rcfra':lion o 20 First-o rd er and Ihird -o rder rcfraction 40 60 80 100 Frequeney 1. 1-1 Fig. 8.56a-c. Simulations ofIPI technique for a water droplet in air. a In-focus image with three glare points. The intensity and the contrast is changed for better visualization, b Out-of-focus image, for a rectangular aperture, c Spatial frequency spectrum from the out-of-focus image corresponding glare point, Fig. 8.56a, can be identified in the experimental result shown in Fig. 2.8. As Fig. 8.57 shows, the linearity between the number of fringes N fr or the fringe frequency fi and particle size is valid for both interference contributions and thus, in principle two values of the particle size can be determined from the image. The second value can be used for validation or for estimating the refractive index. The relation between fringe number or image fringe frequency and particle diameter is not perfectly linear, as can already be expected from Fig. 8.57. In Fig. 8.58a the calculated image frequency is shown for a small particle diameter range with a high resolution. The non-linearity is caused by optical resonances inside the particle. The optical resonances limit the resolution of particle diameter to about l~, as is the case for the phase Doppler technique. In practice, larger particles have weak resonances due to deviations from the ideal homo geneous or spherical case. The resonances influence not only the frequency/diameter relation. Also the intensity ratio between the glare points varies significantly with sm all particle diameter deviations. This is illustrated in Fig. 8.58b. Therefore, the modulation of the defocused image frequency depends on the particle diameter because the intensity ratio between the glare points influence the modulation and a determination of the refractive index by using the intensity ratio is not possible (Schaller 2000). Min and Gomez (1996) have shown the technique to be rather insensitive to intensity amplitude variations of each scattering order and thus to collection angle or to the finite Gaussian beam width in the laser sheet. They also indicate 8.4 Multi-Dimensional Sizing Techniques 477 ~80r--.-,--.--.--.--r-,r-'--'--'-I-r-,--,--r--r-.--.--.--r-. '-' ~ " '2"" 60 - .... ..... ...... Image frequency: First-order and third-order refraction Reflection and first-order refraction •• _ cpo 00 0000 ._ • • - ~ D ODD •••• ] • ••••• •••••• •••• • •- •••• ••••:~.ooDO .......-0 000000 DODD a DD •• - .- a DO 00 0 ••• 20 - - 0000000 a DD •••• ••• 00000000 rP° o .- 40 - _ 00 000 ._ c.b • • • • 00 000 ao D - ~~:DDDD 0 0L-~~__L--L~__L--L~__L-_L-1J--L~L-~-L~__~-L~~ o Particle diameter dp [Ilm] 50 100 Fig. 8.57. Simulations of the dependence of out-of-focus image frequency on the particle diameter for the case visualized in Fig. 8.56. b a "E'" ·0 0.. "... '" Ob '0 ..§ 450 2 ...'" >- .t:: ~ .Ei" 22 24 Particle diameter dp [Ilm] 1 L..l...L..l..-L..1...L..L.L..1--'-..L.-'-'--'-...L..L.JL..L....l....J 25 26 27 Particle diameter dp [Ilm] Fig. 8.58a,b. Dependence of intensity ratio and frequency in an out-of-focus image on particle size. a Highly resolved frequency dependence on particle size, b Dependence of intensity ratio of the glare points of reflection and first-order refraction on the particle diameter that the fringe frequeney is only a weak function of refraetive index, whieh is also evident in Fig. 8.53 for m > 1 and ä r in the region larger than 45 deg. Massoli and Calabria (1999), who investigated the teehnique using inhomogeneous particles, speeifieally reaetive fuel sprays, also drew this eonclusion. Although the high resolution of photo graphie film is desirable, many authors have sueeessfully employed medium resolution CCD eameras. Hess (1998) has 478 8 Phase Doppler Systems implemented the technique using digital holographic recording, thus allowing either near or far-field analysis from a single exposure. No information concerning the sensitivity to non-sphericity is presently available. Another interesting feature of this technique is that the particle image size will depend on the position of the particle on the z axis, as seen from Fig. 8.55 and from Eq. (8.89). Thus, all three position coordinates ofthe particle are available from an image as shown in Fig. 8.54. A logical extension of the IPI technique is to use a double-pulsed laser source and to conduct particle tracking between two successive images. This yields then three components of particle velo city and particle size in three dimensions. Such a combination has been demonstrated by Damaschke et al. (2000) and more recently by Maeda et al. (2000). A further refinement, known as optical compression, has been introduced by Kobayashi et al. (2000) to increase the measurable particle concentration. By placing a cylindricallens in the imaging system, the particle images can be reduced to lines of oscillating intensity, avoiding much of the image overlap. In one plane the image is in focus and in the other it is out of focus. The advantages of such a system for measurements in higher dense sprays are discussed in section 8.4.3 This technique is also the basis of the differential laser Doppler anemometer (DLDA) introduced by Rheims et al. (1999). In their device a CCD line array spans an off-axis angle range and sampIes the IPI fringes in time as one particle passes through the measurement volume of a standard laser Doppler system. In this case the velo city can be determined from the laser Doppler signal and the size from the fringe pattern recorded on the CCD array. 8.4.2 Global Phase Doppler (GPD) Technique The difficulties of choosing a suitable scattering angle for the image recording with the IPI technique, especially using aScheimpflug set-up known from the Particle Image Velocimetry (Raffel et al. 1998), can be partially overcome using the Global Phase Doppler (GPD) technique (Damaschke et al. 1999), as first introduced by Damaschke et al. (2000). In this technique two laser light sheets illuminate the measurement area at an intersection angle ,%, as shown schematically in Fig. 8.59. With this configuration each illuminating laser sheet will contribute a glare point for each visible scattering order/mode. A far-field interference pattern will exist, corresponding exactly to the interference pattern produced using a phase Doppler system. The difference to the phase Doppler technique is that the GPD technique captures simultaneously the fringe patterns arising from many droplets within the illuminated plane and therefore the measurement volume is much larger. Such a GPD image is shown in Fig. 8.60, taken from a water spray. Note that both the IPI and the GPD interference fringes are visible and that they are aligned orthogonal to one another. How intense the IPI fringes appear, depends on the choice of collection angle and the relative intensity of the reflective glare points to the first-order refractive glare points. 8.4 Multi-Dimensional Sizing Techniques Defocuscd camcra 479 Dcfocused parlidc images hCCls Ob crvdlion lidd Lighl shcci opl ics Bcam plillcr unil 1\ Pulscd laser Fig. 8.59. Optical arrangement of a global phase Doppler system Fig. 8.60. Defocused image from a global phase Doppler (GPD) system. The vertical fringes arise from interference between reflection and first-order refraction (IP!). The horizontal fringes correspond to the GPD technique. The image was taken from a water spray (d p = 20 .. .400 11m ) In a mann er similar to IPI, the diameter of the particle can be related to the number of fringes in each particle image. Nfr=Kd p For reflection the conversion factor K becomes (8.93) 480 8 Phase Doppler Systems 1 _d_ " cos9rcos~+ 2z[ - RIi2 1-~sin~ 4z[ 1-~COS9rCOS~-~I- d~2 Sin~l 2z[ (8.94) 4z[ and for first-order refraction (8.95) Note that the GPD technique, as with the phase Doppler technique, requires spatial coherence of the illuminating waves over the entire area of measurement. The GPD technique exhibits several advantages over the IPI technique. The glare points are always the same intensity and thus the fringe modulation is always a maximum. Furthermore, any scattering angle can be used for image capture, especially attractive for implementation in a stereoscopie PTV system. If only one scattering order dominates, like in the phase Doppler technique or for opaque particles, one spatial frequency is imaged. In regions where several scattering orders contribute to the signal, each order can be separated by a frequency analysis. The fringe spacing and also the measurement size range can be adjusted by varying the intersection angle of the laser sheets. If both the IPI technique and the GPD technique are used simultaneously, the size range of the techniques can be combined and the orthogonal fringe patterns allow a verification of sphericity. If the particle is spherical, then the sizes determined from the two interference fringe patterns will be the same. Finally, in contrast to the IPI technique, the GPD technique can be used for measurement of opaque particles by using the two glare points from reflection. Just as the GPD method visualizes the phase Doppler fringes from each particle, the same can be achieved directly in anormal focused beam phase Doppler system for one particle using a CCD line array spanning a range of elevation angles. Such a device has been demonstrated by Rheims et al. (1998) and termed the differential phase Doppler technique. In this case the particle velo city is obtained directly from the laser Doppler signal and the spatial fringe pattern for one particle is reconstructed using sequential signals in time from the CCD array. 8.4 Multi-Dimensional Sizing Techniques 481 8.4.3 Concentration Limits The IPI and GPD sizing techniques are both relatively new and no commercial systems are available yet. Indeed, a need for parameter optimization with respect to • • • • • Measurable size range (also detection ofnon-sphericity) Measurable concentration Field ofview, measurement volume Required optical access Accuracy (in size and in derivative quantities, e.g. concentration, fluxes) can already be recognized. Girasole et al. (1998) and Massoli and Calabria (1999) have presented some systematic computational studies. The work of Kobayashi et al. (2000) and Kawaguchi et al. (2002) has been directed towards conditions of higher particle concentration. In the following discussion, some fundamental relations will be presented concerning the limitations of these techniques. The generic situation for either the IPI technique or the GPD technique is pictured in Fig. 8.61. Normally the given parameters of the system include • L1x"L1Yr • nx,n y • z/ • dp,m;n,dp,max Dimensions of the CCD camera Number of pixels in x and y directions Minimum stand-off distance Minimum and maximum particle diameter to be resolved Given a particle number concentration in the flow, Hp, the task is to choose the following system parameters _-----z, .... _-----z,------+ ,'-----~ ,_ ----Z,.... " ----~ z, Ima ge of Parlide I (d , =cl, ... ,, ) tJ.Yr CC D-Chi Lens In1<lgc 01 partidc 2 Laser Iighl shccl FOall plancs Fig. 8.61. Optical arrangement for IPI or GPD (J, =d._,) y ~, 482 8 Phase Doppler Systems • fx,fy • tz • dax,day • zr • Zz Focallengths of the imaging lens Thickness of the laser sheet Diameters of the aperture Position of the camera behind the lens Stand-off distance under the constraints • The Nyquist criterion is satisfied for the fringes in each particle image, with respect to the pixel resolution. • The degree of image overlap does not exceed some statisticallimit. The degree of overlap will be expressed as an overlap coefficient, Z, defined as the ratio of the area on the camera with two or more overlapping images to the total area with particle images. After choosing these parameters the dimensions of the observation area or detection area, Lix d , Liy d' will then be fixed. Note that the imaging lens has been described using two orthogonal focal lengths, allowing for cylindricallenses, and the aperture has been described using two orthogonal sizes, sometimes employed for image compression (Kawaguchi et al. 2002) Furthermore, the size of the particle image is not a function of particle size but of position of the particle within the laser sheet, as depicted schematically in Fig. 8.61. Thus, the change of image size between two consecutive pictures yields the velo city component normal to the laser sheet. 8.4.3.7 Governing Equations The most important limiting factor for the analysis of IPI or GPD images is the particle image overlap. An overlap coefficient Z is defined as the ratio of the area of particle image overlap to the overall area of particle images. The expectation of Z can be derived as a function of the parameters of the optical arrangement (Fig. 8.61) and the particle density. It is given as the ratio of the prob ability that a specific camera point Q is covered by more than one particle image to the probability that the point Q is covered by any particle image. The overlap coefficient is written as P(N i 2: 2) Z=P(N i 2:I) (8.96) where Ni is the number of particle images covering the point Q. The diameters of a particle image d ix and d iy depend on the particle position z p normal to the laser sheet. The functions are given by d. (z P )=d ax l-z r IX [_1 11+d. f __ zp ,xO x (8.97) 8.4 Multi-Dimensional Sizing Techniques d. (Z )=d lY P ay 1-z r [_1 1) +d. f __ Y zp 'yO 483 (8.98) Because of diffraction and blooming effects d;xo and d;yo can be assumed to be a minimum of 3 pixels, like in partiele image velocimetry (PIV). The aperture must not necessarily be circular or elliptical, also rectangular or any other shape could be used. For the present purposes however, the aperture size has been characterized simply using two orthogonal dimensions, dax and d ay . The values d;x and d;y are the corresponding dimensions of the partiele image depending on the partiele position z p. The image area is given by Aj(zp) = Sa d;x(zp)d;y(zp) (8.99) with the coefficient Sa which depends on the aperture shape. For an elliptical shape Sa = 1t /4 and for a rectangular shape Sa = 1 The transform of the partiele density Hp into the density of partiele images varies with z. For a given z position the density of partiele images is (8.100) The area density of partiele images 71; on the CCD camera can be obtained using (8.101) Z/,mm The number of partiele images covering a specific point Q on the camera is Poisson distributed (8.102) where the Poisson coefficient given by the integral ~ represents the mean imaging density and is (8.103) with ri; as in Eq. (8.100) and the area A; of a partiele image obtained from a partiele at the z position as given in Eq. (8.99). Combining Eqs. (8.97)-(8.100) and (8.103) yields ~~Oi,( ~[U -;},+ ;, d',"][(:, -;}, +:, d,," JZ!"~-Zl"'") 484 8 Phase Doppler Systems (8.104) for zr(!x-1-z-1)<1 and zr(!y-l-z-l)<l in the range [z/,m;n;z/,max]' Otherwise, an appropriate subdivision of the integral is necessary. The overlap coefficient can be expressed using Eq. (8.96) as P(N; X= P(N; ~ 2) ~ 1) l+~- ~ 1-exp(-~ (8.105) with the factor ~ given by Eq. (8.104). The allowable concentration rlp for a given degree of overlap X can be determined from Eqs. (8.104) and (8.105). Closely related to the overlap coefficient is the ratio of the illuminated to the non-illuminated area on the camera. P(N; ~ 1) X = P(N; =0) , exp(~-l (8.106) which can be readily estimated by simply counting the illuminated pixels. Once X' is known, ~ can be computed using (8.107) hence, X can also be readily estimated. In an implementation of the IPI or GPD techniques, these relations are useful for validation, since tolerable limits for X or X' can then be specified. The field of view is now fixed since the magnification z/ I zr is known (8.108) The Nyquist constraint can now be examined. The Nyquist limit will be reached first for the particles resulting in the smallest image, i.e. those in the laser sheet farthest from the focal plane. Their image size is given by d . . =d I,mm a [l-Z [~f __ 1 r zl,min II (8.109) The largest particles, with diameter d p,max' will result in the maximum number of fringes, N Jr,max' where the number of fringes is linearly related to the particle size through (8.110) The value of l( will depend on what scattering order is being exploited by the technique and on the position of the camera and is given for the IPI technique in Eqs. (8.91) and (8.92) and for the GPD technique in Eqs. (8.94) and (8.95). The Nyquist criterion specifies that one fringe must cover at least two pixels, thus 8.4 Multi-Dimensional Sizing Techniques 2Ax di,min nx N fr,max 485 - - r< - - (8.111) and yields the maximum measurable particle diameter d = _xn p,max 2K AX r [l-Z [~f __ 1 zl,min r II The lower particle sizing limit is given when at least (8.112) Nfr,min fringes, e.g. N fr,min = 1, are covered by the aperture N fr,min = Kdadp,m;n (8.113) Therefore, the smallest particle size is given by Nfr . d p,mm.=~ Kd (8.114) a With these parameters for the measurable size range (Eqs. (8.112) and (8.114», the measurable concentration (Eqs. (8.103) and (8.105» and the field of view (Eq. (8.108» some example systems will be analyzed in the next section. 8.4.3.2 Example Systems In practice, the size of the receiving unit, and therefore, the maximum size of the aperture da is given. For the following cases a circular aperture (Sa = 1(; /4) with a maximum diameter of da = 40mm will be assumed. The receiver, generally a camera, observes water droplets (m = 1.33) in the laser light sheet(s) (A. b = 488.0nm) with an observation angle of (lJ r =fftr =90deg), meaning that the camera focal plane is parallel to the laser light sheets. Under these conditions, the minimum measurable droplet size Eq. (8.114) can only be changed by changing the position of the light sheet relative to the aperture. This changes the collection angle for the camera and therefore the minimum number of fringes over the aperture. Using a smallest droplet size of dp,m;n = 311m and a minimum number of fringes of N fr,min = 1, this leads to a laser light sheet position of roughly ZI "" 200mm. Normally ZI is restricted by the experiment (stand-off distance) and therefore the stand-off distance and the aperture size limits the smallest measurable particle diameter. Generally, the distance between the back principal plane and the detector (CCD-Chip) is given by the camera design and the objective. Often an adjustment is possible in connection with aScheimpflug configuration. For the following cases, the position of the chip (Ax r = 8.58 mm, Ay r = 6.86 mm, nx = 1280, ny = 1024) behind the pack principal plane is assumed to be zr "" 100mm. Knowing already the position of the light sheet and the position of the detector, the magnification is fixed at a 486 8 Phase Doppler Systems value of ß = 0.51 (Eq. (3.116» and the size of the observation field is Llx d XLlYd =17mmx14mm (Eq. (8.108». This sm all observation field can be increased by increasing the distance of the laser light sheet from the front lens of the camera, but the smallest detectable partide size increases with this parameter also. The compromise is between smallest measurable partide size and field of view, as in focused imaging techniques. The difference is that the IPI and GPD techniques can resolve much smaller partides compared to direct imaging techniques (13 /lm/Pixel for this example). This is the main advantage of IPI and GPD techniques: observing a larger field of view with the same partide diameter limitations while achieving aresolution comparable to direct imaging techniques. The focallength must be chosen to be Ir = 66.66 mm to ensure that the focal plane for zr '" 100mm is at z/ '" 200mm. This can normally be adjusted using the lens of the camera. The position and thickness of the light sheet determines, together with the Nyquist criterion, the maximum measurable partide size and the resolution of position perpendicular to the light sheet in the z direction. A light sheet at z/ '" 198mm and a thickness of t/ = Imm limits the maximum partide size using the IPI technique to dp,max = 35 /lm (Eq. (8.112». In this case the partide is located at zp '" 198.5mm and produces the smallest defocused image of 22.5 pixels diameter with 11 fringes. By additionally using the GPD technique with an intersection angle between the two light sheets of 8= 15deg, the measurement range can be extended to dp,max = 228 /lm. The resolution in z direction perpendicular to the light sheet, given by the size ofthe defocused images, is 15.2 pixel/mm. In practice, the performance of each system is characterized by the variation of the overlap coefficient X with partide number concentration. The maximum allowable value of the overlap coefficient X without incurring processing errors is not yet dear, but will depend on the image processing software that is available and whether or not the positions of the partide images can be separately established, either with a second camera in focus or through software. A value of 0.1 to 0.3 should, however, be tolerable. The degree of image overlap is mainly determined by the laser light sheet position and the thickness of the laser light sheet. Increasing the distance of the laser light sheet from the focal plane increases the maximum measurable partide diameter; however, the overlap of images increases significantly. By using the GPD technique, this effect can be compensated by using another set of fringes and a shorter distance to the focal plane. Another possibility is to use optical compression. In this case the aperture is reduced in one direction to avoid large defocused images. The same image size reduction but with higher scattered intensity can be achieved by using a cylindricallens in the optical path as shown in Kobayashi et al.(2000). The influence oflaser light sheet position, laser light sheet thickness and optical compression on maximum measurable concentration will be illustrated using the 6 different systems specified in Table 8.6. Each of the systems have been evaluated using the above analysis and their performance has been expressed in 8.4 Multi-Dimensional Sizing Teehniques 487 Table 8.6. Input parameters and specifieations of the example optieal systems Symbols Input Parameters Units Aperture shape A B C D • • • • E I F • 66.66 66.66 66.66 90.0 66.6666.66 fx fy Foeallength (x direetion) mm Foeallength (y direetion) mm d ax Aperture diameter (x direetion) mm 40 20 10 40 40 40 d ay Aperture diameter (y direction) mm 40 20 10 40 3 40 Zjx Camera foeal plane (x direetion) mm 200 200 200 900 200 200 66.66 66.66 66.66 90.0 66.6666.44 Zjy Camera foeal plane (y direction) mm 200 200 200 900 200 198 Zz Laser sheet position 198 198 198 880 198 198 ß Magnifiea tion Llx d Observation field (x direction) mm LlYd Observation field (y direetion) mm dixtnin Smallest image size (x direetion) pixel 22.5 11.3 5.6 14.7 22.5 22.5 diymin Smallest image size (y direetion) pixel 22.5 11.3 5.6 14.7 =4 dix,max Largest image size (x direetion) pixel 37.7 18.9 9.44 15.5 37.7 37.7 diy.m ax Largest image size (y direetion) pixel 37.7 18.9 9.44 15.5 =6 z resolution (x direetion) e tz mm 0.51 0.51 0.51 0.11 0.51 0.51 17 17 17 76 17 17 14 14 14 60 14 14 pixel/mm 15.2 7.6 3.8 z resolution (y direetion) pixel/mm 15.2 7.6 3.8 GPD interseetion angle deg 40 15 20 =3 =8 0.77 15.2 15.2 0.77 1.14 =15 15 Conversion faetor IPI fringes/l1m 0.3230.1620.0810.0730.3230.323 Conversion faetor GPD fringes/l1m 0.0490.0330.0320.011 - Min. particle diameter by IPI Max. particle diameter by IP! 11m 11m 34.8 34.8 34.8 101 24.8 34.8 Min. particle diameter by GPD 11m 20.2 30.4 30.6 89.8 - Max. particle diameter by GPD 11m mm mm· 3 228 Laser light sheet thickness Max. eoneen tra tion (% = 10 %) 3.1 1 6.2 171 12.4 13.7 3.1 87 660 3.1 1 1 1.71 6.85 27.4 0.33417.9 13.6 terms of the value of the overlap coefficient as a function of particle concentration, as shown in Fig. 8.62. System A corresponds to the specifications given in the above discussion, using a da = 40 mm circular lens aperture. The maximum measurable concentration, assuming an allowable overlap coefficient of X = 0.1 and using a laser light thickness of 1 mm is Hp < 1. 7 particles I mm 3 • As a comparison, upper limits ofthe phase Doppler technique are considered to be about 102 particles/mm3 • The remaining example systems (B-F) illustrate various means of increasing the measurable concentration. 488 ~ 8 Phase Doppler Systems 100 ~ "E<lJ '0 10 r.;::: .....<lJ 0 u 0. '" -;::: <lJ > 0 ~ 0.1 100 ~ "E<lJ '0 10 r.;::: '"0"' <lJ u 0. '" -;::: <lJ > 0 ~ 0.1 100 ~ "E<lJ '0 10 r.;::: '"0"' <lJ u 0. '" -;::: <lJ > 0 10"' 10' 10 10' - 10"' 10"' 3 10 - 10' 3 Particle concentarion n p [partides x mm"] Particle concentarion np [particles x mm"] Laser sheet thickness t, [mm] =.-cf- 0.1 ~ 1 -- 2 --0-- 5 - 10 Fig. 8.62. Overlap coefficient as a function of number concentration and laser sheet thickness for example systems A to F from Table 8.6 Reducing the size of the imaging aperture. Example B reduces the imaging aperture from 40 mm to 20 mm and example C uses only a 10 mm aperture. The particle images become sm aller and the measurable concentration limits increase to 6.85 and 27.4 particles/mm3 respectively. Since the IPI fringe spacing remains constant, the minimum resolvable particle diameter increases from 3.1/J-m to 6.2/J-m and 12.4/J-m respectively. This increase in minimum measurable size, which also occurs for the GPD technique with reduced aperture size, can be compensated in the GPD technique by using larger intersection angles; however, then the maximum particle diameter is reduced. The GPD intersection angles for the example systems Band C have been chosen to provide a size range overlap between the IPI and the GPD systems. Reducing the Magnification. Example D reduces the magnification with respect to system A by using a larger focallength lens, resulting also in a larger standoff 8.4 Multi-Dimensional Sizing Techniques 489 distance 2/. The field of observation increases more than four times; however, the resolution of z position decreases significantly. Because of the larger standoff distance, less fringes are on the aperture and the smallest and the largest particle diameters increase in comparison to system A. While the image sizes are smaller than in system A, the maximum concentration decreases, since the number of particles on the camera increases with the larger observation field. Rectangular Aperture. Example system E employs a rectangular aperture with an aspect ratio of 40/3 to reduce image overlap. Otherwise the specifications correspond to system A. This greatly increases the maximum concentration limits to 17.9 particles/mm3 for a 1 mm laser light sheet. However, a parallel GPD cannot be implemented and thus, the overall size range is greatly reduced. Optical Compression. Example system F achieves an image compression by introducing astigmatism into the imaging optics. In this example the image size reduction in the y direction is about a factor 5... 7.5. This also increases the maximum concentration limit (l3.6 particles/mm3 ). Like for the rectangular aperture, the GPD technique cannot been used in combination with the IPI technique and therefore the maximum particle size is limited by the IPI technique to dp,max = 35 j.,lm. Reducing the thickness of the laser sheet. This influence is illustrated on all of the diagrams in Fig. 8.62. While a thinner laser sheet allows higher particle concentrations to be measured, the range of the third velocity component is reduced, since for a thinner sheet the particle will more probably exit its bounds between laser pulses. Increasing the allowable overlap coefficient. This measure has been briefly discussed above. If the processing software can tolerate higher overlap coefficients X, the maximum concentration limits can be increased considerably. To date there is little experience with appropriate image processing software. It appears that two principle tasks must be accomplished. The first is to find the position and size of each image and the second is to determine the fringe spacing or frequency. The image overlap renders the first task more difficult. However, there are several system concepts developed to yield the image position optically. For instance a second camera could be operated in focus to identify particle image centers. Under these circumstances, the allowable overlap may increase considerably. The above examples illustrate that proper optical design can significantly improve the performance of the IPI or GPD technique. Both areduction of the laser light sheet thickness and an increase of system magnification (smaller field of observation) increases the maximum allowable concentration. However, it is apparent that the IPI technique or the GPD technique on their own exhibit rather limited measurement size bandwidths. The only non-compromising method of increasing this is by choosing a CCD camera with a larger number of pixels, something which should become more practical in the future. The analysis in this section has considered only the parameters of the imaging optics in a very idealized manner. There are several additional influences 490 8 Phase Doppler Systems which may make it difficult to achieve the theoreticallimits presented above. For instance there are several causes far fringe distortion, which would immediately require so me relaxation of the Nyquist constraint. One possible cause is nonsphericity of the particle. Fringe distortion also occurs in the near and medium fields due to lens aberrations, as discussed by Hess (1998) and Girasole et al. (1998). Still another cause for fringe distortion is multiple scattering, either from within the laser light sheet or in the path between the light sheet and the receiver. This is similar to cases discussed for laser Doppler or phase Doppler systems. As in these systems, the degree of distortion due to multiple scattering is not generally sufficient to noticeably effect the measurement results, because the partides are too far apart. If fringe distortion does occur for any of the above reasons, the image compression technique will be particularly effected. The modulation depth of fringes along the longitudinal image axis will be directly reduced. A fuH image without compression would allow a more effective determination of the fringe number/frequency. The influence of optical compression is studied in more detail in Fig. 8.63. In Fig. 8.63a the influence of reducing one dimension of the aperture on the maximum allowable concentration is shown. Figure 8.63b shows the same dependence when the astigmatism of the receiving lens is varied. In both cases an increase of maximum allowable concentration of up to a factor 100 can be achieved. However, with an aperture dimension reduction, the light intensity on the detector decreases. From this point ofview optical compression using astigmatism would be preferable. Note that the relative gain in maximum allowable concentration reduces for thicker laser light sheets. b a <=I 0 "-§ I..; Ei 10' Ei ;:: '" '"<=I u"€'" u 0 u '" "d '-8 '"0.. '" 10' ~ " 11:: 10 Ei "Ei 'R :;:s'" 10" 10-' 0.01 0.1 1 Aperture ratio d,,,t d= [-J Laser sheet thickness t, [rnrn] = ----;- 0.1 66.4 66.6 66.8 Focallength in y direction ~,[rnrnJ -.....- 1 ------ 2 --- 5 - - 10 Fig. 8.63a,b. Dependence of maximum allowable particle concentration on image aspect ratio. a Aperture ratio, b Receiving lens astigmatism 492 9 Further Partide Sizing Methods Based on the Laser Doppler Technique amplitude occurs when the particle is at the center of the measurement volume and takes the value (Eqs. (3.188), (4.158), (3.30) and (3.62), Albrecht et al. 1993) I ACmax ( d ) - 8 q1J q AbPb (d) (d ) p ---h-d2 k 2 r p G p 1tC wb (9.1) b where Pb is the power of the laser beams. For a given detection threshold I d in the electronics and for a maximum of the scattered intensity from particles with diameter d p , I ACmax (d p ), the detection volume dimensions are given by Eq. (5.42) which, like other parameters such as the number of signal periods, will depend on particle size. The detection area of particles with diameter d p is (Eqs. (5.148) and (5(5.149) Ad (d p ) = ..!..1tcoboln( I Acmax(d p ) ] = A oFk- 1 (d p ) 2 (9.2) Id with (9.3) The contribution of particles with diameter d p to the total flux of particles (number of particles with diameter d p per unit time) is then (9.4) where npd(d p ) is the number concentration and vpx is the velocity of particles with diameter d p •1 The diameter dependent particle flux density (number of particles with diameter d p per unit time and area) is (9.5) This quantity, which is dependent on particle size, is assumed to be constant over the detection area A d (d p). Grouping all particles into 'mono-disperse' classes, d p,l' d p,2' ... ,dp,max, (Holve and Self 1979), the total particle flux is a summation over all the flux densities and their respective detection areas max Qp "" LqPd (dp)A d (d p) (9.6) j::::::l as shown in Fig. 9.1. 1 As in chapter 5 the subscript d indicates that the quantities are par tide diameter dependent because of the detection threshold I d and they could be different for each partide diameter dass 9.1 Techniques Based on Signal Amplitude q,.,(d ,) 493 = " ",, (d ,.,lv,. Cd , ) q... Cd ,.1) ="""Cd , .. )v" (cl,..> Fig. 9.1. Distribution of the signal rate (particle flux) as a function of signal amplitude, dp,l < dp,2 < dp,3, qpAdp,l) < qpAdp,3) < qpd(dp,2) Similarly, the maximum number of signal periods depends on the partide diameter and the particular dimensions of the measurement volume. 2a o _ No iJx~Fk(dp) - ~Fk(dp) (9.7) where No is the number of finges in measurement volume (Eq. (5.50)). Both the amplitude and the period number distributions can be used as the basis for a partide sizing method. In the next step the signal rate of a polydisperse partide flux will be derived for a given amplitude detection threshold. Each partide dass can contribute according to its specific detection volume area. 9.1.1.1 Statistics of the Number of Periods The detection area and the signal rate (partide flux for 100% detection and validation) are functions of the number of periods (Fig. 9.2) A d (d p ,N)=7rboco N!ax(d p )-N 2 (9.8) 2 No and (9.9) The distribution of the number of periods for a given signal rate (see Fig. 9.3) is given by _ dQpd(dp,N) _ _ N ( ) - 2A on pd (d p )v pJd p)-2 H Nd(dp,N) - d -N No (9.10) The maximum prob ability will be determined by the partide flux density vpx(dp)11pd(dp) and the maximum number of periods NmaxCdp) for each partide 494 9 Further Particle Sizing Methods Based on the Laser Doppler Technique x y z y- o I y = y. t I I 1 v, Parlidc Fig. 9.2. Distribution of periods over the detection area 11 M,(N) N Fig. 9.3. Statistics of the signal rate < vpX(dp,3)npAdp,') < vpx (dp,2 )npd(dp,2)) HNd(N), d p" < dp,2 < d p,3' vpx(dp,,)npd(dp,,) size. A change of the minimum required number of periods from N to N + 1 will result in a change of the partide flux density of L1QPd (d p ) = TC boc ovpx (d p)npd (d p ) 2N : 1 = v px (d p)npd (dp)M d (9.11) No where Md is not a function of partide diameter. Practically, a minimum number of periods is prescribed N m;n' According to Eq. (9.7), this is directly related to a minimum amplitude and therefore to a minimum partide diameter, dp,m;n' The signal rate using N = N m;n + 1 periods is influenced by all particles d p > d p,m;n by the amount (9.12) Through Eq. (9.5) there is a direct relation between the partide dass contribution to the partide flux density. The proportionality constant is the difference 9.1 Techniques Based on Signal Amplitude 495 of the cross-sectional areas. For the signal rate based on N m;n for instance, all particle dasses n make a contribution LlQNm;n = n;!02CO (2N m;n + 1)[ Llqpd(dp,l) + Llqpd(d p,2)+ ... + Llqpd(dp,n)] ° (9.13) n LlQNm;n = Md LLlqPd(dp) j~l In the range N m;n ::::; N; ::::; N max=2(N m;n + i) -1, the signal rate is given by the sum over all flux densities LlQN; with i = 1 ... ~ (N max + 1) - N m;n' (9.14) with LlQNl Llqpd (dp,l) LlQN2 Llqpd(dp,2) Qpd(d p)= LlQN3 qpd(d p)= Llqpd(dp,3) (2N m;n 0 AN= LlQm Llqpd(d p) + 1) (2N m;n + 1) (2N m;n + 1) (2N m;n +3) (2N m;n +3) (2N m;n +3) (9.15) (2N m;n +1) 0 0 (2 Nm;n +5) (2N m;n + 5) 0 0 0 2(N m;n +i)-1 (9.16) Each column in the AN matrix represents a partide size dependent detection area. The width of the matrix is given by the number of dasses. The example above demonstrates the highest possible dass resolution according to period number. An integer scaling factor k > 1 can be used to group dass es and expand the width of each dass qpd, reducing also the rank of the matrix. Inverting the matrix AN leads to an expression for the individual partide flux densities qpd (d p) = C;IA~QPd(dp) (9.17) The inverted matrix AN takes the form 1 A-N- (2N m;n +lt (2N m;n +3t 0 0 (2N m;n +3t (2N m;n +5t 0 0 (2N m;n +5t o o o o o o (9.18) 496 9 Further Particle Sizing Methods Based on the Laser Doppler Technique If the velo city of all partides is equal, then Eq. (9.14) yields also directly the partide size distribution (9.19) The individual steps necessary to apply the technique are summarized as follows: • Over a time period At the partide statistic AQ(N) is recorded • According to the maximum number of periods observed, and possibly applying a scaling (grouping) factor k, the size of the AN matrix is chosen • The link to partide size is achieved through the integral scattering function G(d p), thus the technique requires calibration with partides of known size, dp,c. For these partides, the maximum signal amplitude IAcmax(dp,J and/or the maximum number ofperiods Nmax(dp,J is determined. Furthermore, the scattering function must be unique in the investigated size range (monotonie increasing with size). Ifthis is not the case, a courser dass grouping may solve the problem. The integral scattering function in the Rayleigh range is given approximately by (9.20) where the constant A is given by the partide properties and the constant B could be between 6 and 10 (see Fig. 4.11). With Eqs. (9.1) and (9.7) a relation between the calibration partides and any given measurement value can be found r(dp)G(d p) (9.21) r(dpJG(dpJ The diameter follows from (9.22) It can be shown that the dynamic range of the technique with B = 2 ... 3 and N max / No =0 ... 1.9 is dp / dp,c = 1...10. 9.1.1.2 Statistics of the Signal Amplitude The processing using amplitude statistics turns out to be somewhat more flexible (Hintze 1993). The maximum signal amplitude IAcmax(dp) is determined and the distribution is then divided into i =1,2, .. .,n dasses I max ,;- Using Eq. (9.4), and similar to the Eqs. (9.13) and (9.17), a relation between signal rate and partidesize dependent partide flux density can be established. The proportionality factor is again the difference in cross-sectional area of the detection surface, i.e. the matrix of these differences (9.23) 9.1 Techniques Based on Signal Amplitude 497 with In I max ,2 Imax,l 0 InI max ,2 In I max,2 -Imax,l InI max,3 -I max ,2 AA= 0 0 InI max ,2 -Imax,l Imax,l InI max ,3 -- InI max ,3 -- I max ,2 I max ,2 InI max ,4 -- InI max ,4 -- I max ,3 0 0 (9.24) I max ,3 : : 0 InImax,n --- 0 lmax,n-l 1 -1 InI max ,2 -- InI max,3 -- Imax,l 0 A-Al 1 -1 InI max ,3 -- InI max ,4 -- I max ,2 0 0 I max ,2 - - 0 0 0 I max ,3 1 0 (9.25) In I max,4 I max ,3 : 0 0 0 1 0 In Imax,n Imax,n-l For equal velocities in each partide dass, the partide size distribution is available directly from the distribution of the signal rates. The principle of the technique is illustrated in Fig. 9.4. The distribution of signal amplitudes QA (IACmax (d p )) (3 rd quadrant) is available as a measurement quantity. Over a duration of Llt the distribution of burst amplitudes is accumulated. Through an inversion of the AA matrix, a distribution of partide size dependent maximum amplitudes is obtained, H(IAcmax(d p )) (2 nd quadrant). Since in the inverted matrix only the diagonal and the elements above are non-zero, only two dasses of burst amplitude contribute to this partide-size dependent, maximum signal amplitude. The distribution H(IAcmax(d p )) already corresponds to the partide size distribution (4th quadrant). The equivalence is established through the scattering function G(d p ) (Ist quadrant). In principle the scattering function can be computed, the incident laser intensity in the measurement volume can be measured and a given output voltage (amplitude) can be related to a particular partide size. However the amplification stages are not easily quantified and therefore a calibration with a known partide is necessary (Fig. 9.4). The division ofthe total amplitude range I d :5, I max :5, I ACmax into dasses usually follows one of two possibilities. The first is an equal division along the x axis 498 9 Further Particle Sizing Methods Based on the Laser Doppler Technique Maximum burst amplitude IACm,/d p ) G) Y(d p)G(d P.') Unknown distribution Range of A-matrix Measured particle size distribution particles Irn,,(d p ) Burst amplitude Fig. 9.4. Method of statistics of signal amplitude with dass width .dk x ' This corresponds to the situation when counting signal periods, but is somewhat more flexible since the divisions are not restricted to an integer number of periods. This leads to ~= exp(-2(.dk x i)2) (9.26) I ACmax The second possibility is a non-linear division, for instance a logarithmic spacing. ~ = exp( -2.dk)) (9.27) I ACmax For an equal division, the differences of cross-sectional area take the form LlAü = 1t boco.dk~(2i + 1) (9.28) and the A matrix with its inversion becomes 3 0 3 5 3 5 .1.. _.1.. 5 0 1 "5 -7 0 , A-1 = 0 0 .1.. 0 0 0 0 3 A= 0 0 7 7 0 0 0 2i max 5 3 +1 0 0 1 7 (2(i max +1)t (9.29) 9.1 Techniques Based on Signal Amplitude 499 The non-linear division leads to (9.30) M<LL = 11: bOcOL1k a and the matrix takes the form 1 -1 1 1 1 1 0 1 1 1 0 A= 0 0 1 1 A-'·= 0 0 0 0 1 0 0 0 1 -1 0 0 1 0 0 0 1 (9.31) Thus, two techniques are available for partide sizing in the micron and submicron range. An extension to the case of a distributed velocity function is possible in that the entire procedure is repeated for each velo city dass. In Fig. 9.5 a typical measurement result using this technique, taken from an aerosol generator, is shown. ~ . ~.. IOOO CI 200 500 Burst amplitude Ei Ei 600 r- I lm~ [a.u.] Maximum burst amplitude I IAcm" I [a.u] - 200 '- Fig. 9.5a-c. Particle size distribution in an aerosol. a Histogram of burst amplitudes, b Histogram of the particle size dependent maximum amplitudes, c Particle size distribution (8000 particles) 500 9 Further Particle Sizing Methods Based on the Laser Doppler Technique 9.1.2 Combined Laser Doppler and White Light Sizer Besides the necessity for calibration, one of the main difficulties with amplitudebased techniques is that over a large size range the scattering amplitude does not increase monotonically with size for a point detector, thus, the amplitude signal is ambiguous. The ambiguity can be reduced by using a finite sized receiving aperture, but not sufficiently. A second approach is to use a combination of several wavelengths, most easily realized with a white light source. The superposition of scattering functions of different wavelengths and for a given scattering angle and particle material may result in a more mono tonic change of scattered intensity with particle size. Combining such a device with a conventionallaser Doppler system yields a single particle counting device for both size and velocity. Such a device was first suggested by Durst and Umhauer (1975) and later realized and applied by Kleine et al. (1982) and Ruck and Pavlowski (1984). The optical arrangement is pictured in Fig. 9.6. The laser Doppler system was a conventional forward scatter arrangement. The white light source is a halogen lamp, focused using a microscope objective and imaged to a 100 fJlll spot in the measurement volume of the laser Doppler system. A photomultiplier positioned at an off-a~:is scattering angle of 90deg was fitted with a quadratic pinhole to truncate the measurement volume at 100 11m, thus effectively forming a cubic measurement volume. The focallength of the white light system was 15 mm. The 90deg scattering angle was chosen more out of mechanical convenience than due to optical considerations. After calibration with spherical particles, the instrument yields an optically equivalent particle diameter, through the amplitude of the white light detector. Transmi11 ing optics Pinhole Photo- Lens 1 Laser DOPllcr signal processing Fig. 9.6. Optical arrangement of combined laser Doppler and white light sizer 9.2 Time-Shift Technique 501 9.2 Time-Shift Technique The time-shift technique was introduced already in section 2.3 and equations giving quantitative expressions of the time shift in terms of optical, particle and flow parameters have been presented in sections 5.1.3 and 5.3.4. In the present section possibilities of using the time-shift technique for particle sizing will be explored, first for forward scatter arrangements and then for backscatter configurations. The time-shift technique is only possible with shaped beams and exploits the measurement volume displacement. This displacement is also used by other techniques, like the dual-burst technique (Onofri et al. 1996), which is discussed in section 8.2.4 and the pulse displacement technique (Pavlovski and Semidetnov 1991, Lin et al. 2000). Also Hess and Wood 1993 use the tempo rally separated signals of different scattering orders for particle characterization. The essence of all these techniques lies in the realization that with a shaped beam, each scattering order/mode exhibits its own virtual image of each incident laser beam for every detector. The virtual images are defined over the scattered intensity as a function of the particle center position for a specific receiver location. These images all have the same structure as the incident beams but are displaced in space (see Fig. 5.22). The magnitude of the displacement depends on the scattering order/mode, the receiving location, the relative refractive index, the particle diameter and the particle shape. Thus, if the different scattering orders/modes are identifiable in the received signals at specific detector angles and the relative refractive index is known, the diameter of a spherical particle can be estimated from the time shift between them. Using only one receiver, these technique are restricted to particles which are large compared to the beam diameter, typically for particles larger than one third the beam diameter. With two or more receivers much smaller particles can be measured, as shown below in section 9.2.1. Particle sizing using the time-shift between fractional parts of the signal necessarily requires a measurement of the particle speed. The time shift between scattering orders/modes is measured and this must be related to the volume displacement (section 5.1.3), hence to the particle size, through speed. Several authors (Pavlovski and Semidetnov 1991, Lin et al. 2000) achieved this using two laser beams in a time-of-flight fashion, which is called the pulse displacement technique. Alternatively two beams can be brought to intersection as in laser Doppler systems and the velo city can be measured from the signal modulation frequency. Hess and Wood (1993) and Onofri et al. (1996) used such a configuration with only one detector and measured parameters of large particles by using the time-shift between different scattering orders. The time-shift technique introduced by Albrecht et al. (1993) uses two detectors which are symmetrically placed about the optical axis z. This optical arrangement corresponds exactly to the phase Doppler arrangement as illustrated in Fig. 5.44. In keeping with the notation for standard phase Doppler systems, the angle lf/r is then known as the elevation angle and the off-axis angle is 9r' The particles traverse the beam intersection volume along the x axis and the 502 9 Further Particle Sizing Methods Based on the Laser Doppler Technique time-shift between signals from different receivers but from the same scattering order is used. This technique will be considered below in investigating various approaches for particle sizing in forward and in backscatter. Especially for the backscatter configuration several scattering orders contribute to the signal and the dual-burst technique can also be used. Before continuing further with specific examples of optical configurations for the time-shift technique some general expressions for the magnitude of the time shift will be given. A receiver in the y-z plane (Ij/ r = 0 deg), as shown Fig. 5.1, detects the signal maximum when the x co ordinate of particle position is equal to zero. For a receiver position with Ij/r 7= 0 deg, the signal maximum is shifted in time relative to the particle position at x Op = o. The magnitude of the time shift in reflection mode for one arbitrarily placed receiver is ~t (1) r dp [ coslj/rcostPrtan~-sinlj/r 4.J2 V x ~l- cos Ij/r costPr cos~ - sin Ij/r sin ~ =----- -r======================== 1 ~l-COSlj/rCOstPrcos~+sinlj/rsin~ coslj/rcostPrtan~+sinlj/r - dp sinlj/r 2.J2vx ~l-coslj/rcostPrcos~ for (9.32) Ij/r>1.1e, tPr>1.6e e<30deg,Error<lOO/O and for first-order refraction ~t(2) =~[ r 4.J2vx C, + sin Ij/r ~1+m2-m~2(1+Cc-C,)~l+Cc-Cs J C, -,inV", dp 2.J2vx m sin Ij/r ~l+m2-m~2(l+CJ ~l+Cc (9.33) for tane", e ' with (9.34) Using two detectors, like in the phase Doppler technique, the time-shift between the detectors for signals of the same scattering order is linearly related to the particle diameter by (9.35) The negative sign is for uniformity with the phase difference and has already been explained in section 5.3.4. The time conversion factor Fi N ) and the diame- 9.2 Time-Shift Technique 503 ter conversion factor for the time-shift technique ß~N) relate the measured timeshift to the partide diameter and are system parameter dependent. Comparing the approximations in Eqs. (9.32) and (9.33) with the approximations in Eqs. (8.2) and (8.3), it can be seen that for special angular regions the time shift is proportional to the phase shift 41t (j);N) "" -;-vx sin ~ t~Nl = OJ D t~Nl for ""'b tan8",,8 {rfJ»O forN=l d /\ r c (9.36) 'l'r < 60 eg tanrjJr > 1 lor N = 2 with a proportionality constant equal to the Doppler frequency OJ D= 21tfD of the signal. This conversion is also valid between the phase difference L1(j)\~1 and time-shift difference L1tlfl of the same optical configurations. However, this proportionality does not necessarily mean that the phase difference and the time shift are equivalent. As explained in section 5.1.3, the phase difference is mainly caused by separation of the glare points, the separation of the incident points and the path lengths inside the partide, while the time shift is only caused by separation of the incident points. Furthermore, Eq. (9.36) is only valid for the above given restricted angular regions and small intersection angles. As an example, the relative deviation while converting time-shift difference into phase difference by using Eq. (9.36) for a refractive index of m = 1.3, an intersection angle of 8= 30 deg and first-order refraction is shown in Fig. 9.7. Nevertheless, the angular and refractive index dependencies of the time conversion factor are dosely related to that of the phase conversion factor and the dorninance of one scattering order is the same for the time-shift and for the phase Doppler technique. Therefore, the angular dependencies and relations given in section 8.1 and 8.2.1 to 8.2.6 can be approximately applied for the timeshift technique. 1% 45 90 Iß 180 Off-3X is angle ; , [degl Fig. 9.7. Relative error by converting the time shift into the phase difference by using Eq. (9.35) (m=1.3, e=30deg , fIrst-order refraction). In the shaded region no geometrical optics solution for fIrst-order refraction is possible (z'J, > z'JJ 504 9 Further Partide Sizing Methods Based on the Laser Doppler Technique 9.2.1 Time-Shift Technique in Forward Scatter Typically in forward scatter the phase Doppler technique would be appropriate for particle sizing. The following examples are presented as an illustration of the time-shift technique and for comparison purposes to the phase Doppler technique. For particle motion in the x direction, Eq. (5.258) can be used directly to compute particle size from the time shift for either reflection (N = 1, Eq. (5.259» or first-order refraction (N=2, Eq. (5.259» dominated collected light. To attain aresolution comparable to the phase Doppler technique, higher elevation angles must be chosen, such that the time shift becomes some significant fraction of the total signal duration. This becomes more important for particles smaller than the diameter of the measurement volume. Note that no size ambiguity arises for larger particles and larger elevation angles, as is the case for phase Doppler with only two detectors. Two measurement examples will be presented in the next section to illustrate the above remarks. In accordance with the above mentioned requirements, the dimensions of the measurement volume are chosen small (d w = 33!lJll, Ab =488nm, B=13.7deg) and the elevation angles are large (±10deg::;ljI, ::; ±30deg). Figure 9.8a illustrates a measured pair of signals obtained from a polystyrol sphere (n = 1.59) of diameter d p = 65 /-lm ± 3 /-lm. From the time shift of the received signals a displacement of the measurement volume of one third of the beam waist can be estimated. Using the signal analysis according to sections 6.3 and 6.5.2.2, the frequency is found to be Iv = 580.3 kHz and the time shift to be L:ltg) = 11.44 /-ls. This corresponds to a particle velocity of b a ~ 0. \ - - Receiver \ -------. Receiver 2 ~ j: , ! ~. ~ :: _h::::::f:~. j "=n I' U, ! c.. t, E '" ~ ~ 0,0 : I I~ I' 11 I J :~ :: :: :: :: :': " ,I "'1 .. ' I I $ ' I I , I "f.nH::i::::" Vi 0, I Ow...w...I....L.J...J....J...J....J...J....J...J....'-:L.LL-LLJ'-:-'-'L.LJ'-::'SO Time Ill'sl o 20 40 60 80 Parlicl c diamclcr d. 100 Il'ml Fig. 9.8a,b. Polystyrol sphere in air measured with the time-shift technique (d p = 65 f..lm ± 31lJll, m = 1.59, Ab = 488 nm, e = 13.7 deg, ~,= 30 deg, ljI, = ±10 deg, d w = 33 f..lm ). a Signals from the two detectors, b Measured size distribution after passing the polystyrol sphere through the measurement volume 100 times (counts:100, dass width: 21lJll, dp = 65.2 f..lm, a dp = 004 f..lill) 9.2 Time-Shift Technique 505 V x = 1.18 m S-1 and a volume displacement of LixpACmax = v x L1tg) = 13.5 flm. Equation (9.33) then yields a size/volume displacement factor of 4.82 for first-order refraction, using the system parameters no ted in Fig. 9.8. The polystyrol sphere has a diameter of 65.1 flm according to this measurement. Like in the phase Doppler technique, these measurements are highly reproducible, as seen in Fig. 9.8b, which shows the measured size distribution after 100 repetitions of the particle passing through the measurement volume. An overall mean diameter of 65.2 flm with a standard deviation of 0.41 flm was measured. A second example involves much smaller particles and a comparison to sizes measured using the phase Doppler technique. The individual distribution functions for particle velocity and particle diameter are shown in Fig. 9.9. These a c .s: b 1 1 :; ~ ·C ;; 'Ö ""0 ~ tl '" ~ "" I 1 o 2 Parliclc vclocil)' v, 0 Im s 'I 10 20 30 Partide diameIer d, 1111111 c " .... ... ,' ••• ~~ ~ • • ~ "'.-" h'~" .' .' .' I o I 10 20 ~ ,.. 3.0 ........ ., .'. ' " .".'.', ' .... " " ,~ ."~" .... o jO • •' 1.5 Partide diameter • d , ll1m] Fig. 9.9a-c. Velocity and size distribution functions of drops generated using an ultrasonic nebulizer and measured using the time-shift technique (m = 1.33, Ab = 488nm, 19=13.7deg, d w =33J.Lm, ~,=30deg, V',=±30deg, counts: 5000). aVelocity distribution function (dass width: 0.06 m S-1, lIx = 2.40 m s-', (]' vx = 0.04 m S-1 ), b Partide diameter distribution function (dass width: 0.6 J.Lm, dp = 6.24 J.Lm, (]' dp = 2.21 J.Lm), c Joint velocity/diameter probability density function 506 9 Further Particle Sizing Methods Based on the Laser Doppler Technique measurements deviate in the mean average by less than 1% from the phase Doppler measurements. The droplets were generated using an ultrasonic nebulizer and were significantly smaller than the measurement volume size. In comparisonto the dual-burst technique (section 8.2.4), the time-shift technique can also measure particles much smaller than the beam diameter. While the dual-burst technique is based on the time shift between two different scattering orders in the same signal and the scattering orders cannot be separated for small particles, the time-shift technique separates the time-shifted signals with the two detectors. Therefore, this procedure requires only one dominant scattering order, as with the phase Doppler technique. 9.2.2 Time-Shift Technique in Backscatter For various reasons the phase Doppler technique is not workable in the far backscatter region. This is most easily seen from Figs. 8.3 and 8.4. Some attempts at exploiting scattered light in the second-order refractive modes were presented by Bultynck (1998) and Bultynck et al. (1996); however, the absolute scattering intensity is low and mixing of scattering orders leads to large sizing errors. Moreover, their instrument design was suitable only for a very restricted range of relative refractive index. On the other hand, the time-shift technique offers several possibilities for realizing a particle-sizing instrument in backscatter. In this section some of the fundamental signal dependencies and also possible instrument configurations will be presented. The situation for the backscatter range is illustrated in Fig. 9.10 in which a one-dimensional Gaussian intensity distribution is shown for a single incident beam. A particle moving through the beam in the scattering plane will result, like in the dual-burst technique (section 8.2.4), in various fractional signals arriving sequentially at the detector, hence the name time-shift technique. The main components in order of occurrence for m> 1 will be: surface wave or edge ray (long path), reflection, second-order refraction (inner path), second-order Ineidenl po inls uler palh (N =3.2) ___ _ In"idcnl shapcd bcam Parlid" posilion xo, Fig. 9.10. Scattering orders/modes contributing to the signal in the near backscatter region for m> 1 9.2 Time-Shift Technique 507 refraction (outer path), surface wave or edge ray (short path). Note that there exist two modes for second-order refraction (N = 3), creating the rainbow (section 4.1.3.3). These have been designated N = 3.1 (inner path) and N = 3.2 (outer path). The relative amplitude between each of the fractional signals will depend on the specific scattering order/mode, and the absolute amplitude scales with the incident power and particle size. The width and shape of each fractional signal is given by the width and shape of the incident beam. Basically, the incident beam is being sampled by the incident points of each scattering order/mode on the surface of the particle and it is being imaged through the respective glare points onto the detector. The separation of the fractional signals in time will be determined by the particle size, the relative refractive index, the particle shape and the scattering angles to the receiver. Overlapping of fractional signals from different scattering orders/mo des in one signal is reduced by keeping the ratio of the particle diameter to the incident beam width large. For practical applications this means a highly focused beam should be used, insuring good separation of the fractional signals even for small particles. The centers of the virtual images of the measurement volumes for each scattering order/mode lie approximately on the line connecting the intersection point of the laser beams and the detector direction (see Fig. 5.22). All fractional signals will only be seen if the particle velo city vector intersects all detection volumes. Therefore, the measurement volumes and the receivers have to be 10cated in the x-z plane if the main tlow is in x direction. This optical arrangement corresponds exactly to the planar backscatter phase Doppler arrangement, in which the detectors lie in the same plane as the incident beams and as illustrated in Fig. 9.11. 9.2.2.1 Signal Characteristics A typical signal received at a single detector of the system shown in Fig. 9.11 is y Fig. 9.11. Optical arrangement for a time-shift system in backscatter 508 9 Further Particle Sizing Methods Based on the Laser Doppler Technique b a FLi\IT simu lation Experimenl SWLP SWI.P 50 8 Parlide positi n X, lfll11) c Time IIJ.lSI FLi\ IT \~ilh Dcb)'c scrics dccomposilon for rencction with Debye scries N- I O~LU~~~~~W=~ so o 50 Part icle posit ion x, 1J.lm I o 50 Pa rlide position x, lJ.lml Fig. 9.12a-d. Signal received at the photodetector of a planar phase Doppler system in backscatter (d p =80/-Lm, m=1.333, 19=7.4deg, lf/,=25deg, ~,=180deg, Ab = 514.5 nm, fsh = 40 MHz, d w = 20 /-Lm, SWSP: surface wave short path, SWLP: surface wave long path). a Calculated with FLMT, b Measured, c Reflection from Debye series decomposition, d Second-order refraction from Debye se ries decomposition illustrated in Fig. 9.12, computed using FLMT and the Debye decomposition of the FLMT result and measured in the laboratory using a transient recorder to record the signal. For small intersection angles the AC and DC parts of the signal coincide (section 5.1.3.1) and the signal consists of the same four distinct fractional signals corresponding to the scattering orders/modes shown in Fig. 9.12. Note that the short path surface wave and the refraction mode N = 3.2 overlap almost completely and cannot be individually distinguished. The dependencies of fractional signal separation are illustrated in Fig. 9.13 using simulations for three different scattering angles in the backscatter range. These simulations have been computed using FLMT. The separation of the reflected signal fraction from the refractive fraction N = 3.1 decreases with decreasing elevation angle. Furthermore, the second-order refraction (N = 3.2) increases in amplitude for larger elevation angles. Below about If/ r = 14 deg (m = 1.33), only the N = 3.1 mode contributes to second-order refraction. As expected, the measurement volume displacement and therefore the time shift for reflection and refraction reduces when the receiver is doser to the y-z plane. However, in the backscatter configuration the surface waves dearly delineate the partide borders. The best fractional signal separation is found for If/ r = 20 deg. 9.2 Time-Shift Technique VI, 509 10 deg SWLP VI, = 20 dcg SWLP '1', 30 dcg SWLP 6 8 10 Time I h.ls) Fig. 9.13. Influence of scattering angle on fractional signal separation for three different elevation angles (dp=80~m, 9,=180deg, e=4deg, Ab=s14.5nm, J,I,=40MHz, d w =20 ~m) For a particle trajectory in the x direction, and if the relative refractive index of the particle is known, the particle diameter can be measured by measuring the time shift between selected fractional signals, as in the dual-burst technique (section 8.2.4). This time shift is transformed into a volume displacement using the particle velo city in the x direction, found from the Doppler modulation frequency. For smaller particles the fractional signals overlap increasingly and the estimation of the time shift becomes virtually impossible. Therefore, an alternative approach is to add a second detector, as shown already in Fig. 9.12. Another two sets oflaser beam images - one for each incident beam will now be created. In total two laser beam images for every scattering order and every receiver will exist. Pairs of these images create the measurement volumes. Assuming the intersection angle to be small, all measurement volum es lie along the x axis. The situation has been exemplary pictured in Fig. 9.14, in which the dominating three measurement volumes for each detector lie along the x axis. Note that the spacing between the volumes depends on particle size, refractive index ete. and that this pictorial is only an example. Furthermore, the signal intensity from each volume will vary according to scattering order. This means that while the measurement volumes for all the scattering orders/mo des are the same size, the detection volumes will vary according to intensity of the scattered light and the detection criterion used by the processing electronics. An important feature is that the volumes appear in reverse sequence for each detector because of the symmetric placement of the de- e 510 9 Further Particle Sizing Methods Based on the Laser Doppler Technique y r.leasuremen~ ~ (rece iver J) Receiver I (recei ver 2) Fig. 9.14. Planar optical configuration with separated measurement volumes Rcceiv r J; 1/1, -25 deg " "'C ::l :aE '" Jf'" ~~~~~~~~~~~~~~~~~~~~~~c=r=~~=c~~ Recc i ver 2: o lI'l 25 deg Time I [flsJ Fig. 9.15. Simulated signals from two receivers of a plan ar backscatter configuration (e= 4 deg, A. b = 514.5 nm, J,,, = 40 MHz, d p = 100 J.1m, d w =20 J.1m, (Pr = 180 deg) tectors about the incident beams, as illustrated by the signals in Fig. 9.15. This means that the time shift between the two signals from the two receivers can also be measured for smaller particles, because the shifted signals on the two detectors are now better separated. The influence of the refractive index on the time-shift signals is illustrated in Fig.9.16, showing a simulated detector signal for the values m = 1.25 and m =1.42. As the refractive index increases, the position of the N = 3.1 fractional signal exhibits a monotonie but non-linear increase of time shift, whereas the 9.2 Time-Shift Technique a 511 b .2 " -0 :::> Parliclc border Gcomclricaloplics C. E a '" -.; g c ~ r++1~~+1~t+~~+1~~ Symbols: FLJ\lT N= I o N :\.I .. o Debyc SWSP .nd (N 10 Timc I [Ilsl I 3.2) 1.40 Relative rcfracl ive index /11 [ - ) Fig. 9.16. Change of the signal structure and time shifts due to refractive index changes (dp=80l-lm, 4J, =180deg, lJI,=20deg, Ab =514.5nm, dw=20l-lm, ! ' h=40MHz) reflective fractional signal remains unaffected. The time difference between reflection and N = 3.1 can be used for determining refractive index as already discussed in section 8.2.4.1. A diameter change of 5 .. .6 % corresponds to a 0.025 change in m if the surface waves are not considered. Both the amplitude and position of the p = 3.2 fractional signal change with the refractive index, also influencing the short path surface wave signal due to the strong overlap. Note that for larger particle diameters, all fractional signal dependencies correspond to values predicted by geometrical optics. These dependencies are shown explicitly in Fig. 9.16b calculated with geometrical optics and Debye decomposition of FLMT results. The surface waves are assumed to be on the circurnference of the particle. 9.2.2.2 Particle Sizing Using the Time-Shift Technique The possibility of using the time-shift technique for particle sizing in backscatter will now be investigated quantitatively for the various scattering orders involved. Size information can be extracted by examining the time shift between signals oflike scattering order/mode at different detectors and, using the particle velocity, converting this to a volume displacement. The particle diameter can be measured by the time shift of the signal maxima. For trajectories parallel to the x axis, the time shift is a direct measure for the measurement volume displacement. For oblique trajectories, the time shift leads to a systematic error of the volume displacement, as given in section 5.1.3.2 when only the x component of the velo city is used. General expressions for the time shift are given in Eq. (5.113) for the DC parts and in Eqs. (5.114) to (5.116) for the AC parts of the signals. In the region of the beam intersection, the DC pulses are additive and only one DC amplitude maximum occurs for each scattering order (Fig. 5.20). A 512 9 Further Particle Sizing Methods Based on the Laser Doppler Technique closed solution for arbitrary interseetion angles is not possible, but for small intersection angles the particle position for maximum amplitude position of the DC part is equal to the maximum amplitude position of the AC part as mentioned in seetion 5.1.3.1. For small intersection angles (tan~=~, ~«1 and cos~=1), and therefore small incident point distances (:x r «Xr> Zr «z" see Eqs. (5.101) and (5.102)), receiver locations far from the direct backscatter (lfIr» 0 deg, zrmz %« xr) and a planar configuration (ifJr = 180 deg), the signal maxima for the DC and AC parts occur at (9.37) where the particle trajectory is given by Eq. (5.61) and m y , mz> y po' zpo. By using two detectors, the time shift between the signals becomes independent of the particle trajectory intersection with the plane x = 0 and independent of the z component of the particle trajectory. A (N) (N) (N) At(N) _~_ X max ,1 -X max ,2 12 - 1 - Vx Vx Vx x: +xi N) l+m y N) (9.38) and depends linearly on particle diameter (X~N) - d p , Eqs. (4.79) and (5.101). The y component must be measured for further corrections with e.g. a twovelocity component laser Doppler system. For symmetrie receiver locations (lfIl = -lfI2) the volume displacement or time shift between the signal maxima for reflection (N = 1) and first-order refraction (N = 2) is given as a closed solution for the AC part in Eqs. (9.32) and (9.33). For the planar configuration (ifJ r = 180 deg) and for the small intersection angles (sin % = 0, cos % = 1) considered here, the incident points of the two beams coincide and the time shift for reflection (Eqs. (5.255) and (9.32» simplifies to (9.39) The respective simplified scattering geometries are pictured in Fig. 9.17, in which Bi denotes the angle of incidence and B, is an angle of refraction for the respective scattering order N. A normalized displacement or normalized incident point position independent of particle size can be defined by normalizing Eq. (9.39) with the particle radius rp = d p /2 (9.40) 9.2 Time-Shift Technique a 513 b x Z X 1" .\2l x !"U) Fig. 9.17. Definition of normalized incident points. a Reflection, b Second-order refraction where O\N) and 0~N) are the respective relative displacements for detectors 1 and 2 and scattering order N. The border of the particle in the x direction then corresponds with the coordinates 1 and -1. For reflection, this relation between normalized volume displacement and receiver location becomes Oll) r = sin ljI (9.41) 2 and is pictured in Fig. 9.18. For second-order refraction the incident point shift will depend additionally on mode (N = 3.1 or N = 3.2) and on the relative refractive index m. The angular -;- -~_ 1 ; , == 180 deg -90 o 90 180 Eleva tion angle lfI; Idegl Fig. 9.18. Normalized displacement of the measurement volume on the x axis for a planar backscatter configuration in reflection mode and small interseetion angle 514 9 Further Particle Sizing Methods Based on the Laser Doppler Technique relationship between angle of incident and scattering angle (for the plan ar configuration and a small intersection angle the elevation angle can be approximated by the scattering angle, lfI r = 1.9,) is given in Eq. (4.20) in section 4.1.1.2 for N=3 as 1.9, = 1t + 2( 8; - 28,) = 1t + 28i - 4 arCSin( Si:8; ) (9.42) For a given scattering angle Eq. (9.42) must be solved for Bi iteratively. Solutions are given in Fig. 9.19 for m = 1.33 and m = 1.5 in terms of normalized incident point position as a function of the scattering angle. Note that the N = 3.2 mode appears only for elevation angles IlfIrl>lSdeg for the refractive index m = 1.33 and that the incident point remains near the periphery of the particle. For larger relative refractive indexes (m > 1.4) the situation becomes more complex and the number of fractional signals may even reach the number of the mode N. For instance in Fig. 9.19b a third mode, N = 3.3 can be identified. Physica1ly, this mode corresponds to the ray designated N = 3.3 in the ray paths in Fig. 9.19b. Expressions for the relative displacement of high er scattering orders become somewhat more complex but can always be solved numerically using an iteration (see section 4.1.1.2). The particle diameter is found by measuring the velocity v x and the time shift of the considered scattering order/mode and solving Eq. (9.40) for d p and for particles moving parallel to the x axis. For other trajectories with vy "* 0, corrections of the time shift must been made by measuring the vy velo city component and using Eq. (9.38). For particles moving near the x axis and for small intersection angles, the AC a b AI--==--------'=lO: U1 c: " E "u ..-----------:10: '" C. .~ 1) ~ 0 " .~ -;; E o z 10 20 0 20 40 Eleval ion angle I{/, [d eg[ 40 20 0 20 '10 Elcvali on angle I{/, [degl Fig. 9.19a,b. Normalized volume displacement for second -order refraction (tjJ, = 180 deg, A. b = 514.5 nm). a m = 1.33, b m = 1.5 and corresponding ray pa ths for 1jI, = 10 deg 9.2 Time-Shift Technique 515 and DC parts of the signal will be coineident and the time shift can be measured between the signal maxima of the non-filtered bursts. For trajectories suffieiently displaced along the z axis, or for larger intersection angles, the DC part of each detector signal may exhibit two maxima corresponding to each of the in eident beams (Fig. 5.20 and Eqs. (5.97) and (5.98». The required time shift should be the time between DC maxima from the same beam glare point on each detector. This is somewhat impractical because the DC part is more susceptible to narrow band noise sources. A more robust measurement can be achieved by using the time shift between AC maxima of the two detectors signals (of like scattering order/mode). The accuracy and resolution of the time-shift technique will be in part dependent on the expectation and variance of the estimator used to find the AC part maxima. Note that the estimation procedure must also identify and separate the fractional signals before the estimation of the maxima is performed. Possibilities for estimating the time shift have been discussed in section 6.5.2.2 and a more detailed discussion can be found in Damaschke et al. (2002). 9.2.2.3 Time Shift / Partic/e Diameter Relation The response of a particle sizing system using the time-shift technique and the optical configuration shown in Fig. 9.14 can be investigated with the help of signals generated using the FLMT and signal processing for identifying and isolating the time shifts of the different scattering orders. The results are shown as solid symbols in Fig. 9.20 for the different fractional signals: surface wave (short path) and second-order refraction (N = 3.2) (Fig. 9.20a), second-order refraction (N=3.1) (Fig. 9.20b) and reflection (Fig. 9.20c). The simulations were performed for the detector elevation angles IJI r = ±20 deg and for a measurement volume diameter of 2rw = 2a o = 20 11m. a b c o Geometrical optics prediction Signal processing of complete signal Signal processing of Debye orders .......--L.....L.-L...... I...-L....L.....L.....I--'-SO.....L.-L....L.....1...1-'O""'O.L.-J ~ SO-'--L.....1...-.L1 ol...o....L.....J o I , , , ~o' , , '1 ~o' I Expected particle diameter dp [[lm] Fig. 9.20a-c. Particle size estimates from simulated signals using various fractional signals and a 20 llJI1 measurement volume diameter (1fI, = ±20 deg, e= 4 deg, Ab = 514.5 nm, m = 1.33). a Surface wave short path (SWSP) and second-order refraction (N = 3.2), b Second -order refraction (N = 3.1 ), c Reflection 516 9 Further Particle Sizing Methods Based on the Laser Doppler Technique A second result is shown in Fig. 9.20 by the solid line. These results were obtained using geometrical optics to compute the position of the incident points of each order and showaperfect linear relation. The results for the full signals follow closely the linear curves for isolated orders, with small deviations at small particle sizes. Finally, Fig. 9.20 includes results shown by the open symbols. In this case the signal was simulated with the FLMT using only the respective scattering order. This is possible using the Debye series decomposition in the FLMT computations, as described in seetions 4.1.2.2 and 4.2.2.3. In this simulation, no signal overlapping occurs for reflected light, by definition, and the result is a perfect linear relation between size and time shift. Surface waves and the different modes in a single scattering order cannot be calculated separately using the Debye decomposition and therefore they appear altogether in the calculations for second-order refraction. The deviation of the Debye decomposition from the linear relation of geometrical optics in Fig. 9.20b is due to the mixing inside the second-order refraction between surface waves and different second-order modes. For particle diameters sm aller than 40/lm no separate maximum for N = 3.1 can be identified with the signal processing. For the same diameter range, the scatter for N = 3.2 increases because the interference between all scattering orders results in higher uncertainty of the dominant maximum in the bursts. The calculations confirm that the scatter in the full-signal results originates from order/mode mixing. Nevertheless, so me systematic trends resulting in a non-linear but monotonie diameter/time-shift dependency can be predicted and considered in converting the time shift to particle diameter (Fig. 9.20b). As shown in Eq. (9.40), the time shift is a good approximation for size, independent of the diameter of the incident beam. The best results are achieved for the surface wave short path and refraction (N = 3.2) and the accuracy increases for larger particles. For larger beam diameters or smaller particles, the signals are broader relative to the signal duration and only astronger mixing of orders occurs. For particle sizes near the focal size of the laser beams or the measurement volume, the scatter in the dominant order (N = 3.2) increases, and for higher scattering orders with smaller signal amplitudes (N =3.1 and N =1) the particle diameter limit is already reached, because the maxima can no longer be identified. For even smaller particles, the scatter in the dominant mode/order reduces because the amplitudes of non-dominant orders decrease and the maxima of all scattering orders are closer together. This is the same as for normal phase Doppler configurations in forward scatter, when using first-order refraction. For such a configuration, the time shift of the dominant order is not disturbed by signals from other scattering orders and the technique is limited by the accuracy of determination of the maximum signal amplitude. This is mainly determined by the signal-to-noise ratio of the signal. The time-shift technique works for particle sizes down to 1/10 of the beam diameter quite weIl if only one scattering order is used (section 9.2.1). 9.3 RainbowRefractometry 517 9.3 Rainbow Refractometry Rainbow refractometry refers to a measurement technique used to determine the real part of the refractive index of spherical partieies through analysis of the primary rainbow scattering. This is of practical interest for the case of liquid droplets, for which a unique relationship between temperature and refractive index can be established. Thus, the rainbow refractometer is a nonspectroscopic technique for measuring drop temperature. To date there have been two instrument implementations realized for practical measurements in sprays. One integrates the rainbow refractometer/thermometer into an existing phase Doppler system, thus providing also size and velocity data of each droplet (Sankar et al. 1993). The second concept has been termed a single-beam velocimeter based on rainbow-interferometry and is a particle counting technique, providing also size and velo city information (van Beeck and Riethmuller 1996a, 1996b). Commercial distribution ofthe technique has been very limited, no doubt due both to its complexity and its sensitivity to the non-sphericity of the droplets and to refractive index inhomgeneities within the droplet (Massoli 1998). The rainbow phenomenon has been discussed in sections 4.1.1.3 and 4.1.3.3, both in terms of geometrical optics and using the Lorenz-Mie theory. The scattering intensity displays a maximum intensity and interference pattern in the angular region of the rainbow and this is physieally related to the fact that many rays entering the particle at different incident points exit with approximately the same angle. Depending on the number of internal reflections, one speaks of primary (p = 2) secondary (p = 3) or even higher order rainbows. Nussensveig (1979) has given an excellent description of rainbows and their characteristics. Rainbow refractometry utilizes the primary rainbow and its detailed structure, as computed using LMT and shown in Fig. 9.21 for a water droplet of diameter 300 JlIll. There is a main maximum peak at the scattering angle 131' followed by supernumerary bows generated by the second-order refraction. Superimposed on this structure are higher frequency ripples. The main maximum and the supernumerary bows together are referred to as the Airy rainbow pattern, although the Airy theory (Airy 1838) is not exact and does not account for the ripple structure. Van Beeck and Riethmuller (l996b) give a very illustrative interpretation of the rainbow scattering pattern based on geometrieal optics and the spectrum of the scattering intensity oscillations over scattering angle. Their illustrations are reproduced in Fig. 9.22. Each of the peaks in the spectrum corresponds to interference between two geometrie rays, whereby the respective glare point separations, (6\ to 6 s ) can be interpreted in the sense of Young's classieal double slit experiment, yielding the spatial frequency (see also section 8.4) /; = ~j Ab 1t 180deg -d (9.43) P 1\: Rays arising from internal reflection (p = 2) at aseparation distance 6\ leading to the Airy fringe pattern and 518 9 Further Particle Sizing Methods Based on the Laser Doppler Technique Lorcnz-M ic solulion ccond-o rdcr rcfra":lion (p - 3) ] .. ~ ~ Inßc":lion poin l 140 135 Scal lcri ng angle Fig. 9.21. The rainbow according to LMT and second-order refraction from Debye-series decompostion for a 300 llID water droplet (Ab = 488 nm, m = 1.333, XM = 1931.3, perpendicular polarization) ....,. 8 rr----.--,-..,--,---,-----,--,-- ..'" 15, Lorcnz-Mic calcu lalion Second-order rcfraCl io n (p - 3) Inlcrfcrc n..:c ofrcncCl ion (p - 1) and sccond-o rdcr rcfracl io n (p -. ) 4 15, 0 ~~~~~~=I~~~~~~~==~~ o 5 10 15 Angula r frcqucn..:y of lobcs IR' Idcg 'I Fig. 9.22. Rays contributing to the primary rainbow and the rainbow spectrum genera ted by the interference of rays from reflection, second-order refraction and surface waves (300f,.lm, A b =488nm, m=1.333, angular region used for calculating the spectrum: tJ, = l39...144.1 deg) 9.3 Rainbow Refractometry 519 f2: Inner partial ray from second-order refraction and ray from reflection with separation 8 2 f3: Outer partial ray from second-order refraction and ray from reflection with separation 8 3 The relation fl = f3 - f2 holds and the latter two contributions lead to the ripple structure. Further contributions from the edge rays (f4' 8 4 and f5' 8 5 ) are very weak and of no significance for practical rainbow measurement systems. To realize a measurement instrument it is necessary to relate these measurable spatial frequencies and the rainbow position to the size and refractive index of the particle. An exact computation of these dependencies using Lorenz-Mie theory and Lorenz-Mie theory with Debye series decomposition is shown graphically in Figs. 4.26 and 4.28 in section 4.1.3.3, where their origin and computation are discussed in detail. The refractive index deterrnines mainly the angular position of the rainbow, as can be seen in Figs. 4.28 and 4.33. The particle diameter influences the angular frequency of the intensity maxima, as already expected from Eq. (9.43) . The Airy theory (section 4.1.1.3) predicts a size and refractive index dependence of both the frequency and the position of the main maximum. For the determination of the refractive index, the position of the rainbow should not change with particle diameter. This is not the case for the first maximum of the rainbow as can be seen in Fig. 4.26 computed for a constant refractive index. Recently, Roth et al. (1996) have shown that the position of the inflection point of the main rainbow peak, indicated in Fig. 9.21, is virtually size independent. In fact, this position corresponds closely to the position of the rainbow established by Rene Descartes (1637). This inflection point position has recently been used in a measurement instrument to determine refractive index (V an Beeck et al. 2000). Another approach to measure refractive index is to combine a measurement of the rainbow position (13 1 ) and rainbow frequency (fl) with a conventional phase Doppler instrument, which yields independently the droplet size. Using this strategy of size determination with a phase Doppler instrument and refractive index determination from the rainbow pattern, Sankar et al. (1993) have realized the measurement system shown schematically in Fig. 9.23. The rainbow receiver images the rainbow pattern through a cylindricallens onto a high-speed, high-resolution CCD array. One photomultiplier monitors the received light for gating purposes and synchronization with the phase Doppler acquisition. A second photomultiplier monitors signal intensity to control the CCD. The rainbow angle is quite insensitive to the position of the particle in a Gaussian beam. Nevertheless, in the instrument of Sankar et al. (1993), the beam for the rainbow refractometer was made larger than the phase Doppler beams in the measurement volume by a factor of about 2, since the requirement of total particle illumination was found to be more stringent in the case of rainbow refractometry than with the phase Doppler sizing. A calibration of the instrument was used to establish linearity of the CCD pixel values with the scattering angle and linearity of the rainbow angle with refractive index. An evaluation of the 520 9 Further Particle Sizing Methods Based on the Laser Doppler Technique Fig. 9.23. Schematic of an integrated phase Doppler/rainbow refractometry instrument (adapted from Sankar et al. 1993) achievable accuracy of this instrument has been given in Heukelbach et al. (1998), Heukelbach (1998), Damaschke et al. (1998) and Horn (2000). They show also a predictable sensitivity of the measured refractive index to the trajectory of the particle in the z direction by using monodisperse droplets. This corresponds to the simple geometric change of receiver angle over the length of the measurement volume and amounts to a variation in the refractive index m of approximately 0.1 %. At the far extremes of the measurement volume, the particle trajectories lead to larger measurement errors, associated with the effect of the receiver slit aperture on the collected scattered light. However, the presence of the ripple structure leads to uncertainties in locating the angle of the main rainbow peak, especially for smaller particles. For instance, for a 10 11m water droplet, the ripple structure amplitude reaches 30% of the main peak. One solution, proposed by Roth et al. (1991), is to applyan appropriate low-pass filter and a second approach, adopted by Sankar et al. (1993) for a commercial instrument, is to curve fit the primary fringe of the rainbow. Nevertheless, the accuracy of the refractive index measurement of small spherical particles is mainly limited by the uncertainty in peak detection, caused by the ripple structure (Heukelbach 1998). Already in 1980, Marsten recognized that the angular dependence of the rainbow was quite sensitive to the shape of the drop. Practical applications were seen in detecting micron amplitude changes in the shape of millimeter diameter drops. In the same vein, non-sphericity quickly leads to size and especially refractive index errors using the rainbow technique. Van Beeck and Riethmuller (1995) propose a sphericity validation based on a comparison of the diameter found by the supernumerary bows with the diameter found from the ripple frequency. In principle this should be quite effective, since the optical interference patterns employed originate from different local curvatures of the droplet sur- 9.3 Rainbow Refractometry 521 face. However van Beeck (1997) demonstrates that this condition is not sufficient. More recently van Beeck et al. (2000) have proposed the global rainbow thermometry (GRT), which superimposes the rainbow pattern of many individual particles onto the image. This technique yields the average size and temperature of a particle ensemble. The non-spherical droplets and liquid ligaments result in a uniform background and thus do not influence the interference pattern. Even with perfectly spherical droplets sm all variations in droplet size lead to a vanishing of the ripple structure. Van Beeck and Riethmuller (1995) recognized that the frequency f2 was quite insensitive to refractive index. Therefore, both size and refractive index can be measured using scattered light from a single beam. As a measure of the frequency, they used the angular separation between the main peak and the next peak (tJ 2 - tJ I) and gave a relation for the diameter as d . =~ ( p,A"y 4 COSTRB • 3 Sin T'RB 1/2( 2.37959 180deg J312 J .0.0 U 2 - U1 ' 1t .~ =~~-3- SIll 7' RB (9.44) The position of the main maximum is then given by Airy theory to be tJ I =ItJ RB I+ 1.08728 . SIll T' RB [ A.~ COST RB ) 2 16 dp 113 180deg (9.45) 1t The dependency on the refractive index lies mainly in the geometrie rainbow angle and must be solved implicitly from this equation. The result is that both diameter and refractive index can be obtained from the scattered light of a single beam. Similar algorithms were used by Roth et al. (1991) and Sankar et al. (1996) in an improved version of the integrated phase Doppler/rainbow refractometer presented in Fig. 9.23. The achievable accuracy using the Airy theory (Eqs. (9.44) and (9.45» can be checked by Lorenz-Mie computations with Debye series decompostion, as demonstrated in Fig. 9.24. As shown in section 4.1.2.2 the Debye series decompostion of the Lorenz-Mie result can be interpreted as scattering orders. For second-order refraction alone, the smooth curve in Fig. 9.21 gives the exact positions of the first main maximum and supernumerary bows. The refractive index determination from Eqs. (9.44) and (9.45) based on Airy theory can now be applied to such curves. As seen in Fig. 9.24a, the Airy theory yields satisfactory results down to particle diameters of 50l1m (at m = 1.33). Heukelbach (1998) used an interpolated lookup table of the second-order maxima based on Lorenz-Mie calculations with Debye series decompostion and could extend this to 20l1m (Fig. 9.24b). Important to note is that these methods are based on the rainbow intensity distribution without ripple structure for aperfect spherical particle. In areal rainbow signal the ripple structure prevents the exact determination of the rainbow maxima and reduces the accuracy by at least one order of magnitude. One solution is to illurninate only the incident points of second-order refraction with a shaped beam, but this is impracticable in most cases. 522 9 Further Particle Sizing Methods Based on the Laser Doppler Technique b a ~ 1.3 32 r-r-'-"-'-""I-"-"""-"-"""-"-""" Ir-.-,..., E • >< " "Cl • Reconstructed refractive index using Airy theory • .5 • ~ '.;::J Jlu . 1.330 I - .-.-...........:-•.....,..--. _ _- ~ Recons~ructed refractivel index using LMT theorywith Debye series decompostion . ...... -_---l ..... • - 1.328 Expected refractive index • 0 (m I 200 co - - - 1.330) I 400 0 Particle diameter dp [flm] I I 200 400 Particle diameter dp [flm] Fig. 9.24a,b. Predicted refractive index from the primary rainbow pattern (Ab = 488 nm). a Using Airy theory, b Using Lorenz-Mie Theorywith Debye series decompostion Still further information about the partide size is contained in the ripple frequency. Anders et al. (1993) found that the angular spacing of the ripple structure (LllJ R ) could be fit very weIl to an expression oithe form C LllJ =C +_1 (9.46) Rad p where Co and CI are constants, depending on the refractive index. Van Beeck (1997) gives an equivalent expression !,;PPle = :lb (COST RB +cos lJ;B ) 180:eg , f2 = fr;pple - ~ , f3 = fr;pple + ~ (9.47) which establishes a linear relation between droplet diameter and the spatial frequencies. The dependency on refractive index is weak, thus it can be used for a preliminary estimate of drop diameter. This discussion indicates that there are several avenues to determine size and refractive index of droplets using only one illuminating beam. However, the velocity is also important and such an approach, without reverting to a laser Doppler based system, has been demonstrated by Van Beeck and Riethmuller (1996b). The optical arrangement used is pictured in Fig. 9.25. This system overcomes the problem of the finite integration time of the CCD array, which causes the ripple structure to disappear, by using a photomultiplier instead. As a droplet moves through the illuminating beam, the rainbow pattern moves across the first lens and is imaged onto the photomultiplier. The pinhole 1 acts as a spatial filter to limit the observed section of the measurement volume. Pinhole 2 determines the angular resolution of the system. The virtual measurement plane is adjusted through the second lens to lie dose to the measurement volume in order to capture a large angular portion of the rainbow, at least 9.4 Shadow Doppler Technique 523 Lcns ,, ,, : Rainbow :image plane Beam adju I mcnl !\Iirror I Laser Fig. 9.25. Optical arrangement for measuring size, velocity and refractive index of droplets using one illuminating beam and one photomultiplier (adapted from van Beeck and Riethmuller 1996) the first two Airy fringes. A wire is placed in the center of pinhole 1, thus the signal should fall to zero when the droplet image coincides with the wire position, which is true except for diffractive effects. If a geometric 'shadow' is considered, the velo city of the particle can be expressed as (9.48) where d wire is the diameter of the wire, L1t wire is the time it takes for the image of the droplet to pass the wire and ß is the magnification factor. This velo city estimate is size and shape independent. Once the velocity is known, the relations given above relating size and refractive index to spatial frequencies can be converted to relations in terms of temporal frequencies. Details can be found in van Beeck and Riethmuller (1996a,1996b) or van Beeck (1997). One final remark is directed towards the use of rainbow refractometry for determining temperature. A relation between refractive index and temperature is required and this is not always linear for liquids over large temperature ranges. An example is given in Fig. 9.26 for water (Thormälen et al. 1985). 9.4 Shadow Doppler Technique The shadow Doppler velocimeter (SDV) is a combination of a particle imaging technique and the laser Doppler technique and provides particle velocity and size simultaneously. The technique is applicable for sizing irregular, nonhomogeneous particles. The technique was first introduced by Hardalupas et al. (1993, 1994) and refined processing algorithms were presented in Morikita et al. (1994). Since then, several publications have documented the instrument's performance in sizing pulverized coal (Maeda et al. 1997) and paint sprays (Morikita and Taylor 1998). 524 9 Further Particle Sizing Methods Based on the Laser Doppler Technique o Water '" .,>< -:.- 1..1.15 "C c: '" .~ ti ,~ '-' 1.330 e::: 1.325 1.320 L....-....I....---'_-'---'-_.l---'-_.l..-....L.---'L....-....I....---'_-'---'-_.l---'-_.l..--'----''--....L..--' o 50 100 Tempcrature T ["Cl Fig. 9.26. Dependence of refractive index on temperature for water, (,1 = 514 nm, p=l bar) The optical arrangement of the shadow Doppler velocimeter and also its operating principle is shown in Fig. 9.27. The particle in the measurement volume is illuminated by the two laser beams from the transmitting unit of a conventionallaser Doppler instrument and the image, used for obtaining size information, is a direct projection (or shadow) ofthe particle shape onto the observation plane. The photodiode array is placed at this observation plane. As the particle moves through the measurement volume, the shadow image passes over the photodiode array. The instantaneous output signals from the segments of the photodiode array provide a one-dimensional slice of the particle cross-section, which is read out at high speed and can be converted to 2 bit information in hardware, representing the light/dark part of the shadow. The two-dimensional Bragg ccll partidc motion Fig. 9.27. The optical arrangement of the shadow Doppler velocimeter 9.4 Shadow Doppler Technique 525 particle image is reconstructed from successive readings from the array, employing the particle velo city to scale the dimension in the plane of the beams. The velo city of the particle is obtained from the frequency of the Doppler signal, as with a conventionallaser Doppler system. For this, the Doppler signal pro cessor is fully synchronized with the readout of the photodiode array. With this optical arrangement, transparent and optically inhomogeneous particles provide almost identical shadows. In principle, glare points or diffraction fringes might be expected; however, these are negligibly small due to the spatial averaging caused by the finite size of the detectors and the aberration of the lenses. With the present state-of-the-art, a 35 channel linear diode (Hamamatsu S.4114-35) is used as an image detector, whereby only 32 segments are activated. A readout of the diode array at 20 MHz translates into a processing rate of about 10 particlesls. The dynamic range of particle size is about 12, so that a magnification of 230 corresponds to a size range of about 10 /llll to 120 flm . All of these specifications can be expected to improve rapidly when new hardware components become available. Clearly however, the instrument will always display limitations regarding measurable concentration, since it is a forward scattering, amplitude-based technique and thereby more susceptible to obscuration than the phase Doppler technique. Furthermore, multiple particle images will lead to erroneous results. First estimates of measurable number density is given in Morikita and Taylor (1998) as 1000 droplets/cm-3 in the case of droplets with an average size of 20 flm. The signal and data processing of the SDV must account for at least two additional effects not yet mentioned in the above description. One is the out-offocus problem and the other is a size distribution bias due to image truncation by the photodiode array. The out-of-focus effect is illustrated in Fig. 9.28, in which idealized (2 bit) detector signals are shown for two different particle trajectories through the measurement volume. This figure indicates that from a 2 bit signal, the two separate images from the two laser beams are identifiable and IUuminaied Trajeelor ies Output signal for the indicated photodiode: Trajcelory A: Beam2 11 U Shadow at the deteetor (diode array) plane: A: in foeus D(A)=O Pholodiodes , e• , • -limc lIajcclory B: B: ou 1 of foeus u "" D(B) , • ~ B ~-----------------+ Fig. 9.28. Trajectory dependenee of shadow images and resulting photodiode signal 526 9 Further Particle Sizing Methods Based on the Laser Doppler Technique separable. Using a suitably low trigger ensures that only when the images exhibit an overlap will the data be processed, thus defining a maximum tolerable defocus distance. The active defocus distance Z def is given by D =--- Z def 2tane (9.49) where D is the measured distance between out-of-focus shadows. The value of Zdef can be used as validation criterion. A minimum shape correlation between the two images can also be included as a validation criterion. The horizontal shape displacement can even provide information about the second velocity component perpendicular to the optical axis of the diode array. The equivalent particle size based on the average area of the projected shadows is computed using (9.50) where SH and SL are defined in Fig. 9.21 Generally, the center diode array segment is used for triggering. If the particle is small, the probability that it will not trigger an acquisition increases. If the particle is large, the prob ability that a portion of the shadow image will be truncated at the edges of the photodiode array increases. The measured particle size distribution must, therefore, be corrected for these detection/rejection probabilities. Further details can be found in Morikita et al. (1994). PART 111 DATA PROCESSING 530 10 Fundamentals ofData Processing in particular, for g(x) = x, the mean value of x(k) is obtained by E[x(k)]=,ux = fXP(x)dx (10.8) and for g(x) = x 2 , the mean square value of x(k) is obtained by (10.9) The quantities defined in Eqs. (10.8) and (10.9) are also known as the first and second moments of the random variable x(k). However, often the variance of x(k), is used rather than the mean square value, 0";, f(X -,ux = = If/ 2x - 0" 2x ,ux2 = )2 p(x) dx (10.10) The standard deviation O"x of x(k) is the square root of the variance. Equation (10.10) is one example ofthe more general r th order central moment ,ur = f (x -,uJ p(x) dx (10.11) which quantifies deviations of x(k) about its mean value. Similar expressions can be written for the bivariate case, in which two random variables x(k) and y(k) are considered. The joint prob ability function is defined by P(x,y) = Prob[ x(k)::; xand y(k)::; y] (10.12) and the associated joint prob ability density function by . (prob[x<X(k)::;X+LlX and y<y(k)::;Y+LlY]) p(x,y) = lim dx-->O LlxLly (10.13) dy-->O yielding also p(x,y) ~ 0 (10.14) f f p(x,y) dxdy y x P(x,y) = f fp(~,1])d~d1], =1 d 2P(x,y) _ ( ) dxdy - P x,y (10.15) (10.16) The two random variables are said to be statistically independent if p(x,y) = p(x)p(y) (10.17) 10.1 Sta tistical Principles 531 The expected value of any real, single-valued, continuous function g(x,y) of two random variables x(k) and y(k) is given by E[g(x,y)] = JJg(x,y) p(x,y)dxdy (10.18) One special example is when g(x,y) = [x(k) -,lLx J[y(k) -,lLy]' where ,lLx and ,lL y are the respective mean values. The expected value is known as the covariance function Cxy =E[(x(k)-,lLJ(y(k)-,lLy)] = E[ x(k) y(k)] - E[ x(k)]E[y(k)] = (10.19) JJ(x(k)- ,lLx)(y(k)-,lLy )P(x,y)dxdy The correlation coefficient is then defined by (10.20) and lies between -1 and + 1. Data processing with the laser Doppler and phase Doppler techniques deals with the estimation of relevant flow properties from the primary measurement quantities. The term estimation, rather than determination or computation, is used, since in almost all cases, the physical process has a stochastic part, meaning that the result of an estimation is a random variable (even an exact replication of the experiment would yield a slightly different answer). The procedure or computational algorithm used to obtain the estimation is known as the estimator. Estimators are evaluated on the basis of three properties. First, the expected value of the estimation should be equal to the parameter being estimated (10.21) If this is true, the estimator is unbiased. Note that an estimator is often signified by the hat symbol. Second, the mean square error of the estimator should be smaller than for any other possible estimator. (10.22) In this case the estimator ~1 is said to be efficient. Finally, the estimate should converge to the parameter being estimated for a large sampie number or for a long observation time. H~prob[(~- f/J) 2 c] = 0 (10.23) 532 10 Fundamentals ofData Processing For an arbitrarily sm all c > 0, the estimator is said to be consistent. A sufficient condition to meet this requirement is (10.24) The mean square error used above can be expanded to yield E[(~- 9)2J= E[(~- E[~] + E[~] - 9)2 ] = E[(~- E[~]fJ + E[(E[~] - 9fJ (10.25) Hence, the mean square error is the sum of two parts: the first part is a variance term that describes the random part of the error (10.26) which can be made arbitrarily small by increasing the sampIe size. The second part is the square of a bias term describing the systematic portion of the error (10.27) This part is not influenced directly by the sampIe size and can arise from many sourees, often found outside of the data processing. For example, if the intersection angle of a laser Doppler system is improperly measured, a systematic error of all velo city related quantities would result. Often special calibration pro cedures are required to quantify such errors; however, these will not be considered further in this chapter. In fact, the bias error will be assumed to be negligible in the following discussion. Under these conditions and for a small normalized random error a[~] ~var[~] c=--=---'----- 9 (10.28) 9 the probability density function for the estimates, p[~l, can often be approximated by a Gaussian distribution with the mean value E[~l = ljJ and a standard deviation O"[~l = t:I/J p9 A () = 1 [-(~-9r) c9~ exp 2{c9) 2 (10.29) Prob ability statements about the bounds in which future estimates ~ willlie can thus be made as follows 10.2 Stationary Random Processes 533 Prob[ q>(1- &):::; ~ < q>(l+ &)] "" 0.68 (10.30) Prob[ q>(1- 2&):::; ~ < q>(1 + 2&)] "" 0.95 since for a Gaussian distribution ±a or ±2 a about the mean contains respectively 68 % or 95 % of the probability mass, as sketched in Fig. 1O.l. This leads directly to the concept of confidence intervals, i.e. the interval in which the true value williie with a given probability (valid for small E). ~(1-&):::;q>:::;~(1+&) with 68% confidence ~(1- 2&):::; q>:::; ~l+ 2&) with 95% confidence (10.31) The value of & can be estimated directly from the sampled data, as is discussed in the next section. a p($) ~ 0.4 " '- - - - - - -03989-+-- - - - 0.3 \- - - - - -0.2420 0.2 0.1 0.1585 - 20 • o Fig. 10.1. Gaussian (normal) distribution illustrating confidence limits. a For ±a (68%), b For ±2a (95%) 10.2 Stationary Random Processes Given so me random phenomena, such as a turbulent flow field, any single time history of this function is called a sampie function. The collection of all possible sampie functions, possibly an infinite number, is known as a random process or stochastic process. The mean value (first moment) of the ensemble of sampie functions at time tj is then the arithmetic mean over the instantaneous values of the sampie functions at time tl' as illustrated in Fig. 10.2. A correlation or joint moment of the process at two different times can be computed by taking the ensemble average 534 10 Fundamentals ofData Processing Timet Fig. 10.2. Ensemble of sampie functions defming a random process of the product of instantaneous values at two times t 1 and t 1 + -r. These values can be written as (10.32) (10.33) R xx is known as the autocorrelation function. A random process is known as weakly stationary when the value defined by Eq. (10.32) is independent of t1 and the autocorrelation is only a function -r. The process is known as strongly stationary when the autocorrelation is also independent of t1 • Otherwise the process is instationary. For stationary random processes, the autocorrelation is only a function of -r. Generally, however, statistics of a stationary random process are not computed over an ensemble of sampie functions hut over a time average. For exampIe, 1 f1x (k) =limf x k (t)dt = f1x T~~T T (10.34) 0 (10.35) Ifthese values do not differ from those in Eqs. (10.32) and (10.33), then the process is said to he ergo die, in which case the index k is dropped. All stationary processes encountered in fluid mechanics can be considered ergodic. 10.3 Estimator Expectation and Variance 535 Note that the covariance function is simply the autocorrelation function with the mean removed and the cross-covariance function is the cross-correlation function with the product of the me ans removed. Cxx (r) = R xx (r) - 11~ Cxy (r) = R xy (r)- I1 x l1 y (10.36) 10.3 Estimator Expectation and Variance In many cases the expectation and variance of an estimator can be derived analyticallyand several examples are given below. For more complicated quantities, this is not always possible and other strategies can be followed. The jackknife algorithm will be introduced as one such approach. 10.3.1 Estimators for the Mean The first estimator to be examined is the mean value. The most common sampIe mean estimator is given by (10.37) where x; are individual sampIes of the process x. This estimator is non-biased, since E[x] = x (Bendat and Piersol 1986). The mean square error, or variance, of this estimator is then given by I1 var[x] = O"i = E[(X - I1S] (10.38) Substituting Eq. (10.37) into (10.38) leads to (10.39) If the condition E[x; x j 1= 0 is satisfied, i.e. consecutive sampIes are uncorrelated or statistically independent, Eq. (10.39) can be further reduced to (10.40) N which states that the variance of the mean. estimator decreases with increasing number of sampIes. 536 10 Fundamentals ofData Processing This analysis has been performed for an estimator based on discrete sampies x;; however, a similar analysis could be made for a mean estimator based on the continuous signal x(t) 1T J (10.41) fix = - x(t)dt Ta which differs from the true mean I1x' since the integral is performed only over a finite time T. The variance of this estimator becomes (10.42) In terms of the autocovariance function, this can be written as (Bendat and Pier- sol 1985 ch. 8.2) (52 ", =.!.T JT(I_ El )C T (r) d r xx (10.43) -T for a stationary random process. For small r only C xx remains in the integral and for large r, Cxx goes to zero, thus the integral can be expressed in terms of the integral time scale Tx ' as defined in section 7.1.1.2 (10.44) with (10.45) As pointed out by George (1978), if the results given by Eqs. (10.40) and (10.44) are equated, the condition for statistically independent sampies can be obtained, namely N=~ 2Tx (10.46) This is graphically represented in Fig. 10.3 and leads to two very insightful interpretations • Sampies are statistically independent if they are separated by aperiod of the least 2 Tx in time. • Segments of the continuous signal 2 Tx in length contribute to the mean estimate as one, statistically independent sam pie. The manifestation of this relation is that sampling a signal with time intervals less than 2 Tx will not accelerate the convergence of the mean estimator. At this point, the difference between data and information should become very dear. New information (with respect to the mean estimate), comes only every 2Tx time periods! 10.3 Estimator Expectation and Variance 537 xCI) 2T, 2T, 2T, Timet Fig. 10.3. Graphical interpretation of statistical independence of consecutive sampies of a continuous process In section 7.1.2.2 the term data density for the laser Doppler technique was introduced, meaning the number of detected particles per integral time scale, or N -L1tl D - (10.47) p where L1t p is the mean (inter-arrival) time between particles. It is the data density, not the data rate, which determines how weH turbulent fluctuations of a flow field can be resolved in time using the laser Doppler technique. Equation (10.44) makes a statement about the necessary observation or measurement time to achieve a given statistical uncertainty (variance of the mean estimator). However, to use this equation the integral time seale, as defined using the autocovariance function, must be known beforehand. Moreover, the integral time scale may change by orders of magnitude between different points of a single velo city profIle. Often, however, a simple estimate of Tx suffices. This will be illustrated with the foHowing example of how Eq. (10.44) can be used in practice. The example chosen is a velocity measurement in the recireulation zone of a baekward facing step water flow. In a preliminary measurement the loeal varianee of the velocity fluetuations is estimated to be 0.2 m "S-" at point A (Fig. 10.4). The requirement is that the mean velocity at point A be deterrnined to within ±0.04 m S-1 with 95% confidenee. The integral time scale of the velo city fluctuations can be estimated from appropriate velocity and length scales, in this case U 0 = 2 m S-1 and x R ' whieh is H=5em x x . "' 0.4 m Fig. 10.4. Sketch of example backward facing step flow. length XR is the mean reattachment 538 10 Fundamentals ofData Processing approximately SH or 0.4 m. Thus, T" = X R / U 0 = 0.2 s. Note that the subscript u for the integral time scale is used, since the process being measured is the velocity u. Assuming a normal distribution for the scatter of the estimates, the probability of being within ±O"u of the true mean value would be about 68%. This would increase to the required 95% for ±20"u. 2 o"u O"~ = 0.04, = 0.0004. (10.48) Equation (10.44) can now be solved for the required measurement time to fulfil this condition T= 20"2 T u O"~u u = 200 s (10.49) Note that this calculation has been performed independent of the choice of measurement technique. In fact, no measurement technique can shorten the necessary observation time given in Eq. (10.49), since this describes the fundamental statistical behavior of a random process. In practice, it is unusual to make such calculations prior to every measurement. It is more convenient to display the current measured mean velocity online, accumulated over all sampies up to that time, and then to allow the user to terminate the measurement when the fluctuations of the mean are below an acceptable level. Indeed, from the necessary measurement duration, and from the fluctuation level of the mean, a rough estimate of the integral time scale can often be made. This technique of user intervention does not lend itself to automation, so that still a third approach is often used, in which a fixed number of sampIes is used for each point, whereby the number is chosen very large, to ensure sufficient convergence for all measurement points. In many flows there are regions, where data rates decrease dramatically, e.g. near walls. In such cases there is often no choice but to accept a high er degree of statistical uncertainty, since otherwise the data collection time becomes exorbitant. Alternatively, Eq. (10.40) could have been used if the velocity data were available in discrete form at regular time intervals. Assuming the sampie rate was not faster than every 2Tu ' the number of sampies required to insure the requested accuracy would be O"~ 0.2 N=-=--=500 O"~ 0.0004 (10.50) This discussion puts into perspective expressions like 'high' or 'low' data rates or 'many' or 'few' sampies. The data rate, or the number of sam pIes, must always be considered with respect to the integral time scale of the process at the particular measurement point. This explains the preferred use of data density rather than data rate. It should also be apparent that for the same Reynolds number, measurements performed in air flows will typically be much shorter in duration than in water flows, given the same target accuracy. The reason for this lies in the fact that for the same Reynolds number, the integral time scale of an air flow is generally shorter. 10.3 Estimator Expectation and Variance 539 Further guidelines for reporting measurement uncertainties can be found in Kline and McClintock (1953), Kline (1985) or Moffat (1985,1988). In the particular case of laser Doppler measurements, the velocity sampling function is determined by the presence of tracer particles in the measurement volume. Thus the velocity sampies are not regularly spaced in time. Furthermore, the particle density is usually high enough, such that consecutive velocity sampies are correlated with one another, i.e. not statistically independent. Thus Eq. (10.40) cannot be used to estimate uncertainty of the mean estimate. Rather Eq. (10.44) must be used, meaning the integral time scale must be estimated prior to the measurement, as was illustrated in the above example of the backward facing step flow. The situation is somewhat more subtle if, for instance, the mean particle diameter is to be estimated from an ensemble of individual particle diameter measurements performed with a phase Doppler instrument. This is not a continuous process and, therefore, there can exist differences between the time mean and the ensemble mean diameter. Here it is important to first establish whether there is any correlation between consecutive sampies. In a spray for example, the atomization process may lead to several consecutive droplets of similar size, whereas in a large spray dryer, such a correlation between droplets may be completely lost. Such a correlation is necessary to investigate in order to establish whether an integral time scale for the process exists or is zero. The case of a zero integral time scale is simply the case of statistical independence between sampies (Markov process) and then Eq. (10.40) applies. The correlation between consecutive sampies cannot be estimated from length and velo city scales but must be estimated directly using a preliminary measurement and Eq. (10.33). This integral time scale (Eq. (7.31) can then be used with Eq. (10.44) to estimate the uncertainty in the mean diameter estimate. 10.3.2 Estimators for Higher Order Correlations In the study of turbulence, statistics of not only the mean velo city but also of higher order moments are required, as outlined in section 7.1.1. General formula for the estimator variance for higher order statistics have been given by Stuart and Ord (1994) and Kendall and Stuart (1958). Benedict and Gould (1996) have summarized their results in the following manner. An unbiased estimator1 ofthe r th central moment f.1 r (Eq. (10.11» is given by 1 N _ mr = - ~)x; -x) N ;~l r (10.51) in which the true mean has been replaced by the sampie mean, Eq. (10.37). The sampling variance of m r is given by 1 Strictly this estimator is unbiased only for r= 1, however this also applies for higher moments when N is large. 540 10 Fundamentals ofData Processing var[ mr1= (}~r = ~ (fl2r - fl; + r 2 flr-l fl2 - 2 r flr+l + flr-l) (10.52) where terms of order N- 2 and high er have been neglected. 95% confidence intervals are then mr ± 2 () m,' Note that Eq. (10.52) uses the exact central moments fl r , which are actually unknown. However if N is suitably large, typically N = 1000, these can be replaced by the central moment sampling statistics, m r , for practical computations. Similarly, the mixed central moment flr,s = JJ(x-flJ(Y-fl y )' p(x)p(y)dxdy (10.53) can be estimated using mr,s _)r (x; - Y_)s = -1 ~( L.. x; - x N (10.54) ;=1 which exhibits the variance var[m r,s ] = (}2m,,1 = ~(" ,,2 N fA'2r,2s _ rr,s + r 2 r2,0 + 52" ,,2 " " r'r-l,s rO,2 rr,s-t (10.55) Note that fl lO =f.L Ol =0, flr,-l =f.L-l,s =0, fl 20 =var[x;] and f.L 02 =var[y;]. Eq. (10.55) can be simplified for normally distributed processes, since then all odd moments are zero and the 2nd, 4th, 6th and 8th moments are 1, 3, 15 and 105 times ()~ respectively. The variances of the most common statistics in turbulence research are summarized in Table 10.1, for both an arbitrary and a normal distribution of the process. Note that the formulas given in Table 10.1 are multiplied by N. The u and v velo city components have been used for illustration. As an example, the variance of the mean estimator is given as 0 N -1, which agrees with Eq. (10.40). The expressions in Table 10.1 all assurne statistical independence between sampies, as specified by Eq. (10.46). If the sampie rate is too high to insure statistical independence, the total number of sampies N must be adjusted so that the total observation time yields the desired confidence bounds, according to Eq. (10.44). Furthermore, it shoUld be no ted that turbulence quantities are seldom normally distributed, so that the simplifications given in Table 10.1 can lead to significant errors if the normality is not previously established. For more complex estimators, there exist several resampling algorithms with which the uncertainty of the measured quantity can be estimated. In particular the jackknife algorithm will be discussed, as first introduced by Tukey (1958). Notes on its practical implementation are given by Efron and Tibshirani (1993) and an evaluation of its potential with laser Doppler data is given by Benedict and Gould (1996). This algorithm also assurnes statistical independence in the data set x = (Xl ,x 2 '''''X N ) when computing some statistical estimator. The jackknife sampies 10.3 Estimator Expectation and Variance 541 Table 10.1. Estimator variances multiplied by N (Benedict and Gould, 1996) Statistic Variance for anydistribution Normal assumption U '2 U '2 U '2 2 4U '2 -12 U V'2 - u'v' R .fz!2.r;a =--;==-;= uv (u'V') 2 R;.[-(~:v-':;, ~[-( ::'4f :':i2 ~/~- -,-;2(:- ,-':,)1 + -[ + - ( + -( (U'~") (U'~") II + f U '2 2(U '2 U '3 6(U '2 )' 2 (1 + 2 R~v ) (U '2 f V '2 U '4 X jack,i == (Xl,X2,···,Xi_l,Xi+l)···,XN) (10.56) are obtained by leaving out in turn one of the data sampies. The jackknife sampIes are then used to compute N estimates ~ jack,i' The jackknife variance for ~ is then given by var [A] t/l jack,i = N-l~(A N ~ t/ljack,i - ,,)2 t/ljaCk (10.57) where " t/ljack 1~ = N~t/ljaCk,i A (10.58) The 95% confidence interval for the estimator is given by ~ ± 2var[ (~) jack jI/2. The jackknife algorithm requires N 2 calculations per variance estimate. This computationalload can be greatly reduced if the programming is modified specifically for each statistic to be studied. For example, if the mean square of the velo city fluctuations, i7i, , is being studied, the jackknife sampie can be written 542 10 Fundamentals ofData Processing 1 -'-2- A 9jack,; N = Ujack ,; = N -1 I( Uj - -- 2 (10.59) UjaCk ,;) J~l j:f:.i This equation can be rewritten as (10.60) Each term in the brackets is summed only once over all j = 1, ... N and then decremented by u" and uj respectively for each jackknife replication. It can be showri theoretically that the jackknife is biased high on its estimation of uncertainty and thus, it will never underestimate the uncertainty of a statistic. 10.3.3 Estimators for Transient Processes If the flow process being studied is not statistically stationary, for instance a transient process, then the concept of a mean value and central moments must be modified. For experiments which are periodic or repeatable, the ensemble velocity average defined in Eq. (7.15) is not replaced by a time mean (Eq. (7.16» but rather by a phase average or ensemble average u({}) Phase average 1 N N k~l = lim- Id k )({}) N-->= (10.61) (10.62) Ensemble average where in the phase average, k is aperiod index and in the ensemble average k is the repetition index. The phase average is typical of processes in rotating or reciprocating machinery, while the ensemble average would be used for repeatable, one-shot experiments. (t- t o) is the lag time from some reference time t o in each repetition. Central moments in terms of these averages are dependent on phase or time (t- t o) and can be computed according to m r ({}) = lim N-->= 1 N N I( U(k) ({}) - u({}») r 1 N N k~l mr(t-tO )= lim- I[U(k)(t-to)-u(t-t o)] N-->= (10.63) k~l r (10.64) In such cases, statistical independence between samples is generally automatie. A good example of a periodic process is the flow in the cylinder of an internal combustion engine. The quantity u({}) then represents the mean flow velocity over crank angle, averaged over many cycles. The moments in Eqs. (10.63) 10.4 Propagation of Errors 543 and (10.64) refer to deviates from this mean. It is arguable however, wh ether these moments can be related to turbulence behavior in the same manner as for statistically stationary processes. In any case, the uncertainty estimate of the mean can be computed directly from Eq. (10.40), since samples entering each of the moments in Eqs. (10.63) and (10.64) are statistically independent from one another. The variance can be estimated using the second moment estimator (r = 2). 10.4 Propagation of Errors The concepts of stochastic and systematic errors for a given measurement quantity have already been introduced in Eqs. (10.26) and (10.27) respectively. If a derived quantity, y, depends on several individual measurement quantities Xi' the question arises as to the measurement error in y. (10.65) The propagation of errors from the quantities Xi to y is treated separately for systematic and stochastic errors. The resulting systematic error in y is found by using a first-order Taylor expansion ~ df dX) df dX 2 df dX n &=-&) +-&2+···+-&n (10.66) where 8x i are the systematic errors for each measurement quantity Xi and öy is the overall systematic error. Note that all 8x i quantities are signed and as such, systematic errors may be compensating in nature. Stochastic errors are treated in the mean square, leading to the relation (j y = (10.67) where the individual estimator variances (j~i have been evaluated using techniques described in the previous section. This formula assurnes that an of the individual stochastic errors are normally distributed and that the standard deviations are all evaluated with the same confidence intervals. An extensive discussion of error propagation can be found in Kline and McClintock (1953), Kline (1985) and Moffat (1985). 546 11 Processing of Laser Doppler Da ta Constant velocity 40 % turbulence 0.01 o!:-'---'-.J.......J----'---'-J........L--'--7-'----'---'-J........L--'--'-J........L--'2!:-'---'-.J.......J----'---'-.L.....I----'---:!3----'---'--J........L--'--'-J........L~4 Interarrival time t; - t;.1 [ms] Fig. 11.1. The probability density function of the time between two particles for constant velocity flow and flow with 40% turbulence (mean rate 900 Hz, integral time scale 10 ms) The striking feature of the distribution pictured in Fig. 11.1 and expressed by Eq. (11.1) is that the most probable time between two particles is zero. This is a well-known property of random (Markov) processes and has direct consequences on signal processing hardware. Even at modest mean particle arrival rates, particles will quite often appear in rapid succession. Either the signal processor must be able to evaluate the signals on-line or suitable input buffering must be available to avoid loss of information and to prevent processor 'dead time'. Many early signal processors exhibited such inherent dead times and this influences most moment and spectral estimators considerably. More recent processors achieve on-line processing speeds. On the other hand, the fact that velo city information is often available over the very short time spans of consecutive particles suggests that, principally, information about very high frequency velo city fluctuations is contained in the data. This is in strong contrast to data sampled at equal time intervals, for which the sampling theorem applies and for which no information above the Nyquist frequency f Nyq = 1/ (2Llt,) is available, where Llt, is the sampling interval. In fact, with randomly sampled data, there is no equivalent to the sampling theorem or the Nyquist frequency and with suitable estimators, it is possible to achieve alias-free and unbiased estimates of signal power at frequencies far exceeding the mean particle arrival rate. This is the topic of seetion 11.2. Arrival time information is sometimes used in data processing as a basis for validation. Knowing the particle velo city and the dimensions of the measurement volume, an estimate of mean transit times can be made. Measured arrival times lying far below these estimates indicate that the signal processor may be delivering more than one velo city value per particle. Since moment estimators generally assurne the single realization condition, such multiple values per parti- 11.1 Estimation ofMoments 547 cle are unacceptable. These can often be recognized as peaks in the prob ability density function of the interarrival time, which also indicate suitable thresholds for validating each individual velo city before further pracessing. 11.1 Estimation of Moments Moments of a random variable or of two random variables have been defined in section 10.1 and when the random variable is a flow velocity, the moments become irnportant quantities in the equations governing fluid flow. The most important moments of velo city are the first moment, yielding the mean flow velocity, and the second central moment, giving the variance of velo city fluctuations, or when normalized with the square of the mean velo city, the turbulence intensity squared. Covariance and correlation functions are examples of bivariate functions and correspond to Reynolds shear stress terms in a flow field. Some general estimators for these moments have been introduced in section 10.3 and their statistical variance has been discussed. In the present section the expectation of various estimators for the specific case of laser Doppler data will be addressed. The particle-rate/velocity correlation mentioned in the introduction to this chapter is the main physical reason for requiring special moment estimators for laser Doppler data. The particle rate through the measurement volume is determined by the volume flux of fluid through the measurement volume and this is, in general, correlated with the measured velo city component. Therefore, the sampIe rate of the velocity increases also with velo city. For a given observation time, high er velocities will be sampled more frequently than lower velocities and a simple arithmetic mean of all sampIes will be positively biased over the true time mean of the velocity. The degree ofbias will depend on how strang the correlation between the particle arrival rate and the measured velocity component IS. A correct estimator for the mean of the u flow velocity component must therefore weight each sampIe with a factor g, which is inversely proportional to the conditional prob ability density of a particle arrival at a time t;, given the velocity u;' (11.2) i=l where the index i refers to the arrival time t; and the hat (!\) signifies that this is onlyan estimation of the mean. One possible weighting factor would be 1 (11.3) 548 11 Processing of Laser Doppler Da ta which uses the magnitude of the vector velo city and assumes a spherical measurement volume. For this, a three-velocity component laser Doppler system is necessary. For ellipsoidal measurement volumes, the expression 1 (ll.4) is more appropriate (McDougalI1980), where ao is the measurement volume radius and Co is the measurement volume half-Iength. Nonetheless, three velo city components must be measured to implement this weighting. Furthermore, many authors have pointed out that this estimator is very sensitive to signal noise. Even low noise levels already result in significant systematic errors in the estimated mean. Alternatively, the residence time (or transit time) of the particle, Ti' can be used as a weight factor, since this will be inversely proportional to the vector velo city magnitude (Buchhave 1975, Buchhave et al. 1979). ( 11.5) This is only possible if the signal processor provides residence time information and if it is reliable. An estimator for the second moment is given in a similar manner (11.6) i=l as are estimators for joint moments N _ L(U; -ft)(v; -~)g; u' v' = --,,;~::.cl_--'-'N---- (11.7) Lg; i=l The last estimator assurnes the U and vvelocities are available at the same instant in time: meaning time coincidence was demanded during the acquisition. For independent time series, i.e. data collection without coincidence, Eq. (11.7) must be modified as follows N M LL(U; -ft)(v; -~)guigvi u' v' = _i~_l--,-j_~l_-----,-, _ _ _ _ __ N Lgu;gv; i=l (11.8) 11.1 Estimation ofMoments 549 As an example, if a residence time weighting is being used, g u; corresponds to the residence time of the i th particle, found from the u component signal. All of the above estimators assurne that the seeding in the flow field is homogeneous. A large body of literature studies the expectation bias of these and other estimators, often using simulated data sets (Erdmann and Tropea 1982, Edwards and Jensen 1983, Edwards 1987, Winter et al. 1991a, Winter et al. 1991b, Fuchs et al. 1994). The main reason for examining many different estimators is that not all laser Doppler systems provide the necessary measurement quantities (e.g. residence time) to formulate bias-free estimators. The most common of these alternative mean velo city estimators are as follows: • Inverse velocity (McLaughlin and Tiedermann 1973) 1 g;=~ (11.9) Applied when only one velo city component is available. This estimator cannot be recommended for anyapplication. • Arrival time (11.10) The arrival time is strongly correlated with the instantaneous particle rate only at high data densities, NT" > 5 and its application is therefore limited to such situations. This weighting scheme is suitable for moment estimation but fails for estimation of correlation functions. More appropriate estimators are given in the next section. Noteworthy is the fact that this is the only recommendation for non-homogeneously seeded flow fields. • Free-running processor (11.11) This estimator is exactly the arithmetic mean of all sam pies. Although not recommended for use, this estimator represents an upper bound on the expectation bias and furthermore, is readily analyzed theoretically. If the normalized error in the mean is expressed as ß)-- u - flu (11.12) fl u where fl u is the true mean velo city, Erdmann and Tropea (1982) have evaluated ß) for one-dimensional flows exhibiting turbulence levels up to about 40% as (11.13) where Tu) is the turbulence level defined by Eq. (7.20). Thus, for a flow field with 20% turbulence, the maximum error in the mean estimation will be approximately 4% using a free-running, mean estimator. This maximum error 550 11 Processing oi Laser Doppler Da ta will decrease for three-dimensional turbulence, because the sampie function becomes less correlated with the measured velocity component. Several mean velocity estimators which are based on an arithmetic average over all sampie values, but which use a modified sampling procedure, have been theoreticallyanalyzed. • Sample-and-Hold (S+H). The sample-and-hold procedure uses a regular time sampling of the flow velo city, always using the velo city of the last validated particle signal. This sampling procedure originated with early frequency tracking processors, which provided an analog signal output, updated with every new validated particle signal. Conventional sampling hardware (A/D convertor) was then used to acquire discrete velo city values. More recently, the S+H procedure has been shown to be a viable means of acquiring velocity data when estimating velocity spectra (section 11.2.2). Thus, moment estimators using a S+ H sampling scheme are of some interest. Fuchs et al. (1994) have shown that the statistics of arithmetic averages for the first moment or central moments are virtually identical to the arrival-time weighted estimators. As such, the condition NTu > 5 is again required to achieve unbiased estimates. The difference is that the S+ H procedure uses simple arithmetic averages but not all validated signal values, while the arrival time estimate uses the arrival time as a weighting factor, but includes all acquired velocityvalues in the computation. • Controlled Processor. The controlled processor has its origins in the work of Erdmann and Tropea (1982) and results when the processor has long dead times after acquiring a signal. Once the processor is again enabled, the next particle in the detection volume is measured. Since the processor dead time is independent of all flow phenomena, a degree of de-coupling between the sampling and the velocity results. Other authors generalized the approach and even prescribed a periodic enabling of the processor at times completely unrelated to particle arrival statistics. However, as Erdmann and Tropea (1982) correctly derived in the original analysis, and as Winter et al. (1991a) and later Fuchs et al. (1994) confirmed, a complete de-coupling of the sampie statistics from the velo city is not possible. After enabling the processor, the statistics of the waiting times until the next particle are still highly correlated with the flow velo city. Two conditions are required to achieve unbiased estimators using a controlled processor and arithmetic averaging (no weighting factor). Winter et al. (l991b) demonstrate clearly that S+H estimates are a special case of the controlled processor. Not only must the condition NT" > 5 be met for unbiased estimators of the mean, but also the condition NT, > 5 should be met. T, is the time between periodic enabling pulses of the processor. Some results using experimental and simulated data are used to illustrate the behavior of the various mean velo city estimators discussed above. The first example is taken from Fuchs et al. (1994) and involves a three-dimensional flow field with a turbulence intensity of TU j "" 100 %. The data density (NT" ) is varied 11.1 Estimation ofMoments 551 between 0.1 and 10. In Fig. 11.2 the performance of the mean velo city estimators using a free-running processor estimate, an arrival time estimate and a transit time estimate are compared. The estimates are given both as velo city values and as a normalized error (ßl)' For the normalized error, the true value of the mean velocity was found by a post-diction of the measurements using the simulation model. Both experiments and simulations confirm the expected behavior. For the mean velo city, the free-running processor yields a maximum error, independent of the data density. The error is less than the upper bound ßl = (Tu l )2 '" 1, since the turbulence is three-dimensional and not one-dimensional. The arrival time estimate shows acceptable results only for data densities exceeding about NTu > 5. The transit time estimator is reliable at all data densities. In Fig. 11.2the simulation results obtained without consideration of the processor dead time have been included as lines. Clearly, this aspect can also be important. A second result, adapted from Winter et al. (1991a), is shown in Fig. 11.3. Here the transition of a controlled processor from a free-running processor (NT, «1) to a sample-and-hold processor (NT, > 5) is demonstrated using simulations. Note that the statistics of the S+H procedure are equivalent to the arrival time estimator. The error shown in Fig. 11.3 is normalized with the error ofthe free-running processor. Far fewer studies have investigated the performance of variance estimators. Erdmann and Tropea (1982) demonstrate that the bias of the unweighted variance estimate is negative for turbulence levels up to approximately 60%, after which it becomes positive. Fuchs et al. (1994) confirm this result and also show .,-0.7 I '" ~ 0 <I;:: " >- :';:: I I • - - - - Free runnung processor '" -------- Arrival time • . . - .. - Transit time u Open symbols: Simulation o:t - 50 ... Solid symbols: Experiment ......o Lines: Without Processor simulation " o ~ "0 0.61- 'l '" ~ 0- -- '" " , ""t:;- - " 0.51- p-O__ 0.4 1 0.1 0 - - ts - - - 0 0 • -~--------- - o - - - - - - o o o --------------- ------------:_-- ---~--------~- .~ @ o Z ~~o-·--~O~----O.w-~-o----~o~-~"~'-~'~~--~'~--~'-~"-~---_- _- _-_-~_-_- _- _- _p_- _- _- _- ~-~ -.~o 1 1 10 Data density NTu [-I Fig. 11.2. Comparison of experimental and simulated mean velocity estimators: freerunning processor, arrival time estimator, transit time estimator (adapted from Fuchs et al. 1994) 11 Processing ofLaser Doppler Data 552 Normalized sampie interval IV T, co 0.6 1.0 0.5 1.5 2.0 3.0 0.1 10 100 Integral scale data density NT" [degl Fig. 11.3. Effect of normalized sample internal (NT,) on the mean velocity bias (adapted from Winter et al. (1991a» that, as with the mean velo city estimator, the error is a maximum for the freerunning processor, independent of data density. The arrival time estimator becomes reliable for NT" > 5 and the transit time variance estimator exhibits a very low bias, independent of data density. The issue of choosing an appropriate moment estimator is certainly less critical now than in the past, simply because almost all commercial signal processors now provide reliable estimates of the transit time for each particle passage through the measurement volume. Transit time weighting is the recommended estimator for all measurement situations involving a spatially homogeneous particle seeding. The necessity to use a weighted estimator can be checked by cross-correlating the measured velo city magnitude with the interarrival time fluctuations between particles. This is especially reliable if all three velo city components are available. A weighted estimator is only required if a significant correlation exists between these quantities. 11.2 Estimation of Turbulent Velocity Spectra In section 7.1 the three-dimensional energy spectrum of turbulent velo city fluctuations E(k) (Eq. (7.49», the simplified one-dimensional energy spectra, f/J;j (k), (Eqs. (7.52), (7.53» and their corresponding Fourier pairs, the correlation functions, were introduced. Although these are functions ofwavenumber k, they can be approximated directly from frequency spectra by assuming that Taylor's hypo thesis is valid (u:/u«l) and by substituting t=x/u or 11.2 Estimation ofTurbulent Velocity Spectra 553 k = 2n! tu. Thus, the foHowing discussion will investigate the estimation of frequency spectra (power spectral density, PSD) and their Fourier transform pairs, the temporal correlation functions. The PSD function and its related correlation function allow integrallengthttime scales to be deterrnined, the rate of dissipation of mechanical energy to be estimated and, generally, the small scales of turbulence to be studied. Furthermore, flow periodicity becomes evident in these functions. In contrast to the rather straightforward computation of power spectral density functions for signal processing (section 6.3.1), now the input data is randornly sampled in time and no obvious equivalent to the fast Fourier transform (FFT) is available as a computational algorithm. However, as anticipated at the beginning of this chapter, some estimators are able to exploit the random sampling to achieve estimates of turbulent kinetic energy at frequencies much high er than the mean particle arrival rate. The most common of these are presen ted below. A general classification of spectral estimation techniques is presented in Fig. 1104 and shows the three principle approaches • direct transform • slot correlation foHowed by a eosine transform • reconstruction with equidistant re-sampling and FFT In each of the algorithmic routes, additional steps (shown as dashed boxes in Fig. llA) are possible, representing various enhancements of these basic algorithms. A comprehensive review and evaluation of these techniques are given in Benedict et al. (2000). They come to the conclusion that two main algorithms for estimation of the PSD and autocorrelation function (ACF) can be recommended: the fuzzy slotting technique in combination with local normalization (van Maanen et al. 1999) and the refined sample-and-hold (S+H) reconstruction (Nobach et al. 1998). The direct Fourier transform, first introduced by Gaster and Roberts (1975, 1977) and later modified by Roberts et al. (1980) and others (Scargle 1982, Marquardt and Acuff 1983, Rajpal 1985, Saarenrinne et al.1997), did not meet expectations and has been included here only for historical reasons. The variability of this estimator increases too rapidly with frequency, hence, the estimate becomes quite unreliable, even for very long observation times. The optional pre-filtering step shown in Fig. 11.4 is also discussed fuHy in Benedict et al. (2000) and is used to reduce the variability of the power spectral density estimate. This is particularly appropriate if long-scale periodicities or trends are to be first removed from the data set with a minimum of systematic errors to the PSD or ACF. Also Kalman filters have been used in a pre-filtering step to suppress noise components in the data set. (van Maanen and TuHeken 1994, Benedict and Gould 1995). However, such schemes are only applicable if the data density is sufficiently high. 554 11 Processing of Laser Doppler Da ta r----------- ------------, -r -----------' : Pre-filler SlolI ing Icchniquc .----- - - - - - - : --- • .----~----!...._------I+!-----------, "'. ::l ' ,sr: c E : .g~ u' ~: ~I ..§ : .~ ~ - ~i ] tZ : l ___ I Reconstruction a nd equidistant re-sampling I J r-------t 51 : g : 1 - __ I ' - -_ _ _ _---1~ Filter : '--T --' t AUlOcorrclation cstimatc I r-----------+------------. ,------------r------------' : Rcfincmcnt orcurvc fittin g: Spectral est imate e osine tra nsform I ________ J________ . . : Noisc supprcssion : '- ------- --- --- --' Fig. 11.4. Power spectral density estimation methods for laser Doppler data 11.2.1 The Siotting Technique The slotting technique, generally credited to Mayo et al. (1974), is pictured in Fig. 11.5 and computed as N N I~>juh{tj -ti) Rk = R(kAr)=_i~_lN"-:-j~.:....JN~---IIbk{t j -ti) i~l (11.14) j~J 1 for _t !-t _ - ' -k I<- I Ar 2 otherwise with the velo city sampIes uj = u(t) and u j = u(tj)' The velocity product of all sampIe pairs with time separations falling within a given lag time bin is added to that bin's sum as another estimate of the ACF for that lag time. After processing all possible signal pairs, each sum is divided by the number of accumulated products in that bin. The original algorithm of Mayo uses J = 1 which means that all sampIe pairs occur for k and -k. Calculating only positive lag times (k;;:::O) Eq. (11.14) can be modified to J=i. Note that the first ACF coefficient 11.2 Estimation of Turbulent Velo city Spectra 11(1) i j=i+l Vclocily j=i+2 ~ I .................. ~ f-ft-r I-I--r ~ + ++ + + + + + 555 () Vdocily mC'dsurcmCnI x Produci 11,11, limc 1 Summation umofproducts lvi ton Number 01 products Autocorrclal ion function T k= 0 I 2 3 4 5 6 7 8 9 Fig. 11.5. Time series of laser Doppler velocity data and slot correlation with equidistant lag time bins Llr (r = 0) eontains also self-products. With J = i the self-products are overrepresented compared to J =1. Therefore, an additional weighting factor of 1/2 is often used for self-products when (J = i). However, since noise in each of the velocity estimates is unavoidable, the first slot will be increased by the variance of the noise. Since the ACF eoefficient at T = 0 normally corresponds also to the variance of the process, this variance will be overestimated using the estimator given by Eq. (11.14). This also leads to a biased PSD estimate. Using only crossproducts (J = i + 1) leads to a spectral estimate with a noise-independent expectation. In fact, only the estimation variance increases in the case of noise. However, the first slot can also be under-represented due to processor dead times. This again, can lead to a bias of the k = 0 ACF eoefficient and thus, the PSD estimate. The ACF coefficient estimates are usually considered valid for time lags in the middle of each bin. A one-sided PSD estimator is computed from the slot correlation by taking its discrete eosine transform Sj = SVj) = s( 2/L1r) =2L1{ Ro +2~RkeoS(21tfjkL1r)+(-I)jRK) (11.15) where K is the index of the maximum time lag of the ACF and is chosen by the user. A severe limitation of the standard slotting technique is its high variance (roughly constant up to moderate lag times), which leads to poor estimates of turbulence spectra. In order to reduce the variance of the slotting technique, van 556 11 Processing of Laser Doppler Da ta Maanen and Tummers (1996) employ an ACF normalized by a variance estimate particular to each slot, called the local normalization. This results in an estimate ofthe correlation coefficient N N L~>;ujbk(tj -t;) Pk = p(kAr) = ----,====;~=l=j~J========= [t~U;bk(tj -t;))[t~U~bk(tj -t;)) (11.16) and is used as the basis for the cosine transform. The corresponding one-sided, real PSD estimator is S =S(f.)=s(-j-) 2KAr J J =2o-:A{I+2~Pk cos(21tfjkAr)+(-I)j PK) (11.17) where o-~ is an estimate of the velo city variance. While Eq. (11.16) has been shown to have significantly lower variance for small lag times than does Eq. (11.15) normalized by Ra' the variance at large lag tim es is unchanged. Another method for reducing variance in the slotting technique has been dubbed the fuzzy slotting technique by Nobach et al. (1998). In this estimator, a lag-product weighting scheme is defined as for It.JA-t r I - I k <1 (11.18) otherwise and is used instead ofthe top-hat function in the original algorithm, Eq. (11.14). This estimator allows lag products to contribute to two slots simultaneously and weights lag products that lie dose to the slot centers more heavily, as depicted in Fig. 11.6. The combination of this fuzzy slotting technique with local normalization is one of the recomrnended methods of computing the ACF. Buchhave et al. (1979) argued that the ACF can be corrected for the partiderate/velocity correlation by weighting each cross-product u; U j according to N N LLu;ujgig j Rk = R(kAr) = _i~_l7:-~~_J-;-:-N_ __ LLgigj ;~l (11.19) j~J where the weight g; is the transit time of the i th partide. This correction is combined with the previous estimator to yield a final general expression given as (Benedict et al. 2000) 11.2 Estimation ofTurbulent Velocity Spectra 11(1) Vclocily o i j;i+l j;i+2 II I o VclQ(.i ty measurcment x Producl - - ~- - - -x t x_ . . . . . . . . . ,. I *---*--X--r---t--T xt I . Ir () 11,(1 - I) J f x x x Time I X x xx + ++ + + + + ++ ++ ++ + ':1 x 557 •ummation Sum of wcighlcd produClS .,. .,. .,. .,. .,. + + + Division lot Wcighling • 1J - I , Wcighted numbcr of products function k= 2 0 6 5 ] 7 8 9 Fig. 11.6. The weighting scheme of the fuzzy slotting technique N N &:L~>;Ujg;gjbk(tj -ti) R(kLl ..) = -;=0====;==1~j=~J= = = = = = = = = = = = [t~u~gigjbk(tj -( )It~U~gigjbk(tj -t;)] b,(t;-t)={: (11.20) t. - t -k I<1 for I-}--' LI.. 2 otherwise with the velo city variance estimated as (11.21) i= l The weighting factors as introduced in section 11.1 can be used, but in the case of arrival time weighting the forward-backward weighting is advantageous (Nobach 1999b) 558 11 Processing of Laser Doppler Da ta g; =t; - tH (11.22) gj =t j +1 -tj because of the correlation between the time lag and the inter arrival time distribution. 11.2.2 Reconstruction with FFT Reconstruction methods create equidistantly spaced time series by re-sampling according to various interpolation schemes, thereby allowing a FFT to be used in making PSD or ACF estimates. The most common scheme by far is the sampleand-hold (zeroth-order, S+H) reconstruction. This is the simplest of the polynomial dass of reconstruction algorithms and is depicted schematically in Fig. 11. 7. It can be written as for U(SH)(t)=U(tJ t; <:::;t<t;+1 and i=l, ... ,N (11.23 ) where N is the total number of sampIes in a given block. The reconstruction can be performed either over the entire data set or with single data blocks. The equidistant re-sampling with time steps of Llt, is performed by U,(SH) = U(SH)(I' At,) s: lor LJ ° . 1= , ... , N R- 1 (11.24) and leads to a data set of N R sampIes that can be processed by a Fourier transform. The Fourier transform with the imaginary unit j is given by U (SH) _n (2' in ) - 1tJ- = ~1 L.. u;(SH) exp ;~O NR for n=O, ... ,N R -1 (11.25) and leads to the full block PSD SA(SH) n = Llt, IU(SH)1 2 NR _TI s: lor n=O, ... ,N R -1 (11.26) u(t) Particles x Re-sampled values Timet Fig. 11.7. The concept of zeroth-order reconstruction (S+H) and re-sampling at equal time intervals 11.2 Estimation of Turbulent Velocity Spectra 559 and, through the inverse FFT ~(SH) Kk ~l ~(SH) 1 = - ~ S; N R ;~O ( ik) • exp 2nJNR for k=O, ... ,N R -l (11.27) to the full block ACF (SH) R~ k(SH) = _1_X~ L1t k c lor k 0 =, ... , N R- 1 (11.28) s To reduce the variance of the final PSD estimate, only K + 2< N R values of the ACF are used for further calculations. K is the maximum desired lag time for the ACF. In the refinement step outlined below, K + 1 values of Rk are required and for the subsequent Fourier transform to the PSD one further sampie in the ACF is required, i.e. K + 2 sampies in total. Unquestionably the work of Adrian and Yao (1987) was a major step forward in understanding the fundamental content of a S+H reconstructed signal. They derived an expression for the expectation of the PSD [ ES~(SH) (0) ] = 12 ·2 ( S(W)+-·-3-2 2()~ ) 1 + 0) I N N Au '----y------l ~ Filter (11.29) Step noise where E[S(SH) (0)] is the expectation ofthe spectral estimate and S(w) is the true spectrum «()~ is the variance of the velo city fluctuations and Au is the Taylor microscale as defined by Eq. (7.46». The second term in parentheses was termed step noise and corresponds to the spectral contribution necessary to account for the step-like jumps in a S+H signal. This contribution vanishes with the inverse of the third power of the data rate, N- 3 • The factor in front of the parentheses, operating on both the true spectrum and the step noise, is a first-order, low-pass filter with cut-off frequency NI (2n). This was subsequently named the particlerate fIlter and at higher frequencies, dominates the spectrum. Actually the step noise is a form of aliasing, in which the signal energy above the fIlter cut-off frequency is distributed evenly over the entire spectrum. One result from Adrian and Yao (1987) is reproduced here because of the clarity it brings to how all reconstruction methods effect spectral estimators. Simulated laser Doppler data from a white noise process are submitted to a S+ H reconstruction, re-sampling and a conventional FFT PSD estimation. The result is shown in Fig. 11.8. The effects ofboth the additive step noise and the low-pass fIlter are clearly evident at low particle rates, indeed the completely falsified spectrum for the lowest data rate begins to uncannily resemble that of turbulence! The conclusion of Adrian and Yao was that such spectra are reliable only up to a frequency of NI (2n), a substitute for the Nyquist frequency rule concerning regularly sampled data. Their assumption that the ACF of turbulent velocity fluctuations is exponential was not instrumental to this conclusion. Of course the computed spectrum below the cut-off frequency is still falsified by the 560 11 Processing of Laser Doppler Da ta - Sl>l< - - - - 1 _1- _ _ - 1000 I- "- "- ----------------------------------------~ -'~----, " "\. \ 100 r- \ --N--->= ------- N = 1 "\. "\. "\. N=0.3 1 0.01 ... ... . 1 0.1 Frequency f[a.u.] Fig. 11.8. The PSD of a white noise process computed using a S+H reconstruction and resamplingwith FFT PSD estimation step noise or any other aliased energy from high er frequency components in the signal. Even the nature of the interpolation curve in the reconstruction does not alter the basic limitation of the particle-rate filter. This was demonstrated by Müller et al. (1994), who applied numerous other reconstruction schemes to simulated and measured laser Doppler data. These included higher order polynomials, projection onto convex sets (POCS) (Lee and Sung 1992), fractal reconstruction (Chao and Leu 1992) and the so-called Shannon reconstruction (Veynante and Candel 1988, Clark et al. 1985). Although the reconstructed signal in time domain was visually more appealing than a S+ H signal in many cases, the spectral content was altered surprisingly litde. Nobach et al. (1996) developed a refinement that cancels the particle-rate filter effect associated with S+ H reconstruction. The approach is to derive an expression for the re-sampled ACF in terms of the true ACF. The relation is then inverted to improve the ACF estimation. The inversion is given as for k =0 for k=l, ... K c= exp( - N,1t,) (11.30) ----'----'--.,,- [l-exp(-N,1t,)f where ft.. is the refined ACF estimate based on the ACF of the reconstructed and re-sampled time signal, ft..(SH). A full derivation of this relation is given by Nobach et al. (1998). The PSD follows from a eosine transform of R. In principle, a refinement can be derived for any reconstruction algorithm, but for the S+ H re- 11.2 Estima tion ofTurbulent Velo city Spectra 561 construction the refinement becomes the very simple algorithm given above and is effective enough that the advantages of other reconstruction schemes become negligible. 11.2.3 Post-Processing Steps 77.2.3.7 Model-based Variance and ACF EstimationlNoise Removal To remove the noise and the effect of the processor delay from the ACF estimate, a model-based estimation of Ro can be used. Principally speaking, a convenient modellike that ofvan Maanen and Oldenziel (1998) or Müller et al.(l998b) can be used. van Maanen and Oldenziel (1998) introduce an eight-parameter autocorrelation model, which is extremely flexible and can be analytically Fourier transformed. Nevertheless, the parameter optimization is difficult and costly. The use of a weighting function with strong coefficients dose to the time lag zero allows simpler models to be used. van Maanen and Tummers (1996) used a Gaussian function as a model of the ACF, corresponding to the Taylor microscale estimation (with parabolic behavior of R near r = 0). Good results were obtained using the more flexible model Rk = aexp(-bk c ) (11.31) which is equivalent to the Gaussian function for c = 2 and to the exponential function for c = 1. However, even this model is not able to describe periodic components, so that the weighting function should strongly decrease with the time lag r, i.e. 1/ r or a similar function. The figure of merit (11.32) gives the weighted deviation of the model ACF R k relative to the estimated ACF Rk • Note that the value Ro is not used because of the distorting effects. Minimizing the distance value d leads to an optimal parameter set [a;b;c], which is used to obtain a new ACF estimate at time lag zero Ro = a. 77.2.3.2 Variable Width Windowing From the ACF estimation, a set of K + 1 values (k = 0, .. .,K) is obtained, which can be transformed to the PSD using the discrete eosine transform (Eq. (11.15». Alternatively, Tummers and Passehier (1996) recommend a variable windowing of the ACF for the transform to the PSD: (11.33) with windowing coefficients dk(j), which varywith the frequency f Good experience was obtained using the Tukey-Hanning window 562 11 Processing of Laser Doppler Da ta I 1 -+-cos { dk(J)= 2 2 (1tf kAts ) for IfkAtsl < K K o (11.34) otherwise The parameter K can be chosen arbitrarily, e.g. K = 6 was found to yield good results. This technique reducesthe estimation variance espeeially for higher frequeneies. Although a leakage effect arises because of the windowing, this effect is constant over all frequeneies, hence no spectral distortion occurs. However, now the spectrum can be calculated at any frequency. This could reduce the number of spectrallines required in the case of a logarithmic axis scaling, which is often used to present turbulence spectra. This is important because the FFT cannot be used for this transform and every spectral value has to be calculated separately. An example improvement, achieved with the variable windowing of the ACF, is shown in Fig. 11.9, where, in particular, the variance displays a strong decrease. These results were obtained using simulated data as outlined in Benedict et al. (2000). b a "--T""TTTTmr-r"TTTTTTIr-rTTTTnrr----r"T"13..,- 10- 1 ..------.-.--rTTTTTI---.--,-,-rrTTTI-,--"" ~ 10' o g; 10' 0) u .~ 10-' :>'" 10-5 .... Rectangular window Variable 10-60.Ll---'-..l....L.LWlL-.L.L.J...J..LllJILO----L.I..l....L.J..WJ1O"-0---L..J...L.J Frequency J[kHzl 10 Frequency J[kHzl Fig. 11.9a,b. Comparison of the variable windowing technique with a rectangular window. ACF estimation using the fuzzy slotting technique with local normalization. a Mean spectrum (for illustration the variable windowing is shifted by one decade on the x axis.) b Variance 77.2.3.3 Block Averaging Each block of laser Doppler data yields an independent ACF and PSD estimate. For NB blocks, each of duration TB' the mean ACF estimate (R) and the mean PSD estimate (5) and the corresponding variances a-~ and a-~ of the single block estimates can be calculated using 11.3 Correlation Estimates [rom Multi-Point Systems 563 (11.35) (5;=_1 ISr;) NB (11.36) ;=\ (11.37) (11.38) where the upper index (i) represents the estimates of the i th block. The variance of the averaged ACF and PSD can be estimated through (11.39) 1 N A 2 A 2 (J'- = - ( J ' S S (11.40) B This result suggests that a block splitting of the input data record prior to processing could be advantageous, since the overall variances of the estimates decrease with increasing number of blocks. This is not actually true for the two estimators discussed here. Equations (11.39) and (11.40) are correct and can be used for estimating variances but in fact, the variances are determined alone by the total record length, T, for the averaged ACF and additionally by the maximum lag time for the PSD. A 2 1 (J'- - - - . - R rBNB (11.41) If an FFT is being used in the estimation procedure, the block length should be large relative to the correlation interval (integral time scale) and at least 2KAt to reduce edge effects. Effective use of Eqs. (11.39) and (11.40) also requires a minimum of 10 blocks. (K is the ACF index at the maximum time lag) To summarize, the slotting technique requires no block splitting and the reconstruction technique with refinement benefits from block splitting if the FFT is used to obtain the initial ACF estimate RrSH ). 11.3 Correlation Estimates from Multi-Point Systems Two or multi-channellaser Doppler systems are used when the space-time or spatial correlation function defined in Eq. (7.28) is to be measured. However, as pointed out in seetion 7.4.4, estimators must be carefully chosen to avoid bias errors at small spatial separations due to hardware coincidence effects. Fur- 564 11 Processing of Laser Doppler Da ta thermore, demanding coincidence on two channels separated in space greatly reduces the data collection rate, often far below the data rate on either of the individual channels. The duration of the measurement to achieve a reasonable number of sam pIes N, may become intolerable, or if fewer sampies are accepted, the variance of the estimator increases. The discussion below is therefore restricted to the case of non-coincidence data acquisition; meaning that data is acquired on each laser Doppler channel independent of the other but with a common time base. Und er these circumstances, two estimators can be recommended: • Slot correlation • Reconstruction with re-sampling and refinement Müller et al. (1998) have demonstrated that both estimators perform well, although the slot correlation does exhibit a larger bias at short time lags when a particle-rate/velocity correlation is present in the data set. Denoting the two laser Doppler channels as A and B, the slot correlation can be computed using NA NB L~>A (t; )uB(tj )bk(t j - t;) (kLir) = _;~_l-,j_~l_N:-O--'-'N------AB R !~bk(tj-t;) (11.42) ;=1 j=l 1 t - t 1 for k __ ~_J_ _' <k+2 Lir 2 otherwise where Lir is a prescribed lag time window or slot. As with ACF estimators, this cross-correlation estimator may be extended using various weights, a fuzzy slotting scheme or local normalization. The effect of these modifications have to date not been studied systematically. The method using reconstruction and re-sampling is pictured schematically in Fig. 11.10 for the case of a sample-and-hold (S+H) reconstruction. The conventional cross-correlation function is obtained from the re-sampled data (Eq. (11.23» using for k? 0 (11.43 ) for k < 0 where N r is the number of re-sampled points. This estimate of R AB (r) will have a systematic error associated with the S+ H reconstruction, similar in nature to that derived for a single channel system, as discussed in the previous seetion. To estimate and correct this error, a statistical relationship between the velo city 11.3 Correlation Estimates from Multi-Point Systems U A (t) - Channel A QiI x 565 Flowvelocity Reconstruction Particle Resampling Timet Timet Fig. 11.10. Sample-and-hold reconstruction and re-sampling of two laser Doppler ehannels values at the sampIe points r A = iArand r B = (i + k)Ar to the velocities at the previous particle passages t A and t B must be found. Details of this statistical relationship are lengthy and involve careful consideration of the particle arrival statistics in each of the measurement volumes. The solution is presented only conceptually here and the reader is referred to originalliterature for a complete solution (Müller et al. 1998). The re-sampled correlation function is related to the true correlation function through a matrix F (11.44) which, when inverted, yields an improved estimate of the correlation function (11.45) The matrix is a function of the total number of events in each of the channels, NA and NB' as weIl as the total observation time T. The performance of these two estimators has been evaluated using simulated data sets and can be summarized in Fig. 11.1l. The data density was 1.6, i.e. on average 1.6 particle arrivals per integral time seale. The slot correlation exhibits a high er bias due to the particle-rate/velocity correlation. This bias will increase at high er data densities. An application of suitable weights in estimating the slot correlation can decrease this bias. Having estimated the cross-correlation, the eross-spectral density can also be computed by performing one further FFT. As discussed in section 7.4.4, one purpose for conducting two-point laser Doppler measurements is to estimate Taylor mieroscales, Äii,k (Eq. (7.46». To do this, a parabolic curve-fit is performed on the measured spatial correlation function, near a spatial separation of zero (for isotropie turbulence). Llx 2 1 Rll (Ax,O,O) = 1 - 2 - ,111,1 (I 1.46) 566 11 Processing of Laser Doppler Da ta b a 0.8 r-T--r--,--,---,.,.-,...--r--'-T"""T--r--,--,---,.,.-,r-r......---. Refined estimator Simulation Slot correlation Simulation -10 -5 o 5 10-10 Timelag T[sl -5 o 5 10 Time lag T[sl Fig. l1.11a,b. Performance of cross-correlation estimators when a particle-rate/velocity correlation exists in input data. a Slot correlation without weights, b S+H estimator with refinement The importance and means of achieving unbiased estimates of Rll for low Llx j was the topic of section 7.4.4, nevertheless, the correlation estimator will generally exhibit some error at Llx j = 0 and will begin to deviate from a parabolic behavior at larger separations. Furthermore, as both Trimis and Melling (1995) and Belmabrouk and Michard (1998) demonstrate, the correlation function will exhibit a bias value at zero separation due to noise. Therefore, the range of separations used for the curve-fit must be carefully chosen. This issue has been addressed by Belmabrouk and Michard (1998). The minimum separation entering the curve-fit is determined by the noise influence and the influence of the measurement volume overlap. It is generally unambiguous from the measurements and should not depend on characteristics of the flow. The maximum separation to be included is