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Principles of Photonics With this self-contained and comprehensive text, students will gain a detailed understanding of the fundamental concepts and major principles of photonics. Assuming only a basic background in optics, readers are guided through key topics such as the nature of optical ﬁelds, the properties of optical materials, and the principles of major photonic functions regarding the generation, propagation, coupling, interference, ampliﬁcation, modulation, and detection of optical waves or signals. Numerous examples and problems are provided throughout to enhance understanding, and a solutions manual containing detailed solutions and explanations is available online for instructors. This is the ideal resource for electrical engineering and physics undergraduates taking introductory, single-semester or single-quarter courses in photonics, providing them with the knowledge and skills needed to progress to more advanced courses on photonic devices, systems, and applications. Jia-Ming Liu is Distinguished Professor of Electrical Engineering and Associate Dean for Academic Personnel of the Henry Samueli School of Engineering and Applied Science at the University of California, Los Angeles. Professor Liu has published over 250 scientiﬁc papers and holds 12 US patents, and is the author of Photonic Devices (Cambridge, 2005). He is a fellow of the Optical Society of America, the American Physical Society, the IEEE, and the Guggenheim Foundation. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:12:45 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 ENDORSEMENTS FOR LIU, PRINCIPLES OF PHOTONICS “With much thoughtfulness and a rigorous approach, Prof. Jia-Ming Liu has put together an excellent textbook to introduce students to the principles of photonics. This book covers a comprehensive list of subjects that allow students to learn the fundamental properties of light as well as key phenomena and functions in photonics. Compared to other textbooks in classical optics, this book places the necessary emphasis on photonics for readers who want to learn about this ﬁeld. Compared to other textbooks introducing photonics, this book is carefully and well written, with ample examples, illustrations, and well-designed homework problems. Instructors will ﬁnd this book very helpful in teaching the subjects, and students will ﬁnd themselves gaining solid understanding of the materials by reading and working through the book.” Lih Lin, University of Washington “For a long while the photonics community has been waiting for a new textbook which is informative, comprehensive, and also contains practical examples for students; in other words, one which describes fundamental concepts and provides working principles in optics. Professor Jia-Ming Liu’s book, Principles of Photonics, serves very well for these purposes – it covers optical phenomena and optical properties of materials, as well as the basic principles behind light emitting, modulation, ampliﬁcation and detection devices that are commonly used nowadays in communications, displays, and sensing. A distinguishing feature of this book is its seamless use of “additional space” to ensure that each concept is sufﬁciently explained in words, coupled with mathematics, simple yet illustrative ﬁgures, and/or examples. Each chapter ends with questions/problems followed by key references, making it very self-contained and very easy to follow.” Paul Yu, University of California, San Diego “A pedagogical tour-de-force. Professor Liu covers the principles of photonics with extreme attention to notation, completeness of derivations, and clear examples matched to the concepts being taught. This is a book one can really learn from.” Jeffrey Tsao, Sandia National Lab Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:12:45 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Principles of Photonics JIA-MING LIU University of California Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:12:45 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107164284 © Jia-Ming Liu 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalog record for this publication is available from the British Library Library of Congress Cataloging-in-Publication data Names: Liu, Jia-Ming, 1953- author. Title: Principles of photonics / Jia-Ming Liu. Description: Cambridge, United Kingdom : Cambridge University Press, [2016] | Includes bibliographical references and index. Identiﬁers: LCCN 2016011758 | ISBN 9781107164284 (Hard back : alk. paper) Subjects: LCSH: Photonics. Classiﬁcation: LCC TA1520 .L58 2016 | DDC 621.36/5–dc23 LC record available at https://lccn.loc.gov/2016011758 ISBN 978-1-107-16428-4 Hardback Additional resources for this publication at www.cambridge.org/9781107164284 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:12:45 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 To Vida and Janelle Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:12:57 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:12:57 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 CONTENTS Preface Partial List of Symbols 1 Basic Concepts of Optical Fields 1.1 Nature of Light 1.2 Optical Fields and Maxwell’s Equations 1.3 Optical Power and Energy 1.4 Wave Equation 1.5 Harmonic Fields 1.6 Polarization of Optical Fields 1.7 Optical Field Parameters Problems Bibliography 2 Optical Properties of Materials 2.1 Optical Susceptibility and Permittivity 2.2 Optical Anisotropy 2.3 Resonant Optical Susceptibility 2.4 Optical Conductivity and Conduction Susceptibility 2.5 Kramers–Kronig Relations 2.6 External Factors 2.7 Nonlinear Optical Susceptibilities Problems Bibliography 3 Optical Wave Propagation 3.1 Normal Modes of Propagation 3.2 Plane-Wave Modes 3.3 Gaussian Modes 3.4 Interface Modes 3.5 Waveguide Modes 3.6 Phase Velocity, Group Velocity, and Dispersion 3.7 Attenuation and Ampliﬁcation Problems Bibliography Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:08 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 page xi xiii 1 1 4 8 10 11 13 18 20 21 22 22 24 32 38 44 44 55 60 65 66 66 73 86 92 108 122 129 132 139 viii Contents 4 Optical Coupling 4.1 Coupled-Mode Theory 4.2 Two-Mode Coupling 4.3 Codirectional Coupling 4.4 Contradirectional Coupling 4.5 Conservation of Power 4.6 Phase Matching Problems Bibliography 5 Optical Interference 5.1 Optical Interference 5.2 Optical Gratings 5.3 FabryPérot Interferometer Problems Bibliography 6 Optical Resonance 6.1 Optical Resonator 6.2 Longitudinal Modes 6.3 Transverse Modes 6.4 Cavity Lifetime and Quality Factor 6.5 FabryPérot Cavity Problems Bibliography 7 Optical Absorption and Emission 7.1 Optical Transitions 7.2 Transition Rates 7.3 Attenuation and Ampliﬁcation of Optical Fields Problems Bibliography 8 Optical Ampliﬁcation 8.1 Population Rate Equations 8.2 Population Inversion 8.3 Optical Gain 8.4 Optical Ampliﬁcation 8.5 Spontaneous Emission Problems Bibliography Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:08 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 141 141 147 154 156 159 160 165 168 169 169 183 191 200 203 204 204 207 211 214 216 221 223 224 224 234 241 245 248 249 249 251 259 265 267 270 273 Contents 9 Laser Oscillation 9.1 Conditions for Laser Oscillation 9.2 Mode-Pulling Effect 9.3 Oscillating Laser Modes 9.4 Laser Power Problems Bibliography 10 Optical Modulation 10.1 Types of Optical Modulation 10.2 Modulation Schemes 10.3 Direct Modulation 10.4 Refractive External Modulation 10.5 Absorptive External Modulation Problems Bibliography 11 Photodetection ix 274 274 277 279 285 293 296 297 297 298 308 319 344 353 361 11.1 Physical Principles of Photodetection 11.2 Photodetection Noise 11.3 Photodetection Measures Problems Bibliography 362 362 375 382 391 395 Appendix A Appendix B Appendix C Appendix D Index 396 403 405 406 409 Symbols and Notations SI Metric System Fundamental Physical Constants Fourier-Transform Relations Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:08 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:08 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter Preface pp. xi-xii Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.001 Cambridge University Press PREFACE The ﬁeld of photonics has matured into an important discipline of modern engineering and technology. Its core principles have become essential knowledge for all undergraduate students in many engineering and scientiﬁc ﬁelds. This fact is fully recognized in the new curriculum of the Electrical Engineering Department at UCLA, which makes the principles of photonics a required course for all electrical engineering undergraduate students. Graduate students studying in areas related to photonics also need this foundation. The most fundamental concepts in photonics are the nature of optical ﬁelds and the properties of optical materials because the entire ﬁeld of photonics is based on the interplay between optical ﬁelds and optical materials. Any photonic device or system, no matter how simple or sophisticated it might be, consists of some or all of these functions: the generation, propagation, coupling, interference, ampliﬁcation, modulation, and detection of optical waves or signals. The properties of optical ﬁelds and optical materials are addressed in the ﬁrst two chapters of this book. The remaining nine chapters cover the principles of the major photonic functions. This book is written for a one-quarter or one-semester undergraduate course for electrical engineering or physics students. Only some of these students might continue to study advanced courses in photonics, but at UCLA we believe that all electrical engineering students need to have a basic understanding of the core knowledge in photonics because it has become an established key area of modern technology. Many universities already have departments that are entirely devoted to the ﬁeld of photonics. For the students in such photonics-speciﬁc departments or institutions, the subject matter in this book is simply the essential foundation that they must master before advancing to other photonics courses. Based on this consideration, this book emphasizes the principles, not the devices or the systems, nor the applications. Nevertheless, it serves as a foundation for follow-up courses on photonic devices, optical communication systems, biophotonics, and various subjects related to photonics technology. Because this book is meant for a one-quarter or one-semester course, it is kept to a length that can be completed in a quarter or a semester. Because it likely serves the only required undergraduate photonics course in the typical electrical engineering curriculum, it has to cover most of the essential principles. The chapters of this book are organized based on the major principles of photonics rather than based on device or system considerations. These attributes are the key differences between this book and other books in this ﬁeld. Through my teaching experience on this subject over many years, I ﬁnd a need for a textbook that has the following features. 1. It is self-contained, and its prerequisites are among the required core courses in the typical electrical engineering curriculum. 2. It covers the major principles in a single book that can be completely taught in a one-quarter or one-semester course. And it treats these subjects not superﬁcially but to a sufﬁcient depth Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:21 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.001 Cambridge Books Online © Cambridge University Press, 2016 xii Preface for a student to gain a solid foundation to move up to advanced photonics courses, if the student stays in the photonics ﬁeld, or for a student to gain a useful understanding of photonics, if the student moves on to a different ﬁeld. 3. It has ample examples that illustrate the concepts discussed in the text, and it has plenty of problems that are closely tied to these concepts and examples. This book is written with the above features to serve the need for a book covering a core photonics course in a modern electrical engineering curriculum. There are two prerequisites for a course that uses this book: (1) basic electromagnetics up to electromagnetic waves and (2) basic solid-state physics or solid-state electronics. No advanced background in optics beyond what a student normally learns in general physics is required. At UCLA, this course is taught as a required course in the Electrical Engineering Department to undergraduate juniors and seniors. The materials of this book have been test taught for a few years in this one-quarter course, which has 38 hours of lectures, excluding the time for the midterm and ﬁnal exams. This course is followed by elective courses on photonic devices and circuits, photonic sensors and solar cells, and biophotonics. Carefully designed examples are given at proper locations to illustrate the concepts discussed in the text and to help students apply what they learn to solving problems. Each example is tied closely to one or more concepts discussed in the text and is placed right after that text; its solution does not simply give the answer but presents a detailed explanation as part of the teaching process. An ample number of problems are given at the end of each chapter. The problems are labeled with the corresponding section numbers and are arranged in the sequence of the material presented in the text. The entire book has 100 examples and 247 problems. The materials in this book are selected and structured to suit the purpose of a course on the principles of photonics. Besides the newly written materials, text and ﬁgures are adopted from my book Photonic Devices wherever suitable. All examples and problems, except for the very few that illustrate key concepts, are newly designed speciﬁcally to meet the pedagogical purpose of this book. This book was developed through test teaching a course in the new curriculum at UCLA. In this process, I received much feedback from my colleagues and my students. I would like to thank my editor, Julie Lancashire, for her help at every stage during the development of this book, and my content manager, Jonathan Ratcliffe, for taking care of the production matters of this book. I would like to express my loving appreciation to my daughter, Janelle, who took a special interest in this project and shared my excitement in it. Special thanks are due to my wife, Vida, who gave me constant support and created an original oil painting for the cover art of this book. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:21 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.001 Cambridge Books Online © Cambridge University Press, 2016 PARTIAL LIST OF SYMBOLS Symbol Unit Meaning; derivatives References1 a none round-trip intracavity ﬁeld ampliﬁcation factor (6.4) aE , aM none asymmetry factors for TE and TM modes (3.130) ~ A, A W1=2 mode amplitude (4.23), (4.26) Av W1=2 amplitude of mode ν (4.3) A21 s1 Einstein A coefﬁcient (7.21) A m2 area (11.59) b m confocal parameter of Gaussian beam (3.69)f b none normalized guide index (3.129) b none linewidth enhancement factor (9.39) ~ B, B W1=2 mode amplitude (4.24), (4.27) B Hz bandwidth (11.1) B12 , B21 m3 J1 s1 Einstein B coefﬁcients (7.19), (7.20) B T real magnetic induction in the time domain (1.3) B, B T complex magnetic induction (1.41) c m s1 speed of light in free space (1.1)b, (1.39) cvμ none overlap coefﬁcient between modes v and μ (4.19) cijkl m2 A2 quadratic magneto-optic coefﬁcient (2.77) d m thickness or distance; d g , dQW (3.127) d, d0 m beam spot size diameter, d ¼ 2w, d 0 ¼ 2w0 (3.69)b dE , dM m effective waveguide thicknesses for TE and TM modes (3.138), (3.143) D none group-velocity dispersion; D1 , D2 , Dβ (3.167) D W1 detectivity (11.58) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 xiv Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 Dλ s m2 group-velocity dispersion; Dλ ¼ D=cλ (3.168) D∗ m Hz1=2 W1 speciﬁc detectivity (11.59) D C m2 real electric displacement in the time domain (1.2) D, D C m2 complex electric displacement; De , Do , Dþ , D (1.42), (1.51) D, D C m2 slowly varying amplitudes of D and D; De , D0 (3.57) DR dB dynamic range (11.62) e C electronic charge (2.30)f ^e none unit vector of electric ﬁeld polarization; ^e e , ^e o , ^e þ , ^e (1.61) E1 , E2 eV energies of levels j1i and j2i (7.1) Ec , Ev eV conduction-band and valence-band edges (10.106) EF eV Fermi energy (11.5)b Eg eV bandgap (10.105), (11.7) Eth eV threshold photon energy (11.5) E V m1 real electric ﬁeld in the time domain (1.2) E0 , E 0 V m1 static or low-frequency electric ﬁeld (2.54) Ee , Eh V m1 electric ﬁelds seen by electrons and holes (10.106) E, E V m1 complex electric ﬁeld (1.40) Ev , E v V m1 complex electric ﬁeld of mode v (3.1) E, E V m1 slowly varying amplitudes of E and E; E e , E o , E þ , E (1.52) Ev, E v V m1 complex electric ﬁeld proﬁle of mode v (3.1) ^v E V m1 W1=2 normalized electric mode ﬁeld distribution, E v ¼ Av E^ v (3.18) ER dB extinction ratio (10.18) f Hz acoustic or modulation frequency, f ¼ Ω=2π (2.79)b, (10.27)b f 3dB Hz 3-dB modulation bandwidth or cutoff frequency (10.31), (11.64) fK m Kerr focal length (10.115) f ijk m A1 linear magneto-optic coefﬁcient, Faraday coefﬁcient (2.76), (10.77) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Partial List of Symbols xv (cont.) Symbol Unit Meaning; derivatives References1 fr Hz relaxation resonance frequency, f r ¼ Ωr =2π (10.41) F none excess noise factor (11.38) F none ﬁnesse of interferometer or optical cavity (5.49), (6.12) Fðz; z0 Þ none forward-coupling matrix for codirectional coupling (4.48) g, g ðvÞ m1 gain coefﬁcient, ampliﬁcation coefﬁcient (3.183)f, (7.46) g0 m1 unsaturated gain coefﬁcient (8.22) g th m1 threshold gain coefﬁcient (9.9), (9.19) g^ ðvÞ s lineshape function (7.2) g none degeneracy factor; g1 , g2 (7.1)f, (7.28) g s1 gain parameter (9.18) g0 s1 unsaturated gain parameter (9.22) gn m3 s1 differential gain parameter (10.36) gp m3 s1 nonlinear gain parameter (10.36) gth s1 threshold gain parameter (9.20), (10.34) G none cavity round-trip ﬁeld gain; Gc , Gmn , Gcmn (6.4) G none photodetector current gain (11.4)f, (11.36) G, G0 none optical ampliﬁer power gain, G0 for unsaturated gain (8.39) h, ℏ Js Planck’s constant, ℏ ¼ h=2π (1.1) h1 , h2 , h3 m1 transverse oscillation parameters of mode ﬁeld (3.104), (3.133) H m height of acousto-optic transducer (10.89) H ðÞ none Heaviside step function (2.24) H m ðÞ none Hermite function (3.73)f H A m1 real magnetic ﬁeld in the time domain (1.3) H0 , H 0 A m1 static or low-frequency magnetic ﬁeld (2.68) H, H A m1 complex magnetic ﬁeld (1.42) Hv , H v A m1 complex magnetic ﬁeld of mode v (3.2) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 xvi Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 H, H A m1 slowly varying amplitudes of H and H (3.5) Hv , Hv A m1 complex magnetic ﬁeld proﬁle of mode v (3.2) ^ν H A m1 W1=2 normalized electric mode ﬁeld distribution, ^ H ¼AH (3.18) v v v i none pﬃﬃﬃﬃﬃﬃﬃ 1 i A current; ib , id , in , iph , is (11.4)f I A injection current; I 0 , I m , I th (10.22) I W m2 optical intensity; I 0 , I i , I in , I out , I r , I t (1.56) I0 A reverse current (11.15) I ðvÞ W m2 Hz1 optical spectral intensity distribution (7.17) Ip, Is W m2 optical pump and signal intensities (8.36) I sat W m2 saturation intensity (8.22) J, J A m2 real current density (1.5) J A m2 complex current density (2.35) k m1 propagation constant, wavenumber; k 0 , k i , k r , k t (1.84) kB J K1 Boltzmann constant (7.14), (7.25) ke , ko m1 propagation constants of extraordinary and ordinary waves (3.57) k 0 , k 00 m1 real and imaginary parts of k, k ¼ k 0 þ ik 00 (3.180) kx , ky , kz m1 propagation constants of x, y, and z polarized ﬁelds (2.15) kX , kY , kZ m1 propagation constants of X, Y, and Z polarized ﬁelds (2.67) kþ , k m1 propagation constants of circularly polarized ﬁelds (2.21) k^ none unit vector in the k direction (1.84) k m1 wavevector; ki , kr , kt , kq (1.1)b, (1.52) ke , ko m1 wavevectors of extraordinary and ordinary waves (3.56)f, (3.57) kx , ky , kz m1 wavevectors of x, y, and z polarized ﬁelds (3.48)f kþ , k m1 wavevectors of circularly polarized ﬁelds (10.74) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Partial List of Symbols xvii (cont.) Symbol Unit Meaning; derivatives References1 K m1 wavenumber of acoustic wave or grating, K ¼ 2π=Λ (2.79)f, (4.35) K s K factor of a semiconductor laser (10.43) K m1 wavevector of acoustic wave (2.79) l m length or distance (3.185) lc m coupling length; lPM c (4.56) lRT m round-trip optical path length (6.1) lλ=4 , lλ=2 m quarter-wave and half-wave lengths (3.49), (3.50) L m length of acousto-optic transducer (10.89) m none transverse mode index associated with x (3.1)f m none modulation index (10.27) m0 kg free electron rest mass Fig. 11.1 m∗ kg effective mass of carriers (2.31) ∗ m∗ e , mh kg effective masses of electrons and holes (10.107) M kg atomic or molecular mass (7.14) M TE , M TM none numbers of guided TE and TM modes (3.152), (3.153) Ms A m1 saturation magnetization (10.78) M A m1 real magnetic polarization in the time domain (1.3) M0 , M 0 A m1 static or low-frequency magnetization (2.70) n none transverse mode index associated with y (3.1)f n none index of refraction; nβ , n (1.84) n m3 electron concentration (11.9) n0 m3 equilibrium concentration of electrons (11.9)f n1 , n2 , n3 none refractive indices of waveguide layers, n1 > n2 > n3 (3.125) n2 m2 W1 coefﬁcient of intensity-dependent index change (10.101) ne , no none extraordinary and ordinary indices of refraction (2.15)f, (3.56) nx , ny , nz none principal indices of refraction (2.14) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 xviii Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 nX , nY , nZ none new principal indices of refraction (2.66) nþ , n none principal indices of refraction for circular polarized modes (2.20) n? , njj none indices of second-order magneto-optic effect (2.16) n0 , n00 none real and imaginary parts of refractive index, n ¼ n0 þ in00 (3.181) n^ none unit normal vector (1.23) N none some number (5.21) N none group index; N 1 , N 2 , N β (3.171) N m3 carrier density (2.31) N m3 effective population inversion (8.4) N1, N2, Nt m3 population densities in levels j1i, j2i, and all levels (7.26), (8.12) N sp none spontaneous emission factor (9.14) N none number of charge carriers (11.3) NEP W noise equivalent power (11.55) p none probability (11.18) p none cross-section ratio for pumping (8.13) p m3 hole concentration (11.9) p0 m3 equilibrium concentration of holes (11.9)f pijkl p0ijkl none elasto-optic and rotation-optic coefﬁcients (2.83) pðvk Þ Hz1 probability density function (7.10) P W power; Pa , Pin , Pout , Ppk , Pth , Pv (3.17) Pp , Ps W tr in out pump and signal powers; Pth p , Pp , Ps , Ps (9.27), (8.37) Psat W saturation power (8.37) Psp W spontaneous emission power (8.44) Ptrsp W critical ﬂuorescence power (8.46) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Partial List of Symbols xix (cont.) Symbol Unit Meaning; derivatives References1 ^ sp P W m3 spontaneous emission power density (8.43) p^trsp W m3 critical ﬂuorescence power density (8.45) P C m2 real electric polarization in the time domain (1.2) P, P C m2 complex electric polarization (1.50) PðnÞ C m2 nth-order nonlinear real electric polarization (2.91) PðnÞ , PðnÞ C m2 nth-order nonlinear complex electric polarization (2.91) Pres C m2 complex electric polarization from resonant transition (7.47)b q none longitudinal mode index (5.47), (6.9) q none order of coupling or diffraction (4.36), (5.24) q C charge (2.30) qðzÞ m complex radius of curvature of a Gaussian beam (3.75) Q none quality factor of resonator; Qmnq (6.26), (6.30) Q none acousto-optic diffraction parameter (10.83) r m radial coordinate, radial distance r none reﬂection coefﬁcient; r1 , r 2 , rp , rs (3.91), (4.67) r none pumping ratio of a laser (9.26) rijk , rαk m V1 linear electro-optic coefﬁcients, Pockels coefﬁcients (2.58), (2.60) rðf Þ, rðΩÞ none complex modulation response function (10.29), (10.40) r m spatial vector (1.2) R none reﬂectance, reﬂectivity; R1 , R2 , Rp , Rs (3.93) R Ω resistance; Ri , RL (11.16) R m3 s1 effective pumping rate for population inversion (8.6) R1 , R2 m3 s1 pumping rates for levels j1i and j2i (8.1), (8.2) Rðf Þ none electrical power spectrum of modulation response (10.30), (10.44) Rðz; 0; l Þ none R, Rij none reverse-coupling matrix for contradirectional coupling (4.59) (2.82) rotation tensor and elements, R ¼ Rij Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 xx Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 R m radius of curvature; R1 , R2 (3.71), (6.31) R none amplitude of rotation; Rij (2.87) R A W1 responsivity of photodetector with current output; R0 (11.50) R V W1 responsivity of photodetector with voltage output (11.51) s m separation, spacing; se Fig. 4.2, (10.69) s none signal; sn (11.18) sijkl , sαkl m2 V2 quadratic electro-optic coefﬁcients, Kerr coefﬁcients (2.58), (2.60) S m3 photon density (9.21) Ssat m3 saturation photon density (9.24) S W m2 real Poynting vector (1.32) S W m2 (1.54) S, Sij none complex Poynting vector; Se , So strain tensor and elements, S ¼ Sij (2.81) S none amplitude of strain; S ij (2.87) S none number of photons (11.2) SNR none, dB signal-to-noise ratio (11.26) t s time t none transmission coefﬁcient; tp , ts (3.92) tr , tf s risetime and falltime (11.63)b T K temperature (7.14) T s time interval (1.53) T s round-trip time of optical cavity (6.1) T none transmittance, transmissivity; T p , T s (3.94), (10.108) u, u0 J m3 electromagnetic energy density (7.16), (1.33) uðvÞ J m3 Hz1 spectral energy density (7.16) u, ui m elastic deformation wave and its components (2.79), (2.81) U J optical energy; U mode (9.28) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Partial List of Symbols xxi (cont.) Symbol Unit Meaning; derivatives References1 U m amplitude of elastic wave (2.79) v V voltage; v n , v out , v s (11.16) v m s1 velocity Fig. 11.1 va m s1 acoustic wave velocity (2.80)b vg m s1 group velocity; v gβ (3.165) vp m s1 phase velocity; v pβ (3.162) V none normalized frequency and waveguide thickness, V number (3.128) V rad A1 Verdet constant (10.77) V V voltage; V m , V π , V π=2 (10.51), (11.15) Vc none cutoff V number; V cm (3.147) V m3 volume; V gain , V mode (1.31)b, (6.2) w, w0 m Gaussian beam radius, spot size (3.69), (3.70) W m width of acousto-optic cell (10.91) W s1 transition probability rate; W 12 , W 21 , W p , W sp (7.22)(7.24) W p, W m W m3 power densities expended by EM ﬁeld on P and M (1.34), (1.35) W ðvÞ none transition rate per unit frequency; W 12 ðvÞ, W 21 ðvÞ,W sp ðvÞ (7.19)(7.21) x m spatial coordinate ^x none unit coordinate vector or principal dielectric axis X m ^ spatial coordinate along X ^ X none new principal dielectric axis y m spatial coordinate ^y none unit coordinate vector or principal dielectric axis Y m spatial coordinate along Y^ Y^ none new principal dielectric axis z m spatial coordinate Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 (1.62), (2.13)b (2.65)b (1.62), (2.13)b (2.65)b xxii Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 ^z none unit coordinate vector or principal dielectric axis (3.16), (2.13)b zR m Rayleigh range of a Gaussian beam (3.69) Z m spatial coordinate along Z^ Z^ none new principal dielectric axis (2.65)b α rad ﬁeld polarization angle (1.64) α rad walk-off angle of extraordinary wave (3.60) α, αðvÞ m1 attenuation coefﬁcient, absorption coefﬁcient (3.180), (7.45) α0 m1 unsaturated absorption coefﬁcient (10.110) αc m1 propagation parameter for contradirectional coupling (4.61) β none bottleneck factor (8.7) β m1 propagation constant of a mode; βmn , βTE , βTM (3.1) β0 , β00 m1 real and imaginary parts of β, β ¼ β0 þ iβ00 (3.184) βc m1 propagation parameter for codirectional coupling (4.50) γ s1 relaxation rate, decay rate; γ21 , γi , γout (2.23) γ1 , γ2 , γ3 m1 transverse decay parameters of mode ﬁeld (3.118), (3.131) γa s1 acoustic decay rate (10.93) γc s1 cavity decay rate, photon decay rate; γcmnq (6.25) γn s1 differential carrier relaxation rate (10.37) γp s1 nonlinear carrier relaxation rate (10.37) γr s1 total carrier relaxation rate (10.42) γs s1 spontaneous carrier relaxation rate (10.42) Γ none overlap factor (6.2) δ m1 phase mismatch parameter for phase mismatch of 2δ (4.31) δωmnq rad s1 frequency shift of mode pulling (9.12) Δn, Δp m3 excess electron and hole concentrations (11.10) ΔP C m2 change in electric polarization (4.8) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Partial List of Symbols xxiii (cont.) Symbol Unit Meaning; derivatives References1 Δt s pulsewidth or time duration; Δt ps (10.117), (11.1) Δϵ, Δϵ F m1 change or variation in electric permittivity (2.55), (4.12) ~ Δϵ , Δ~ϵ F m1 amplitudes of Δϵ and Δϵ (2.88) Δη, Δηij none change or variation in relative impermeability (2.58) Δθ rad divergence angle of a Gaussian beam (3.72) Δλ m spectral width; Δλg Table 7.1 Δv Hz optical linewidth, bandwidth; ΔvD , Δvg , Δvinh , Δvh (7.4) Δvc Hz longitudinal mode linewidth (6.18) ΔvL Hz longitudinal mode frequency spacing (6.17) Δvmnq Hz oscillating laser mode linewidth (9.13) ΔvST Hz SchawlowTownes linewidth of laser mode; ΔvST mnq (9.14) Δφ rad phase shift or phase retardation (10.13) Δφc rad phase width of a cavity resonance peak (6.11) ΔφL rad phase spacing between cavity resonance peaks (6.10) Δχ, Δχ none change or variation in electric susceptibility (2.54) Δω rad s1 optical linewidth, bandwidth, Δω ¼ 2πΔv; Δωinh , Δωh (7.3)f, (7.13) ϵ F m1 electric permittivity (2.11), (3.4) ϵ0 F m1 electric permittivity of free space (1.2) ϵ 0 , ϵ 00 F m1 real and imaginary parts of ϵ, ϵ ¼ ϵ 0 þ iϵ 00 (3.179) ϵx, ϵy, ϵz F m1 principal dielectric permittivities (2.13) ϵX , ϵY , ϵZ F m1 new principal dielectric permittivities (2.65) ϵþ, ϵ F m1 principal dielectric permittivities of circular polarizations (2.17) ϵðr, tÞ F m4 s1 real permittivity tensor in the real space and time domain (1.21) ϵ ðωÞ, ϵ ij F m1 complex permittivity tensor in the frequency domain (1.60) ϵ res ðωÞ F m1 permittivity of resonant transition (6.36) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 xxiv Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 ε rad ellipticity of polarization ellipse (1.68) ζ mn ðzÞ rad phase variation of Gaussian mode ﬁeld; ζ RT mn (3.76) η none coupling efﬁciency; ηPM (4.55) ηc none power conversion efﬁciency (9.37) ηcoll none collection efﬁciency (11.48) ηe none external quantum efﬁciency (10.24), (11.48) ηi none internal quantum efﬁciency (11.48) ηinj none injection efﬁciency (10.22) ηs none slope efﬁciency, differential power conversion efﬁciency (9.38) ηt none transmission efﬁciency (11.48) η, ηij , ηα none relative impermeability tensor and its elements, η ¼ ½ηij (2.57) θ rad angle, spherical angular coordinate (3.51) θ rad orientation of the polarization ellipse (1.69) θB rad Brewster angle or Bragg angle (3.100), (10.88) θc rad critical angle (3.102) θd rad angle of diffraction (10.87) θdef rad deﬂection angle Example 10.9 θF rad Faraday rotation angle (10.75) θi , θr , θt rad angles of incidence, reﬂection, and refraction (transmitted) (3.88) κ m1 coupling coefﬁcient; κvμ (4.13) ~κ m1 coupling coefﬁcient; κ~vμ (4.20) λ m optical wavelength in free space (1.1) λc m cutoff wavelength; λcm (3.151) λth m threshold wavelength (11.5) Λ m acoustic wavelength or grating period (2.79)b, (4.35) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 μ0 H m1 magnetic permeability of free space (1.3) μe , μh m2 V1 s1 electron and hole mobilities (11.9) v Hz optical frequency (1.1) v0 Hz center optical frequency (2.27)f, (7.12) v21 Hz resonance frequency between levels j1i and j2i (7.1) ξ none duty factor Fig. 4.3 ξ, ξ ðM 0z Þ none permittivity tensor elements for circular birefringence (2.16), (2.78) ρ C m3 charge density (1.6) ρF rad m1 speciﬁc Faraday rotation (10.79) σ S m1 conductivity; σ 0 (2.33), (11.9) σ 12 , σ 21 m2 transition cross sections (7.36), (7.37) σa , σe m2 absorption and emission cross sections (7.38), (7.39) σ 2s none variance of s (11.19) τ s lifetime, decay time, or time constant (2.30), (7.6) τ1, τ2 s ﬂuorescence lifetimes of levels j1i and j2i (7.6), (7.8) τc s photon lifetime; τ cmnq (6.23) τs s saturation lifetime or spontaneous carrier lifetime (8.23), (10.23) τ sp s spontaneous radiative lifetime (7.32) ϕ rad azimuthal angle, azimuthal angular coordinate (3.52) ϕ V work function potential; eϕ ¼ work function (11.6) φ rad phase or phase shift (1.63), (1.83) χ none electric susceptibility (2.11) χ V electron afﬁnity potential; eχ ¼ electron affinity (11.7) χ res none resonant electric susceptibility (2.25), (2.26) χx, χy, χz none principal dielectric susceptibilities (2.15)f χ 0 , χ 00 none real and imaginary parts of χ, χ ¼ χ 0 þ 1χ 00 (2.7)b Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 xxv xxvi Partial List of Symbols (cont.) Symbol Unit Meaning; derivatives References1 χðr; tÞ m3 s1 real susceptibility tensor in the real space and time domain (1.20) χðωÞ, χ ij none complex susceptibility tensor in the frequency domain (1.59) ð2 Þ m V1 second-order nonlinear susceptibility in the frequency (2.98), (2.100) domain χð3Þ , χ ijkl ð3 Þ m2 V2 third-order nonlinear susceptibility in the frequency domain (2.99), (2.101) ψ rad spatial phase of mode ﬁeld distribution (3.107) ψe rad angle between Se and optical axis of crystal (3.60) ω rad s1 optical angular frequency; ω ¼ 2πv (1.1)b ω0 rad s1 center optical angular frequency; ω0 ¼ 2πv0 (2.22), (7.13) ω21 rad s1 resonance angular frequency between levels j1i and j2i (2.22) ωc rad s1 cutoff frequency; ωcm (3.151) Ω rad s1 acoustic or modulation angular frequency; Ω ¼ 2πf (2.79), (10.27) Ωr rad s1 relaxation resonance frequency; Ωr ¼ 2πf r (10.41) χð2Þ , χ ijk 1 Sufﬁxes, f “forward” and b “backward,” on the equation number indicate symbols explained for the ﬁrst time in the text immediately after or before the equation cited. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:32 BST 2016. http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781316687109 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 1 - Basic Concepts of Optical Fields pp. 1-21 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge University Press 1 1.1 Basic Concepts of Optical Fields NATURE OF LIGHT .............................................................................................................. Photonics addresses the control and use of light for various applications. Light is electromagnetic radiation of frequencies in the range from 1 THz to 10 PHz, corresponding to wavelengths between 300 μm and 30 nm in free space, which is generally divided into the infrared, visible, and ultraviolet regions. In this spectral region, the electromagnetic radiation exhibits the dual nature of photon and wave. The photon nature has to be considered in the generation, ampliﬁcation, frequency conversion, or detection of light, whereas the wave nature is important in all processes but especially in the propagation, transmission, interference, modulation, or switching of light. 1.1.1 Photon Nature of Light The energy of a photon is determined by its frequency ν or, equivalently, its angular frequency ω ¼ 2πν. Associated with its particle nature, a photon has a momentum determined by its wavelength λ or, equivalently, its wavevector k. These characteristics are summarized below for a photon in free space: speed energy momentum c ¼ λν; hν ¼ ℏω ¼ pc; p ¼ hν=c ¼ h=λ, p ¼ ℏk. The energy of a photon that has a wavelength of λ in free space can be calculated using the formula: hν ¼ 1:2398 1239:8 μm eV ¼ nm eV: λ λ (1.1) The photon energy at the optical wavelength of 1 μm is 1.2398 eV, and its frequency is 300 THz. EXAMPLE 1.1 The visible spectrum ranges from 700 nm wavelength at the red end to 400 nm wavelength at the violet end. What is the frequency range of the visible spectrum? What are the energies of visible photons? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 2 Basic Concepts of Optical Fields Solution: The 700 nm optical wavelength at the red end has a frequency of νred ¼ c λred ¼ 3 108 m s1 ¼ 429 THz 700 nm and a photon energy of hνred ¼ 1239:8 1239:8 nm eV ¼ eV ¼ 1:77 eV: λred 700 The 400 nm optical wavelength at the violet end has a frequency of νviolet ¼ c λviolet ¼ 3 108 m s1 ¼ 750 THz 400 nm and a photon energy of hνviolet ¼ 1239:8 1239:8 nm eV ¼ eV ¼ 3:10 eV: λviolet 400 Therefore, the frequency range of the visible spectrum is from 429 THz to 750 THz. Visible photons have energies in the range from 1.77 eV to 3.10 eV. The energy of a photon is determined only by its frequency or, equivalently, by its free-space wavelength, but not by the light intensity. The intensity, I, of monochromatic light is related to the photon ﬂux density, or the number of photons per unit time per unit area, by photon flux density ¼ I I ¼ : hν ℏω The photon ﬂux, or the number of photons per unit time, of a monochromatic optical beam is related to the beam power P by photon flux ¼ P P ¼ : hν ℏω EXAMPLE 1.2 Find the photon ﬂux of a monochromatic optical beam that has a power of P ¼ 1 W by taking its wavelength at either end of the visible spectrum. What are the momentum carried by a red photon and the momentum carried by a violet photon? What is the total momentum carried by the beam in a time duration of Δt ¼ 1 s? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.1 Nature of Light 3 Solution: From Example 1.1, the photon energy of the 700 nm wavelength at the red end is hνred ¼ 1:77 eV, and that of the 400 nm wavelength at the violet end is hνviolet ¼ 3:10 eV. Therefore, the photon ﬂux of a beam that has a power of P ¼ 1 W at the 700 nm red wavelength is red photon flux ¼ P 1 ¼ s1 ¼ 3:53 1018 s1 , hνred 1:77 1:6 1019 and the photon ﬂux of a beam that has a power of P ¼ 1 W at the 400 nm violet wavelength is violet photon flux ¼ P hνviolet ¼ 1 s1 ¼ 2:02 1018 s1 : 3:10 1:6 1019 The momentum carried by a red photon is pred ¼ hνred 1:77 1:6 1019 N s ¼ 9:44 1028 N s, ¼ c 3 108 and that carried by a violet photon is pviolet ¼ hνviolet 3:10 1:6 1019 ¼ N s ¼ 1:65 1027 N s: c 3 108 The total momentum carried by an optical beam that has a power of P during a time duration of Δt is independent of the optical wavelength: total momentum ¼ ðphoton fluxÞpΔt ¼ P hν PΔt Δt ¼ : hν c c Therefore, irrespective of whether the wavelength of the beam is at the red or the violet end, the total momentum carried by the beam in a time duration of Δt ¼ 1 s is total momentum ¼ PΔt 11 ¼ 3:33 109 N: ¼ c 3 108 1.1.2 Wave Nature of Light An optical wave is characterized by the space and time dependence of the optical ﬁeld, which is composed of coupled electric and magnetic ﬁelds governed by Maxwell’s equations. It varies with time at an optical carrier frequency, and it propagates in a spatial direction determined by a wavevector. The behavior of an optical wave is strongly dependent on the optical properties of the medium. An optical ﬁeld is a vectorial ﬁeld characterized by ﬁve parameters: polarization, magnitude, phase, wavevector, and frequency. Polarization and wavevector are vectorial quantities; magnitude, frequency, and phase are scalar quantities. The general properties of optical ﬁelds are described in the following sections. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 4 Basic Concepts of Optical Fields 1.2 OPTICAL FIELDS AND MAXWELL’S EQUATIONS .............................................................................................................. An electromagnetic ﬁeld in a medium is characterized by four vectorial ﬁelds: electric ﬁeld electric displacement magnetic ﬁeld magnetic induction Eðr; t Þ Dðr; t Þ H ðr; t Þ Bðr; t Þ V m1 , C m2 , A m1 , T or Wb m2 : The response of a medium to an electromagnetic ﬁeld generates the polarization and the magnetization: polarization (electric polarization) magnetization (magnetic polarization) Pðr; tÞ C m2 , M ðr; t Þ A m1 : The electric ﬁeld Eðr; tÞ and the magnetic induction Bðr; t Þ are the macroscopic forms of the microscopic ﬁelds seen by the charge and current densities in the medium. The polarization Pðr; t Þ and the magnetization M ðr; t Þ are the macroscopically averaged densities of microscopic electric dipoles and magnetic dipoles that are induced by the presence of the electromagnetic ﬁeld in the medium. These macroscopic forms are obtained by averaging over a volume that is small compared to the dimension of the optical wavelength but is large compared to the atomic dimension. The electric displacement Dðr; tÞ and the magnetic ﬁeld H ðr; t Þ are macroscopic ﬁelds deﬁned as Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; tÞ, (1.2) and H ðr; tÞ ¼ 1 Bðr; t Þ M ðr; t Þ, μ0 (1.3) where ϵ 0 1=36π 109 F m1 ¼ 8:854 1012 F m1 is the electric permittivity of free space and μ0 ¼ 4π 107 H m1 is the magnetic permeability of free space. In addition to the induced charge density and current density that respectively generate electric dipoles and magnetic dipoles for Pðr; t Þ and M ðr; t Þ, an independent charge or current density, or both, from external sources may exist: charge density current density ρðr; tÞ C m3 , J ðr; t Þ A m2 : The behavior of a space- and time-varying electromagnetic ﬁeld in a medium is governed by space- and time-dependent macroscopic Maxwell’s equations: ∇E¼ ∇H ¼ ∂B , ∂t ∂D þ J, ∂t Faraday’s law; Ampère’s law; Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 (1.4) (1.5) 1.2 Optical Fields and Maxwell’s Equations ∇ D ¼ ρ, ∇ B ¼ 0, 5 Gauss’s law; Coulomb’s law; (1.6) absence of magnetic monopoles: (1.7) Note that Gauss’s law in the form of (1.6) is equivalent to Coulomb’s law because one can be derived from the other. The current and charge densities are constrained by the continuity equation: ∇J þ ∂ρ ¼ 0, ∂t conservation of charge: (1.8) The total current density in an optical medium has two contributions: the polarization current from the bound charges of the medium and the current from free charge carriers, thus Jtotal ¼ J bound þ J free . The free-carrier current has two possible origins, one from the response of the conduction electrons and holes of the medium to the optical ﬁeld and the other from an external current source: J free ¼ J cond þ J ext . Both J bound and J cond are induced by the optical ﬁeld; thus J total ¼ J bound þ J free ¼ J bound þ Jcond þ J ext ¼ Jind þ J ext , (1.9) where J ind ¼ J bound þ J cond : Similarly, the total charge density can be decomposed as ρtotal ¼ ρbound þ ρfree ¼ ρbound þ ρcond þ ρext ¼ ρind þ ρext : (1.10) In an optical medium, charge conservation requires that an increase of charge density induced by an optical ﬁeld at a location is always accompanied by a reduction at another location, resulting in no net macroscopic induced charge density. Therefore, ρind ¼ 0 and ρtotal ¼ ρext for a macroscopic optical ﬁeld. By contrast, an induced macroscopic current density of J ind 6¼ 0 can exist in an optical medium. In an optical medium that is free of external sources, J ext ¼ 0 and ρtotal ¼ ρext ¼ 0, but Jtotal ¼ J bound þ J cond ¼ J ind 6¼ 0: Both J bound and Jcond are induced currents in response to an optical ﬁeld. The bound-electron polarization current J bound is a displacement current that is always included in the ∂D=∂t term but not in the J term in (1.5). The conduction current J cond is also an induced current, but it is carried by free charge carriers in the medium. In the case when both external current and external charge are absent, the form of Maxwell’s equations depends on how the conduction current is treated. There are generally two alternatives. 1. Being an induced current, J cond can be considered as a displacement current to be included in the ∂D=∂t term so that J ¼ 0 in (1.5). Then, Maxwell’s equations are ∇E¼ ∇H ¼ ∂B , ∂t ∂D , ∂t (1.11) (1.12) ∇ D ¼ 0, (1.13) ∇ B ¼ 0, (1.14) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 6 Basic Concepts of Optical Fields where D is the electric displacement that includes optical-ﬁeld-induced responses from all bound and conduction charges in the medium. 2. Being a current carried by free charge carriers, J cond can be separated from the ∂D=∂t term so that J ¼ J cond in (1.5). Then, Maxwell’s equations have the form: ∇E¼ ∇H ¼ ∂B , ∂t ∂Dbound þ J cond , ∂t (1.15) (1.16) ∇ Dbound ¼ 0, (1.17) ∇ B ¼ 0, (1.18) with ∇ J cond ¼ 0, where Dbound is the electric displacement that includes only the contribution from bound charges and excludes that from the conduction current. These two alternative forms of Maxwell’s equations are equivalent. The form using (1.16) is taken only when a speciﬁc effect of the conduction current is considered, as in Section 2.4. Otherwise, the form using (1.12) is generally taken. Therefore, we use the general form given in (1.11)–(1.14) unless the situation calls for speciﬁc attention to a conduction current. 1.2.1 Transformation Properties Maxwell’s equations and the continuity equation are the basic physical laws that govern the behavior of electromagnetic ﬁelds. They are invariant under the transformation of space inversion, in which the spatial vector r is changed to r0 ¼ r, i.e., r ! r, or ðx; y; zÞ ! ðx; y; zÞ, and under the transformation of time reversal, in which the time variable t is changed to t 0 ¼ t, i.e., t ! t: This means that the form of these equations is not changed when we perform the space-inversion transformation or the time-reversal transformation, or both together. The ﬁeld quantities that appear in Maxwell’s equations, however, do not have to be invariant under space inversion or time reversal. Their transformation properties are summarized as follows. 1. Electrical ﬁelds: The electric ﬁeld vectors E, D, and P are polar vectors associated with the charge-density distribution. They change sign under space inversion but not under time reversal. 2. Magnetic ﬁelds: The magnetic ﬁeld vectors B, H, and M are axial vectors associated with the current-density distribution. They change sign under time reversal but not under space inversion. 3. Charge density: The charge density ρ is a scalar. It does not change sign under either space inversion or time reversal. 4. Current density: The current density J is a polar vector that is the product of charge density and velocity: J ¼ ρv. It changes sign under either space inversion or time reversal following the sign change of the velocity vector under either transformation. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.2 Optical Fields and Maxwell’s Equations 7 1.2.2 Optical Response of a Medium Polarization and magnetization are generated in a medium by the response of the medium to the electric and magnetic ﬁelds, respectively: Pðr; tÞ depends on Eðr; t Þ, and M ðr; t Þ depends on Bðr; tÞ: At an optical frequency, the magnetization vanishes: M ¼ 0: Therefore, it is always true for an optical ﬁeld that Bðr; t Þ ¼ μ0 Hðr; tÞ: (1.19) Because μ0 is a constant that is independent of the medium, the magnetic induction Bðr; t Þ can be replaced by μ0 H ðr; t Þ for any equations that describe optical ﬁelds, including Maxwell’s equations, thus effectively eliminating one ﬁeld variable. Note that this is not true at DC or low frequencies, however, because a nonzero DC or low-frequency magnetization, M 6¼ 0, can exist in any material. Indeed, it is possible to change the optical properties of a medium through a magnetization induced by a DC or low-frequency magnetic ﬁeld, leading to the functioning of magneto-optics. It should be noted that even for magneto-optics, the magnetization is induced by a DC or low-frequency magnetic ﬁeld that is separate from the optical ﬁeld. No magnetization is induced by the magnetic component of the optical ﬁeld. The optical properties of a material are completely determined by the relation between Pðr; tÞ and Eðr; tÞ: This relation is generally characterized by an electric susceptibility tensor, χ, through the following deﬁnition for electric polarization, ðt ððð Pðr; t Þ ¼ ϵ 0 χðr r0 ; t t 0 Þ Eðr0 , t 0 Þdr0 dt 0: (1.20) ∞ all r0 The relation between Dðr; t Þ and Eðr; t Þ is characterized by the electric permittivity tensor, ϵ, of the medium: ðt ððð Dðr; t Þ ¼ ϵ 0 Eðr; tÞ þ Pðr; t Þ ¼ ϵ ðr r0 ; t t 0 Þ Eðr0 , t 0 Þdr0 dt 0: (1.21) ∞ all r0 From (1.20) and (1.21), the relationship between χ and ϵ in the real space and time domain is ϵ ðr; t Þ ¼ ϵ 0 ½δðrÞδðt ÞI þ χðr; tÞ, (1.22) where I is the identity tensor that has the form of a 3 3 unit matrix and the delta functions are ÐÐÐ Ð∞ Dirac delta functions: all r δðrÞdr and ∞ δðtÞdt ¼ 1. The relation in (1.22) indicates that χ and ϵ contain exactly the same information about the medium: one is known when the other is known. Because χ and, equivalently, ϵ represent the response of a medium to an optical ﬁeld and thus completely characterize the macroscopic electromagnetic properties of the medium, (1.20) and (1.21) can be regarded as the deﬁnitions of Pðr; t Þ and Dðr; t Þ, respectively. 1.2.3 Boundary Conditions At the interface of two media of different optical properties, as shown in Fig. 1.1, the optical ﬁeld components must satisfy certain boundary conditions. These boundary conditions can be Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 8 Basic Concepts of Optical Fields Figure 1.1 Boundary between two media of different optical properties. derived from Maxwell’s equations given in (1.11)–(1.14). From (1.11) and (1.12), the tangential components of the ﬁelds at the boundary satisfy n^ E1 ¼ n^ E2 , (1.23) n^ H 1 ¼ n^ H2 , (1.24) where n^ is the unit vector normal to the interface as shown in Fig. 1.1. From (1.13) and (1.14), the normal components of the ﬁelds at the boundary satisfy n^ D1 ¼ n^ D2 , (1.25) n^ B1 ¼ n^ B2 : (1.26) The tangential components of E and H are continuous across an interface, while the normal components of D and B are continuous. Because B ¼ μ0 H at an optical frequency, as discussed above, (1.24) and (1.26) also imply that the tangential component of B and the normal component of H are also continuous. Consequently, all of the magnetic ﬁeld components in an optical ﬁeld are continuous across a boundary. Possible discontinuities in an optical ﬁeld exist only in the normal component of E or in the tangential component of D. 1.3 OPTICAL POWER AND ENERGY .............................................................................................................. Taking the dot product of H and (1.4) and that of E and (1.5) yields H ð∇ EÞ ¼ H E ð∇ H Þ ¼ E ∂B , ∂t ∂D þ E J: ∂t (1.27) (1.28) Using the vector identity B ð∇ AÞ A ð∇ BÞ ¼ ∇ ðA BÞ, (1.27) and (1.28) can be combined to give Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.3 Optical Power and Energy 9 Figure 1.2 Boundary surface enclosing a volume element. ∇ ðE H Þ ¼ E J þ E ∂D ∂B þH : ∂t ∂t Using (1.2) and (1.3) and rearranging (1.29), we obtain ∂P ∂ ϵ 0 2 μ0 ∂M 2 E J ¼ ∇ ðE HÞ þ μ0 H : jEj þ jH j E 2 ∂t 2 ∂t ∂t (1.29) (1.30) Recall that power in an electric circuit is given by voltage times current and has the unit of W ¼ V A (watts = volts amperes). Similarly, in an electromagnetic ﬁeld E J is the power density and has the unit of V A m3 , or W m3 . From (1.30), the total power dissipated by an electromagnetic ﬁeld in a volume of V is simply the integral of E J over the volume: ð þ ð ð ∂ ϵ 0 2 μ0 ∂P ∂M 2 E JdV ¼ E H n^da E þ μ0 H dV , (1.31) jEj þ jH j dV 2 2 ∂t ∂t ∂t V A V V where the ﬁrst term on the right-hand side is a surface integral over the closed surface A of the volume V and n^ is the outward-pointing unit normal vector of the surface, as shown in Fig. 1.2. Each term in (1.31) has the unit of power, and each has an important physical meaning. 1. The vectorial quantity S¼EH (1.32) is called the Poynting vector of the electromagnetic ﬁeld. It represents the instantaneous magnitude and direction of the power ﬂow of the ﬁeld. 2. The scalar quantity u0 ¼ ϵ 0 2 μ0 jEj þ jH j2 2 2 (1.33) has the unit of energy per unit volume and is the energy density stored in the propagating ﬁeld. It consists of two components, thus accounting for energies stored in both electric and magnetic ﬁelds at any instant of time. 3. The last term in (1.31) also has two components associated with electric and magnetic ﬁelds, respectively. The quantity Wp ¼ E ∂P ∂t Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 (1.34) 10 Basic Concepts of Optical Fields is the power density expended by the electromagnetic ﬁeld on the polarization. It is the rate of energy transfer from the electromagnetic ﬁeld to the medium on inducing the electric polarization in the medium. Similarly, the quantity W m ¼ μ0 H ∂M ∂t (1.35) is the power density expended by the electromagnetic ﬁeld on the magnetization. With these physical meanings attached to the terms in (1.31), it can be seen that (1.31) simply states the law of conservation of energy in any arbitrary volume element V in the medium. The total electromagnetic energy in the medium equals that contained in the propagating ﬁeld plus that stored in the electric and magnetic polarizations. For an optical ﬁeld, E J ¼ 0 and W m ¼ 0 because J ¼ 0 and M ¼ 0, as discussed above. Then, (1.31) becomes þ ð ð ∂ S n^da ¼ u0 dV þ W p dV , (1.36) ∂t V A V which states that the total optical power ﬂowing into volume V through its boundary surface A is equal to the rate of increase with time of the energy stored in the propagating ﬁelds in V plus the power transferred to the polarization of the medium in this volume. 1.4 WAVE EQUATION .............................................................................................................. By applying ∇ to (1.11) and using (1.19) and (1.12), we obtain the wave equation: ∇ ∇ E þ μ0 ∂2 D ¼ 0: ∂t 2 (1.37) By using (1.2), the wave equation can be expressed as ∇∇Eþ 1 ∂2 E ∂2 P ¼ μ , 0 c2 ∂t 2 ∂t 2 (1.38) where 1 c ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 108 m s1 μ0 ϵ 0 (1.39) is the speed of light in free space. The wave equation in (1.38) describes the space-and-time evolution of the electric ﬁeld of the optical wave. Its right-hand side can be regarded as the driving source for the optical wave; that is, the polarization in a medium drives the evolution of an optical ﬁeld. This wave equation can take on various forms depending on the characteristics of the medium, as will be seen on various occasions later. Here we leave it in this general form. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.5 Harmonic Fields 1.5 11 HARMONIC FIELDS .............................................................................................................. Optical ﬁelds are harmonic ﬁelds that vary sinusoidally with time. The ﬁeld vectors deﬁned in the preceding section are all real quantities. For harmonic ﬁelds, it is always convenient to use complex ﬁelds. We deﬁne the space- and time-dependent complex electric ﬁeld, Eðr; tÞ, through its relation to the real electric ﬁeld, Eðr; t Þ:1 Eðr; t Þ ¼ Eðr; tÞ þ E∗ ðr; tÞ ¼ Eðr; t Þ þ c:c:, (1.40) where c.c. means the complex conjugate. In our convention, Eðr; tÞ contains the complex ﬁeld components that vary with time as exp ðiωtÞ with ω having a positive value, while E∗ ðr; tÞ contains those components that vary with time as exp ðiωt Þ with positive ω. The complex ﬁelds of other ﬁeld quantities are similarly deﬁned. With this deﬁnition for the complex ﬁelds, all of the linear ﬁeld equations retain their forms. In terms of complex optical ﬁelds, Maxwell’s equations in the form of (1.11)–(1.14) are ∇E¼ ∇H¼ ∂B , ∂t ∂D , ∂t (1.41) (1.42) ∇ D ¼ 0, (1.43) ∇ B ¼ 0; (1.44) and those in the form of (1.15)–(1.18) are ∇E¼ ∇H¼ ∂B , ∂t ∂Dbound þ Jcond , ∂t (1.45) (1.46) ∇ Dbound ¼ 0, (1.47) ∇ B ¼ 0: (1.48) The wave equation in terms of the complex electric ﬁeld is ∇∇Eþ 1 1 ∂2 E ∂2 P ¼ μ , 0 c2 ∂t2 ∂t 2 (1.49) In some literature, the complex ﬁeld is deﬁned through a relation with the real ﬁeld as Eðr; t Þ ¼ ½Eðr; tÞ þ E∗ ðr; tÞ=2, which differs from our deﬁnition in (1.40) by the factor 1=2. The magnitude of the complex ﬁeld deﬁned through this alternative relation is twice that of the complex ﬁeld deﬁned through (1.40). As a result, expressions for many quantities may be different under the two different deﬁnitions. An example is the time-averaged Poynting vector given in (1.53), which would be changed to S ¼ ReðE H∗ Þ=2 in this alternative deﬁnition of the complex ﬁeld. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 12 Basic Concepts of Optical Fields while ðt ððð Pðr; t Þ ¼ ϵ 0 χðr r0 ; t t 0 Þ Eðr0 , t 0 Þdr0 dt0 (1.50) ∞ all r0 and ðt ððð Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; t Þ ¼ ∞ all ϵ ðr r0 ; t t0 Þ Eðr0 , t0 Þdr0 dt0: (1.51) r0 It is important to note that while E, D, and P are complex, χðr r0 ; t t 0 Þ and ϵ ðr r0 ; t t0 Þ in (1.50) and (1.51) are always real functions of space and time and are the same as those in (1.20) and (1.21). The complex electric ﬁeld of a harmonic optical ﬁeld that has a carrier wavevector of k and a carrier angular frequency of ω can be further expressed as Eðr; tÞ ¼ E ðr; t Þ exp ðik r iωtÞ ¼ ^e E ðr; t Þ exp ðik r iωt Þ, (1.52) where E ðr; t Þ is the space- and time-dependent amplitude of the ﬁeld, and ^e is the unit polarization vector of the ﬁeld. The vectorial ﬁeld amplitude E ðr; t Þ is generally a complex vectorial quantity that has a magnitude, a phase, and a polarization. Other complex ﬁeld quantities, such as Dðr; t Þ, Bðr; t Þ, and Hðr; t Þ, can be similarly expressed. The space- and time-dependent phase factor in (1.52) indicates the direction of wave propagation: ik r iωt for a wave propagating in the k direction; ik r iωt for a wave propagating in the k direction. 1.5.1 Light Intensity The light intensity, or irradiance, is the power density of the harmonic optical ﬁeld. It can be calculated by time averaging the Poynting vector over one wave cycle: ðT 1 S¼ Sdt ¼ 2Re E H∗ , T (1.53) 0 where Reð Þ means taking the real part. We can deﬁne a complex Poynting vector: S ¼ E H∗ (1.54) so that ∗ S ¼ S þ S∗ ¼ S þ S , (1.55) which has the same form as the relation between the real and complex ﬁelds deﬁned in (1.40) except that the Poynting vector in this relation is time averaged. In the case of a coherent monochromatic wave, E H∗ ¼ E H∗ ; then, (1.55) can be written as S ¼ S þ S∗ . The Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.6 Polarization of Optical Fields 13 light intensity, I, on a surface is simply the magnitude of the real time-averaged Poynting vector projected on the surface: ∗ I ¼ S n^ ¼ ðS þ S Þ n^, (1.56) where n^ is the unit normal vector of the projected surface and I is in watts per square meter. 1.5.2 Fields in Momentum Space and Frequency Domain For harmonic optical ﬁelds, it is often useful to consider the complex ﬁelds in the momentum space and frequency domain deﬁned by the following Fourier-transform relations: ð∞ ððð Eðr; t Þ exp ðik r þ iωtÞdrdt, Eðk; ωÞ ¼ for ω > 0, (1.57) ∞ all r Eðr; t Þ ¼ 1 ð2π Þ4 ð∞ ððð Eðk; ωÞ exp ðik r iωt Þdkdω: (1.58) 0 all k Note that Eðk; ωÞ in (1.57) is only deﬁned for ω > 0; therefore, the integral for the time dependence of Eðr; t Þ in (1.58) only extends over positive values of ω. This is in accordance with the convention we used to deﬁne the complex ﬁeld Eðr; t Þ in (1.40). All other space- and time-dependent quantities, including other ﬁeld vectors and the permittivity and susceptibility tensors, are transformed in a similar manner. Through the Fourier transform, the convolution integrals in real space and time become simple products in the momentum space and frequency domain. Consequently, we have Pðk; ωÞ ¼ ϵ 0 χðk; ωÞ Eðk; ωÞ (1.59) Dðk; ωÞ ¼ ϵ 0 ½1 þ χðk; ωÞ Eðk; ωÞ ¼ ϵ ðk; ωÞ Eðk; ωÞ: (1.60) and Note that in the real space and time domain Pðr; t Þ and Dðr; tÞ are connected to Eðr; t Þ through convolution integrals in space and time, whereas in the momentum space and frequency domain Pðk; ωÞ and Dðk; ωÞ are connected to Eðk; ωÞ through direct products. 1.6 POLARIZATION OF OPTICAL FIELDS .............................................................................................................. The polarization state of an optical ﬁeld is determined by the vectorial nature of the optical ﬁeld. It is characterized by the unit polarization vector ^e of the complex electric ﬁeld expressed in (1.52). Consider a monochromatic plane optical wave that has a complex electric ﬁeld of Eðr; t Þ ¼ E exp ðik r iωt Þ ¼ ^e E exp ðik r iωtÞ, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 (1.61) 14 Basic Concepts of Optical Fields where E is a constant independent of r and t, and ^e is its unit vector. The polarization state of the optical ﬁeld is characterized by the unit vector ^e . The optical ﬁeld is linearly polarized, also called plane polarized, if ^e can be expressed as a constant, real vector. Otherwise, the optical ﬁeld is elliptically polarized in general, and is circularly polarized in some special cases. For the convenience of discussion, we take the direction of wave propagation to be the z direction so that k ¼ k^z and assume that both E and H lie in the xy plane. Then, we have E ¼ ^x E x þ ^y E y ¼ ^x jE x jeiφx þ ^y E y eiφy , (1.62) where E x and E y are space- and time-independent complex amplitudes, with phases φx and φy , respectively. The polarization state of the wave is completely characterized by the phase difference and the magnitude ratio between the two ﬁeld components E x and E y : φ ¼ φ y φx , π < φ π, (1.63) π 0α : 2 (1.64) and 1 α ¼ tan E y , jE x j Because only the relative phase φ matters, we can set φx ¼ 0 and take E ¼ jE j to be real in the following discussion. Then E from (1.62) can be written as E ¼ ^e E, with ^e ¼ ^x cos α þ ^y eiφ sin α: (1.65) Using (1.40), the space- and time-dependent real ﬁeld is Eðz; t Þ ¼ 2E ½^x cos α cos ðkz ωtÞ þ ^y sin α cos ðkz ωt þ φÞ: (1.66) At a ﬁxed z location, say z ¼ 0, we see that the electric ﬁeld varies with time as Eðt Þ ¼ 2E ½^x cos α cos ωt þ ^y sin α cos ðωt φÞ: (1.67) 1.6.1 Elliptic Polarization In general, E x and E y have different phases and different magnitudes. Therefore, the values of φ and α can be any combination. At a ﬁxed point in space, both the direction and the magnitude of the ﬁeld vector E in (1.67) can vary with time. Except when the values of φ and α fall into one of the special cases discussed below, the tip of this vector generally describes an ellipse, and the wave is said to be elliptically polarized. Note that we have assumed that the wave propagates in the positive z direction. When we view the ellipse by facing against this direction of wave propagation, we see that the tip of the ﬁeld vector rotates counterclockwise, or left handedly, if φ > 0; and it rotates clockwise, or right handedly, if φ < 0: Figure 1.3 shows the ellipse traced by the tip of the rotating ﬁeld vector at a ﬁxed point in space. Also shown in the ﬁgure are the relevant parameters that characterize elliptic polarization. In the description of the polarization characteristics of an optical ﬁeld, it is sometimes convenient to use, in place of φ and α, a set of two other parameters, θ and ε, which specify the orientation and ellipticity of the ellipse, respectively. The orientational parameter θ is the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.6 Polarization of Optical Fields 15 Figure 1.3 Ellipse described by the tip of the ﬁeld of an elliptically polarized optical wave at a ﬁxed point in space. Also shown are relevant parameters characterizing the state of polarization. The propagation direction is assumed to be the positive z direction, and the ellipse is viewed by facing against this direction. directional angle measured from the x axis to the major axis of the ellipse. Its range is taken to be 0 θ < π for convenience. The ellipticity ε is deﬁned as ε¼ b tan1 , a π π ε , 4 4 (1.68) where a and b are the major and minor semiaxes, respectively, of the ellipse. The plus sign for ε > 0 is taken to correspond to φ > 0 for left-handed polarization, whereas the minus sign for ε < 0 is taken to correspond to φ < 0 for right-handed polarization. The two sets of parameters ðα; φÞ and ðθ; εÞ have the following relations: tan 2θ ¼ tan 2α cos φ, (1.69) sin 2ε ¼ sin 2α sin φ: (1.70) Either set is sufﬁcient to completely characterize the polarization state of an optical ﬁeld. Elliptic polarization can be considered as the general polarization state for any combination of α and φ values, whereas linear polarization and circular polarization are special cases of elliptic polarization for speciﬁc combinations of α and φ values. 1.6.2 Linear Polarization An optical ﬁeld is linearly polarized when φ ¼ 0 or π for any value of α. It is also characterized by ε ¼ 0 and θ ¼ α, if φ ¼ 0; or by ε ¼ 0 and θ ¼ π α, if φ ¼ π. Clearly, the ratio E x =E y is real in this case; therefore, linear polarization is described by a constant, real unit vector as ^e ¼ ^x cos θ þ ^y sin θ: (1.71) It follows that Eðt Þ described by (1.67) reduces to Eðt Þ ¼ 2E^e cos ωt: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 (1.72) 16 Basic Concepts of Optical Fields Figure 1.4 Field of a linearly polarized optical wave. The tip of this vector traces a line in space at an angle of θ with respect to the x axis, as shown in Fig. 1.4. 1.6.3 Circular Polarization An optical ﬁeld is circularly polarized when φ ¼ π=2 or π=2, and α ¼ π=4. It is also characterized by ε ¼ π=4 or π=4, and θ ¼ 0. Because α ¼ π=4, we have jE x j ¼ E y ¼ pﬃﬃﬃ E= 2. There are two different circular polarization states. 1. Left-circular polarization: For φ ¼ π=2, also ε ¼ π=4, the wave is left circularly polarized if it propagates in the positive z direction. The complex ﬁeld amplitude in (1.65) becomes ^x þ i^y E ¼ E pﬃﬃﬃ ¼ E^e þ , 2 and Eðt Þ described by (1.67) reduces to pﬃﬃﬃ EðtÞ ¼ 2E ð^x cos ωt þ ^y sin ωt Þ: (1.73) (1.74) As we view against the direction of propagation ^z , we see that the ﬁeld vector EðtÞ rotates counterclockwise at an angular frequency of ω. The tip of this vector describes a circle. This is shown in Fig. 1.5(a). This left-circular polarization is also called positive helicity. Its unit vector is ^e þ ^x þ i^y pﬃﬃﬃ : 2 (1.75) 2. Right-circular polarization: For φ ¼ π=2, also ε ¼ π=4, the wave is right circularly polarized if it propagates in the positive z direction. We then have ^x i^y E ¼ E pﬃﬃﬃ ¼ E^e 2 (1.76) and EðtÞ ¼ pﬃﬃﬃ 2E ð^x cos ωt ^y sin ωt Þ: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 (1.77) 1.6 Polarization of Optical Fields 17 Figure 1.5 (a) Field of a left circularly polarized wave. (b) Field of a right circularly polarized wave. The tip of this ﬁeld vector rotates clockwise in a circle, as shown in Fig. 1.5(b). This rightcircular polarization is also called negative helicity. Its unit vector is ^e ^x i^y pﬃﬃﬃ : 2 (1.78) As can be seen, neither ^e þ nor ^e is a real vector. Note that the identiﬁcation of ^e þ , deﬁned in (1.75), with left-circular polarization and that of ^e , deﬁned in (1.78), with right-circular polarization are based on the assumption that the wave propagates in the positive z direction. For a wave that propagates in the negative z direction, the handedness of these unit vectors changes: ^e þ becomes right-circular polarization, while ^e becomes left-circular polarization. 1.6.4 Orthogonal Polarizations Two polarizations are orthogonal if they are normal to each other. The unit polarization vector ^e can be either a real vector, for a linearly polarized wave, or a complex vector, for a circularly or elliptically polarized wave. Each unit polarization vector is normalized to be a unit vector according to the relation: ^e ^e ∗ ¼ 1: (1.79) Two polarizations, ^e 1 and ^e 2 , are orthogonal if ^e 1 ^e ∗ 2 ¼ 0: (1.80) Note that normalization is not performed by ^e ^e ¼ 1, and orthogonality is not deﬁned by ^e 1 ^e 2 ¼ 0. EXAMPLE 1.3 Consider the two circularly polarized unit vectors ^e þ and ^e that are given in (1.75) and (1.78), respectively. Show that they are normalized unit vectors that are orthogonal to each other. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 18 Basic Concepts of Optical Fields Solution: Using (1.79) for normalization, we ﬁnd that ^x þ i^y ^x þ i^y ∗ ^x þ i^y ^x i^y ∗ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ ^e þ ^e þ ¼ ¼ ¼1 2 2 2 2 and ^e ^e ∗ ^x i^y ^x i^y ∗ ^x i^y ^x þ i^y pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ ¼ ¼ ¼ 1: 2 2 2 2 Therefore, both ^e þ and ^e are normalized unit vectors. Using (1.80) for orthogonality, we ﬁnd that ^x þ i^y ^x i^y ∗ ^x þ i^y ^x þ i^y ∗ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ ^e þ ^e ¼ ¼0 ¼ 2 2 2 2 and ^e ^e ∗ þ ^x i^y ^x þ i^y ∗ ^x i^y ^x i^y pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ ¼ 0: ¼ ¼ 2 2 2 2 Therefore, ^e þ and ^e are normalized unit vectors that are orthogonal to each other. The two circular polarizations are orthogonal to each other. Note that ^e þ ^e þ ¼ ^e ^e ¼ 0 6¼ 1 and ^e þ ^e ¼ ^e ^e þ ¼ 1 6¼ 0, which can be easily veriﬁed. 1.7 OPTICAL FIELD PARAMETERS .............................................................................................................. As stated in Section 1.1, an optical ﬁeld is characterized by the ﬁve parameters of polarization ^e , magnitude jE j, phase φE , wavevector k, and frequency ω: Eðr; t Þ ¼ E ðr; t Þ exp ðik r iωt Þ ¼ ^e E ðr; tÞ exp ðik r iωt Þ (1.81) ¼ ^e jE ðr; t ÞjeiφE ðr;tÞ exp ðik r iωtÞ, where E ¼ ^e E is the vectorial complex ﬁeld amplitude that contains the ﬁeld polarization ^e and the scalar complex ﬁeld amplitude E. The scalar complex ﬁeld amplitude E ¼ jEjeiφE has a magnitude of jE j and a phase of φE . Note that in general, jE j and φE can vary with space and time, as indicated above in (1.81). Among the ﬁve parameters, ^e and k are vectors, while jE j, φE , and ω are scalars. The unit polarization vector ^e fully characterizes the polarization state of an optical ﬁeld. It can be real, for linearly polarized light, or complex, for elliptically or circularly polarized light. The details are discussed in the preceding section. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 1.7 Optical Field Parameters 19 The magnitude jE j of the complex ﬁeld amplitude deﬁnes the strength of the optical ﬁeld. For simplicity of discussion, consider a linearly polarized wave so that the unit polarization vector ^e is a real vector. Then the complex ﬁeld given in (1.81) yields the following real ﬁeld, (1.82) Eðr; t Þ ¼ Eðr; t Þ þ E∗ ðr; t Þ ¼ 2jE ðr; t Þj^e cos k r ωt þ φE ðr; t Þ : Therefore, under our deﬁnition of the complex ﬁeld through (1.40), the amplitude of the real ﬁeld is 2jE ðr; t Þj. Note that this ﬁeld amplitude can be a function of space and time to describe the modulation on the ﬁeld strength in space and time. It describes an envelope of the ﬁeld on the optical carrier. The phase φE of the complex ﬁeld amplitude is the phase shift with respect to the space- and time-varying phase factor, k r ωt. As seen in (1.82), the total phase of the ﬁeld is φðr; tÞ ¼ k r ωt þ φE ðr; t Þ: (1.83) In the case when φE is a constant that is independent of both space and time, it has physical meaning only when it is compared to a reference, such as the phase of another ﬁeld. An unreferenced constant phase can be eliminated by redeﬁning the origin of the space or time coordinate. Nevertheless, as expressed in (1.81) and (1.82), this phase can be a function of space or time, or both: φE ðr; tÞ: The spatial dependence of φE ðr; t Þ leads to a shift of the wavevector from the carrier wavevector k; the temporal dependence of φE ðr; t Þ leads to a shift of the frequency from the carrier frequency ω: The wavevector k deﬁnes the spatial variation and the propagation direction of the optical carrier ﬁeld. Its value, k, known as the propagation constant or the wavenumber, is determined by the wavelength, or equivalently the frequency, of the optical wave and the refractive index of the medium: 2πn ^ nω ^ k ¼ kk^ ¼ k¼ k, λ c (1.84) where n is the refractive index of the medium. From (1.82), it can be seen that k deﬁnes the spatial variation of the optical carrier ﬁeld. The propagation direction of a wave is deﬁned as the direction normal to the wavefront of the wave, and a wavefront is the surface of a constant phase: φðr; t Þ ¼ constant: With φðr; tÞ ¼ k r ωt þ φE ðr; t Þ from (1.83), the space-dependent wavevector is kðrÞ ¼ ∇φ ¼ k þ ∇φE : (1.85) Thus, the space-dependent wave propagation direction can be found as kðrÞ k^ðrÞ ¼ : k ðrÞ (1.86) In the case when φE is independent of space so that ∇φE ¼ 0, such as the case of a plane wave, the wave propagates with a space-independent propagation constant k in a space-independent propagation direction deﬁned by the constant unit vector k^ ¼ k=k. In the case when φE varies across space so that ∇φE 6¼ 0, such as the case of a spatially diverging or converging wave, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 20 Basic Concepts of Optical Fields either one or both of the propagation constant kðrÞ and the propagation direction deﬁned by k^ðrÞ ¼ kðrÞ=k ðrÞ vary from one spatial location to another. The frequency ω deﬁnes the temporal variation of the optical carrier ﬁeld. It is the optical angular frequency that is related to the ﬁeld oscillation frequency ν as ω ¼ 2πν; ν has the unit of hertz ðHzÞ while ω has the unit of radians per second ðrad s1 Þ. As an optical wave propagates through different media of different refractive indices, its wavelength, thus the value of k, changes with the changing refractive indices, but its frequency remains unchanged. The angular frequency of a wave is deﬁned by the temporal variation of its phase. With φðr; t Þ ¼ k r ωt þ φE ðr; tÞ from (1.83), the angular frequency can be found as ωðt Þ ¼ ∂φ ∂φ ¼ω E: ∂t ∂t (1.87) The frequency of the wave is the constant ω in the case when φE is independent of time so that ∂φE =∂t ¼ 0, such as the case of a monochromatic wave. In the case when φE varies with time, such as the case of a phase-modulated wave, the frequency ωðtÞ is a function of time with a shift of ∂φE =∂t from the constant frequency ω. Problems 1.1.1 At room temperature, diamond transmits optical waves of wavelengths longer than 227 nm but absorbs shorter wavelengths. What is the bandgap energy of diamond at room temperature? 1.1.2 At room temperature, the bandgap energy of Ge is 0.66 eV. It absorbs photons of energies above its bandgap and transmits those of energies below its bandgap. What is the cutoff wavelength for light to be transmitted through a thick piece of pure Ge? 1.1.3 Find the wavelength and photon energy of a terahertz wave at a frequency of 5 THz. 1.1.4 The optical window for long-distance optical communications is at the 1.55 μm wavelength. What are the optical frequency and the photon energy? 1.1.5 A red laser pointer emits a red beam of P ¼ 1 mW power at the λ ¼ 635 nm wavelength. What are the photon energy, the photon momentum, and the photon ﬂux of this beam? If it illuminates a totally absorbing surface, what is the force exerted by the beam on the absorbing surface? If it illuminates a totally reﬂecting surface, what is the force exerted by the beam on the reﬂecting surface? 1.2.1 Verify that Maxwell’s equations and the continuity equation, given in (1.4)–(1.8), are invariant under (a) the transformation of space inversion, (b) the transformation of time reversal, and (c) the simultaneous transformation of space inversion and time reversal. 1.4.1 Derive the optical wave equation given in (1.37) in the case when J ¼ 0 so that Maxwell’s equations take the form of (1.11)–(1.14). Show that in this case the optical wave equation can be expressed in the form of (1.38). Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 Bibliography 21 1.4.2 In the case when a conduction current Jcond is explicitly separated from the ∂D=∂t term so that Maxwell’s equations take the form of (1.15)–(1.18), rewrite the optical wave equation given in (1.37) and that given in (1.38) to explicitly account for Jcond . 1.5.1 By taking the Fourier transform on the relation given in (1.50) between Pðr; t Þ and Eðr; tÞ in the real space and time domain, verify the relation given in (1.59) between Pðk; ωÞ and Eðk; ωÞ in the momentum space and frequency domain. 1.6.1 As discussed in the text, any polarization state in the xy plane can be generally considered as elliptic polarization represented by the unit polarization vector ^e ¼ ^x cos α þ ^y eiφ sin α given in (1.65) with proper choices of α and φ for a particular polarization state. Because the xy plane is a two-dimensional space, a basis set of unit polarization vectors consists of two orthonormal vectors. Find the other unit polarization vector ^e ⊥ that forms a basis together with ^e . 1.6.2 The circularly polarized unit vectors ^ e þ and ^e given in (1.75) and (1.78) are each expressed in terms of the linearly polarized unit vectors ^x and ^y . Each pair form a basis for representing any polarization state in the xy plane. Show that each of the linearly polarized unit vectors ^x and ^y can be represented in terms of a linear superposition of two circularly polarized components on the basis of ^e þ and ^e . 1.6.3 Express the general linearly polarized unit vector ^ e ¼ ^x cos θ þ ^y sin θ given in (1.71) as a linear superposition of two circularly polarized components on the basis of the circularly polarized unit vectors ^e þ and ^e given in (1.75) and (1.78), respectively. Bibliography Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999. Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975. Iizuka, K., Elements of Photonics in Free Space and Special Media, Vol. I. New York: Wiley, 2002. Jackson, J. D., Classical Electrodynamics, 3rd edn. New York: Wiley, 1999. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:13:50 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.002 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 2 - Optical Properties of Materials pp. 22-65 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge University Press 2 2.1 Optical Properties of Materials OPTICAL SUSCEPTIBILITY AND PERMITTIVITY .............................................................................................................. The electric susceptibility, χ, and the electric permittivity, ϵ, of an optical medium characterize the intrinsic response of the medium to an optical ﬁeld. They are respectively deﬁned in (1.20) for the relation between Pðr; t Þ and Eðr; t Þ and in (1.21) for the relation between Dðr; t Þ and Eðr; tÞ: ðt ððð Pðr; tÞ ¼ ϵ 0 χðr r0 ; t t 0 Þ Eðr0 ; t 0 Þdr0 dt 0, (2.1) ∞ all r0 ðt ððð Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; t Þ ¼ ϵ ðr r0 ; t t0 Þ Eðr0 ; t 0 Þdr0 dt0: (2.2) ∞ all r0 These relations can be expressed in terms of the complex ﬁeld: ðt ððð Pðr; t Þ ¼ ϵ 0 χðr r0 ; t t0 Þ Eðr0 ; t 0 Þdr0 dt 0 (2.3) ∞ all r0 ðt ððð Dðr; t Þ ¼ ϵ 0 Eðr; t Þ þ Pðr; t Þ ¼ ϵ ðr r0 ; t t0 Þ Eðr0 ; t 0 Þdr0 dt0: (2.4) ∞ all r0 The relations in the momentum space and frequency domain, obtained by taking the Fourier transform on (2.3) and (2.4), are direct products, given in (1.59) and (1.60): Pðk; ωÞ ¼ ϵ 0 χðk; ωÞ Eðk; ωÞ (2.5) Dðk; ωÞ ¼ ϵ 0 ½1 þ χðk; ωÞ Eðk; ωÞ ¼ ϵ ðk; ωÞ Eðk; ωÞ: (2.6) The real-space and time-domain relations given in (2.1)(2.4) are convolution integrals over real space and time. The convolution in time accounts for the fact that the response of a medium to the stimulation by an electric ﬁeld is generally not instantaneous, or local, in time and does not vanish for some time after the stimulation is over. Because time is unidirectional, causality exists in physical processes. An earlier stimulation can inﬂuence the property of a medium at a later time, whereas a later stimulation does not have any effect on the medium at an earlier time. Therefore, the upper limit in the time integral is t, not inﬁnity. By contrast, the convolution in space accounts for the spatial nonlocality of the material response. Stimulating a medium at a Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.1 Optical Susceptibility and Permittivity 23 Figure 2.1 Nonlocal responses in (a) time and (b) space. location r0 can result in a change in the property of the medium at another location r: For example, the property of a semiconductor at one location can be changed by electric or optical excitation at another location through carrier diffusion. There is generally no spatial causality because space is not unidirectional; therefore, spatial convolution is integrated over the entire space. Figure 2.1 shows the temporal and spatial nonlocality of responses to electromagnetic excitations. The temporal nonlocality of the optical response of a medium makes the optical property of the medium dependent on the optical frequency, a phenomenon known as frequency dispersion, whereas the spatial nonlocality makes the optical property of the medium dependent on the optical wavevector, a phenomenon known as momentum dispersion. The frequency dispersion and the momentum dispersion of a medium are respectively characterized by the frequency dependence and the momentum dependence of χðk; ωÞ and ϵ ðk; ωÞ. Because χðk; ωÞ and ϵ ðk; ωÞ are respectively the Fourier transforms of χðr; tÞ and ϵ ðr; t Þ, it is clear that the frequency dispersion and the momentum dispersion of a medium respectively originate from the temporal nonlocality and the spatial nonlocality of its response to an optical stimulation. The susceptibility tensor χðr; t Þ and the permittivity tensor ϵ ðr; tÞ of real space and time are always real quantities though the optical ﬁelds in the real space and time domain can be expressed either as real ﬁelds, as in (2.1) and (2.2), or as complex ﬁelds, as in (2.3) and (2.4). This statement is true even when the medium exhibits an optical loss or gain. However, the susceptibility tensor χðk; ωÞ and the permittivity tensor ϵ ðk; ωÞ in the momentum space and frequency domain are generally complex. If an eigenvalue χ i of χðk; ωÞ is complex, the corresponding eigenvalue ϵ i of ϵ ðk; ωÞ is also complex, and their imaginary parts have the same sign because ϵ ðk; ωÞ ¼ ϵ 0 ½1 þ χðk; ωÞ. The signs of the imaginary parts of such eigenvalues tell whether the medium provides an optical gain or loss. In our convention, we write, for example, χ i ¼ χ 0i þ iχ 00i in the frequency domain. Then, χ 00i ðωÞ > 0 indicates an optical loss or absorption, while χ 00i ðωÞ < 0 represents an optical gain or ampliﬁcation. The fact that χðr; t Þ and ϵ ðr; t Þ are real quantities leads to the following symmetry relations for the tensor elements of χðk; ωÞ and ϵ ðk; ωÞ: ∗ χ∗ ij ðk; ωÞ ¼ χ ij ðk; ωÞ and ϵ ij ðk; ωÞ ¼ ϵ ij ðk; ωÞ, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 (2.7) 24 Optical Properties of Materials which are called the reality condition. The reality condition implies that χ 0ij ðk; ωÞ ¼ χ 0ij ðk; ωÞ and χ 00ij ðk; ωÞ ¼ χ 00ij ðk; ωÞ: The real and imaginary parts of ϵ ij ðk; ωÞ have similar properties. Therefore, the real parts of χ ij ðk; ωÞ and ϵ ij ðk; ωÞ are even functions of k and ω, whereas the imaginary parts are odd functions of k and ω. If a tensor element χ ij ðk; ωÞ or ϵ ij ðk; ωÞ has any constant term that is independent of k and ω, the constant term can only appear in its real part because a constant value is an even function of k and ω. As a result, the imaginary part is always a function of k or ω, or both. The optical loss, or gain, in a medium is associated with the imaginary part of an eigenvalue of χðk; ωÞ or ϵ ðk; ωÞ; consequently, a medium that absorbs or ampliﬁes light is inherently dispersive. Any other effect that can be described by the imaginary part of an eigenvalue of χðk; ωÞ or ϵ ðk; ωÞ is also inherently dispersive in either momentum or frequency, or both. In addition to the nonlocality of medium response, it is also important to consider the inhomogeneity of a medium, in both space and time. Spatial inhomogeneity exists in every optical structure, such as an optical waveguide, where the optical property is a function of space. Temporal inhomogeneity exists when the optical property of a medium varies with time, for example, because of modulation by a low-frequency electric ﬁeld or by an acoustic wave. The space and time variables characterizing nonlocality are relative space and time of the medium response with respect to an optical stimulation, whereas those characterizing inhomogeneity are absolute space and time measured with respect to a reference point in space and a reference point in time. When both response nonlocality and medium inhomogeneity are considered, the response nonlocality is commonly characterized in the momentum space and frequency domain as a function of k and ω by taking the Fourier transform on the relative space and time, whereas the medium inhomogeneity is characterized in the real space and time domain as a function of the absolute space and time variables r and t; therefore, χðk; ω; r; t Þ and, correspondingly, ϵ ðk; ω; r; t Þ. In a linear medium, changes in the wavevector of an optical wave, or coupling between waves of different wavevectors, can occur only if the optical property of the medium in which the wave propagates is spatially inhomogeneous such that χðk; ω; r; t Þ is a function of space. Likewise, changes in the frequency of an optical wave, or coupling between waves of different frequencies, are possible in a linear medium only if the optical property of the medium is time varying such that χðk;ω;r;tÞ varies with time. A change in the wavevector of an optical wave ^ as in the case of reﬂection or can take the form of a change in the wave propagation direction k, diffraction of an optical wave, or in the propagation constant k through a change in the optical wavelength, as in the case when a wave propagates from one part of the medium to another part of a different refractive index. A change in the frequency of an optical wave results in the generation of other frequencies or the conversion to a completely different frequency. 2.2 OPTICAL ANISOTROPY .............................................................................................................. In general, both χ and ϵ are tensors because the P and D vectors are not necessarily parallel to the E vector due to material anisotropy. In the case of an isotropic medium, both χ and ϵ reduce to the scalars χ and ϵ, respectively. In the case of a linear anisotropic medium, both χ and ϵ are second-order tensors. They can be expressed in the matrix form: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.2 Optical Anisotropy 0 χ 11 @ χ ¼ χ 21 χ 31 χ 12 χ 22 χ 32 1 0 χ 13 ϵ 11 A @ and ϵ ¼ ϵ 21 χ 23 χ 33 ϵ 31 1 ϵ 13 ϵ 23 A: ϵ 33 ϵ 12 ϵ 22 ϵ 32 25 (2.8) Each of the relationships P ¼ ϵ 0 χ E and D ¼ ϵ E is carried out as the product between a tensor and a column vector: 0 1 0 P1 χ 11 @ P2 A ¼ ϵ 0 @ χ 21 P3 χ 31 χ 12 χ 22 χ 32 10 1 0 1 0 E1 D1 ϵ 11 χ 13 χ 23 A@ E 2 A and @ D2 A ¼ @ ϵ 21 χ 33 E3 D3 ϵ 31 ϵ 12 ϵ 22 ϵ 32 10 1 E1 ϵ 13 ϵ 23 A@ E 2 A: (2.9) ϵ 33 E3 In general, the matrices in (2.8) representing the χ and ϵ tensors are not diagonal when they are expressed using an arbitrarily chosen coordinate system. When optical ﬁeld vectors are projected on the axes of this coordinate system, a component of P or D does not necessarily contain only the corresponding component of E but can also contain one or both of the other two E components. For example, P1 and D1 are functions of E 2 or E3 , or both, unless χ 12 ¼ χ 13 ¼ 0, in which case ϵ 12 ¼ ϵ 13 ¼ 0 as well, because P1 ¼ ϵ 0 ðχ 11 E 1 þ χ 12 E 2 þ χ 13 E 3 Þ and D1 ¼ ϵ 11 E 1 þ ϵ 12 E 2 þ ϵ 13 E 3 . Because χ and ϵ are physical quantities, they are diagonalizable matrices that can always be diagonalized by a proper set of eigenvectors, yielding 0 χ1 χ¼@0 0 0 χ2 0 1 0 ϵ1 0 0 A and ϵ ¼ @ 0 χ3 0 0 ϵ2 0 1 0 0 A: ϵ3 (2.10) Here χ i and ϵ i are, respectively, the eigenvalues of χ and ϵ with corresponding eigenvectors ^e i such that χ ^e i ¼ χ i ^e i and ϵ ^e i ¼ ϵ i ^e i , for i ¼ 1, 2, 3: (2.11) The characteristics of the eigenvalues χ i and ϵ i , as well as their eigenvectors ^e i , depend on the symmetry properties of χ and ϵ. The two matrices representing χ and ϵ have the same symmetry properties because ϵ ¼ ϵ 0 ð1 þ χÞ, where 1 has the form of a 3 3 identity matrix when it is added to the χ tensor. Therefore, χ and ϵ are diagonalized by the same set of eigenvectors. When an optical ﬁeld is projected on these eigenvectors, each component of P or D depends only on the corresponding component of E but not on the other two E components; that is, Pi ¼ ϵ 0 χ i E i and Di ¼ ϵ i E i . The three eigenvectors ^e i deﬁne the principal polarization states for proper decomposition of optical ﬁeld vectors so that each component has a well-deﬁned susceptibility χ i and permittivity ϵ i . They are the principal normal modes of polarization satisfying the orthonormality condition: 1, for i ¼ j; ∗ ^e i ^e j ¼ δij ¼ (2.12) 0, for i 6¼ j: As discussed in Section 1.6, a real eigenvector represents linear polarization, while a complex eigenvector represents elliptic or circular polarization. The characteristics of these eigenvectors Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 26 Optical Properties of Materials are determined by the symmetry properties of χ and ϵ, which are determined by the properties of the medium. Because χ and ϵ have the same properties and the same eigenvectors, only ϵ is mentioned in the following discussion while all conclusions apply equally to χ. 2.2.1 Reciprocal Media Nonmagnetic materials that are not subject to an external magnetic ﬁeld are reciprocal media. In a reciprocal medium, the Lorentz reciprocity theorem of electromagnetics holds; consequently, the source and the detector of an optical signal can be interchanged for the same function of an optical system. If such a material is not optically active, its optical properties are described by a symmetric ϵ tensor: ϵ ij ¼ ϵ ji . The eigenvectors ^e i of a symmetric tensor are always real vectors. They can be chosen to be ^x , ^y , and ^z of a rectilinear coordinate system in real space. This is true even when ϵ is complex. 1. If a nonmagnetic medium does not have any optical loss or gain, its ϵ tensor is Hermitian, ∗ ∗ i.e., ϵ ij ¼ ϵ ∗ ji . A symmetric Hermitian tensor is real and symmetric: ϵ ij ¼ ϵ ij ¼ ϵ ji ¼ ϵ ji : The eigenvectors ^e i are real vectors representing linear polarization states, and all three eigenvalues ϵ i have real values. 2. If a nonmagnetic medium has an optical loss or gain, its ϵ tensor is still symmetric but is complex and non-Hermitian: ϵ ij ¼ ϵ ji but ϵ ij 6¼ ϵ ∗ e i are real ji : Then, the eigenvectors ^ vectors representing linear polarization states, but at least one of the eigenvalues ϵ i is complex. The sign of the imaginary part, ϵ 00i , indicates whether the medium has a loss or gain for an ^e i -polarized optical wave: ϵ 00i > 0 for a loss and ϵ 00i < 0 for a gain, as discussed in Section 2.1 in terms of χ 00i . 3. If a nonmagnetic medium is optically active, it is still reciprocal although its ϵ tensor is not symmetric. The eigenvectors ^e i are complex vectors representing elliptic or circular polarization states, but the eigenvalues can be real, if the medium has no loss or gain, or complex, if the medium has an optical loss or gain. 2.2.2 Nonreciprocal Media Magnetic materials, and nonmagnetic materials that are subject to an external magnetic ﬁeld, are nonreciprocal media. In such a medium, no symmetry exists when the source and the detector of an optical signal are interchanged. The ϵ tensor describing the optical properties of such a material is not symmetric: ϵ ij 6¼ ϵ ji . The eigenvectors ^e i of a nonsymmetric matrix are generally complex vectors. Therefore, they are not ordinary coordinate axes in real space. 1. For a magnetic medium that has no optical loss or gain, ϵ is Hermitian: ϵ ij ¼ ϵ ∗ ji : The eigenvalues ϵ i are real even though the eigenvectors ^e i are complex vectors representing elliptic or circular polarization states. 2. For a magnetic medium that has an optical loss or gain, ϵ is nonsymmetric and nonHermitian. The eigenvectors ^e i and the eigenvalues ϵ i are generally complex. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 27 2.2 Optical Anisotropy EXAMPLE 2.1 At a given optical wavelength, the permittivity tensors of several optical materials are obtained with respect to an arbitrary set of rectilinear coordinates in real space. From each of the permittivity tensors shown below, identify each material as being (i) reciprocal or nonreciprocal and (ii) lossless or lossy. Here “lossless” means having no loss or gain, and “lossy” means having a loss or gain. 0 0 1 2:3 þ i0:3 0 C A; 0 3:2 þ i0:1 3:4 þ i0:2 0:7 i0:1 B A : ϵ ¼ ϵ 0 @ 0:7 þ i0:1 0 0 2:25 B C : ϵ ¼ ϵ 0 @ i0:35 0 i0:35 2:20 0 1 0 C 0 A; 0 4:79 0:17 B B : ϵ ¼ ϵ 0 @ 0:17 4:49 0 0 B D : ϵ ¼ ϵ 0 @ 0:02 4:88 2:30 0 4:91 0:02 B E : ϵ ¼ ϵ 0 @ 0:20 þ i0:18 0:20 i0:18 0 2:72 i0:22 1 C 0:05 A; 0:05 5:01 0 1 0:01 C A; 0:01 4:58 þ i0:02 0 2:74 0 0 1 C i0:22 A: 2:38 Solution: The permittivity tensor of a reciprocal material is symmetric with ϵ ij ¼ ϵ ji , and that of a lossless medium is Hermitian with ϵ ij ¼ ϵ ∗ ji . The properties of each material can be determined by examining its permittivity tensor using these two characteristics. A, nonreciprocal and lossy; B, reciprocal and lossless; C, nonreciprocal and lossless; D, reciprocal and lossy; E, nonreciprocal and lossless. 2.2.3 Linear Birefringence and Linear Dichroism For a reciprocal material that is not optically active, the eigenvectors ^e i of ϵ for proper decomposition of optical ﬁeld vectors are real unit vectors representing three linearly polarized principal normal modes. These three orthogonal real unit vectors can be labeled as ^x , ^y , and ^z , which can be used to deﬁne the axes of a rectilinear coordinate system in real space. Noncrystalline materials are generally isotropic, for which the choice of the orthogonal coordinate axes ^x , ^y , and ^z is arbitrary. For a crystal, these unique ^x , ^y , and ^z coordinate axes are called the principal dielectric axes, or simply the principal axes, of the crystal. In the coordinate system deﬁned by these principal axes, ϵ is diagonalized with eigenvalues ϵ x , ϵ y , and ϵ z , known as the principal permittivities. The properly decomposed components of D and E along these axes have the following simple relations, Dx ¼ ϵ x E x , Dy ¼ ϵ y E y , Dz ¼ ϵ z E z : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 (2.13) 28 Optical Properties of Materials The values ϵ x =ϵ 0 , ϵ y =ϵ 0 , and ϵ z =ϵ 0 are the eigenvalues of the dielectric constant tensor, ϵ=ϵ 0 , and are called the principal dielectric constants. They deﬁne three principal indices of refraction: rﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ ϵx ϵy ϵz nx ¼ , ny ¼ , nz ¼ : (2.14) ϵ0 ϵ0 ϵ0 The propagation constants for the ^x , ^y , and ^z principal normal modes of polarization are, respectively, kx ¼ nx ω ny ω nz ω , ky ¼ , kz ¼ : c c c (2.15) When ϵ is diagonalized, χ is also diagonalized along the same principal axes with corresponding principal dielectric susceptibilities, χ x , χ y , and χ z . The principal dielectric susceptibilities of any dielectric material of no loss or gain always have real, positive values; therefore, the principal dielectric constants of a lossless dielectric material are always greater than unity. In an anisotropic crystal, the properly decomposed optical ﬁeld components in two different principal normal modes of polarization deﬁned by two different eigenvectors ^e i and ^e j have different indices of refraction, i.e., ni 6¼ nj , and thus different propagation constants, i.e., k i 6¼ kj , when the eigenvalues ϵ i and ϵ j are different for the two polarization states. This phenomenon is known as birefringence. A crystal that shows birefringence is a birefringent crystal. Two principal normal modes of polarization experience different degrees of optical loss or gain when their principal dielectric constants have different imaginary parts. This phenomenon is known as dichroism. The birefringence of an anisotropic nonmagnetic crystal causes two different linearly polarized principal normal modes to propagate with different propagation constants; this is known as linear birefringence. The dichroism of an anisotropic nonmagnetic crystal appears between two linearly polarized principal normal modes; this is known as linear dichroism. The state of polarization of an optical wave generally varies along its path of propagation through an anisotropic crystal unless it is linearly polarized in the direction of a principal axis. However, in an anisotropic crystal with nx ¼ ny 6¼ nz , a wave propagating in the z direction does not see the anisotropy of the crystal because in this situation the x and y components of the ﬁeld have the same propagation constant. This wave maintains its original polarization as it propagates through the crystal. Evidently, this is true only for propagation along the z axis in such a crystal. Such a unique axis in a crystal along which an optical wave can propagate with an index of refraction that is independent of its polarization state is called the optical axis of the crystal. An anisotropic crystal that has only one distinctive principal index among its three principal indices is called a uniaxial crystal because it has only one optical axis, which coincides with the axis of the distinctive principal index of refraction. It is customary to assign ^z to this unique principal axis such that nz is the distinctive index with nx ¼ ny 6¼ nz . The two identical principal indices of refraction are called the ordinary index, no , and the distinctive principal index of refraction is called the extraordinary index, ne . Thus, nx ¼ ny ¼ no and nz ¼ ne . The crystal is called positive uniaxial if ne > no ; it is negative uniaxial if ne < no . A birefringent crystal of Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.2 Optical Anisotropy 29 three distinct principal indices of refraction is called a biaxial crystal because it has two optical axes, neither of which coincides with any of the principal axes. EXAMPLE 2.2 At the 1 μm optical wavelength, the permittivity tensor of the KDP crystal represented in an arbitrarily chosen Cartesian coordinate system deﬁned by ^x 1 , ^x 2 , and ^x 3 unit vectors, with ^x 1 ^x 2 ¼ ^x 3 to satisfy the right-hand rule, is found to be 0 1 2:19 0 0:05196 A: 0 2:28 0 ϵ ¼ ϵ0@ 0:05196 0 2:25 Find the principal indices of refraction and the corresponding principal axes ^x , ^y , and ^z in terms of the coordinate axes ^x 1 , ^x 2 , and ^x 3 . Is KDP uniaxial or biaxial? If it is uniaxial, is it positive or negative uniaxial? Solution: The given ϵ tensor is symmetric and Hermitian because KDP is a nonmagnetic dielectric crystal that has a negligible optical loss at the 1 μm optical wavelength. Diagonalization of the matrix yields the eigenvalues 2.28, 2.28, and 2.16 for the principal dielectric constants. Thus, the crystal is uniaxial. By convention we assign the distinctive dielectric constant of 2.16 to be associated with the z principal axis. The principal indices of refraction and the corresponding principal axes are pﬃﬃﬃﬃﬃﬃﬃﬃﬃ nx ¼ 2:28 ¼ 1:51, ^x ¼ 0:500^x 1 0:866^x 3 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ny ¼ 2:28 ¼ 1:51, ^y ¼ ^x 2 ; pﬃﬃﬃﬃﬃﬃﬃﬃﬃ nz ¼ 2:16 ¼ 1:47, ^z ¼ 0:866^x 1 þ 0:500^x 3 : Note that ^x ^y ¼ ^z to satisfy the right-hand rule. The KDP crystal is negative uniaxial because nx ¼ ny > nz so that no > ne . The optical anisotropy of a crystal depends on its structural symmetry. Crystals are classiﬁed into seven systems according to their symmetry. The linear optical properties of these seven systems are summarized in Table 2.1. Some important remarks regarding the relation between the optical properties and the structural symmetry of a crystal are as follows. 1. A cubic crystal does not have an isotropic structure although its linear optical properties are isotropic. For example, most III–V semiconductors, such as GaAs, InP, InAs, AlAs, etc., are cubic crystals with isotropic linear optical properties. Nevertheless, they have well-deﬁned ^ and ^c . They are also polar semiconductors, which have anisotropic crystal axes, a^, b, nonlinear optical properties. 2. Although the principal axes may coincide with the crystal axes in certain crystals, they are ^ and not the same concept and are not necessarily the same. The crystal axes, denoted by a^, b, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 30 Optical Properties of Materials Table 2.1 Linear optical properties of crystals Crystal symmetry Optical property Cubic Isotropic: nx ¼ ny ¼ nz Trigonal, tetragonal, hexagonal Uniaxial: nx ¼ ny 6¼ nz Orthorhombic, monoclinic, triclinic Biaxial: nx 6¼ ny 6¼ nz ^c , are deﬁned by the structural symmetry of a crystal, whereas the principal axes, denoted by ^x , ^y , and ^z , are determined by the symmetry of ϵ. The principal axes of a crystal are orthogonal to one another, but the crystal axes are not necessarily so. 2.2.4 Circular Birefringence and Circular Dichroism For a nonreciprocal material or an optically active reciprocal material, the eigenvectors ^e i of ϵ for proper decomposition of optical ﬁeld vectors are generally complex unit vectors representing orthogonal elliptic polarization states, which may reduce to linear or circular polarization states in particular cases. Optical activity is the phenomenon that a linearly polarized optical wave remains linearly polarized but with its plane of polarization rotating about the direction of propagation as it travels through a material. Natural optical activity that appears in a nonmagnetic reciprocal material not subject to a magnetic ﬁeld was ﬁrst discovered in quartz. It occurs in many organic materials such as solutions of sugar or amino acids. Nonreciprocal materials of interest in photonics can be magnetic with an intrinsic magnetization, M 0 , or nonmagnetic but subject to a static or low-frequency external magnetic ﬁeld, H 0 ; these materials exhibit magnetically induced optical activity for magneto-optics applications, such as optical isolation and optical circulation. Consider, for simplicity, a nonsymmetric ϵ that has only two off-diagonal elements: 0 n2⊥ ϵ ¼ ϵ 0 @ iξ 0 iξ n2⊥ 0 1 0 0 A, n2k where n⊥ , nk , and ξ can be real or complex. The eigenvalues are ϵ þ ¼ ϵ 0 n2⊥ ξ , ϵ ¼ ϵ 0 n2⊥ þ ξ , ϵ z ¼ ϵ 0 n2k ; (2.16) (2.17) and the corresponding eigenvectors are 1 1 ^e þ ¼ pﬃﬃﬃ ð^x þ i^y Þ, ^e ¼ pﬃﬃﬃ ð^x i^y Þ, ^z : 2 2 (2.18) The complex eigenvectors, ^e þ and ^e are respectively the left and right circularly polarized unit vectors deﬁned in (1.75) and (1.78). These two eigenvectors are complex unit vectors because the ϵ tensor in (2.16) is not symmetric. If n⊥ , nk , and ξ are all real, the eigenvalues are all real Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.2 Optical Anisotropy 31 because then ϵ is Hermitian. If n⊥ , nk , or ξ is complex, the eigenvalues are also complex because then ϵ is non-Hermitian. It is clearly not possible to attach the meaning of the principal axes in real space to the complex eigenvectors given in (2.18). Nonetheless, these eigenvectors still deﬁne the principal normal modes of polarization for proper decomposition of optical ﬁeld components: Dþ ¼ ϵ þ E þ , D ¼ ϵ E , Dz ¼ ϵ z E z : (2.19) Therefore, ϵ þ =ϵ 0 , ϵ =ϵ 0 , and ϵ z =ϵ 0 are the principal dielectric constants for the three normal modes. They deﬁne the following three principal indices of refraction: nþ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ξ ξ n2⊥ ξ n⊥ , n ¼ n2⊥ þ ξ n⊥ þ , nz ¼ nk , 2n⊥ 2n⊥ (2.20) where the approximate expansion of the square root is valid for ξ=2n⊥ n⊥ : The propagation constants for the principal normal modes of polarization are kþ ¼ nþ ω n ω nz ω , k ¼ , kz ¼ : c c c (2.21) When an optical wave propagates along the z axis, in either the positive z or the negative z direction, the principal normal modes of polarization are the circularly polarized modes ^e þ and ^e , which have different propagation constants k þ and k , respectively. This phenomenon that the two circularly polarized modes have different propagation constants is called circular birefringence. In the presence of an optical loss or gain, both nþ and n become complex no matter whether the optical loss or gain is characterized by the nonzero imaginary part of a complex n⊥ or ξ, or both. When the imaginary parts of nþ and n have different values, the two circularly polarized normal modes experience different degrees of optical loss or gain. This phenomenon is called circular dichroism, as distinct from the linear dichroism between two linearly polarized modes. Circular birefringence caused by the magneto-optic effect in a magnetic material or in a nonmagnetic material subject to a magnetic ﬁeld is known as magnetic circular birefringence. Circular birefringence in a nonmagnetic reciprocal material that has natural optical activity is known as natural circular birefringence. Circular dichroism caused by a loss or gain associated with the magneto-optic effect in a magnetic material or in a nonmagnetictic material subject to a magnetic ﬁeld is known as magnetic circular dichroism. Circular dichroism due to a loss or gain in a nonmagnetic reciprocal material that has natural optical activity is known as natural circular dichroism. The similarities between the two phenomena of natural optical activity and magnetically induced optical activity are that both have circularly polarized normal modes and both can cause circular birefringence and circular dichroism. In both cases, the plane of polarization of a linearly polarized wave can be rotated as the wave travels through the material. The fundamental difference between the two phenomena is that natural optical activity is reciprocal, so that a round trip through the medium cancels the polarization rotation, whereas magnetically induced optical activity is nonreciprocal, so that a round trip through the medium does not cancel but doubles the polarization rotation. In the simplest case of the nonsymmetric ϵ tensor of the form Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 32 Optical Properties of Materials given in (2.16), natural optical activity can be described by ξ ¼ γk^ ^z , which depends on the propagation direction k^ and on a characteristic constant γ of the medium, whereas magnetically induced optical activity is described by ξ ðM 0z Þ or ξ ðH 0z Þ, which is a linear function of M 0z or ^ Whereas all materials exhibit magneticH 0z but is independent of the propagation direction k. ally induced optical activity in the presence of a magnetization or a magnetic ﬁeld, natural optical activity cannot exist in centrosymmetric materials. In an otherwise centrosymmetric medium, such as a liquid, the addition of molecules, such as sugar molecules, that cause optical activity breaks the centrosymmetry of the system. 2.3 RESONANT OPTICAL SUSCEPTIBILITY .............................................................................................................. Frequency dispersion of a medium is caused by the fact that the response of the medium to an optical ﬁeld does not end instantaneously but relaxes over time after the optical stimulation. The root of the optical response is the interaction between the electrons in the material and the optical ﬁeld. The electrons in a material can be either valence electrons, which are localized bound electrons, or conduction electrons, which are nonlocalized free electrons. The electrons in atoms and molecules are bound electrons that have discrete energy levels. In a condensed matter, such as a solid material, the electronic states form energy bands. Separate impurity atoms or molecules that are embedded in an insulator or a semiconductor as dopants can have discrete energy levels inside an energy band or in the gap between two energy bands of the host solid. The electrons in a valence band of a solid material, which can be an insulator, a semiconductor, or a metal, are bound electrons. An electron in a conduction band of a semiconductor or a metal behaves like a free electron, but it has an effective mass that is determined by the structure of the conduction band and is different from the electron mass in free space. A hole in a valence band of a semiconductor behaves like a free positive charge carrier with an effective mass that is determined by the structure of the valence band. Resonant interaction involves the transition of an electron, stimulated by an optical ﬁeld, between two discrete energy levels or between two energy bands. Nonresonant interaction can take place between an electron in a conduction band, or a hole in a valence band, and an optical ﬁeld while the electron or hole stays in the same band without making a transition to another band. Both resonant and nonresonant interactions contribute to material dispersion, but their characteristics are different. In this section, the salient characteristics of resonant interactions involving valence electrons are considered. The dispersion characteristics of nonresonant interactions involving free charge carriers are considered in the next section. A given material generally has many transition resonances across the electromagnetic spectrum; each resonance is characterized by a resonance frequency, ω0 , and a relaxation rate, γ. A resonant interaction involves two separate energy states: a lower energy state j1i of energy E1 and population density N 1 , and an upper energy state j2i of energy E2 and population density N 2 . The energy states j1i and j2i are discrete energy levels in an atom or molecule, or speciﬁc states in different energy bands of a condensed matter. The population densities N 1 and N 2 are the number of electrons per unit volume in states j1i and j2i, respectively. When a material is in thermal equilibrium with its background environment, i.e., in its normal state, the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 33 2.3 Resonant Optical Susceptibility Figure 2.2 Discrete energy levels for resonant interaction. laws of population distribution require that its lower energy level be more populated than its upper energy level such that N 1 > N 2 : Population inversion with N 2 > N 1 is possible only when a material is sufﬁciently pumped to bring it far away from thermal equilibrium. Because the focus of this section is on the salient features of resonant susceptibility, we consider the simple case of the resonant interaction involving two discrete energy levels as shown in Fig. 2.2. The transition resonance frequency is determined by the energy separation of the two levels, ω0 ¼ ω21 ¼ E2 E1 , ℏ (2.22) and the relaxation rate is the total susceptibility relaxation rate contributed by various relaxation mechanisms involving the two energy levels, γ ¼ γ21 : (2.23) Note that the susceptibility relaxation rate γ ¼ γ21 discussed here is the rate of relaxation of the optical polarization induced by the optical ﬁeld, which is generally different from the population decay rates of the two energy levels. The details of such differences are discussed in Section 7.1. The resonant susceptibility associated with two discrete energy levels can be obtained by quantum mechanical calculation through the density matrix formalism. Quantum mechanical calculation allows the accurate treatment of the susceptibility as a tensor; it can be extended to a complex system that has multiple energy levels or energy bands. A classical Lorentz model that describes the single-resonance system as a one-dimensional damped oscillator is often used to obtain the key features of the resonant susceptibility. (See Problem 2.3.1.) The quantum mechanical result of the resonant susceptibility tensor as a function of the response time t with respect to an optical excitation at time zero is 2ðN 1 N 2 Þp12 p12 γ21 t e sin ω21 t H ðt Þ ϵ0ℏ 8 < 2ðN 1 N 2 Þp12 p12 γ21 t sin ω21 t, t 0; e ¼ ϵ0ℏ : 0, t < 0; χres ðt; ω21 Þ ¼ (2.24) where the Heaviside step function H ðt Þ has the values of H ðtÞ ¼ 1 for t 0 and H ðt Þ ¼ 0 for t < 0; and p12 ¼ h1jp^j2i is the matrix element of the electric-dipole operator p^ ¼ e^ x for the transition between states j1i and j2i, where e is the electronic charge and x^ is the displacement Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 34 Optical Properties of Materials operator. We consider the eigenvalue of the susceptibility tensor for a normal mode of polarization ^e . For simplicity, we express it in terms of ω0 and γ by applying (2.22) and (2.23): 8 < 2ΔNp2 γt 2 2ΔNp γt (2.25) χ res ðt; ω0 Þ ¼ e sin ω0 t H ðt Þ ¼ ϵ 0 ℏ e sin ω0 t, t 0; : ϵ0ℏ 0, t < 0; where ΔN ¼ N 2 N 1 is the population difference between the upper and the lower energy levels, and p ¼ p12 ^e is the electric-dipole strength of the resonant transition. Note that χ res ðt Þ ¼ 0 for t < 0 because a medium can respond only after, but not before, an excitation. This is the causality condition, which applies to all physical systems. The Fourier transform of (2.25) to the frequency domain yields ð∞ χ res ðω; ω0 Þ ¼ χ res ðt; ω0 Þeiωt dt ∞ ΔNp2 1 1 ¼ ϵ 0 ℏ ω ω0 þ iγ ω þ ω0 þ iγ ΔNp2 1 : ϵ 0 ℏ ω ω0 þ iγ (2.26) In (2.26), we have taken the so-called rotating-wave approximation by keeping only the resonant term that contains ω ω0 in the denominator and dropping the nonresonant term that contains ω þ ω0 in the denominator because for a frequency ω in the optical spectral region it is always valid that ω þ ω0 jω ω0 j near resonance. The real and imaginary parts of this resonant susceptibility are χ 0res ðωÞ ¼ ΔNp2 ω ω0 ΔNp2 γ 00 , χ ð ω Þ ¼ , res 2 2 ϵ 0 ℏ ðω ω0 Þ þ γ ϵ 0 ℏ ðω ω0 Þ2 þ γ2 (2.27) which are plotted in Fig. 2.3. Figure 2.3 Real and imaginary parts, χ 0res ðωÞ and χ 00res ðωÞ, respectively, of susceptibility for a medium that shows (a) a loss and (b) a gain near a resonance frequency at ω0 . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.3 Resonant Optical Susceptibility 35 The imaginary part χ 00res ðωÞ of the resonant susceptibility has a Lorentzian lineshape, which has a full width at half-maximum (FWHM) of Δω ¼ 2γ. In terms of the frequency ν ¼ ω=2π, the lineshape has a center frequency at ν0 ¼ ω0 =2π and a FWHM of Δν ¼ Δω=2π ¼ γ=π. The sign of χ 00res ðωÞ depends on that of ΔN. When the material is in its normal state in thermal equilibrium with the surrounding, the lower energy level is more populated than the upper level so that ΔN < 0; thus, the material shows an optical loss near resonance with χ 00res ðωÞ > 0. This characteristic results in the absorption of light at the resonance frequency ω ¼ ω0 when the material is in thermal equilibrium with its background environment. When population inversion is accomplished so that ΔN > 0, the material shows an optical gain with χ 00res ðωÞ < 0, resulting in the ampliﬁcation of light at ω ¼ ω0 due to stimulated emission, such as in the case of an optical ampliﬁer or a laser. Note that both χ 0res ðωÞ and χ 00res ðωÞ are proportional to ΔN. Therefore, when χ 00res ðωÞ changes sign with ΔN, χ 0res ðωÞ also changes sign. When χ 00res ðωÞ > 0, for ΔN < 0, χ 0res ðωÞ is positive for ω < ω0 and negative for ω > ω0 , as is shown in Fig. 2.3(a); when χ 00res ðωÞ < 0, for ΔN > 0, χ 0res ðωÞ is negative for ω < ω0 and positive for ω > ω0 , as is shown in Fig. 2.3(b). A medium generally has many resonance frequencies, each corresponding to an absorption frequency for the medium in its normal state. The permittivity of the medium due to all bound electrons is the sum of all resonance susceptibilities: " # X X ΔN i p2 1 1 i : (2.28) χ res ðω; ω0i Þ ¼ ϵ 0 þ ϵ bound ðωÞ ¼ ϵ 0 1 þ ℏ ω ω0i þ iγi ω þ ω0i þ iγi i i Note that the rotating-wave approximation is not taken in the above expression because a frequency ω near one resonance frequency can be very far away from another resonance frequency. For this reason, the rotating-wave approximation is not generally valid across a broad spectrum. The characteristics of the real and imaginary parts of ϵ bound ðωÞ for a medium in its normal state as a function of ω over a spectral range covering a few resonances are illustrated in Fig. 2.4. Some important dispersion characteristics of χ res ðωÞ and ϵ bound ðωÞ are summarized below. 1. It can be seen from Fig. 2.3(a) that for a material in its normal state, χ 0res ðω < ω0 Þ is always larger than χ 0res ðω > ω0 Þ. Therefore, around any single resonance frequency, ϵ 0bound ðωÞ at any frequency on the low-frequency side has a value greater than that at any frequency on the high-frequency side. 2. From (2.28), it is found that ϵ bound ð0Þ ¼ ϵ 0 X ΔN i p2 2ω0i i > ϵ 0 and ϵ bound ð∞Þ ¼ ϵ 0 : ℏ ω20i þ γ2i i (2.29) We see that because ΔN i < 0 for a material in thermal equilibrium, the DC susceptibility contributed by all bound electrons in a material is real and positive so that the DC permittivity ϵ bound ð0Þ due to all bound electrons is always real and larger than ϵ 0 . At a very high frequency that is well above all resonance frequencies, such as one in the hard X-ray region, all bound electrons stop responding to the high-frequency ﬁeld so that the medium behaves much like free space to the high-frequency ﬁeld; thus ϵ bound ð∞Þ ¼ ϵ 0 . At a ﬁnite frequency of ω that is far away from any resonance frequency, ϵ 00bound ðωÞ 0 so that ϵ bound ðωÞ ϵ 0bound ðωÞ and ϵ bound ð0Þ > ϵ bound ðωÞ. Therefore, the permittivity of an insulator, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 36 Optical Properties of Materials Figure 2.4 Real and imaginary parts of ϵ bound as a function of ω for a medium in its normal state over a spectral range covering a few resonance frequencies. which does not have free charge carriers, at a frequency that is far away from all resonances is always smaller than its DC permittivity. 3. A medium is said to have normal dispersion in a spectral region where ϵ 0 ðωÞ increases with frequency so that dϵ 0 =dω > 0. It is said to have anomalous dispersion in a spectral region where ϵ 0 ðωÞ decreases with increasing frequency so that dϵ 0 =dω < 0. Because dn=dω and dϵ 0 =dω have the same sign, the index of refraction also increases with frequency in a spectral region of normal dispersion and decreases with frequency in a spectral region of anomalous dispersion. 4. It can be seen from Fig. 2.4 that when a material is in its normal state in thermal equilibrium, normal dispersion appears everywhere except in the immediate neighborhood within the FWHM of a resonance frequency, where anomalous dispersion occurs. This characteristic can be reversed near a resonance frequency where resonant ampliﬁcation, rather than absorption, takes place due to population inversion. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.3 Resonant Optical Susceptibility 37 5. In most materials that are transparent in the visible spectral region, such as glass and water, normal dispersion appears in the visible region and may extend to the near-infrared and nearultraviolet regions. Only transitions between discrete energy levels are considered above. In a solid material where electronic states form energy bands, transitions between separate energy bands, called band-to-band transitions or interband transitions, contribute to the resonant susceptibility of the material. The susceptibility is found by integrating over the electronic states in the two bands involved in the transitions; the integration takes into account the population distribution probability of electrons in each band. The general concepts described above are still valid, except that the susceptibility contributed by band-to-band transitions does not show the characteristic sharp resonance peaks of transitions between discrete energy levels seen in Figs. 2.3 and 2.4. EXAMPLE 2.3 An atomic absorption spectral line associated with an optical transition from the ground state to an excited state is found to appear at a center wavelength of λ ¼ 800 nm with a FWHM spectral width of Δλ ¼ 0:48 nm. Find the energy of the excited state above the ground state. Find the resonance frequency and the polarization relaxation rate associated with this transition. Where can we ﬁnd anomalous dispersion caused by this atomic transition when the atoms are in their normal state in thermal equilibrium with the surrounding? Solution: The energy of the excited state above the ground state is the photon energy of the absorption wavelength at λ ¼ 800 nm: E2 E1 ¼ hν ¼ 1239:8 1239:8 nm eV ¼ eV ¼ 1:55 eV: λ 800 The resonance frequency is c 3 108 m s1 ν0 ¼ ¼ ¼ 375 THz ; λ 800 109 m Because λ ω0 ¼ 2πν0 ¼ 2:36 1015 rad s1 : Δλ, we can use the approximation Δν=ν0 Δλ=λ to ﬁnd that Δν ¼ Δλ 0:48 ν0 ¼ 375 THz ¼ 225 GHz: λ 800 Thus, the relaxation rate is γ ¼ πΔν ¼ 7:07 1011 s1 : When the atoms are in their normal state in thermal equilibrium with the surrounding, the ground state is more populated than the excited state. In this situation, anomalous dispersion caused by this transition is found within the FWHM of the spectral line, in the wavelength range of λ Δλ=2 ¼ 800 0:24 nm, corresponding to the frequency range of ν0 Δν=2 ¼ 375 THz 112:5 GHz. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 38 Optical Properties of Materials 2.4 OPTICAL CONDUCTIVITY AND CONDUCTION SUSCEPTIBILITY .............................................................................................................. An electron in a conduction band of a semiconductor or a metal behaves like a free electron with an effective mass, while a hole in a valence band of a semiconductor behaves like a free positive charge carrier with an effective mass. The response of these free charge carriers to an optical ﬁeld can be treated using quantum mechanics by considering induced transitions within a band, known as intraband transitions, or using a classical Drude model by considering an induced conduction current J cond as discussed in Section 1.1. Because the quantum mechanical approach involves the consideration of the band structure, we use the classical Drude model for simplicity. In this classical approach, the effective mass m∗ of the charge carrier accounts for the effect of the energy band; clearly, the value of m∗ depends on the structure of the energy band on which the charge carrier lies. In the Drude model, conduction electrons, and holes in a semiconductor, are treated as independent free charge carriers. The momentum, p, of a charge carrier is driven by the force of an electric ﬁeld, F ¼ qE, and is damped by random collisions with the ions of the medium, characterized by an average momentum relaxation time τ. Therefore, dp p ¼ qE , dt τ (2.30) where q ¼ e for an electron and q ¼ e for a hole. The conduction current density is J cond ¼ Nqv ¼ Nqp , m∗ (2.31) where N is the density of the free charge carriers. By combining (2.30) and (2.31), we have the equation for the conduction current that is induced by an electric ﬁeld: dJ cond J cond Ne2 þ ¼ ∗ E, dt τ m (2.32) where q2 ¼ e2 is used for the charge carriers to be either electrons or holes. The general solution of (2.32) can be expressed as a convolution integral: ðt J cond ðtÞ ¼ σ ðt t 0 ÞEðt 0 Þdt 0, (2.33) ∞ where 8 2 < Ne t=τ Ne t=τ e , σ ðtÞ ¼ ∗ e H ðt Þ ¼ m∗ : m 0, 2 for t 0; (2.34) for t < 0: Note that Jcond ðt Þ and Eðt Þ are real ﬁelds in the real space and time domain. The relation in (2.33) deﬁnes the optical conductivity σ ðt Þ in the real space and time domain, as seen in (2.34). For simplicity, their spatial dependence is ignored. In terms of the complex ﬁeld, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.4 Optical Conductivity and Conduction Susceptibility ðt Jcond ðt Þ ¼ σ ðt t 0 ÞEðt0 Þdt 0, 39 (2.35) ∞ where σ ðt Þ is the same as that in (2.34). The frequency domain relation is obtained by taking the Fourier transform on (2.35): Jcond ðωÞ ¼ σ ðωÞEðωÞ, (2.36) where ð∞ σ ðtÞeiωt dt ¼ σ ðω Þ ¼ ∞ Ne2 τ 1 : m∗ 1 iωτ (2.37) This frequency-dependent optical conductivity, also known as the AC conductivity, can be expressed in terms of the DC conductivity: σ ðωÞ ¼ σ ð0Þ , 1 iωτ (2.38) Ne2 τ : m∗ (2.39) where σ ð0Þ is the DC conductivity, σ ð0Þ ¼ As discussed in Section 1.1, there are two alternative, but equivalent, ways to described the optical response of free charge carriers: (1) by treating it as part of the total susceptibility and total permittivity in the total displacement D, as in (1.12); or (2) by treating it as an optical conductivity through an explicit conduction current Jcond , as in (1.16). The discussion above follows the second alternative, which allows us to ﬁnd the optical conductivity in (2.38). By equating the two alternative approaches, the conduction susceptibility, χ cond , due to the free charge carriers can be found. Equating (1.12) and (1.16) but expressing them in complex ﬁelds, we have ∂D ∂Dbound þ Jcond : ¼ ∂t ∂t (2.40) Converting this relation to the frequency domain, we ﬁnd iωDðωÞ ¼ iωDbound ðωÞ þ Jcond ðωÞ: (2.41) By using the relations DðωÞ ¼ ϵ ðωÞEðωÞ, DðωÞbound ¼ ϵ bound ðωÞEðωÞ, and Jcond ðωÞ ¼ σ ðωÞEðωÞ from (2.36), we ﬁnd the total permittivity that includes all contributions from bound and free charges in a material: ϵ ðωÞ ¼ ϵ bound ðωÞ þ iσ ðωÞ σ ð0Þ ¼ ϵ bound ðωÞ , ω ωðωτ þ iÞ (2.42) where ϵ bound ðωÞ is the permittivity from bound charges discussed in Section 2.3. Therefore, we can identify the conduction susceptibility due to the free charge carriers: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 40 Optical Properties of Materials Figure 2.5 Real and imaginary parts, χ 0cond ðωÞ and χ 00cond ðωÞ, respectively, of the conduction susceptibility, normalized to σ ð0Þτ=ϵ 0 , as a function of ωτ. χ cond ðωÞ ¼ iσ ðωÞ σ ð0Þτ 1 ¼ : ϵ0ω ϵ 0 ωτ ðωτ þ iÞ (2.43) The real and imaginary parts of this conduction susceptibility are χ 0cond ðωÞ ¼ σ ð0Þτ 1 σ ð0Þτ 1 , χ 00cond ðωÞ ¼ , 2 2 2 ϵ0 ω τ þ 1 ϵ 0 ωτ ðω τ 2 þ 1Þ (2.44) which are plotted in Fig. 2.5. At an optical frequency that is far away from any resonance transition frequency, 00 ϵ bound ðωÞ 0 so that ϵ bound ðωÞ ϵ 0bound ðωÞ. In this case, the real and imaginary parts of the total permittivity given in (2.42) are ϵ 0 ðωÞ ¼ ϵ bound ðωÞ σ ð0Þτ , þ1 ω2 τ 2 ϵ 00 ðωÞ ¼ σ ð0Þτ : ωτ ðω2 τ 2 þ 1Þ (2.45) We ﬁnd that due to the effect of the conduction electrons, the real part of the total susceptibility vanishes, i.e., ϵ 0 ðωÞ ¼ 0, at the frequency ωp , known as the plasma frequency: ω2p ¼ σ ð0Þ ϵ bound τ 1 Ne2 1 σ ð0Þ Ne2 ¼ ¼ : τ 2 ϵ bound m∗ τ 2 ϵ bound τ ϵ bound m∗ (2.46) Because it is almost always true that ωp τ 1 for most conducting materials, the plasma frequency is generally deﬁned by neglecting the 1=τ 2 term in (2.46). The permittivity ϵ bound in (2.46) is taken to be a constant that has the value in the frequency range of interest. In terms of ω2p , the total permittivity can be expressed as " ϵ ðωÞ ¼ ϵ bound 1 ω2p τ 2 ωτ ðωτ þ iÞ # " ¼ ϵ bound 1 ω2p τ 2 ω2 τ 2 þ 1 þi ω2p τ 2 ωτ ðω2 τ 2 þ 1Þ The real and imaginary parts of this total permittivity are plotted in Fig. 2.6. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 # : (2.47) 2.4 Optical Conductivity and Conduction Susceptibility 41 Figure 2.6 Real and imaginary parts, ϵ 0 ðωÞ and ϵ 00 ðωÞ, respectively, of the total permittivity, normalized to ϵ bound , as a function of frequency ω showing (a) low-frequency characteristics and (b) high-frequency characteristics. The value of ωp τ ¼ 10 is used for this plot. Some important characteristics are summarized below. 1. For all frequencies, the real part χ 0cond ðωÞ of the conduction susceptibility is negative, and the imaginary part χ 00cond ðωÞ is positive. Thus the conduction susceptibility only contributes to optical loss and never contributes to optical gain, and it makes possible a negative real part for the permittivity, as discussed below. 2. At low frequencies for which ωτ 1, ϵ 0 ðωÞ=ϵ bound 1 ω2p τ 2 approaches a constant but ϵ 00 ðωÞ=ϵ bound ω2p τ=ω becomes inversely proportional to frequency so that jϵ 00 ðωÞj jϵ 0 ðωÞj. Then, ! 2 ω τ p : (2.48) ϵ ðωÞ ϵ bound 1 ω2p τ 2 þ i ω These low-frequency characteristics are seen in Fig. 2.6(a). 3. At high frequencies for which ωτ 1, ϵ 0 ðωÞ=ϵ bound 1 ω2p =ω2 and ϵ 00 ðωÞ 0 so that jϵ 0 ðωÞj jϵ 00 ðωÞj. Then, ! ω2p ϵ ðωÞ ϵ bound 1 2 : (2.49) ω These high-frequency characteristics are seen in Fig. 2.6(b). 4. At all frequencies, the imaginary part of the permittivity is positive because χ 00cond ðωÞ is positive: ϵ 00 ðωÞ > 0 for all ω. 5. At frequencies below the plasma frequency, the real part of the permittivity is negative: ϵ 0 ðωÞ < 0 for ω < ωp . This leads to high reﬂectivity on the surface and low penetration of the optical ﬁeld in the medium, which are the common properties of metallic surfaces. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 42 Optical Properties of Materials 6. At frequencies above the plasma frequency, the real part of the permittivity is positive while the positive imaginary part decreases quickly with increasing frequency. Consequently, the contribution of the conduction susceptibility quickly diminishes. Then the medium behaves optically like an insulator, allowing a high-frequency optical ﬁeld to penetrate through with little attenuation except when the optical frequency comes close to a transition resonance. 7. For a perfect conductor, only free conduction electrons contribute to the optical response so that the permittivity has no contribution from bound electrons; thus, ϵ bound ¼ ϵ 0 . For this reason, it is a good approximation to take ϵ bound ¼ ϵ 0 for a metal that has a high conductivity, such as Ag, Au, Cu, and Al. For such a metal, it is also a good approximation to take the effective electron mass as the free electron mass, m∗ ¼ m0 , when applying (2.46). 8. For a semiconductor where electrons and holes both contribute to the conduction susceptibility, the total permittivity is ϵ ðωÞ ¼ ϵ bound ðωÞ σ e ð0Þ σ h ð0Þ , ωðωτ e þ iÞ ωðωτ h þ iÞ (2.50) where σ e ð0Þ ¼ N e e2 τ e N h e2 τ h and σ ð0Þ ¼ : h m∗ m∗ e h (2.51) The plasma frequency is found at ϵ 0 ðωÞ ¼ 0 to be ω2p ¼ σ e ð0Þ 1 σ h ð0Þ 1 N e e2 N h e2 2þ 2 þ : ϵ bound m∗ ϵ bound τ e τ e ϵ bound τ h τ h ϵ bound m∗ e h (2.52) EXAMPLE 2.4 Silver is one of the best conductors such that the free-electron Drude model describes its optical properties reasonably well. In this model, the free electron density of Ag is found to be N ¼ 5:86 1028 m3 . The DC conductivity of Ag at T ¼ 273 K is σ ð0Þ ¼ 6:62 107 S m1 . Find the plasma frequency ωp and the relaxation time τ for Ag at T ¼ 273 K. Also ﬁnd the cutoff optical frequency νp and the cutoff wavelength λp . For what optical wavelengths is Ag expected to be highly reﬂective? For what wavelengths is it expected to become transmissive? Solution: For Ag, it is a good approximation to take ϵ bound ¼ ϵ 0 and m∗ ¼ m0 . Then, using (2.46), we ﬁnd that ω2p 2 5:86 1028 1:6 1019 Ne2 ¼ ¼ rad2 s2 ¼ 1:86 1032 rad2 s2 12 31 ϵ 0 m∗ 8:854 10 9:1 10 ) ωp ¼ 1:36 1016 rad s1 , τ¼ σ ð0Þ 6:62 107 ¼ s1 ¼ 4:02 1014 s ¼ 40:2 fs: ϵ 0 ω2p 8:854 1012 1:86 1032 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.4 Optical Conductivity and Conduction Susceptibility 43 The cutoff frequency and cutoff wavelength are those at the plasma frequency: νp ¼ ωp ¼ 2:17 PHz, 2π λp ¼ c ¼ 138 nm: νp Ag is highly reﬂective for λ > λp , corresponding to ν < νp ; it becomes transmissive for λ < λp , corresponding to ν > νp . EXAMPLE 2.5 GaAs is a direct-gap semiconductor that has an electron effective mass of m∗ e ¼ 0:067m0 and a hole effective mass of m∗ ¼ 0:52m , where m is the mass of a free electron. Its low-frequency 0 0 h dielectric constant is 10.9. Find the plasma frequency, the cutoff frequency, and the cutoff wavelength for (a) an n-type GaAs sample that has an electron density of N e ¼ 1 1024 m3 , (b) a p-type GaAs sample that has a hole density of N h ¼ 1 1024 m3 , and (c) a GaAs sample that is injected with an equal electron and hole density of N e ¼ N h ¼ 1 1024 m3 . Solution: As we will see below, the plasma frequency is much lower than the bandgap frequency of GaAs, which corresponds to a wavelength of λg ¼ 871 nm. Therefore, the low-frequency dielectric constant is used for ϵ bound ¼ 10:9ϵ 0 . Then, the plasma frequency is found using (2.52). (a) For the n-type GaAs with N e ¼ 1 1024 m3 , the hole density is negligibly small so that ω2p N e e2 ϵ bound m∗ e 2 1 1024 1:6 1019 ¼ rad2 s1 ¼ 4:35 1027 rad2 s2 : 10:9 8:854 1012 0:067 9:1 1031 Therefore, ωp ¼ 6:60 1013 rad s1 , νp ¼ 10:5 THz, and λp ¼ 28:6 μm. (b) For the p-type GaAs with N h ¼ 1 1024 m3 , the electron density is negligibly small so that ω2p N h e2 ϵ bound m∗ h 2 1 1024 1:6 1019 ¼ rad2 s1 ¼ 5:60 1026 rad2 s2 : 10:9 8:854 1012 0:52 9:1 1031 Therefore, ωp ¼ 2:37 1013 rad s1 , νp ¼ 3:77 THz, and λp ¼ 79:6 μm. (c) For the injected GaAs with N e ¼ N h ¼ 1 1024 m3 , ω2p N e e2 N h e2 þ ϵ bound m∗ ϵ bound m∗ e h ¼ 4:35 1027 rad2 s2 þ 5:60 1026 rad2 s2 ¼ 4:91 1027 rad2 s2 : Therefore, ωp ¼ 7:01 1013 rad s1 , νp ¼ 11:2 THz, and λp ¼ 26:8 μm. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 44 Optical Properties of Materials 2.5 KRAMERS–KRONIG RELATIONS .............................................................................................................. It can be seen from the above discussion that the real and imaginary parts of χ ðωÞ, or those of ϵ ðωÞ, are not independent of each other. The susceptibility of any physical system has to satisfy the causality requirement in the time domain. This requirement leads to a general relationship between χ 0 ðωÞ and χ 00 ðωÞ in the frequency domain: ð∞ ð∞ 2 ω0 χ 00 ðω0 Þ 0 2 ωχ 0 ðω0 Þ 00 dω , χ ð ω Þ ¼ dω0 , P χ ðωÞ ¼ P 0 2 π ω ω2 π ω0 2 ω2 0 0 (2.53) 0 where the principal values are taken for the integrals. These relations are known as the Kramers– Kronig relations. They are valid for any χ ðωÞ that represents a physical process, such as the resonant susceptibility χ res ðωÞ in Section 2.3 and the conduction susceptibility χ cond ðωÞ in Section 2.4. Therefore, once the real part of χ ðωÞ for any physical process is known over the entire spectrum, its imaginary part can be found, and vice versa. Note that the relations in (2.53) are consistent with the fact that χ 0 ðωÞ is an even function of ω and χ 00 ðωÞ is an odd function of ω, as discussed in Section 2.1. The contradiction to this statement seen in (2.27) for χ 0res ðωÞ and χ 00res ðωÞ is only apparent but not real; it is caused by the rotating-wave approximation taken in (2.26). There is no contradiction when the rotating-wave approximation is removed and exact expressions are used for χ 0res ðωÞ and χ 00res ðωÞ. For χ 0cond ðωÞ and χ 00cond ðωÞ given in (2.44), it is clear that χ 0cond ðωÞ is an even function of ω and χ 00cond ðωÞ is an odd function of ω. 2.6 EXTERNAL FACTORS .............................................................................................................. The optical property of a material is inﬂuenced by external factors, such as an electric ﬁeld, a magnetic ﬁeld, an acoustic wave, an injection current, a pressure, or a temperature change. The dependence of the optical property on an externally controllable factor allows the active control and modulation of an optical wave; this is the basis for active photonic devices. On the other hand, this characteristic is passively used in a photonic sensor which optically senses the parameter that causes a change in the optical property of the sensor material. Some of the major effects that cause changes in the permittivity of an optical material are discussed below. 2.6.1 Electro-optic Effect The optical property of a dielectric material can be changed by a static or low-frequency electric ﬁeld E0 through an electro-optic effect. The result is a ﬁeld-dependent susceptibility and thus a ﬁeld-dependent permittivity: Pðω; E0 Þ ¼ ϵ 0 χðω; E0 Þ EðωÞ ¼ ϵ 0 χðωÞ EðωÞ þ ϵ 0 Δχðω; E0 Þ EðωÞ (2.54) Dðω; E0 Þ ¼ ϵ ðω; E0 Þ EðωÞ ¼ ϵ ðωÞ EðωÞ þ Δϵ ðω; E0 Þ EðωÞ, (2.55) and Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.6 External Factors 45 where the ﬁeld-independent susceptibility χðωÞ ¼ χðω; E0 ¼ 0Þ and permittivity ϵ ðωÞ ¼ ϵ ðω; E0 ¼ 0Þ represent the intrinsic linear response of the material at the optical frequency ω, whereas Δχðω; E0 Þ and Δϵ ðω; E0 Þ represent the changes induced by the electric ﬁeld E0 . We can deﬁne the electric-ﬁeld-induced polarization change as ΔPðω; E0 Þ ¼ ϵ 0 Δχðω; E0 Þ EðωÞ to express the total ﬁeld-dependent displacement as Dðω; E0 Þ ¼ DðωÞ þ ΔPðω; E0 Þ. The total permittivity of the material in the presence of the electric ﬁeld is then ϵ ðω; E0 Þ ¼ ϵ ðωÞ þ Δϵ ðω; E0 Þ ¼ ϵ ðωÞ þ ϵ 0 Δχðω; E0 Þ: (2.56) In the discussion of electro-optic effects, it is necessary to introduce the relative impermeability tensor, which is the inverse of the dielectric constant tensor: 1 ϵ : (2.57) η¼ ϵ0 The reason for considering the relative impermeability tensor is historical because electro-optic effects are traditionally not expressed as Δϵ ðω; E0 Þ or Δχðω; E0 Þ but are deﬁned in terms of the changes in the elements of the relative impermeability tensor as ηðE0 Þ ¼ η þ ΔηðE0 Þ, which is expanded as X X ηij ðE0 Þ ¼ ηij þ Δηij ðE0 Þ ¼ ηij þ r ijk E 0k þ sijkl E 0k E 0l þ , (2.58) k k, l where the ﬁrst term ηij is the ﬁeld-independent component, the elements of the third-order rijk tensor are the linear electro-optic coefﬁcients known as the Pockels coefﬁcients, and those of the fourth-order sijkl tensor are the quadratic electro-optic coefﬁcients known as the electrooptic Kerr coefﬁcients. The ﬁrst-order electro-optic effect characterized by the linear dependence of ηij ðE0 Þ on E0 through the coefﬁcients r ijk is called the linear electro-optic effect, also known as the Pockels effect. The second-order electro-optic effect characterized by the quadratic ﬁeld dependence through the coefﬁcients sijkl is called the quadratic electro-optic effect, also known as the electro-optic Kerr effect. In (2.58), indices i and j are associated with optical ﬁelds, whereas indices k and l are associated with the low-frequency applied ﬁeld. Because the ϵ tensor of a nonmagnetic electro-optic material is symmetric, the η tensor as deﬁned in (2.57) is also symmetric; thus ηij ¼ ηji and Δηij ¼ Δηji . The symmetric indices i and j can be contracted to reduce the double index ij to a single index α using the index contraction rule: ij : or ij : α: 11 22 33 23, 32 31, 13 12, 21 xx yy zz yz, zy zx, xz xy, yx 1 2 3 4 5 6 Using index contraction, (2.58) is expressed as X X ηα ðE0 Þ ¼ ηα þ Δηα ðE0 Þ ¼ ηα þ r αk E 0k þ sαkl E 0k E 0l þ , k k, l (2.59) (2.60) where α ¼ 1, 2, . . . , 6 and k, l ¼ 1, 2, 3 or x, y, z: The Pockels effect does not exist in a centrosymmetric material, which is a material that possesses inversion symmetry. The structure and properties of such a material remain Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 46 Optical Properties of Materials unchanged under the transformation of space inversion, which changes the signs of all rectilinear spatial coordinates from ðx; y; zÞ to ðx; y; zÞ, and the signs of all polar vectors. As discussed in Section 1.1, an electric ﬁeld vector is a polar vector that changes sign under the transformation of space inversion. By simply considering the effect of space inversion, it is clear that the electro-optically induced changes in the optical property of a centrosymmetric material are not affected by the sign change in the applied ﬁeld from E0 to E0 , meaning that ηij ðE0 Þ ¼ ηij ðE0 Þ. As can be seen from (2.58), this condition requires that the Pockels coefﬁcients r ijk vanish, but it does not require the electro-optic Kerr coefﬁcients sijkl to vanish. Consequently, the Pockels effect exists only in noncentrosymmetric materials, whereas the electro-optic Kerr effect exists in all materials, including centrosymmetric ones. Structurally isotropic materials, including all gases, liquids, and amorphous solids such as glass, show no Pockels effect because they are centrosymmetric. The majority of electro-optic devices are based on the Pockels effect because the electro-optic Kerr coefﬁcients are generally very small. For this reason, practical electro-optic applications usually require noncentrosymmetric crystals in order to make use of the Pockels effect. Among the 32 point groups in the 7 crystal systems, 11 are centrosymmetric, and the remaining 21 are noncentrosymmetric. It is important to note that the linear optical property of a crystal is determined only by its crystal system, as mentioned in Section 2.2 and summarized in Table 2.1, but the electro-optic property further depends on its point group. Because the electro-optic coefﬁcients are traditionally deﬁned through the changes in the relative impermeability tensor, as expressed in (2.58), the ﬁeld-induced changes in the permittivity tensor have to be found through the relation between Δϵ ij ðE0 Þ and Δηij ðE0 Þ. Using the relation η ϵ=ϵ 0 ¼ 1, the relation between Δϵ and Δη can be found: Δϵ ¼ 1 1 ϵ Δη ϵ and Δη ¼ η Δϵ η: ϵ0 ϵ0 (2.61) As discussed in Section 2.2, the intrinsic permittivity tensor ϵ ðωÞ of a crystal in the absence of the electric ﬁeld is diagonal with eigenvalues ϵ x , ϵ y , and ϵ z in the coordinate system deﬁned by the intrinsic principal dielectric axes ^x , ^y , and ^z , which are determined by the structural symmetry of the crystal lattice. In this coordinate system, the relations in (2.61) can be written explicitly as Δηij Δϵ ij Δϵ ij ¼ ϵ 0 n2i n2j Δηij and Δηij ¼ ϵ 0 ¼ , (2.62) ηi ηj ϵiϵj ϵ 0 n2i n2j pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where ηi ¼ ϵ 0 =ϵ i are the eigenvalues of the η tensor and ni ¼ ϵ i =ϵ 0 are the principal indices of refraction. Δϵ ij ¼ ϵ 0 EXAMPLE 2.6 LiNbO3 is a negative uniaxial crystal with nx ¼ ny ¼ no > nz ¼ ne . Being a crystal of the 3m symmetry group, it has eight nonvanishing Pockels coefﬁcients of four distinct values: r 12 ¼ r22 , r 13 , r 22 , r 23 ¼ r 13 , r 33 , r42 , r 51 ¼ r42 , r 61 ¼ r22 . Find the ﬁeld-induced permittivity change Δϵ ðE0 Þ for an applied DC electric ﬁeld of E0 ¼ E 0x ^x þ E 0y ^y þ E 0z^z . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.6 External Factors 47 Solution: According to (2.58), the ﬁeld-induced impermeability change due to the Pockels effect is X Δηα ðE0 Þ ¼ rαk E 0k , k which can be expressed in the matrix form as 0 Δη1 1 0 r 12 r 11 B C B B Δη2 C B r 21 B C B B Δη C B r 31 B 3C B B C¼B B Δη4 C B r 41 B C B B C B @ Δη5 A @ r 51 Δη6 1 r 52 C r 33 C0 1 C E0x C r 33 CB C C@ E 0y A: r 43 C C E 0z C r 53 A r 62 r 63 r 22 r 32 r 42 r 61 r 13 Using the given nonvanishing Pockels coefﬁcients for LiNbO3 , we have 0 Δη1 1 0 B C B B Δη2 C B B C B B Δη C B B 3C B B C¼B B Δη4 C B B C B B C B @ Δη5 A @ Δη6 0 r22 0 r22 0 0 0 r42 r 42 0 r 22 0 r 13 1 0 r 22 E 0y þ r13 E 0z B C r 13 C0 1 B r 22 E 0y þ r13 E 0z B C E 0x B r 33 C r 33 E 0z C B CB C@ E 0y A ¼ B B 0 C r 42 E 0y B C E 0z B C 0 A r 42 E 0x @ 1 C C C C C C: C C C A r 22 E 0x 0 By the index contraction rule, Δη1 ¼ Δηxx , Δη2 ¼ Δηyy , Δη3 ¼ Δηzz , Δη4 ¼ Δηyz ¼ Δηzy , Δη5 ¼ Δηzx ¼ Δηxz , Δη6 ¼ Δηxy ¼ Δηyx . Using (2.62), we ﬁnd Δϵ xx ¼ ϵ 0 n4x Δηxx ¼ ϵ 0 n4o r 22 E 0y ϵ 0 n4o r 13 E 0z , Δϵ yy ¼ ϵ 0 n4y Δηyy ¼ ϵ 0 n4o r 22 E 0y ϵ 0 n4o r 13 E 0z , Δϵ zz ¼ ϵ 0 n4z Δηzz ¼ ϵ 0 n4e r 33 E 0z , Δϵ yz ¼ Δϵ zy ¼ ϵ 0 n2y n2z Δηyz ¼ ϵ 0 n2o n2e r 42 E 0y , Δϵ zx ¼ Δϵ xz ¼ ϵ 0 n2x n2z Δηyz ¼ ϵ 0 n2o n2e r 42 E 0x , Δϵ xy ¼ Δϵ yx ¼ ϵ 0 n2x n2y Δηxy ¼ ϵ 0 n4o r22 E 0x : Expressed in the matrix form, the ﬁeld-induced permittivity change is 0 4 1 no r 22 E 0y n4o r 13 E 0z n4o r22 E 0x n2o n2e r 42 E 0x B C Δϵ ðE0 Þ ¼ ϵ 0 @ n4o r 22 E 0x n4o r22 E 0y n4o r13 E 0z n2o n2e r 42 E 0y A: n2o n2e r42 E 0x n2o n2e r 42 E 0y Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 n4e r 33 E 0z 48 Optical Properties of Materials The electric-ﬁeld-induced changes represented by Δϵ ðω; E0 Þ usually generate off-diagonal elements besides changing the diagonal elements: 0 1 0 1 Δϵ xy Δϵ xz ϵ x þ Δϵ xx ϵx 0 0 ϵ y þ Δϵ yy Δϵ yz A (2.63) ϵ ðωÞ ¼ @ 0 ϵ y 0 A while ϵ ðω; E0 Þ ¼ @ Δϵ yx Δϵ zx Δϵ zy ϵ z þ Δϵ zz 0 0 ϵz in the coordinate system of the principal ^x , ^y , and ^z axes. As discussed in Section 2.2, ϵ of a nonmagnetic material is a symmetric tensor. This remains true for a nonmagnetic material subject to an applied electric ﬁeld; thus, for ϵ ðω; E0 Þ in (2.63), ϵ ij ðω; E0 Þ ¼ ϵ ji ðω; E0 Þ and Δϵ ij ðω; E0 Þ ¼ Δϵ ji ðω; E0 Þ: (2.64) Because the ﬁeld-dependent nondiagonal permittivity tensor is symmetric, it can be diagonalized to ﬁnd a new set of eigenvalues ϵ X , ϵ Y , and ϵ Z with corresponding real ^ , Y^ , and Z^ , which represent a new set of linearly polarized principal normal eigenvectors X modes for deﬁning the new principal dielectric axes of the material in the presence of the electric ﬁeld E0 . In general, the new principal axes depend on the direction and, in certain cases, the magnitude of E0 . Thus, 0 1 ϵX 0 0 (2.65) ϵ ðω; E0 Þ ¼ @ 0 ϵ Y 0 A: 0 0 ϵZ The propagation characteristics of an optical wave in the presence of an electro-optic effect are then determined by ϵ X , ϵ Y , and ϵ Z , which deﬁne the principal indices of refraction, rﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ ϵX ϵY ϵZ , nY ¼ , nZ ¼ , (2.66) nX ¼ ϵ0 ϵ0 ϵ0 and the propagation constants, kX ¼ nX ω nY ω nZ ω , kY ¼ , kZ ¼ , c c c (2.67) ^ Y^ , and Z^ principal normal modes of polarization. Note that these three new principal for the X, normal modes of polarization are linearly polarized. Therefore, electrically induced birefringence and dichroism due to an electro-optical effect are linear birefringence and linear dichroism. EXAMPLE 2.7 At the 1 μm optical wavelength, LiNbO3 has the refractive indices of no ¼ 2:238 and ne ¼ 2:159. The four distinct values of its Pockels coefﬁcients are r 13 ¼ 8:6 pm V1 , r 22 ¼ 3:4 pm V1 , r 33 ¼ 30:8 pm V1 , and r 42 ¼ 28 pm V1 . Use the results from Example 2.6 to answer the following questions. Is it possible to apply a DC electric ﬁeld to change the principal indices of refraction through the Pockels effect without rotating the principal axes? If this is possible, ﬁnd the changes in the principal indices of refraction caused by an applied electric ﬁeld of E 0 ¼ 5 MV m1 . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.6 External Factors 49 Solution: For the Pockels effect to cause only changes in the principal indices of refraction without rotating the principal axes, an applied electric ﬁeld has to generate changes only in the diagonal elements, but not in the off-diagonal elements, of Δϵ ðE0 Þ. By examining Δϵ ðE0 Þ obtained in Example 2.6 for LiNbO3 , we ﬁnd that this is possible if the DC electric ﬁeld is applied only along the direction of the z principal axis such that E0 ¼ E 0^z for E 0z ¼ E 0 and E0x ¼ E 0y ¼ 0. Then, 0 2 1 no n4o r13 E 0 0 0 A: ϵ ðE0 Þ ¼ ϵ þ Δϵ ðE0 Þ ¼ ϵ 0 @ 0 n2o n4o r 13 E 0 0 2 4 0 0 ne ne r33 E 0 Because ϵ ðE0 Þ is diagonal in the coordinate system of the original principal axes, all principal axes remain unchanged: ^ ¼ ^x , Y^ ¼ ^y , Z^ ¼ ^z : X Using (2.65) and (2.66), we ﬁnd the new principal indices of refraction: nX ¼ nY ¼ ðn2o n4o r 13 E 0 Þ1=2 no n3o r 13 n3 r 33 E 0 , nZ ¼ ðn2e n4e r 33 E 0 Þ1=2 ne e E 0 : 2 2 Clearly, the crystal remains negative uniaxial. The changes in the principal indices of refraction caused by an applied electric ﬁeld of E 0 ¼ 5 MV m1 are ΔnX ¼ ΔnY ¼ Δno ¼ n3o r13 2:2283 8:6 1012 E0 ¼ 5 106 ¼ 2:41 104 2 2 for the ordinary index and ΔnZ ¼ Δne ¼ n3e r 33 2:1593 30:8 1012 E0 ¼ 5 106 ¼ 7:75 104 2 2 for the extraordinary index. 2.6.2 Magneto-optic Effect A material can be either diamagnetic or paramagnetic. A diamagnetic material does not contain intrinsic magnetic dipole moments; a paramagnetic material consists of atoms or ions that have intrinsic magnetic dipole moments. A paramagnetic material can be either magnetically disordered, when its intrinsic magnetic dipole moments are randomly oriented, or magnetically ordered. A magnetically ordered material is ferromagnetic if all of its intrinsic dipole moments line up in the same direction; it is ferrimagnetic if it contains different types of intrinsic dipole moments that line up in alternating antiparallel directions but do not cancel each other; it is antiferromagnetic, also called antiferrimagnetic, if different types of intrinsic dipole moments line up in alternating antiparallel directions and cancel each other. Below a critical temperature, known as the Curie temperature for a ferromagnetic material and the Néel temperature for a ferrimagnetic Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 50 Optical Properties of Materials material, the magnetic ordering in a ferromagnetic or ferrimagnetic material generates a spontaneous magnetization M 0 . No spontaneous magnetization exists in a diamagnetic material, in a magnetically disordered paramagnetic material, or in an antiferromagnetic material. As mentioned in Section 1.1, at an optical frequency μ ¼ μ0 and thus BðωÞ ¼ μ0 HðωÞ; the response of a material, irrespective of whether it is magnetic or nonmagnetic, to an optical ﬁeld at an optical frequency of ω is fully described by its electric susceptibility χðωÞ and, equivalently, by its electric permittivity ϵ ðωÞ. Nevertheless, a nonmagnetic material that does not have a spontaneous magnetization still responds to a static or low-frequency magnetic ﬁeld, H 0 . Its optical property can be changed by H 0 , resulting in a magnetic-ﬁeld-dependent susceptibility and a magnetic-ﬁeld-dependent permittivity: Pðω; H 0 Þ ¼ ϵ 0 χðω; H 0 Þ EðωÞ ¼ ϵ 0 χðωÞ EðωÞ þ ϵ 0 Δχðω; H 0 Þ EðωÞ (2.68) Dðω; H 0 Þ ¼ ϵ ðω; H 0 Þ EðωÞ ¼ ϵ ðωÞ EðωÞ þ Δϵ ðω; H 0 Þ EðωÞ, (2.69) and where χðωÞ ¼ χðω; H 0 ¼ 0Þ and ϵ ðωÞ ¼ ϵ ðω; H 0 ¼ 0Þ represent the intrinsic properties of the material in the absence of the static or low-frequency magnetic ﬁeld. In the case of a ferromagnetic or ferrimagnetic material, in which a static magnetization M 0 exists, the properties of the material at an optical frequency are dependent on M 0 . Then, instead of (2.68) and (2.69), we have magnetization-dependent susceptibility and magnetization-dependent permittivity: Pðω; M 0 Þ ¼ ϵ 0 χðω; M 0 Þ EðωÞ ¼ ϵ 0 χðωÞ EðωÞ þ ϵ 0 Δχðω; M 0 Þ EðωÞ (2.70) Dðω; M 0 Þ ¼ ϵ ðω; M 0 Þ EðωÞ ¼ ϵ ðωÞ EðωÞ þ Δϵ ðω; M 0 Þ EðωÞ: (2.71) and While χ and ϵ are changed by H 0 or M 0 , the magnetic permeability of the material at an optical frequency remains the constant μ0 , and the relation between BðωÞ and HðωÞ remains independent of H 0 or M 0 : BðωÞ ¼ μ0 HðωÞ. Therefore, magneto-optic effects are completely characterized by ϵ ðω; H 0 Þ, if no internal magnetization is present, or by ϵ ðω; M 0 Þ, if an internal magnetization is present. In general, these effects are weak perturbations to the optical properties of the material. The ﬁrst-order magneto-optic effect, or linear magneto-optic effect, is characterized by a linear dependence of ϵ on H 0 or M 0 , and the second-order magneto-optic effect, or quadratic magneto-optic effect, causes a quadratic dependence of ϵ on H 0 or M 0 . We ﬁrst consider the magneto-optic effects in a material that has no spontaneous magnetization, i.e., a diamagnetic material, a magnetically disordered paramagnetic material, or an antiferromagnetic material. In such a material, operation of the time-reversal transformation yields ϵ ij ðω; H 0 Þ ¼ ϵ ji ðω; H0 Þ (2.72) when the material is subject to an external magnetic ﬁeld H 0 . This relation is generally true regardless of the symmetry property of the material. If the material is lossless, then its dielectric permittivity tensor is Hermitian: ϵ ij ðω; H 0 Þ ¼ ϵ ∗ ji ðω; H 0 Þ: (2.73) If we express the real and imaginary parts of ϵ explicitly by writing ϵ ij ¼ ϵ 0ij þ iϵ 00ij , we ﬁnd by combining these two relations that Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.6 External Factors 51 ϵ 0ij ðω; H 0 Þ ¼ ϵ 0ij ðω; H 0 Þ ¼ ϵ 0ji ðω; H 0 Þ ¼ ϵ 0ji ðω; H 0 Þ, (2.74) ϵ 00ij ðω; H 0 Þ ¼ ϵ 00ij ðω; H0 Þ ¼ ϵ 00ji ðω; H 0 Þ ¼ ϵ 00ji ðω; H 0 Þ: (2.75) As a result, the magneto-optic effects in a lossless material that has no spontaneous magnetization can be generally described as X X f ijk H 0k þ ϵ 0 cijkl H 0k H 0l þ , (2.76) ϵ ij ðH 0 Þ ¼ ϵ ij þ Δϵ ij ðH 0 Þ ¼ ϵ ij þ iϵ 0 k k, l where f ijk and cijkl are real quantities that satisfy the following relations: f ijk ¼ f jik , cijkl ¼ cjikl ¼ cijlk ¼ cjilk : (2.77) Because magnetic ﬁelds have transformation symmetry properties that are very different from those of electric ﬁelds, magneto-optic effects also have properties very different from those of electro-optic effects. 1. Because a magnetic ﬁeld does not change sign under space inversion, the linear magnetooptic effect does not vanish, thus f ijk 6¼ 0, in a centrosymmetric material. By comparison, the linear electro-optic effect vanishes, thus r ijk ¼ 0, in a centrosymmetric material because an electric ﬁeld changes sign under space inversion. 2. Because a magnetic ﬁeld changes sign under time reversal, the linear magneto-optic effect is nonreciprocal, thus f ijk ¼ f jik . By comparison, the linear electro-optic effect is reciprocal, thus rijk ¼ r jik , because an electric ﬁeld does not change sign under time reversal. 3. Because the product of two electric ﬁeld components, E 0k E 0l , and the product of two magnetic ﬁeld components, H 0k H 0l , both do not change sign under space inversion or time reversal, the quadratic electro-optic effect and the quadratic magneto-optic effect both exist in centrosymmetric materials and are both reciprocal, thus sijkl ¼ sjikl ¼ sijlk ¼ sjilk and cijkl ¼ cjikl ¼ cijlk ¼ cjilk . 4. Both linear and quadratic magneto-optic effects exist in all materials, i.e., f ijk 6¼ 0 and cijkl 6¼ 0 in all materials, including all solids, liquids, and gases. 5. When a magnetically induced optical loss exists in the linear magneto-optic effect, f ijk becomes complex with an imaginary part that characterizes the loss. When it exists in the quadratic magneto-optic effect, cijkl becomes complex with an imaginary part that characterizes the loss. The magneto-optic effects in magnetically ordered crystals have the same general properties as discussed above, but their details can be rather complicated due to the magnetic symmetry properties of such crystals. In reality, the magneto-optic effects are relatively weak in comparison to, and tend to be obscured by, any natural or structural birefringence that might exist in a material. Fortunately, both ﬁrst- and second-order magneto-optic effects exist in nonbirefringent materials, which have isotropic linear optical properties, including noncrystals and cubic crystals. For these reasons, materials of particular interest and practical importance for magneto-optic effects and their applications are those in which birefringence originating from other effects, such as material anisotropy or inhomogeneity, does not exist or, if it exists, does not dominate the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 52 Optical Properties of Materials particular magneto-optic effect of interest. Such materials include isotropic materials and, in some cases, uniaxial crystals subject to a magnetic ﬁeld or a magnetization that is parallel to the optical axis. For magneto-optic effects in these materials, we can take the direction of H 0 or M 0 to be the z direction without loss of generality, i.e., H0 ¼ H 0z^z or M 0 ¼ M 0z^z . Then, ϵ ðH 0 Þ or ϵ ðM 0 Þ can be generally expressed in the form of (2.16): 0 n2⊥ ϵ ðH 0 Þ or ϵ ðM 0 Þ ¼ ϵ 0 @ iξ 0 iξ n2⊥ 0 1 0 0 A, n2k (2.78) where ξ represents the ﬁrst-order effect, and n2⊥ and n2k account for the second-order effect. In the case of ϵ ðH 0 Þ, ξ ¼ f 123 H 0z , n2⊥ ¼ n2o þ c1133 H 20z ¼ n2o þ c2233 H 20z , and n2k ¼ n2o þ c3333 H 20z . In the case of ϵ ðM 0 Þ, ξ is linearly proportional to M 0z with the symmetry of ξ ðM 0z Þ ¼ ξ ðM 0z Þ, and n2⊥ and n2k are functions of M 20z . The linear dependence of ϵ ij ðH 0 Þ on the magnetic ﬁeld, or that of ϵ ij ðM 0 Þ on the magnetization, appears only as antisymmetric components in the off-diagonal elements of the permittivity tensor. In the absence of a magnetically induced optical loss, these off-diagonal elements are purely imaginary; then this ﬁrst-order magneto-optic effect results in a magnetically induced circular birefringence, discussed in Section 2.2. When this ﬁrst-order magneto-optic effect induces an optical loss, these off-diagonal elements become complex, resulting in a magnetically induced circular dichroism, also discussed in Section 2.2. The linear magneto-optic effect has two notable phenomena: the Faraday effect and the magneto-optic Kerr effect. The Faraday effect is manifested in the propagation and transmission of an optical wave through a material subject to a magnetic ﬁeld or a magnetization; the magneto-optic Kerr effect is manifested in the reﬂection of an optical wave from the surface of such a material. The ﬁrst-order magnetooptic effect and these phenomena resulting from it are nonreciprocal. By contrast, the quadratic dependence on the magnetic ﬁeld or the magnetization appears as symmetric components in the permittivity tensor elements. This second-order magneto-optic effect is reciprocal and is called the Cotton–Mouton effect. In the absence of a magnetically induced optical loss, it causes a magnetically induced linear birefringence in the material and is analogous to, but much weaker than, the electro-optic Kerr effect. When this second-order magneto-optic effect causes an optical loss, the symmetric permittivity tensor elements are complex, resulting in a magnetically induced linear dichroism. 2.6.3 Acousto-optic Effect An acoustic wave in a medium is an elastic wave of space- and time-dependent periodic deformation in the medium. A traveling plane acoustic wave of a wavelength Λ ¼ 2π=K and a frequency f ¼ Ω=2π can be expressed as uðr; t Þ ¼ U cos ðK r Ωt Þ, (2.79) where U is the amplitude vector of the deformation that deﬁnes the polarization of the acoustic ^ is the acoustic wavevector wave, Ω is the angular frequency of the acoustic wave, and K ¼ K K Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.6 External Factors 53 ^ describes the propagation direction and K ¼ 2π=Λ ¼ Ω=v a is the propagation constant where K with v a being the acoustic velocity. A standing plane acoustic wave is a combination of two contrapropagating traveling waves of equal amplitude, wavelength, and frequency: uðr; tÞ ¼ U cos ðK rÞ cos Ωt: (2.80) An acoustic wave polarized in the direction of K is known as a longitudinal wave, while one with a polarization perpendicular to K is called a transverse wave. For any given direction of propagation in a medium, there are three orthogonal acoustic normal modes of polarization: one longitudinal or quasi-longitudinal mode, and two transverse or quasi-transverse modes. The mechanical strains associated with deformation are described by a symmetric strain tensor, S ¼ Sij , deﬁned by 1 ∂ui ∂uj , (2.81) þ Sij ¼ 2 ∂xj ∂xi where the indices i, j ¼ x, y, z. The three tensor elements Sxx , Syy , and Szz are tensile strains, while the other elements Syz ¼ Szy , Szx ¼ Sxz , and Sxy ¼ Syx are shear strains. In addition, there is an antisymmetric rotation tensor, R ¼ Rij , deﬁned by 1 ∂ui ∂uj Rij ¼ : (2.82) 2 ∂xj ∂xi Clearly, Rxx ¼ Ryy ¼ Rzz ¼ 0, while Ryz ¼ Rzy , Rzx ¼ Rxz , and Rxy ¼ Ryx . For elastic deformation caused by an acoustic wave, all of the strain and rotation tensor elements are space- and time-dependent quantities. Mechanical strain in a medium causes changes in the optical property of the medium due to the photoelastic effect. The basis of acousto-optic interaction is the dynamic photoelastic effect in which the periodic time-dependent mechanical strain and rotation caused by an acoustic wave induce periodic time-dependent variations in the optical properties of the medium. The photoelastic effect is traditionally deﬁned in terms of changes in the elements of the relative impermeability tensor: X ηij ðS; RÞ ¼ ηij þ Δηij ðS; RÞ ¼ ηij þ (2.83) pijkl Skl þ p0ijkl Rkl , k, l where pijkl are dimensionless elasto-optic coefﬁcients, also called strain-optic coefﬁcients or photoelastic coefﬁcients, and p0ijkl are dimensionless rotation-optic coefﬁcients. Both are fourth order tensors. Because ηij ¼ ηji and Skl ¼ Slk , the pijkl tensor is symmetric in i and j and in k and l. Because ηij ¼ ηji and Rkl ¼ Rlk , the ½p0ijkl tensor is symmetric in i and j but is antisymmetric in k and l. The photoelastic effect exists in all matter, including centrosymmetric crystals and isotropic materials, because the pijkl tensor never vanishes in any material though the ½ p0ijkl tensor vanishes in isotropic materials and cubic crystals. Acousto-optic interactions are not precluded by any symmetry property of a material. The tensor form of pijkl for a crystal is determined by the point group of the crystal. The ½p0ijkl tensor elements of a crystal are determined by the birefringence of the crystal. If the indices i, j, k, are l referenced to the principal axes of a crystal, we have Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 54 Optical Properties of Materials p0ijkl ! 1 1 1 ¼ δ δ δ , δ ik jl il jk 2 n2i n2j (2.84) where ni and nj represent the principal indices of refraction of the crystal. It is clear that p0ijkl vanishes in an isotropic material or a cubic crystal. It is desirable to formally express the photoelastic effect caused by strain and rotation in a medium in terms of a change in the permittivity of the medium as ϵ ðω; S; RÞ ¼ ϵ ðωÞ þ Δϵ ðω; S; RÞ ¼ ϵ ðωÞ þ ϵ 0 Δχðω; S; RÞ, (2.85) where ϵ ðωÞ is the dielectric permittivity tensor of the medium in the absence of strain and rotation ﬁelds. Using (2.62), the elements of Δϵ can be found from those of Δη in (2.83): X Δϵ ij ¼ ϵ 0 n2i n2j Δηij ¼ ϵ 0 n2i n2j pijkl Skl þ p0ijkl Rkl , (2.86) k, l where for an acoustic wave, Skl and Rkl are functions of space and time. For a traveling wave characterized by a wavevector of K and a frequency of Ω as described by (2.79), Skl and Rkl can be found by using (2.81) and (2.82), respectively. They have the form: Skl ¼ S kl sin ðK r ΩtÞ, Rkl ¼ Rkl sin ðK r ΩtÞ, (2.87) where S kl is the amplitude of the strain and Rkl is the amplitude of the rotation. Therefore, the photoelastic permittivity tensor is a function of space and time: Δϵ ¼ Δe ϵ sin ðK r Ωt Þ, where Δe ϵ is the amplitude of Δϵ, and its elements are X Δe ϵ ij ¼ ϵ 0 n2i n2j pijkl S kl þ p0ijkl Rkl : k, l (2.88) (2.89) EXAMPLE 2.8 Silica glass is an isotropic material. An acoustic wave propagating in any direction in silica glass can have two transverse modes and one longitudinal mode. The two transverse modes have the same acoustic wave velocity of v Ta ¼ 5:97 km s1 , whereas the longitudinal mode has an acoustic wave velocity of v La ¼ 3:76 km s1 . Take the acoustic wave propagation direction to be the z direction. How does each mode of an acoustic wave at a frequency of 500 MHz modulate the optical permittivity in space and time? Solution: All three modes modulate the optical permittivity at the same frequency of f ¼ 500 MHz, thus Ω ¼ 1 109 π rad s1 , in time, but they modulate the optical permittivity differently in space. Because the wave propagates in the z direction, the longitudinal mode is polarized in the z direction while the two transverse modes are polarized in the x and y directions, respectively. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.7 Nonlinear Optical Susceptibilities 55 For the longitudinal mode, v La ¼ 3:76 km s1 . Thus, ΛL ¼ v La 3:76 103 2π m ¼ 7:52 μm and K L ¼ ¼ 8:36 105 m1 : ¼ 6 f Λ 500 10 L The wavevector of the longitudinal mode is K ¼ K L^z . The optical permittivity that is modulated by the longitudinal acoustic wave varies in space and time with K L ¼ 8:36 105 m1 and Ω ¼ 1 109 π rad s1 as Δϵ ðz; t Þ ¼ Δe ϵ sin ðK L z Ωt Þ: For both transverse modes, v Ta ¼ 5:97 km s1 . Thus, ΛT ¼ v Ta 5:97 103 2π m ¼ 11:94 μm and K T ¼ ¼ 5:26 105 m1 : ¼ 6 f ΛT 500 10 The wavevectors of both transverse modes are K ¼ K T^z . The optical permittivity that is modulated by either of the transverse acoustic waves varies in space and time with K T ¼ 5:26 105 m1 and Ω ¼ 1 109 π rad s1 as Δϵ ðz; t Þ ¼ Δe ϵ sin ðK T z Ωt Þ: The permittivity tensor Δe ϵ is a constant that does not vary with space or time, but it has different forms for different acoustic modes. 2.7 NONLINEAR OPTICAL SUSCEPTIBILITIES .............................................................................................................. The origin of optical nonlinearity is the nonlinear response of electrons in a material to an optical ﬁeld as the strength of the ﬁeld is increased. Macroscopically, the nonlinear optical response of a material is described by a polarization that is a nonlinear function of the optical ﬁeld. In general, such nonlinear dependence on the optical ﬁeld can take a variety of forms. In particular, it can be very complicated when the optical ﬁeld becomes extremely strong. In most situations of interest, with the exception of saturable absorption, the perturbation method can be used to expand the total optical polarization in terms of a series of linear and nonlinear polarizations: Pðr; t Þ ¼ Pð1Þ ðr; tÞ þ Pð2Þ ðr; t Þ þ Pð3Þ ðr; t Þ þ , (2.90) where Pð1Þ is the linear polarization, and Pð2Þ and Pð3Þ are the second- and third-order nonlinear polarizations, respectively. Except in some special cases, nonlinear polarizations of the fourth and higher orders are usually not important and thus can be ignored. Note that the space- and time-dependent polarizations in (2.90) are complex polarizations deﬁned with respect to the corresponding real polarizations according to the deﬁnition of the complex ﬁeld in (1.40): PðnÞ ðr; t Þ ¼ PðnÞ ðr; t Þ þ PðnÞ∗ ðr; tÞ ¼ PðnÞ ðr; t Þ þ c:c:, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 (2.91) 56 Optical Properties of Materials where PðnÞ ðr; t Þ is the nth-order real nonlinear polarization and PðnÞ ðr; t Þ is the nth-order complex polarization. The optical ﬁeld involved in a nonlinear interaction usually contains multiple, distinct frequency components. Such a ﬁeld can be expanded in terms of its frequency components: X X Eðr; t Þ ¼ Eq ðrÞ exp iωq t ¼ E q ðrÞ exp ikq r iωq t , (2.92) q q where E q ðrÞ is the slowly varying amplitude and kq is the wavevector of the ωq frequency component. The nonlinear polarizations also contain multiple frequency components and can be expanded as X PðnÞ ðr; t Þ ¼ PðqnÞ ðrÞ exp iωq t : (2.93) q Note that we do not attempt to further express PðqnÞ ðrÞ in terms of a slowly varying polarization amplitude multiplied by a fast varying spatial phase term, as is done for Eq ðrÞ. The reason is that the wavevector that characterizes the fast varying spatial phase of a nonlinear polarization PðqnÞ ðrÞ is not simply determined by the frequency ωq but is dictated by the ﬁelds that generate the nonlinear polarization. In the discussion of nonlinear polarizations, we also use the notations E ωq and PðnÞ ωq deﬁned respectively as E ωq ¼ Eq ðrÞ and PðnÞ ωq ¼ PðqnÞ ðrÞ: (2.94) Field and polarization components of negative frequencies are interpreted as E ωq ¼ E∗ ωq and PðnÞ ωq ¼ PðnÞ∗ ωq : (2.95) The frequency-domain nth-order nonlinear susceptibility characterizing the nonlinear response of a material to optical ﬁelds at frequencies ω1 , ω2 , . . . , ωn is a function of these optical frequencies: χðnÞ ðω1 ; ω2 ; ; ωn Þ. In the momentum space and frequency domain, the nonlinear susceptibility is also a function of wavevectors: χðnÞ ðk1 ; ω1 ; k2 ; ω2 ; ; kn ; ωn Þ. The reality condition discussed in Section 2.1 and expressed explicitly in (2.7) for the linear susceptibility can be generalized for each nonlinear susceptibility. This reality condition leads to the following relation for a nonlinear susceptibility: χðnÞ∗ ðk1 ; ω1 ; k2 ; ω2 ; ; kn ; ωn Þ ¼ χðnÞ ðk1 ; ω1 ; k2 ; ω2 ; ; kn ; ωn Þ: (2.96) When expressing the nonlinear polarization that is generated at a frequency of ωq ¼ ω1 þ ω2 þ þ ωn by the nonlinear optical interaction of the optical ﬁelds at frequencies ω1 , ω2 , . . . , ωn , the following notation for the nonlinear susceptibility is used: χðnÞ ωq ¼ ω1 þ ω2 þ þ ωn ¼ χðnÞ ðω1 ; ω2 ; ; ωn Þ: (2.97) Using the expansions of the complex ﬁelds and polarizations in (2.92) and (2.93), we have the expressions for the second- and third-order nonlinear polarizations: X Pð2Þ ωq ¼ ϵ 0 χð2Þ ωq ¼ ωm þ ωn : Eðωm ÞEðωn Þ m, n Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 (2.98) 57 2.7 Nonlinear Optical Susceptibilities and X χð3Þ ωq ¼ ωm þ ωn þ ωp : Eðωm ÞEðωn ÞE ωp : Pð3Þ ωq ¼ ϵ 0 m, n, p (2.99) The summation is performed for a given ωq over all positive and negative values of frequencies that satisfy the constraint of ωm þ ωn ¼ ωq in the case of (2.98) and the constraint of ωm þ ωn þ ωp ¼ ωq in the case of (2.99). More explicitly, by performing the summation over positive frequencies only and by expanding the tensor products, we have X X h ð2Þ ð2Þ χ ijk ωq ¼ ωm þ ωn E j ðωm ÞE k ðωn Þ Pi ωq ¼ ϵ 0 j, k ωm , ωn >0 ð2 Þ þ χ ijk ωq ¼ ωm ωn E j ðωm ÞE ∗ k ðωn Þ i ð2Þ þχ ijk ωq ¼ ωm þ ωn E ∗ ð ω ÞE ð ω Þ (2.100) m k n j and ð3Þ Pi X X h ð3Þ ωq ¼ ϵ 0 χ ijkl ωq ¼ ωm þ ωn þ ωp Ej ðωm ÞE k ðωn ÞE l ωp j, k , l ωm , ωn , ωp >0 ð3Þ þ χ ijkl ωq ¼ ωm þ ωn ωp Ej ðωm ÞE k ðωn ÞE ∗ l ωp ð3Þ þ χ ijkl ωq ¼ ωm ωn þ ωp Ej ðωm ÞE ∗ k ðωn ÞE l ωp ð3Þ þ χ ijkl ωq ¼ ωm þ ωn þ ωp E ∗ j ðωm ÞE k ðωn ÞE l ωp ð3Þ ∗ þ χ ijkl ωq ¼ ωm ωn ωp Ej ðωm ÞE ∗ k ðωn ÞE l ωp ð3Þ ∗ þ χ ijkl ωq ¼ ωm þ ωn ωp E ∗ j ðωm ÞE k ðωn ÞE l ωp i ð3Þ ∗ ð ω ÞE ð ω ÞE ωp : þχ ijkl ωq ¼ ωm ωn þ ωp E ∗ m n l j k (2.101) Usually only a limited number of frequencies participate in a given nonlinear optical interaction. Consequently, only one or a few terms among those listed in (2.100) or (2.101) contribute to a particular nonlinear polarization. EXAMPLE 2.9 Three optical ﬁelds at the wavelengths of λ1 ¼ 300 nm, λ2 ¼ 750 nm, and λ3 ¼ 1500 nm, corresponding to the frequencies of ω1 ¼ 2πc=λ1 , ω2 ¼ 2πc=λ2 , and ω3 ¼ 2πc=λ3 , respectively, are involved in second-order nonlinear optical interactions. The optical ﬁelds at the three pﬃﬃﬃ frequencies are E ðω1 Þ ¼ E 1 ð^x þ ^y Þ= 2, E ðω2 Þ ¼ E 2^z , and Eðω3 Þ ¼ E 3^z , where ^x , ^y , and ^z are the x, y, and z principal axes of the nonlinear crystal. Find the nonlinear polarization Pð2Þ ðω4 Þ at the frequency of ω4 ¼ 2πc=λ4 where λ4 ¼ 375 nm. Express the components of Pð2Þ ðω4 Þ explicitly in terms of the elements of χð2Þ and the given magnitudes, E 1 , E2 , and E3 , of the three optical ﬁelds. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 58 Optical Properties of Materials Solution: 1 1 1 1 Because λ1 1 λ3 ¼ λ2 þ λ2 ¼ λ4 , we ﬁnd that ω4 ¼ ω1 ω3 ¼ ω2 þ ω2 . Therefore, the second-order nonlinear polarization at the frequency ω4 is Pð2Þ ðω4 Þ ¼ ϵ 0 χð2Þ ðω4 ¼ ω1 ω3 Þ : Eðω1 ÞE∗ ðω3 Þ þ χð2Þ ðω4 ¼ ω3 þ ω1 Þ : E∗ ðω3 ÞEðω1 Þ i þχð2Þ ðω4 ¼ ω2 þ ω2 Þ : Eðω2 ÞEðω2 Þ : Note that there are two terms from the mixing of ω1 and ω3 because of permutation, but there is only one term from ω2 mixing with itself. Using the given ﬁelds at the three frequencies, we can express the components of Pð2Þ ðω4 Þ as E1 E∗ E1 E∗ 2Þ 2Þ ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 þ χ ðxyz ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 Pðx2Þ ðω4 Þ ¼ ϵ 0 χ ðxxz 2 2 E∗ E∗ ð2Þ ð2Þ 3 E1 3 E1 ﬃﬃﬃ þ χ xzx ðω4 ¼ ω3 þ ω1 Þ pﬃﬃﬃ þ χ xzy ðω4 ¼ ω3 þ ω1 Þ p 2 i 2 ð2Þ 2 þχ xzz ðω4 ¼ ω2 þ ω2 ÞE 2 , E1 E∗ E1 E∗ 2Þ 2Þ ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 þ χ ðyyz ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 Pðy2Þ ðω4 Þ ¼ ϵ 0 χ ðyxz 2 2 ∗ E E E∗ 2Þ 2Þ 3 1 3 E1 ﬃﬃﬃ þ χ ðyzy ﬃﬃﬃ ðω4 ¼ ω3 þ ω1 Þ p ðω4 ¼ ω3 þ ω1 Þ p þ χ ðyzx 2 2 i ð2Þ 2 þχ yzz ðω4 ¼ ω2 þ ω2 ÞE 2 , E1 E∗ E1 E∗ 2Þ 2Þ Pðz2Þ ðω4 Þ ¼ ϵ 0 χ ðzxz ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 þ χ ðzyz ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 2 2 ∗ E3 E1 E∗ 2Þ 2Þ 3 E1 ﬃﬃﬃ þ χ ðzzy ﬃﬃﬃ þ χ ðzzx ðω4 ¼ ω3 þ ω1 Þ p ðω4 ¼ ω3 þ ω1 Þ p 2 2 i 2Þ þχ ðzzz ðω4 ¼ ω2 þ ω2 ÞE 22 : As discussed in Section 2.2, the form of the linear susceptibility tensor is determined by the symmetry property of the material. The forms of the nonlinear susceptibility tensors of a material also reﬂect the spatial symmetry property of the material structure. As a result, some elements in a nonlinear susceptibility tensor may be zero and others may be related in one way or another, greatly reducing the total number of independent tensor elements. The linear susceptibility tensor has its form determined only by the crystal system of a material, whereas the form of a nonlinear susceptibility tensor further depends on the point group of the material. Within the 7 crystal systems, there are 32 point groups. Among the 32 point groups, 21 are noncentrosymmetric and 11 are centrosymmetric. All gases, liquids, and amorphous solids are centrosymmetric. Centrosymmetric materials possess space-inversion symmetry. In the electricdipole approximation, nonlinear optical effects of all even orders, but not those of the odd Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 2.7 Nonlinear Optical Susceptibilities 59 orders, vanish identically in a centrosymmetric material. Therefore, χð2Þ contributed by electricdipole interaction is identically zero in a centrosymmetric material, whereas a nonzero χð3Þ exists in any material. This fact can be easily veriﬁed by considering the effect of space inversion on the nonlinear polarizations Pð2Þ and Pð3Þ given in (2.98) and (2.99), respectively. The space-inversion transformation can be performed on a centrosymmetric material without changing the properties of the material. Being polar vectors, Pð2Þ , Pð3Þ , and E all change sign under such a transformation. From (2.98), we then ﬁnd that Pð2Þ ¼ Pð2Þ . Therefore, Pð2Þ cannot exist and χð2Þ has to vanish identically in a centrosymmetric material. No such conclusion is drawn for Pð3Þ or χð3Þ as we examine (2.99) following the same procedure. Comparing the above discussion with that in Section 2.6 for the Pockels coefﬁcients r ijk , which vanish in centrosymmetric materials, and the electro-optic Kerr coefﬁcients sijkl , which exist in any material, we ﬁnd the similarity between the χð2Þ and r ijk , and that between χð3Þ and sijkl . Indeed, they are directly related: r ijk ¼ 2 ð2Þ χ ðω 2 ni n2j ijk ¼ ω þ 0Þ ¼ 2 ð2 Þ χ ð0 2 ni n2j kij ¼ ω ωÞ (2.102) and sijkl ¼ 3 ð3Þ χ ðω 2 ni n2j ijkl ¼ ω þ 0 þ 0Þ: (2.103) EXAMPLE 2.10 The BBO crystal structure belongs to the 3m point group, for which the only nonvanishing 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ χð2Þ elements are χ ðxzx ¼ χ ðyzy , χ ðxxz ¼ χ ðyyz , χ ðyyy ¼ χ ðyxx ¼ χ ðxxy ¼ χ ðxyx , χ ðzxx ¼ χ ðzyy , and χ ðzzz . If the nonlinear interaction considered in Example 2.9 takes place in a BBO crystal, what are the expressions of the components of Pð2Þ ðω4 Þ in terms of the nonvanishing elements of χð2Þ ? Solution: By keeping the terms that contain only the nonvanishing χð2Þ elements in each of the components of Pð2Þ ðω4 Þ obtained in Example 2.9, we ﬁnd that E1 E∗ E∗ 2Þ 2Þ 3 E1 ﬃﬃﬃ , Pðx2Þ ðω4 Þ ¼ ϵ 0 χ ðxxz ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 þ χ ðxzx ðω4 ¼ ω3 þ ω1 Þ p 2 2 E1 E∗ E∗ 2Þ 2Þ 3 E1 ﬃﬃﬃ , Pðy2Þ ðω4 Þ ¼ ϵ 0 χ ðyyz ðω4 ¼ ω1 ω3 Þ pﬃﬃﬃ3 þ χ ðyzy ðω4 ¼ ω3 þ ω1 Þ p 2 2 2Þ ðω4 ¼ ω2 þ ω2 ÞE22 : Pðz2Þ ðω4 Þ ¼ ϵ 0 χ ðzzz Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 60 Optical Properties of Materials Problems 2.1.1 Verify the relations given in (2.7) that are required by the reality condition. 2.2.1 At a given optical frequency, the optical susceptibility tensors of several materials are measured with respect to an arbitrary rectilinear coordinate system in space, as listed below. Identify each material as (1) a dielectric or magnetic material and (2) an optically lossless or lossy material. 0 1 0 1 2:3 0:1 þ i0:2 0 2:0 þ i0:1 i0:3 0 1 þ i0:2 0 A; A : χ ¼ @ 0:1 þ i0:2 2:7 i0:2 A; B : χ ¼ @ i0:3 0 i0:2 2:4 0 0 1:5 0 1 0 1 1:59 0:13 0:16 1:9 0:2 0:3 @ A @ A; C:χ¼ 0:13 1:59 0:11 ; D : χ ¼ 0:2 2:8 0:1 0:16 0:11 1:41 0:3 0:1 2:5 þ i0:2 0 1 1:30 i0:35 0 E : χ ¼ @ i0:35 1:25 0:15 A: 0 0:15 1:40 2.2.2 Represented in an arbitrarily chosen right-handed Cartesian coordinate system deﬁned by the unit vectors ^x 1 , ^x 2 , and ^x 3 , with ^x 1 ^x 2 ¼ ^x 3 , the permittivity tensor of a crystal at λ ¼ 0:50 μm is 0 1 5:481 0 0 ϵ ¼ ϵ0@ 0 5:267 0:214 A: 0 0:214 5:267 (a) Find the principal indices and the corresponding principal axes of the crystal at this wavelength. (b) Is this crystal birefringent or nonbirefringent? If it is birefringent, is it uniaxial or biaxial? If it is uniaxial, is it positive or negative uniaxial? 2.2.3 At the λ ¼ 1:300 μm optical wavelength, the permittivity tensor of a LiNbO3 crystal represented in an arbitrarily chosen ðx1 ; x2 ; x3 Þ rectilinear coordinate system with ^x 1 ^x 2 ¼ ^x 3 is found to be 0 1 4:938 0 0 4:770 0:168 A: ϵ ¼ ϵ0@ 0 0 0:168 4:770 (a) Find the principal indices and the corresponding principal axes of the LiNbO3 crystal at this wavelength. (b) Is it possible to send an optical wave at this wavelength through a LiNbO3 crystal of arbitrary thickness in such a manner that the polarization of the wave is maintained throughout its path no matter how the wave is initially polarized? If the answer is yes, how can this be arranged? If the answer is no, why is it not possible? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 Problems 61 2.2.4 Represented in an arbitrarily chosen right-handed ðx1 ; x2 ; x3 Þ Cartesian coordinate system with ^x 1 ^x 2 ¼ ^x 3 , the permittivity tensor of a KTP crystal at λ ¼ 1:0 μm is 0 1 3:035 0 0 3:210 0:147 A: ϵ ¼ ϵ0@ 0 0 0:147 3:210 (a) Find the principal indices and the corresponding principal axes of the crystal at this wavelength. (b) Is the crystal birefringent or nonbirefringent? If it is birefringent, is it uniaxial or biaxial? If it is uniaxial, is it positive or negative uniaxial? 2.2.5 What is the difference between linear birefringence and circular birefringence? 2.2.6 What is the difference between linear birefringence and linear dichroism? What is the difference between circular birefringence and circular dichroism? 2.2.7 In a properly chosen xyz Cartesian coordinate system, a particular medium has a symmetric but non-Hermitian permittivity tensor of the form: 0 2 1 n þ iς iξ þ γ 0 (2.104) ϵ ¼ ϵ 0 @ iξ þ γ n2 þ iς 0 A, 2 0 0 nz where n, ς, ξ, and γ are all real and positive numbers with n ς, ξ, γ. Find the principal refractive indices and the corresponding principal normal modes of polarization. Show that this medium is linearly birefringent and linearly dichroic. 2.2.8 In a properly chosen xyz Cartesian coordinate system, a particular medium has an asymmetric and non-Hermitian permittivity tensor of the form: 0 2 1 n þ iς iξ þ γ 0 (2.105) ϵ ¼ ϵ 0 @ iξ γ n2 þ iς 0 A, 0 0 n2z where n, ς, ξ, and γ are all real and positive numbers with n ς, ξ, γ. Find the principal refractive indices and the corresponding principal normal modes of polarization. Show that this medium is circularly birefringent and circularly dichroic. 2.3.1 Lorentz model: The resonant susceptibility given in (2.26) for an atomic system that has a single resonance frequency at ω0 and a relaxation rate of γ can be obtained using a classical Lorentz model by considering a one-dimensional damped oscillator for the bound electrons of this system. The system consists of N oscillating electrons, each of which has a charge of q ¼ e and an effective mass of m∗ . The displacement of the oscillating electron in response to the force of an optical ﬁeld is described by the Lorentz oscillator equation: d2 x dx F þ 2γ þ ω20 x ¼ ∗ , dt2 dt m (2.106) where xðt Þ ¼ xðt Þ^x and FðtÞ ¼ qEðt Þ ¼ eEeiωt þ c:c: ¼ eE^x eiωt þ c:c: The electricdipole polarization due to the electron displacement induced by the optical ﬁeld is deﬁned as Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 62 Optical Properties of Materials Pðt Þ ¼ NexðtÞ: (2.107) The induced electron displacement and the corresponding optical polarization in response to the optical ﬁeld at the frequency ω can be expressed as xðt Þ ¼ xðt Þ^x ¼ xðωÞ^x eiωt þ c:c: and PðtÞ ¼ PðωÞ^x eiωt þ c:c: (a) Solve the Lorentz oscillator equation by using the above expression for xðtÞ to ﬁnd xðωÞ. (b) Use the deﬁnition of the electric-dipole polarization and the frequency-domain relation PðωÞ ¼ ϵ 0 χ ðωÞE ðωÞ, as given in (1.59), between the optical polarization and the optical ﬁeld to ﬁnd χ ðωÞ, which is the resonant susceptibility χ res ðω; ω0 Þ. (c) Compare the result obtained in (b) with the resonant susceptibility given in (2.26) by taking ΔN ¼ N 2 N 1 N because the atomic system considered here is in the thermal-equilibrium state so that it is almost all populated in the ground level. Identify the electric-dipole moment p in (2.26) and express it in terms of the parameters of the Lorentz oscillator model. 2.3.2 All susceptibilities and permittivities of physical materials have to satisfy the reality condition given in (2.7). (a) Show that the real and imaginary parts of the resonant susceptibility given in (2.27) do not satisfy the reality condition. Explain this apparent issue. (b) Show that the resonant susceptibility given in (2.26) satisﬁes the reality condition before the rotating-wave approximation is applied but not after that. 2.3.3 The absorption spectral line of Yb3þ : Al2 O3 due to the optical transition from the 2 F7=2 ground level to the 2 F5=2 upper level appears at a center wavelength of λ ¼ 974:5 nm with a FWHM spectral width of Δλ ¼ 7:4 nm. Find the energy separation between the two levels. Find the resonance frequency and the polarization relaxation rate associated with this transition. Where is anomalous dispersion caused by this transition found when the Yb3þ ions are in their normal state in thermal equilibrium with the surrounding? 2.4.1 Drude model: The Drude model considers free-moving electrons or holes that, unlike bound electrons, do not have resonant oscillation frequencies. (a) Show that the Drude model given in (2.30) can be obtained by setting ω0 ¼ 0 and 2γ ¼ 1=τ for the Lorentz model in Problem 2.3.1. (b) Show that χ cond ðωÞ given in (2.43) can be obtained from the expression of χ res ðωÞ found in Problem 2.3.1 by setting ω0 ¼ 0 and 2γ ¼ 1=τ. 2.4.2 Show that the conduction susceptibility given in (2.43) and its real and imaginary parts given in (2.44) all satisfy the reality condition. 2.4.3 Aluminum is a good conductor. The free-electron Drude model describes its optical properties reasonably well with a free electron density of N ¼ 1:81 1029 m3 . The DC conductivity of Al at T ¼ 273 K is σ ð0Þ ¼ 4:08 107 S m1 . Find the plasma frequency ωp and the relaxation time τ for Al at T ¼ 273 K. Also ﬁnd the cutoff optical frequency νp and the cutoff wavelength λp . For what wavelengths is Al expected to be highly reﬂective? For what wavelengths is it expected to become transmissive? ∗ 2.4.4 Si has an electron effective mass of m∗ e ¼ 1:08m0 and a hole effective mass of mh ¼ 0:56m0 , where m0 is the mass of a free electron. Its low-frequency dielectric constant is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 Problems 63 11.8. Find the plasma frequency, the cutoff frequency, and the cutoff wavelength for (a) an n-type Si sample that has an electron density of N e ¼ 1 1024 m3 , (b) a p-type Si sample that has a hole density of N h ¼ 1 1024 m3 , and (c) a Si sample that is injected with an equal electron and hole density of N e ¼ N h ¼ 1 1024 m3 . 2.5.1 Show that the Kramers–Kronig relations given in (2.53) satisfy the reality condition. 2.5.2 Do the real part χ 0res ðωÞ and the imaginary part χ 00res ðωÞ of the exact χ res ðωÞ given in (2.26) before making the rotating-wave approximation satisfy the Kramers–Kronig relations? Do the real and imaginary parts, given in (2.27), of the χ res ðωÞ obtained under the rotating-wave approximation satisfy the Kramers–Kronig relations? 2.5.3 Do the real part χ 0cond ðωÞ and the imaginary part χ 00cond ðωÞ of the conduction susceptibility given in (2.44) satisfy the Kramers–Kronig relations? 2.6.1 LiNbO3 is a negative uniaxial crystal with nx ¼ ny ¼ no > nz ¼ ne . Being a crystal of the 3m symmetry group, it has eight nonvanishing Pockels coefﬁcients of four distinct values: r12 ¼ r 22 , r 13 , r 22 , r 23 ¼ r13 , r 33 , r42 , r 51 ¼ r 42 , and r61 ¼ r 22 . At the 1 μm optical wavelength, no ¼ 2:238 and ne ¼ 2:159, and the four distinct values of its Pockels coefﬁcients are r13 ¼ 8:6 pm V1 , r 22 ¼ 3:4 pm V1 , r 33 ¼ 30:8 pm V1 , and r 42 ¼ 28 pm V1 . Use the results from Example 2.6 to ﬁnd the new principal axes and the changes in the principal indices of refraction caused by an electric ﬁeld of E 0 ¼ 5 MV m1 that is applied along the y principal axis. 2.6.2 InP is a cubic crystal of the 43m symmetry group with nx ¼ ny ¼ nz ¼ no and three nonvanishing Pockels coefﬁcients of the same value: r 41 ¼ r52 ¼ r 63 . At the 1:55 μm optical wavelength, no ¼ 3:166 and r 41 ¼ 1:6 pm V1 . Because of the symmetry among the three principal axes, an electric ﬁeld applied along any principal axis results in a similar effect. Consider a DC electric ﬁeld of E0 ¼ 10 MV m1 applied along the z principal axis. Find the new principal axes and the changes in the principal indices of refraction caused by the applied ﬁeld due to the Pockels effect. 2.6.3 KTP is a biaxial crystal of the mm2 symmetry group with nx 6¼ ny 6¼ nz and ﬁve nonvanishing Pockels coefﬁcients of distinct values: r 13 , r 23 , r 33 , r42 , and r 51 . Find the ﬁeld-induced permittivity change Δϵ ðE0 Þ for an applied DC electric ﬁeld of E0 ¼ E 0x ^x þ E 0y ^y þ E 0z^z . 2.6.4 At the 1 μm optical wavelength, the principal indices of KTP are nx ¼ 1:742, ny ¼ 1:750, and nz ¼ 1:832; the nonvanishing Pockels coefﬁcients are r 13 ¼ 8:8 pm V1 , r 23 ¼ 13:8 pm V1 , r 33 ¼ 35 pm V1 , r 42 ¼ 8:8 pm V1 , and r51 ¼ 6:9 pm V1 . Is it possible to apply a DC electric ﬁeld to change the principal indices of refraction through the Pockels effect without rotating the principal axes? If this is possible, ﬁnd the changes in the principal indices of refraction caused by an applied electric ﬁeld of E 0 ¼ 12 MV m1 . 2.6.5 Magneto-optic effect can lead to circular birefringence and circular dichroism. For simplicity, consider a material for which the only optical loss is magnetically induced so that ϵ ij ¼ ϵ ∗ ji in the absence of a magnetic ﬁeld or a magnetization but ∗ ϵ ij ðH0 Þ 6¼ ϵ ∗ ji ðH 0 Þ in the presence of a magnetic ﬁeld and ϵ ij ðM 0 Þ 6¼ ϵ ji ðM 0 Þ in the presence of a magnetization. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 64 Optical Properties of Materials (a) Show for the case of a magnetic-ﬁeld-induced loss that the relations in (2.76) and (2.77) are still valid but f ijk or cijkl , or both, are complex. Thus, the magneto-optic permittivity tensor given in (2.78) can be generalized to the form: 0 n2⊥ þ iς ϵ ¼ ϵ 0 @ iξ 0 þ ξ 00 0 iξ 0 ξ 00 n2⊥ þ iς 0 1 0 0 A, n2k (2.108) where ξ 0 ¼ f 0123 H 0z , ξ 00 ¼ f 00123 H 0z , n2⊥ ¼ n2o þ c01234 H 20z , and ς ¼ c001234 H 20z . The same concept is applicable to a magnetization-induced optical loss for which ξ 0 and ξ 00 are linearly proportional to M 0z , and n2⊥ and ς are functions of M 20z . (b) Show that the ﬁrst-order magneto-optic effect results in circular birefringence and, in the situation when ξ 00 6¼ 0 with a magnetically induced loss, circular dichroism. (c) Show, by setting ξ 0 ¼ ξ 00 ¼ 0 to mathematically turn off the ﬁrst-order magneto-optic effect, that the second-order magneto-optic effect does not cause circular birefringence, or circular dichroism, but only linear birefringence or linear dichroism. 2.7.1 Three optical ﬁelds at the wavelengths of λ1 ¼ 1200 nm, λ2 ¼ 600 nm, and λ3 ¼ 800 nm, corresponding to the frequencies of ω1 ¼ 2πc=λ1 , ω2 ¼ 2πc=λ2 , and ω3 ¼ 2πc=λ3 , respectively, are involved in second-order nonlinear optical interactions. The pﬃﬃﬃoptical ﬁelds at the three frequencies are E ðω1 Þ ¼ E 1 ^x , E ðω2 Þ ¼ E 2 ð^y þ ^z Þ= 2, and E ðω3 Þ ¼ E 3^z , where ^x , ^y , and ^z are the x, y, and z principal axes of the nonlinear crystal. (a) Find the nonlinear polarization Pð2Þ ðω4 Þ at the frequency of ω4 ¼ 2πc=λ4 where λ4 ¼ 400 nm. Express each of the components of Pð2Þ ðω4 Þ explicitly in terms of the elements of χð2Þ and the given magnitudes, E1 , E2 , and E 3 , of the three optical ﬁelds. (b) If the nonlinear interaction takes place in a KTP crystal, what are the expressions of the components of Pð2Þ ðω4 Þ in terms of the nonvanishing elements of χð2Þ ? Note that KTP belongs to the mm2 point group, for which the only nonvanishing χð2Þ elements 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ are χ ðxzx , χ ðxxz , χ ðyyz , χ ðyzy , χ ðzxx , χ ðzyy , and χ ðzzz . 2.7.2 Three optical ﬁelds at the wavelengths of λ1 ¼ 1200 nm, λ2 ¼ 600 nm, and λ3 ¼ 800 nm, corresponding to the frequencies of ω1 ¼ 2πc=λ1 , ω2 ¼ 2πc=λ2 , and ω3 ¼ 2πc=λ3 , respectively, are involved in second-order nonlinear optical interactions. The pﬃﬃﬃoptical ^ ^ ﬁelds at the three frequencies are E ðω1 Þ ¼ E 1 x , E ðω2 Þ ¼ E 2 ðy þ ^z Þ= 2, and E ðω3 Þ ¼ E 3^z , where ^x , ^y , and ^z are the x, y, and z principal axes of the nonlinear crystal. (a) Find the nonlinear polarization Pð2Þ ðω4 Þ at the frequency of ω4 ¼ 2πc=λ4 where λ4 ¼ 2400 nm. Express each of the components of Pð2Þ ðω4 Þ explicitly in terms of the elements of χð2Þ and the given magnitudes, E1 , E2 , and E 3 , of the three optical ﬁelds. (b) If the nonlinear interaction takes place in a KTP crystal, what are the expressions of the components of Pð2Þ ðω4 Þ in terms of the nonvanishing elements of χð2Þ ? Note that KTP belongs to the mm2 point group, for which the only nonvanishing χð2Þ elements 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ are χ ðxzx , χ ðxxz , χ ðyyz , χ ðyzy , χ ðzxx , χ ðzyy , and χ ðzzz . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 Bibliography 65 2.7.3 Two optical ﬁelds at the wavelengths of λ1 ¼ 500 nm and λ2 ¼ 1500 nm, corresponding to the frequencies of ω1 ¼ 2πc=λ1 and ω2 ¼ 2πc=λ2 , respectively, are involved in second-order nonlinear optical interactions. The optical ﬁelds at the two frequencies are E ðω1 Þ ¼ E 1 ^x and E ðω2 Þ ¼ E 2 ^y , where ^x , ^y , and ^z are the x, y, and z principal axes of the nonlinear crystal. (a) Find the nonlinear polarization Pð2Þ ðω3 Þ at the frequency of ω3 ¼ 2πc=λ3 where λ3 ¼ 750 nm. Express each of the components of Pð2Þ ðω3 Þ explicitly in terms of the elements of χð2Þ and the given magnitudes, E 1 and E 2 , of the two optical ﬁelds. (b) If the nonlinear interaction takes place in a LiNbO3 crystal, what are the expressions of the components of Pð2Þ ðω3 Þ in terms of the nonvanishing elements of χð2Þ ? Note that LiNbO3 belongs to the 3m point group, for which the only nonvanishing 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ 2Þ ¼ χ ðyzy , χ ðxxz ¼ χ ðyyz , χ ðyyy ¼ χ ðyxx ¼ χ ðxxy ¼ χ ðxyx , χ ðzxx ¼ χ ðzyy , χð2Þ elements are χ ðxzx 2Þ and χ ðzzz . Bibliography Altman, C. and Suchy, K., Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, 2nd edn. Dordrecht: Springer, 2001. Bloembergen, N., Nonlinear Optics, 4th edn. Singapore: World Scientiﬁc, 1996. Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999. Boyd, R. W., Nonlinear Optics, 3rd edn. Boston, MA: Academic Press, 2008. Butcher, P. N. and Cotter, D., The Elements of Nonlinear Optics. Cambridge: Cambridge University Press, 1990. Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975. Fox, M., Optical Properties of Solids, 2nd edn. Oxford: Oxford University Press, 2010. Iizuka, K., Elements of Photonics in Free Space and Special Media, Vol. I. New York: Wiley, 2002. Jackson, J. D., Classical Electrodynamics, 3rd edn. New York: Wiley, 1999. Korpel, A., Acousto-Optics, 2nd edn. New York: Marcel Dekker, 1997. Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media. Oxford: Pergamon, 1960. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Nye, J. F., Physical Properties of Crystals. London: Oxford University Press, 1957. Post, E. J., Formal Structure of Electromagnetics. Amsterdam: North-Holland, 1962. Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991. Sapriel, J., Acousto-Optics. New York: Wiley, 1979. Shen, Y. R., The Principles of Nonlinear Optics. New York: Wiley, 1984. Sugano, S. and Kojima, N., eds., Magneto-Optics. Berlin: Springer, 2000. Wooten, F., Optical Properties of Solids. New York: Academic Press, 1972. Zernike, F. and Midwinter, J. E., Applied Nonlinear Optics. New York: Wiley, 1973. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:13 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.003 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 3 - Optical Wave Propagation pp. 66-140 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge University Press 3 3.1 Optical Wave Propagation NORMAL MODES OF PROPAGATION .............................................................................................................. The propagation of an optical wave is governed by Maxwell’s equations. The propagation characteristics depend on the optical property and the physical structure of the medium. They also depend on the makeup of the optical wave, such as its frequency content and its temporal characteristics. In this chapter, we discuss the basic propagation characteristics of a monochromatic optical wave in three basic categories of medium: an inﬁnite homogeneous medium, two semi-inﬁnite homogeneous media separated by an interface, and an optical waveguide deﬁned by a transverse structure. Some basic effects of dispersion and attenuation on the propagation of an optical wave are discussed in Sections 3.6 and 3.7. The optical property of a medium at a frequency of ω is fully described by its permittivity ϵ ðωÞ, which is a tensor for an anisotropic medium but reduces to a scalar for an isotropic medium. For a homogeneous medium, ϵ ðωÞ is a constant of space; for an optical structure, it is a function of space variables. Without loss of generality, we designate the z coordinate axis to be the direction of optical wave propagation in an isotropic medium; thus the longitudinal axis of an optical waveguide that is fabricated in an isotropic medium is the z axis. For this reason, ϵ ðωÞ has only transverse spatial variations that are functions of the transverse coordinates, which are x and y in the rectilinear coordinate system, or ϕ and r in the cylindrical coordinate system. We use the rectilinear coordinates for our general discussion. The exception is optical wave propagation in an anisotropic crystal, for which the natural coordinate system is that deﬁned by its principal axes but an optical wave does not have to propagate along its principal z axis. For the following discussion in this section, we consider propagation in an isotropic medium, which is not necessarily homogeneous in space. The wave propagates in the z direction, and the possible inhomogeneity characterizing the optical structure is described by a scalar permittivity ϵ ðx; yÞ, as illustrated in Fig. 3.1. If the medium is homogeneous, then ϵ ðx; yÞ ¼ ϵ is a constant of space, as shown in Fig. 3.1(a). If the medium is inhomogeneous in only one transverse dimension, then it has a planar optical structure, such as a planar interface shown in Fig. 3.1(b) or a planar waveguide shown in Fig. 3.1(c); in these cases, we take the structural variation to be in the x direction for ϵ ðx; yÞ ¼ ϵ ðxÞ to be independent of the y variable. If structural variations exist in two dimensions, then the medium has a nonplanar optical structure with ϵ ðx; yÞ being a function of both x and y, such as the single-core nonplanar waveguide shown in Fig. 3.1(d). In any event, there is no structural variation in the direction of propagation; therefore, ϵ ðx; yÞ is never a function of the z variable. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.1 Normal Modes of Propagation 67 Figure 3.1 (a) Homogeneous medium. (b) Planar interface. (c) Planar waveguide. (d) Nonplanar waveguide. The normal modes of propagation for an optical wave in a medium are the characteristic solutions of Maxwell’s equations subject to the boundary conditions that are deﬁned by the physical structure of the medium and are fully described by ϵ ðx; yÞ. Each characteristic solution has an eigenvalue, which gives the propagation constant, and an eigenfunction, which gives the ﬁeld pattern of the normal mode. Therefore, each normal mode is deﬁned by a speciﬁc propagation constant β and a pair of speciﬁc electric and magnetic mode ﬁeld proﬁles E ðx; yÞ and Hðx; yÞ. It is possible for two or more degenerate normal modes to have the same propagation constant but different ﬁeld proﬁles. By contrast, two normal modes of different propagation constants cannot share the same ﬁeld proﬁle. Because electric and magnetic ﬁelds are vectorial ﬁelds, a mode ﬁeld is deﬁned by a speciﬁc amplitude and polarization pattern of E ðx; yÞ and Hðx; yÞ. A mode index ν is used to label a mode when the optical structure supports multiple normal modes. Therefore, the space- and time-dependent electric and magnetic ﬁelds of a normal mode at a frequency of ω are expressed as Eν ðr; tÞ ¼ E ν ðx; yÞ exp ðiβν z iωt Þ, (3.1) Hν ðr; t Þ ¼ Hν ðx; yÞ exp ðiβν z iωtÞ, (3.2) where βν is the propagation constant of the mode. If the cylindrical coordinate system is used, then the mode ﬁelds in (3.1) and (3.2) are expressed as functions of ϕ and r: E ν ðϕ; rÞ and Hν ðϕ; r Þ. The characteristic of the mode index ν depends on the transverse boundary conditions imposed on the mode ﬁeld. For an optical medium that imposes two-dimensional boundary conditions in the transverse xy plane, the mode ﬁeld proﬁles are functions of two transverse spatial variables: E ν ðx; yÞ and Hν ðx; yÞ. Therefore, the mode index ν consists of two parameters for characterizing the variations of the mode ﬁelds in these two transverse dimensions. Then ν represents two mode numbers or symbols: ν ¼ mn. This is the case for an optical structure that Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 68 Optical Wave Propagation provides two-dimensional transverse optical conﬁnement, such as the nonplanar waveguide in Fig. 3.1(d). Another example is a collimated Gaussian mode in a homogeneous medium, which has a two-dimensional transverse proﬁle. For an optical medium that imposes boundary conditions in only one transverse direction, such as that in Fig. 3.1(b) or (c), the mode ﬁeld proﬁles are functions of only one transverse spatial variable: E ν ðxÞ and Hν ðxÞ. In this case, the mode index ν consists of only one parameter for characterizing the variations of the mode ﬁelds in the transverse dimension x. Then ν represents only one mode number or symbol: ν ¼ m. For discrete modes, i.e., modes of discrete propagation constants, the mode index numbers are discrete numbers, which are normally integers. For continuous modes, i.e., modes of continuously distributed propagation constants, the mode index numbers are continuously distributed numbers. 3.1.1 Mode Types For an optical structure in an isotropic medium, which is characterized by a spatial permittivity distribution of scalar ϵ ðx; yÞ, Maxwell’s equations for wave propagation take the form: — E ¼ μ0 —H¼ϵ ∂H , ∂t ∂E : ∂t (3.3) (3.4) For the mode ﬁelds of the form of (3.1) and (3.2), these two equations can be expressed in terms of the components of the mode ﬁeld proﬁles as ∂E z iβE y ¼ iωμ0 Hx , ∂y ∂E z ¼ iωμ0 Hy , ∂x ∂E y ∂E x ¼ iωμ0 Hz , ∂x ∂y iβE x (3.5) (3.6) (3.7) and ∂Hz iβHy ¼ iωϵE x , ∂y (3.8) ∂Hz ¼ iωϵE y , ∂x (3.9) ∂Hy ∂Hx ¼ iωϵE z : ∂x ∂y (3.10) iβHx From these equations, the transverse components of the electric and magnetic mode ﬁelds can be expressed in terms of the longitudinal components: ∂E z ∂Hz k2 β2 E x ¼ iβ þ iωμ0 , ∂x ∂y Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.11) 3.1 Normal Modes of Propagation 69 2 ∂E z ∂Hz iωμ0 , k β2 E y ¼ iβ ∂y ∂x (3.12) 2 ∂Hz ∂E z k β2 Hx ¼ iβ iωϵ , ∂x ∂y (3.13) 2 ∂Hz ∂E z k β2 Hy ¼ iβ þ iωϵ , ∂y ∂x (3.14) k 2 ¼ ω2 μ0 ϵ ðx; yÞ (3.15) where is a function of x and y to account for the transverse spatial inhomogeneity of the structure. The relations in (3.11)–(3.14) are generally valid for a longitudinally homogeneous structure of any transverse geometry and any transverse index proﬁle, for which ϵ ðx; yÞ is not a function of z. In a structure of cylindrical symmetry, such as an optical ﬁber, the x and y coordinates of the rectilinear system can be transformed to the ϕ and r coordinates of the cylindrical system for similar relations. It is clear from (3.11)–(3.14) that once the longitudinal mode ﬁeld components, E z and Hz , are known, all mode ﬁeld components can be obtained. Therefore, a normal mode can be classiﬁed based on the characteristics of its longitudinal ﬁeld components, as follows. 1. 2. 3. 4. A transverse electromagnetic mode, or TEM mode, has E z ¼ 0 and Hz ¼ 0. A transverse electric mode, or TE mode, has E z ¼ 0 and Hz 6¼ 0. A transverse magnetic mode, or TM mode, has Hz ¼ 0 and E z 6¼ 0. A hybrid mode has both E z 6¼ 0 and Hz 6¼ 0. Several comments can be made. 1. Any dielectric optical structure that has an inhomogeneous transverse proﬁle does not support TEM modes. For such an optical structure, k2 ¼ ω2 μ0 ϵ ðx; yÞ is not a constant of space but β2 is always a constant; therefore, all ﬁeld components vanish when E z ¼ 0 and Hz ¼ 0, as can be seen from (3.11)–(3.14). 2. TEM modes exist in (a) a homogeneous dielectric medium without any conductors, (b) the outside of a single-conductor transmission line in a homogeneous dielectric medium, and (c) a waveguide consisting of multiple separate conductors in a homogeneous dielectric medium. For a TEM mode to exist, (3.11)–(3.14) require that ϵ ðx; yÞ ¼ ϵ be a constant of space so that k2 ¼ β2 . Therefore, the propagation constant of a TEM mode is simply that pﬃﬃﬃﬃﬃﬃﬃ of the dielectric medium: β ¼ k ¼ ω μ0 ϵ . 3. Only TE and TM modes are allowed in (a) a planar dielectric structure of ϵ ðx; yÞ ¼ ϵ ðxÞ and (b) the inside of a hollow metallic waveguide. 4. TE and TM modes are allowed but are not the only modes in (a) a planar metallic waveguide consisting of two parallel plates, which also supports TEM modes, and (b) a nonplanar dielectric waveguide, which also supports hybrid modes. 5. Hybrid modes are allowed in nonplanar dielectric waveguides, but not in planar dielectric structures. The HE and EH modes of optical ﬁbers are hybrid modes. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 70 Optical Wave Propagation 6. From the above discussion, planar dielectric optical structures only have TE and TM modes, whereas nonplanar dielectric optical structures only have TE, TM, and hybrid modes. None of them have TEM modes. EXAMPLE 3.1 Find the general relations between the transverse components of the electric ﬁeld and those of the magnetic ﬁeld for (a) a TEM mode, (b) a TE mode, (c) a TM mode, and (d) a hybrid mode. Solution: The general relations between the transverse electric-ﬁeld components, E x and E y , and the transverse magnetic-ﬁeld components, Hx and Hy , for each type of mode can be found from (3.5)–(3.10). (a) TEM modes: For a TEM mode, E z ¼ 0 and Hz ¼ 0. Therefore, Hx ¼ β ωϵ E y ¼ E y, ωμ0 β β ωϵ Ex ¼ E x: ωμ0 β pﬃﬃﬃﬃﬃﬃﬃ From these relations, it is always true that β ¼ ω ϵμ0 ¼ k for a TEM mode. (b) TE modes: For a TE mode, E z ¼ 0 but Hz 6¼ 0. Therefore, Hy ¼ Hx ¼ β ωϵ E y 6¼ E y , ωμ0 β β ωϵ E x 6¼ E x: ωμ0 β pﬃﬃﬃﬃﬃﬃﬃ From these relations, it is always true that β 6¼ ω ϵμ0 for a TE mode. (c) TM modes: For a TM mode, Hz ¼ 0 but E z 6¼ 0. Therefore, Hy ¼ Hx ¼ ωϵ β E y, E y 6¼ β ωμ0 ωϵ β E x: E x 6¼ β ωμ0 pﬃﬃﬃﬃﬃﬃﬃ From these relations, it is always true that β 6¼ ω ϵμ0 for a TM mode. (d) Hybrid modes: For a hybrid mode, E z 6¼ 0 and Hz 6¼ 0. Therefore, Hy ¼ Hx 6¼ β ωϵ E y 6¼ E y , ωμ0 β β ωϵ E x 6¼ E x: ωμ0 β pﬃﬃﬃﬃﬃﬃﬃ From these relations, it is always true that β 6¼ ω ϵμ0 for a hybrid mode. Hy 6¼ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.1 Normal Modes of Propagation 71 3.1.2 Power and Orthonormalization of Modes The intensity distribution of a normal mode ν projected on a transverse plane, which has a surface normal of n^ ¼ ^z , is given by ∗ z ¼ E ν H∗ z, I ν ¼ S ν ^z ¼ Sν þ S∗ ν ^ ν þ E ν Hν ^ (3.16) which is a function of x and y. The power, Pν , of the mode is obtained by integrating I ν ðx; yÞ over the entire transverse cross-sectional plane. It can be seen from (3.16) that the longitudinal components, E z and Hz , of the mode ﬁelds do not contribute to the mode intensity or the mode power. Because different normal modes are orthogonal to each other, the mode ﬁelds of a lossless isotropic structure satisfy the orthogonality relation: ð∞ ð∞ ∗ ^z dxdy ¼ Pν δνμ : E ν H∗ þ E H ν μ μ (3.17) ∞ ∞ where δνμ is the Kronecker delta function for discrete modes, with ν and μ representing discrete numbers; but δνμ is the Dirac delta function δðν μÞ for continuous modes, with ν and μ representing continuous numbers. For a nonplanar structure, ν ¼ mn and μ ¼ m0 n0 ; hence δνμ ¼ δmm0 δnn0 . For a planar structure, ν ¼ m and μ ¼ m0 ; then, δνμ ¼ δmm0 . The normal mode ﬁelds are normalized according to the following orthonormality relation: ð∞ ð∞ ^∗ H ^ν H ^∗þE ^ ν ^z dxdy ¼ δνμ : E μ μ (3.18) ∞ ∞ This orthonormality relation deﬁned in terms of cross products based on the form of the Poynting vector is valid for all types of modes. Simpliﬁed relations in terms of dot products exist for TE, TM, and TEM modes. For TE modes, (3.17) can be reduced to 2βν ωμ0 ð∞ ð∞ TE Eν E∗ μ dxdy ¼ Pν δνμ : (3.19) ∞ ∞ Therefore, as an alternative to (3.18), the orthonormality relation among TE modes can also be written as 2βν ωμ0 ð∞ ð∞ ^ ∗ dxdy ¼ δνμ : ^ν E E μ (3.20) 1 TM H ν H∗ μ dxdy ¼ Pν δνμ : ϵ ðx; yÞ (3.21) ∞ ∞ For TM modes, (3.17) can be reduced to 2βν ω ð∞ ð∞ ∞ ∞ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 72 Optical Wave Propagation As an alternative to (3.18), the orthonormality relation among TM modes can also be written as 2βν ω ð∞ ð∞ ∞ ∞ 1 ^ ^∗ H ν H μ dxdy ¼ δνμ : ϵ ðx; yÞ (3.22) The simpliﬁed relations for TE modes and those for TM modes are both valid for TEM modes because a TEM mode is both TE and TM. As discussed above, a TEM mode exists only when ϵ ðx; yÞ ¼ ϵ is a constant of space. Therefore, for TEM modes, 2βν ωμ0 ð∞ ð∞ Eν E∗ μ dxdy ∞ ∞ 2β ¼ ν ωϵ ð∞ ð∞ TEM Hν H∗ δνμ : μ dxdy ¼ Pν (3.23) ∞ ∞ There are two equivalent dot-product orthonormality relations among TEM modes: 2βν ωμ0 ð∞ ð∞ ^ ∗ dxdy ^ν E E μ ¼ δνμ ∞ ∞ 2βν and ωϵ ð∞ ð∞ ^ ∗ dxdy ¼ δνμ : ^ ν H H μ (3.24) ∞ ∞ The orthogonality relation in (3.17) and the orthonormality relation in (3.18) indicate that power cannot be transferred between different normal modes in a linear, lossless structure of isotropic dielectric medium. For anisotropic or lossy structures, (3.17) and (3.18) do not apply, neither do the other simpliﬁed relations for TE, TM, and TEM modes. The orthogonality conditions and orthonormality relations for modes of such structures have other forms. 3.1.3 Mode Expansion The normal modes are orthogonal and can be normalized with the general orthonormality relation given in (3.18). They form a basis for linear expansion of any optical ﬁeld at a frequency of ω propagating in the optical medium: X ^ ν ðx; yÞ exp ðiβ z iωt Þ, Eðr; t Þ ¼ Aν E (3.25) ν ν Hðr; t Þ ¼ X ν ^ ν ðx; yÞ exp ðiβν z iωt Þ, Aν H (3.26) where the summation symbol sums over all discrete indices of the discrete modes and integrates over all continuous indices of the continuous modes. In a linear structure where the normal modes are deﬁned, these modes propagate independently without exchanging power. Therefore, the expansion coefﬁcients Aν are constants that are independent of x, y, and z. According to (3.17) and (3.18), the normal modes are normalized such that the mode power is simply P ν ¼ jA ν j2 : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.27) 3.2 Plane-Wave Modes 3.2 73 PLANE-WAVE MODES .............................................................................................................. A plane wave has wavefronts of inﬁnite parallel planes. As deﬁned in Section 1.7, a wavefront is the surface of a constant phase, and the wavevector is the gradient of the phase, which is normal to the wavefront. Therefore, a monochromatic plane wave that propagates in a homogeneous medium is deﬁned by one constant frequency ω and one constant wavevector k: Eðr; tÞ ¼ E exp ðik r iωt Þ, (3.28) Hðr; tÞ ¼ H exp ðik r iωt Þ, (3.29) where both E and H are constants of space and time. The electric displacement and the magnetic induction of the plane wave have similar forms: Dðr; t Þ ¼ ϵ Eðr; t Þ ¼ D exp ðik r iωt Þ and Bðr; tÞ ¼ μ0 Hðr; t Þ ¼ B exp ðik r iωt Þ, where D and B are constants of space and time. When operating on the ﬁelds of a plane wave, the space operator — always yields ik and the time operator ∂=∂t always yields iω. Therefore, for a plane wave propagating in a homogeneous medium, the following replacements can be made: — ! ik, ∂ ! iω: ∂t (3.30) A monochromatic plane wave is a normal mode of propagation in a homogeneous medium because it has a well-deﬁned wavevector, thus a well-deﬁned propagation constant. In an isotropic medium, the propagation constant of a plane wave does not depend on the polarization of the wave; therefore, a plane wave of any polarization has the same well-deﬁned propagation constant and is a normal mode. In an anisotropic medium, only ﬁelds of certain polarizations have well-deﬁned propagation constants, as discussed in Section 2.2. Plane-wave normal modes in a homogeneous anisotropic medium have speciﬁc polarization characteristics and polarization-dependent propagation constants that are determined by both the property of the medium and the direction of wave propagation. In any event, for a monochromatic plane-wave normal mode, Maxwell’s equations as given in (1.41)–(1.44) can be expressed in the algebraic form: k E ¼ ωμ0 H, (3.31) k H ¼ ωD, (3.32) k D ¼ 0, (3.33) k H ¼ 0: (3.34) Note that the relation B ¼ μ0 H, as is always true for optical ﬁelds, is used for the above equations. The wave propagation direction is deﬁned by the wavevector k, whereas the power ﬂow direction is deﬁned by the Poynting vector from (1.54): S ¼ E H∗ : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.35) 74 Optical Wave Propagation By combining (3.31) and (3.32) to eliminate the magnetic ﬁeld H, the algebraic form of the wave equation for a plane wave is obtained: k k E þ ω2 μ0 D ¼ 0: (3.36) A plane-wave normal mode is characterized by six vectors: E, D, H, B, k, and S. Their relations found from (3.31)–(3.35) are summarized as follows. 1. From (3.31) and (3.35), the three vectors E, H, and S are always mutually orthogonal for a plane wave in any homogeneous medium. 2. From (3.32)–(3.34), the three vectors D, H, and k are always mutually orthogonal for a plane wave in any homogeneous medium. 3. In any optical medium BkH is always true because B ¼ μ0 H. Both are orthogonal to all of the other four vectors E, D, k, and S. 4. In a homogeneous isotropic medium, DkE because D ¼ ϵE. Both are orthogonal to all of the other four vectors H, B, k, and S. 5. In a homogeneous anisotropic medium, D is not necessarily parallel to E because D ¼ ϵ E. Both D and E are orthogonal to H and B, but E is not necessarily orthogonal to k while D is not necessarily orthogonal to S. As expressed in (3.28) and (3.29), a true plane wave transversely extends to inﬁnity in space, which is unrealistic. It is a good approximation if a medium is homogeneous in all directions over dimensions that are very large compared to the wavelength. Because the ﬁeld amplitude of every plane wave is a constant of space, the difference between two plane waves of the same frequency that propagate in the same direction is only in their polarization characteristics. Orthogonality between two such plane-wave modes is determined only by the orthogonality of their polarization states but not by the spatial integral of their ﬁeld overlap. Therefore, for a given wave propagation direction, there are only two orthogonally polarized plane-wave modes. Furthermore, because a plane wave has a constant amplitude extending throughout the transverse plane, the integrals that deﬁne mode normalization in Section 3.1 cannot be performed. For these reasons, the actual amplitude of each wave is used in the ﬁeld expansion though a unit polarization vector is often used to represent the polarization state of a plane wave. The plane wave basis for linear expansion of any optical ﬁeld that has a frequency of ω and propagates in the k^ direction through a homogeneous optical medium consists of only two orthogonally polarized elements: Eðr; t Þ ¼ E1 ðr; t Þ þ E2 ðr; t Þ ¼ E 1 exp iβ1 k^ r iωt þ E 2 exp iβ2 k^ r iωt , (3.37) Hðr; t Þ ¼ H1 ðr; t Þ þ H2 ðr; tÞ ¼ H1 exp iβ1 k^ r iωt þ H2 exp iβ2 k^ r iωt , (3.38) where E 1 , H1 , E 2 , and H2 are constants of space; β1 and β2 are the propagation constants of the two plane-wave modes; and the two modes satisfy the polarization orthogonality relations: ∗ ∗ ∗ E1 E∗ 2 ¼ E 1 E 2 ¼ 0 and H1 H2 ¼ H1 H2 ¼ 0: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.39) 3.2 Plane-Wave Modes 75 Figure 3.2 Relationships among the directions of E, D, H, B, k, and S in free space or in an isotropic medium. In a homogeneous medium, the propagation constants are determined by the material properties and the polarization states of the waves but not by any optical structure. Therefore, β1 ¼ k1 and β2 ¼ k2 . The two propagation constants are the same if the medium is isotropic, but they are generally different if the medium is anisotropic, as discussed below. 3.2.1 Isotropic Medium The permittivity tensor of a homogeneous isotropic medium reduces to a scalar ϵ that is independent of spatial location and direction. Free space is a special case of homogeneous isotropic medium with ϵ ¼ ϵ 0 . Figure 3.2 shows the relations among the six vectors E, D, H, B, k, and S of a plane wave that propagates in a homogeneous isotropic medium. For this plane wave, EkD⊥k because D ¼ ϵE. A plane-wave normal mode of a homogeneous isotropic medium is a TEM wave because its E and H ﬁelds are both orthogonal to its wavevector k. With E⊥k, we ﬁnd that k k E ¼ k2 E. By using this relation and D ¼ ϵE, the wave equation in (3.36) is reduced to k2 E þ ω2 μ0 ϵE ¼ 0, (3.40) which yields the eigenvalue equation: k 2 ¼ ω2 μ0 ϵ: (3.41) Therefore, the propagation constant of the wave in the medium is pﬃﬃﬃﬃﬃﬃﬃ nω 2πnν 2πn k ¼ ω μ0 ϵ ¼ ¼ ¼ , c c λ (3.42) where ν is the frequency of the optical wave, λ is its wavelength, 1 c ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ μ0 ϵ 0 (3.43) rﬃﬃﬃﬃﬃ ϵ ¼ ðdielectric constantÞ1=2 n¼ ϵ0 (3.44) is the speed of light in free space, and Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 76 Optical Wave Propagation is the index of refraction, or refractive index, of the isotropic medium. Because k is proportional to 1=λ, it is also called the wavenumber. In a medium that has an index of refraction of n, the optical frequency is still ν, but the optical wavelength is λ=n, and the speed of light is v ¼ c=n. Regardless of the propagation direction or the polarization state, all plane waves of the same frequency ω in a homogeneous isotropic medium are degenerate and have the same propagation ^ any two orthogonally polarized constant k found in (3.42). For any given propagation direction k, plane waves that propagate in the k^ direction can be used as the basis for linear expansion. Both ^ as is seen in Fig. 3.2. Because are TEM waves and are orthogonal to the propagation direction k, the medium is isotropic, the coordinates can be chosen such that the z axis is in the direction of ^ Then the ﬁeld expansion of (3.37) and (3.38) takes the form: wave propagation, i.e., ^z ¼ k. Eðr; tÞ ¼ E 1 exp ðikz iωt Þ þ E 2 exp ðikz iωt Þ ¼ ðE 1 þ E 2 Þ exp ðikz iωtÞ, (3.45) Hðr; tÞ ¼ H1 exp ðikz iωtÞ þ H2 exp ðikz iωt Þ ¼ ðH1 þ H2 Þ exp ðikz iωtÞ: (3.46) For propagation in the z direction with k^ ¼ ^z as considered here, any two orthogonal polarization states in the xy plane can be used as the basis set for the ﬁeld expansion. For example, the basis set can be formed by the two linearly polarized waves E x ^x and E y ^y , by the two circularly polarized waves E þ ^e þ and E ^e , or by any two orthogonal elliptically polarized waves. It can be seen from (3.45) and (3.46) that the linear superposition of two plane-wave normal modes of a homogeneous isotropic medium is also a normal mode of the same propagation constant. Hence any plane wave of a given frequency ω traveling in a homogeneous isotropic medium is a normal mode with the same propagation constant k. This is not true for plane waves traveling in a homogeneous anisotropic medium, which is discussed below. EXAMPLE 3.2 GaAs is a cubic crystal. At the λ ¼ 900 nm wavelength, its principal indices of refraction are nx ¼ ny ¼ nz ¼ 3:593. A circularly polarized wave and a linearly polarized wave at this wavelength propagate along the z and x principal axes, respectively. What are the propagation constants and the wavelengths of these two waves in the GaAs crystal? Solution: Though GaAs has well-deﬁned principal axes, it is optically isotropic because nx ¼ ny ¼ nz ¼ n. Therefore, a plane wave of any polarization state propagating in any direction is a normal mode that has a refractive index of n. At λ ¼ 900 nm, n ¼ 3:593. For both waves, we ﬁnd the propagation constant to be k¼ 2πn 2π 3:593 ¼ ¼ 2:51 107 m1 λ 900 nm and the wavelength in GaAs to be λGaAs ¼ λ 900 nm ¼ ¼ 250:5 nm: n 3:593 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.2 Plane-Wave Modes 77 3.2.2 Anisotropic Medium As discussed in Sections 2.2, 2.6, and 2.7, the anisotropy of a medium can be intrinsic, such as that of an anisotropic crystal, or it can be induced by an external factor, such as that caused by an electro-optic, magneto-optic, acousto-optic, or nonlinear optical effect. The principal normal modes associated with linear or circular birefringence have already been discussed in Section 2.2. Here we consider only linear birefringence of an anisotropic crystal characterized by a symmetric dielectric tensor ϵ whose eigenvectors deﬁne the principal axes ^x , ^y , and ^z with eigenvalues ϵ x , ϵ y , and ϵ z , respectively. Plane-wave normal modes still exist for wave propagation in a homogeneous anisotropic medium. However, their characteristics depend on the direction of propagation with respect to the principal axes of the medium. In contrast to plane-wave normal modes in an isotropic medium, all of which are degenerate with the same propagation constant, plane-wave normal modes in an anisotropic medium are generally nondegenerate. Their polarization states and propagation constants are speciﬁc to each propagation direction. Three general cases are discussed in the following. Propagation along an Optical Axis In the special case of propagation along an optical axis, the crystal appears to be isotropic to the wave. For a uniaxial crystal, the optical axis is one of the principal axes, taken to be the z principal axis by convention. For a biaxial crystal, neither of the two optical axes is a principal axis. In any event, by the deﬁnition of optical axis, a wave does not experience any birefringence when it propagates along an optical axis. Then the plane-wave normal modes have the same characteristics as those discussed above for an isotropic medium. All plane waves polarized in the plane normal to an optical axis are normal modes of propagation along this optical axis, and any two of them that are orthogonally polarized can be used as the basis for linear expansion. EXAMPLE 3.3 LiNbO3 is a negative uniaxial crystal that has principal refractive indices of nx ¼ ny ¼ no ¼ 2:238 and nz ¼ ne ¼ 2:159 at the λ ¼ 1 μm wavelength. Find the possible arrangements for (a) a linearly polarized wave and (b) a circularly polarized wave to propagate through LiNbO3 with a propagation constant deﬁned by either no or ne . In each case, ﬁnd the propagation constant and the wavelength for the wave in LiNbO3 . Solution: The refractive index seen by a wave is determined by the polarization of the wave. Then, the possible direction of propagation is constrained by a given polarization. Because the z principal axis of the uniaxial LiNbO3 crystal is an optical axis, a wave that propagates along the z direction with its polarization in the xy plane sees the crystal as optically isotropic with no without seeing ne . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 78 Optical Wave Propagation (a) A linearly polarized wave at λ ¼ 1 μm sees no ¼ 2:238 if it is polarized in any direction in the xy plane. This is always true when the wave propagates along the z principal axis. Then, it has ko ¼ 2πno 2π 2:238 ¼ ¼ 1:41 107 m1 λ 1 μm and λo ¼ λ 1 μm ¼ ¼ 446:8 nm: no 2:238 A linearly polarized wave sees ne ¼ 2:159 if it is polarized along the z principal axis. This is possible only when the wave propagates in a direction that lies in the xy plane. Then, it has ke ¼ 2πne 2π 2:159 ¼ ¼ 1:36 107 m1 λ 1 μm and λe ¼ λ 1 μm ¼ ¼ 463:2 nm: ne 2:159 (b) A circularly polarized wave at λ ¼ 1 μm sees no ¼ 2:238 if its circular polarization lies in the xy plane. For this to happen, the wave has to propagate along the z principal axis. It has ko ¼ 2πno 2π 2:238 ¼ ¼ 1:41 107 m1 λ 1 μm and λo ¼ λ 1 μm ¼ ¼ 446:8 nm: no 2:238 There is no possible arrangement for a circularly polarized wave to propagate in a uniaxial crystal with a propagation constant deﬁned by ne . Propagation along a Principal Axis When an optical wave propagates in a direction other than that along an optical axis, the index of refraction depends on the direction of its polarization. In this situation, there exist two normal modes of linearly polarized waves, each of which has a unique index of refraction. If the propagation direction is along a principal axis that is not an optical axis, the two normal modes are simply the principal modes of polarization that are linearly polarized along the other two principal axes. Each principal mode of polarization has its characteristic principal index of refraction. Without loss of generality, take the principal axis along which the wave propagates to be the z ^ z . In the case when the z principal axis is not an optical axis, the other principal axis so that kk^ two principal axes ^x and ^y , which are orthogonal to the propagation direction, are birefringent with different principal permittivities, ϵ x 6¼ ϵ y , thus different propagation constants: k x 6¼ k y , where kx ¼ nx ω=c and ky ¼ ny ω=c as deﬁned in (2.15). Note that kx and ky are the propagation constants of the x- and y-polarized principal normal modes, respectively, not to be confused with the x and y components of a wavevector k, which are normally expressed as kx and ky : These two plane wave principal normal modes are E 1 ¼ ^x E 1 ¼ ^x E x , E 2 ¼ ^y E 2 ¼ ^y E y , H1 ¼ ^y H1 ¼ ^y Hy , H2 ¼ ^x H2 ¼ ^x Hx , k1 ¼ β1 k^ ¼ k x ^z , k2 ¼ β2 k^ ¼ k y ^z : (3.47) In the form of (3.37) and (3.38), these two normal modes form the basis for linear decomposition of any plane wave that propagates along the z principal axis. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.2 Plane-Wave Modes 79 Figure 3.3 Evolution of the polarization state of an optical wave propagating along the principal axis ^z of an anisotropic crystal that has nx 6¼ ny . Only the evolution over one half-period is shown here. (a) The optical wave is initially linearly polarized at an arbitrary angle θ with respect to the principal axis ^x . (b) The optical wave is initially polarized at 45 with respect to ^x . For a plane wave propagating along ^z , the electric ﬁeld can be expressed as Eðr; t Þ ¼ E1 ðr; t Þ þ E2 ðr; t Þ ¼ ^x E x exp ðikx z iωt Þ þ ^y E y exp ðiky z iωt Þ: (3.48) Because the wave propagates in the z direction, the wavevectors are kx ¼ kx ^z for the x-polarized ﬁeld and ky ¼ k y ^z for the y-polarized ﬁeld. The ﬁeld expressed in (3.48) has the following propagation characteristics. 1. If Eðr; t Þ is originally linearly polarized along one of the principal axes, i.e., E y ¼ 0 for Eðr; t Þ ¼ E1 ðr; t Þk^x or E x ¼ 0 for Eðr; t Þ ¼ E2 ðr; t Þk^y , it remains linearly polarized in the same direction as it propagates. 2. If Eðr; t Þ is originally linearly polarized at an angle of θ ¼ tan1 E y =E x with respect to the x axis with E1 ðr; t Þ 6¼ 0 and E2 ðr; t Þ 6¼ 0, its polarization state varies periodically along z with a period of 2π=jk y kx j because the two normal modes propagate with different propagation constants. In general, its polarization follows a sequence of variations from linear to elliptic to linear in the ﬁrst half-period and then reverses the sequence back to linear in the second half-period. At the half-period position, it is linearly polarized at an angle of θ on the other side of the x axis. Thus the polarization is rotated by 2θ from the original direction, as shown in Fig. 3.3(a). In the special case when θ ¼ 45 , the wave is circularly polarized at the quarter-period point and is linearly polarized at the half-period point with its polarization rotated by 90 from the original direction, as shown in Fig. 3.3(b). These characteristics have very useful applications. A plate of an anisotropic material that has a quarter-period thickness of lλ=4 ¼ 1 2π λ y x ¼ 4 jk k j 4 ny nx Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.49) 80 Optical Wave Propagation is called a quarter-wave plate. It can be used to convert a linearly polarized wave to circular or elliptic polarization, and vice versa. A plate that has a thickness of 3lλ=4 or 5lλ=4 , or any odd integral multiple of lλ=4 , also has the same function. By contrast, a plate that has a half-period thickness of lλ=2 ¼ 1 2π λ y x ¼ 2 jk k j 2 ny nx (3.50) is called a half-wave plate. It can be used to rotate the polarization direction of a linearly polarized wave by any angular amount by properly choosing the angle θ between the direction of the incident linear polarization and the principal axis ^x , or ^y , of the crystal. A plate of a thickness that is any odd integral multiple of lλ=2 has the same function. Note that though the output from a quarter-wave or half-wave plate can be linearly polarized, the wave plates are not polarizers. Wave plates and polarizers are based on different principles and have completely different functions. For the quarter-wave and half-wave plates discussed here, nx 6¼ ny . Between the two principal axes ^x and ^y , the one with the smaller index is called the fast axis, while the other, with the larger index, is the slow axis. EXAMPLE 3.4 At λ ¼ 1 μm, the principal indices of refraction of the KTP crystal are nx ¼ 1:742, ny ¼ 1:750, and nz ¼ 1:832. Is the crystal uniaxial or biaxial? If you want to propagate a linearly polarized wave through it, how do you arrange it so that its linear polarization is maintained throughout the propagation path in the crystal? If the crystal is used to make a half-wave plate for λ ¼ 1 μm, what is the minimum thickness of the plate? In which direction must the wave propagate to use this half-wave plate? Note that there is only one possible minimum thickness. Solution: Because nx 6¼ ny 6¼ nz , the crystal is biaxial. To maintain linear polarization throughout, the wave has to be linearly polarized along one of the principal axes while propagating along a direction that is perpendicular to its polarization direction. Its propagation constant is determined by its polarization direction but not by its propagation direction. For example, it can be polarized in the x direction while propagating in any direction in the yz plane. In this case, the wave sees nx and has a propagation constant of kx ¼ 2πnx =λ. Because the largest difference between two principal refractive indices is nz nx ¼ 1:832 1:742 ¼ 0:09, the wave must propagate along the y axis of the crystal and have its polarization in the zx plane, but not along the x or z axis, to utilize this birefringence for the minimum thickness of the half-wave plate: lλ=2 ¼ λ 1:00 μm ¼ 5:56 μm: ¼ 2jnz nx j 2j1:832 1:742j Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.2 Plane-Wave Modes 81 Figure 3.4 Relationships among the direction of wave propagation and the polarization directions of the ordinary and extraordinary waves. Propagation in a General Direction In the general case when the propagation direction is neither along an optical axis nor along a principal axis, there still exist two linearly polarized normal modes. For simplicity, the propagation in a uniaxial crystal is considered. The z principal axis of the uniaxial crystal is the optical axis, and the wave propagation direction k^ is at an angle of θ with respect to the z principal axis and at an angle of ϕ with respect to the x principal axis, as shown in Fig. 3.4. One of the normal modes is the polarization that is perpendicular to the optical axis. This normal mode is called the ordinary wave. We use ^e o to indicate its direction of polarization. The other normal mode is clearly perpendicular to ^e o because the two normal-mode polarizations are orthogonal to each other. This normal mode is called the extraordinary wave, and we use ^e e to indicate its direction of polarization. Note that these are the directions of D rather than those of E. For the ordinary wave, ^e o kDo kEo . For the extraordinary wave, ^e e kDe = kEe except when ^e e is parallel to a principal axis. Both ^e o and ^e e , being the unit vectors of Do and De , are perpendicular to the propagation direction k^ because D is always perpen^ From this understanding, both ^e o and ^e e can be found if both k^ and the optical dicular to k. axis ^z are known: ^e o ¼ 1 ^ k ^z , sin θ ^ ^e e ¼ ^e o k: (3.51) These vectors are illustrated in Fig. 3.4. They can be expressed as k^ ¼ ^x sin θ cos ϕ þ ^y sin θ sin ϕ þ ^z cos θ, (3.52) ^ ϕ, ^e o ¼ ^x sin ϕ y cos (3.53) ^e e ¼ ^x cos θ cos ϕ ^y cos θ sin ϕ þ ^z sin θ: (3.54) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 82 Optical Wave Propagation Figure 3.5 Determination of the indices of refraction for the ordinary and extraordinary waves in a uniaxial crystal using index ellipsoid. The indices of refraction associated with the ordinary and extraordinary waves can be found by using the index ellipsoid deﬁned as x2 y2 z2 þ þ ¼ 1: n2x n2y n2z (3.55) The index ellipsoid for the uniaxial crystal under consideration is illustrated in Fig. 3.5 with nx ¼ ny ¼ no and nz ¼ ne . The intersection of the index ellipsoid and the plane normal to k^ at the origin of the ellipsoid deﬁnes an index ellipse. The principal axes of this index ellipse are in the directions of ^e o and ^e e , and their half-lengths are the corresponding indices of refraction. For a uniaxial crystal, the index of refraction for the ordinary wave is simply no . The index of refraction for the extraordinary wave depends on the angle θ and is given by 1 cos2 θ sin2 θ ¼ þ 2 , n2e ðθÞ n2o ne (3.56) which can be seen from Fig. 3.5. We see that ne ð0 Þ ¼ no and ne ð90 Þ ¼ ne . For θ ¼ 0 , the propagation direction k^ is along the optical axis. For θ ¼ 90 , the propagation direction k^ lies in the plane perpendicular to the optical axis; in a uniaxial crystal, this situation is the same as when k^ is along a principal axis that is not the optical axis. Each of the two normal modes has a well-deﬁned propagation constant; the ordinary wave has k o ¼ no ω=c and the extraordinary wave has ke ¼ ne ðθÞω=c. Maxwell’s equations in the form of (3.31)–(3.34) have to be separately written with different values of k for the ordinary and the extraordinary normal modes; no such form applies to a wave that is a ^ for the extraordinary way, mixture of the two modes. For the ordinary way, k ¼ ko ¼ ko k; ^ k ¼ ke ¼ k e k. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.2 Plane-Wave Modes 83 EXAMPLE 3.5 LiNbO3 is a negative uniaxial crystal that has principal refractive indices of nx ¼ ny ¼ no ¼ 2:238 and nz ¼ ne ¼ 2:159 at the λ ¼ 1 μm wavelength. Find the polarization directions ^e o and ^e e , and the corresponding propagation constants k o and ke , of the ordinary and extraordinary normal modes for a propagation direction k^ that makes an angle of ϕ ¼ 30 with respect to the x principal axis and an angle of θ ¼ 45 with respect to the z principal axis. Solution: With ϕ ¼ 30 and θ ¼ 45 , we ﬁnd by using (3.52)–(3.54) that pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 3 1 6 2 2 6 2 2 ^x þ ^y þ ^e o ¼ ^x ^x ^y þ ^y , ^e e ¼ ^z , ^z : k^ ¼ 4 4 2 4 4 2 2 2 At θ ¼ 45 , we ﬁnd by using (3.56) that 2 cos 45 sin2 45 þ ne ð45 Þ ¼ 2:2382 2:1592 1=2 ¼ 2:197: Therefore, the propagation constants of the two normal modes are, respectively, ko ¼ ke ¼ 2πno 2π 2:238 ¼ ¼ 1:41 107 m1 , λ 1 μm 2πne ð45 Þ 2π 2:197 ¼ ¼ 1:38 107 m1 : λ 1 μm Because D is always perpendicular to the propagation direction, D⊥k for both ordinary and extraordinary waves. For an ordinary wave, Eo ⊥ko because Eo kDo . Therefore, the relationships shown in Fig. 3.6(a) among the ﬁeld vectors for an ordinary wave in an anisotropic medium are the same as those shown in Fig. 3.2 for a wave in an isotropic medium. For an extraordinary wave, in general Ee ⊥k = e because Ee = kDe ; thus Se is not necessarily parallel to ke . This means that Ee is not transverse to ke but has a longitudinal component in the ke direction. The only exception is when ^e e is parallel to a principal axis. As a result, the direction of power ﬂow, which is that of Se , is not the same as the direction of wave propagation, which is that of ke and is normal to the wavefronts, i.e., the planes of constant phase. Their relationship is shown in Fig. 3.6(b) together with the relationships among the directions of the ﬁeld vectors. Note that Ee , De , ke , and Se lie in the plane normal to He because Be kHe . Though it is still true that Ee ⊥He because ke Ee kHe according to (3.31), ke He = kEe because ke He kDe according to (3.32). These two plane-wave normal modes have the following characteristics: E o ¼ ^e o E o , Do ¼ ^e o Do , Ho ¼ ^e e Ho , ^ ko ¼ ko k; ^ k E e ¼ ^e e E ⊥ e þ kE e , De ¼ ^e e De , He ¼ ^e o He , ^ ke ¼ ke k; Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.57) 84 Optical Wave Propagation Figure 3.6 Relationships among the directions of E, D, H, B, k, and S in an anisotropic medium for (a) an ordinary wave and (b) an extraordinary wave. In both cases, the vectors E, D, k, and S lie in a plane normal to H. where E ⊥ e e and E ke ¼ E e k^ are, respectively, the transverse and longitudinal compone ¼ Ee ^ ents of the electric ﬁeld of the extraordinary wave. Note that only E e has a longitudinal component, and this component vanishes when ^e e is parallel to a principal axis. Note also that Ho kk^ ^e o ¼ ^e e and He kk^ ^e e ¼ ^e o because ωμ0 H ¼ k E for each mode, according to (3.31). In the form of (3.37) and (3.38), these two normal modes form the basis for the linear expansion of any plane wave propagating along the k^ direction: Eðr; t Þ ¼ Eo ðr; tÞ þ Ee ðr; t Þ ¼ E o exp ik o k^ r iωt þ E e exp ik e k^ r iωt , (3.58) Hðr; t Þ ¼ Ho ðr; t Þ þ He ðr; tÞ ¼ Ho exp iko k^ r iωt þ He exp ik e k^ r iωt : (3.59) If the electric ﬁeld of an extraordinary wave is not parallel to a principal axis, its Poynting vector is not parallel to its propagation direction because Ee is not parallel to De . As a result, its energy ﬂows away from its direction of propagation. This phenomenon is known as spatial beam walk-off. If this characteristic appears in one of the two normal modes of an optical wave propagating in an anisotropic crystal, the optical wave splits into two beams that have parallel wavevectors but separate, nonparallel traces of energy ﬂow. Consider a plane wave that propagates in a uniaxial crystal along a general direction k^ at an angle of θ with respect to the optical axis ^z ; this wave consists of both ordinary and extraordinary waves, as described by (3.58) and (3.59). Clearly, there is no walk-off for the ordinary wave because ^ For the extraordinary wave, Se is not parallel to k^ but points in a direction at an Eo kDo so that So kk. angle of ψ e with respect to the optical axis. Figure 3.7(a) shows the relationships among these ^ which is deﬁned as α ¼ ψ e θ, is called the walk-off angle vectors. The angle α between Se and k, of the extraordinary wave. Note that α is also the angle between Ee and De , as is seen in Fig. 3.7(a). Because neither Ee nor De is parallel to any principal axis, their relationship is found through their projections on the principal axes: Dez ¼ ϵ 0 n2e E ez and Dex, y ¼ ϵ 0 n2o E ex, y . Using these two relations and the deﬁnition of α in Figs. 3.6(b) and 3.7(a), it is found that the walk-off angle is given by 2 no α ¼ ψ e θ ¼ tan tan θ θ: n2e 1 (3.60) If the crystal is negative uniaxial, α as deﬁned in Fig. 3.6(b) is positive. This means that k^ is between Se and ^z for a negative uniaxial crystal. If the crystal is positive uniaxial, α is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.2 Plane-Wave Modes 85 Figure 3.7 (a) Wave propagation and walk-off in a uniaxial crystal. (b) Birefringent plate acting as a polarizing beam splitter for a normally incident wave. The ^x , ^y , and ^z unit vectors indicate the principal axes of the birefringent plate. negative and Se is between k^ and ^z . No walk-off appears if an optical wave propagates along any of the principal axes of a crystal. A birefringent crystal can be used to construct a simple polarizing beam splitter by taking advantage of the walk-off phenomenon. For such a purpose, a uniaxial crystal is cut into a plate whose surfaces are at an oblique angle with respect to the optical axis, as shown in Fig. 3.7(b). When an optical wave is normally incident on the plate, it splits into ordinary and extraordinary waves in the crystal if its original polarization contains components of both polarizations. The extraordinary wave is separated from the ordinary wave because of spatial walk-off, creating two orthogonally polarized beams. Because of normal incidence, both ke and ko are parallel to k^ although they have different magnitudes. When both beams reach the other side of the plate, they are separated by a distance of d ¼ l tan jαj, where l is the thickness of the plate. After leaving the plate, the two spatially separated beams propagate parallel to each other in the same k^ direction because the directions of their wavevectors have not changed, as also shown in Fig. 3.7(b). EXAMPLE 3.6 LiNbO3 is a negative uniaxial crystal that has principal refractive indices of nx ¼ ny ¼ no ¼ 2:238 and nz ¼ ne ¼ 2:159 at the λ ¼ 1 μm wavelength. Find the walk-off angle of α of the extraordinary wave in LiNbO3 for a propagation direction k^ that makes an angle of ϕ ¼ 30 with respect to the x principal axis and an angle of θ ¼ 45 with respect to the z principal axis. If a collimated optical beam that consists of both ordinary and extraordinary components at this wavelength propagates in this direction through a LiNbO3 plate, how thick must the plate be for the ordinary and extraordinary beams to be separated by at least 100 μm? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 86 Optical Wave Propagation Solution: The walk-off angle for θ ¼ 45 is found by using (3.60) to be 2 1 2:238 tan 45 45 ¼ 2:06 : α ¼ tan 2:1592 For the ordinary and extraordinary beams to be separated by at least 100 μm, d ¼ l tan α > 100 μm ) l > 100 μm ¼ 2:78 mm: tan 2:06 Thus, the thickness of the plate has to be at least 2:78 mm. 3.3 GAUSSIAN MODES .............................................................................................................. A monochromatic optical wave propagating in a homogeneous isotropic medium is governed by Maxwell’s equations for wave propagation given in (3.3) and (3.4). In this situation, ϵ is a scalar constant so that D ¼ ϵE and — E ¼ — D=ϵ ¼ 0: Then, — — E ¼ — — E r2 E ¼ r2 E. By using this relation while combining (3.3) and (3.4), we obtain the simple wave equation that is speciﬁc for the propagation of a monochromatic wave in a homogeneous isotropic medium: r2 E þ ω2 μ0 ϵE ¼ 0, (3.61) where the substitution of ∂=∂t ! iω is taken for the monochromatic wave at the frequency ω. Because every term in (3.61) has the same constant unit vector, the vectorial wave equation can be reduced to the scalar Helmholtz equation: r2 E þ k2 E ¼ 0, (3.62) where k2 ¼ ω2 μ0 ϵ, as deﬁned in (3.41). A similar equation can be written for the magnetic ﬁeld. Clearly, a monochromatic plane wave of the form in (3.28) and (3.29) is a solution of the equations for wave propagation given in (3.3) and (3.4), which in this case reduce to the simple form of (3.31) and (3.32) with D ¼ ϵE; thus, it is a solution of the wave equation in (3.61). Therefore, plane waves are normal modes of propagation in a homogeneous isotropic medium. They are not the only normal modes, however, as the equations that govern wave propagation in such a medium have other normal-mode solutions. One important set of modes is the Gaussian modes. Like plane waves, Gaussian modes are normal modes of wave propagation in a homogeneous isotropic medium. Different from a plane wave, a Gaussian mode has a ﬁnite cross-sectional ﬁeld distribution deﬁned by its spot size. Being an unguided ﬁeld that has a ﬁnite spot size, a Gaussian mode differs from a waveguide mode, discussed in Section 3.5, in that its spot size varies along its longitudinal axis, taken to be the z axis, of propagation though its pattern remains unchanged. Its transverse ﬁeld distribution also changes with z though the ﬁeld pattern does not change. The beam has a ﬁnite divergence angle, Δθ. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.3 Gaussian Modes 87 For a collimated Gaussian beam that has a small divergence angle such that the paraxial approximation sin Δθ Δθ 1 (3.63) is valid, the propagation constant of the Gaussian normal mode is β ¼ k. Therefore, rather than those in (3.1) and (3.2), the electric and magnetic ﬁelds of a monochromatic Gaussian mode at a frequency of ω can be expressed as Emn ðr; t Þ ¼ E mn ðx; y; zÞ exp ðikz iωt Þ ¼ ^e E mn ðx; y; zÞ exp ðikz iωtÞ, (3.64) Hmn ðr; tÞ ¼ Hmn ðx; y; zÞ exp ðikz iωtÞ ¼ k^ ^e Hmn ðx; y; zÞ exp ðikz iωt Þ, (3.65) where m and n are mode indices associated with the two transverse dimensions x and y, respectively. The paraxial approximation requires that 2 ∂ E ∂E k and ∂E , ∂E , ∂E jkE j (3.66) ∂z2 ∂z ∂x ∂y ∂z for the electric ﬁeld amplitude, and there are similar relations for the magnetic ﬁeld amplitude. In this approximation, the Helmholtz equation in (3.62) reduces to ∂2 E ∂2 E ∂E þ 2 þ i2k ¼0 2 ∂x ∂y ∂z (3.67) for the electric ﬁeld amplitude in (3.64). The magnetic ﬁeld amplitude in (3.65) satisﬁes an equation in H of the same form. In the paraxial approximation, a Gaussian mode ﬁeld is a TEM mode that has only transverse electric and magnetic ﬁeld components; it has neither longitudinal electric nor longitudinal magnetic ﬁeld components. Then, the unit polarization vector ^e for the electric mode ﬁeld in (3.64) is polarized in the transverse xy plane; the unit vector k^ ^e for the magnetic mode ﬁeld in (3.65) is also polarized in the transverse xy plane because k^ ¼ ^z . The paraxial approximation is not valid when a Gaussian beam is very tightly focused to the extent that its spot size is on the order of its optical wavelength. In this situation, the longitudinal electric and magnetic ﬁeld components cannot be ignored; such a Gaussian mode ﬁeld is not truly TEM. The electric mode ﬁelds of Gaussian modes in the paraxial approximation are eigenfunctions of (3.67); the corresponding magnetic mode ﬁelds have the same form because they are eigenfunctions of an equation of H that has the same form as (3.67). As TEM modes, they can be normalized by the dot-product orthonormality relations given in (3.24): 2k ωμ0 ð∞ ^ ∗0 0 ðx; y; zÞdxdy ^ mn ðx; y; zÞ E E mn ∞ 2k ¼ ωϵ ð∞ ^ mnðx; y; zÞ H ^ ∗0 0 ðx; y; zÞdxdy ¼ δmm0 δnn0 : H mn (3.68) ∞ The Gaussian beam eigenfunctions of (3.67) in the paraxial approximation have several salient characteristics. A Gaussian beam has a ﬁnite spot size that varies with location along the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 88 Optical Wave Propagation Figure 3.8 Gaussian beam characteristics. propagation axis. The location where the smallest spot size of the beam occurs is known as the waist of the Gaussian beam. This beam waist location is taken to be z ¼ 0 for a beam that propagates in the direction along the z axis. The minimum Gaussian beam spot size, w0 , is deﬁned as the e1 radius of the Gaussian beam electric ﬁeld magnitude proﬁle, i.e., the e2 radius of the Gaussian beam intensity proﬁle, at the beam waist. The diameter of the beam waist is d 0 ¼ 2w0 : As illustrated in Fig. 3.8, a Gaussian beam has a plane wavefront at its beam waist. The beam remains well collimated within a distance of zR ¼ kw20 πnw20 ¼ , 2 λ (3.69) pﬃﬃﬃﬃﬃﬃﬃ known as the Rayleigh range, on either side of the beam waist. In (3.69), k ¼ ω μ0 ϵ ¼ 2πn=λ is the propagation constant of the optical beam in a medium of a refractive index n. The parameter b ¼ 2zR is called the confocal parameter of the Gaussian beam. Because of diffraction, a Gaussian beam diverges away from its waist and acquires a spherical wavefront at a far-ﬁeld distance, where jzj zR . As a result, both its spot size, wðzÞ, and the radius of curvature, RðzÞ, of its wavefront are functions of the distance z from its beam waist: " #1=2 1=2 z2 2z 2 wðzÞ ¼ w0 1 þ 2 ¼ w0 1 þ (3.70) zR kw20 and " 2 2# z2R kw0 RðzÞ ¼ z 1 þ 2 ¼ z 1 þ : (3.71) z 2z pﬃﬃﬃ We see from (3.70) that w ¼ 2w0 at z ¼ zR . At jzj zR , far away from the beam waist, we ﬁnd that RðzÞ z and wðzÞ 2jzj=kw0 . Therefore, the far-ﬁeld beam divergence angle is Δθ ¼ 2 wðzÞ 4 2λ ¼ : ¼ kw0 πnw0 jzj Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.72) 3.3 Gaussian Modes 89 For the far ﬁeld at jzj zR , we ﬁnd that the beam spot size wðzÞ is inversely proportional to the beam waist spot size w0 but is linearly proportional to the distance jzj from the beam waist. This characteristic does not exist for the near ﬁeld at jzj zR : From (3.72), it can be seen that the paraxial approximation sin Δθ Δθ 1 expressed in (3.63) is valid when the beam is well collimated so that the spot size is much larger than the optical wavelength in the medium: w0 λ=n. Then the Gaussian mode ﬁelds are TEM modes. This is normally the case for Gaussian wave propagation. The Gaussian mode ﬁelds are not TEM when the beam is tightly focused such that the spot size is on the order of the optical wavelength. In this situation, w0 λ=n, and the paraxial approximation is invalid. EXAMPLE 3.7 A Gaussian beam from a Nd:YAG laser at the λ ¼ 1:064 μm wavelength propagates in free space with a beam divergence of 1 mrad. Find the beam waist spot size, the Rayleigh range, and the confocal parameter of the beam. What are the spot sizes and the radii of curvature of the beam at the distances of 10 cm, 1 m, 10 m, and 1 km, respectively? Solution: Given λ ¼ 1:064 μm and Δθ ¼ 1 mrad, we ﬁnd from (3.72) that the beam waist spot size is w0 ¼ 2λ 2 1:064 μm ¼ 677 μm: ¼ πΔθ π 1 103 From (3.69), the Rayleigh range and the confocal parameter are found: 2 πw20 π 677 106 zR ¼ m ¼ 1:35 m and b ¼ 2zR ¼ 2:7 m: ¼ λ 1:064 106 By using (3.70) and (3.71), the spot sizes and the radii of curvature at different locations are found: w ¼ 695 μm w ¼ 843 μm w ¼ 5:06 mm w ¼ 50:1 cm R ¼ 18:33 m at z ¼ 10 cm, R ¼ 2:82 m at z ¼ 1 m, R ¼ 10:18 m at z ¼ 10 m, R ¼ 1 km at z ¼ 1 km: Within the Rayleigh range, both the spot size and the radius of curvature vary nonlinearly with distance; the spot size increases slowly, whereas the radius of curvature decreases with distance. At a large distance, both the spot size and the radius of curvature increase approximately linearly with distance as the Gaussian beam approaches a spherical wave. A complete set of Gaussian modes in the paraxial approximation includes the fundamental TEM00 mode and high-order TEMmn modes. The speciﬁc forms of the mode ﬁelds depend on the transverse coordinates of symmetry: the mode ﬁelds are described by a set of Hermite–Gaussian functions in the rectilinear coordinates, whereas they are described by the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 90 Optical Wave Propagation Laguerre–Gaussian functions in the cylindrical coordinates. Both sets are equally valid in free space or in a homogeneous isotropic medium because there is no structurally deﬁned symmetry. Usually the Hermite–Gaussian functions in the rectilinear coordinates are used. In a transversely isotropic and homogeneous medium, a normalized TEMmn Hermite–Gaussian mode ﬁeld propagating along the z axis can be expressed as pﬃﬃﬃ pﬃﬃﬃ 2x 2y Cmn k x2 þ y2 ^ Hm exp ½iζ mn ðzÞ Hn exp i E mn ðx; y; zÞ ¼ wðzÞ wðzÞ wðzÞ 2 qðzÞ (3.73) pﬃﬃﬃ pﬃﬃﬃ 2 2 2 2 2x 2y C mn x þy kx þy ¼ Hm exp i exp ½iζ mn ðzÞ , Hn exp 2 wðzÞ w ðzÞ wðzÞ wðzÞ 2 RðzÞ ^ mn ðx; y; zÞ ¼ k E^ mn ðx; y; zÞ, H ωμ0 (3.74) 1=2 is the normalization constant, H m is the Hermite where Cmn ¼ ðωμ0 =πk Þ1=2 ð2mþn m!n!Þ polynomial of order m, qðzÞ is the complex radius of curvature of the Gaussian wave given by qðzÞ ¼ z izR or 1 1 2 ¼ þi 2 , qðzÞ RðzÞ kw ðzÞ and ζ mn ðzÞ is a mode-dependent on-axis phase variation along the z axis given by 2z 1 z 1 ζ mn ðzÞ ¼ ðm þ n þ 1Þtan ¼ ðm þ n þ 1Þ tan : zR kw20 (3.75) (3.76) The Hermite polynomials can be obtained using the following relation: 2 dm eξ : H m ðξ Þ ¼ ð1Þ e dξ m m ξ2 (3.77) Some low-order Hermite polynomials are H 3 ðξ Þ ¼ 8ξ 3 12ξ: (3.78) We see from (3.73) and (3.78) that the transverse ﬁeld distribution E^ 00 ðx; yÞ of the fundamental TEM00 Gaussian mode at a ﬁxed longitudinal location z is simply a Gaussian 1=2 function of the transverse radial distance r ¼ ðx2 þ y2 Þ and that the spot size wðzÞ is the e1 radius of this Gaussian ﬁeld distribution at z. The transverse ﬁeld distribution of a high-order TEMmn mode is the Gaussian function spatially modulated by the Hermite polynomials H m ðxÞ and H n ðyÞ in the x and y directions, respectively. As a result, its ﬁeld distribution spreads out radially farther than that of the fundamental TEM00 mode. In general, the higher the order of a mode is, the farther its transverse ﬁeld distribution spreads out. The intensity patterns of some low-order Hermite–Gaussian modes are shown in Fig. 3.9. The Hermite–Gaussian modes are deﬁned in the rectilinear ðx; y; zÞ coordinates. Because a homogeneous isotropic medium is also cylindrically symmetric with respect to the wave propagation direction, it is also possible to deﬁne a complete set of the TEM Gaussian modes, known as the Laguerre–Gaussian modes, in the cylindrical ðr; ϕ; zÞ coordinates with z being the longitudinal wave propagation direction. The Hermite–Guassian modes have rectilinear symmetry in the transverse plane, whereas the H 0 ðξ Þ ¼ 1, H 1 ðξ Þ ¼ 2ξ, H 2 ðξ Þ ¼ 4ξ 2 2, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.3 Gaussian Modes 91 Figure 3.9 Intensity patterns of some low-order Hermite–Gaussian modes. Laguerre–Gaussian modes have circular and radial symmetry in the transverse plane. Each set is a complete set of modes for ﬁeld expansion, and one set can be mathematically transformed to the other set by linear expansion. EXAMPLE 3.8 Find the transverse intensity distribution of the fundamental Gaussian mode as a function of the distance z from the beam waist. Given a fundamental Gaussian beam of a power P, ﬁnd the intensity I 0 ðzÞ at the beam center as a function of the distance z. Express P and I 0 ðzÞ in terms of the beam spot sizes w0 at the beam waist and wðzÞ at the location z. Solution: For the fundamental Guassian mode, m ¼ n ¼ 0. Because the zeroth-order Hermite function is a constant, H 0 ðxÞ ¼ H 0 ðyÞ ¼ 1, we ﬁnd from (3.73) that the fundamental Guassian mode ﬁeld 1=2 varies with x and y as x2 þ y2 so that E^ 00 ðx; y; zÞ ¼ E^ 00 ðr; zÞ, where r ¼ ðx2 þ y2 Þ is the transverse radial coordinate variable. Because a Guassian mode is a TEM mode, its ﬁeld 2 intensity is I ðr; zÞ / E^ 00 ðr; zÞ . Then, using (3.73), we can express I ðr; zÞ as I ðr; zÞ ¼ I 0 ðzÞexp 2r2 , w2 ðzÞ where I 0 ðzÞ is the intensity at the beam center r ¼ 0. The power of the beam is found by integrating the intensity distribution over the transverse plane: ð∞ ð∞ 2r 2 πw2 ðzÞ P ¼ I ðr; zÞ2πrdr ¼ I 0 ðzÞ exp 2 2πrdr ¼ I 0 ðzÞ: w ðzÞ 2 0 0 Note that the power of a beam is a constant that does not vary with the propagation distance z. By contrast, the intensity at the beam center varies with z as I 0 ðzÞ ¼ 2P : πw2 ðzÞ In terms of the parameters at the beam waist, P¼ πw20 w2 I 0 ð0Þ and I 0 ðzÞ ¼ 2 0 I 0 ð0Þ: 2 w ðzÞ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 92 Optical Wave Propagation For Gaussian beam propagation in a homogeneous isotropic medium along the longitudinal coordinate axis ^z , any two mutually orthogonal unit polarization vectors ^e 1 and ^e 2 in the transverse xy plane can be chosen as the polarization basis for linear decomposition of the wave polarization. Thus, the linear expansion of a Gaussian beam ﬁeld can be expressed as X X Eðr; tÞ ¼ ^e 1 Amn, 1 E^ mn ðx;y;zÞ exp ðikz iωt Þ þ ^e 2 Amn, 2 E^ mn ðx;y;zÞ exp ðikz iωtÞ, (3.79) m, n m, n Hðr; tÞ ¼ k ^ k ^z Eðr; tÞ, k Eðr; tÞ ¼ ωμ0 ωμ0 (3.80) where ^e 1 ^z ¼ ^e 2 ^z ¼ 0 and ^e i ^e ∗ j ¼ δij . The concept discussed above can be extended to Gaussian beam propagation in a homogeneous anisotropic crystal. For simplicity, consider the case when the propagation direction k^ is along a principal axis ^z that is not an optical axis so that nx 6¼ ny . As discussed in Section 3.2, the two principal modes of polarization, ^x and ^y , form the unique basis for polarization decomposition of TEM waves propagating along the z axis, when the x and y principal axes are birefringent. In this situation, the Gaussian ﬁeld is decomposed into two linearly polarized components that propagate with different propagation constants: k x ¼ nx ω=c and ky ¼ ny ω=c for the x and y polarizations, respectively. The linear expansion of such a Gaussian beam ﬁeld can be expressed as Eðr; t Þ ¼ Ex ðr; t Þ þ Ey ðr; t Þ X X ¼ ^x Amn, x E^ mn, x ðx; y; zÞ exp ðik x z iωtÞ þ ^y Amn, y E^ mn, y ðx; y; zÞ exp ðiky z iωt Þ, m, n m, n (3.81) Hðr; t Þ ¼ kx ky ^z Ex ðr; tÞ þ ^z Ey ðr; tÞ: ωμ0 ωμ0 (3.82) Because all of the characteristic parameters deﬁned in (3.69)–(3.72) for a Gaussian mode ﬁeld are functions of the refractive index n, the two polarization modes in (3.81) have different Gaussian beam parameters besides having different propagation constants. Therefore, in addition to changing its polarization state along the propagation axis as was the case for the plane wave discussed in Section 3.2, a Gaussian beam that propagates in an anisotropic medium can have two different spot sizes, two different divergence angles, and two different radii of curvature between the two principal polarization modes. The beam typically has an elliptic cross-sectional proﬁle. When focused by a spherical lens, the two polarization modes are focused at different focal points with different beam waist spot sizes. 3.4 INTERFACE MODES .............................................................................................................. The simplest optical structure is a planar interface separating two semi-inﬁnite homogeneous media, as shown in Fig. 3.1(b). The coordinates are chosen as shown in Fig. 3.1(b), with the interface located at x ¼ 0 such that ϵ ðxÞ ¼ ϵ 1 for x > 0 and ϵ ðxÞ ¼ ϵ 2 for x < 0. The Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.4 Interface Modes 93 permittivities ϵ 1 and ϵ 2 of the two media are scalar constants, whereas the permeabilities are simply μ0 at optical frequencies. As discussed in Section 3.1, only TE and TM modes are possible for this structure. Take the z axis to be the wave propagation direction. Then, because the index proﬁle is independent of the y coordinate and the wavevector has no y component, all ﬁeld components have no variations in the y direction: ∂E=∂y ¼ 0 and ∂H=∂y ¼ 0. 1. TE mode: For any TE mode of a planar structure, E z ¼ 0. It can be seen from (3.11)–(3.14) that E x ¼ 0, and Hy ¼ 0 as well because ∂Hz =∂y ¼ 0. The only nonvanishing ﬁeld components are Hx , E y , and Hz . Once the only nonvanishing electric ﬁeld component E y is found for a TE mode, the two nonvanishing magnetic ﬁeld components can be obtained by using (3.5) and (3.7): β E y, ωμ0 (3.83) 1 ∂E y : iωμ0 ∂x (3.84) Hx ¼ Hz ¼ 2. TM mode: For any TM mode of a planar structure, Hz ¼ 0. It can be seen from (3.11)– (3.14) that Hx ¼ 0, and E y ¼ 0 as well because ∂E z =∂y ¼ 0. The only nonvanishing ﬁeld components are E x , Hy , and E z . Once the only nonvanishing magnetic ﬁeld component Hy is found for a TM mode, the two nonvanishing electric ﬁeld components can be obtained by using (3.8) and (3.10): Ex ¼ Ez ¼ β Hy , ωϵ (3.85) 1 ∂Hy : iωϵ ∂x (3.86) In the case of a planar structure, it is convenient to solve for the unique transverse ﬁeld component ﬁrst: E y for a TE mode and Hy for a TM mode. The other ﬁeld components, including the longitudinal component, then follow directly. 3.4.1 Reﬂection and Refraction We ﬁrst consider the simple case of reﬂection and refraction of plane waves at the planar interface of two media as shown in Fig. 3.1(b). With the coordinates described above, the interface is located at x ¼ 0 and the plane of incidence is the xz plane so that all wavevectors have no y component. We assume that the optical wave is incident from the medium of ϵ 1 with a wavevector of ki , while the reﬂected wave has a wavevector of kr and the transmitted wave has a wavevector of kt . Because an optical wave varies with exp ðik r iωt Þ, the condition ki r ¼ kr r ¼ kt r Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.87) 94 Optical Wave Propagation Figure 3.10 Reﬂection and refraction of a TE-polarized wave at the interface of two isotropic dielectric media. The three vectors ki , kr , and kt lie in the plane of incidence. The relationship between θi and θt shown here is for the case of n1 < n2 : Figure 3.11 Reﬂection and refraction of a TM-polarized wave at the interface of two isotropic dielectric media. The three vectors ki , kr , and kt lie in the plane of incidence. The relationship between θi and θt shown here is for the case of n1 < n2 : is required at the interface x ¼ 0 for the boundary conditions described by (1.23)–(1.26) to be satisﬁed at all points along the interface at all times. This condition implies that the three vectors ki , kr , and kt lie in the same plane known as the plane of incidence, as shown in Figs. 3.10 and 3.11. The projections of these three wavevectors on the interface are all equal so that ki sin θi ¼ kr sin θr ¼ kt sin θt (3.88) where θi is the angle of incidence, and θr and θt are the angle of reﬂection and the angle of refraction, respectively, for the reﬂected and transmitted waves. All three angles are measured with respect to the normal n^ of the interface, as is shown in Figs. 3.10 and 3.11. Because ki ¼ kr and ki =kt ¼ n1 =n2 , (3.88) yields the relation θi ¼ θr Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.89) 3.4 Interface Modes 95 for reﬂection, and the familiar Snell’s law for refraction: n1 sin θi ¼ n2 sin θt : (3.90) By expressing H in terms of k E in the form of (3.31) with appropriate values of k for the incident, reﬂected, and refracted ﬁelds, respectively, the amplitudes of the reﬂected and transmitted ﬁelds can be obtained from the boundary conditions n^ E1 ¼ n^ E2 and n^ H1 ¼ n^ H2 given in (1.23) and (1.24). There are two different modes of ﬁeld polarization. TE Polarization (s Wave, σ Wave) For the transverse electric (TE) polarization, or the perpendicular polarization, the electric ﬁeld is linearly polarized in a direction perpendicular to the plane of incidence while the magnetic ﬁeld is polarized parallel to the plane of incidence, as shown in Fig. 3.10. This wave is also called s polarized, or σ polarized. For the TE-polarized wave, the reﬂection coefﬁcient, r, and the transmission coefﬁcient, t, of the electric ﬁeld are respectively given by the following Fresnel equations: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E r n1 cos θi n2 cos θt n1 cos θi n22 n21 sin2 θi pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , rs ¼ ¼ (3.91) ¼ E i n1 cos θi þ n2 cos θt n1 cos θi þ n22 n21 sin2 θi ts ¼ Et 2n1 cos θi 2n1 cos θi pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1 þ r s : ¼ ¼ E i n1 cos θi þ n2 cos θt n1 cos θi þ n22 n21 sin2 θi (3.92) The intensity reﬂectance and transmittance, R and T, which are also known as reﬂectivity and transmissivity, respectively, are given by I r Sr n^ n1 cos θi n2 cos θt 2 Rs ¼ ¼ ¼ jr s j2 , (3.93) ¼ Ii n1 cos θi þ n2 cos θt Si n^ I t St n^ ¼ 1 Rs 6¼ jt s j2 : (3.94) Ts ¼ ¼ I i S n^ i TM Polarization (p Wave, π Wave) For the transverse magnetic (TM) polarization, or the parallel polarization, the electric ﬁeld is linearly polarized in a direction parallel to the plane of incidence while the magnetic ﬁeld is polarized perpendicular to the plane of incidence, as shown in Fig. 3.11. This wave is also called p polarized, or π polarized. For the TM-polarized wave, the reﬂection and transmission coefﬁcients of the electric ﬁeld are respectively given by the following Fresnel equations: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E r n2 cos θi n1 cos θt n22 cos θi n1 n22 n21 sin2 θi pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , rp ¼ ¼ (3.95) ¼ E i n2 cos θi þ n1 cos θt n22 cos θi þ n1 n22 n21 sin2 θi tp ¼ Et 2n1 cos θi 2n1 n2 cos θi n1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ¼ ¼ 2 1 þ r : p E i n2 cos θi þ n1 cos θt n2 cos θi þ n1 n22 n21 sin2 θi n2 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.96) 96 Optical Wave Propagation The intensity reﬂectance and transmittance for the TM polarization are given, respectively, by I r Sr n^ n2 cos θi n1 cos θt 2 2 ¼ Rp ¼ ¼ ¼ rp , I i Si n^ n2 cos θi þ n1 cos θt (3.97) I t St n^ ¼ 1 Rp 6¼ t p 2 : Tp ¼ ¼ I i Si n^ (3.98) Several important characteristics of the reﬂection and refraction of an optical wave at an interface between two media are summarized below. 1. For both TE and TM polarizations, R ¼ jrj2 and R þ T ¼ 1, but T 6¼ jt j2 : 2. In the case when n1 < n2 , light is incident from a rare medium upon a dense medium; then, the reﬂection is called external reﬂection. In the case when n1 > n2 , light is incident from a dense medium on a rare medium; then, the reﬂection is called internal reﬂection. 3. Normal incidence: In the case of normal incidence, θi ¼ θt ¼ 0: Then, there is no difference between TE and TM polarizations, and n1 n2 2 , T ¼ 1 R ¼ 4n1 n2 : R ¼ n1 þ n2 ðn1 þ n2 Þ2 (3.99) In the case when both media are lossless so that the values of n1 and n2 are both real, there is a π phase change for the reﬂected electric ﬁeld with respect to the incident ﬁeld for external reﬂection at normal incidence, but the phase of the reﬂected ﬁeld is not changed for internal reﬂection at normal incidence. A phase change of a value between 0 and π is possible when either or both media have an optical loss or gain so that n1 or n2 or both have complex values. In any event, the values of R and T do not depend on the side of the interface from which the incident wave comes from. 4. Brewster angle: For a TE wave, Rs increases monotonically with the angle of incidence. For a TM wave, Rp ﬁrst decreases then increases as the angle of incidence increases. For the interface between two lossless media, Rp ¼ 0 at an angle of incidence of θi ¼ θB , where θB ¼ tan1 n2 n1 (3.100) is known as the Brewster angle. When θi ¼ θB , the angle of refraction for the transmitted wave is θt ¼ π θB : 2 (3.101) It can be shown that this angle is the Brewster angle for the same wave incident from the other side of the interface. Thus, the Brewster angles from the two sides of an interface are complementary angles. Figure 3.12 shows, for both the external reﬂection and the internal reﬂection, the reﬂectances of TE and TM waves as functions of the angle of incidence at the interface between two media of refractive indices of 1 and 3.5. These characteristics are very useful in practical applications. At θi ¼ θB , a TM-polarized incident wave is totally Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.4 Interface Modes 97 Figure 3.12 Reﬂectances of TE and TM waves at an interface of lossless media as functions of the angle of incidence for (a) external reﬂection and (b) internal reﬂection. The reﬂective indices of the two media used for these plots are 1 and 3.5. transmitted, resulting in a perfect transmitting window for the TM polarization. Such windows are called Brewster windows and are useful as laser windows. For a wave of any polarization that is incident at θi ¼ θB , the reﬂected wave is completely TE polarized. Linearly polarized light can be produced by a reﬂection-type polarizer based on this principle. 5. Critical angle: In the case of internal reﬂection with n1 > n2 , total internal reﬂection occurs if the angle of incidence θi is larger than the angle θc ¼ sin1 n2 , n1 (3.102) which is called the critical angle. The reﬂectances of TE and TM waves as functions of the angle of incidence for internal reﬂection at the interface between two media of refractive indices of 1 and 3.5 are shown in Fig. 3.12(b). Note that the Brewster angle for internal reﬂection is always smaller than the critical angle. 6. At the interface of two lossless dielectric media, both of which have real refractive indices, the transmitted ﬁeld has the same phase as the incident ﬁeld for both TE and TM polarizations because both ts and tp have positive, real values. For external reﬂection of a TE wave, the reﬂected ﬁeld has a π phase change at any incident angle. For internal reﬂection of a TE wave, the reﬂected ﬁeld has no phase change at any incident angle smaller than the critical angle. For external reﬂection of a TM wave, the reﬂected ﬁeld has no phase change at any incident angle smaller than the Brewster angle, θi < θB , but has a π phase change at any incident angle larger than the Brewster angle, θi > θB . For internal reﬂection of a TM wave, the reﬂected ﬁeld has a π phase change at any incident angle smaller than the Brewster angle, θi < θB , but it has no phase change at any incident angle larger than the Brewster angle but smaller than the critical angle, θB < θi < θc . (See Problem 3.4.1.) 7. The relations for the reﬂection and transmission coefﬁcients and those for the reﬂectance and transmittance, given in (3.91)–(3.98), remain valid if one or both media have an optical loss Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 98 Optical Wave Propagation or gain so that the refractive indices have complex values. In this situation, each of the reﬂection and transmission coefﬁcients of TE and TM waves has a phase that is different from 0 or π. 8. If one or both media have a loss or gain, the indices of refraction become complex. In this situation, the reﬂectance of the TM wave has a minimum value that does not reach zero. This minimum value is determined by the imaginary parts of the refractive indices of both media. 9. For wave propagation in a general direction in an anisotropic medium, there are two normal modes that have different indices of refraction. The refracted ﬁelds of these two normal modes can propagate in different directions, resulting in the phenomenon of double refraction. Meanwhile, the Poynting vector of a normal mode in the anisotropic medium does not have to be in the plane of incidence. 10. Optical media are generally dispersive. Therefore, reﬂectance and transmittance, as well as the direction of the refracted wave, are generally frequency dependent. EXAMPLE 3.9 The index of refraction of water is n ¼ 1:33. The index of refraction of ordinary glass depends on its composition and the optical wavelength but is approximately n ¼ 1:5. The refractive indices of semiconductors, such as Si, GaAs, and InP, vary signiﬁcantly with the optical wavelength and the material composition, as well as with temperature, but they usually fall in the range between 3 and 4. Take a nominal value of n ¼ 3:5 for the typical semiconductor. For each material at its interface with air, ﬁnd the reﬂectivity at normal incidence, the Brewster angle for external reﬂection, and the critical angle. Solution: Using (3.99), the reﬂectivities at normal incidence are found to be R ¼ 0:02 for water, R ¼ 0:04 for glass, and R ¼ 0:31 for the semiconductor. Using (3.100), the Brewster angles for external reﬂection are found to be θB ¼ 53:1 for water, θB ¼ 56:3 for glass, and θB ¼ 74 for the semiconductor. Using (3.102), the critical angles are found to be θc ¼ 48:8 for water, θc ¼ 41:8 for glass, and θc ¼ 16:6 for the semiconductor. 3.4.2 Radiation Modes In the above, we considered the reﬂection and refraction at a planar interface. Here we consider the mode ﬁelds of this structure in the form of (3.1) and (3.2) with the characteristic propagation constants βν in the z direction along the interface but with the mode ﬁeld proﬁles E ν ðxÞ and Hν ðxÞ being functions of only the x coordinate. The normal modes of a single interface are radiation modes that have a continuous spectrum of eigenvalues, i.e., continuously distributed values of propagation constants. From (3.87), we ﬁnd that the propagation constant in the z direction is that of the common longitudinal z component of ki , kr , and kt : β ¼ k i, z ¼ k r , z ¼ k t, z : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.103) 3.4 Interface Modes 99 We assume that the two media are dielectric with ϵ 1 > ϵ 2 so that k1 ¼ n1 ω=c > k 2 ¼ n2 ω=c: There are two different cases: (1) k 1 > β > k 2 and (2) k 1 > k2 > β, discussed below. One-Sided Radiation Modes: k1 > β > k2 This is the case when total internal reﬂection occurs with θi > θc ¼ sin1 n2 =n1 , as discussed above. Because ki, z ¼ k1 sin θi ¼ β and kr, z ¼ k1 sin θr ¼ β, the condition k2i, x þ k2i, z ¼ k2r, x þ k2r, z ¼ k 21 requires that the transverse x components of ki and kr have the same real value: h1 ¼ ki, x ¼ kr, x ¼ k1 cos θi . However, no real solution of θt exists for kt, z ¼ k2 sin θt ¼ β and k t, x ¼ k 2 cos θt to be valid because β > k2 in this case; therefore, no real value for the transverse x component of kt can be found. Instead, the condition k 2t, x þ k2t, z ¼ k22 requires that k t, x ¼ iγ2 be purely imaginary. Therefore, positive real parameters h1 and γ2 can be deﬁned for the transverse ﬁeld proﬁles in media 1 and 2, respectively, as h21 ¼ k21 β2 , γ22 ¼ β2 k22 : (3.104) Using the two parameters h1 and γ2 , the reﬂection coefﬁcients found in (3.91) and (3.95) for the TE and TM polarizations can be expressed respectively as n22 h1 in21 γ2 : (3.105) n22 h1 þ in21 γ2 2 As expected for total internal reﬂection, Rs ¼ jr s j2 ¼ 1 and Rp ¼ r p ¼ 1. However, from (3.105), it is found that total internal reﬂection has the following phase shifts for the TE and TM polarizations, respectively, r TE ¼ r s ¼ h1 iγ2 , h1 þ iγ2 φTE ¼ φs ¼ 2 tan1 γ2 , h1 r TM ¼ r p ¼ φTM ¼ φp ¼ 2 tan1 n21 γ2 : n22 h1 (3.106) As commented in the preceding subsection, for external reﬂection at any incident angle or internal reﬂection at an incident angle smaller than the critical angle, the reﬂection coefﬁcient of a TE or TM wave at an interface between two lossless dielectric media can only have a phase of either 0 or π. By contrast, (3.106) indicates that total internal reﬂection of a TE or TM wave can have a phase shift between 0 and π. The fact that ki, x and kr, x both have the real value of k i, x ¼ kr, x ¼ h1 means that the transverse ﬁeld proﬁle in medium 1 has sinusoidal variations extending to inﬁnity in the positive x direction. By contrast, k t, x ¼ iγ2 means that the transverse ﬁeld proﬁle in medium 2 decays exponentially in the negative x direction away from the interface. This is a one-sided radiation mode which is a radiation wave in medium 1 but is evanescent in medium 2, as illustrated in Fig. 3.13. The penetration depth of the evanescent tail into medium 2 is γ1 2 . For the TE mode, it is only necessary to ﬁnd E y ; then the other two nonvanishing components Hx and Hz can be found by using (3.83) and (3.84), respectively. The boundary conditions require that E y , Hx , and Hz be continuous at the interface, which dictates that E y and ∂E y =∂x be both continuous at x ¼ 0. The ﬁeld proﬁle satisfying these boundary conditions is E y ðxÞ ¼ cos ðh1 x ψ Þ, x > 0, cos ψ exp ðγ2 xÞ, x < 0, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.107) 100 Optical Wave Propagation Figure 3.13 Total internal reﬂection and transverse ﬁeld proﬁle of one-sided radiation mode. The fact that θr ¼ θi is shown. where ψ ¼ tan1 γ2 1 ¼ φTE : h1 2 (3.108) Note that the mode ﬁeld proﬁle E y given in (3.107) is not normalized because it extends to inﬁnity in the positive x direction. For x > 0, E y in (3.107) is the superposition of an incident ﬁeld of an amplitude E i ¼ ^y eiψ =2 and a wavevector ki ¼ h1 ^x þ β ^z and a totally reﬂected ﬁeld of an amplitude E r ¼ E i eiφTE and a wavevector kr ¼ h1 ^x þ β ^z so that the total space- and time-varying electric ﬁeld is Eðr; tÞ ¼ E i exp ðiki r iωt Þþ E r exp ðikr r iωt Þ ¼ ^y E y ðxÞ exp ðiβz iωtÞ. For the TM mode, it is only necessary to ﬁnd Hy ; then the other two nonvanishing components E x and E z can be found by using (3.85) and (3.86), respectively. The boundary conditions require that Hy , E x , and E z be continuous at the interface, which dictates that Hy and ϵ 1 ∂Hy =∂x, i.e., n2 ∂Hy =∂x, be both continuous at x ¼ 0. The ﬁeld proﬁle satisfying these boundary conditions is Hy ðxÞ ¼ x > 0, cos ðh1 x ψ Þ, cos ψ exp ðγ2 xÞ, x < 0, (3.109) n21 γ2 1 ¼ φTM : 2 2 n2 h1 (3.110) where ψ ¼ tan1 Again, the mode ﬁeld proﬁle Hy given in (3.109) is not normalized because it extends to inﬁnity in the positive x direction. For x > 0, Hy in (3.109) is the superposition of an incident ﬁeld of an amplitude Hi ¼ ^y eiψ =2 and a wavevector ki ¼ h1 ^x þ β ^z and a totally reﬂected ﬁeld of an amplitude Hr ¼ Hi eiφTM and a wavevector kr ¼ h1 ^x þ β ^z so that the total space- and time-varying magnetic ﬁeld is Hðr; tÞ ¼ Hi exp ðiki r iωt Þ þ Hr exp ðikr r iωt Þ ¼ ^y Hy ðxÞ exp ðiβz iωt Þ. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.4 Interface Modes 101 EXAMPLE 3.10 A glass plate has a refractive index of 1.5 at the λ ¼ 1 μm wavelength. Find the parameters of the radiation modes at the air–glass interface corresponding to internal reﬂection at the two different incident angles of 45 and 75 , respectively. What is the penetration depth of the evanescent tail into the air if a radiation mode is found to be a one-sided radiation mode at a particular incident angle? What are the phase shifts on reﬂection at the interface for TE and TM waves, respectively? Solution: In this problem, n1 ¼ 1:5 and n2 ¼ 1 so that the critical angle of the interface is θc ¼ sin1 ð1=1:5Þ ¼ 41:8 . Because θi > θc for both incident angles, the radiation modes for both cases are one-sided radiation modes. At λ ¼ 1 μm, k1 ¼ 2πn1 ¼ 9:42 106 m1 λ and k2 ¼ 2πn2 ¼ 6:28 106 m1 : λ For θi ¼ 45 > θc , the radiation mode is a one-sided radiation mode; the parameters of this radiation mode are β ¼ k1 sin θi ¼ 6:66 106 m1 , qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 1 h1 ¼ k 1 cos θi ¼ 6:66 10 m , γ2 ¼ β2 k22 ¼ 2:22 106 m1 : The penetration depth of the evanescent tail into the air is γ1 2 ¼ 451 nm. The phase shifts on reﬂection at the interface for TE and TM waves are φTE ¼ 2 tan1 γ2 n2 γ ¼ 0:64 rad ¼ 0:20π, φTM ¼ 2 tan1 21 2 ¼ 1:29 rad ¼ 0:41π: h1 n2 h1 For θi ¼ 75 > θc , the radiation mode is a one-sided radiation mode; the parameters of this radiation mode are β ¼ k1 sin θi ¼ 9:10 106 m1 , qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 1 h1 ¼ k1 cos θi ¼ 2:44 10 m , γ2 ¼ β2 k22 ¼ 6:59 106 m1 : The penetration depth of the evanescent tail into the air is γ1 2 ¼ 152 nm. The phase shifts on reﬂection at the interface for TE and TM waves are φTE ¼ 2 tan1 γ2 n2 γ ¼ 2:43 rad ¼ 0:77π, φTM ¼ 2 tan1 21 2 ¼ 2:82 rad ¼ 0:90π: h1 n2 h1 Two-Sided Radiation Modes: k1 > k2 > β This is the case when partial reﬂection accompanied by refracted transmission occurs for an incident angle of θi < θc . In this case, k i, z ¼ k1 sin θi ¼ β and kr, z ¼ k1 sin θr ¼ β so that the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 102 Optical Wave Propagation condition k 2i, x þ k2i, z ¼ k2r, x þ k2r, z ¼ k 21 requires that the transverse x components of ki and kr have the same real value: h1 ¼ k i, x ¼ kr, x ¼ k 1 cos θi . Meanwhile, because k 2 > β, a real solution of θt exists for k t, z ¼ k 2 sin θt ¼ β so that the transverse x component of kt also has a real value: h2 ¼ kt, x ¼ k2 cos θt . Therefore, positive real parameters h1 and h2 can be deﬁned for the transverse ﬁeld proﬁles in media 1 and 2, respectively, as h21 ¼ k21 β2 , h22 ¼ k 22 β2 : (3.111) Note that h1 > h2 because k1 > k 2 . Using the two parameters h1 and h2 , the reﬂection coefﬁcients found in (3.91) and (3.95) for the TE and TM polarizations can be respectively expressed as n22 h1 n21 h2 : (3.112) n22 h1 þ n21 h2 2 As expected for partial reﬂection, Rs ¼ jr s j2 6¼ 1 and Rp ¼ r p 6¼ 1. Because h1 > h2 , there is no phase shift in reﬂection for the TE polarization: φTE ¼ φs ¼ 0. The phase shift in reﬂection for the TM polarization ﬂips at the Brewster angle: φTM ¼ φp ¼ π for θi < θB , but φTM ¼ φp ¼ 0 for θi > θB . (See Problem 3.4.1.) The real parameters h1 ¼ ki, x ¼ kr, x and h2 ¼ kt, x characterize a two-sided radiation mode ﬁeld proﬁle that has sinusoidal variations extending to inﬁnity in both positive and negative x directions, as illustrated in Fig. 3.14. This ﬁeld pattern is the superposition of the incident, reﬂected, and transmitted ﬁelds on each side from two incident waves, one from medium 1 and the other from medium 2, as also illustrated in Fig. 3.14 and discussed below. For the TE mode, the E y ﬁeld proﬁle satisfying the boundary conditions that E y and ∂E y =∂x are continuous at x ¼ 0 is r TE ¼ r s ¼ E y ðxÞ ¼ h1 h2 , h1 þ h2 r TM ¼ r p ¼ cos ψ 2 cos ðh1 x ψ 1 Þ, cos ψ 1 cos ðh2 x ψ 2 Þ, x > 0, x < 0, (3.113) where the two phase factors ψ 1 and ψ 2 are related by h1 tan ψ 1 ¼ h2 tan ψ 2 : (3.114) Figure 3.14 Partial reﬂection and transmission, and transverse ﬁeld proﬁle of two-sided radiation mode. The fact that θr ¼ θi and θt > θi for incidence from medium 1 is shown. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.4 Interface Modes 103 The nonvanishing magnetic ﬁeld components Hx and Hz of the TE mode are found from E y by using (3.83) and (3.84), respectively. The mode ﬁeld E y in (3.113) is not normalized because it extends to inﬁnity in both positive and negative x directions. For all x, E y in (3.113) is the superposition of the incident, reﬂected, and transmitted ﬁelds resulting from two incident waves: one from medium 1 that has a ﬁeld amplitude of E i1 ¼ ^y cos ψ 2 eiψ 1 =2 and a wavevector of ki1 ¼ h1 ^x þ β^z , and the other from medium 2 that has E i2 ¼ ^y cos ψ 1 eiψ 2 =2 and ki2 ¼ h2 ^x þ β^z . Note that (3.114) eliminates one free phase parameter so that the phase relation between the two incident waves in the composition of the TE mode ﬁeld is determined. For the TM mode, the Hy ﬁeld proﬁle satisfying the boundary conditions that Hy and 2 n ∂Hy =∂x are continuous at x ¼ 0 is Hy ðxÞ ¼ cos ψ 2 cos ðh1 x ψ 1 Þ, cos ψ 1 cos ðh2 x ψ 2 Þ, x > 0, x < 0, (3.115) where the two phase factors ψ 1 and ψ 2 are related by h1 h2 tan ψ 1 ¼ 2 tan ψ 2 : 2 n1 n2 (3.116) The nonvanishing electric ﬁeld components E x and E z of the TM mode are found from Hy by using (3.85) and (3.86), respectively. The mode ﬁeld Hy in (3.115) is not normalized because it extends to inﬁnity in both positive and negative x directions. For all x, Hy in (3.115) is the superposition of the incident, reﬂected, and transmitted ﬁelds resulting from two incident waves: one from medium 1 that has a ﬁeld amplitude of Hi1 ¼ ^y cos ψ 2 eiψ1 =2 and a wavevector of ki1 ¼ h1 ^x þ β^z , and the other from medium 2 that has Hi2 ¼ ^y cos ψ 1 eiψ 2 =2 and ki2 ¼ h2 ^x þ β^z . The relation in (3.116) eliminates one free phase parameter so that the phase relation between the two incident waves in the composition of the TM mode ﬁeld is determined. EXAMPLE 3.11 The glass plate with a refractive index of 1.5 at the λ ¼ 1 μm wavelength given in Example 3.10 is now immersed in water, which has a refractive index of 1.33. Find the parameters of the radiation modes at the water–glass interface corresponding to internal reﬂection at the two different incident angles of 45 and 75 , respectively. What is the penetration depth of the evanescent tail into the water if a radiation mode is found to be a one-sided radiation mode at a particular incident angle? What are the phase shifts on reﬂection at the interface for TE and TM waves, respectively? Solution: In this problem, n1 ¼ 1:5 and n2 ¼ 1:33 so that the critical angle of the interface is θc ¼ sin1 ð1:33=1:5Þ ¼ 62:5 and the Brewster angle for internal reﬂection is θB ¼ tan1 ð1:33=1:5Þ ¼ 41:6 < θc . At λ ¼ 1 μm, k1 ¼ 2πn1 2πn2 ¼ 9:42 106 m1 and k2 ¼ ¼ 6:28 106 m1 : λ λ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 104 Optical Wave Propagation For θi ¼ 45 < θc , the radiation mode is a two-sided radiation mode; the parameters of this radiation mode are β ¼ k 1 sin θi ¼ 6:66 106 m1 , qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 1 h1 ¼ k1 cos θi ¼ 6:66 10 m , h2 ¼ k 22 β2 ¼ 5:05 106 m1 : Because this mode is a two-sided radiation mode, it extends to inﬁnity on both the glass and water sides. Because θi ¼ 45 > θB , the phase shifts of the internal reﬂection at the interface for TE and TM waves are φTE ¼ 0, φTM ¼ 0: For θi ¼ 75 > θc , the radiation mode is a one-sided radiation mode; the parameters of this radiation mode are β ¼ k 1 sin θi ¼ 9:10 106 m1 , qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 1 h1 ¼ k 1 cos θi ¼ 2:44 10 m , γ2 ¼ β2 k22 ¼ 3:59 106 m1 : The penetration depth of the evanescent tail into the water is γ1 2 ¼ 278 nm. The phase shifts on reﬂection at the interface for TE and TM waves are φTE ¼ 2 tan1 γ2 ¼ 1:95 rad ¼ 0:62π, h1 φTM ¼ 2 tan1 n21 γ2 ¼ 2:16 rad ¼ 0:69π: n22 h1 3.4.3 Surface Plasmon Mode In the above, we have seen that an interface between two isotropic dielectric media supports only radiation modes. At most, it supports a one-sided radiation mode that has a localized transverse ﬁeld distribution on only one side of the interface. No localized, guided surface mode is supported by this type of interface. Guided surface modes do exist in certain types of interface, such as that between an isotropic dielectric medium and an anisotropic dielectric medium or that between an isotropic dielectric medium and a plasma medium. We consider the interface between an isotropic dielectric medium of a permittivity ϵ 1 and an isotropic plasma medium of a permittivity ϵ 2 , as shown in Fig. 3.15. For simplicity, we take the limit that ωτ 1 so that the permittivity of the plasma medium is that given in (2.49): ! ω2p ϵ2 ¼ ϵb 1 2 , (3.117) ω where ϵ b ¼ ϵ bound is the background permittivity due to bound electrons and ωp is the plasma frequency deﬁned in (2.46). The plasma medium can be any medium that has free charge carriers, such as a doped semiconductor or a metal. For simplicity, we neglect the absorption Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.4 Interface Modes 105 Figure 3.15 Surface plasmon mode at the interface between a dielectric medium of ϵ 1 and a plasma medium of ϵ 2 . loss in the dielectric medium and that due to bound electrons in the plasma medium so that both ϵ 1 and ϵ b are real and positive: ϵ 1 > 0 and ϵ b > 0. However, as discussed in Section 2.4 and seen from (3.117), at any frequency below the plasma frequency, the permittivity of the plasma medium is negative: ϵ 2 < 0 for ω < ωp . The opposite signs of ϵ 1 and ϵ 2 in this situation create the possibility of a guided surface plasmon mode that is supported by the interface. The surface plasmon mode between a dielectric medium and a plasma medium is a TM mode. To be guided by the interface, it has to be transversely localized near the interface. Thus, it has to decay exponentially away from the interface in both positive and negative x directions with characteristic parameters γ1 and γ2 , respectively: γ21 ¼ β2 k21 , γ22 ¼ β2 k 22 : (3.118) Because the surface plasmon mode is a TM mode, we ﬁnd Hy with the boundary conditions that Hy and ϵ 1 ∂Hy =∂x are continuous at the interface located at x ¼ 0. The guided TM mode can be normalized using (3.22). The normalized ﬁeld proﬁle of the surface plasmon mode that satisﬁes the boundary condition for the continuity of Hy is ^ y ðxÞ ¼ C exp ðγ1 xÞ, H exp ðγ2 xÞ, x > 0, x < 0, (3.119) where ω C¼ β 1=2 γ1 γ2 ϵ 1 ϵ 2 γ1 ϵ 1 þ γ2 ϵ 2 1=2 : (3.120) The boundary condition for the continuity of ϵ 1 ∂Hy =∂x at x ¼ 0 yields the eigenvalue equation: γ1 γ2 þ ¼ 0: ϵ1 ϵ2 (3.121) ^y The nonvanishing mode electric ﬁeld components are E^ x and E^ z , which can be found from H by using (3.85) and (3.86), respectively. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 106 Optical Wave Propagation Figure 3.16 Dispersion curve for surface plasmon mode showing (a) propagation constant as a function of frequency and (b) frequency as a function of propagation constant. At a low frequency, the surface plasmon propagation constant β approaches the propagation constant k 1 in the dielectric medium. As the frequency increases towards ωsp , β becomes much larger than k 1 and approaches inﬁnity. The example in this ﬁgure ispplotted ﬃﬃﬃ with ϵ 1 ¼ ϵ 0 and ϵ b ¼ ϵ 0 for the surface of a perfect metal in free space. In this special case, ωsp ¼ ωp = 2: Because γ1 > 0, γ2 > 0, and ϵ 1 > 0, it is necessary that ϵ 2 < 0 for the eigenvalue equation to have a solution. Using the relations in (3.118), with k21 ¼ ω2 μ0 ϵ 1 and k22 ¼ ω2 μ0 ϵ 2 , the eigenvalue equation (3.121) can be solved to ﬁnd μ ϵ1ϵ2 β¼ω 0 ϵ1 þ ϵ2 1=2 , μ0 ϵ 21 γ1 ¼ ω ϵ1 þ ϵ2 1=2 , γ2 ¼ ω μ0 ϵ 22 ϵ1 þ ϵ2 1=2 : (3.122) The condition for γ1 , γ2 , and β in (3.122) to have real and positive solutions is that ϵ 2 < 0 and ϵ 1 þ ϵ 2 < 0 ) ϵ 2 < ϵ 1 < 0: This condition limits the surface plasmon mode to the frequency range: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ϵb ω < ωsp ¼ ωp , ϵ1 þ ϵb (3.123) (3.124) where ωsp is known as the surface plasma frequency. Figure 3.16 shows the relation between β and ω for the surface plasmon mode. At a low pﬃﬃﬃﬃﬃﬃﬃﬃﬃ frequency such that ω ωsp , β ω μ0 ϵ 1 ¼ k1 so that the surface plasmon propagation constant β approaches the propagation constant k1 in the dielectric medium. As the frequency increases, β increases and gradually becomes much larger than k 1 , β k1 , approaching inﬁnity as the frequency approaches ωsp . Note that ωsp < ωp , as is also shown in Fig. 3.16. The cutoff frequency and cutoff wavelength of a surface plasmon mode are νsp ¼ ωsp =2π and λsp ¼ c=νsp ¼ 2πc=ωsp , respectively. The surface plasmon mode can be excited only by a TM-polarized wave of ν < νsp and λ > λsp . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.4 Interface Modes 107 EXAMPLE 3.12 A surface plasmon mode can exist at the interface between a silver plate and free space. The plasma frequency of Ag found in Example 2.4 is ωp ¼ 1:36 1016 rad s1 . What is the surface plasma frequency of this interface? What are the cutoff frequency and cutoff wavelength of the surface plasmon mode? Does the surface plasmon mode exist at the λ ¼ 500 nm wavelength? If it exists, ﬁnd its propagation constant and characteristic parameters. Find the penetration depths of the mode into the free space and into the silver to ﬁnd its conﬁnement at the interface. Solution: At the interface between free space and Ag, ϵ 1 ¼ ϵ 0 for free space and ϵ 2 is that of Ag. For Ag, ϵ b ¼ ϵ 0 so that ! ! ! ω2p ω2p λ2 ϵ2 ¼ ϵb 1 2 ¼ ϵ0 1 2 ¼ ϵ0 1 2 : ω ω λp Given ωp ¼ 1:36 1016 rad s1 for Ag, the surface plasma frequency is rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωp ϵb ϵ0 ωp ¼ ωp ¼ pﬃﬃﬃ ¼ 9:62 1015 rad s1 : ωsp ¼ ϵ0 þ ϵb ϵ0 þ ϵ0 2 Therefore, the cutoff frequency and cutoff wavelength are, respectively, ωsp c ¼ 196 nm: ¼ 1:53 1015 Hz ¼ 1:53 PHz, λsp ¼ νsp ¼ 2π νsp The surface plasmon mode exists at the λ ¼ 500 nm wavelength because λ > λsp . For ωp ¼ 1:36 1016 rad s1 , we ﬁnd λp ¼ 138 nm. Therefore, for λ ¼ 500 nm, ! λ2 5002 ¼ 12:13ϵ 0 : ϵ2 ¼ ϵ0 1 2 ¼ ϵ0 1 1382 λp Then, by using (3.122), we ﬁnd μ ϵ1ϵ2 β¼ω 0 ϵ1 þ ϵ2 1=2 2π ðϵ 1 =ϵ 0 Þðϵ 2 =ϵ 0 Þ ¼ λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0 1=2 2π 12:13 ¼ 9 1 12:13 500 10 ¼ 1:31 107 m1 , " #1=2 1=2 μ0 ϵ 21 2π ðϵ 1 =ϵ 0 Þ2 2π 1 ¼ ¼ γ1 ¼ ω 9 ϵ1 þ ϵ2 λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0 1 12:13 500 10 1=2 1=2 m1 m1 ¼ 3:77 106 m1 , " #1=2 2π ðϵ 2 =ϵ 0 Þ2 2π 12:132 ¼ ¼ λ ϵ 1 =ϵ 0 þ ϵ 2 =ϵ 0 500 109 1 12:13 ¼ 4:57 107 m1 : μ0 ϵ 22 γ2 ¼ ω ϵ1 þ ϵ2 1=2 1=2 m1 1 The penetration depths are γ1 1 ¼ 265 nm into the free space and γ2 ¼ 22 nm into the silver. 1 Therefore, the conﬁnement of the surface plasmon mode at the interface is γ1 1 þ γ2 ¼ 287 nm. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 108 Optical Wave Propagation 3.5 WAVEGUIDE MODES .............................................................................................................. The basic structure of a dielectric optical waveguide consists of a longitudinally extended high-permittivity, thus high-index, optical medium, called the core, which is transversely surrounded by low-permittivity, thus low-index, media, called the cladding. We consider a straight waveguide whose longitudinal direction is taken to be the z direction, as shown in Figs. 3.1(c) and (d). In a planar waveguide, which has optical conﬁnement in only one transverse dimension, the core is sandwiched between cladding layers in only one dimension, designated the x dimension, with a permittivity proﬁle of ϵ ðxÞ, thus an index proﬁle of nðxÞ, as shown in Fig. 3.1(c). The core of a planar waveguide is also called the ﬁlm, while the upper and lower cladding layers are called the cover and the substrate, respectively. Optical conﬁnement is provided only in the x dimension by the planar waveguide. A waveguide in which the index proﬁle has abrupt changes between the core and the cladding is called a step-index waveguide, while one in which the index proﬁle varies gradually is called a graded-index waveguide. Figure 3.17 shows examples of step-index and graded-index planar waveguides. In a nonplanar waveguide of two-dimensional transverse optical conﬁnement, the core is surrounded by the cladding in all transverse directions, with ϵ ðx; yÞ and nðx; yÞ being functions of both x and y coordinates. A nonplanar waveguide can also have a step-index or graded-index proﬁle. As discussed in Section 3.1, a planar dielectric waveguide supports only TE and TM modes, whereas a nonplanar dielectric waveguide supports TE, TM, and hybrid modes. No TEM modes exist in dielectric waveguides. To get a general idea of the modes of a dielectric waveguide, it is instructive to consider the qualitative behavior of an optical wave in the asymmetric planar step-index waveguide shown in Fig. 3.17(a), where n1 > n2 > n3 . For an optical wave of an angular frequency ω and a free-space wavelength λ, the media in the three different regions of the waveguide deﬁne three propagation constants: k1 ¼ n1 ω 2πn1 , ¼ λ c k2 ¼ n2 ω 2πn2 , ¼ λ c k3 ¼ n3 ω 2πn3 , ¼ λ c (3.125) where k1 > k 2 > k3 . Figure 3.17 Index proﬁles of (a) a step-index planar waveguide and (b) a graded-index planar waveguide. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.5 Waveguide Modes 109 An intuitive picture of waveguide modes can be obtained from studying ray optics by considering the path of an optical ray, or a plane optical wave, in the waveguide, as shown in the central column of Fig. 3.18. There are two critical angles associated with the internal reﬂections at the lower and upper interfaces: θc2 ¼ sin1 n2 n1 and θc3 ¼ sin1 n3 , n1 (3.126) respectively, where θc2 > θc3 because n2 > n3 . The characteristics of the reﬂection and refraction of the ray at the interfaces depend on the incident angle θ and the polarization of the wave. Guided Modes For a ray that has an incident angle of θ > θc2 > θc3 at the interfaces of the waveguide, the wave inside the core is totally reﬂected at both interfaces and is trapped by the core, resulting in a guided mode when the resonance condition described below is satisﬁed. As the wave is reﬂected back and forth between the two interfaces, it interferes with itself. A guided mode can exist only when a transverse resonance condition is satisﬁed so that the repeatedly reﬂected wave constructively interferes with itself. In the core region, the x component of the wavevector is h1 ¼ k1 cos θ, and the z component is β ¼ k 1 sin θ. The phase shift caused by a round-trip transverse passage of the ﬁeld in the core that has a thickness of d is 2h1 d ¼ 2k1 dcos θ. In addition, the internal reﬂection at the lower interface causes a localized phase shift of φ2 as given in (3.106), and that at the upper interface causes a phase shift of φ3 , which can be found by replacing γ2 with γ3 in (3.106). The phase shifts φ2 and φ3 are functions of the incident angle θ; for a given θi ¼ θ > θc2 > θc3 , each of them has different values for TE and TM waves. The transverse resonance condition for constructive interference is that the total phase shift in a round-trip transverse passage is 2h1 d þ φ2 ðθÞ þ φ3 ðθÞ ¼ 2k1 d cos θ þ φ2 ðθÞ þ φ3 ðθÞ ¼ 2mπ, (3.127) where m is an integer. Because m takes only integral values, only certain discrete values of θ satisfy (3.127). This condition results in discrete values of the propagation constant βm for guided modes identiﬁed by the mode number m. From (3.106), we ﬁnd that π < φ2 < 0 and π < φ3 < 0 so that 2π < φ2 þ φ3 < 0. Therefore, the smallest value of m for (3.127) to have a solution is m ¼ 0; no negative values of m are allowed. The guided mode with m ¼ 0 is the fundamental mode, and those with m 6¼ 0 are high-order modes. Though the critical angles, θc2 and θc3 , do not depend on the polarization of the wave, the phase shifts, φ2 ðθÞ and φ3 ðθÞ, caused by internal reﬂection at a given angle θ depend on the polarization, as seen in (3.106). Therefore, (3.127) have different solutions for TE and TM waves, resulting in different values of βm and different mode characteristics for TE and TM modes of a given mode number m. Because φTM < φTE < 0 as seen from (3.106), the TM solution of (3.127) yields θTE > θTM for a given value of m; thus, βTE m > βm . For a given polarization, the solution of (3.127) yields a smaller value of θ and a correspondingly smaller value of βm for a larger value of m. Therefore, among guided modes of different Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 110 Optical Wave Propagation Figure 3.18 Modes of an asymmetric planar step-index waveguide where n1 > n2 > n3 . The range of the propagation constants, the zig-zag ray pictures, and the ﬁeld patterns are shown correspondingly for (a) the guided fundamental mode, (b) the guided ﬁrst high-order mode, (c) a substrate radiation mode for β ¼ 1:3k3 , and (d) a substrate–cover radiation mode for β ¼ 0:3k3 . The waveguide structure is chosen so that it supports only two guided modes. The mode ﬁeld proﬁles are calculated mode ﬁeld distributions that are normalized to their respective peak values. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.5 Waveguide Modes 111 orders but of the same polarization that are supported by a waveguide, the fundamental mode has the largest propagation constant β0 ; that is, β0 > β1 > . . . for a given polarization, as shown in Figs. 3.18(a) and (b). Substrate Radiation Modes When θc2 > θ > θc3 , total reﬂection occurs only at the upper interface and not at the lower interface. As a result, an optical wave incident from either the core or the substrate is refracted and transmitted at the lower interface. This wave is not conﬁned to the core, but is transversely extended to inﬁnity in the substrate. It is called a substrate radiation mode. In this case, the angle θ is not dictated by a resonance condition like (3.127) but can take any value in the range of θc2 > θ > θc3 . As a result, the allowed values of β form a continuum between k2 and k3 such that the modes are not discrete. The characteristics of a substrate radiation mode are illustrated in Fig. 3.18(c). Substrate–Cover Radiation Modes When θc2 > θc3 > θ, no total reﬂection occurs at either interface. An optical wave incident from either side is refracted and transmitted at both interfaces; thus, it transversely extends to inﬁnity on both sides of the waveguide, resulting in a substrate–cover radiation mode. These modes are not discrete; their values of β form a continuum between k 3 and 0. The characteristics of a substrate–cover radiation mode are illustrated in Fig. 3.18(d). In addition to the three types of modes discussed above, there are also evanescent radiation modes, which have purely imaginary values of β that are not discrete. Their ﬁelds decay exponentially along the z direction. Because the dielectric waveguide considered here is lossless and does not absorb energy, the energy of an evanescent mode transversely radiates away from the waveguide. A lossless waveguide cannot generate energy, either. Therefore, evanescent modes do not exist in a perfect, longitudinally inﬁnite waveguide. They exist at a longitudinal junction or imperfection of a waveguide, as well as at the terminals of a realistic waveguide that has a ﬁnite length. By comparison, a substrate radiation mode or a substrate–cover radiation mode has a real β; therefore, its energy does not diminish as it propagates. Like a plane wave, its power ﬂows in the z direction, though its ﬁeld transversely extends to inﬁnity because the power ﬂowing away from the center of the waveguide in the transverse direction is equal to that ﬂowing toward the center. The approach of ray optics used above gives an intuitive picture of the waveguide modes and their key characteristics. Nevertheless, this approach has many limitations. In more sophisticated waveguide geometries such as that of a circular ﬁber, the idea of using the resonance condition based on total internal reﬂection to ﬁnd the allowed values of β for the guided modes does not necessarily yield correct results. For a complete description of the waveguide ﬁelds, rigorous electromagnetic analyses as illustrated below are required. 3.5.1 Step-Index Planar Waveguides A step-index planar waveguide is also called a slab waveguide. The general structure and parameters of a three-layer slab waveguide are shown in Fig. 3.17(a), which has a core Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 112 Optical Wave Propagation thickness of d and a step-index proﬁle of n1 > n2 > n3 . In the above, the approach of ray optics was used to illustrate an intuitive picture and some basic mode characteristics of a slab waveguide. Further understanding requires quantitative analyses of the mode ﬁelds discussed below. Normalized Waveguide Parameters The mode properties of a waveguide are commonly characterized in terms of a few dimensionless normalized waveguide parameters. The normalized frequency and waveguide thickness, also known as the V number, of a step-index planar waveguide is deﬁned as V¼ 2π d λ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ω n21 n22 ¼ d n21 n22 , c (3.128) where d is the thickness of the waveguide core. The propagation constant β can be represented by the following normalized guide index, β2 k22 n2β n22 b¼ 2 ¼ , k1 k22 n21 n22 (3.129) where nβ ¼ cβ=ω ¼ βλ=2π is the effective refractive index of the waveguide mode that has a propagation constant of β. The measure of the asymmetry of the waveguide is represented by an asymmetry factor a, which depends on the polarization of the mode under consideration: aE ¼ n22 n23 for TE modes, n21 n22 aM ¼ n41 n22 n23 for TM modes: n43 n21 n22 (3.130) Note that aM > aE for a given asymmetric structure. For a symmetric waveguide, aM ¼ aE ¼ 0 because n3 ¼ n2 . Mode Parameters For a guided mode, positive real parameters h1 , γ2 , and γ3 exist such that h21 ¼ k21 β2 , γ22 ¼ β2 k22 , γ23 ¼ β2 k23 (3.131) because k1 > β > k2 > k3 . From the ray-optics approach discussed above and from (3.131), the transverse component of the wavevector in the core region of a refractive index n1 is h1 ¼ k 1 cos θ. For a guided mode, the transverse components of the wavevectors in the 1=2 1=2 substrate and cover regions are h2 ¼ k22 β2 ¼ iγ2 and h3 ¼ k 23 β2 ¼ iγ3 , respectively, which are purely imaginary because β > k 2 > k3 . Thus, the ﬁeld of the guided mode has to exponentially decay in the transverse direction with decay constants γ2 and γ3 in the substrate and cover regions, respectively. For a substrate radiation mode, h2 can be chosen to be real and positive because k 1 > k 2 > β > k 3 ; thus, (3.131) is replaced by h21 ¼ k 21 β2 , h22 ¼ k22 β2 , γ23 ¼ β2 k23 : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.132) 3.5 Waveguide Modes 113 For a substrate–cover radiation mode, both h2 and h3 are real and positive because k1 > k2 > k 3 > β; thus, (3.131) is replaced by h21 ¼ k21 β2 , h22 ¼ k22 β2 , h23 ¼ k23 β2 : (3.133) The transverse ﬁeld pattern of a mode is characterized by the transverse parameters h1 , γ2 (or h2 ), and γ3 (or h3 ). Because k1 , k 2 , and k3 are speciﬁed parameters of a given slab waveguide, the only parameter that has to be determined for a particular waveguide mode is the longitudinal propagation constant β. Once the value of β is found, all parameters that characterize the transverse ﬁeld pattern are completely determined. Therefore, a waveguide mode is completely speciﬁed by its β. Alternatively, because of the deﬁnite relations between β and the transverse parameters, a mode is completely speciﬁed, and the value of its β determined, if any one of the transverse parameters is known. In most cases, rather than directly solving for β, it is more convenient to solve an eigenvalue equation for h1 , as seen below. EXAMPLE 3.13 A step-index planar waveguide of the structure shown in Fig. 3.17(a) is made of glass of slightly different compositions for the core and the substrate so that n1 ¼ 1:54 for the core and n2 ¼ 1:47 for the substrate. The cover is simply air so that n3 ¼ 1:00. The exact values of the parameters for the guided modes depend on the core thickness, but the propagation constant of any guided mode at a given wavelength is bounded within a range irrespective of the core thickness. In what range can the propagation constant of a guided mode, if it exists, be found at the λ ¼ 1 μm wavelength? For what wavelengths can a guided mode be found to have a propagation constant of β ¼ 1:5 107 m1 ? What will happen to the answers if the structure is immersed in water so that n3 ¼ 1:33? What will happen if it is immersed in benzene so that n3 ¼ 1:50? What will happen if it is immersed in CS2 so that n3 ¼ 1:63? Solution: With n1 ¼ 1:54, n2 ¼ 1:47, and n3 ¼ 1:00, we have k 1 > k2 > k3 so that the propagation constant β of any guided mode, if it exists, has to be in the range of k1 > β > k2 . At λ ¼ 1 μm, we ﬁnd that 2πn1 2πn2 >β> λ λ ) 9:68 106 m1 > β > 9:24 106 m1 : The wavelength of a guided mode that has a propagation constant of β ¼ 1:5 107 m1 falls in the range: 2πn1 2πn2 >λ> β β ) 645:1 nm > λ > 615:8 nm: If the structure is immersed in water so that n3 ¼ 1:33, we still ﬁnd that k1 > k 2 > k3 because n1 > n2 > n3 . Therefore, there are no changes in the answers obtained above. If the structure is immersed in benzene so that n3 ¼ 1:50, then k 1 > k3 > k2 because n1 > n3 > n2 . Then, at λ ¼ 1 μm, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 114 Optical Wave Propagation 2πn1 2πn3 >β> λ λ ) 9:68 106 m1 > β > 9:43 106 m1 : And the wavelength of a guided mode that has a propagation constant of β ¼ 1:5 107 m1 falls in the range: 2πn1 2πn3 >λ> β β ) 645:1 nm > λ > 632:8 nm: If the structure is immersed in CS2 so that n3 ¼ 1:63, then k 3 > k 1 > k2 because n3 > n1 > n2 . In this situation, the structure does not have any guided mode because the core has a lower refractive index than the cover. Only cover radiation modes and substrate–cover radiation modes can be found for this structure. Guided TE Modes For a TE mode, it is only necessary to ﬁnd E y ; then the other two nonvanishing ﬁeld components Hx and Hz can be found by using (3.83) and (3.84), respectively. The boundary conditions require that E y , Hx , and Hz be continuous at the interfaces at x ¼ d=2 between layers of different refractive indices. From (3.83) and (3.84), it can be seen that these boundary conditions are equivalent to requiring E y and ∂E y =∂x be continuous at these interfaces. For a guided mode, we know that the transverse ﬁeld patterns in the core, substrate, and cover regions are respectively characterized by the transverse ﬁeld parameters h1 , γ2 , and γ3 , given in (3.131). A guided TE mode ﬁeld distribution that satisﬁes the boundary conditions for the continuity of E y at x ¼ d=2 has the form: 8 < cos ðh1 d=2 ψ Þ exp ½γ3 ðd=2 xÞ , x > d=2, (3.134) d=2 < x < d=2, E^ y ¼ CTE cos ðh1 x ψ Þ, : cos ðh1 d=2 þ ψ Þ exp ½γ3 ðd=2 þ xÞ , x < d=2: Application of the other two boundary conditions for the continuity of ∂E y =∂x at x ¼ yields two eigenvalue equations: d=2 h1 ðγ2 þ γ3 Þ h21 γ2 γ3 (3.135) h1 ðγ2 γ3 Þ : h21 þ γ2 γ3 (3.136) tan h1 d ¼ and tan 2ψ ¼ A guided TE mode can be normalized using the orthonormality relation in (3.20) for rﬃﬃﬃﬃﬃﬃﬃﬃ ωμ0 , (3.137) C TE ¼ βd E where dE ¼ d þ 1 1 þ γ2 γ3 is the effective waveguide thickness for a guided TE mode. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.138) 3.5 Waveguide Modes 115 Guided TM Modes For a TM mode, it is only necessary to ﬁnd Hy ; then the other two nonvanishing ﬁeld components E x and E z can be found by using (3.85) and (3.86), respectively. The boundary conditions require that Hy , ϵE x , and E z be continuous at the interfaces at x ¼ d=2 between layers of different refractive indices. From (3.85) and (3.86), it can be seen that these boundary conditions are equivalent to requiring Hy and ϵ 1 ∂Hy =∂x, or n2 ∂Hy =∂x, be continuous at these interfaces. For a guided mode, we know that the transverse ﬁeld patterns in the core, substrate, and cover regions are respectively characterized by the transverse ﬁeld parameters h1 , γ2 , and γ3 , given in (3.131). A guided TM mode ﬁeld distribution that satisﬁes the boundary conditions for the continuity of Hy at x ¼ d=2 has the form: 8 < cos ðh1 d=2 ψ Þ exp ½γ3 ðd=2 xÞ , x > d=2, ^ y ¼ C TM cos ðh1 x ψ Þ, d=2 < x < d=2, H : cos ðh1 d=2 þ ψ Þ exp ½γ3 ðd=2 þ xÞ , x < d=2: (3.139) Application of the other two boundary conditions for the continuity of n2 ∂Hy =∂x at x ¼ d=2 yields two eigenvalue equations: h1 =n21 γ2 =n22 þ γ3 =n23 (3.140) tan h1 d ¼ 2 h1 =n21 γ2 γ3 =n22 n23 and h1 =n21 γ2 =n22 γ3 =n23 tan 2ψ ¼ : 2 h1 =n21 þ γ2 γ3 =n22 n23 (3.141) A guided TM mode can be normalized using the orthonormality relation in (3.22) for CTM sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωμ0 n21 , ¼ βd M (3.142) where the effective waveguide thickness for a guided TM mode is dM ¼ d þ 1 1 β2 β2 þ , where q2 ¼ 2 þ 2 1 and γ2 q2 γ3 q3 k1 k2 q3 ¼ β2 β2 þ 1: k 21 k23 (3.143) Modal Dispersion Guided modes have discrete allowed values of β. They are determined by the allowed values of h1 because β and h1 are directly related to each other through (3.131). Because γ2 and γ3 are uniquely determined by β through (3.131), they are also uniquely determined by h1 : γ22 d 2 ¼ β2 d 2 k22 d 2 ¼ V 2 h21 d 2 , (3.144) γ23 d 2 ¼ β2 d 2 k23 d 2 ¼ ð1 þ aE ÞV 2 h21 d 2 : (3.145) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 116 Optical Wave Propagation Figure 3.19 Allowed values of normalized guide index b as a function of the V number and the asymmetry factor aE for the ﬁrst three guided TE modes. The cutoff value V c for a mode is the value of V at the intersection of its dispersion curve with the horizontal axis. Figure 3.20 Propagation constants of guided modes as functions of optical frequency for a given step-index dielectric waveguide. Therefore, there is only one independent variable h1 in the eigenvalue equations. The solutions of (3.135) yield the allowed parameters for guided TE modes, while those of (3.140) yield the parameters for guided TM modes. A transcendental equation such as (3.135) or (3.140) is usually solved numerically, or graphically by plotting its left- and right-hand sides as a function of h1 d while using (3.144) and (3.145) to replace γ2 and γ3 by expressions in terms of h1 d. The solutions yield the allowed values of β, or the normalized guide index b, as a function of the parameters a and V. The results for the ﬁrst three guided TE modes are shown in Fig. 3.19. For a given waveguide, a guided TE mode has a larger propagation constant than the TM mode of the same order: TM βTE m > βm : (3.146) TM However, the difference between βTE m and βm is very small for modes of an ordinary dielectric waveguide, where n1 n2 n1 . Then Fig. 3.19 can be used approximately for TM modes with a ¼ aM . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.5 Waveguide Modes 117 For a given waveguide, the values of aE and aM , as well as those of d and n21 n22 , are completely speciﬁed. Then, β of any guided mode is a function of the optical frequency ω because V is a function of ω. Figure 3.20 illustrates the typical relation between β and ω for guided modes of different orders. Comparing β, k 1 , and k2 in Fig. 3.20, it is seen that the propagation constant of a waveguide mode has a frequency dependence that is contributed by the structure of the waveguide besides that due to material dispersion. This extra contribution also causes different modes to have different dispersion properties, resulting in the phenomenon of modal dispersion. Polarization dispersion also exists because TE and TM modes generally have different propagation constants. Polarization dispersion is very small in a weakly guiding waveguide for which n1 n2 n1 . Cutoff Conditions As discussed above, γ2 and γ3 of a guided mode are real and positive so that the mode ﬁeld exponentially decays in the transverse direction outside the core region and remains bound to the core. This characteristic of a guided mode is equivalent to the condition that θ > θc2 > θc3 in the ray optics picture illustrated in Fig. 3.18 so that the ray in the core is totally reﬂected by both interfaces. Because θc2 > θc3 , the transition from a guided mode to an unguided radiation mode occurs when θ ¼ θc2 . This transition point corresponds to the condition that β ¼ k2 and γ2 ¼ 0. As can be seen from the mode ﬁeld solutions given in (3.134) and (3.139), the ﬁeld extends to inﬁnity on the substrate side when γ2 ¼ 0. This deﬁnes the cutoff condition for a guided mode. The cutoff condition is determined by γ2 ¼ 0, rather than by γ3 ¼ 0, because γ3 > γ2 so that γ2 reaches zero ﬁrst as their values are reduced. At cutoff, V ¼ V c . The cutoff value V c of a particular guided mode is the value of V at the point where the curve of its b versus V dispersion relation, shown in Fig. 3.19, intersects with the horizontal axis b ¼ 0. From (3.144) and (3.145), we ﬁnd by setting γ2 ¼ 0 that, at cutoff, pﬃﬃﬃﬃﬃ h1 d ¼ V c and γ3 d ¼ aE V c : (3.147) Substituting (3.147) and γ2 ¼ 0 into (3.135) for a guided TE mode yields pﬃﬃﬃﬃﬃ tanV c ¼ aE : (3.148) Therefore, the cutoff condition for the mth guided TE mode is pﬃﬃﬃﬃﬃ V cm ¼ mπ þ tan1 aE , m ¼ 0, 1, 2, . . . : (3.149) Substituting (3.147) and γ2 ¼ 0 into (3.140) yields the cutoff condition for the mth guided TM mode: pﬃﬃﬃﬃﬃﬃ V cm ¼ mπ þ tan1 aM , m ¼ 0, 1, 2, . . . : (3.150) Using the deﬁnition of the V number given in (3.128), we can write V cm ¼ 2π d λcm qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωc n21 n22 ¼ m d n21 n22 c (3.151) where λcm is the cutoff wavelength and ωcm is the cutoff frequency of the mth mode. The mth mode is not guided at a wavelength longer than λcm , or a frequency lower than ωcm . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 118 Optical Wave Propagation For given waveguide parameters, (3.149) and (3.150) can be used, respectively, to determine the cutoff wavelengths, and the corresponding cutoff frequencies, of TE and TM modes from (3.151). For a given optical wavelength, they can be used to determine the waveguide parameters that allow the existence of a particular guided mode. For given waveguide parameters and optical wavelength, they can be used to determine the number of guided modes for the waveguide. Therefore, the total number of guided TE modes supported by a given waveguide at a given optical wavelength is M TE ¼ pﬃﬃﬃﬃﬃ V 1 tan1 aE π π int pﬃﬃﬃﬃﬃﬃ V 1 tan1 aM π π int , (3.152) , (3.153) and that of guided TM modes is M TM ¼ where ½ int takes the nearest integer larger than the value in the bracket. Because aM > aE 6¼ 0 for an asymmetric waveguide, the value of V cm for the mth-order TM mode is larger than that for the mth-order TE mode. Furthermore, both TE0 and TM0 modes pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ have cutoff: V cTE0 ¼ tan1 aE for the TE0 mode and V cTM0 ¼ tan1 aM for the TM0 mode, with V cTM0 > V cTE0 . An asymmetric waveguide of a V number such that V cTM0 > V cTE0 > V supports no guided modes, neither TE nor TM. An asymmetric waveguide of a V number such that V cTM0 > V > V cTE0 supports the TE0 mode but not the TM0 mode. For V > V cTM0 > V cTE0 , both TE0 and TM0 modes are supported. As the V number increases, additional high-order modes are supported in the sequence: TE1 , TM1 , TE2 , TM2 , . . .. As the V number decreases, the highest order TM mode is cut off before the TE mode of the same order. A waveguide that supports only one mode is called a single-mode waveguide. A waveguide that supports more than one mode is a multimode waveguide. From the above discussion, a truly single-mode asymmetric waveguide is one that supports only the TE0 mode but not the TM0 mode. However, a waveguide that supports only the fundamental TE0 and TM0 modes is often called a single-mode waveguide, particularly in the situation of a symmetric waveguide, for which the two fundamental modes both have no cutoff, as discussed below. EXAMPLE 3.14 The step-index planar glass waveguide considered in Example 3.13 has n1 ¼ 1:54 for the core, n2 ¼ 1:47 for the substrate, and n3 ¼ 1:00 for the cover. Consider the λ ¼ 1 μm wavelength. What is the range of core thickness for the waveguide to support the TE0 mode but not the TE1 mode? What is the range of core thickness for the waveguide to support the TM0 mode but not the TM1 mode? What is the range of core thickness for the waveguide to support the TE0 mode but not the TM0 mode? Solution: With n1 ¼ 1:54, n2 ¼ 1:47, and n3 ¼ 1:00, we ﬁnd that qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2π V ¼ d n21 n22 ¼ 2:884d, where d is in μm; λ aE ¼ n22 n23 ¼ 5:51, n21 n22 aM ¼ n41 n22 n23 ¼ 31: n43 n21 n22 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 119 3.5 Waveguide Modes For the waveguide to support the TE0 mode but not the TE1 mode, pﬃﬃﬃﬃﬃ V 1 tan1 aE 1 ) π π 1:168 μm < d 1:168 < V 4:310 ) 2:884 405 nm < d 1:494 μm: M TE ¼ 1 ) ) ) 0< pﬃﬃﬃﬃﬃ tan1 aE < V 4:310 μm 2:884 pﬃﬃﬃﬃﬃ π þ tan1 aE For the waveguide to support the TM0 mode but not the TM1 mode, pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ V 1 tan1 aM 1 ) tan1 aM < V π π 1:393 4:535 1:393 < V 4:535 ) μm < d μm 2:884 2:884 483 nm < d 1:572 μm: M TM ¼ 1 ) ) ) 0< pﬃﬃﬃﬃﬃﬃ π þ tan1 aM For the waveguide to support the TE0 mode but not the TM0 mode, pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ M TE ¼ 1 and M TM ¼ 0 ) tan1 aE < V < tan1 aM ) 405 nm < d 483 nm: 3.5.2 Symmetric Slab Waveguides For a symmetric slab waveguide, n3 ¼ n2 , aE ¼ aM ¼ 0, and γ3 ¼ γ2 . Then, it can be seen from (3.136) and (3.141) that for both TE and TM modes,tan 2ψ ¼ 0 so that ψ¼ mπ , 2 m ¼ 0, 1, 2, . . . : (3.154) Therefore, the mode ﬁeld patterns of a symmetric waveguide given by (3.134) and (3.139) are either even functions of x, varying in space as cos h1 x in the core region d=2 < x < d=2, for even values of m, or odd functions of x, varying in space as sin h1 x in the core region d=2 < x < d=2, for odd values of m. This characteristic is expected because the mode ﬁeld pattern in a symmetric structure is either symmetric or antisymmetric. Figure 3.21 shows the ﬁeld patterns and the corresponding intensity distributions of the ﬁrst few guided modes of a symmetric slab waveguide. By using the identity tan 2θ ¼ 2 tan θ=ð1 tan2 θÞ ¼ 2 cot θ=ð cot2 θ 1Þ while equating γ3 to γ2 , the eigenvalue equation in (3.135) for guided TE modes can be transformed to two equations: tan h1 d γ2 ¼ , h1 2 for even modes; cot h1 d γ2 ¼ , h1 2 for odd modes: (3.155) These two equations can be combined in one eigenvalue equation for all guided TE modes: h1 d mπ tan 2 2 ¼ γ2 ¼ h1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V 2 h21 d 2 h1 d , m ¼ 0, 1, 2, . . . , Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.156) 120 Optical Wave Propagation Figure 3.21 (a) Field patterns and (b) intensity distributions of the ﬁrst few guided modes of a symmetric slab waveguide. Figure 3.22 Graphic solutions for the eigenvalues of guided TE and TM modes of a symmetric waveguide of V ¼ 5π. The intersections of dashed and solid curves yield the values of h1 d for eigenmodes. where m is the same mode number as the one in (3.154). Using (3.140), a similar procedure yields the eigenvalue equation for all guided TM modes: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 h1 d mπ n1 γ2 n1 V h1 d tan , m ¼ 0, 1, 2, . . . : (3.157) ¼ 2 ¼ 2 2 2 h1 d n2 h1 n2 For a given value of the waveguide parameter V, the solutions of (3.156) yield the allowed values of h1 d for both even and odd TE modes, and those of (3.157) yield the allowed values of h1 d for both even and odd TM modes. Figure 3.22 shows an example for V ¼ 5π. Because n1 > n2 , it can be seen from comparing (3.156) with (3.157) and from the graphic solution shown in Fig. 3.22 TE TM TM that for modes of the same order, hTE 1 < h1 ; thus βm > β m . This observation is consistent with the conclusion obtained from the above general discussion on asymmetric waveguides. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.5 Waveguide Modes 121 Because aE ¼ aM ¼ 0, TE and TM modes of a symmetric waveguide have the same cutoff condition: V cm ¼ mπ (3.158) for the mth TE and TM modes. This can also be seen in Fig. 3.22. Because m ¼ 0 for the fundamental modes, neither the fundamental TE mode nor the fundamental TM mode of a symmetric waveguide has cutoff. Any symmetric planar dielectric waveguide supports at least one TE and one TM mode. The number of TE modes supported by a given symmetric waveguide is the same as that of the TM modes, which is simply M TE ¼ M TM ¼ V π int : (3.159) For this reason, a symmetric waveguide is never truly single mode because it supports at least both TE0 and TM0 modes no matter how small its V number is, as long as V > 0. Often, a symmetric slab waveguide that has V < π is loosely called a single-mode waveguide because it supports only the fundamental TE0 and TM0 modes. These conclusions are unique to symmetric waveguides. They are not true for an asymmetric waveguide. For example, an asymmetric slab waveguide might not support any guided mode at a given optical wavelength because both its fundamental TE and TM modes have a nonzero cutoff. EXAMPLE 3.15 The step-index planar glass waveguide considered in Example 3.14 is made symmetric by using the substrate material for the cover so that n2 ¼ n3 ¼ 1:47 for the substrate and the cover while keeping n1 ¼ 1:54 for the core. Consider the λ ¼ 1 μm wavelength. What is the range of core thickness for the waveguide to support the TE0 mode but not the TE1 mode? What is the range of core thickness for the waveguide to support the TM0 mode but not the TM1 mode? What is the range of core thickness for the waveguide to support the TE0 mode but not the TM0 mode? Solution: With n1 ¼ 1:54 and n2 ¼ n3 ¼ 1:47, we ﬁnd that qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2π V ¼ d n21 n22 ¼ 2:884d, where d is in μm; aE ¼ 0, λ For the waveguide to support the TE0 mode but not the TE1 mode, aM ¼ 0: V 0< 1 ) 0<V π π π ) 0<d μm ) 0 < d 1:089 μm: 2:884 For the waveguide to support the TM0 mode but not the TM1 mode, M TE ¼ 1 ) V 0< 1 ) 0<V π π π ) 0<d μm ) 0 < d 1:089 μm: 2:884 It is not possible for a symmetric waveguide to support the TE0 mode but not the TM0 mode because they both have no cutoff. M TM ¼ 1 ) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 122 Optical Wave Propagation 3.6 PHASE VELOCITY, GROUP VELOCITY, AND DISPERSION .............................................................................................................. Phase velocity, group velocity, and dispersion are important parameters that characterize the propagation of an optical wave. Phase velocity determines the rate of phase variation in wave propagation. Group velocity determines the speed of transmission of an optical signal. Dispersion is the primary cause of limitation on the bandwidth of the transmission of optical signals. As discussed in Chapter 2, the susceptibility χðωÞ, thus the permittivity ϵ ðωÞ, of a medium is a function of the optical frequency. This is the origin of material dispersion. In a homogeneous anisotropic medium, normal modes of different polarizations have different characteristic refractive indices, and thus different propagation constants, resulting in polarization dispersion. In an optical structure, there are waveguide dispersion and modal dispersion besides material dispersion. Both material dispersion and waveguide dispersion are examples of chromatic dispersion because both are frequency dependent. Waveguide dispersion is caused by the frequency dependence of the propagation constant of a speciﬁc mode due to the waveguiding effect. The combined effect of material dispersion and waveguide dispersion for a particular mode alone is called intramode dispersion. Modal dispersion is also called intermode dispersion because it is caused by the variation in propagation constant between different modes. Modal dispersion appears only when more than one mode is excited in a multimode waveguide; it exists even when chromatic dispersion disappears. To illustrate the concepts of phase velocity, group velocity, and dispersion, we ﬁrst consider a plane-wave normal mode of a homogeneous medium that has a characteristic propagation constant of k ðωÞ ¼ nðωÞω=c, where nðωÞ is the frequency-dependent characteristic refractive index of the normal mode. Without loss of generality, the z coordinate direction is taken to be along the propagation direction. The electric ﬁeld of such a monochromatic plane optical wave can be written as E ¼ E exp ðikz iωt Þ, (3.160) where E is a constant vector independent of space and time. The ﬁeld expressed in (3.160) represents a sinusoidal wave that has a phase varying with z and t as φ ¼ kz ωt: (3.161) A point of constant phase on the space- and time-varying ﬁeld is deﬁned by φ ¼ constant, thus dφ ¼ kdz ωdt ¼ 0. If we track this point of constant phase as the wave propagates, we ﬁnd that it moves with a velocity of vp ¼ dz ω ¼ : dt k (3.162) This is called the phase velocity of the wave. Note that the phase velocity is a function of the optical frequency because the refractive index nðωÞ is a function of frequency. There is phase-velocity dispersion due to the fact that dn=dω 6¼ 0. In the case of normal dispersion, dn=dω > 0 and dn=dλ < 0; in the case of anomalous dispersion, dn=dω < 0 and dn=dλ > 0. As discussed in Section 2.3, normal dispersion and anomalous dispersion are associated with resonant transitions in a material. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.6 Phase Velocity, Group Velocity, and Dispersion 123 Figure 3.23 Wave packet composed of two frequency components showing the carrier and the envelope. The carrier travels at the phase velocity, whereas the envelope travels at the group velocity. In practice, a propagating optical wave rarely contains only one frequency. It usually consists of many frequency components that are grouped around some center frequency, ω0 . For the simplicity of illustration, we consider a wave packet traveling in the z direction that is composed of two plane waves of equal real amplitude E. The frequencies and propagation constants of the two components are ω1 ¼ ω0 þ dω, k1 ¼ k0 þ dk, ω2 ¼ ω0 dω, k2 ¼ k0 dk: (3.163) The space- and time-dependent total real ﬁeld of the wave packet is then given by E ¼ E exp ðik 1 z iω1 t Þ þ c:c: þ E exp ðik 2 z iω2 t Þ þ c:c: n o ¼ 2E cos ðk0 þ dkÞz ðω0 þ dωÞt þ cos ðk 0 dk Þz ðω0 dωÞt (3.164) ¼ 4E cos ðdkz dωtÞ cos ðk0 z ω0 tÞ: As illustrated in Fig. 3.23, the resultant wave packet has a carrier, which has a frequency of ω0 and a propagation constant of k0 , and an envelope, which varies in space and time as cosðdkz dωtÞ. Therefore, a ﬁxed point on the envelope is deﬁned by dkz dωt ¼ constant, which travels with a velocity of dω vg ¼ : (3.165) dk This is the velocity of the wave packet and is called the group velocity. Because the energy of a harmonic wave is proportional to the square of its ﬁeld amplitude, the energy carried by a wave packet that is composed of many frequency components is concentrated in the regions where the amplitude of the envelope is large. Therefore, the energy in a wave packet is transported at the group velocity v g . Because a wave package carries an optical signal, thus information, optical signals and optical information are transmitted at the group velocity. The constant-phase wavefront travels at the phase velocity, but optical energy and information are transmitted at the group velocity. In reality, the group velocity is usually a function of the optical frequency. Then, d2 k d 1 ¼ v 6¼ 0, 2 dω dω g Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.166) 124 Optical Wave Propagation which represents group-velocity dispersion. A dimensionless coefﬁcient for group-velocity dispersion is deﬁned as D ¼ cω d2 k 2πc2 d2 k ¼ : dω2 λ dω2 (3.167) Group-velocity dispersion is an important consideration in the propagation of optical pulses, which can represent information bits of an optical signal. It can cause broadening of an individual pulse, as well as changes in the time delay between pulses of different frequencies. The sign of the groupvelocity dispersion can be either positive or negative. In the case of positive group-velocity dispersion, d2 k=dω2 > 0 and D > 0, a long-wavelength, or low-frequency, pulse travels faster than a short-wavelength, or high-frequency, pulse. By contrast, a short-wavelength pulse travels faster than a long-wavelength pulse in the case of negative group-velocity dispersion, d2 k=dω2 < 0 and D < 0. In a given material, the sign of D generally depends on the spectral region of concern. Group-velocity dispersion and phase-velocity dispersion discussed above have different meanings. When measuring the transmission delay or the broadening of optical signals or pulses due to the dispersion in a medium that has a large transmission length, such as an optical ﬁber, another group-velocity dispersion coefﬁcient deﬁned as Dλ ¼ 2πc d2 k D ¼ 2 dω2 cλ λ (3.168) is usually used. This coefﬁcient is generally expressed as a function of wavelength in the unit of picoseconds per kilometer per nanometer ps km1 nm1 . It is a direct measure of the chromatic pulse transmission delay over a unit transmission length. To summarize, the propagation constant of a plane-wave normal mode is k¼ ω nðωÞ: c (3.169) vp ¼ ω c ¼ , k n (3.170) dω c ¼ , dk N (3.171) dn dn ¼nλ dω dλ (3.172) Therefore, the phase velocity is and the group velocity is vg ¼ where N ¼nþω is called the group index. Using (3.167) and (3.168), the group-velocity dispersion coefﬁcient can be expressed as DðλÞ ¼ λ2 d2 n λ d2 n or D ð λ Þ ¼ : λ c dλ2 dλ2 (3.173) Figure 3.24 shows, as an example, the dispersion properties of pure silica glass and germania– silica glass. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.6 Phase Velocity, Group Velocity, and Dispersion 125 Figure 3.24 (a) Index of refraction n and group index N and (b) group-velocity dispersion D as functions of wavelength for pure silica (solid curves) and germania–silica containing 13.5 mol% GeO2 (dashed curves). Zero group-velocity dispersion appears at 1:284 μm for pure silica. EXAMPLE 3.16 The index of refraction of pure silica in the wavelength range between 1:0 and 1:6 μm varies with wavelength approximately as n ¼ 1:4507 þ 0:00301λ2 0:00332λ2 : (a) Within this wavelength range, where does silica have normal dispersion? Where does it have anomalous dispersion? (b) Within this wavelength range, where does silica have positive group-velocity dispersion? Where does it have negative group-velocity dispersion? (c) Find the refractive index, the group index, and the group-velocity dispersion of silica at the three wavelengths of λ ¼ 1:0 μm, 1:3 μm, and 1:6 μm. (d) Express the group-velocity dispersion as Dλ in the unit of ps km1 nm1 . Solution: With the given wavelength dependence of the refractive index, we ﬁnd dn ¼ 0:00602λ3 0:00664λ, dλ N ¼nλ dn ¼ 1:4507 þ 0:00903λ2 þ 0:00332λ2 , dλ D ¼ λ2 d2 n ¼ 0:01806λ2 0:00664λ2 : 2 dλ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 126 Optical Wave Propagation (a) From the above, we ﬁnd that dn=dλ < 0 for all wavelengths in the wavelength range between 1:0 and 1:6 μm. Therefore, silica has normal dispersion throughout this wavelength range. (b) The wavelength dependence of D obtained above indicates that it can be zero at the wavelength: D ¼ 0 ) λ ¼ 1:284 μm: It is found that silica has positive group-velocity dispersion with D > 0 for λ < 1:284 μm, and it has negative group-velocity dispersion with D < 0 for λ > 1:284 μm. (c) Using the wavelength dependence of each parameter obtained above, we ﬁnd λ 1:0 μm 1:3 μm 1:6 μm n N D 1:450 1:463 0:01142 1:447 1:462 0:00054 1:443 1:463 0:00994: (d) Using (3.168) and the values of D obtained in (c), we ﬁnd λ 1:0 μm 1:3 μm 1:6 μm D 0:01142 0:00054 0:00994 Dλ 38 ps km1 nm1 1:4 ps km1 nm1 21 ps km1 nm1 : 3.6.1 Waveguide Dispersion The propagation constant β of a mode of an optical structure is determined both by the parameters of the optical structure and by the material properties. As seen in Figs. 3.16 and 3.20, due to the waveguiding effect, the frequency dependence of β can be very different from that of the k constants of the materials that form the optical structure. Therefore, β of a mode has mixed contributions from both material dispersion and waveguide dispersion. It is in fact more convenient to directly consider the combined effect. To do so, we only have to replace k of a plane-wave normal mode in all of the formulas obtained in the above by β of the waveguide mode under consideration, thus deﬁning the effective refractive index nβ , the effective group index N β , and the effective group-velocity dispersion Dβ for the mode: nβ ¼ Nβ ¼ c cβ , ω (3.174) dnβ dβ , ¼ nβ λ dλ dω (3.175) 2 d2 β 2 d nβ ¼ λ : dω2 dλ2 (3.176) Dβ ¼ cω Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.6 Phase Velocity, Group Velocity, and Dispersion 127 Figure 3.25 (a) Effective index of refraction and group index and (b) group-velocity dispersion of the fundamental mode as a function of wavelength. The solid curves show the effective parameters of the mode with both material and waveguide contributions. The dashed curves show only the material contribution to the core and cladding regions, labeled 1 and 2, respectively. The phase velocity and group velocity of the mode are, respectively, ω c ¼ , β nβ (3.177) dω c : ¼ dβ N β (3.178) v pβ ¼ and v gβ ¼ As an example of the contributions of the waveguiding effect to the dispersion parameters, Fig. 3.25 shows nβ , N β , and Dβ of the fundamental mode of a circular optical ﬁber in comparison to the parameters of its core and cladding materials. 3.6.2 Modal Dispersion The frequency dependence of the propagation constant β of a mode discussed above is the total intramode dispersion that includes material and waveguide contributions for the mode. Different normal modes of an anisotropic medium or an optical structure have different propagation constants at a given optical frequency. Such differences lead to modal dispersion among different modes, which is intermode dispersion. For plane waves or Gaussian modes propagating in a homogeneous anisotropic medium, modal dispersion exists due to different propagation constants for normal modes of different polarizations, such as k x , ky , and k z of the linearly birefringent principal normal modes of polarization given in (2.15), k þ and k of the circularly birefringent principal normal modes of polarization given in (2.21), or k o and ke of the ordinary and extraordinary waves in (3.57). Such modal dispersion causes polarization dispersion. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 128 Optical Wave Propagation For normal modes in an optical structure, such as an interface or a waveguide, modal dispersion exists among modes of the same polarization but of different order, such as the different propagation constants of TE modes of different orders shown in Fig. 3.19. In general, at a given frequency, a lower order mode has a larger propagation constant, as seen in Figs. 3.19 and 3.20. This dispersion is not caused by polarization or frequency but is purely imposed by the optical structure. This type of modal dispersion is mode-order dispersion. Modal dispersion in an optical structure also exists among modes of the same order but of different polarizations, such as that between TEm and TMm modes of a planar waveguide. As discussed in Section 3.5 TM and expressed in (3.146), βTE m > βm for any given order m. This type of modal dispersion is polarization-mode dispersion. EXAMPLE 3.17 An optical pulse has a pulse duration of Δt ps ¼ 20 ps and a spectral width of Δλps ¼ 0:1 nm. It is transmitted through a silica ﬁber over a distance of 10 km. Use the data of silica obtained in Example 3.16 for the silica ﬁber to ﬁnd the transmission time and the temporal broadening of the pulse due to group-velocity dispersion at the transmission end in the case when the center wavelength of the pulse is at λ ¼ 1:0 μm, 1:3 μm, or 1:6 μm. How does the group-velocity dispersion temporally spread the pulse spectrum in each case? Solution: For a transmission distance of l, the transmission time ttr is t tr ¼ l N ¼ l vg c and the temporal pulse broadening ΔtGVD due to group-velocity dispersion is Δt GVD ¼ jDλ jΔλps l: At λ ¼ 1:0 μm, N ¼ 1:463 and Dλ ¼ 38 ps km1 nm1 . Thus, for l ¼ 10 km, ttr ¼ N 1:463 10 103 s ¼ 48:8 μs, l¼ c 3 108 ΔtGVD ¼ jDλ jΔλps l ¼ 38 0:1 10 ps ¼ 38 ps: At λ ¼ 1:3 μm, N ¼ 1:462 and Dλ ¼ 1:4 ps km1 nm1 . Thus, for l ¼ 10 km, ttr ¼ N 1:462 10 103 s ¼ 48:7 μs, l¼ 8 c 3 10 ΔtGVD ¼ jDλ jΔλps l ¼ 1:4 0:1 10 ps ¼ 1:4 ps: At λ ¼ 1:6 μm, N ¼ 1:463 and Dλ ¼ 21 ps km1 nm1 . Thus, for l ¼ 10 km, ttr ¼ N 1:463 10 103 s ¼ 48:8 μs, l¼ c 3 108 ΔtGVD ¼ jDλ jΔλps l ¼ 21 0:1 10 ps ¼ 21 ps: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.7 Attenuation and Ampliﬁcation 129 We ﬁnd that the transmission time is about the same for all three wavelengths because the group index is about the same for all three wavelengths. However, the temporal pulse broadening varies much among the three wavelengths because of the different values of group-velocity dispersion. At the low group-velocity dispersion point of 1:3 μm, the pulse is only slightly broadened. At the other two wavelengths, the broadening is larger than the original pulse duration. Group-velocity dispersion causes frequency chirping in an optical pulse. At λ ¼ 1:0 μm, the broadening causes the long-wavelength component of the pulse to move to the temporal leading edge of the pulse because of positive group-velocity dispersion with D > 0 and Dλ < 0, making the pulse positively chirped with its frequency increasing with time within the pulse. At λ ¼ 1:3 μm and 1:6 μm, the broadening causes the short-wavelength component of the pulse to move to the temporal leading edge of the pulse because of negative groupvelocity dispersion with D < 0 and Dλ > 0, making the pulse negatively chirped with its frequency decreasing with time within the pulse. 3.7 ATTENUATION AND AMPLIFICATION .............................................................................................................. As discussed in Section 2.1, a complex eigenvalue of χðωÞ, thus that of ϵ ðωÞ, signiﬁes an optical loss or gain for the corresponding principal mode of polarization of the medium, with χ 00 > 0 and ϵ 00 > 0 for optical loss, and χ 00 < 0 and ϵ 00 < 0 for optical gain. For a plane-wave normal mode characterized by a complex eigenvalue ϵ, k 2 ¼ ω2 μ0 ϵ ¼ ω2 μ0 ðϵ 0 þ iϵ 00 Þ: (3.179) Therefore, the propagation constant k becomes complex: α k ¼ k0 þ ik00 ¼ k0 þ i : 2 (3.180) The index of refraction also becomes complex: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ϵ 0 þ iϵ 00 : (3.181) n ¼ n þ in ¼ ϵ0 The relation k ¼ nω=c between k and n is still valid. If we choose k0 to be positive, the sign of α is the same as that of ϵ 00 . Then, k0 and n0 are both positive, and k 00 and n00 also have the same sign as ϵ 00 . Taking the z coordinate direction to be along the propagation direction, the electric ﬁeld of a monochromatic plane optical wave as expressed in (3.160) is 0 00 E ¼ E exp ðikz iωt Þ ¼ E eαz=2 exp ðik0 z iωt Þ: (3.182) It can be seen that the wave has a phase that varies sinusoidally with a period of 2π=k0 along z. However, because of the nonvanishing imaginary part k00 ¼ α=2 of the propagation constant, the magnitude jEj of the electric ﬁeld is not constant but varies exponentially with z. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 130 Optical Wave Propagation The intensity of an optical ﬁeld projected on a surface is deﬁned in (1.56): ∗ I ¼ S n^ ¼ S þ S n^, where n^ is the unit normal vector of the projected surface. Note that the intensity of a given optical ﬁeld depends on the projected surface on which the intensity is measured. Note further that for an extraordinary wave in an anisotropic medium, the ^ These factors have Poynting vector S is not generally parallel to the propagation direction k. to be considered when calculating the intensity. For a monochromatic plane-wave normal mode that has an optical ﬁeld given in (3.182), we can use the relation k E ¼ ωμ0 H given in (3.31) to ﬁnd that its intensity projected on the surface that is normal to the propagation direction k^ can be expressed as I¼ 2k 0 jE⊥ j2 2k 0 jE ⊥ j2 αz ¼ e , ωμ0 ωμ0 (3.183) where E⊥ ¼ E E k^ k^ is the component of the optical ﬁeld that is transverse to the ^ For a plane wave in an isotropic propagation direction deﬁned by k^ and E ⊥ ¼ E E k^ k. medium or an ordinary wave in an anisotropic medium, E⊥ ¼ E because E k^ ¼ 0. For an extraordinary wave in an anisotropic medium, E⊥ 6¼ E because E k^ 6¼ 0. In any event, the optical intensity varies exponentially with z when α 6¼ 0. Clearly, k 0 is the wavenumber in this situation, and the sign of α determines the attenuation or ampliﬁcation of the optical wave. 1. If χ 00 > 0, then ϵ 00 > 0 and α > 0. As the optical wave propagates, its ﬁeld amplitude and intensity decay exponentially along the direction of propagation. Therefore, α is called the absorption coefﬁcient or attenuation coefﬁcient. 2. If χ 00 < 0, then ϵ 00 < 0 and α < 0. The ﬁeld amplitude and intensity of the optical wave grow exponentially. Then, we deﬁne g ¼ α as the gain coefﬁcient or ampliﬁcation coefﬁcient. Both α and g have the unit of per meter, often also quoted per centimeter. EXAMPLE 3.18 A Si crystal has a complex refractive index of n ¼ 4:30 þ i0:073 at the λ ¼ 500 nm wavelength. Find the absorption coefﬁcient and the absorption depth of Si at this wavelength. What is the complex susceptibility? Solution: From (3.180), the absorption coefﬁcient is α ¼ 2k00 ¼ 4πn00 4π 0:073 1 m ¼ 1:835 106 m1 : ¼ λ 500 109 The absorption depth is α1 ¼ 545 nm. Because 1 þ χ ¼ ϵ=ϵ 0 ¼ n2 , the complex susceptibility is χ ¼ n2 1 ¼ ð4:30 þ i0:073Þ2 1 ¼ 17:48 þ i0:628: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 3.7 Attenuation and Ampliﬁcation 131 3.7.1 Attenuation and Ampliﬁcation of Waveguide Modes Several factors contribute to the attenuation of the power of an optical wave propagating in an optical structure. Besides the loss or gain contributed by the material as discussed above, the imperfections of an optical structure, such as the roughness of its interfaces and the irregularity of its geometric shape, cause additional losses. Furthermore, the distribution of optical loss or gain might not be uniform across an optical structure because different regions of an optical structure generally have different optical properties. In any event, the normal mode of an optical structure is characterized by a unique, well-deﬁned propagation constant β. The attenuation or ampliﬁcation of the normal mode while it propagates through the structure is characterized by a complex β in the same manner as the complex k for a plane wave. Thus, α β ¼ β0 þ iβ00 ¼ β0 þ i : 2 (3.184) As described above, a positive α is the absorption coefﬁcient or attenuation coefﬁcient of the mode, whereas g ¼ α is the gain coefﬁcient or ampliﬁcation coefﬁcient of the mode. For a guided mode, attenuation or ampliﬁcation affects the mode across its entire proﬁle even though it does not have a uniform ﬁeld proﬁle across the transverse plane. Therefore, the attenuation or ampliﬁcation of a guided mode is measured with respect to the change of its mode power rather than its intensity: PðzÞ / eαz . The attenuation of optical power over a propagation distance of l in an optical structure for a mode that has an attenuation coefﬁcient of α is given by Pout ¼ Pin eαl : (3.185) The input and output powers of the mode, Pin and Pout , respectively, are measured in watts, while α is given per meter. The power is often measured in milliwatts or microwatts in lowpower applications, and in kilowatts or megawatts in high-power applications. In practical applications, α is also measured per centimeter or per kilometer when l is measured in centimeters or kilometers. In practical engineering applications, it is convenient to use decibels (dB) as a measure of relative changes of quantities. The attenuation coefﬁcient α is then measured in decibels per meter or decibels per kilometer when l is measured in meters or kilometers: 1 Pout 1 Pout α dB m1 ¼ , α dB km1 ¼ , 10 log 10 log Pin Pin lðmÞ lðkmÞ (3.186) where Pin and Pout are measured in the same unit which can be watts, milliwatts, or microwatts. In the case of a low-loss ﬁber, the propagation length l in the ﬁber is usually measured in kilometers, and α is conventionally given in decibels per kilometer. Comparing (3.185) with (3.186), we ﬁnd that α dB km1 ¼ 4:32α km1 and α km1 ¼ 0:23α dB km1 : (3.187) Power can also be measured in decibels and has the unit of decibel-watts (dBW), decibelmilliwatts (dBm), or decibel-microwatts (dBμ), deﬁned as PðdBWÞ ¼ 10 log PðWÞ, PðdBmÞ ¼ 10 log PðmWÞ, PðdBμÞ ¼ 10 log PðμWÞ: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 (3.188) 132 Optical Wave Propagation When power is given in decibel-watts or decibel-milliwatts and the attenuation coefﬁcient is in decibels per kilometer, (3.185) can be expressed as Pout ðdBWÞ ¼ Pin ðdBWÞ α dB km1 lðkmÞ (3.189) Pout ðdBmÞ ¼ Pin ðdBmÞ α dB km1 lðkmÞ: (3.190) or, equivalently, A similar formula can be written for power measured in decibel-microwatts. These formulas are convenient and useful in practical applications as they relate the input power, output power, and attenuation in a simple arithmetic relation. EXAMPLE 3.19 An optical ﬁber has an attenuation coefﬁcient of α ¼ 0:4 dB km1 at λ ¼ 1:3 μm. An optical signal at an input power level of Pin ¼ 10 mW is transmitted through this ﬁber over a distance of l ¼ 100 km. What is the output power? If the attenuation coefﬁcient is slightly reduced to α ¼ 0:35 dB km1 , what is the output power? Solution: The input power is Pin ¼ 10 mW ¼ 10 dBm. With α ¼ 0:4 dB km1 , the output power is Pout ¼ Pin αl ¼ 10 dBm 0:4 dB km1 100 km ¼ 30 dBm ¼ 103 mW ¼ 1 μW: If the attenuation coefﬁcient is slightly reduced to α ¼ 0:35 dB km1 , the output power is Pout ¼ Pin αl ¼ 10 dBm 0:35 dB km1 100 km ¼ 25 dBm ¼ 102:5 mW ¼ 3:16 μW: For a transmission distance of 100 km, the output power is increased by more than 200% when the attenuation coefﬁcient is reduced by only 0:05 dB km1 . Problems 3.1.1 Explain why a TEM mode ﬁeld can exist only in an optically homogeneous space where ϵ is a constant of space, and not in an optically inhomogeneous space where ϵ varies in space. 3.1.2 Can a dielectric waveguide support TEM modes? Explain. 3.1.3 Can a planar optical structure support hybrid modes? Explain. 3.1.4 What types of guided modes does each of the following structure support: (a) a planar metallic structure, (b) a planar dielectric structure, (c) a hollow cylindrical metallic structure, and (d) a cylindrical dielectric structure? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 Problems 133 3.1.5 Show that (a) the dot-product orthonormality relation of (3.20) applies to TE modes, (b) the dot-product orthonormality relation of (3.22) applies to TM modes, and (c) both relations apply to TEM modes. 3.2.1 The principal indices of refraction of InP, which is a cubic crystal, at the λ ¼ 1:3 μm wavelength are nx ¼ ny ¼ nz ¼ 3:205. Find the propagation constant and the wavelength in the crystal for an optical wave at λ ¼ 1:3 μm that propagates through an InP crystal under each of the following conditions. In each case, does the polarization state change as the wave propagates through the crystal? (a) Linearly polarized along ^x , propagating along ^y . (b) Linearly polarized along ^y , propagating along ^z . (c) Linearly polarized along ^z , propagating along ^x . (d) Circularly polarized in the xy plane, propagating along ^z . (e) Circularly polarized in the yz plane, propagating along ^x . 3.2.2 The principal indices of refraction of LiNbO3 , which is a negative uniaxial crystal, at the λ ¼ 1:3 μm wavelength are nx ¼ ny ¼ no ¼ 2:222 and nz ¼ ne ¼ 2:145. Find the propagation constant and the wavelength in the crystal for an optical wave at λ ¼ 1:3 μm that propagates through a LiNbO3 crystal under each of the following conditions. In each case, does the polarization state change as the wave propagates through the crystal? (a) Linearly polarized along ^x , propagating along ^y . (b) Linearly polarized along ^y , propagating along ^z . (c) Linearly polarized along ^z , propagating along ^x . (d) Circularly polarized in the xy plane, propagating along ^z . (e) Circularly polarized in the yz plane, propagating along ^x . 3.2.3 The principal indices of refraction of KTP, which is a biaxial crystal, at the λ ¼ 1:3 μm wavelength are nx ¼ 1:734, ny ¼ 1:742, and nz ¼ 1:822. Find the propagation constant and the wavelength in the crystal for an optical wave at λ ¼ 1:3 μm that propagates through a KTP crystal under each of the following conditions. In each case, does the polarization state change as the wave propagates through the crystal? (a) Linearly polarized along ^x , propagating along ^y . (b) Linearly polarized along ^y , propagating along ^z . (c) Linearly polarized along ^z , propagating along ^x . (d) Circularly polarized in the xy plane, propagating along ^z . (e) Circularly polarized in the yz plane, propagating along ^x . 3.2.4 The principal indices of refraction of LiNbO3 at λ ¼ 1:3 μm are nx ¼ ny ¼ no ¼ 2:222 and nz ¼ ne ¼ 2:145. Design a waveplate based on LiNbO3 for rotating the polarization direction of a linearly polarized wave at λ ¼ 1:3 μm by 30o . Give the possible thicknesses of the plate and the arrangement for this purpose. 3.2.5 The principal indices of refraction of LiNbO3 at λ ¼ 1:3 μm are nx ¼ ny ¼ no ¼ 2:222 and nz ¼ ne ¼ 2:145. Design a waveplate based on LiNbO3 for converting a linearly polarized wave into a circularly polarized wave at λ ¼ 1:3 μm. Give the possible thicknesses of the plate and the arrangement for this purpose. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 134 Optical Wave Propagation 3.2.6 The permittivity tensor of a KDP crystal at λ ¼ 1 μm in an arbitrarily chosen Cartesian coordinate system is found to be 0 1 2:174 0 0:039 ϵ ¼ ϵ0@ 0 2:280 0 A: 0:039 0 2:266 3.2.7 3.2.8 3.2.9 3.2.10 (a) Is the KDP crystal birefringent or nonbirefringent? If it is birefringent, is it uniaxial or biaxial? What are its principal indices of refraction? (b) If it is used to make a half-wave plate at λ ¼ 1 μm, what is the thickness of the plate? (c) If it is used to make a quarter-wave plate at λ ¼ 1 μm, what is the thickness of the plate? The principal indices of refraction of quartz at the λ ¼ 600 nm wavelength are nx ¼ ny ¼ 1:544 and nz ¼ 1:553. (a) Quartz is clearly a birefringent crystal, is it positive or negative uniaxial? (b) What kind of quartz plate can be used to rotate the polarization direction of a linearly polarized wave by 90 to its orthogonal linear polarization? Describe the arrangement for this function and ﬁnd the thickness of the plate. (c) What kind of quartz plate can be used to convert a circularly polarized wave into a linearly polarized wave? Describe the arrangement for this function and ﬁnd the thickness of the plate. How is the direction of the output linear polarization determined? The principal indices of refraction of BBO, which is a negative uniaxial crystal, are nx ¼ ny ¼ no ¼ 1:677 and nz ¼ ne ¼ 1:557 at the λ ¼ 500 nm wavelength. Consider a propagation direction k^ that makes an angle of ϕ ¼ 45 with respect to the x principal axis and an angle of θ ¼ 60 with respect to the z principal axis. (a) Find the polarization directions ^e o and ^e e , and the corresponding propagation constants k o and ke , of the ordinary and extraordinary normal modes. (b) Find the walk-off angle α of the extraordinary wave. What is the separation of the ordinary and extraordinary beams if an optical wave that consists of both ordinary and extraordinary components at this wavelength propagates in this direction through a BBO crystal over a distance of 3 mm? The principal indices of refraction of quartz, which is a positive uniaxial crystal, are nx ¼ ny ¼ no ¼ 1:544 and nz ¼ ne ¼ 1:553 at the λ ¼ 600 nm wavelength. Consider a propagation direction k^ that makes an angle of ϕ ¼ 60 with respect to the x principal axis and an angle of θ ¼ 30 with respect to the z principal axis. (a) Find the polarization directions ^e o and ^e e , and the corresponding propagation constants k o and ke , of the ordinary and extraordinary normal modes. (b) Find the walk-off angle α of the extraordinary wave. What is the separation of the ordinary and extraordinary beams if an optical wave that consists of both ordinary and extraordinary components at this wavelength propagates in this direction through a quartz crystal over a distance of 5 mm? Show that there is no walk-off for an extraordinary wave when it propagates in any direction that lies in the xy plane of a uniaxial crystal, for which the z principal axis is the unique optical axis. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 Problems 135 3.3.1 Give two examples of TEM modes that are not plane waves: (a) one example in purely dielectric medium and (b) another example not in purely dielectric medium. 3.3.2 A fundamental Gaussian beam from an Er:ﬁber laser at the λ ¼ 1:53 μm wavelength exits the ﬁber with a spot size of w0 ¼ 8 μm, which is determined by the ﬁber core radius. The beam then propagates in free space without being collimated. Find the beam divergence angle, the Rayleigh range, and the confocal parameter of the beam. What are the spot sizes and the radii of curvature of the beam at the distances of 1 mm, 1 cm, 10 cm, and 1 m, respectively, from the end of the ﬁber? 3.3.3 A Gaussian beam of an unknown wavelength in free space is found to have spot sizes of w0 ¼ 100 μm at the beam waist and wðzÞ ¼ 300 μm at a distance of z ¼ 15 cm from the beam waist. Find the wavelength, the Rayleigh range, and the divergence angle of the beam. 3.3.4 A fundamental Gaussian laser beam that has a power of P ¼ 10 W at a wavelength of λ ¼ 600 nm is focused to a small spot size for an intensity at the beam center of I 0 ¼ 2:5 MW cm2 at its beam waist. What is the beam-waist radius w0 of the beam? What is the divergence angle of the beam? What are its spot size and beam-center intensity at a distance of 5 m from the beam waist? If the spot size is increased to w0 ¼ 50 μm at the beam waist, what are the changes in the beam-center intensities at the beam waist and at 5 m from the waist, respectively? 3.4.1 Consider reﬂection and transmission of TE and TM waves at the interface of two lossless dielectric media that have real refractive indices of n1 and n2 , respectively. Use (3.91) and (3.95) to show the following facts. (a) For external reﬂection of a TE wave, the reﬂected ﬁeld has a π phase change at any incident angle. For internal reﬂection of a TE wave, the reﬂected ﬁeld has no phase change at any incident angle that is smaller than the critical angle. (b) For external reﬂection of a TM wave, the reﬂected ﬁeld has no phase change at any incident angle that is smaller than the Brewster angle, θi < θB , but has a π phase change at any incident angle that is larger than the Brewster angle, θi > θB . For internal reﬂection of a TM wave, the reﬂected ﬁeld has a π phase change at any incident angle that is smaller than the Brewster angle, θi < θB , but has no phase change at any incident angle that is larger than the Brewster angle and smaller than the critical angle, θB < θi < θc . 3.4.2 When a collimated beam of broadband white light covering the spectrum from red to violet is incident at an oblique angle from free space on a ﬂat surface of ordinary glass, the transmitted beam is no longer collimated. Sketch how the spectral components of the transmitted beam spread from red to violet. Give a brief explanation why they spread in that manner. 3.4.3 The refractive index of a glass plate is 1.5. It can be used as a reﬂection-type polarizer so that if a beam is incident on its surface at a proper angle, the reﬂected beam is always linearly polarized no matter what the polarization of the incident beam is. If the glass plate is placed in air, what is this proper incident angle from the air? What is the polarization of the reﬂected beam at this incident angle? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 136 Optical Wave Propagation 3.4.4 The refractive index of diamond at λ ¼ 1:0 μm is n ¼ 2:39. What is the reﬂectivity of the diamond surface at normal incidence? At a particular incident angle, a speciﬁc linearly polarized optical wave at λ ¼ 1:0 μm is completely transmitted through a diamond surface exposed to air. What are this incident angle and the speciﬁc polarization of the incident wave that make this happen? 3.4.5 The refractive index of water is 1.33. For the λ ¼ 600 nm wavelength, ﬁnd the parameters of the radiation modes at the air–water interface for internal reﬂection at the two different incident angles of 45 and 75 , respectively. What is the penetration depth of the evanescent tail into the air if a radiation mode is found to be a one-sided radiation mode at a particular incident angle? What are the phase shifts on reﬂection at the interface for TE and TM waves, respectively? 3.4.6 At the λ ¼ 1:5 μm wavelength, the refractive index of intrinsic GaAs is 3.38. Find the parameters of the radiation modes at the air–GaAs interface for internal reﬂection at the two different incident angles of 30 and 60 , respectively. What is the penetration depth of the evanescent tail into the air if a radiation mode is found to be a one-sided radiation mode at a particular incident angle? What are the phase shifts on reﬂection at the interface for TE and TM waves, respectively? 3.4.7 Consider the interface between SiO2 and silver. The refractive index of SiO2 is 1.46 in the visible spectral region. Use the plasma frequency ωp ¼ 1:36 1016 rad s1 of Ag to ﬁnd the surface plasma frequency of this interface. What are the cutoff frequency and cutoff wavelength for the surface plasmon mode? Does the surface plasmon mode exist at the λ ¼ 500 nm wavelength? If it exists, ﬁnd its propagation constant and characteristic parameters. Find the penetration depths of the mode into the SiO2 and the silver to ﬁnd its conﬁnement at the interface. 3.4.8 Consider the interface between GaAs and silver. The refractive index of GaAs varies with optical wavelength, increasing with decreasing wavelength. For simplicity, take the refractive index of GaAs to be 3.51 at λ ¼ 1 μm. Use the plasma frequency ωp ¼ 1:36 1016 rad s1 of Ag to ﬁnd the surface plasma frequency of this interface. What are the cutoff frequency and cutoff wavelength for the surface plasmon mode? Does the surface plasmon mode exist at the λ ¼ 500 nm and λ ¼ 1 μm wavelengths, respectively? If it exists, ﬁnd its propagation constant and characteristic parameters. Find the penetration depths of the mode into the GaAs and the silver to ﬁnd its conﬁnement at the interface. 3.5.1 A step-index planar GaAs=AlGaAs waveguide has a GaAs core and AlGaAs cover and substrate. At λ ¼ 900 nm, the GaAs core has n1 ¼ 3:593, the AlGaAs substrate has n2 ¼ 3:409, and the AlGaAs cover of a different composition has n3 ¼ 3:261. In what range can the propagation constant of a guided mode, if it exists, be found at the λ ¼ 900 nm wavelength? Ignoring wavelength-dependent changes in the refractive indices, for what wavelengths can a guided mode be found to have a propagation constant of β ¼ 2:5 107 m1 ? What happens to the answers if the AlGaAs composition for the cover is changed so that n3 ¼ 3:453? 3.5.2 A step-index planar glass waveguide has a glass core of n1 ¼ 1:54, a glass substrate of a different composition of n2 ¼ 1:47, and a free-space cover of n3 ¼ 1:00. The core Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 Problems 3.5.3 3.5.4 3.5.5 3.5.6 3.5.7 3.5.8 3.5.9 137 thickness is d ¼ 1:5 μm. What is the range of optical wavelength for the waveguide to support the TE0 mode but not the TE1 mode? What is the range of optical wavelength for the waveguide to support the TM0 mode but not the TM1 mode? What is the range of optical wavelength for the waveguide to support the TE0 mode but not the TM0 mode? A step-index planar glass waveguide has a glass core of n1 ¼ 1:54 and a substrate and a cover of n2 ¼ n3 ¼ 1:47. The core thickness is d ¼ 1:5 μm. What is the range of optical wavelength for the waveguide to support the TE0 mode but not the TE1 mode? What is the range of optical wavelength for the waveguide to support the TM0 mode but not the TM1 mode? What is the range of optical wavelength for the waveguide to support the TE0 mode but not the TM0 mode? What is the most outstanding difference between symmetric and asymmetric waveguides in terms of ﬁnding guided modes? A planar dielectric waveguide supports exactly three modes among all types of modes. Name these modes. Which mode has the largest propagation constant? Which one has the smallest propagation constant? An asymmetric InGaAsP=InP waveguide has a refractive index of n1 ¼ 3:432 for its core, and indices of n2 ¼ 3:354 and n3 ¼ 3:166 for its two cladding layers. What is the required core thickness for the waveguide to have one and only one guided mode at λ ¼ 1:55 μm, including modes of all different polarizations? A symmetric step-index planar InGaAsP=InP waveguide has the high-index InGaAsP for its core and the low-index InP for its cladding layers. At λ ¼ 1:55 μm, the core index is n1 ¼ 3:432 and the cladding index is n2 ¼ n3 ¼ 3:166. If a single-mode waveguide is desired, what is the required core thickness? Is the waveguide truly single-mode if this requirement is met? Name the mode or modes. A symmetric step-index planar InGaAsP=InP waveguide has a core index of n1 ¼ 3:438 and a cladding index of n2 ¼ 3:205. The core thickness is d ¼ 0:60 μm. (a) At the λ ¼ 1:30 μm wavelength, how many guided modes are supported by the waveguide? What are they? (b) At what wavelengths does the waveguide support only one TE mode and one TM mode? A symmetric step-index planar GaAs=Al0:3 Ga0:7 As waveguide has the high-index GaAs for its core and the low-index Al0:3 Ga0:7 As for its two cladding layers. At λ ¼ 1:5 μm, the core index is n1 ¼ 3:38 and the cladding index is n2 ¼ 3:22. (a) If a single-mode waveguide is desired, what is the required core thickness? Is the waveguide truly single-mode if this requirement is met? Name the mode or modes. (b) If the core thickness is chosen to be d ¼ 2 μm, how many guided modes are supported by the waveguide? What are they? (c) If the waveguide thickness is kept at d ¼ 2 μm, but its structure is made asymmetric by lowering the index of only one cladding layer, would existing modes start disappearing or new modes start appearing if that index is sufﬁciently reduced? What is the ﬁrst mode to disappear or appear if this happens? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 138 Optical Wave Propagation 3.6.1 The effective index of refraction of a single-mode optical ﬁber as a function of optical wavelength around λ ¼ 1:3 μm is found to be approximated as nβ ¼ 1:465 0:0114 ðλ 1:3Þ 0:004ðλ 1:3Þ3 , where λ is in micrometers. (a) Characterize the phase-velocity dispersion of this ﬁber at λ ¼ 1:2 μm and λ ¼ 1:5 μm, respectively. (b) Find and characterize the group-velocity dispersion of this ﬁber at λ ¼ 1:2 μm and λ ¼ 1:5 μm, respectively. (c) Express the group-velocity dispersion as Dλ in the unit of ps km1 nm1 at λ ¼ 1:2 μm and λ ¼ 1:5 μm, respectively. 3.6.2 The ﬁber described in Problem 3.6.1 is used to transmit two optical pulses at λ ¼ 1:2 μm and λ ¼ 1:5 μm, respectively. Each pulse has a pulse duration of Δt ps ¼ 5 ps and a spectral width of Δλps ¼ 1 nm. Find the temporal widths of these two pulses after propagating over a distance of 5 km in the ﬁber. 3.6.3 How far can the pulse at each of the three wavelengths described in Example 3.17 propagate through that ﬁber before the pulse broadening caused by group-velocity dispersion is larger than the original pulse duration? 3.6.4 The ordinary and extraordinary indices of refraction of LiNbO3 in the wavelength range between 1:0 and 2:0 μm vary with wavelength approximately as no ¼ 2:2158 þ 0:00286λ2 0:0062λ2 , ne ¼ 2:1395 þ 0:00247λ2 0:0052λ2 : (3.191) Answer each of the following questions for the ordinary and extraordinary waves, respectively. (a) Within this wavelength range, where does LiNbO3 have normal dispersion? Where does it have anomalous dispersion? (b) Within this wavelength range, where does LiNbO3 have positive group-velocity dispersion? Where does it have negative group-velocity dispersion? (c) Find the refractive index, the group index, and the group-velocity dispersion of LiNbO3 at the three wavelengths of λ ¼ 1:0 μm, 1:5 μm, and 2:0 μm. (d) Express the group-velocity dispersion as Dλ in the unit of fs cm1 nm1 . 3.6.5 An optical pulse has a pulse duration of Δt ps ¼ 100 fs and a spectral width of Δλps ¼ 75 nm. Use the values of Dλ obtained in Problem 3.6.4(d) for LiNbO3 to ﬁnd the pulse broadening caused by group-velocity dispersion after the pulse propagates over 1 cm in LiNbO3 . Find also the distance that the pulse can propagate in LiNbO3 before its pulse duration doubles. Answer both questions for the pulse polarized in the ordinary and extraordinary axes, respectively, and for its center wavelength at λ ¼ 1:0 μm, 1:5 μm, and 2:0 μm, respectively. ^j given in (1.56) 3.7.1 By using the deﬁnition of the optical intensity I ¼ jS n^j ¼ jðS þ S Þ n for a coherent wave and the equation k E ¼ ωμ0 H given in (3.31), show that the optical intensity of a plane-wave mode projected on the surface that is normal to its propagation direction k^ is given by the expression in (3.183). Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 Bibliography 139 3.7.2 Show that under the condition that ϵ 00 ϵ 0 , so that χ 00 χ 0 , n00 n0 , and α k 0 , the absorption coefﬁcient can be approximated as α k0 00 00 ϵ 00 2π χ 00 0 χ 0 χ ¼ k k ¼ : ϵ0 1 þ χ0 n02 λ n0 (3.192) 3.7.3 At the λ ¼ 300 nm wavelength, Si has a complex refractive index of n ¼ 5:0 þ i4:16, and GaAs has n ¼ 3:73 þ i2:0. Find the absorption coefﬁcients and the absorption depths of Si and GaAs at this wavelength. What is the complex susceptibility for each material at this wavelength? 3.7.4 The complex susceptibility of GaAs is χ ¼ 17:31 þ i3:70 at λ ¼ 500 nm and χ ¼ 12:55 þi0:63 at λ ¼ 800 nm. Find the absorption coefﬁcient and the absorption depth of GaAs at these wavelengths. 3.7.5 At λ ¼ 800 nm, Si has an absorption depth of α1 ¼ 9:8 μm and a reﬂectivity of 32:9% at normal incidence on its surface exposed to air. Find its complex refractive index and complex susceptibility at this wavelength. 3.7.6 An optical ﬁber of a length l ¼ 120 km has an attenuation coefﬁcient of 0:3 dB km1 at λ ¼ 1:3 μm and 0:15 dB km1 at λ ¼ 1:55 μm. If 2 mW of optical power at each wavelength is launched into the ﬁber, what is the output power at each wavelength? 3.7.7 An optical ﬁber has an attenuation coefﬁcient of 0:5 dB km1 at λ ¼ 1:3 μm and 0:2 dB km1 at λ ¼ 1:55 μm. If 1 mW of optical power at each wavelength is launched into the ﬁber and the detection limit of a detector at each wavelength is 1 μW, what is the maximum length of the ﬁber for the power at each wavelength to be detectable by the detector? Bibliography Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999. Buckman, A. B., Guided-Wave Photonics. Fort Worth, TX: Saunders College Publishing, 1992. Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975. Ebeling, K. J., Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors. Berlin: SpringerVerlag, 1993. Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Hunsperger, R. G., Integrated Optics: Theory and Technology, 5th edn. New York: Springer-Verlag, 2002. Iizuka, K., Elements of Photonics, Vols. I and II. New York: Wiley, 2002. Jackson, J. D., Classical Electrodynamics, 3rd edn. New York: Wiley, 1999. Kasap, S. O., Optoelectronics and Photonics: Principles and Practices, 2nd edn. Upper Saddle River, NJ: Prentice-Hall, 2012. Liu, J.M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Marcuse, D., Theory of Dielectric Optical Waveguides, 2nd edn. Boston, MA: Academic Press, 1991. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 140 Optical Wave Propagation Nishihara, H., Haruna, M., and Suhara, T., Optical Integrated Circuits. New York: McGraw-Hill, 1989. Pollock, C. R. and Lipson, M, Integrated Photonics. Boston, MA: Kluwer, 2003. Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991. Syms, R. and Cozens, J., Optical Guided Waves and Devices. London: McGraw-Hill, 1992. Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University Press, 2007. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:14:46 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.004 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 4 - Optical Coupling pp. 141-168 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge University Press 4 4.1 Optical Coupling COUPLED-MODE THEORY .............................................................................................................. Coupled-mode theory deals with the coupling of normal modes of propagation due to spatially dependent perturbations. The theory has broad applicability. It applies to the coupling of spatial modes in various optical structures, including Gaussian spatial modes in a homogeneous medium, interface modes, and waveguide modes. The space- and time-dependent electric and magnetic ﬁelds of a normal mode at a given frequency ω are expressed in the form of (3.1) and (3.2). Because the coupled-mode theory describes mode coupling caused by spatially dependent perturbations, no temporal changes are involved. Therefore, the time dependence of all ﬁelds remains exp ðiωtÞ throughout the interaction so that it can be ignored in the expressions of the ﬁelds while ∂=∂t is replaced by iω in Maxwell’s equations. Then, the two Maxwell equations for wave propagation can be written in the form: ∇ E ¼ iωμ0 H, (4.1) ∇ H ¼ iωϵ E: (4.2) The normal modes of an unperturbed optical structure are governed by (4.1) and (4.2). They are mutually orthogonal and are normalized through the orthonormality relation given in (3.18). These normal modes form a basis for linear expansion of any optical ﬁeld at the frequency ω in the optical structure: X ^ ν ðx; yÞ exp ðiβ zÞ, EðrÞ ¼ Aν E (4.3) ν ν HðrÞ ¼ X ν ^ ν ðx; yÞ exp ðiβν zÞ, Aν H (4.4) ^ ν and H ^ ν are normalized mode ﬁelds; the linear expansion sums over all discrete where E indices of the guided modes and integrates over all continuous indices of the radiation and evanescent modes. In the original, unperturbed structure where these modes are deﬁned, the normal modes do not couple because they are mutually orthogonal. Then, the expansion coefﬁcients Aν are constants that are independent of x, y, and z, as discussed in Section 3.1. In the presence of a spatially dependent perturbation to an optical structure, the modes deﬁned by the original structure are not exact normal modes of the perturbed structure. For this reason, the perturbation can cause coupling of these modes as they propagate. As a result, if an optical ﬁeld in the perturbed structure is expanded in terms of the normal modes of the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 142 Optical Coupling unperturbed structure, the expansion coefﬁcients are not constants of propagation but vary with z as the optical ﬁeld propagates through the structure: EðrÞ ¼ X ν HðrÞ ¼ X ν ^ ν ðx; yÞ exp ðiβ zÞ, Aν ðzÞE ν (4.5) ^ ν ðx; yÞ exp ðiβν zÞ: Aν ðzÞH (4.6) Because the power in a normal mode is given by Pν ¼ jAν j2 , according to (3.27), the z dependence of Aν ðzÞ in the above indicates that the power of a mode that is coupled to another mode does not remain a constant of propagation. Thus, coupling of modes leads to exchange of mode power. 4.1.1 Single-Structure Mode Coupling We ﬁrst consider the coupling between normal modes in a single optical structure, such as a single waveguide, that is subject to some perturbation. By single structure, we mean that the entire optical structure is considered in deﬁning the normal modes characterized by normalized ^ ν, H ^ ν of propagation constants βν . The structure can be a simple structure, such mode ﬁelds E as a homogeneous medium, a single interface, or a single waveguide; or it can be a compound structure that consists of multiple interfaces or multiple waveguides. In any event, no matter how complicated the structure might be, it is considered as a single entity and is described with a single ϵ ðrÞ to deﬁne the normal modes. A spatially dependent perturbation to the structure at a frequency of ω can be represented by a single perturbing polarization, ΔPðrÞ, so that the equations in (4.1) and (4.2) are modiﬁed as ∇ E ¼ iωμ0 H, (4.7) ∇ H ¼ iωϵ E iωΔP: (4.8) Any optical ﬁeld propagating in this perturbed structure can be expanded as (4.5) and (4.6) while its propagation is governed by these two equations with ΔP 6¼ 0. Meanwhile, the normal mode ﬁelds deﬁned by the unperturbed structure, which are deﬁned by (4.1) and (4.2), also satisfy these two equations with ΔP ¼ 0. Applying (4.7) and (4.8) to two arbitrary sets of ﬁelds, ðE1 ; H1 Þ and ðE2 ; H2 Þ, with respective perturbations of ΔP1 and ΔP2 , we ﬁnd the Lorentz reciprocity theorem: ∗ ∗ ∗ ∇ E1 H ∗ 2 þ E2 H1 ¼ iω E1 ΔP2 E2 ΔP1 , (4.9) which holds for any two sets of ﬁelds that are respectively associated with two arbitrary perturbations. To derive the couple-mode equation, we take ðE1 ; H1 Þ to be the optical ﬁeld propagating in the perturbed structure with ΔP1 ¼ ΔP, which can be expanded as (4.5) and ^ ν, H ^ ν deﬁned by the unperturbed structure (4.6), and ðE2 ; H2 Þ to be the normal mode ﬁelds E Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.1 Coupled-Mode Theory 143 with ΔP2 ¼ 0. By substituting these into (4.9) and integrating both sides of the resultant equation over the cross section of the waveguide, we ﬁnd ð∞ ð∞ ð∞ ð∞ Xd iðβν βμ Þz ∗ ∗ iβμ z ^ ^ ^ ∗ ΔPdxdy: ^ ^ E ν H μ þ E μ H ν ^z dxdy ¼ iωe E Aν ðzÞe μ dz ν ∞ ∞ ∞ ∞ (4.10) By applying the orthonormality relation given in (3.18) to (4.10), we ﬁnd the general form of the coupled-mode equations: dAν ¼ iωeiβν z dz ð∞ ð∞ ^ ∗ ΔPdxdy, E ν (4.11) ∞ ∞ where the plus sign is used when βν > 0 for mode ν to be forward propagating in the positive z direction, and the minus sign is used when βν < 0 for mode ν to be backward propagating in the negative z direction. The general form of the coupled-mode equations expressed in (4.11) is applicable to mode coupling caused by any kind of spatially dependent perturbation on any feature of the optical structure. For example, ΔP can be a perturbing polarization at the frequency ω on the ﬁelds in a waveguide due to any of the external effects discussed in Section 2.6 or due to any nonlinear optical susceptibility discussed in Section 2.7. For the simple case where the perturbation can be represented by a change in the linear polarization as X ^ ν eiβν z , ΔP ¼ Δϵ E ¼ Δϵ Aν E (4.12) ν the coupled-mode equations can be expressed in the form: dAν X iκνμ Aμ eiðβμ βν Þz , ¼ dz μ (4.13) where ð∞ ð∞ κνμ ¼ ω ^ μdxdy ^ ∗ Δϵ E E ν (4.14) ∞ ∞ is the coupling coefﬁcient between mode ν and mode μ. This result is applicable to isotropic and anisotropic structures. For an optical structure made of isotropic media, Δϵ simply reduces to a ^ ∗ Δϵ E ^ μ ¼ ΔϵE ^∗ E ^ μ in (4.14). For a lossless optical structure, the scalar Δϵ so that E ν ν dielectric tensor is a Hermitian matrix so that Δϵ ij ¼ Δϵ ∗ ji , as discussed in Section 2.2. Consequently, mode coupling in a lossless dielectric single structure is symmetric with κνμ ¼ κ∗ μν : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 (4.15) 144 Optical Coupling EXAMPLE 4.1 Any physical mechanism that creates a change in the optical permittivity of a material can possibly be a perturbation for the coupling of two modes in a waveguide. Is the mode coupling caused by the electro-optic Pockels effect symmetric? Is that caused by optical absorption in a semiconductor due to current injection symmetric? Solution: The Pockels effect mainly changes the permittivity tensor without causing additional optical loss. The permittivity change is Hermitian: Δϵ ¼ Δϵ † . Thus the mode coupling caused by this effect is symmetric: ð∞ ð∞ κνμ ¼ ω ^ E^ ∗ ν Δϵ E μ dxdy ∞ ∞ 0 1∗ ð∞ ð∞ ¼ @ω A E^ ν Δϵ ∗ E^ ∗ μ dxdy ∞ ∞ 0 ¼ @ω ð∞ ð∞ 1∗ † ^ A E^ ∗ μ Δϵ E ν dxdy ∞ ∞ 0 ¼ @ω ð∞ ð∞ ) κνμ ¼ κ∗ μν : 1∗ ^ A E^ ∗ μ Δϵ E ν dxdy ∞ ∞ ¼ κ∗ μν The permittivity change associated with optical absorption is not Hermitian: Δϵ 6¼ Δϵ † . Thus the mode coupling caused by this effect is not symmetric: ð∞ ð∞ κνμ ¼ ω ^ E^ ∗ ν Δϵ E μ dxdy ∞ ∞ 0 ¼ @ω 0 ∞ ∞ ð∞ ð∞ ¼ @ω 0 ð∞ ð∞ ∞ ∞ 6¼ @ω ð∞ ð∞ 1∗ A E^ ν Δϵ ∗ E^ ∗ μ dxdy 1∗ † ^ A E^ ∗ μ Δϵ E ν dxdy ) κνμ 6¼ κ∗ μν : 1∗ ^ A E^ ∗ μ Δϵ E ν dxdy ∞ ∞ ¼ κ∗ μν Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 145 4.1 Coupled-Mode Theory 4.1.2 Multiple-Structure Mode Coupling In a compound optical structure, such as a structure that consists of more than one waveguide, there are two alternative approaches to the analysis of the characteristics of optical ﬁelds that propagate in the structure. One approach is to treat the compound structure as a single super structure by expanding any optical ﬁeld in terms of its normal modes, known as the super modes, which are found by solving Maxwell’s equations directly with the boundary conditions deﬁned by the entire super structure. The alternative approach is to divide the compound structure into separate substructures, expand the ﬁelds in terms of the normal modes of the individual substructures, and treat the problem with a coupledmode approach. The ﬁrst approach can yield exact solutions and is sometimes desirable. However, it is not generally possible to obtain the exact super-mode solutions for complicated structures. The coupled-mode approach yields approximate solutions, but it can be applied to most structures without difﬁculty. In addition, it gives an intuitive picture of how optical waves interact in a compound structure. Here we consider the coupled-mode formulation for multiple substructures. The concept of dividing a super structure into a combination of individual substructures is illustrated in Fig. 4.1. In this illustration, the individual waveguides are the substructures of the multiple-waveguide super structure. The multiple-waveguide super structure is described by ϵ ðx; yÞ, whereas the individual waveguides are described by ϵ a ðx; yÞ, ϵ b ðx; yÞ, ϵ c ðx; yÞ, and so on. The normal modes are solved for each individual substructure. The ﬁelds in the entire structure can be expanded in terms of these normal modes in the same form as (4.5) and (4.6) but with the summation over the index ν covering all the modes of every substructure. From the standpoint of any substructure, the presence of other substructures is a perturbation to it. Thus, for substructure i that is described by ϵ i ðx; yÞ, the entire structure looks like ϵ i ðx; yÞ plus a perturbation of Δϵ i ðx; yÞ ¼ ϵ ðx; yÞ ϵ i ðx; yÞ: (4.16) The coupled-mode equations for the multiple-structure scenario can be obtained by using the reciprocity theorem of (4.9) and then following a procedure similar to that taken above to obtain Figure 4.1 Schematic diagram of three coupled waveguides showing the decomposition into individual waveguides, in solid curves, plus the corresponding perturbation, in dashed curves, for each of them. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 146 Optical Coupling the coupled-mode equations for the single structure. Because the mathematics is quite involved, only the results are given in the following without detailed derivation. The coupled-mode equations for multiple substructures can still be written in the same form as that of (4.13): dAν X iκνμ Aμ eiðβμ βν Þz , ¼ dz μ (4.17) where the plus sign is taken if mode ν is forward propagating, and the minus sign is used if it is backward propagating. It is noted that the summation over the index μ runs through the modes of every substructure, not just the modes of one single substructure. In contrast to that for single-structure coupling discussed above, the coupling coefﬁcients κνμ for multiple-structure coupling have a complicated form and are best expressed in terms of matrix elements: ~ νμ , (4.18) κνμ ¼ cνν c1 κ where cνν ¼ 1 if mode ν is forward propagating and cνν ¼ 1 if it is backward propagating, as ~ ¼ κ~νμ are given, can be seen from (4.19) below. The elements of the matrices c ¼ cνμ and κ respectively, by cνμ ¼ ð∞ ð∞ ∗ ∗ ^ ^ ^ ^ E ν H μ þ E μ H ν ^z dxdy ¼ c∗ μν (4.19) ∞ ∞ and ð∞ ð∞ κ~νμ ¼ ω ^ μdxdy: ^ ∗ Δϵ μ E E ν (4.20) ∞ ∞ Note that Δϵ μ in (4.20) is the perturbation, deﬁned in (4.16), to the substructure that deﬁnes the ^ μ, H ^ μ of normal mode μ. The coefﬁcient cνμ represents the overlap coefﬁcient of ﬁelds E ^ ν, H ^ μ, H ^ ν and E ^ μ , which can be the mode ﬁelds of different substructures in the super E structure. In general, cνμ 6¼ 0 because modes of different substructures are not necessarily orthogonal to each other. Because the mode ﬁelds used in (4.19) are normalized, we have cνν ¼ 1 or cνν ¼ 1, depending on whether mode ν is forward or backward propagating as mentioned above, and cνμ 1 for any ν and μ. Note also the difference between the form of κ~νμ expressed in (4.20) and that of the single-structure coupling coefﬁcients κνμ given in (4.14). As discussed above and expressed in (4.15), the coupling between modes of a single structure is always symmetric with κνμ ¼ κ∗ μν if the structure is dielectric and lossless. By contrast, the coupling between modes of different substructures in a super structure, such as those of different individual waveguides in a multiple-waveguide structure, is generally asymmetric: ∗ κ~νμ 6¼ κ~∗ μν and κνμ 6¼ κμν (4.21) where ν and μ refer to modes of two different substructures. Indeed, it can be shown by using the reciprocity theorem that Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.2 Two-Mode Coupling κ~νμ κ~∗ μν ¼ cνμ þ c∗ μν βν βμ ¼ cνμ βν βμ : 2 147 (4.22) This relation indicates that there is a direct relationship between the coupling coefﬁcients and the propagation constants. It has the following implications. 1. Unless βν ¼ βμ or cνμ ¼ c∗ μν ¼ 0, coupling between two modes is not symmetric, i.e., ∗ κνμ 6¼ κμν , because the normal modes of different substructures are not necessarily orthogonal to each other. 2. The coupling of modes of the same order between two identical substructures is always ∗ symmetric because βν ¼ βμ , resulting in κ~νμ ¼ κ~∗ μν and κνμ ¼ κμν . 3. The relation in (4.22) applies to modes of a single structure as well. In this situation, ~νμ ¼ κνμ . Therefore, κνμ ¼ κ∗ cνμ ¼ c∗ μν ¼ 0 if ν 6¼ μ, and κ μν in (4.15) holds true for the normal modes of the same structure because they are mutually orthogonal. 4. It is not possible to change the coupling between two modes without simultaneously changing their overlap coefﬁcient or their propagation constants. 4.2 TWO-MODE COUPLING .............................................................................................................. The coupling between two modes is the simplest and most common situation of mode coupling. It includes coupling between two modes of the same structure, such as mode coupling in a single waveguide that is modulated by a grating, or coupling between modes of two substructures, such as mode coupling in a directional coupler that is formed by two parallel waveguides. For two-mode coupling, the coupled-mode equations can be written in a simple form that can be analytically solved. In this section, we consider the general formulation of two-mode coupling. We have shown that both coupling among modes of a single structure and coupling among modes of different substructures can be described by coupled-mode equations of the same form as given in (4.13) and (4.17). The only difference is that the coupling coefﬁcients in (4.17) for multiple-structure mode coupling are deﬁned differently from those in (4.13) for singlestructure mode coupling. This commonality is convenient because the general solutions of the coupled-mode equations can be applied to both cases. For a particular problem, we only have to calculate the coupling coefﬁcients that are speciﬁc to the problem under consideration. For two-mode coupling either in a single structure or between two different substructures, the ﬁeld expansion in (4.5) and (4.6) consists of only two modes, designated as mode a and mode b of amplitudes A and B, respectively. Thus, coupled-mode equations of the form given in (4.13) or (4.17) reduce to the following two coupled equations: dA ¼ iκaa A þ iκab Beiðβb βa Þz , dz (4.23) dB ¼ iκbb B þ iκba Aeiðβa βb Þz : dz (4.24) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 148 Optical Coupling For coupling between two modes of a single structure, the coupling coefﬁcients in these equations are given by (4.14), which are always symmetric with κab ¼ κ∗ ba if the structure is dielectric and lossless. For coupling between modes of two different substructures, the coupling coefﬁcients are given by (4.18), which can be explicitly expressed as κaa ¼ κba κ~aa cab κ~ba =cbb κ~ab cab κ~bb =cbb , κab ¼ , 1 cab cba =caa cbb 1 cab cba =caa cbb κ~ba cba κ~aa =caa κ~bb cba κ~ab =caa ¼ , κbb ¼ : 1 cab cba =caa cbb 1 cab cba =caa cbb (4.25) As discussed earlier and expressed in (4.21), in general κab 6¼ κ∗ ba for coupling between modes of two different substructures. The iκaa A and iκbb B terms in the coupled equations (4.23) and (4.24) are self-coupling terms. These terms are caused by the fact that the normal modes see in the perturbed structure an index proﬁle that is different from the index proﬁle of the unperturbed original structure where the modes are deﬁned. They can be removed from the equations by expressing the normal-mode expansion coefﬁcients as 2 z 3 ð ~ ðzÞ exp 4i κaa ðzÞdz5, (4.26) AðzÞ ¼ A 0 2 3 ðz ~ ðzÞ exp 4i κbb ðzÞdz5, BðzÞ ¼ B (4.27) 0 where a plus or minus sign is chosen for a forward-propagating or backward-propagating mode, respectively. Then (4.23) and (4.24) can be transformed into two coupled equations in terms of ~ and B ~ to remove the self-coupling terms: A ~ dA ~ iφðzÞ , ¼ iκab ðzÞBe dz (4.28) ~ dB ~ iφðzÞ , ¼ iκba ðzÞAe dz (4.29) where 2 ðz 3 2 ðz 3 φðzÞ ¼ 4βb z κbb ðzÞdz5 4βa z κaa ðzÞdz5: 0 (4.30) 0 As shown in (4.28)(4.30), we have to consider the fact that each coupling coefﬁcient can be a function of z because Δϵ can be a function of z but the integration in (4.14) and (4.20) is carried out only over x and y. In the case when κab ðzÞ and κba ðzÞ are arbitrary functions of z, the coupled-mode equations cannot be analytically solved. In this situation, there is no need to further simplify the coupled-mode equations because they can only be numerically solved. However, for optical structures of practical interest that are designed for two-mode coupling, Δϵ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.2 Two-Mode Coupling 149 is usually either independent of z or periodic in z. Then, the coupling coefﬁcients are either independent of z or periodic in z. In either case, (4.28) and (4.29) can be reduced to the ~ and B ~ with κab and κba being constants that are independfollowing general form in terms of A ent of z: ~ dA ~ i2δz , ¼ iκab Be dz (4.31) ~ dB ~ i2δz : ¼ iκba Ae dz (4.32) The parameter 2δ is the phase mismatch between the two modes. Perfectly phase-matched coupling of two modes with δ ¼ 0 is always symmetric with κab ¼ κ∗ ba irrespective of whether these two modes belong to the same structure or two different substructures. The general form of (4.31) and (4.32) applies to both cases of uniform and periodic perturbations, but the details of the parameters vary between the two cases. 4.2.1 Uniform Perturbation In this case, Δϵ is only a function of x and y but is not a function of z. Then all of the coupling coefﬁcients κaa , κbb , κab , and κba in (4.28)(4.30) are constants that are independent of z. We then ﬁnd that ~ ðzÞeiκaa z , AðzÞ ¼ A ~ ðzÞeiκbb z , BðzÞ ¼ B (4.33) and 2δz ¼ φðzÞ ¼ ½ðβb κbb Þ ðβa κaa Þz for (4.30) so that 2δ ¼ ðβb κbb Þ ðβa κaa Þ: (4.34) The choice of sign in each in (4.33) and (4.34) is consistent with that in (4.26) and (4.27) discussed above. The physical meaning of the self-coupling coefﬁcients, κaa and κbb , is a change in the propagation constant of each normal mode. While the propagation constants of the normal modes in the original unperturbed structure are βa and βb , their values are changed by the perturbation characterized by Δϵ. These modes now propagate with the modiﬁed propagation constants βa κaa and βb κbb , respectively, which take into account the effect of the perturbation on the structure. In addition, they couple to each other through κab and κba . With the simple transformation of (4.33) and the phase mismatch 2δ given in (4.34), twomode coupling due to a uniform perturbation is described by the general form of (4.31) and (4.32) with constant values of κab and κba . A good example of two-mode coupling due to a uniform perturbation is that in a two-channel directional coupler, which consists of two parallel single-mode waveguides, as shown schematically in Fig. 4.2. This is the case of multiplestructure coupling. If the two waveguides are not identical, the directional coupler is not symmetric. Then, in general κba 6¼ κ∗ ab , as discussed in Section 4.1. Furthermore, 2δ 6¼ 0 except for a certain possible phase-matched optical frequency because κaa 6¼ κbb and βa 6¼ βb in general. If the two waveguides are identical, the directional coupler is symmetric. Then, κba ¼ κ∗ ab , κaa ¼ κbb , and βa ¼ βb so that 2δ ¼ 0 for all frequencies. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 150 Optical Coupling Figure 4.2 Schematic diagram of (a) a two-channel directional coupler of a length l consisting of two parallel waveguides and (b) its index proﬁle assuming two step-index waveguides on the same substrate. The coupler is symmetric if na ¼ nb ¼ n1 and d a ¼ d b ¼ d. 4.2.2 Periodic Perturbation In this case, Δϵ is a periodic function of z, and so are the coupling coefﬁcients κaa ðzÞ, κbb ðzÞ, κab ðzÞ, and κba ðzÞ in (4.28)(4.30). The periodic perturbation has a period of Λ and a wavenumber of 2π : (4.35) Λ Each coupling coefﬁcient, being periodic in z with a period of Λ, can be expanded in a Fourier series: X X κνμ ðqÞ exp ðiqKzÞ ¼ κνμ ðqÞ exp ðiqKzÞ (4.36) κνμ ðzÞ ¼ K¼ q q where q represents the order of coupling, the summation over q runs through all integers, and ðΛ 1 κνμ ðqÞ ¼ κνμ ðzÞ exp ðiqKzÞdz: Λ (4.37) 0 Using (4.36) for κab ðzÞ and κba ðzÞ, (4.28) and (4.29) can be expressed as X ~ dA ~ iφðzÞþiqKz , κab ðqÞBe ¼i dz q (4.38) X ~ dB ~ iφðzÞiqKz : κba ðqÞAe ¼i dz q (4.39) For κaa ðzÞ and κbb ðzÞ, we ﬁnd that ðz X κνν ðqÞ κνν ðzÞdz ¼ κνν ð0Þz þ eiqKz 1 : iqK q6¼0 0 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 (4.40) 4.2 Two-Mode Coupling 151 The κνν ð0Þ term represents a possible uniform perturbation that might exist due to a uniform bias in the periodic Δϵ. It can be removed by redeﬁning Δϵ or by considering it separately. In any event, for Kz 1, X κ ðqÞ νν iqKz 1 Kz: (4.41) e q6¼0 iqK Therefore, the contributions of the q 6¼ 0 terms of κaa ðzÞ and κbb ðzÞ to the z-dependent phases in (4.38) and (4.39) are negligible so that 2 3 2 3 ðz ðz φðzÞ þ qKz ¼ 4βb z κbb ðzÞdz5 4βa z κaa ðzÞdz5 þ qKz (4.42) 0 0 f½βb κbb ð0Þ ½βa κaa ð0Þ þ qK gz: With this approximation, the coupled-mode equations in the case of a periodic perturbation can be expressed as X ~ dA ~ iφðzÞþiqKz κab ðqÞBe ¼i dz q X ~ dB ~ iφðzÞiqKz κba ðqÞAe ¼i dz q ~ i2δz , iκab ðqÞBe (4.43) ~ i2δz , iκba ðqÞAe (4.44) where 2δ ¼ ½βb κbb ð0Þ ½βa κaa ð0Þ þ qK: (4.45) Note that only one term in the Fourier series that yields a minimum value for jδj is kept in each of the two coupled-mode equations expressed in (4.43) and (4.44) because only this term will effectively couple the two modes. Thus, the coupled-mode equations in (4.43) and (4.44) have the general form of (4.31) and (4.32) with κab ¼ κab ðqÞ and κba ¼ κba ðqÞ being constants that are independent of z, where q is the integer chosen to minimize the phase mismatch given in (4.45). The most common periodic perturbations are gratings. The simplest gratings are onedimensional gratings. For our purpose, such one-dimensional gratings are structures that are periodic only in the longitudinal direction, which is taken to be the z direction. Grating waveguide couplers have many useful applications and are one of the most important kinds of waveguide couplers. They consist of periodic ﬁne structures that form gratings in waveguides. The grating in a waveguide can take the form of either periodic index modulation or periodic structural corrugation. Periodic index modulation can be permanently written in a waveguide by periodically modulating the doping concentration in the waveguide medium, for example, or it can be created by an electro-optic, acousto-optic, or nonlinear optical effect. Figure 4.3 shows some examples of planar grating waveguide couplers in single waveguides. In these examples, there is no uniform perturbation apart from the periodic perturbation; therefore, κaa ð0Þ ¼ κbb ð0Þ ¼ 0 in (4.45) for these single-waveguide grating couplers. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 152 Optical Coupling Figure 4.3 Structures of planar grating waveguide couplers with (a) and (b) periodic index modulation, (c), (d), (e), and (f) periodic structural corrugation. EXAMPLE 4.2 Find the qth-order coupling coefﬁcient κνμ ðqÞ for a sinusoidal grating that has a period of Λ, as shown in Fig. 4.3(c), such that κνμ ðzÞ ¼ a cos Kz, where K ¼ 2π=Λ. Find it for a squarefunction grating that has a period of Λ and a duty factor of ξ, as shown in Fig. 4.3(d), such that κνμ ðzÞ ¼ a for 0 < z < ξΛ and κνμ ðzÞ ¼ a for ξΛ < z < Λ within each period. In each case, which orders are useful for mode coupling? Solution: For the sinusoidal grating, we ﬁnd by using (4.37) that ðΛ 1 κνμ ðqÞ ¼ κνμ ðzÞ exp ðiqKzÞdz Λ 0 ðΛ ¼ 1 a cos Kz exp ðiqKzÞdz Λ 0 ¼ a Λ ðΛ 0 exp ðiKz iqKzÞ þ exp ðiKz iqKzÞ dz: 2 a ¼ δq, 1 þ δq, 1 , 2 where δq, 1 and δq, 1 are the Kronecker delta functions. Therefore, only the order q ¼ 1 and q ¼ 1 the order are useful for mode coupling because only these two orders have a nonzero coupling coefﬁcient of κνμ ð1Þ ¼ κνμ ð1Þ ¼ a=2. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.2 Two-Mode Coupling 153 For the square-function grating, we ﬁnd by using (4.37) that ðΛ 1 κνμ ðqÞ ¼ κνμ ðzÞ exp ðiqKzÞdz Λ 0 ξΛ ð ðΛ 1 1 ¼ a exp ðiqKzÞdz a exp ðiqKzÞdz: Λ Λ 0 ¼ 2a sin ξqπ iξqπ : e qπ ξΛ We ﬁnd that κμν ðqÞ for a given value of q can be made nonzero by an appropriate choice of the duty factor ξ. Therefore, any order can be used if the value of ξ is properly chosen to maximize the value of κνμ ðqÞ for a given q. However, it is possible to have κνμ ðqÞ ¼ 0 for certain combinations of the values of q and ξ, such as q ¼ 2 and ξ ¼ 1=2, or q ¼ 3 and ξ ¼ 1=3, etc. The largest value of κνμ ðqÞ appears when q ¼ 1 or q ¼ 1 while ξ ¼ 1=2 so that κνμ ðqÞ ¼ 2a=π. A grating can also be used in a multiple-structure coupler. Figure 4.4 shows an example of a grating placed in a dual-channel coupler that consists of two waveguides. The two waveguides can be either identical, as in a symmetric structure, or nonidentical, as in an asymmetric structure. In both cases, the phase mismatch of this dual-channel coupler with a grating is that given in (4.45) with κaa ð0Þ 6¼ 0 and κbb ð0Þ 6¼ 0 due to the uniform perturbation on one waveguide by the other waveguide, as in the directional coupler shown in Fig. 4.2. EXAMPLE 4.3 Find the grating period for perfect phase matching of two modes a and b. Solution: For perfect phase matching, the phase mismatch given in (4.45) between two modes a and b of propagation constants βa and βb has to be made zero by the perturbation of a grating: 2δ ¼ ½ βb κbb ð0Þ ½ βa κaa ð0Þ þ qK ¼ 0 ) ) ) qK ¼ ½ βa κaa ð0Þ ½ βb κbb ð0Þ 2π ¼ ½ βa κaa ð0Þ ½ βb κbb ð0Þ Λ 2qπ Λq ¼ , ½ βa κaa ð0Þ ½ βb κbb ð0Þ q where ½ βa κaa ð0Þ ½ βb κbb ð0Þ is the total phase mismatch including all uniform perturbations on the structure, and the sign of q is chosen to be the sign of ½ βa κaa ð0Þ ½ βb κbb ð0Þ so that the grating period Λ has a positive value. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 154 Optical Coupling Figure 4.4 Dual-channel directional coupler with a grating of period Λ. With the above general considerations, (4.31) and (4.32) represent the most general coupled equations for two-mode coupling in structures of practical interest. They can be analytically solved; their solutions apply to various two-mode coupling problems. 4.3 CODIRECTIONAL COUPLING .............................................................................................................. First, we consider the coupling of two modes that propagate in the same direction, taken to be the positive z direction, over a length of l, as is shown in Fig. 4.5. In this case, βa > 0 and βb > 0. The coupled equations are ~ dA ~ i2δz , ¼ iκab Be dz (4.46) ~ dB ~ i2δz : ¼ iκba Ae dz (4.47) The equations for codirectional coupling are generally solved as an initial-value problem with ~ ðzÞ and B ~ ðz0 Þ and B ~ ðz0 Þ at z ¼ z0 to ﬁnd the values of A ~ ðzÞ at any other given initial values of A location z. The general solution can be expressed in the matrix form: " # " # ~ ðzÞ ~ ðz0 Þ A A ¼ Fðz; z0 Þ , (4.48) ~ ðzÞ ~ ðz0 Þ B B where the forward-coupling matrix Fðz; z0 Þ relates the ﬁeld amplitudes at the location z0 to those at the location z. It has the form: 2 3 βc cos βc ðzz0 Þiδ sin βc ðzz0 Þ iδðzz0 Þ iκab iδðzþz0 Þ e sin β ð zz Þe 0 c 6 7 βc βc 7 Fðz;z0 Þ ¼ 6 4 iκba βc cos βc ðzz0 Þþiδ sin βc ðzz0 Þ iδðzz0 Þ 5 iδðzþz0 Þ sin βc ðzz0 Þe e βc βc (4.49) where 1=2 : βc ¼ κab κba þ δ2 (4.50) We consider a simple case when power is launched only into mode a at z ¼ 0. Then the initial ~ ð0Þ 6¼ 0 and B ~ ð0Þ ¼ 0. By applying these conditions to (4.48) and taking z0 ¼ 0 in values are A (4.49), we ﬁnd that Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.3 Codirectional Coupling 155 Figure 4.5 Codirectional coupling between two modes of propagation constants βa and βb (a) in the same waveguide and (b) in two parallel waveguides. A perturbation is required for codirectional coupling in the same waveguide but is not required for codirectional coupling between two waveguides. Figure 4.6 Periodic power exchange between two codirectionally coupled modes for (a) the phase-mismatched condition δ 6¼ 0 and (b) the phase-matched condition δ ¼ 0. The solid curves represent Pa ðzÞ=Pa ð0Þ, and the dashed curves represent Pb ðzÞ=Pa ð0Þ. ~ ðzÞ ¼ A ~ ð0Þ cos βc z iδ sin βc z eiδz , A βc (4.51) ~ ðzÞ ¼ B ~ ð0Þ iκba sin βc z eiδz : B βc (4.52) The power in the two modes varies with z as ~ ðzÞ 2 κab κba Pa ðzÞ A δ2 2 ¼ cos β z þ , ¼ c ~ ð0Þ Pa ð0Þ β2c β2c A (4.53) ~ ðzÞ 2 jκba j2 Pb ðzÞ B ¼ ¼ sin2 βc z: 2 ~ ð0Þ Pa ð0Þ A βc (4.54) The coupling efﬁciency for codirectional coupling over a length of l is η¼ Pb ðlÞ jκba j2 2 ¼ 2 sin βc l: Pa ð0Þ βc Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 (4.55) 156 Optical Coupling Thus, power is exchanged periodically between two modes with a coupling length of lc ¼ π , 2βc (4.56) where maximum power transfer occurs. Figure 4.6 shows the periodic power exchange between the two coupled modes as a function of z. As can be seen from Fig. 4.6, complete power transfer can occur only in the phase-matched condition when δ ¼ 0. EXAMPLE 4.4 Find the maximum coupling efﬁciency for codirectional coupling and the length of a codirectional coupler that reaches this efﬁciency. What happens if the phase mismatch is large such that δ2 > κab κba ? Solution: From (4.55), the maximum efﬁciency for codirectional coupling is ηmax ¼ jκba j2 jκba j2 ¼ , β2c κab κba þ δ2 which is reached when sin2 βc l ¼ 1. Because sin2 βc l is periodic, sin2 βc l ¼ 1 has many solutions. The length to reach the maximum efﬁciency is any of lmax ¼ ð2m þ 1Þ π ¼ ð2m þ 1Þlc for m ¼ 0, 1, 2, . . . 2βc The formulas obtained above remain valid for δ2 > κab κba . There are no qualitative changes, but only quantitative changes, when the phase mismatch is large such that δ2 > κab κba . The maximum coupling efﬁciency decreases with increasing phase mismatch because βc increases with δ2 . The length lmax to reach the maximum efﬁciency also decreases with increasing phase mismatch because the coupling length lc decreases with increasing βc . 4.4 CONTRADIRECTIONAL COUPLING .............................................................................................................. We now consider the coupling of two modes that propagate in opposite directions over a length of l, as is shown in Fig. 4.7 where mode a is forward propagating in the positive z direction and mode b is backward propagating in the negative z direction. In this case, βa > 0 and βb < 0. Thus, the coupled equations are ~ dA ~ i2δz , ¼ iκab Be dz (4.57) ~ dB ~ i2δz : ¼ iκba Ae dz (4.58) Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.4 Contradirectional Coupling 157 Figure 4.7 Contradirectional coupling between two modes of propagation constants βa and βb (a) in the same waveguide and (b) in two parallel waveguides. A signiﬁcant perturbation is required for contradirectional coupling in both cases. The equations for contradirectional coupling are generally solved as a boundary value problem with ~ ð0Þ at one end and B ~ ðzÞ and B ~ ðlÞ at the other end to ﬁnd the values of A ~ ðzÞ given boundary values of A at any location z between the two ends. The general solution can be expressed in the matrix form: " # " # ~ ðzÞ ~ ð0Þ A A ¼ Rðz; 0; lÞ (4.59) ~ ðzÞ ~ ðlÞ B B ~ ð0Þ at z ¼ 0 and B ~ ðlÞ where the reverse-coupling matrix Rðz; 0; lÞ relates the ﬁeld amplitudes A at z ¼ l to those at any location z. It has the form: 2 3 αc cosh αc ðl zÞ þ iδ sinh αc ðl zÞ iδz iκab sinh αc z iδðlþzÞ e e 6 7 αc cosh αc l þ iδ sinh αc l αc cosh αc l þ iδ sinh αc l 6 7 Rðz; 0; lÞ ¼ 6 7 4 iκba sinh αc ðl zÞ αc cosh αc z þ iδ sinh αc z iδðlzÞ 5 iδz e e αc cosh αc l þ iδ sinh αc l αc cosh αc l þ iδ sinh αc l (4.60) where 1=2 : αc ¼ κab κba δ2 (4.61) We consider a simple case when power is launched only into mode a at z ¼ 0 but not into ~ ð0Þ 6¼ 0 and B ~ ðlÞ ¼ 0. By applying these mode b at z ¼ l. Then the boundary values are A conditions to (4.59), we ﬁnd that ~ ðzÞ ¼ A ~ ð0Þ αc cosh αc ðl zÞ þ iδ sinh αc ðl zÞ eiδz , A αc cosh αc l þ iδ sinh αc l ~ ð0Þ ~ ðzÞ ¼ A B iκba sinh αc ðl zÞ eiδz : αc cosh αc l þ iδ sinh αc l The power in the two contradirectionally coupled modes varies with z as ~ ðzÞ 2 cosh2 αc ðl zÞ δ2 =κab κba Pa ðzÞ A ¼ , ¼ ~ ð0Þ Pa ð0Þ A cosh2 αc l δ2 =κab κba ~ ðzÞ 2 κ∗ Pb ðzÞ B sinh2 αc ðl zÞ ¼ ba ¼ : ~ ð0Þ κab cosh2 αc l δ2 =κab κba Pa ð0Þ A (4.62) (4.63) (4.64) (4.65) Because mode b is propagating backward with no input at z ¼ l but with an output at z ¼ 0, the coupling efﬁciency for contradirectional coupling over a length of l is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 158 Optical Coupling Figure 4.8 Power exchange between two contradirectionally coupled modes for (a) the phase-mismatched condition δ 6¼ 0 and (b) the phase-matched condition δ ¼ 0. The solid curves represent Pa ðzÞ=Pa ð0Þ, and the dashed curves represent Pb ðzÞ=Pa ð0Þ. ~ ð0Þ2 κ∗ Pb ð0Þ B sinh2 αc l ¼ ba η¼ : ¼ ~ ð0Þ Pa ð0Þ κab cosh2 αc l δ2 =κab κba A (4.66) Figure 4.8 shows the power exchange between the two contradirectionally coupled modes as a 2 function of z. Power transfer approaches 100% as l ! ∞ if κab ¼ κ∗ ba and δ < κab κba . ~ ð0Þ 6¼ 0 and B ~ ðlÞ ¼ 0, as considered above, contradirectional coupling can In the case when A ~ ð0Þ at z ¼ 0 with a reﬂection coefﬁcient of be viewed as reﬂection of the ﬁeld amplitude A ~ ð0Þ iκba sinh αc l B r ¼ jr jeiφ ¼ : (4.67) ¼ ~ ð0Þ αc cosh αc l þ iδ sinh αc l A The reﬂectivity is R ¼ jr j2 ¼ η as is given in (4.66). The phase shift is φ¼ π δ tanh αc l þ φκba tan1 2 αc ¼ φPM tan1 δ tanh αc l , αc (4.68) where φκba is the phase angle of κba , and φPM ¼ π=2 þ φκba is the phase shift at the phasematched point where δ ¼ 0. EXAMPLE 4.5 Find the maximum coupling efﬁciency for contradirectional coupling and the length of a contradirectional coupler that reaches this efﬁciency. What happens if the phase mismatch is large such that δ2 > κab κba ? Solution: In the case when δ2 < κab κba , the parameter αc given in (4.61) has a real, positive value. Then, sinh αc l and cosh αc l are both monotonic functions with sinh αc l ! 1 and cosh αc l ! 1 as l ! ∞. From (4.66), the maximum efﬁciency for contradirectional coupling in the case when δ2 < κab κba is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.5 Conservation of Power ηmax ¼ 159 κ∗ ba , κab which can only be asymptotically reached when l ! ∞. Therefore, lmax ¼ ∞ when δ2 < κab κba . In the case when δ2 > κab κba , we ﬁnd that the parameter αc given in (4.61) becomes purely imaginary: 1=2 1=2 ¼ iγc with γc ¼ δ2 κab κba : αc ¼ κab κba δ2 Then the coupling efﬁciency given in (4.66) becomes η¼ κ∗ sin2 γc l ba : κab δ2 =κab κba cos2 γc l We ﬁnd that η varies with l periodically. By taking dη=dðγc lÞ ¼ 0, the maximum value of η is found when 2γc l ¼ ð2m þ 1Þπ. Thus, it takes place when sin2 γc l ¼ 1 and cos2 γc l ¼ 0 with ηmax ¼ jκba j2 : δ2 The length to reach this maximum efﬁciency is any of lmax ¼ ð2m þ 1Þ π ð2m þ 1Þπ ¼ 2γc 2 δ2 κab κba 1=2 for m ¼ 0, 1, 2, . . . For contradirectional coupling, there is a qualitative change in the coupling efﬁciency when the phase mismatch becomes large so that δ2 > κab κba . 4.5 CONSERVATION OF POWER .............................................................................................................. Conservation of power requires that in a lossless structure the net power ﬂowing across any cross section of the structure be a constant that does not vary along the longitudinal direction of the structure. For codirectional coupling between two modes with the power initially launched into only one mode such that Pa ð0Þ 6¼ 0 but Pb ð0Þ ¼ 0, this requirement suggests that the sum of power in the two waveguides, Pa ðzÞ þ Pb ðzÞ, be a constant independent of z because the power in the two modes ﬂows in the same direction. For contradirectional coupling with the power launched into only one mode such that Pa ð0Þ 6¼ 0 and Pb ðlÞ ¼ 0, this requirement suggests that Pa ðzÞ Pb ðzÞ be a constant independent of z because the power in mode b ﬂows in the backward direction while that in mode a ﬂows in the forward direction. These conclusions are correct for mode coupling in a single structure, such as a single waveguide, but they do not generally hold for coupling between modes of two different substructures, such as two separate waveguides. It can be seen from (4.53) and (4.54) that Pa ðzÞ þ Pb ðzÞ is not a constant of z for codirectional coupling unless κab ¼ κ∗ ba . Similarly, from (4.64) and (4.65), it is also found that Pa ðzÞ Pb ðzÞ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 160 Optical Coupling is not a constant of z for contradirectional coupling when κab 6¼ κ∗ ba . It seems that the total power is not conserved in a lossless structure in the case of asymmetric coupling with κab 6¼ κ∗ ba . A close examination reveals that because cab 6¼ 0 in the case of asymmetric coupling, the two interacting modes are not orthogonal to each other. For this reason, the total power ﬂow cannot be fully accounted for by gathering the power in each individual mode as if the modes were mutually orthogonal. Indeed, by expanding the total electric and magnetic ﬁelds in the structure as a linear superposition of the two modes in the form of (4.5) and (4.6) to calculate the power of the entire structure, we ﬁnd that the total power as a function of space is PðzÞ ¼ caa jAðzÞj2 þ cbb jBðzÞj2 þ 2Re cab A∗ ðzÞBðzÞeiΔβz (4.69) ¼ caa Pa ðzÞ þ cbb Pb ðzÞ þ Pab ðzÞ, where Pab ðzÞ ¼ 2Re cab A∗ ðzÞBðzÞeiΔβz can be considered as the power residing between the two nonorthogonal modes of the two different substructures. As deﬁned in Section 4.1, cνν ¼ 1 if mode ν is forward propagating and cνν ¼ 1 if mode ν is backward propagating. It can be shown, using (4.53) and (4.54) for the case of codirectional coupling and using (4.64) and (4.65) for the case of contradirectional coupling, that PðzÞ given in (4.69) is a constant ∗ independent of z no matter whether κab ¼ κ∗ ba or κ ab 6¼ κba . Therefore, conservation of power holds as expected. It can be shown simply by applying conservation of power that the coupling is symmetric with κab ¼ κ∗ ba when Pab ðzÞ ¼ 0. Conversely, if the coupling is symmetric, Pab ðzÞ always vanishes even when mode a and mode b are not orthogonal to each other. Two conclusions can thus be made. 1. If mode a and mode b are orthogonal to each other with cab ¼ 0, then Pab ðzÞ ¼ 0 and κab ¼ κ∗ ba even when the two modes are not phase matched so that δ 6¼ 0. 2. If mode a and mode b are phase matched with δ ¼ 0, then Pab ðzÞ ¼ 0 and κab ¼ κ∗ ba even when the two modes are not orthogonal to each other with cab 6¼ 0. Consequently, coupling between two modes a and b is symmetric with κab ¼ κ∗ ba if these two modes are orthogonal to each other or if they are phase matched. 4.6 PHASE MATCHING .............................................................................................................. As can be seen from Figs. 4.6 and 4.8, power transfer is most efﬁcient when δ ¼ 0. The parameter δ is a measure of phase mismatch between the two modes being coupled. For the simple case when 2δ ¼ Δβ ¼ βb βa , the phase-matching condition δ ¼ 0 is achieved when βa ¼ βb . Then, the two modes are synchronized to have the same phase velocity. In the case when δ includes a contribution from additional structural perturbation, such as a periodic grating, phase matching of the two modes being coupled can be accomplished by compensating for the difference Δβ ¼ βb βa with a perturbation phase factor to make δ ¼ 0, as can be seen in (4.34) for a uniform perturbation and in (4.45) for a periodic perturbation. When considering phase matching between two modes, it is important to always include all sources of contribution to the phase-mismatch parameter δ. When all contributions to the phase mismatch are Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.6 Phase Matching 161 considered and their effects on the coupling coefﬁcients are accounted for, the coupling coefﬁcients and the phase mismatch have a relation similar to (4.22): ∗ κab κ∗ (4.70) ba ¼ cab þ cba δ ¼ cab 2δ: Phase-matched coupling is always symmetric because κab ¼ κ∗ ba whenever δ ¼ 0, as seen in (4.70). This statement is true even when cab 6¼ 0 and βa 6¼ βb . However, symmetric coupling does not necessarily imply a phase-matched condition because symmetric coupling can be accomplished by having cab ¼ 0 when δ 6¼ 0, as also seen in (4.70). Therefore, though δ ¼ 0 ∗ always implies κab ¼ κ∗ ba , the converse is not true; it is possible to have κab ¼ κ ba when δ 6¼ 0. The clearest example of this situation is the coupling between two phase-mismatched modes in the same waveguide. 4.6.1 Phase-Matched Coupling When perfect phase matching is accomplished so that δ ¼ 0, we can take iφ κab ¼ κ∗ ba ¼ κ ¼ jκje : (4.71) βc ¼ αc ¼ jκj: (4.72) Because δ ¼ 0, we ﬁnd that With these relations, the matrix Fðz; z0 Þ for codirectional coupling is reduced to FPM ðz; z0 Þ ¼ ieiφ sin jκjðz z0 Þ cos jκjðz z0 Þ , ie sin jκjðz z0 Þ cos jκjðz z0 Þ iφ and the matrix Rðz; 0; lÞ for contradirectional coupling is reduced to 2 3 cosh jκjðl zÞ iφ sinh jκjz ie 6 cosh jκjl cosh jκjl 7 6 7 RPM ðz; 0; lÞ ¼ 6 7: 4 iφ sinh jκjðl zÞ cosh jκjz 5 ie cosh jκjl cosh jκjl (4.73) (4.74) For perfectly phase-matched codirectional coupling, the coupling efﬁciency is ηPM ¼ sin2 jκjl, (4.75) as shown in Fig. 4.9(a), and the coupling length is lPM c ¼ π : 2jκj (4.76) PM By choosing the interaction length to be l ¼ lPM c , or any odd multiple of lc , 100% power transfer from one mode to the other with ηPM ¼ 1 can be accomplished. For perfectly phase-matched contradirectional coupling, the coupling efﬁciency is ηPM ¼ tanh2 jκjl, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 (4.77) 162 Optical Coupling Figure 4.9 Coupling efﬁciency ηPM as a function of the normalized coupling length jκjl for (a) perfectly phasematched codirectional coupling and (b) perfectly phase-matched contradirectional coupling. as shown in Fig. 4.9(b). For an interaction length of l ¼ lPM deﬁned in (4.76), phase-matched c contradirectional coupling has a coupling efﬁciency of ηPM ¼ 84%. Although complete power transfer with 100% efﬁciency cannot be accomplished for contradirectional coupling, most power is transferred in a length comparable to the coupling length of codirectional coupling if perfect phase matching is accomplished. EXAMPLE 4.6 The coupling efﬁciency of a contradirectional coupler never reaches 100% but only approaches 100% as the length of the coupler approaches inﬁnity: η ! 1 as l ! ∞. For a practical application, η ¼ 99% might be as good. Find the length of a perfectly phase-matched contradirectional coupler that has η ¼ 99%. Solution: The length for a perfectly phase-matched contradirectional coupler that has η ¼ 99% is found as 2 η99% ¼ tanh jκjl99% ¼ 0:99 ) l99% pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3:0 1 1 ¼ tanh 0:99 ¼ : jκ j jκ j EXAMPLE 4.7 A 3-dB coupler is one that has a coupling efﬁciency of η ¼ 50%. Consider a 3-dB codirectional coupler and a 3-dB contradirectional coupler. Both have perfect phase matching and have the same coupling coefﬁcient of κ. Find the length l3dB of each phase-matched 3-dB coupler in terms of jκj? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 4.6 Phase Matching 163 Solution: Using (4.75), the length of a phase-matched 3-dB codirectional coupler is found to be one of the many values: η3dB ¼ sin2 jκjl3dB ¼ 1 2 ) l3dB ¼ 1 1 1 π sin1 pﬃﬃﬃ ¼ m þ for m ¼ 0, 1, 2, . . . 2 2jκj jκ j 2 Using (4.77), the length of a phase-matched 3-dB contradirectional coupler is found to have only one value: η3dB ¼ tanh2 jκjl3dB ¼ 1 2 ) l3dB ¼ 1 1 0:88 tanh1 pﬃﬃﬃ ¼ : jκ j jκ j 2 The values of l3dB found above for codirectional and contradirectional coupling can be seen in Figs. 4.9(a) and (b), respectively. 4.6.2 Phase-Mismatched Coupling In the presence of phase mismatch with δ 6¼ 0, symmetric coupling with κab ¼ κ∗ ba is still true for coupling between two modes in the same structure but is not necessarily true for coupling between two different substructures. Nevertheless, to illustrate the effect of phase mismatch on the coupling efﬁciency between two modes, we consider the simple case that κ ¼ κab ¼ κ∗ ba , as expressed in (4.71). For codirectional coupling with a phase mismatch of δ, the coupling efﬁciency obtained in (4.55) can be written in terms of jκjl and jδ=κj as η¼ 1 1 þ jδ=κj2 2 sin qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jκjl 1 þ jδ=κj2 : (4.78) The maximum efﬁciency is ηmax ¼ 1 1 þ jδ=κj2 (4.79) at a coupling length of lPM c : lc ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 þ jδ=κj (4.80) The maximum coupling efﬁciency is clearly less than unity when δ 6¼ 0. As shown in Fig. 4.10(a), both lc and ηmax decrease as jδ=κj increases. If the interaction length is ﬁxed at l ¼ lPM c , the efﬁciency drops quickly as jδ=κj increases, as shown in Fig. 4.10(b). For contradirectional coupling with a phase mismatch of δ, the coupling efﬁciency can be expressed in terms of jκjl and jδ=κj as Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 164 Optical Coupling Figure 4.10 Effect of phase mismatch on codirectional coupling showing, as a function of jδ=κj, (a) the coupling length lc , normalized as lc =lPM c , and the maximum coupling efﬁciency ηmax and (b) the coupling PM efﬁciency for ﬁxed interaction lengths of l ¼ lPM , 3lPM c c ,5lc . Figure 4.11 Effect of phase mismatch on contradirectional coupling showing the coupling efﬁciency for a few different values of jκjl as a function of jδ=κj. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh jκjl 1 jδ=κj2 η¼ : qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 cosh jκjl 1 jδ=κj jδ=κj 2 (4.81) The coupling efﬁciency decreases as phase mismatch increases, as seen in Fig. 4.11. It decreases monotonically with increasing jδ=κj for jδ=κj < 1; it decreases nonmonotonically but oscillatorily for jδ=κj > 1. In summary, to accomplish efﬁcient coupling between two waveguide modes, the following three parameters have to be considered. 1. Coupling coefﬁcient: The coupling coefﬁcient κ has to exist and be sufﬁciently large. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 Problems 165 2. Phase matching: The phase mismatch has to be minimized so that jδ=κj is made as small as possible. Ideally, perfect phase matching with δ ¼ 0 is desired. 3. Interaction length: For codirectional coupling, because the efﬁciency oscillates with interaction length, the length has to be properly chosen. An overly large length is neither required nor beneﬁcial. For contradirectional coupling, because the efﬁciency monotonically increases with the interaction length, the length has to be sufﬁciently large but does not have to be critically chosen. A very large length is not necessary, either. Problems 4.1.1 Is the mode coupling caused by introducing an optical gain to a single waveguide symmetric? Is the mode coupling caused by a slight structural change in the waveguide symmetric? 4.1.2 Show that the general formulation for multiple-structure mode coupling is applicable to the coupling of modes in a single waveguide. 4.2.1 Show that symmetric mode coupling in a single waveguide remains symmetric when a lossless grating is introduced for phase matching. 4.2.2 Find the qth-order coupling coefﬁcient κνμ ðqÞ for a saw-tooth grating, as shown in Fig. 4.3(f), that has a period of Λ and a duty factor of ξ such that 8 > > 2z ξΛ a, for 0 < z < ξΛ; < ξΛ κνμ ðzÞ ¼ ð1 þ ξ ÞΛ 2z (4.82) > > a, for ξΛ < z < Λ; : ð1 ξ ÞΛ with K ¼ 2π=Λ. Which orders are useful for mode coupling? 4.2.3 A single-mode GaAs/AlGaAs waveguide supports a mode that has a propagation constant of β ¼ 2:5 107 m1 at λ ¼ 900 nm. To make a waveguide reﬂector, the forwardpropagating wave in this mode has to be coupled to the backward-propagating wave of the same mode. A grating is incorporated into the waveguide for phase matching. Ignore any zeroth-order effect of the grating. Find the ﬁrst-order grating period and the secondorder grating period for this purpose. 4.2.4 A dual-channel directional coupler consists of two parallel InGaAsP/InP waveguides for the two channels. A grating is fabricated in the space between the two channels to phase match the waveguide modes of the two channels, as shown in Fig. 4.4. At λ ¼ 1:55 μm, the modes have effective indices of nβa ¼ 3:40 and nβb ¼ 3:35, respectively. Ignore any zeroth-order effect of the grating. Find the ﬁrst-order grating period and the second-order grating period for phase matching the modes of the two channels in the same direction. Find those values for phase matching the modes in the two channels for them to propagate in opposite directions. 4.3.1 Find the length of a codirectional coupler that has a coupling efﬁciency of half of the maximum possible efﬁciency for given coupling coefﬁcients of κab and κba and phase Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 166 Optical Coupling mismatch of δ between two modes in the case when the phase mismatch is small such that δ2 < κab κba . What happens if the phase mismatch is large such that δ2 > κab κba ? 4.3.2 Find the length of a codirectional coupler that has a coupling efﬁciency of 25% of the maximum possible efﬁciency for given coupling coefﬁcients of κab and κba and phase mismatch of δ between two modes in the case when the phase mismatch is small such that δ2 < κab κba . What happens if the phase mismatch is large such that δ2 > κab κba ? 4.4.1 Find the length of a contradirectional coupler that has a coupling efﬁciency of half of the maximum possible efﬁciency for given coupling coefﬁcients of κab and κba and phase mismatch of δ between two modes in the case when the phase mismatch is small such that δ2 < κab κba . 4.4.2 Find the length of a contradirectional coupler that has a coupling efﬁciency of half of the maximum possible efﬁciency for given coupling coefﬁcients of κab and κba and phase mismatch of δ between two modes in the case when the phase mismatch is large such that δ2 > κab κba . 4.5.1 Show that in the case of symmetric coupling with κab ¼ κ∗ ba , the powers of the two codirectionally coupled modes given in (4.53) and (4.54) for the condition of Pa ð0Þ 6¼ 0 and Pb ð0Þ ¼ 0 satisfy the power conservation relation PðzÞ ¼ Pa ðzÞ þ Pb ðzÞ ¼ Pa ð0Þ with Pab ðzÞ ¼ 0. 4.5.2 Show that in the case of symmetric coupling with κab ¼ κ∗ ba , the powers of the two contradirectionally coupled modes given in (4.64) and (4.65) for the condition of Pa ð0Þ 6¼ 0 and Pb ðlÞ ¼ 0 satisfy the power conservation relation PðzÞ ¼ Pa ðzÞ Pb ðzÞ ¼ Pa ð0Þ Pb ð0Þ with Pab ðzÞ ¼ 0. Show also that Pa ðlÞ þ Pb ð0Þ ¼ Pa ð0Þ for the total power to be conserved. 4.6.1 Two optical waves of exactly the same wavelength and the same power are respectively launched into the two input ports of a perfectly phase-matched 3-dB directional coupler at the same time, as shown in Fig. 4.12. What are the possible power ratios between the two output ports? What factor determines this ratio? 4.6.2 If the length of the coupler shown in Fig. 4.12 is doubled so that it becomes a coupler of 100% efﬁciency, what are the possible power ratios between the two output ports? What factor determines this ratio? Figure 4.12 3-dB directional coupler. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 Problems 167 4.6.3 A waveguide distributed Bragg reﬂector (DBR) has a grating of square corrugation as shown in Fig. 4.3(d). The period of the grating is Λ, and its duty factor is ξ. It is found that the propagation constant of the fundamental TE0 mode of the waveguide at the λ ¼ 1:0 μm optical wavelength is β ¼ 1:0 107 m1 . It is also found that the maximum absolute value of the coupling coefﬁcient of this grating is jκjmax ¼ 1:0 104 m1 , which is obtained when the parameters of the grating are properly chosen. Assume that the waveguide structural parameters and the grating depth are ﬁxed. Only the period Λ and the duty factor ξ of the grating are varied. (a) What are the optimal choices of the period Λ and the duty factor ξ for the grating to have the maximum coupling coefﬁcient jκjmax ? What is the length of the DBR if 50% reﬂectivity is desired? (b) If a second-order grating has to be used, what are the best choices of its period Λ and its duty factor ξ for the highest efﬁciency? What is the length of the DBR if 50% reﬂectivity is desired in this case? 4.6.4 A waveguide Bragg reﬂector is fabricated with a grating of a period Λ in a symmetric planar semiconductor waveguide, which has a core index of 3.25 and a cladding index of 3.20 for the wavelength of λ ¼ 1:55 μm. (a) Estimate the required grating period for a ﬁrst-order grating and that for a secondorder grating. (b) Between the sinusoidal and the square gratings, choose a combination of shape and duty factor for a ﬁrst-order grating that has a maximized coupling efﬁciency for a given modulation depth. (c) If the grating chosen in (b) has a coupling coefﬁcient of jκj ¼ 1:0 104 m1 , what is the required length of the grating for the Bragg reﬂector to have a 90% reﬂectivity? 4.6.5 A ﬁber-optic frequency ﬁlter is made of two single-mode ﬁbers of different mode propagation constants. They are placed in close contact over a length of l, as shown in Fig. 4.13. At the λ ¼ 1:55 μm optical wavelength, the effective indices for the two ﬁber modes are βa ¼ 5:959 106 m1 and βb ¼ 5:849 106 m1 , respectively, and the coupling coefﬁcient between the two ﬁber modes is κ ¼ κab κba ¼ 2 103 m1 . A grating that has a period of Λ is built into the ﬁbers in the coupling section. The input port of the device is port 1. The device is to function as an optical ﬁlter for separating the 1:55 μm wavelength from other wavelengths. (a) If the device is to direct all of the optical power at the 1:55 μm wavelength to port 4 and to dump all other wavelengths to port 3, what is the maximum possible coupling efﬁciency for the 1:55 μm wavelength without the grating? (b) With a ﬁrst-order grating, what are the values of Λ and l that have to be selected to obtain the best efﬁciency for directing the power at the 1:55 μm wavelength to port 4? What is the maximum efﬁciency if the parameters of the grating are properly chosen? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 168 Optical Coupling (c) If the device is to direct the power at the 1:55 μm wavelength to port 2, what is the maximum possible coupling efﬁciency without the grating? (d) With a ﬁrst-order grating, what should the choice of the grating period Λ be in order to get the highest efﬁciency for directing the power at the 1:55 μm wavelength to port 2? In this case, if the length l of the coupler remains the same as that found in (b), what is the efﬁciency of directing the 1:55 μm light from port 1 to port 2? Figure 4.13 Fiber-optic frequency ﬁlter consisting of two single-mode ﬁbers and a grating. 4.6.6 In designing an efﬁcient waveguide coupler of any geometry, what are the three major parameters that have to be considered in order to have a good efﬁciency? In what order of priority do they have to be considered? Bibliography Buckman, A. B., Guided-Wave Photonics. Fort Worth, TX: Saunders College Publishing, 1992. Chuang, S. L., Physics of Photonic Devices, 2nd edn. New York: Wiley, 2009. Hunsperger, R. G., Integrated Optics: Theory and Technology, 5th edn. New York: Springer-Verlag, 2002. Liu, J.M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Marcuse, D., Theory of Dielectric Optical Waveguides, 2nd edn. Boston, MA: Academic Press, 1991. Nishihara, H., Haruna, M., and Suhara, T., Optical Integrated Circuits. New York: McGraw-Hill, 1989. Pollock, C. R. and Lipson, M., Integrated Photonics. Boston, MA: Kluwer, 2003. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:15:31 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.005 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 5 - Optical Interference pp. 169-203 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge University Press 5 5.1 Optical Interference OPTICAL INTERFERENCE .............................................................................................................. An optical ﬁeld is a sinusoidal wave that has a space- and time-varying phase. The complex electric ﬁeld of an optical wave that propagates in a homogeneous medium can be generally expressed in the form of (1.81): Eðr; t Þ ¼ E ðr; t Þ exp ðik r iωt Þ ¼ ^e jE ðr; t ÞjeiφE ðr;tÞ exp ðik r iωt Þ, (5.1) which has a total space- and time-dependent phase as given in (1.83): φðr; tÞ ¼ k r ωt þ φE ðr; t Þ: (5.2) For a waveguide mode that propagates along the longitudinal waveguide axis, taken to be the z axis, the complex electric ﬁeld takes the form of (3.1): Eν ðr; tÞ ¼ E ν ðr; t Þ exp ðiβν z iωtÞ ¼ ^e jE ν ðr; t ÞjeiφE ν ðz;tÞ exp ðiβν z iωt Þ, (5.3) which has a total space- and time-dependent phase of φν ðz; t Þ ¼ βν z ωt þ φE ν ðz; tÞ: (5.4) The wave nature of an optical ﬁeld is fully characterized by its total space- and time-dependent phase factor. Because φν ðz; tÞ in (5.4) for a waveguide mode is mathematically a special form of φðr; tÞ in (5.2), by taking k to be βν^z and φE ðr; t Þ to be φE ν ðz; t Þ, in the following discussion we consider only optical waves in a homogeneous medium. The general concept applies equally to waveguide modes. Unless otherwise speciﬁed, we also consider a lossless medium for simplicity so that the propagation constant k has a real value. One phenomenon that clearly demonstrates the wave nature of optical ﬁelds is optical interference of two or more ﬁelds of different phases. In this section, we consider the interference of two ﬁelds that are superimposed only once. In Section 5.2, the concept of an optical grating based on the interference of multiple waves that emerge from a periodic optical structure is discussed. Multiple interference leading to optical resonance and optical ﬁltering is discussed in Section 5.3. Consider the superposition of two optical ﬁelds, E1 and E2 . The total ﬁeld is the linear vector sum of the two: E ¼ E1 þ E2 ¼ ^e 1 jE 1 jeiφ1 þ ^e 2 jE 2 jeiφ2 , (5.5) where φ1 ¼ k1 r ω1 t þ φE 1 and φ2 ¼ k2 r ω2 t þ φE 2 are the total phases of the two ﬁelds E1 and E2 , respectively. According to (3.183), the intensity of an optical ﬁeld is proportional to Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 170 Optical Interference jE⊥ j2 . Though (3.183) is strictly only applicable to a plane-wave normal mode that has a unique wavevector of k and a unique frequency of ω while the composite ﬁeld E in (5.5) might not be a normal mode because k1 and k2 might not be the same and ω1 and ω2 might not be the same, it is clear that the intensity of the composite ﬁeld E is not simply the sum of the intensities of the component ﬁelds E1 and E2 because ∗ jEj2 ¼ jE1 j2 þ jE2 j2 þ E1 E∗ 2 þ E1 E2 iðφ1 φ2 Þ ¼ jE 1 j2 þ jE 2 j2 þ 2jE 1 jjE 2 jRe ^e 1 ^e ∗ : 2e (5.6) The interference between the two ﬁelds E1 and E2 arises from the term iðφ1 φ2 Þ 2jE 1 jjE 2 jRe ^e 1 ^e ∗ in (5.6). Clearly, interference does not exist between two orthog2e onally polarized ﬁelds for which ^e 1 ^e ∗ 2 ¼ 0. Note that the orthogonality between two optical ﬁelds is deﬁned by ^e 1 ^e ∗ ¼ 0, as given in (1.80), but not by ^e 1 ^e 2 ¼ 0. This is important for 2 circularly polarized or elliptically polarized ﬁelds, which have complex unit polarization vectors. Interference occurs only when two ﬁelds are not orthogonally polarized so that ^e 1 ^e ∗ 2 6¼ 0. Using the time-averaged Poynting vector S deﬁned in (1.53) and the deﬁnition of the light intensity I ¼ S n^j while assuming that the angle between k1 and k2 is small, the intensity of the total ﬁeld can be expressed as I ¼ I 1 þ I 2 þ I 12 cos ðφ1 φ2 þ φ^e 1 ^e ∗2 Þ ¼ I 1 þ I 2 þ I 12 cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 þ φ^e 1 ^e ∗2 , (5.7) where I 1 ¼ 2k 1 jE 1⊥ j2 =ω1 μ0 and I 2 ¼ 2k 2 jE 2⊥ j2 =ω2 μ0 are respectively the intensities of E1 and E2 alone, k1 k2 I 12 ¼ 2 (5.8) þ jE 1⊥ E 2⊥ j^e 1 ^e ∗ 2 0 ω1 μ0 ω2 μ0 is the intensity magnitude of the interference between the two ﬁelds, φ^e 1 ^e ∗2 is the phase of ^e 1 ^e ∗ Þ is the time average of cos ðφ1 φ2 þ φ^e 1 e^∗2 Þ over one e∗ e1 ^ 2 , and cos ðφ1 φ2 þ φ^ 2 wave cycle, as deﬁned in (1.53) for the time-averaged Poynting vector S. The phase factor φ^e 1 ^e ∗2 matters only when the two polarizations ^e 1 and ^e 2 are not mutually orthogonal and at least one of them is not linearly polarized because φ^e 1 ^e ∗2 ¼ 0 when ^e 1 ^e ∗ e 1 and 2 ¼ 0 or both ^ ^e 2 are real vectors. With this understanding, in the following we consider for simplicity only the case when the two component ﬁelds have the same polarization, i.e., ^e 1 ¼ ^e 2 , so that ^e 1 ^e ∗ ¼ 0. Then, 2 ¼ 1 and φ^e 1 ^e ∗ 2 I ¼ I 1 þ I 2 þ I 12 cos ðφ1 φ2 Þ ¼ I 1 þ I 2 þ I 12 cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 , (5.9) and I 12 k1 k2 ¼2 þ jE 1⊥ E 2⊥ j > 0: ω1 μ0 ω2 μ0 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 (5.10) 5.1 Optical Interference 171 As seen from (5.9), I 1 þ I 2 I 12 I I 1 þ I 2 þ I 12 . Depending on the total phase difference φ1 φ2 , the total intensity I of the composite ﬁeld can be higher or lower than, or equal to, the sum of the intensities I 1 and I 2 of the individual component ﬁelds. Because I 12 I 1 þ I 2 , maximum interference takes place when the two component ﬁelds have the same polarization and the same amplitude so that I 12 ¼ I 1 þ I 2 . 1. Constructive interference occurs when the phase difference φ1 φ2 is such that the total intensity I is higher than the sum of the intensities I 1 and I 2 of the individual component ﬁelds: I 1 þ I 2 < I I 1 þ I 2 þ I 12 . Complete constructive interference happens when the two component ﬁelds are in phase, i.e., φ1 φ2 ¼ 2qπ, where q is an integer, so that I ¼ I 1 þ I 2 þ I 12 . Partial constructive interference happens when the phase difference is such that 2qπ π=2 < φ1 φ2 < 2qπ þ π=2 but φ1 φ2 6¼ 2qπ so that I 1 þ I 2 < I < I 1 þ I 2 þ I 12 . These concepts of constructive interference are illustrated in Fig. 5.1 for the case when the two component ﬁelds have the same frequency. 2. Destructive interference occurs when the phase difference φ1 φ2 is such that the total intensity I is lower than the sum of the intensities I 1 and I 2 of the individual component ﬁelds: 0 I 1 þ I 2 I 12 I < I 1 þ I 2 . Complete destructive interference happens when Figure 5.1 Constructive interference between two ﬁelds of the same frequency but of different amplitudes showing the individual ﬁelds (dashed curves) and the composite ﬁeld (solid curve). The two component ﬁelds have amplitudes of jE 1 j ¼ E 0 and jE 2 j ¼ 0:8E 0 in this example. (a) Complete constructive interference for φ1 φ2 ¼ 0. In this case, I 1 ¼ I 0 , I 2 ¼ 0:64I 0 , and I ¼ 3:24I 0 > I 1 þ I 2 because the amplitude of the composite ﬁeld is jE j ¼ 1:8E 0 . (b) Partial constructive interference for φ1 φ2 ¼ π=4 as an example. In this case, I 1 ¼ I 0 , I 2 ¼ 0:64I 0 , and I 2:77I 0 > I 1 þ I 2 because the amplitude of the composite ﬁeld is jE j 1:665E 0 . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 172 Optical Interference the two component ﬁelds are completely out of phase, i.e., φ1 φ2 ¼ ð2q þ 1Þπ, and they have the same amplitude to completely cancel each other so that I ¼ I 1 þ I 2 I 12 ¼ 0. Partial destructive interference happens when the two ﬁelds cancel each other only partially but not completely so that I 6¼ 0 but 0 < I < I 1 þ I 2 . Partial destructive interference occurs under one of the two following different situations. The two ﬁelds are completely out of phase, φ1 φ2 ¼ ð2q þ 1Þπ, but they do not have the same amplitude, jE 1⊥ j 6¼ jE 2⊥ j, so that I 12 < I 1 þ I 2 ; or the phase difference is such that ð2q þ 1Þπ π=2 < φ1 φ2 < ð2q þ 1Þπ þ π=2 but φ1 φ2 6¼ ð2q þ 1Þπ. These concepts of destructive interference are illustrated in Fig. 5.2 for the case when the two component ﬁelds have the same frequency. Figure 5.2 Destructive interference between two ﬁelds of the same frequency showing the ﬁelds and intensities of the individual ﬁelds (dashed curves) and the composite ﬁeld (solid curve). (a) Complete destructive interference for φ1 φ2 ¼ π and jE 1⊥ j ¼ jE 2⊥ j so that I ¼ 0. (b) Partial destructive interference for φ1 φ2 ¼ π but jE 1⊥ j 6¼ jE 2⊥ j so that I 6¼ 0 but 0 < I < I 1 þ I 2 . (c) Partial destructive interference for φ1 φ2 ¼ 3π=4 and jE 1⊥ j ¼ jE 2⊥ j so that I 6¼ 0 but 0 < I < I 1 þ I 2 . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.1 Optical Interference 173 Interference between two optical ﬁelds can create intensity patterns that vary in space or time, or both, because the phase difference φ1 φ2 can be a function of space or time, or both. As seen in (5.9), the phase difference φ1 φ2 ¼ ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 has three components. 1. When k1 ¼ 6 k2 , the spatially varying phase factor ðk1 k2 Þ r creates periodic spatial interference fringes that have a period of Λ ¼ 2π=jk1 k2 j along the k1 k2 direction. These interference fringes disappear when k1 ¼ k2 : When ω1 ¼ ω2 and φE 1 φE 2 is time independent, these interference fringes in space are stationary patterns that do not vary with time. Figure 5.3 shows the stationary periodic fringes produced by the interference between two ﬁelds of the same polarization, same amplitude, and same frequency, but different wavevectors. 2. When ω1 6¼ ω2 , the temporally varying phase factor ðω1 ω2 Þt causes periodic temporal beats that have a frequency of f ¼ jω1 ω2 j=2π. In the case when jω1 ω2 j ω1 and jω1 ω2 j ω2 , these beats create a detectable temporal intensity variation at the frequency f . This periodic temporal intensity variation disappears when ω1 ¼ ω2 . When k1 ¼ k2 and φE 1 φE 2 is space independent, these periodic beats in time are spatially uniform patterns that do not vary in space. Figure 5.4 shows the periodic temporal beats produced by the interference between two ﬁelds of the same polarization, same amplitude, and same wavevector, but different frequencies. 3. The phase factor φE 1 φE 2 depends on the phases of the two optical ﬁelds E 1 and E 2 . It deﬁnes the coherence between the two ﬁelds. The two ﬁelds are temporally coherent with each other if φE 1 φE 2 is a constant of time; they are spatially coherent if φE 1 φE 2 is a constant of space. The two ﬁelds are temporally incoherent if φE 1 φE 2 varies randomly with time on the scale of the optical cycle; they are spatially incoherent if φE 1 φE 2 varies randomly with space on the scale of the optical wavelength. Between the extremes of complete coherence and complete incoherence, the two ﬁelds can be partially coherent to different degrees in time, space, or both. Figure 5.3 Stationary periodic fringes produced by the interference between two optical ﬁelds of the same polarization, same amplitude, and same frequency, but different wavevectors. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 174 Optical Interference Figure 5.4 Periodic temporal beats (sold curve) produced by the interference between two ﬁelds (dashed curves) of the same polarization, same amplitude, and same wavevector, but different frequencies. The envelope of the beat notes is shown in dashed gray curves. The time average cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 depends strongly on the degree of coherence. When the two ﬁelds are coherent, φE 1 φE 2 does not vary on the time scale of the optical cycle or on the space scale of the optical wavelength, but it can still vary in time or space slowly so that cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 ¼ cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 : The phase factor φE 1 φE 2 is a constant of both space and time when the phases of the ﬁeld amplitudes E 1 and E 2 are constants or vary in the same manner with space and time. It varies with space or time when the phases of the two ﬁeld amplitudes vary differently with space or time; it varies with both space and time when the phases of the ﬁeld amplitudes have different spatial variations and different temporal variations. Thus, a modulation on the total intensity I in space or time, or both, can be accomplished by properly modulating this phase factor. The principles of most interferometers are based on this concept. EXAMPLE 5.1 A glass wedge of a refractive index n has a small wedge angle of α as shown in Fig. 5.5. It has a length of l in the x direction and a height of h in the y direction. A monochromatic plane optical wave at the wavelength λ vertically illuminates the wedge from above. If the optical wave is coherent, ﬁnd the locations of the bright and dark fringe lines when viewed from above. What is the period of the fringes? How many periods of interference fringes appear on the top surface of the wedge? What happens to the fringes if the light is not completely coherent? Solution: The incident wave propagates in the negative y direction with a wavevector of ki ¼ k^y . When viewed from above, there are two reﬂected waves, from the two surfaces of the glass wedge, respectively. The ﬁrst is reﬂected from the top wedge surface; it has a wavevector of k1 ¼ k sin 2α^x þ k cos 2α^y at an angle of 2α from the y direction. The second is reﬂected from the bottom wedge surface; it has a wavevector of k2 ¼ k^y in the y direction. Thus, k1 k2 ¼ k sin 2α^x þ kð cos 2α 1Þ^y : Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.1 Optical Interference 175 Figure 5.5 Interference fringes formed by reﬂected waves from the two surfaces of a glass wedge. Because the two reﬂected waves are from the same source, they have the same frequency: ω1 ¼ ω2 . However, the two reﬂected waves have different phases because the top reﬂection is external reﬂection at nearly normal incidence with a phase change of π for the electric ﬁeld, whereas the bottom reﬂection is internal reﬂection at normal incidence with no phase change. If the incident optical wave is coherent, the phase of the two reﬂected waves does not vary with time so that φE 1 φE 2 ¼ π. Then, cos ðk1 k2 Þ r ðω1 ω2 Þt þ φE 1 φE 2 ¼ cos ð2kx sin α þ π Þ ¼ cos ð2kx sin αÞ: Therefore, I ¼ I 1 þ I 2 I 12 cos ð2kx sin αÞ: Bright fringe lines appear at the locations where cos ð2kx sin αÞ ¼ 1 so that I ¼ I 1 þ I 2 þ I 12 ; dark fringe lines appear where cos ð2kx sin αÞ ¼ 1 so that I ¼ I 1 þ I 2 I 12 . We ﬁnd that a dark fringe line appears at the tip of the wedge at x ¼ 0. Therefore, the dark and bright fringe lines appear, respectively, at the locations: π λ λl ¼m m , m ¼ 0, 1, 2 . . . k sin α 2n sin α 2nh 1 π 1 λ 1 λl b xm ¼ m þ ¼ mþ mþ , m ¼ 0, 1, 2 . . . 2 k sin α 2 2n sin α 2 2nh xdm ¼ m where we take sin α h=l for a small angle of α. The period Λ of the fringes is found for 2kΛ sin α ¼ 2π: Λ¼ π λ λl ¼ : k sin α 2n sin α 2nh The number of periods over the length is M¼ l 2nl sin α 2nh ¼ : Λ λ λ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 176 Optical Interference If the incident optical wave is not coherent, then φE 1 φE 2 is not a constant of time. Because the two reﬂected waves are from the same source, whether they will create interference fringes or not depends on the coherence time of the incident wave, i.e., the degree of coherence or incoherence of the wave. The difference in the optical path lengths between the two reﬂected waves depends on the location of the fringe. It is Δy ¼ 2nh for the last fringe located at the end of the wedge at x ¼ l, and it is 8 mλ, for the mth dark fringe, < xm Δym ¼ Δy ¼ 1 : m þ λ, for the mth bright fringe: l 2 The corresponding time difference of the two waves for the last fringe located at the end of the wedge is Δt ¼ Δy 2nh M ¼ ¼ , c c ν and it is 8m > , Δym <ν Δt m ¼ ¼ 1 1 > c : mþ , 2 ν for the mth dark fringe, for the mth bright fringe: For the mth fringe to appear, the coherence time τ coh of the incident optical wave has to be such that τ coh > Δtm , which means that τ coh is longer than m optical cycles for the mth dark fringe and longer than m þ 1=2 cycles for the mth bright fringe. If the coherence time is sufﬁciently long such that τ coh > Δt, then all fringes on the surface of the wedge appear. 5.1.1 Double-Slit Interference Young’s double-slit experiment established the wave nature of light. Figure 5.6 illustrates the double-slit interference. We consider a monochromatic plane wave of a frequency ω and a wavevector ki ¼ k^x , which is normally incident on two identical slits separated at a spacing of Λ in the z direction. The observation point is in the direction that makes an angle of θ with respect to the x axis and is on a plane at a distance of l from the plane of the slits. The optical path lengths from the two slits to the observation point are r 1 and r 2 , respectively. In the limit that l Λ, the path difference is r 2 r 1 Λ sin θ: (5.11) Because the incoming wave is normally incident on the plane of the slits, the ﬁelds that emerge from the two slits have the same phase at the exit plane of the slits. Because the two slits have the same geometrical dimensions, these ﬁelds have the same polarization and the same amplitude such that E 1 ¼ E 2 ¼ ^e E 0 . The total ﬁeld at the observation point is the linear superposition of the ﬁelds from the two slits: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.1 Optical Interference 177 Figure 5.6 Double-slit interference. E ¼ E 1 eikr1 iωt þ E 2 eikr2 iωt ¼ ^e E 0 eikr1 iωt 1 þ eiδ , (5.12) δ ¼ k ðr 2 r 1 Þ kΛ sin θ (5.13) where is the phase difference at the observation point between the two ﬁelds that come from the two slits. The intensity at the observation point is I ¼ 4I 0 cos2 δ , 2 (5.14) where I 0 / jE 0 j2 is the intensity contributed by a single slit alone. This result can be obtained from (5.9) because I 1 ¼ I 2 ¼ I 0 , I 12 ¼ I 1 þ I 2 ¼ 2I 0 , and φ1 φ2 ¼ δ. EXAMPLE 5.2 Find the angles at which the double-slit interference from normal incidence of a plane wave shows bright interference fringes. Find the locations of the bright fringes on a screen that is at a distance of l from the slits. Solution: The intensity pattern of the double-slit interference from normal incidence of a plane wave is that given in (5.14). A bright interference fringe appears when cos 2 δ ¼1 2 ) δ ¼ 2qπ for q ¼ 0, 1, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 2, . . . 178 Optical Interference Using (5.13), the qth-order bright interference fringe appears at the angles θq : kΛ sin θq ¼ 2qπ ) sin θq ¼ 2qπ qλ ¼ kΛ nΛ ) θq ¼ sin1 qλ , nΛ where n is the refractive index of the medium. On a screen that is located at a distance of l from the slits, the qth-order bright fringe is found at a distance of zq ¼ l sin θq ¼ q λl nΛ from the zeroth-order bright fringe, which is located at z ¼ 0. 5.1.2 Optical Interferometers Optical interference has been developed into many advanced concepts and applications. One important application is interferometry, which uses optical interference to interrogate the characteristics, including the polarization state, the wavevector, the frequency, and the phase, of an optical wave with respect to a reference wave. Many types of interferometers have been developed. The most important ones for photonics applications include the Michelson interferometer, MachZehnder interferometer, and FabryPérot interferometer. The Michelson interferometer and the MachZehnder interferometer are illustrated below. The FabryPérot interferometer is discussed in Section 5.3. Michelson Interferometer The Michelson interferometer was used in the historical MichelsonMorley experiment. Figure 5.7 shows its basic structure. The single beam splitter in this structure deﬁnes four optical paths. The two paths that are respectively on the left of and below the beam splitter Figure 5.7 Michelson interferometer. BS, beam splitter. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.1 Optical Interference 179 deﬁne two ports, each of which serves as a port for both input and output. Input light can be sent into either port or into both ports, but usually only one input is supplied, as shown in Fig. 5.7 where only port 1 receives an input of an intensity I in while port 2 receives no input. By contrast, both ports always function as output ports with output intensities of I out, 1 and I out, 2 , respectively, though the output intensity at a port can be zero when totally destructive interference occurs at the port. The input wave is split by the beam splitter into two waves, each of which enters one of the two internal paths that are respectively above and on the right of the beam splitter. Each internal path ends with a totally reﬂective mirror, which reﬂects the light back to the beam splitter. The beam splitter again divides each returning wave into one reﬂected wave and one transmitted wave for the two output ports. Each output ﬁeld is the combination of one reﬂected ﬁeld from one internal path and one transmitted ﬁeld from the other internal path: The output ﬁeld at port 1 is the linear superposition of the reﬂected ﬁeld from the vertical internal path and the transmitted ﬁeld from the horizontal internal path, whereas the output ﬁeld at port 2 is the linear superposition of the transmitted ﬁeld from the vertical internal path and the reﬂected ﬁeld from the horizontal internal path. Though the two component ﬁelds of each output ﬁeld come from different internal paths, they have the same polarization, the same frequency, and the same wavevector because they both originate from the same input ﬁeld and they propagate in the same direction. Their phase difference depends only on the optical length difference of the two internal paths and the phase change caused by reﬂection or transmission at the beam splitter. Because the phase change at the beam splitter has a ﬁxed value, the output intensity at a port can be varied by varying the optical length difference of the two internal paths. Note that what matters is not the physical length difference of the paths but the optical length difference. The optical length difference can be varied by varying the physical length difference, through moving one or both mirrors, or by varying the refractive index along one or both paths, through modulating the medium using any of the effects discussed in Sections 2.6 and 2.7. The beam splitter is partially reﬂective and partially transmissive. In practice, it has negligible absorption so that R þ T 1. The beam splitter can have any reﬂectance/transmittance ratio, but complete destructive interference is possible only when it is a 50/50 beam splitter so that the reﬂected ﬁeld and the transmitted ﬁeld have the same magnitude though possibly different phases. Conservation of energy requires that I out, 1 þ I out, 2 ¼ I in when there is no loss in the system. Clearly, I out, 1 ¼ 0 and I out, 2 ¼ I in when complete destructive interference occurs at port 1, whereas I out, 2 ¼ 0 and I out, 1 ¼ I in when complete destructive interference occurs at port 2. Thus, complete constructive interference occurs at one output port when complete destructive interference occurs at the other output port. This condition is clearly required by conservation of energy, but it is not trivial if we take a closer look. It implies that the two component ﬁelds for the total output ﬁeld at port 1 are completely in phase when those at port 2 are completely out of phase. This seems puzzling: each output ﬁeld is the combination of one reﬂected ﬁeld and one transmitted ﬁeld through the beam splitter, but one combination is constructive while the other is destructive at the same time. To resolve this puzzle, we have to pay attention to two key properties of the functioning of an optical beam splitter. (1) An optical beam splitter always has a layer of properly designed and Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 180 Optical Interference accurately implemented coating on one of its two surfaces to accomplish the desired reﬂectance/transmittance ratio. The other surface is often antireﬂection coated to eliminate unwanted reﬂection. In any event, reﬂection takes place on only one surface of the beam splitter. Because the two waves returning from the two different internal paths reach the beam splitter from different sides, one undergoes external reﬂection while the other undergoes internal reﬂection. (2) For any polarization, a transmitted ﬁeld through a lossless dielectric interface has no phase change with respect to the incident ﬁeld. A reﬂected ﬁeld may have either no phase change or a phase change of π, depending on its polarization, its incident angle, and whether it undergoes external reﬂection or internal reﬂection; in any case, the phase difference between external reﬂection and internal reﬂection for a given polarization at a given incident angle is always π. (See Problem 3.4.1.) Considering the above two characteristics, it is clear that the phase difference between the two ﬁeld components at one output port is always different by a phase factor of π from that at the other output port because the reﬂected ﬁeld component for one output port comes from external reﬂection and that for the other output port is from internal reﬂection. For this reason, constructive interference happens at one output port when destructive interference takes place at the other output port, ensuring conservation of energy. Assume that the beam splitter has the reﬂective surface on the left side. Then, reﬂection on the left side of the beam splitter is external reﬂection with a phase change of π and reﬂection on the right side of the beam splitter is internal reﬂection with no phase change. If the beam splitter is a 50/50 splitter, the output intensities of the two output ports are I out, 1 ¼ I in cos2 Δφ , 2 I out, 2 ¼ I in sin2 Δφ , 2 (5.15) where Δφ is the phase difference of the two optical paths. In the case when the two paths are ﬁlled with the same uniform medium, Δφ ¼ 2kðla lb Þ, where la and lb are respectively the lengths of the two arms, and the factor 2 accounts for the fact that the wave in each arm travels through the arm twice before returning to the beam splitter. Mach–Zehnder Interferometer Figure 5.8 shows the basic structure of the MachZehnder interferometer. With two beam splitters, this structure is different from that of the Michelson interferometer in two basic features: The output ports are separate from the input ports, and light propagates through each of the two separate internal paths only once. Despite these differences, the fundamental concepts discussed above for the Michelson interferometer are applicable to the MachZehnder interferometer. The output intensity at a given port can be varied by varying the difference of the optical path lengths between the two paths, which can be accomplished by varying the physical length difference between the two paths or by varying the refractive index in the medium along one or both paths. When constructive interference occurs at one output port, destructive interference happens at the other output port. Thus, I out, 1 þ I out, 2 ¼ I in for a lossless system. Assume that each beam splitter has the reﬂective surface on the left side. Then, reﬂection on the left side of each beam splitter is external reﬂection with a phase change of π and reﬂection on the right side of each beam splitter is internal reﬂection with no phase change. If both beam splitters are 50/50 splitters, the output intensities of the two output ports are Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.1 Optical Interference 181 Figure 5.8 MachZehnder interferometer. BS, beam splitter. Figure 5.9 MachZehnder interferometers in the waveguide form using (a) two Y-junction waveguides and (b) two directional couplers. Only one input is supplied in this illustration. In general, the lengths of the two arms are not identical. I out, 1 ¼ I in sin2 Δφ , 2 I out, 2 ¼ I in cos2 Δφ , 2 (5.16) where Δφ is the phase difference of the two optical paths. In the case when the two paths are ﬁlled with the same uniform medium, Δφ ¼ k ðla lb Þ, where la and lb are respectively the lengths of the two arms, and the wave in each arm travels through the arm only once before reaching the output beam splitter. The MachZehnder interferometer can be implemented in various waveguide forms. Figure 5.9 shows two common forms using (a) Y-junctions and (b) directional couplers for Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 182 Optical Interference the beam-splitting function. In the case when the Y-junctions and the directional couplers are all 3-dB couplers such that ξ ¼ 1=2, we ﬁnd that T ¼ cos2 Δφ 2 (5.17) for the interferometer using 3-dB Y-junctions shown in Fig. 5.9(a), and T ¼ sin2 Δφ 2 (5.18) for the interferometer using 3-dB directional couplers shown in Fig. 5.9(b), where Δφ ¼ φa φb is the phase difference of the two optical paths. 5.1.3 Standing Wave In the analysis and discussion presented above following (5.7), we have assumed that the angle between the two wavevectors k1 and k2 of the interfering waves is small, or zero as in the case of the interferometers. In the case when the angle between k1 and k2 is large, the principle of linear superposition expressed as (5.5) is still valid and interference between two ﬁelds still occurs, but the intensity of the combined ﬁeld expressed as (5.7) is not valid. Here we consider the special case when two waves have the same polarization, ^e 2 ¼ ^e 1 ¼ ^e , the same amplitude, E 2 ¼ E 1 ¼ E, and the same frequency, ω2 ¼ ω1 ¼ ω, but they propagate in opposite directions, k2 ¼ k1 ¼ k, so that E1 ¼ ^e 1 E 1 eik1 riω1 t ¼ ^e Eeik riωt and E2 ¼ ^e 2 E 2 eik2 riω2 t ¼ ^e Eeik riωt . The linear superposition of these two ﬁelds yields Eðr; t Þ ¼ E1 ðr; t Þ þ E2 ðr; t Þ ¼ ^e Eeik riωt þ ^e Eeik riωt ¼ 2^e Eeiωt cos ðk rÞ: (5.19) For simplicity of discussion without loss of generality, we assume linear polarization and a ﬁeld amplitude of E ¼ jEj by taking its phase to be zero. Then the real ﬁeld of the combined ﬁeld can be expressed as Eðr; t Þ ¼ Eðr; t Þ þ E ðr; tÞ ¼ 4^e jEj cos ωt cos ðk rÞ: (5.20) The spatial variation of this ﬁeld is decoupled from the temporal variation. We ﬁnd that Eðr; t Þ vanishes for all times at the ﬁxed locations, known as nodes, where k r ¼ ð2q þ 1=2Þπ for integers q so that cos ðk rÞ ¼ 0, as shown in Fig. 5.10. The nodes are periodically distributed along the line deﬁned by k^ at a spacing of π=k ¼ λ=2n, where λ=n is the wavelength of the optical ﬁeld in the medium of a refractive index n. At the locations where k r ¼ 2qπ so that cos ðk rÞ ¼ 1, we ﬁnd that Eðr; tÞ ¼ 4^e jE j cos ðωt Þ; such locations are known as antinodes. The antinodes are also periodically distributed along the line deﬁned by k^ at a spacing of π=k ¼ λ=2n. An antinode is found at the midpoint between two neighboring nodes. Because the nodes and antinodes are ﬁxed in space, the ﬁeld given in (5.20) appears to stand still in space. It does not travel but only oscillates in time. Therefore, the interference of the two contrapropagating waves of the same polarization, the same frequency, and the same amplitude results in a standing wave. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.2 Optical Gratings 183 Figure 5.10 Standing wave. Nodes, labeled with N, are periodically distributed along the line deﬁned by k^ at a spacing of π=k ¼ λ=2n. Each antinode, labeled with A, is located at the midpoint between two neighboring nodes. A standing wave oscillates in time but appears to stand still in space. 5.2 OPTICAL GRATINGS .............................................................................................................. An optical grating is a periodic optical structure. Some waveguide grating structures are illustrated in Fig. 4.3. The functioning of an optical grating can be understood from the viewpoint of phase matching, as discussed in Chapter 4, or from the viewpoint of optical interference. In this section, we make the connection between these two viewpoints. The concept of double-slit interference discussed in the preceding section can be extended to equally spaced multiple slits of identical geometrical parameters, which form a periodic structure of a period Λ in the z direction, as shown in Fig. 5.11. The slits are on the yz plane, which is normal to the x axis. Being a periodic optical structure, this multiple-slit structure can be considered a grating. Indeed, it functions as a transmissive diffraction grating, also called a transmission grating. 5.2.1 Normal Incidence We ﬁrst consider normal incidence of a monochromatic plane wave of a frequency ω and a wavevector ki ¼ k^x on the periodic multiple-slit structure, as shown in Fig. 5.11. Because the incoming plane wave is normally incident on the plane of the slits, the ﬁelds that emerge from all of the slits have the same phase at the exit plane of the slits, which is perpendicular to ki . They also have the same polarization and the same amplitude because the slits have the same geometrical dimensions. Therefore, on the exit plane of the slits, E 1 ¼ E 2 ¼ ¼ E N ¼ ^e E 0 : As seen in Fig. 5.11, at a distant point in the direction at an angle of θ with respect to the x axis, the phases of the rays coming from different slits increase between successive slits by the amount of δ ¼ kΛ sin θ given in (5.13). Following the same reasoning for the double slits, the total ﬁeld at the distant point in this direction is the linear superposition of the ﬁelds coming from all slits to the point: E ¼ E 1 eikr1 iωt þ E 2 eikr2 iωt þ þ E N eikrN iωt ¼ ^e E 0 eikr1 iωt 1 þ eiδ þ þ eiðN1Þδ ¼ ^e E 0 eikr1 iωt iNδ 1e : 1 eiδ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 (5.21) 184 Optical Interference Figure 5.11 Normal incidence of a monochromatic plane wave on a periodic multiple-slit structure. The intensity at the distant observation point is I ¼ I0 sin2 ðNδ=2Þ , sin2 ðδ=2Þ (5.22) where I 0 / jE 0 j2 is the intensity contributed by a single slit alone. Using the mathematical relations lim x!0 sin2 Nx ¼ N 2 and sin2 x sin2 ½N ðx þ qπ Þ sin2 Nx ¼ for q ¼ 0, sin2 ðx þ qπ Þ sin2 x 1, 2, , (5.23) we ﬁnd that the intensity I has maxima of the value N 2 I 0 when δ ¼ kΛ sin θq ¼ 2qπ, (5.24) where q is an integer that represents the order of diffraction. Figure 5.12 shows the intensity distribution given in (5.22) for the multiple-slit structure as a function of the phase factor δ ¼ kΛ sin θ, which varies with the angle θ for ﬁxed values of k and Λ. The primary maxima that have the peak intensity of N 2 I 0 appear at the angles that satisfy the condition given in (5.24). Secondary maxima of lower peak intensities exist between primary maxima. As the number N of the periods in the structure increases, the peak intensity of each primary maximum increases quadratically as N 2 while the width decreases linearly as N 1 ; meanwhile, the peak intensities of all secondary maxima decrease. The periodic multiple-slit structure functions as a transmissive diffraction grating that has a wavenumber of K¼ 2π Λ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 (5.25) 5.2 Optical Gratings 185 Figure 5.12 Intensity distribution as a function of the phase factor δ ¼ kΛ sin θ for a multiple-slit structure functioning as a transmission grating. As the number N of periods increases, the primary maxima representing the diffraction orders have peak intensities increasing as N 2 and widths decreasing as N 1 while the peak intensities of all secondary maxima decrease. in the z direction. Each primary maximum in the spatial intensity distribution of the transmitted light represents a diffraction order. The qth-order diffracted beam has a wavevector of kq ¼ k cos θq ^x þ k sin θq ^z : (5.26) Using the relations in (5.24) and (5.25), we ﬁnd that k sin θq ¼ qK: (5.27) Because the wavevector of the incident wave is ki ¼ k^x , there is a phase mismatch of (5.28) Δkq ¼ kq ki ¼ k cos θq 1 ^x þ k sin θq ^z ¼ k cos θq 1 ^x þ qK^z between the qth-order diffracted beam and the incident wave. A phase mismatch between two waves is a momentum difference between two photons of the two waves. Clearly from (5.28), except for the zeroth order, an incident photon acquires momentum changes in both x and z directions in the process to exit as a diffracted photon. Because of conservation of momentum, any momentum change of a photon has to be compensated by an opposite momentum change of another physical object. The momentum change Δkq, x ¼ k cos θq 1 in the x direction is easily compensated by an opposite momentum Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 186 Optical Interference change of the entire multiple-slit structure in the x direction because it is in the direction normal to the plane of the structure and it is negligibly small for the mass of the structure. In the z direction, however, no such momentum compensation is possible if the slits are absent from the structure because no force in the direction parallel to the plane of the structure can be exerted on the structure. In the presence of the periodic slits, the periodicity along the z direction provides the necessary compensation for the momentum change of Δkq, z ¼ k sin θq in the z direction when the phase-matching condition k sin θq ¼ qK of (5.27) is satisﬁed. Therefore, constructive interference for a diffracted beam is equivalent to phase matching for the beam. 5.2.2 Oblique Incidence In the above, we considered a monochromatic plane wave that is normally incident on the multiple-slit structure at an incident angle of θi ¼ 0 so that ki ¼ k^x . The equivalent concepts of constructive interference and phase matching for the diffraction orders can also be applied to oblique incidence at a nonzero incident angle of θi 6¼ 0 so that ki ¼ k i, x ^x þ k i, z^z ¼ k cos θi ^x þ k sin θi^z 6¼ k^x , as shown in Fig. 5.13. With an incident wave of this wavevector, the ﬁeld emerging from the slits has a phase shift of k i, z Λ ¼ kΛ sin θi from one slit to the next in the z direction so that E 1 ¼ ^e E 0 , E 2 ¼ E 1 eiki, z Λ ¼ ^e E 0 eikΛ sin θi , . . . , E N ¼ E 1 eiðN1Þki, z Λ ¼ ^e E 0 eiðN1ÞkΛ sin θi . Applying these relations to (5.21), we ﬁnd that the phase factor δ ¼ kΛ sin θ for normal incidence at θi ¼ 0 is generalized to δ ¼ kΛ sin θ kΛ sin θi (5.29) for oblique incidence at θi 6¼ 0. Therefore, the condition given in (5.24) for ﬁnding the maxima of the diffracted intensity distribution is generalized to the condition: δ ¼ kΛ sin θq kΛ sin θi ¼ 2qπ, where q is an integer. Figure 5.13 Oblique incidence of a monochromatic plane wave on a periodic multiple-slit structure. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 (5.30) 187 5.2 Optical Gratings From the phase-matching point of view, the condition in (5.30) can be easily obtained from the condition for phase matching assisted by the grating of a wavenumber K in the z direction: Δkz ¼ k q, z k i, z ¼ qK, (5.31) which is identical to (5.30) in the form of k sin θq ¼ k sin θi þ qK: (5.32) As discussed above for the case of normal incidence, it is also true for oblique incidence that phase matching in the x direction normal to the plane of the grating structure does not set a required condition because it is automatically satisﬁed by a compensating momentum change of the massive structure. Note that the zeroth order takes place at θ0 ¼ θi . EXAMPLE 5.3 A monochromatic plane wave at the λ ¼ 651 nm wavelength is normally incident on the plane of an array of equally spaced slits. The 20th-order diffraction peak is found at the angle of θ20 ¼ 10 . Find the spacing Λ between neighboring slits. If a plane wave at λ ¼ 488 nm is normally incident on the slits, what is the diffraction angle of the 20th-order diffraction peak? If it is obliquely incident for the 20th-order diffraction peak to appear at θ20 ¼ 10 , what is the required incident angle? Solution: For normal incidence with λ ¼ 651 nm and θ20 ¼ 10 , (5.27) requires that k sin θq ¼ qK ) sin θq q ¼ λ Λ ) Λ¼ 651 109 m ¼ 75 μm: sin 10 qλ 20 ¼ sin θq For λ ¼ 488 nm at normal incidence, the 20th-order diffraction peak appears at k sin θq ¼ qK ) θq ¼ sin1 qλ Λ ) θ20 ¼ sin1 20 488 109 ¼ 7:48 : 75 106 For oblique incidence, the incident angle is found using (5.32) as qλ 1 sin θq k sin θq ¼ k sin θi þ qK ) θi ¼ sin : Λ For the 20th-order diffraction peak of λ ¼ 488 nm to appear at θ20 ¼ 10 , we ﬁnd 20 488 109 1 ¼ 2:49 : sin 10 θi ¼ sin 75 106 5.2.3 Grating at an Interface When an optical wave is incident on a grating at an interface between two different optical media, as shown in Fig. 5.14, diffraction orders in reﬂection and in transmission can both Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 188 Optical Interference Figure 5.14 (a) Optical grating at an interface. (b) Phase-matching conditions for the reﬂective and transmissive diffraction orders. appear. Assuming that the incident wave comes from medium 1, which has a refractive index of n1 , at an incident angle of θi with respect to the normal of the interface, the diffraction orders on both sides of the interface are determined by the phase-matching conditions: k1 sin θ1q ¼ k1 sin θi þ qK (5.33) for the reﬂective diffraction orders in medium 1, and k2 sin θ2q ¼ k2 sin θi þ qK (5.34) for the transmissive diffraction orders in medium 2. Note that for the zeroth order, θ10 ¼ θi in reﬂection and n2 sin θ20 ¼ n1 sin θi in transmission, which are just those required by Snell’s law for a ﬂat surface when the grating does not exist. Here we only consider the phase-matching conditions that determine the direction of each diffraction order; whether a diffraction order appears or not also depends on the shape and the geometrical parameters of the grating, as discussed in Example 4.2. EXAMPLE 5.4 A grating that has a period of Λ ¼ 2 μm is fabricated on the surface of a glass plate, which has a refractive index of 1:5. It is exposed to air. A laser beam at the wavelength of λ ¼ 850 nm is normally incident on the grating from the air side. How many diffraction orders are possible on each side? What is the diffraction angle of each order? Solution: For normal incidence, θi ¼ 0 . Thus, the phase-matching conditions in (5.33) and (5.34) reduce to k1 sin θ1q ¼ qK and k2 sin θ2q ¼ qK, which can be expressed as sin θ1q ¼ qλ qλ and sin θ2q ¼ : n1 Λ n2 Λ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.2 Optical Gratings 189 Every diffraction angle is required to be within the range between 90 and 90 , i.e., 1 sin θ1q 1 and 1 sin θ2q 1. On the air side, n1 ¼ 1; thus 1 sin θ1q ¼ qλ 1 n1 Λ ) 0 jqj n1 Λ 1 2 106 ¼ 2:35: ¼ λ 850 109 There are ﬁve diffraction orders on the air side for q ¼ 2,1, 0, 1, 2. The diffraction angles with respect to the surface normal are θ1q ¼ sin1 ) qλ q 850 109 ¼ sin1 n1 Λ 1 2 106 θ1q ¼ 58:21 , 25:15 , 0 , 25:15 , 58:21 : On the glass side, n2 ¼ 1:5; thus 1 sin θ2q ¼ qλ 1 n2 Λ ) 0 jqj n2 Λ 1:5 2 106 ¼ 3:52: ¼ λ 850 109 There are seven diffraction orders on the glass side for q ¼ 3, 2, 1, 0, 1, 2, 3. The diffraction angles with respect to the surface normal are θ2q ¼ sin1 ) qλ q 850 ¼ sin1 n2 Λ 1:5 2 109 106 θ1q ¼ 58:21 , 34:52 , 16:46 , 0 , 16:46 , 34:52 , 58:21 : 5.2.4 Surface Grating–Waveguide Coupling A grating fabricated on the surface of a waveguide can couple a radiation ﬁeld that propagates in the homogeneous space on one side of the waveguide into a waveguide mode. In reverse operation, it can also couple a waveguide mode into a radiation ﬁeld from the surface of the waveguide. These concepts are illustrated in Fig. 5.15. For this purpose, it is necessary to phase match the radiation ﬁeld with the waveguide mode in the longitudinal direction of the waveguide, which is taken to be the z direction. For coupling Figure 5.15 Surface grating for (a) input coupling and (b) output coupling of a waveguide mode. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 190 Optical Interference with a waveguide mode that has a propagation constant of β, the incident optical wave has to satisfy the phase-matching condition: k2 sin θ2q þ qK ¼ β (5.35) if the wave is incident from the substrate side of a refractive index n2 at an incident angle of θ2q , or k3 sin θ3q þ qK ¼ β (5.36) if the wave is incident from the cover side of a refractive index n3 at an incident angle of θ3q . The same phase-matching conditions are used to determine the directions of output coupling. Note that the phase-matching conditions given in (5.35) and (5.36) only determine the directions of the radiation ﬁelds that can be coupled into or out from a waveguide mode, but they do not tell us the efﬁciency of the coupling. The coupling efﬁciency is determined by the coupling coefﬁcient, which depends on the shape, the depth, and other geometrical parameters of the grating, as discussed in Example 4.2. EXAMPLE 5.5 A sinusoidal grating that can only serve as a ﬁrst-order grating is fabricated on the surface of a GaAs slab waveguide as shown in Fig. 5.15. The cover of the waveguide is simply air so that n3 ¼ 1. At the wavelength of λ ¼ 1:3 μm, the propagation constant of the TE0 mode of this waveguide is β ¼ 1:62 107 nm, corresponding to an effective index of nβ ¼ 3:35. If it is desired that a laser beam at this wavelength be coupled into this guided mode through the surface grating at an incident angle of θi ¼ 45 , what is the required period of the grating? Solution: Because a sinusoidal grating can be used only as a ﬁrst-order grating, it is necessary that the phase-matching condition is satisﬁed for q ¼ 1 or q ¼ 1. Because the wave is incident from the cover side, the condition is that from (5.36) with q ¼ 1: k 3 sin θ31 þ K ¼ β ) n3 1 nβ sin θ31 þ ¼ λ λ Λ ) Λ¼ λ nβ n3 sin θ31 : With λ ¼ 1:3 μm, nβ ¼ 3:35, n3 ¼ 1, and θ31 ¼ θi ¼ 45 , the required grating period is Λ¼ λ nβ n3 sin θ31 ¼ 1:3 106 m ¼ 492 nm: 3:35 1 sin 45 5.2.5 Flat Interface The phase-matching concept can be applied to reﬂection and refraction at a ﬂat, smooth interface between two media of different indices n1 and n2 to obtain Snell’s law discussed in Section 3.4. For a smooth surface that is not modiﬁed by any periodic structure, we can take the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.3 Fabry–Pérot Interferometer 191 limit of an inﬁnitely large period, Λ ! ∞, thus a zero wavenumber of K ¼ 0. Then, by applying (5.31) with K ¼ 0 to the reﬂected and transmitted waves, we obtain the following phasematching condition: k r, z ki, z ¼ k t, z ki, z ¼ 0, (5.37) which yields the condition of ki sin θi ¼ k r sin θr ¼ k t sin θt given in (3.88) and Snell’s law expressed in (3.89) and (3.90). 5.3 FABRY–PÉROT INTERFEROMETER .............................................................................................................. The basic principle of the Fabry–Pérot interferometer is the interference of multiple reﬂections from two partially reﬂective parallel surfaces. The desired reﬂectivity for each of these two surfaces can be obtained by proper coating. The basic structure of the Fabry–Pérot interferometer takes two different forms. The ﬁrst form shown in Fig. 5.16(a) consists of two partially reﬂective mirrors on the parallel inner surfaces of two dielectric plates; the outer surfaces of the plates are antireﬂection coated and often wedged to prevent unwanted reﬂection from these surfaces. In the second form shown in Fig. 5.16(b), the two partially reﬂective surfaces are the parallel surfaces of a transparent dielectric plate; a Fabry–Pérot interferometer of this form is usually called a Fabry–Pérot etalon. The two structures shown in Figs. 5.16(a) and (b) have the same interferometric characteristics despite the differences in their detailed structures. For both structures, we consider a physical spacing of l that is ﬁlled with a medium of a refractive index n between the two partially reﬂective surfaces, as shown in Fig. 5.16. The direction normal to the reﬂective surfaces is taken to be the z direction. We consider for generality oblique incidence of a Figure 5.16 (a) Fabry–Pérot interferometer. The outer surfaces of the wedged plates are antireﬂection coated. (b) Fabry–Pérot etalon. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 192 Optical Interference monochromatic plane wave of a frequency ω and a wavelength λ. The wavevector of the wave that is transmitted through the ﬁrst partially reﬂective surface makes an angle of θ with respect to the normal of the reﬂective surface; this angle is not necessarily the same as the incident angle of the wave coming from outside because the refractive index of the outside medium is not necessary the same as that inside the interferometer. The ﬁeld-amplitude reﬂection coefﬁcients r 1 and r2 of the left and right mirrors, respectively, can be expressed as 1=2 1=2 r1 ¼ R1 eiφ1 , r 2 ¼ R2 eiφ2 , (5.38) where R1 and R2 are the intensity reﬂectivities of the left and right reﬂective surfaces, respectively, and φ1 and φ2 are the phase changes of the optical ﬁelds upon reﬂection on these surfaces. As discussed in Section 3.4, the reﬂection coefﬁcients r 1 and r 2 are functions of the incident angle θ and the polarization of the optical ﬁeld. Multiple partial reﬂections inside the interferometer take place at the two partially reﬂective surfaces, as seen in Fig. 5.16. Between the two reﬂective surfaces, all forward-propagating waves have the same wavevector at an angle of θ with respect to the z direction so that k z ¼ k cos θ, and all backward-propagating waves have the same wavevector at an angle of π θ with respect to the z direction so that k z ¼ k cos ðπ θÞ ¼ k cos θ. Each forward or backward pass through the spacing of a length l causes a phase shift of kl cos θ. Each time a wave reaches a reﬂective surface, part of it is transmitted and the rest of it is reﬂected; multiple reﬂections by the reﬂective surfaces produce multiple transmitted waves. At a given location on the outside of the interferometer, each successive transmitted ﬁeld is related to the preceding transmitted ﬁeld by a factor of 1=2 1=2 1=2 1=2 r 1 r 2 ei2kl cos θ ¼ R1 R2 eið2kl cos θþφ1 þφ2 Þ ¼ R1 R2 eiφRT , (5.39) where φRT ¼ 2kl cos θ þ φ1 þ φ2 ¼ 4π νnl nl cos θ þ φ1 þ φ2 ¼ 4π cos θ þ φ1 þ φ2 c λ (5.40) is the total phase shift caused by a round-trip passage between the two reﬂective surfaces. This phase shift includes the phase shift of 2kl cos θ from the double passes through the medium in the spacing and the localized phase shifts of φ1 and φ2 from reﬂections at the two reﬂective surfaces. The interferometer has two output ports: one in the forward direction for the total transmitted ﬁeld and the other in the backward direction for the total reﬂected ﬁeld. The total transmitted ﬁeld through the interferometer at the forward output port is the linear sum of all transmitted ﬁelds through the second reﬂective surface: Etout ¼ E 0 eiωt þ E 1 eiωt þ E 2 eiωt þ 1=2 1=2 1=2 1=2 ¼ E 0 eiωt 1 þ R1 R2 eiφRT þ R1 R2 eiφRT ¼ E 0 eiωt 1 1=2 1=2 1 R1 R2 eiφRT 2 þ , Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 (5.41) 5.3 Fabry–Pérot Interferometer 193 where E 0 is the transmitted ﬁeld that directly passes through the two reﬂective surfaces, E 1 is the transmitted ﬁeld after one reﬂection by each reﬂective surface, and E 2 is the transmitted ﬁeld after two reﬂections by each reﬂective surface, and 2 . so forth. 1=2 1=2 iφRT t From (5.41), the total transmitted intensity is I out ¼ I 0 1 R1 R2 e , where the intensity I 0 of the directly transmitted ﬁeld E 0 is related to the input intensity as I 0 ¼ ð1 R1 Þð1 R2 ÞI in . Therefore, the transmittance of a lossless Fabry–Pérot interferometer for the forward output port is T FP ¼ I tout ð1 R1 Þð1 R2 Þ ð1 R1 Þð1 R2 Þ : ¼ 2 ¼ 2 I in 1=2 1=2 1=2 1=2 1=2 1=2 iφ 2 RT R þ 4R R sin ð φ =2 Þ 1 R R e 1 R RT 1 2 1 2 1 2 (5.42) The reﬂectance of the Fabry–Pérot interferometer for the backward output port is RFP ¼ I rout ¼ 1 T FP : I in (5.43) The maximum transmittance of the Fabry–Pérot interferometer is T max FP ¼ ð1 R1 Þð1 R2 Þ 1=2 1=2 1 R1 R2 2 : (5.44) The maximum transmittance is T max FP ¼ 1 for a lossless symmetric Fabry–Pérot interferometer max that has R1 ¼ R2 , but T FP < 1 for an asymmetric Fabry–Pérot interferometer that has R1 6¼ R2 . We can deﬁne a normalized transmittance as T FP T^ FP ¼ max ¼ T FP where 1 2 4F 1 þ 2 sin2 ðφRT =2Þ π , (5.45) 1=4 1=4 F¼ πR1 R2 1=2 1=2 1 R1 R2 (5.46) is the ﬁnesse of the lossless Fabry–Pérot interferometer. As expressed in (5.46) and plotted in Fig. 5.17, the ﬁnesse of a lossless Fabry–Pérot interferometer is a nonlinear function of the product, R1 R2 , of the reﬂectivities of the two reﬂective surfaces that form the interferometer. The normalized transmittance T^ FP of a lossless Fabry–Pérot interferometer expressed in (5.45) is plotted in Fig. 5.18 as a function of the round-trip phase shift φRT for a few values of the ﬁnesse of the interferometer. The strong dependence of T^ FP on φRT is the consequence of the interference of the multiple reﬂections between the two reﬂective surfaces. The transmittance peaks appear at φRT ¼ 2qπ, νq ¼ q φ1 þ φ2 c , 2π 2nl cos θ (5.47) where q is an integer so that all transmitted ﬁelds resulting from multiple reﬂections in the interferometer are in phase for constructive interference. The separation between two Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 194 Optical Interference Figure 5.17 Finesse, F, of a lossless Fabry–Pérot interferometer as a function of the product, R1 R2 , of the reﬂectivities of the two reﬂective surfaces of the interferometer. Figure. 5.18 Normalized transmittance T^ FP of a lossless Fabry–Pérot interferometer as a function of the roundtrip phase shift φRT for a few values of the ﬁnesse of the interferometer. neighboring peaks in the spectrum is called the free spectral range, which has a round-trip phase difference of ΔφFSR and a frequency difference of ΔνFSR : c ΔφFSR ¼ 2π, ΔνFSR ¼ : (5.48) 2nl cos θ Away from the peaks, the transmittance is low because the transmitted ﬁelds are out of phase, resulting in destructive interference. Each transmittance peak has a ﬁnite FWHM linewidth, Δφline , measured in terms of the shift in the round-trip phase, or Δνline , measured in terms of the optical frequency. Actually, the ﬁnesse is deﬁned as the ratio of the free spectral range to the linewidth: F¼ ΔφFSR ΔνFSR ¼ : Δφline Δνline Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 (5.49) 5.3 Fabry–Pérot Interferometer 195 The relation given in (5.46) for a lossless Fabry–Pérot interferometer is a valid approximation for F 1: Therefore, the linewidth decreases with increasing ﬁnesse, which in turn increases nonlinearly with the value of R1 R2 . As seen in (5.40), the round-trip phase φRT is a function of the wavelength λ of the optical wave, the physical spacing l of the interferometer, the refractive index n of the medium between the two reﬂective surfaces, and the angle θ at which the wave propagates inside the interferometer and is incident on the reﬂective surfaces. The transmittance of a Fabry–Pérot interferometer can be varied by varying any of these physical parameters. The strong dependence of the transmittance on the optical wavelength, thus on the optical frequency, allows a high-ﬁnesse Fabry–Pérot interferometer to be used as an optical spectrum analyzer. A high ﬁnesse leads to a narrow linewidth for the transmittance peaks, thus a high resolution for the optical spectrum analyzer. Further detailed characteristics of the Fabry–Pérot interferometer used as an optical resonator are discussed in Chapter 6. EXAMPLE 5.6 What happens to the maximum transmittance T max FP , the ﬁnesse F, the frequencies νq at which the peak transmittance occurs, the free-spectral range ΔνFSR , and the spectral linewidth Δνline of a Fabry–Pérot interferometer in each of the following situations? (a) The reﬂectivity R1 or R2 is increased, or both are increased. (b) The spacing l is increased. (c) The index n of the medium between the reﬂective surfaces is increased. (d) The angle θ at which the wave propagates between the reﬂective surfaces is increased. Solution: The transmittance of a Fabry–Pérot interferometer is a direct function of only three parameters, R1 , R2 , and φRT , as seen in (5.42); however, φRT is a function of the parameters l, n, θ, and the optical frequency ν. Each of the other characteristics of the Fabry–Pérot interferometer depends on some of these parameters but is independent of the other parameters. (a) The reﬂectivity R1 or R2 is increased, or both are increased. From (5.44), we ﬁnd that T max FP does not monotonically vary with R1 or R2 . Indeed, we ﬁnd max max dT FP dT FP ¼ signðR2 R1 Þ and sign ¼ signðR1 R2 Þ: sign dR1 dR2 Therefore, T max FP increases with increasing R1 if R1 < R2 , but it decreases with R1 if max R1 R2 , including when R1 ¼ R2 because T max FP reaches its largest value of T FP ¼ 1 when R1 ¼ R2 . Similarly, T max FP increases with increasing R2 if R1 > R2 , but it decreases with R2 if R1 R2 , including when R1 ¼ R2 . From (5.46), we ﬁnd that the ﬁnesse F monotonically increases with the product R1 R2 ; therefore, it increases when R1 R2 is increased through increasing either R1 or R2 , or both. From (5.47) and (5.48), we ﬁnd that both νq and ΔνFSR do not vary with R1 or R2 . From (5.49), we ﬁnd that Δνline decreases when the product R1 R2 is increased because Δνline ¼ ΔνFSR =F. (b) The spacing l is increased. From (5.44) and (5.46), we ﬁnd that both T max FP and F are independent of the spacing l; they do not change as l is increased. From (5.47) and (5.48), Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 196 Optical Interference we ﬁnd that both νq and ΔνFSR decrease when the spacing l is increased. From (5.49), we ﬁnd that Δνline decreases with increasing l because Δνline ¼ ΔνFSR =F. (c) The index n of the medium between the reﬂective surfaces is increased. From (5.40), (5.47), and (5.48), we ﬁnd that the index n and the spacing l always appear together in the form of their product nl. Indeed what counts is the optical path length nl, rather than the physical length. Therefore, increasing the index n has exactly the same consequences as increasing the spacing l discussed in (b). (d) The angle θ at which the wave propagates between the reﬂective surfaces is increased. From (5.40), (5.47), and (5.48), we ﬁnd that actually the angle θ always appears together with the index n and the spacing l in the form of nl cos θ. Increasing θ reduces the effective optical path length nl cos θ. Therefore, increasing θ is equivalent to reducing the spacing l or the refractive index n: Both T max FP and F do not change with θ; both νq and ΔνFSR increase with increasing θ; Δνline increases with increasing θ. 5.3.1 Optical Thin Films Optical thin ﬁlms are thin layers of optical materials that have thicknesses on the order of the optical wavelength. An optical thin ﬁlm can be either a free-standing layer in a homogeneous medium, such as the ﬁlm of a soap bubble in air, or a layer deposited on a substrate of a different optical property, such as a thin SiO2 layer on a silicon substrate. A sophisticated thinﬁlm structure can be composed of multiple thin layers of different optical properties. Figure 5.19 shows some examples of optical thin ﬁlms. Figure 5.19 Examples of optical thin ﬁlms. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 5.3 Fabry–Pérot Interferometer 197 A single optical thin ﬁlm has the structure, thus the basic optical property, of a Fabry–Pérot interferometer in the etalon form. The two surfaces of the thin ﬁlm act as the two partially reﬂective surfaces of the interferometer. Multiple reﬂections take place in the thin ﬁlm between these two surfaces. Therefore, the reﬂectance and transmittance of an optical thin ﬁlm are functions of the optical wavelength, the incident angle, the thickness and refractive index of the thin ﬁlm, and the refractive indices of the media on the two sides of the thin ﬁlm. An optical thin ﬁlm often exhibits a color because of the strong wavelength dependence of its reﬂectance and transmittance. A thin ﬁlm that has a spatially varying thickness can produce a spectrum of spatially distributed colors, as often seen in soap bubbles or oil slicks. EXAMPLE 5.7 An oil ﬁlm of a uniform thickness l ¼ 100 nm ﬂoats on water. The refractive index of the oil ﬁlm is noil ¼ 1:40 and that of water is nw ¼ 1:33. When it is illuminated by white light at normal incidence, which wavelength in the visible spectral range shows the highest reﬂection? What color does it appear to be? If the same ﬁlm is coated on a glass surface of a refractive index ng ¼ 1:50, does it show the same high reﬂection? Solution: For the oil ﬁlm on water, we ﬁnd that noil > nw > nair . Therefore, for the wave inside the oil ﬁlm as an interferometer, the reﬂection at the air–ﬁlm interface and that at the ﬁlm–water interface are both internal reﬂection with no phase changes so that φ1 ¼ φ2 ¼ 0. Then, according to (5.47), for normal incidence the peak transmittance for dark reﬂection occurs at νq ¼ q φ 1 þ φ2 c c ¼q 2π 2nl 2noil l ) λdark ¼ c 2noil l ¼ , νq q and the minimum transmittance for bright reﬂection occurs at 1 φ 1 þ φ2 c 1 c c 4noil l νq1=2 ¼ q ¼ ) λbright ¼ ¼ q : 2π 2 2nl 2 2noil l νq1=2 2q 1 With noil ¼ 1:40 and l ¼ 100 nm, we ﬁnd that the only λbright that falls within the 400 to 700 nm visible spectral range is found for q ¼ 1 at λbright ¼ 4noil l ¼ 4noil l ¼ 4 2q 1 1:4 100 nm ¼ 560 nm: The next bright reﬂection takes place for q ¼ 2 at 186:7 nm, which is in the deep UV. Therefore, the ﬁlm appears to be green. If the same ﬁlm is coated on a glass surface of a refractive index ng ¼ 1:50, then ng > noil > nair . In this situation, the reﬂection at the air–ﬁlm interface is still internal reﬂection with φ1 ¼ 0, but that at the ﬁlm–glass interface is external reﬂection with φ1 ¼ π. Then, according to (5.47), for normal incidence the peak transmittance for dark reﬂection occurs at φ1 þ φ2 c 1 c c 4noil l νq ¼ q ) λdark ¼ ¼ ¼ q , 2π 2nl 2 2noil l νq 2q 1 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 198 Optical Interference and the minimum transmittance for bright reﬂection occurs at 1 φ1 þ φ2 c c c 2noil l νq1=2 ¼ q ¼ ) λbright ¼ ¼ ðq 1Þ : 2π 2 2nl 2noil l νq1=2 q 1 With noil ¼ 1:40 and l ¼ 100 nm, we ﬁnd that no λbright falls within the 400 to 700 nm visible spectral range because the largest value for λbright is found for q ¼ 2 at 280 nm, which is in the UV. Therefore, this ﬁlm appears to be colorless on glass. A thin ﬁlm on an optical surface can dramatically change the reﬂection and transmission properties of the surface. Thin-ﬁlm coating is an important technology for designing and achieving desired reﬂection and transmission properties of an optical surface, and thin-ﬁlm optics has been developed into an important ﬁeld in optics. Sophisticated thin ﬁlms consisting of multiple layers of different thicknesses and different refractive indices are used for advanced optical coatings. A desired reﬂection property, such as broadband antireﬂection, broadband total reﬂection, narrowband antireﬂection, or narrowband high reﬂection, can be obtained by coating an optical surface with a properly designed thin-ﬁlm structure. Applications of thin-ﬁlm optical coatings range from high-precision coatings for optical ﬁlters and laser mirrors to lowemission glass panes for house windows. EXAMPLE 5.8 A uniform thin ﬁlm of MgF2 , which has a refractive index of nf ¼ 1:38 is deposited on the surface of a glass lens, which has a refractive index of ng ¼ 1:50, to serve as an antireﬂective coating at the wavelength of λ ¼ 552 nm. What is the minimum thickness of the thin ﬁlm? What other thicknesses can be chosen? How effective is this thin ﬁlm as an antireﬂective coating? How can the thin-ﬁlm material be chosen to further increase the effectiveness of the antireﬂective coating? Solution: There are two interfaces: the air–MgF2 interface and the MgF2–glass interface. Because the refractive index increases from one medium to the next with nair ¼ 1, nf ¼ 1:38, and ng ¼ 1:50, for the wave inside the thin ﬁlm as an interferometer, the reﬂection at the air–ﬁlm interface is internal reﬂection with no phase change and that at the ﬁlm–glass interface is external reﬂection with a phase change of π; thus φ1 ¼ 0 and φ2 ¼ π. For the ﬁlm to serve as an antireﬂective coating, it is desired that T FP ¼ T max FP , which takes place at the optical frequencies νq given in (5.47): φ 1 þ φ2 c 1 c νq ¼ q ¼ q 2π 2nl cos θ 2 2nf l for normal incidence. With the given wavelength at λ ¼ c=ν ¼ 552 nm, the acceptable thicknesses are Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 199 5.3 Fabry–Pérot Interferometer lq ¼ 1 c 1 λ 1 552 1 q ¼ q ¼ q nm ¼ 200 q nm: 2 2nf ν 2 2nf 2 2 1:38 2 Therefore, the minimum thickness is lmin ¼ 100 nm for q ¼ 1, and any thickness that is larger than the minimum thickness by an integral multiple of 200 nm, such that l ¼ 100ð2m þ 1Þ nm, also works. Without the coating, the reﬂectivity at the air–glass interface is nair ng 2 1 1:52 ¼ 0:04: R¼ ¼ nair þ ng 1 þ 1:5 With the thin-ﬁlm coating, the reﬂectivities at the two interfaces are nair nf 2 1 1:382 nf ng 2 1:38 1:52 ¼ R1 ¼ 1 þ 1:38 ¼ 0:0255, R2 ¼ nf þ ng ¼ 1:38 þ 1:5 ¼ 1:736 nair þ nf 103 : The reﬂectivity of the coated surface is RFP ¼ 1 T max FP ¼ 1 ð1 R1 Þð1 R2 Þ 1=2 1=2 1 R1 R2 2 ¼ 1 0:986 ¼ 0:014: Therefore, the thin-ﬁlm coating cuts the reﬂectivity by 65% from 0:04 to 0:014. To increase the effectiveness of the antireﬂective coating, the material of the thin ﬁlm has to be chosen so that R1 and R2 have closer values. The coating results in total antireﬂection with RFP ¼ 0 when R1 ¼ R2 so that T max FP ¼ 1. This can be accomplished by choosing the refractive pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ index of the thin ﬁlm to be ﬃnf ¼ nair ng . For this thin ﬁlm to be totally antireﬂective, a material pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ of an index nf ¼ 1 1:5 ¼ 1:225 has to be chosen for the ﬁlm. 5.3.2 Interference Filters A high-ﬁnesse Fabry–Pérot interferometer can be used as an interference ﬁlter to selectively transmit a desired wavelength. The wavelength selectivity of the ﬁlter is determined by its free spectral range; a larger free spectral range allows fewer transmission wavelengths within a given spectrum. For a desired transmission wavelength λ, the largest spectral range for an interference ﬁlter is ΔνFSR ¼ ν ¼ c=λ, which is achieved when the optical path length of the interferometer is half the optical wavelength: nl ¼ λ=2. For such a ﬁlter, the next transmission peak occurs at the second harmonic frequency, 2ν, of the desired transmission frequency ν, i.e., at the wavelength λ=2 that is half the desired transmission wavelength λ, if the dispersion of the refractive index n is negligible between ν and 2ν. The pass band around the transmission frequency is determined by the linewidth of the interferometer. As discussed above, for a given free spectral range the linewidth can be reduced by increasing the ﬁnesse through increasing the product R1 R2 of the reﬂectivities of the reﬂective surfaces. By properly coating the two reﬂective surfaces for high reﬂectivities, an interference ﬁlter of a narrow linewidth on the order of a nanometer or an angstrom can be obtained. Such a highly selective, narrow-linewidth ﬁlter is also called a line ﬁlter. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 200 Optical Interference Problems 5.1.1 Show that in the case when the angles between the wavevectors k1 and k2 of two optical ﬁelds is small, the intensity of the combined optical ﬁeld projected on a plane that is normal to k1 þ k2 is approximately that given in (5.7). 5.1.2 A glass wedge of a refractive index n ¼ 1:5 as shown in Fig. 5.5 has a length of l ¼ 5 cm and a height of h ¼ 1 mm. It is vertically illuminated with coherent light at the λ ¼ 600 nm wavelength. What is the period of the interference fringes? How many dark and bright interference fringes appear on the surface of the wedge? 5.1.3 If the incident light in Problem 5.1.2 is not completely coherent, what is the minimum coherence time of the wave for all of the interference fringes to appear on the wedge? If 1000 periods of interference fringes appear, what is the coherence time of the incident light? 5.1.4 An air wedge is formed between two ﬂat glass plates by making them in contact at one end but separated by the thickness of a piece of paper at the other end. When it is vertically illuminated with monochromatic coherent light at the λ ¼ 500 nm wavelength, exactly 400 periods of interference fringes are seen. What is the thickness of the paper? 5.1.5 A laser beam at the λ ¼ 532 nm wavelength is normally incident on two slits that are spaced at Λ ¼ 200 μm. What is the angle between the two bright interference fringes of the diffraction orders q ¼ 10? On a screen that is at a distance of l ¼ 2 m from the slits, what is the separation of these two fringes? 5.1.6 Two slits separated by Λ ¼ 100 μm are illuminated with a laser beam at normal incidence. On a screen that is at a distance of l ¼ 2:5 m from the slits, it is found that the separation between two neighboring dark fringes is 12:2 mm, what is the wavelength of the laser light? 5.1.7 A laser beam is sent into a Michelson interferometer that is constructed in free space, as shown in Fig. 5.7. (a) When the mirror of one arm is moved to increase the length of the arm by 0:5 mm while the other arm is ﬁxed, the intensity pattern at each output port repeats itself 1880 times. Find the wavelength of the laser beam. (b) The two arms are adjusted such that I out, 1 ¼ I in and I out, 2 ¼ 0. Then, a thin glass plate that has a refractive index of n ¼ 1:46 and a thickness of d ¼ 1 mm is inserted perpendicularly to the beam path into one of the two arms without changing the optical alignment. What are the output intensities I out, 1 and I out, 2 now? 5.1.8 A laser beam is sent into a Mach–Zehnder interferometer that is constructed in free space, as shown in Fig. 5.8. (a) When the mirror of one arm is moved to increase the length of the arm by 0:5 mm while the other arm is ﬁxed, the intensity pattern at each output port repeats itself 940 times. Find the wavelength of the laser beam. (b) The two arms are adjusted such that I out, 1 ¼ I in and I out, 2 ¼ 0. Then, a thin glass plate that has a refractive index of n ¼ 1:46 and a thickness of d ¼ 1 mm is inserted Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 Problems 201 perpendicularly to the beam path into one of the two arms without changing the optical alignment. What are the output intensities I out, 1 and I out, 2 now? 5.1.9 A waveguide Mach–Zehnder interferometer uses Y-junction couplers for its input and output ports, as shown in Fig. 5.9(a). It has a symmetric structure with an equal length of la ¼ lb ¼ l for the two arms. The two Y-junctions are both 3-dB couplers. Thus, Δφ ¼ 0, and the transmittance is T ¼ 1. By changing the refractive index of the medium in one arm with respect to the other through the Pockels effect, for example, the phase shifts through the two arms can be made different for Δφ 6¼ 0 so that T 6¼ 1. Find the minimum necessary index difference Δn between the two arms for T ¼ 0 at an optical wavelength of λ. At λ ¼ 1 μm, what is the minimum value of Δn for an equal arm length of l ¼ 1 mm? If the Mach–Zehnder interferometer has a symmetric structure with la ¼ lb ¼ l using two 3-dB directional couplers, as shown in Fig. 5.9(b), the transmittance is T ¼ 0 with Δφ ¼ 0. Then, what is the minimum necessary index difference Δn between the two arms for T ¼ 1 at an optical wavelength of λ? At λ ¼ 1 μm, what is the minimum value of Δn for an equal arm length of l ¼ 1 mm? 5.2.1 Identical slits in an array are equally spaced at Λ ¼ 20 μm. A plane wave at the λ ¼ 532 nm wavelength is normally incident on the slits. How many diffraction peaks can be found in transmission within the range of angles between 30 and 30 ? If the wave is obliquely incident at an angle of θi ¼ 15 , how many diffraction peaks can be found in transmission within the range of angles between 30 and 30 ? 5.2.2 Three perfectly aligned plane optical waves at λ1 ¼ 450 nm, λ2 ¼ 550 nm, and λ3 ¼ 650 nm are normally incident at the same time on an array of identical slits that are equally spaced at Λ. The diffraction peaks in transmission are examined. It is clear that the zeroth-order peaks for all three wavelengths completely overlap at θq ¼ 0 for q1 ¼ q2 ¼ q3 ¼ 0. (a) What are the lowest nonzero diffraction orders q1 and q2 for λ1 and λ2 , respectively, that have exactly overlapped peaks? What is the minimum slit spacing Λ for this to be possible? (b) Answer the questions in (a) for λ2 and λ3 . (c) Answer the questions in (a) for λ1 and λ3 . (d) What are the nonzero diffraction orders q1 , q2 , q3 for λ1 , λ2 , λ3 , respectively, that have exactly overlapped peaks? What is the smallest slit spacing Λ for this to be possible? 5.2.3 A grating on the surface of a glass plate has a period of Λ ¼ 800 nm. The glass plate has a refractive index of 1:5. A laser beam is normally incident on the grating from the air. Only two nonzero diffraction orders, for q ¼ 1 and q ¼ 1, are allowed on the glass side, but no nonzero diffraction orders are allowed on the air side. What is the possible wavelength of the incident laser light? 5.2.4 A collimated laser beam at λ ¼ 800 nm is incident on a grating at an air–glass interface from the air side. The refractive index of this glass is 1.5. At normal incidence, three diffraction peaks for q ¼ 1, 0, and 1 are found on the glass side. By carefully varying the incident angle of the laser beam, it is found that the q ¼ 1 diffraction peak just disappears when the incident angle is θi ¼ 12:1 . Find the grating period. How many Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 202 Optical Interference diffraction peaks can be found at an incident angle of θi ¼ 10 from the air and glass sides, respectively? At what angles are these diffraction peaks found? 5.2.5 Consider the waveguide and the grating of a period Λ ¼ 492 nm found in Example 5.5. The waveguide supports the TE0 mode at the λ ¼ 1:55 μm wavelength. The effective index of this mode at this wavelength is nβ ¼ 3:33. Find the incident angle for a laser beam at λ ¼ 1:55 μm to be coupled into this guided mode. 5.2.6 A surface grating that has a period of Λ ¼ 300 nm is fabricated on the surface of a GaAs/ AlGaAs slab waveguide as shown in Fig. 5.15. The cover of the waveguide is simply air with n3 ¼ 1. At the wavelength of λ ¼ 900 nm, the GaAs core has n1 ¼ 3:59 and the AlGaAs substrate has n2 ¼ 3:39. The waveguide supports only the TE0 mode of an unknown propagation constant. If it is found that a laser beam at λ ¼ 900 nm can be coupled into this guided mode through the surface grating at an incident angle of θi ¼ 30 , what is the propagation constant of the mode? What grating period will allow coupling of this laser beam into this waveguide mode at normal incidence with θi ¼ 0 ? 5.3.1 A laser beam is sent at normal incidence into a Fabry–Pérot interferometer that is constructed in free space with R1 ¼ R2 ¼ 0:5. (a) When one reﬂective surface is ﬁxed in location but the other is moved to increase the spacing between them by 0:5 mm, the transmitted intensity pattern repeats itself 1880 times. Find the wavelength of the laser beam. (b) The interferometer is adjusted such that T FP ¼ 1. Then, a thin glass plate that has a refractive index of n ¼ 1:46 and a thickness of d ¼ 1 mm is inserted perpendicularly to the beam path into the spacing without changing the optical alignment. What is the transmittance of the interferometer now? 5.3.2 A lossless Fabry–Pérot interferometer consists of two highly reﬂective surfaces with R1 ¼ 95% and R2 ¼ 90%, which are separated by a spacing of l in free space. What are the maximum transmittance and the ﬁnesse of this interferometer? It is used as an optical spectrum analyzer. If a spectral resolution with a linewidth of Δλline ¼ 0:1 nm at the λ ¼ 500 nm wavelength is desired, what is the required spacing l of the interferometer? What is the wavelength separation ΔλFSR between neighboring transmission peaks? If a higher resolution is needed, how should the spacing be changed in order to reduce the spectral linewidth by half to Δλline ¼ 0:05 nm? 5.3.3 A Fabry–Pérot etalon consists of a thin glass plate that has a refractive index of n ¼ 1:50 and a thickness of l ¼ 100 μm. Its surfaces are coated such that its peak transmittance is 100% and it has a spectral linewidth of Δνline 5 GHz for high spectral resolution. Find the values of R1 and R2 that allow the etalon to have these properties. 5.3.4 An oil ﬁlm that has a refractive index of noil ¼ 1:40 ﬂoats on a smooth water surface, which has nw ¼ 1:33. It reﬂects most strongly at the 672 nm red wavelength and appears to have no reﬂection at the 504 nm blue wavelength. What is the thickness of the oil ﬁlm? 5.3.5 A material that has a refractive index of nf ¼ 1:25 is used for the thin ﬁlm discussed in Example 5.8, which is deposited on the surface of a glass lens that has a refractive index of ng ¼ 1:50. To serve as an antireﬂective coating at the wavelength of λ ¼ 552 nm, what Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 Bibliography 203 is the minimum thickness required for the thin ﬁlm? What other thicknesses can be chosen? How effective is this thin ﬁlm as an antireﬂective coating? 5.3.6 The refractive index of Si at the λ ¼ 1:0 μm wavelength is nSi ¼ 3:61. If an antireﬂective thin ﬁlm is to be coated on a smoothly polished Si surface, how should the refractive index of the thin-ﬁlm material be chosen so that the coated surface is totally antireﬂective when exposed to air? What should the refractive index of the thin ﬁlm be chosen if the surface is to become totally antireﬂective in water, which has a refractive index of nw ¼ 1:33? Bibliography Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edn. Cambridge: Cambridge University Press, 1999. Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975. Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Serway, R. A. and Jewett, J. W., Physics for Scientists and Engineers, 9th edn. Boston, MA: Brooks Cole, 2013. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:16:14 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.006 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 6 - Optical Resonance pp. 204-223 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge University Press 6 6.1 Optical Resonance OPTICAL RESONATOR .............................................................................................................. As discussed in Section 5.3, multiple reﬂections take place between the two reﬂective surfaces of a Fabry–Pérot interferometer, resulting in multiple transmitted ﬁelds. A transmittance peak occurs when the round-trip phase shift φRT between the two reﬂective surfaces is an integral multiple of 2π so that all of the transmitted ﬁelds are in phase. From the viewpoint of the ﬁeld inside the interferometer, this condition results in optical resonance between the two reﬂective surfaces. Thus a Fabry–Pérot interferometer behaves as an optical resonator, also called a resonant optical cavity. At resonance, the ﬁeld amplitude inside an optical resonator reaches a peak value due to constructive interference of multiple reﬂections. The optical energy stored in an optical cavity peaks at its resonance frequencies. An optical cavity can take a variety of forms. Figure 6.1 shows the schematic structures of a few different forms of optical cavities. Though an optical cavity has a clearly deﬁned longitudinal axis, the axis can lie on a straight line, as in Fig. 6.1(a), or it can be deﬁned by a folded path, as in Figs. 6.1(b), (c), and (d). A linear cavity deﬁned by two end mirrors, as in Fig. 6.1(a), is known as a Fabry–Pérot cavity because it takes the form of the Fabry–Pérot interferometer. A folded cavity can simply be a folded Fabry–Pérot cavity that supports a standing intracavity ﬁeld, as in Fig. 6.1(b). A folded cavity can also be a non-Fabry–Pérot ring cavity that supports two independent, contrapropagating intracavity ﬁelds, as in Figs. 6.1(c) and (d). An optical cavity provides optical feedback to the optical ﬁeld in the cavity. Optical resonance occurs when the optical feedback is in phase with the intracavity optical ﬁeld. The optical feedback in a Fabry–Pérot cavity is provided simply by the two end mirrors that have the reﬂective surfaces perpendicular to the longitudinal axis, as in Figs. 6.1(a) and (b). In a ring cavity, it is provided by the circulation of the laser ﬁeld along a ring path deﬁned by mirrors, as in Fig. 6.1(c), or a ring path deﬁned by an optical ﬁber, as in Fig. 6.1(d). The cavity can also be constructed with an optical waveguide, as in the case of a semiconductor laser or a ﬁber laser. In the following discussion, we take the coordinate deﬁned by the longitudinal axis to be the z coordinate, and the transverse coordinates that are perpendicular to the longitudinal axis to be the x and y coordinates. In a folded cavity, the z axis is thus also folded along with the longitudinal optical path. Sophisticated optical cavities can use gratings to provide distributed feedback; such advanced cavities are not shown in Fig. 6.1 and are not discussed in this chapter. In a ring cavity, an intracavity ﬁeld completes one round trip by circulating inside the cavity in only one direction. The two contrapropagating ﬁelds that circulate in a ring cavity in opposite directions are independent of each other even when they have the same frequency. In a Fabry–Pérot cavity, an intracavity ﬁeld has to travel the length of the cavity twice in Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.1 Optical Resonator 205 Figure 6.1 Schematics of a few different forms of optical cavities: (a) linear Fabry–Pérot cavity with end mirrors; (b) folded Fabry–Pérot cavity with end mirrors; (c) three-mirror ring cavity with two independent, contrapropagating ﬁelds; and (d) ring cavity with two independent, contrapropagating ﬁelds guided by an optical-ﬁber waveguide. opposite directions to complete a round trip. The time it takes for an intracavity ﬁeld to complete one round trip in the cavity is called the round-trip time, T¼ round-trip optical path length lRT , ¼ c c Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 (6.1) 206 Optical Resonance Figure 6.2 Passive laser cavities with a gain ﬁlling factor Γ under optical injection: (a) a Fabry–Perot cavity and (b) a ring cavity. The refractive index of the gain medium is n, while that of the background medium in the cavity is n0 . A laser cavity is simply a passive optical cavity when its gain medium is absent or is present but not pumped. where the round-trip optical path length lRT takes into account the refractive index of the medium inside the cavity. The space inside an optical cavity can be ﬁlled with a variety of optical media of different properties. For example, a laser cavity contains at least a gain medium. The gain medium may ﬁll up the entire length of the cavity, or it may occupy a fraction of the cavity length. For a laser cavity of a length l that contains a gain medium of a length lg , as shown in Fig. 6.2, we can deﬁne an overlap factor between the gain medium and the intensity distribution of the laser mode as the ratio ððð jEj2 dxdydz V gain lg gain : Γ ¼ ððð (6.2) V mode l 2 jEj dxdydz cavity This ratio is commonly known as the gain ﬁlling factor for a gain medium that takes up only a fraction of the length of the laser cavity, whereas it is related to the mode conﬁnement factor in a waveguide laser, such as a ﬁber laser or a semiconductor laser. When the gain medium ﬁlls up an optical cavity and covers the entire intracavity ﬁeld distribution, Γ ¼ 1; otherwise, Γ < 1. Take the refractive index of the gain medium to be n and that of the intracavity medium excluding the gain medium to be n0 ; then, the round-trip optical path length can be expressed as 2½Γnl þ ð1 ΓÞn0 l ¼ 2nl, for a linear cavity; (6.3) lRT ¼ for a ring cavity; Γnl þ ð1 ΓÞn0 l ¼ nl, where n ¼ Γn þ ð1 ΓÞn0 is the weighted average index of refraction throughout the laser cavity. When an optical cavity contains optical elements other than a gain medium, n is still the weighted average index throughout the cavity with n0 being the weighted average index of the background medium and these optical elements. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.2 Longitudinal Modes 207 Consider an intracavity ﬁeld, Ec ðzÞ, at any location z along the longitudinal axis inside an optical cavity. When this ﬁeld completes a round trip in the cavity and returns back to the location z, it is ampliﬁed or attenuated by a factor a to become aEc ðzÞ. The complex ampliﬁcation or attenuation factor a can be generally expressed as a ¼ GeiφRT , (6.4) where G is the round-trip gain factor for the ﬁeld amplitude, equivalent to the power gain in a single pass through a linear Fabry–Pérot cavity, and φRT is the round-trip phase shift for the intracavity ﬁeld. Both G and φRT have real values, and G 0. For a cavity that has a net optical gain, G > 1, and the intracavity ﬁeld is ampliﬁed. For a cavity that has a net optical loss, G < 1, and the intracavity ﬁeld is attenuated. EXAMPLE 6.1 Consider a linear cavity, as shown in Fig. 6.1(a), and a ring cavity, as shown in Fig. 6.1(c). The linear cavity has two mirrors with R1 ¼ R2 ¼ 0:9, which are separated at l ¼ 1:5 m. The ring cavity has three mirrors with R1 ¼ R2 ¼ 0:9 and R3 ¼ 1, which are separated at l12 ¼ 0:7 m and l23 ¼ l31 ¼ 0:4 m. Find the physical length, the round-trip length lRT , the round-trip time T, and the round-trip gain factor G of each cavity. Solution: For the linear cavity, the physical length is simply l ¼ 1:5 m deﬁned by the separation of the two mirrors. The round-trip length and the round-trip time are, respectively, llinear ¼ 2l ¼ 3 m, T linear ¼ RT ¼ 10 ns: c In a round trip through the linear cavity, the intracavity intensity changes by a factor of R1 R2 because the intracavity light is reﬂected once by each of the two mirrors in each round trip. Therefore, the round-trip gain factor for the ﬁeld amplitude is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Glinear ¼ R1 R2 ¼ 0:9: llinear RT For the ring cavity, the physical length is simply l ¼ l12 þ l23 þ l31 ¼ 1:5 m deﬁned by the ring length. The round-trip length and the round-trip time are, respectively, lring ¼ l ¼ l12 þ l23 þ l31 ¼ 1:5 m, T ring ¼ RT ¼ 5 ns: c In a round trip through the ring cavity, the intracavity intensity changes by a factor of R1 R2 R3 because the intracavity light is reﬂected once by each of the three mirrors in each round trip. Therefore, the round-trip gain factor for the ﬁeld amplitude is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Gring ¼ R1 R2 R3 ¼ 0:9: lring RT 6.2 LONGITUDINAL MODES .............................................................................................................. We ﬁrst consider the resonant characteristics of a passive optical cavity. A passive cavity cannot generate or amplify an optical ﬁeld; thus G < 1. In order to maintain an intracavity ﬁeld Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 208 Optical Resonance in such a cavity, it is necessary to constantly inject an input optical ﬁeld, Ein , into the cavity. As shown in Fig. 6.2, the forward-traveling component of the intracavity ﬁeld at the location z1 just inside the cavity next to the injection point is the sum of the transmitted input ﬁeld and the fraction of the intracavity ﬁeld that returns after one round trip through the cavity: Ec ðz1 Þ ¼ t in Ein þ aEc ðz1 Þ, (6.5) where t in is the complex transmission coefﬁcient for the input ﬁeld. We ﬁnd that t in (6.6) Ec ðz1 Þ ¼ Ein : 1a The transmitted output ﬁeld, Eout , is proportional to the intracavity ﬁeld: Eout / Ec ðz1 Þ. Therefore, the output intensity is proportional to the input intensity through the following relationship, I out / I in j1 aj 2 ¼ I in 2 ð1 GÞ þ 4G sin2 ðφRT =2Þ : (6.7) The proportionality constant of this relationship depends on the transmittance of the output mirror and the intracavity attenuation over the distance from the point at z1 to the output point. The transmittance of the cavity is T c ¼ I out =I in , which is scaled by the value of this proportionality constant. For our discussion in the following, this proportionality constant is irrelevant. Therefore, we only have to consider the normalized transmittance of the passive cavity: T^ c ¼ 1 1 h i h i ¼ , 2 2 2 1 þ 4G=ð1 GÞ sin ðφRT =2Þ 1 þ ð4=GÞ=ð1 1=GÞ sin2 ðφRT =2Þ (6.8) which is obtained by normalizing T c to its peak value. Clearly, T^ c has a peak value of unity, as expected for a normalized quantity. In Fig. 6.3, T^ c is plotted as a function of the round-trip phase shift φRT for a few different values of G. We ﬁnd that Figure 6.3 Normalized transmittance of an optical cavity as a function of the round-trip phase shift in the cavity. In a resonator that has a ﬁxed, frequency-independent optical path length, the round-trip phase shift is directly proportional to the optical frequency. The longitudinal mode frequencies are deﬁned by the frequencies corresponding to the resonance peaks. The spectral shape for a gain factor of G is the same as that for a gain factor of 1=G. Thus, the curve for G ¼ 0:1 is the same as that for G ¼ 10, that for G ¼ 0:5 is the same as that for G ¼ 2, and so on. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.2 Longitudinal Modes 209 the spectral shape for a gain factor of G is the same as that for a gain factor of 1=G. Therefore, a passive cavity that has a gain factor of Gp ¼ G < 1 has the same spectral characteristics as an active cavity that has a gain factor of Ga ¼ 1=G > 1. Note that the characteristics of T^ c shown in Fig. 6.3 are the same as those of T^ FP shown in Fig. 5.18 because a Fabry–Pérot interferometer can be considered as an optical cavity. Clearly, T^ FP given in (5.45) for a Fabry–Pérot interferometer can be identiﬁed with T^ c in (6.8) for a general optical cavity by properly relating the ﬁnesse F of a cavity to the gain factor G, as is given below in (6.12). At a given input ﬁeld intensity, the intracavity ﬁeld intensity of a passive cavity is proportional to T^ c because the transmitted output ﬁeld intensity is directly proportional to the intracavity ﬁeld intensity while it is also proportional to T^ c . Therefore, resonances of the cavity occur at the peaks of T^ c , where the intracavity intensity reaches its maximum level with respect to a constant input ﬁeld intensity. As can be seen from Fig. 6.3, the resonance condition of the cavity is that the round-trip phase shift is an integral multiple of 2π: φRT ¼ 2qπ, q ¼ 1, 2, . . . : (6.9) From (6.9) and Fig. 6.3, we ﬁnd that the separation between two neighboring resonance peaks of T^ c is ΔφL ¼ 2π (6.10) and that the FWHM of each resonance peak is 1G : G1=2 The ﬁnesse, F, of the cavity is the ratio of the separation to the FWHM of the peaks: Δφc ¼ 2 (6.11) ΔφL πG1=2 : (6.12) ¼ Δφc 1 G In the simplest situation that the optical ﬁeld is a plane wave at a frequency of ω, the roundtrip phase shift can be generally expressed as F¼ φRT ¼ ω lRT þ φlocal , c (6.13) where the ﬁrst term on the right-hand side is the phase shift contributed by the propagation of the optical ﬁeld over an optical path length of lRT , and the second term, φlocal , is the sum of all the localized, and usually ﬁxed, phase shifts such as those caused by reﬂection from the mirrors of a cavity. In the case when the frequency of the input ﬁeld is ﬁxed, the resonance condition given in (6.9) can be satisﬁed by varying the optical path length lRT of the cavity, either by varying the physical length of the cavity or by varying the refractive index of the intracavity medium, or both. The optical cavity then functions as an optical interferometer, which is used to accurately measure the frequency and the spectral width of an optical wave. When both the optical path length and the localized phase shifts are ﬁxed, as is typically the case for a laser resonator, the resonance condition of φRT ¼ 2qπ is satisﬁed only if the optical frequency satisﬁes the condition: ωq ¼ c lRT ð2qπ φlocal Þ, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 (6.14) 210 Optical Resonance or νq ¼ c lRT q φlocal : 2π (6.15) These discrete resonance frequencies are the longitudinal mode frequencies of the optical resonator because they are deﬁned by the resonance condition of the round-trip phase shift along the longitudinal axis of the cavity. The frequency spacing, ΔνL , between two neighboring longitudinal modes is known as the free spectral range, also called the longitudinal mode frequency spacing, of the optical resonator. The FWHM of a longitudinal mode spectral peak is Δνc , which is known as the longitudinal mode width of the cavity. If the values of lRT and φlocal are independent of frequency, then ΔνL / ΔφL and Δνc / Δφc . Therefore, the ﬁnesse of an optical resonator is the ratio of its free spectral range to its longitudinal mode width: F¼ ΔφL ΔνL ¼ : Δφc Δνc (6.16) From (6.15), we ﬁnd that the longitudinal mode frequency spacing is related to the round-trip time as ΔνL ¼ νqþ1 νq ¼ c 1 ¼ : lRT T (6.17) The longitudinal mode width of the cavity can be expressed as Δνc ¼ ΔνL 1 G ¼ ΔνL : F πG1=2 (6.18) EXAMPLE 6.2 Find the ﬁnesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc of the linear and ring cavities that are considered in Example 6.1. Solution: For the linear cavity, the ﬁnesse is 1=2 F linear πGlinear π 0:91=2 ¼ ¼ ¼ 29:8: 1 Glinear 1 0:9 The longitudinal mode frequency spacing is ¼ Δνlinear L 1 T linear ¼ 1 Hz ¼ 100 MHz: 10 109 The longitudinal mode width is Δνlinear ¼ c Δνlinear 100 L ¼ MHz ¼ 3:36 MHz: F linear 29:8 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.3 Transverse Modes 211 For the ring cavity, the ﬁnesse is 1=2 F ring ¼ πGring 1 Gring ¼ π 0:91=2 ¼ 29:8: 1 0:9 The longitudinal mode frequency spacing is Δνring L ¼ 1 T ring ¼ 1 Hz ¼ 200 MHz: 5 109 The longitudinal mode width is Δνring ¼ c 6.3 Δνring 200 L ¼ MHz ¼ 6:71 MHz: F ring 29:8 TRANSVERSE MODES .............................................................................................................. Any realistic optical cavity has a ﬁnite transverse cross-sectional area. Therefore, the resonant optical ﬁeld inside a realistic optical cavity cannot be a plane wave. Indeed, there exist certain normal modes for the transverse ﬁeld distribution in a given optical cavity. Such transverse ﬁeld patterns are known as the transverse modes of a cavity. A transverse mode of an optical cavity is a stable transverse ﬁeld pattern that reproduces itself after each round-trip pass in the cavity, except that it might be ampliﬁed or attenuated in magnitude and shifted in phase. The transverse modes of an optical cavity are deﬁned by the transverse boundary conditions that are imposed by the transverse cross-sectional index proﬁle of the cavity. For a cavity that utilizes an optical waveguide for lateral conﬁnement of the optical ﬁeld, the transverse modes are the waveguide modes, such as the TE and TM modes of a slab waveguide or the TE, TM, HE, and EH modes of a cylindrical ﬁber waveguide. For a nonwaveguiding cavity, the transverse modes are TEM ﬁelds determined by the shapes and sizes of the end mirrors of the cavity, as well as by the properties of the medium and any other optical components inside the cavity. The Gaussian modes discussed in Section 3.3 are an important set of such unguided TEM modes. In an optical cavity that supports multiple transverse modes, the round-trip phase shift is generally a function of the transverse mode indices m and n. Therefore, the resonance condition can be explicitly written as φRT mn ¼ 2qπ: (6.19) As a result, the resonance frequencies of the cavity, ωmnq or νmnq , are dependent on both longitudinal and transverse mode indices. When the frequency spacing between neighboring transverse modes is smaller than that between neighboring longitudinal modes, multiple resonance frequencies of different transverse modes can exist for each longitudinal mode, as illustrated schematically in Fig. 6.4. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 212 Optical Resonance Figure 6.4 Cavity resonance frequencies associated with different longitudinal and transverse modes. For clarity, the heights of the transverse modes are made arbitrarily decreasing. In a cavity that consists of an optical waveguide, the propagation constant βmn ðωÞ is a function of the waveguide mode. If the physical length of the waveguide cavity is l, the effective round-trip optical path length of a waveguide mode is 8 c > < 2 βmn ðωÞl, for a linear cavity; ω RT (6.20) lmn ¼ c > : βmn ðωÞl, for a ring cavity: ω The round-trip optical path length lRT mn generally varies from one mode to another due to the modal dispersion of the waveguide. In addition, the localized phase shift can also be mode dependent. Therefore, instead of the resonance frequencies ωq given by (6.14) for a plane wave, the resonance frequencies ωmnq of a waveguide cavity are found by solving, for integral values of q, the following resonance condition, φRT mn ¼ ω RT l þ φlocal mn ¼ 2qπ: c mn (6.21) In a nonwaveguiding cavity, the propagation constant, k, is a property of only the medium and is not mode dependent. Nevertheless, a mode-dependent on-axis phase variation ζ mn ðzÞ does exist, which is given in (3.76) for a Hermite–Gaussian mode as discussed in Section 3.3. The total on-axis phase variation of the TEMmn Gaussian mode is φmn ðzÞ ¼ kz þ ζ mn ðzÞ, which includes the mode-independent phase shift kz and the mode-dependent phase shift ζ mn ðzÞ. Consequently, the cavity resonance condition for a Gaussian mode is a modiﬁcation of that for a plane wave made by adding the round-trip contribution of the mode-dependent phase shift: φRT mn ¼ ω local lRT þ ζ RT mn þ φmn ¼ 2qπ, c (6.22) where the localized phase shift can, in general, also be mode dependent. It is clear from the above discussion that the qth longitudinal mode frequency of a given longitudinal mode index q varies among different transverse modes, as illustrated in Fig. 6.4. For transverse modes deﬁned by a waveguide structure, the longitudinal mode frequency spacing ΔνLmn ¼ νmnðqþ1Þ νmnq between two neighboring longitudinal modes, q and q þ 1, of the same transverse mode mn varies slightly among different transverse modes, as illustrated in Example 6.3. Because a higher-order transverse waveguide mode has a smaller propagation constant, thus a smaller effective index of refraction, ΔνLmn is generally larger for a higher-order Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.3 Transverse Modes 213 transverse mode. By comparison, the longitudinal mode frequency spacing ΔνLmn stays constant for different transverse Gaussian modes deﬁned in free space because all Gaussian modes are TEM modes of the same propagation constant. The mode-dependent phase shift ζ mn ðzÞ only changes the mode frequency νmnq but not the difference ΔνLmn between two neighboring longitudinal modes mnq and mnðq þ 1Þ. EXAMPLE 6.3 A GaAs/AlGaAs semiconductor optical cavity has the longitudinal structure of a linear Fabry– Pérot cavity and the transverse structure of a slab waveguide. The cavity has a physical length of l ¼ 500 μm. The GaAs/AlGaAs slab waveguide supports three TE modes at the λ ¼ 870 nm wavelength, with propagation constants of βTE0 ¼ 2:61 107 m1 , βTE1 ¼ 2:58 107 m1 , and βTE2 ¼ 2:53 107 m1 for the TE0 , TE1 , and TE2 modes, respectively. The end surfaces of the cavity are not coated. Find the effective round-trip optical path length lRT m , the round-trip L time T m , the longitudinal mode frequency spacing Δνm , and the longitudinal mode width Δνcm for each transverse mode. Solution: For the linear cavity, the effective round-trip optical path length of each transverse waveguide mode is found using (6.20): c λβ l RT RT βm l ¼ m ) lRT TE0 ¼ 3614 μm, lTE1 ¼ 3572 μm, lTE2 ¼ 3503 μm: ω π The round-trip time of the cavity for each transverse waveguide mode is lRT m ¼2 lRT m ) T TE0 ¼ 12:05 ps, T TE1 ¼ 11:91 ps, T TE2 ¼ 11:68 ps: c The longitudinal mode frequency spacing for each transverse waveguide mode is Tm ¼ ΔνLm ¼ 1 Tm ) ΔνLTE0 ¼ 83:0 GHz, ΔνLTE1 ¼ 84:0 GHz, ΔνLTE2 ¼ 85:6 GHz: To ﬁnd Δνcm , it is necessary to ﬁnd the ﬁnesse. The effective refractive index for each mode is found, which is used to ﬁnd the reﬂectivities of the cavity and the ﬁnesse: nβm ¼ R1, m ¼ R2, m ¼ RTEm 1=4 Fm ¼ λβm ) nTE0 ¼ 3:61, nTE1 ¼ 3:57, nTE2 ¼ 3:50; 2π 1 nβm 2 ) RTE ¼ 32:1%, RTE ¼ 31:6%, RTE ¼ 30:9%; ¼ 0 1 2 1 þ nβ m 1=4 πR1, m R2, m 1=2 1=2 1 R1, m R2, m 1=2 πRTEm ¼ 1 RTEm ) F TE0 ¼ 2:62, F TE1 ¼ 2:58, F TE2 ¼ 2:53: The longitudinal mode width Δνcm for each transverse waveguide mode is Δνcm ¼ ΔνLm Fm ) ΔνcTE0 ¼ 31:7 GHz, ΔνcTE1 ¼ 32:6 GHz, ΔνcTE2 ¼ 33:8 GHz: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 214 Optical Resonance 6.4 CAVITY LIFETIME AND QUALITY FACTOR .............................................................................................................. Here we consider some important parameters of a passive optical cavity of zero optical gain so that χ res ¼ 0, thus g ¼ 0. Such a passive optical cavity is known as a cold cavity. To be speciﬁc, we identify the round-trip gain factor for the ﬁeld amplitude in a cold cavity as Gc , or as Gcmn for the transverse mode mn. Because there is no optical gain in a cold cavity, the cavity has a net loss from ﬁnite transmission through the end mirrors and various passive loss mechanisms so that Gc < 1. Any optical ﬁeld that initially exists in the cavity gradually decays as it circulates inside the cavity. Because the ﬁeld amplitude is attenuated by a factor of Gc per round trip, the intensity and thus the number of intracavity photons are attenuated by a factor of G2c per round trip. We can deﬁne a photon lifetime, also called cavity lifetime, τ c , and a cavity decay rate, γc , for a cold cavity through the relation: G2c ¼ eT=τc ¼ eγc T : (6.23) Therefore, the cavity lifetime is found as τc ¼ T : 2 ln Gc (6.24) The cavity decay rate is the decay rate of the optical energy stored in a cavity and is given by γc ¼ 1 2 ¼ ln Gc : τc T (6.25) In general, the value of Gc for a given cavity is mode dependent. Usually, the fundamental transverse mode has the lowest loss because its ﬁeld distribution is transversely most concentrated toward the center along the longitudinal axis of the cavity. As the order of a mode increases, its loss in the cavity increases due to the increased diffraction loss caused by the transverse spreading of its ﬁeld distribution. Consequently, both τ c and γc are also mode dependent: τ cmnq and γcmnq . Unless a speciﬁc mode-discriminating mechanism is introduced in a cavity, either intentionally or unintentionally, the fundamental mode generally has the largest τ c and, correspondingly, the lowest γc . The quality factor, Q, of a resonator is generally deﬁned as the ratio of the resonance frequency, ωres , to the energy decay rate, γ, of the resonator: energy stored in the resonator ωres : (6.26) Q ¼ ωres ¼ γ average power dissipation Therefore, the quality factor of a cold cavity is Q¼ ωq ¼ ωq τ c , γc (6.27) where ωq is the longitudinal mode frequency. For a low-loss, high-Q cavity, Gc is not much less than unity; then, it can be shown by using (6.17), (6.18), and (6.23) that Δνc 1 γ ¼ c 2πτ c 2π Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 (6.28) 6.4 Cavity Lifetime and Quality Factor 215 and Q νq : Δνc (6.29) Note that though it is not explicitly spelled out in (6.27) and (6.29), the quality factor is a function of not only the longitudinal-mode index q but also the transverse-mode indices m and n: Q ¼ Qmnq . To be precise, (6.27) should be written as Qmnq ¼ ωmnq ¼ ωmnq τ c : γc (6.30) For an optical cavity, the dependence of Qmnq on the longitudinal-mode index q is generally insigniﬁcant because q is a very large number except in the case of a very short microcavity. By comparison, the dependence of Qmnq on the transverse-mode indices m and n cannot be ignored. Indeed, Q00q for the fundamental transverse mode is generally larger than Qmnq for any highorder transverse mode because the fundamental transverse mode generally has the lowest loss. EXAMPLE 6.4 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼ 500 nm wavelength of the linear and ring cavities that are considered in Example 6.1. Solution: For the linear cavity, the photon lifetime is ¼ τ linear c T linear 10 ¼ ns ¼ 47:5 ns: linear 2 ln 0:9 2 ln Gc The cavity decay rate is ¼ γlinear c 1 τ linear c ¼ 1 s1 ¼ 2:1 107 s1 : 47:5 109 The quality factor Q at λ ¼ 500 nm is 2πc linear 2π 3 108 ¼ 47:5 109 ¼ 1:79 108 : τ λ c 500 109 For the ring cavity, the photon lifetime is Qlinear ¼ ωτ linear ¼ c τ ring ¼ c T ring 5 ¼ ns ¼ 23:7 ns: ring 2 ln 0:9 2 ln Gc The cavity decay rate is γring ¼ c 1 τ ring c ¼ 1 s1 ¼ 4:2 107 s1 : 9 23:7 10 The quality factor Q at λ ¼ 500 nm is Qring ¼ ωτ ring ¼ c 2πc ring 2π 3 108 23:7 109 ¼ 8:93 107 : τ ¼ λ c 500 109 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 216 Optical Resonance 6.5 FABRY–PÉROT CAVITY .............................................................................................................. The most common type of optical cavity is the Fabry–Pérot cavity, which consists of two end mirrors in the form of the Fabry–Pérot interferometer and, in the case when it is used as a laser cavity, an optical gain medium, as shown in Fig. 6.5. The radii of curvature of the left and right mirrors are R1 and R2 , respectively. The sign of the radius of curvature is taken to be positive for a concave mirror and negative for a convex mirror. For example, the cavity shown in Fig. 6.5 has R1 > 0 and R2 > 0 because it is formed with two concave mirrors. 6.5.1 Stability Criterion Most of the important features of a nonwaveguiding Fabry–Pérot cavity can be obtained by applying the following simple concept. For the cavity to be a stable cavity in which a Gaussian mode can be established, the radii of curvature of both end mirrors have to match the wavefront curvatures of the Gaussian mode at the surfaces of the mirrors: Rðz1 Þ ¼ R1 and Rðz2 Þ ¼ R2 , where z1 and z2 are, respectively, the coordinates of the left and right mirrors measured from the location of the Gaussian beam waist. Based on this concept, we have from (3.71) two relations: z1 þ z2R z2 ¼ R1 and z2 þ R ¼ R2 : z1 z2 (6.31) From these relations, we ﬁnd that z2R ¼ lðR1 lÞðR2 lÞðR1 þ R2 lÞ ðR1 þ R2 2lÞ2 , (6.32) where l ¼ z2 z1 is the length of the cavity deﬁned by the separation between the two end mirrors. Given the values of R1 , R2 , and l, stable Gaussian modes exist for the cavity if both relations in (6.31) can be satisﬁed with a real and positive parameter of zR > 0 from (6.32) for a ﬁnite, positive beam -waist spot size w0 according to (3.69). Then the cavity is stable. If the relations in (6.31) cannot be simultaneously satisﬁed with a real and positive value for zR , then the cavity is unstable because no stable Gaussian mode can be established in the cavity. Application of this concept yields the stability criterion for a Fabry–Pérot cavity: Figure 6.5 Fabry–Pérot cavity containing an optical gain medium with a ﬁlling factor Γ. Changes of Gaussian beam divergence at the boundaries of the gain medium are ignored in this plot. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.5 Fabry–Pérot Cavity 0 l 1 R1 l 1 1: R2 217 (6.33) In a stable Fabry–Pérot cavity, the mode-dependent on-axis phase shift in a single pass through the cavity from the left mirror to the right mirror is simply ζ mn ðz2 Þ ζ mn ðz1 Þ for the TEMmn Hermite–Gaussian mode. Therefore, the round-trip mode-dependent on-axis phase shift is ζ RT mn ¼ 2½ζ mn ðz2 Þ ζ mn ðz1 Þ: (6.34) With proper modiﬁcations, the above concept can be used to ﬁnd the characteristics and stability criterion of a cavity that has multiple mirrors, such as a folded Fabry–Pérot cavity or a ring cavity. EXAMPLE 6.5 A two-mirror Fabry–Pérot cavity as shown in Fig. 6.5 has a cavity length of l ¼ 1 m. One mirror has a radius of curvature of R1 ¼ 2 m. Find the condition that the radius of curvature R2 of the other mirror has to satisfy in order for the cavity to be stable. Choose a proper value for R2 so that the cavity is stable and is most symmetric. Find the beam spot size w0 at the beam waist for a Gaussian beam at λ ¼ 600 nm that is stably established in the cavity. Where is the beam waist located? Solution: With l ¼ 1 m and R1 ¼ 2 m, the stability condition in (6.33) requires that l l 1 l 0 1 1 1 ) 0 1 ) jR2 j l ¼ 1 m: 1 R1 R2 2 R2 Under this condition, R2 can be either positive or negative but its magnitude has to be larger than 1 m. For the cavity to be stable and most symmetric, we can choose R2 ¼ R1 ¼ 2 m. Then, using (6.32), we ﬁnd the Rayleigh range: pﬃﬃﬃ 3 lðR1 lÞðR2 lÞðR1 þ R2 lÞ 3 2 2 m: ¼ m ) zR ¼ zR ¼ 2 2 4 ðR1 þ R2 2lÞ The spot size at the beam waist is w0 ¼ λzR π 1=2 pﬃﬃﬃ1=2 600 109 3 ¼ m ¼ 407 μm: 2π Because R2 ¼ R1 , by symmetry the beam waist must be located right at the center of the cavity. 6.5.2 Characteristic Parameters We consider a cavity that contains an isotropic gain medium with a ﬁlling factor of Γ. The surfaces of the gain medium are antireﬂection coated so that there is no reﬂection inside the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 218 Optical Resonance cavity other than the reﬂection at the two end mirrors. If the gain medium ﬁlls up the entire cavity, we simply make Γ ¼ 1 in the results obtained below. The Fabry–Pérot cavity has a physical length of l between the two end mirrors. The ﬁeld reﬂection coefﬁcients are r 1 and r 2 for the left and right mirrors, respectively. They are generally complex to account for the phase changes on reﬂection, φ1 and φ2 , respectively, and can be expressed as 1=2 r 1 ¼ R1 eiφ1 , 1=2 r 2 ¼ R2 eiφ2 , (6.35) where R1 and R2 are the reﬂectivities of the left and right mirrors, respectively. The dielectric property of the intracavity gain medium includes the permittivity of the background material and a resonant susceptibility χ res ðωÞ that characterizes the laser transition. To clearly identify the effect of each contribution, it is instructive to explicitly express the permittivity of the gain medium, including the contribution of the resonant laser transition, as ϵ res ðωÞ ¼ ϵ ðωÞ þ ϵ 0 χ res ðωÞ, (6.36) where ϵ ¼ ϵ 0 n2 is the background permittivity of the gain medium excluding the resonant susceptibility. Because χ res ¼ 0 for a cold cavity, the weighted average of the propagation constant for the intracavity ﬁeld in a cold cavity is k¼ nω ¼ Γk þ ð1 ΓÞk0 , c (6.37) where k ¼ nω=c is the propagation constant in the gain medium and k0 ¼ n0 ω=c is that in the surrounding medium. The round-trip optical path length in this cavity is lRT ¼ 2nl. Usually there is an intracavity background loss contributed by a variety of mechanisms that are irrelevant to the laser transition, such as scattering or absorption. In addition, modedependent diffraction losses exist for the intracavity optical ﬁeld due to the ﬁnite sizes of the end mirrors. The combined effect of these losses can be accounted for by taking a spatially averaged, mode-dependent loss coefﬁcient, α mn , so that the effective propagation constant is complex with a mode-dependent imaginary part: k þ iα mn =2. This loss is known as the distributed loss of the cavity mode. In general, α mn k for a practical optical cavity. By following a mode ﬁeld through one round trip in the cavity, we ﬁnd that a ¼ r 1 r 2 exp i2kl α mn l þ iζ RT mn (6.38) for the TEMmn Hermite–Gaussian mode. Therefore, by using (6.4) and (6.35), we ﬁnd that both the round-trip gain factor and the round-trip phase shift are mode dependent: 1=2 1=2 Gcmn ¼ R1 R2 eα mn l (6.39) RT φRT mn ¼ 2kl þ ζ mn þ φ1 þ φ2 : (6.40) and Using (6.40) for the resonance condition given in (6.19), we ﬁnd the resonance frequencies of the cold Fabry–Pérot cavity: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 6.5 Fabry–Pérot Cavity ωcmnq c ¼ 2qπ ζ RT mn φ1 φ2 , 2nl νcmnq c ζ RT mn þ φ1 þ φ2 ¼ ¼ q , 2π 2π 2nl ωcmnq 219 (6.41) where the superscript c indicates the fact that the frequencies are those for a cold cavity with χ res ¼ 0. These frequencies are clearly functions of the transverse-mode indices because of the RT mode-dependent phase shift ζ RT mn . However, because ζ mn is not a function of the longitudinalmode index q, the frequency separation between two neighboring longitudinal modes of the same transverse mode group is a mode-independent constant: ΔνL ¼ νcmn, qþ1 νcmnq ¼ c 1 ¼ : 2nl T (6.42) Here we assume that the background optical property of the medium is not very dispersive so that the background refractive index n can be considered a constant that is independent of optical frequency in the narrow range between neighboring modes of interest. Using (6.12) and (6.39), the ﬁnesse of the lossy Fabry–Pérot cavity is 1=4 1=4 F¼ πR1 R2 eα mn l=2 1=2 1=2 1 R1 R2 eα mn l , (6.43) which is mode dependent due to the mode-dependent loss α mn . The longitudinal mode width, Δνc ¼ ΔνL =F, is also mode dependent for the same reason. For a cavity that has a negligible loss, we can take α mn ¼ 0; then, (6.43) reduces to the familiar formula for the ﬁnesse of a lossless Fabry–Pérot interferometer as given in (5.46): 1=4 1=4 F¼ πR1 R2 1=2 1=2 1 R1 R2 : (6.44) Therefore, for a nondispersive, lossless Fabry–Pérot cavity, ΔνL , F, and Δνc are all independent of the longitudinal and transverse mode indices though the mode frequency νmnq is a function of all three mode indices. Using (6.24) and (6.39), the mode-dependent photon lifetime of the Fabry–Pérot cavity can be expressed as τ cmnq ¼ nl pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , cðαmn l ln R1 R2 Þ and the mode-dependent cavity decay rate can be expressed as c 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c α mn ln R1 R2 : γmnq ¼ n l (6.45) (6.46) Clearly, both τ cmnq and γcmnq are also mode dependent due to the mode-dependent distributed loss α mn . However, they are independent of the longitudinal mode index q under the assumption that the background refractive index n, the loss α mn , and the mirror reﬂectivities R1 and R2 are not sensitive to the frequency differences among different longitudinal modes. If any of these parameters vary signiﬁcantly within the range of the longitudinal modes of interest, then the dependence of τ cmnq and γcmnq on the index q cannot be ignored. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 220 Optical Resonance A Fabry–Pérot cavity that is used as a laser cavity has a Q value ranging from the order of 103 for a cavity of a high-gain laser that has low mirror reﬂectivities to the order of 108 for a cavity of a low-gain laser that has high mirror reﬂectivities. A Fabry–Pérot cavity that is used as a high-resolution optical spectrum analyzer can have an even higher Q value. EXAMPLE 6.6 The Fabry–Pérot cavity of a high-gain InGaAsP/InP semiconductor laser emitting at the 1.3 μm wavelength has an effective average refractive index of n ¼ nβ ¼ 3:5 deﬁned by the InGaAsP/ InP waveguide mode, a physical length of l ¼ 300 μm, and mirror reﬂectivities of R1 ¼ R2 ¼ 0:3. The structure supports only one transverse mode. Assume a negligibly small α for simplicity. Find the round-trip time, the longitudinal mode frequency spacing, the ﬁnesse, the longitudinal mode width, the photon lifetime, the cavity decay rate, and the quality factor of this cavity as a cold cavity. Solution: The round-trip time of the cavity is T¼ 2nl 2 3:5 300 106 s ¼ 7 ps: ¼ c 3 108 The longitudinal mode frequency spacing is ΔνL ¼ 1 1 Hz ¼ 142:9 GHz: ¼ T 7 1012 Assuming no distributed loss, the ﬁnesse of the cavity is 1=4 1=4 F¼ πR1 R2 1=2 1=2 1 R1 R2 ¼ π 0:31=4 0:31=4 ¼ 2:46: 1 0:31=2 0:31=2 The longitudinal mode width is Δνc ¼ ΔνL 142:9 GHz ¼ 58:1 GHz: ¼ 2:46 F The photon lifetime is τc ¼ nl 3:5 300 106 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s ¼ 2:9 ps: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ c ln R1 R2 3 108 ln 0:3 0:3 The cavity decay rate is γc ¼ 1 1 ¼ s1 ¼ 3:4 1011 s1 : τ c 2:9 1012 To ﬁnd the quality factor, we note that the frequency is found using ω ¼ 2πc=λ for the given optical wavelength of λ ¼ 1:3 μm. Thus, using (6.27), we ﬁnd the quality factor of this cavity to be Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 Problems Q ¼ ωτ c ¼ 221 2πc 2π 3 108 2:9 1012 ¼ 4:2 103 : τc ¼ λ 1:3 106 The approximate relation (6.29) yields a slightly smaller value of Q ¼ 4:0 103 . A Q value on the order of 103 is relatively low for a laser cavity. Even so, the difference between (6.27) and (6.29) is only about 5%. Problems 6.1.1 A folded Fabry–Pérot cavity as shown in Fig. 6.1(b) has two end mirrors with R1 ¼ R2 ¼ 0:8 and a middle mirror with Rm ¼ 0:9 for folding the cavity, which is separated from the two end mirrors at l1m ¼ 0:8 m and l2m ¼ 0:3 m, respectively. A glass rod that has a length of lg ¼ 0:2 m and a refractive index of ng ¼ 1:5 is placed along the beam path between the two mirrors of R1 and Rm . Find the physical length, the round-trip length lRT , the round-trip time T, and the round-trip gain factor G of the cavity. 6.1.2 A ring cavity as shown in Fig. 6.1(c) has three mirrors with R1 ¼ R2 ¼ 0:8 and R3 ¼ 0:9, which are separated at l12 ¼ 0:5 m and l23 ¼ l31 ¼ 0:3 m. A glass rod that has a length of lg ¼ 0:2 m and a refractive index of ng ¼ 1:5 is placed along the beam path between the two mirrors of R1 and R2 . Find the physical length, the round-trip length lRT , the roundtrip time T, and the round-trip gain factor G of the cavity. 6.1.3 An optical-ﬁber ring cavity as shown in Fig. 6.1(d) has one input–output coupler that has a coupling efﬁciency of η ¼ 20%. The ﬁber loop has a length of l ¼ 2 m, and the effective index of the ﬁber mode is n ¼ 1:47. Find the physical length, the round-trip length lRT , the round-trip time T, and the round-trip gain factor G of the cavity. 6.2.1 Find the ﬁnesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc of the folded Fabry–Pérot cavity considered in Problem 6.1.1. 6.2.2 Find the ﬁnesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc of the ring cavity considered in Problem 6.1.2. 6.2.3 Find the ﬁnesse F, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc of the ﬁber ring cavity considered in Problem 6.1.3. 6.3.1 An InP/InGaAsP semiconductor optical cavity has the longitudinal structure of a linear Fabry–Pérot cavity and the transverse structure of a slab waveguide. The cavity has a physical length of l ¼ 400 μm. The slab waveguide supports two TE and two TM modes at the λ ¼ 1:3 μm wavelength, with propagation constants of βTE0 ¼ 1:67 107 m1 , βTM0 ¼ 1:65 107 m1 , βTE1 ¼ 1:57 107 m1 , and βTM1 ¼ 1:56 107 m1 for the TE0 , TM0 , TE1 , and TM1 modes, respectively. The end surfaces of the cavity are not coated. Find the effective round-trip optical path length lRT , the round-trip time T, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc for each transverse mode. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 222 Optical Resonance 6.4.1 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼ 850 nm wavelength of the folded Fabry–Pérot cavity considered in Problems 6.1.1 and 6.2.1. 6.4.2 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼ 850 nm wavelength of the ring cavity considered in Problems 6.1.2 and 6.2.2. 6.4.3 Find the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼ 850 nm wavelength of the ﬁber ring cavity considered in Problems 6.1.3 and 6.2.3. 6.4.4 An optical cavity has two characteristic time constants: the round-trip time T and the photon lifetime τ c . Once they are known, most of the other characteristic parameters of the cavity can be found. Find the cold-cavity ﬁeld-amplitude gain factor Gc , the ﬁnesse F, the longitudinal mode frequency spacing ΔνL , the longitudinal mode width Δνc , the cavity decay rate γc , and the quality factor Q at the λ ¼ 1:3 μm wavelength for an optical cavity that has T ¼ 1 ns and τ c ¼ 20 ns. 6.4.5 An optical cavity has two characteristic spectral parameters: the longitudinal mode frequency spacing ΔνL and the longitudinal mode width Δνc . Once they are known, most of the other characteristic parameters of the cavity can be found. Find the ﬁnesse F, the cold-cavity ﬁeld-amplitude gain factor Gc , the round-trip time T, the photon lifetime τ c , the cavity decay rate γc , and the quality factor Q at the λ ¼ 1:064 μm wavelength for an optical cavity that has ΔνL ¼ 150 MHz and Δνc ¼ 5 MHz. 6.4.6 An optical cavity has two characteristic quality factors: the ﬁnesse F and the quality factor Q at a speciﬁc resonance frequency. Once they are known, most of the other characteristic parameters of the cavity can be found. Find the cold-cavity ﬁeld-amplitude gain factor Gc , the photon lifetime τ c , the cavity decay rate γc , the round-trip time T, the longitudinal mode frequency spacing ΔνL , and the longitudinal mode width Δνc for an optical cavity that has a ﬁnesse of F ¼ 100 and a quality factor of Q ¼ 2 108 at the λ ¼ 532 nm wavelength. 6.5.1 Show for a linear Fabry–Pérot cavity of a length l as shown in Fig. 6.5 that the locations of the left and right end mirrors measured from the beam waist are, respectively, lðR2 lÞ lðR1 lÞ , z2 ¼ , (6.47) R1 þ R2 2l R1 þ R2 2l where R1 and R2 are the radii of curvature of the left and right mirrors, respectively. Show also that the Rayleigh range of a stable Gaussian beam deﬁned by the cavity is that given by (6.32). z1 ¼ 6.5.2 A linear Fabry–Pérot cavity in free space has a concave left mirror that has a radius of curvature of R1 ¼ 2 m and a convex right mirror that has a radius of curvature of R2 ¼ 1 m. The cavity length is l ¼ 1:5 m. Is the cavity stable? If it is stable, where is the Gaussian beam waist located? What is the beam waist spot size? 6.5.3 A symmetric linear Fabry–Pérot cavity in free space has a cavity length of l and two mirrors of the same radius of curvature of R1 ¼ R2 ¼ R ¼ 1 m. (a) In what range can the cavity length be chosen to make the cavity stable? (b) For different choices of the cavity length, where is the location of the beam waist of the Gaussian beam that is deﬁned by the cavity? (c) Find the cavity length that maximizes the waist spot size of the Gaussian beam? What is this spot size for an optical wavelength of λ ¼ 1:064 μm? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 Bibliography 223 (d) For a beam waist spot size of w0 ¼ 350 μm, what is the cavity length that has to be chosen? (e) If the cavity length is chosen to be l ¼ 1:5 m, is the cavity stable? If it is stable, what is the beam waist spot size? 6.5.4 The length of the InGaAsP/InP Fabry–Pérot cavity described in Example 6.6 is doubled to l ¼ 600 μm. At the λ ¼ 1:3 μm wavelength, the effective index of n ¼ nβ ¼ 3:5 and the mirror reﬂectivities of R1 ¼ R2 ¼ 0:3 remain unchanged, while the distributed loss is still negligible. Find the round-trip time, the longitudinal mode frequency spacing, the ﬁnesse, the longitudinal mode width, the photon lifetime, the cavity decay rate, and the quality factor of this cavity. How are these parameters changed as compared to those found in Example 6.6? 6.5.5 The length of the InGaAsP/InP Fabry–Pérot cavity described in Example 6.6 remains l ¼ 300 μm. At the λ ¼ 1:3 μm wavelength, the effective index of n ¼ nβ ¼ 3:5 and the mirror reﬂectivities of R1 ¼ R2 ¼ 0:3 remain unchanged, but the cavity now has a small distributed loss of α ¼ 10 cm1 . Find the round-trip time, the longitudinal mode frequency spacing, the ﬁnesse, the longitudinal mode width, the photon lifetime, the cavity decay rate, and the quality factor of this cavity. How are these parameters changed as compared to those found in Example 6.6? 6.5.6 An optical-ﬁber Fabry–Perot cavity has a physical length of l ¼ 20 m, an averaged intracavity refractive index of n ¼ 1:45, a distributed loss of α ¼ 0:005 m1 , and mirror reﬂectivities of R1 ¼ R2 ¼ 80%. (a) What are the round-trip optical path length, the round-trip time, and the longitudinal mode frequency spacing of this cavity? (b) Find the free spectral range, the ﬁnesse, and the longitudinal mode width of this cavity. (c) What are the cavity decay rate, the photon lifetime, and the quality factor for λ ¼ 1:3 μm? Bibliography Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Fowler, G. R., Introduction to Modern Optics, 2nd edn. New York: Dover, 1975. Haus, H. A., Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984. Iizuka, K., Elements of Photonics in Free Space and Special Media, Vol. I. New York: Wiley, 2002. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010. Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991. Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986. Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996. Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010. Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995. Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University Press, 2007. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:09 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.007 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 7 - Optical Absorption and Emission pp. 224-248 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge University Press 7 7.1 Optical Absorption and Emission OPTICAL TRANSITIONS .............................................................................................................. Optical absorption and emission occur through the interaction of optical radiation with electrons in a material system that deﬁnes the energy levels of the electrons. Depending on the properties of a given material, electrons that interact with optical radiation can be either those bound to individual atoms or those residing in the energy-band structures of a material such as a semiconductor. In any event, the absorption or emission of a photon by an electron is associated with a resonant transition of the electron between a lower energy level j1i of energy E1 and an upper energy level j2i of energy E 2 , as illustrated in Fig. 7.1. The resonance frequency, ν21 , of the transition is determined by the separation between the energy levels: v21 ¼ E2 E1 : h (7.1) In an atomic or molecular system, a given energy level usually consists of a number of degenerate quantum mechanical states that have the same energy. The degeneracy factors g1 and g2 account for the degeneracies in the energy levels j1i and j2i, respectively. There are three basic types of processes associated with resonant optical transitions of electrons between two energy levels: absorption, stimulated emission, and spontaneous emission, which are illustrated in Figs. 7.1(a), (b), and (c), respectively. Absorption and stimulated emission of a photon are both associated with induced transitions between two energy levels caused by the interaction of an electron with existing optical radiation. An electron that is initially in the lower level j1i can absorb a photon to make a transition to the upper level j2i. An electron that is initially in the upper level j2i can be stimulated by the optical radiation to emit a photon while making a downward transition to the lower level j1i. By contrast, spontaneous emission is not induced. Irrespective of the presence or absence of existing optical radiation, an electron initially in the upper level j2i can spontaneously relax to the lower level j1i by emitting a spontaneous photon. Figure 7.1 (a) Absorption, (b) stimulated emission, and (c) spontaneous emission of photons resulting from resonant transitions of electrons in a material. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.1 Optical Transitions 225 A photon that is emitted through stimulated emission has the same frequency, phase, polarization, and propagation direction as the optical radiation that induces the process. By contrast, spontaneously emitted photons are random in phase and polarization, and they are emitted in all directions, though their frequencies are still dictated by the separation between the two energy levels, subject to a degree of uncertainty determined by the linewidth of the transition. Therefore, stimulated emission results in the ampliﬁcation of an optical signal, whereas spontaneous emission merely adds noise to an optical signal. Absorption simply leads to the attenuation of an optical signal. 7.1.1 Spectral Lineshape A resonant transition is selective of the frequency of the interacting optical ﬁeld because the process is associated with the absorption or emission of a photon that has a frequency determined by the energy change of the electron making the transition, as indicated in (7.1). The spectral characteristic of a resonant transition is never inﬁnitely sharp, however. The ﬁnite spectral width of a resonant transition is dictated by the uncertainty principle of quantum mechanics, but it can be intuitively understood using the reasoning in Section 2.3. One important conclusion learned from the discussion in Section 2.3 is that any response that has a ﬁnite relaxation time in the time domain must have a ﬁnite spectral width in the frequency domain. As we shall see later, the induced transition rates of both absorption and stimulated emission between two energy levels in a given system are directly proportional to the spontaneous emission rate from the upper to the lower of the two levels. Therefore, it is a basic law of physics that any allowed resonant transition between two energy levels has a ﬁnite relaxation time because at least the upper level has a ﬁnite lifetime due to spontaneous emission. Consequently, every optical process associated with a resonant transition between two speciﬁc energy levels is characterized by a lineshape function, g^ðvÞ or g^ ðωÞ, of a ﬁnite linewidth. The lineshape function is generally normalized as ð∞ ð∞ g^ðvÞdv ¼ g^ðωÞdω ¼ 1, where g^ðvÞ ¼ 2π^ g ðωÞ: 0 (7.2) 0 7.1.2 Homogeneous Broadening If all of the atoms in a material that participate in a resonant interaction associated with the energy levels j1i and j2i are indistinguishable, their responses to an electromagnetic ﬁeld are characterized by the same transition resonance frequency ν21 and the same relaxation rate γ21 . Note that γ21 is the phase relaxation rate of the resonant interaction between the electromagnetic ﬁeld and the two energy levels. In such a homogeneous system, the physical mechanisms that broaden the linewidth of the transition affect all atoms equally. Spectral broadening caused by such mechanisms is called homogeneous broadening. From the discussion in Section 2.3, the spectral characteristics of a damped response that is characterized by a single resonance frequency and a single relaxation rate, such as that of a Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 226 Optical Absorption and Emission resonant interaction in a homogeneously broadened system, are described by the resonant susceptibility given in (2.26), with its real and imaginary parts given in (2.27). As discussed later in Section 7.2, in the interaction of an optical ﬁeld with a material, the absorption and emission of optical energy are characterized by the imaginary part χ 00res of the resonant susceptibility of the material. Therefore, the spectral characteristics of resonant optical absorption and emission in a homogeneously broadened medium are described by the Lorentzian lineshape function of χ 00res ðωÞ given in (2.27). The normalized Lorentzian lineshape function, which is normalized using (7.2), for the resonant transitions between j1i and j2i has the form: 1 γ21 g^ðωÞ ¼ , π ðω ω21 Þ2 þ γ221 (7.3) which has a FWHM of Δωh ¼ 2γ21 , or g^ ðvÞ ¼ Δvh 2π½ðv v21 Þ2 þ ðΔvh =2Þ2 , (7.4) where Δvh ¼ γ21 π (7.5) is the FWHM of g^ðvÞ. We see that the spectrum has a ﬁnite width that is determined by the relaxation rate γ21 . The fundamental mechanism for homogeneous broadening is lifetime broadening due to the ﬁnite lifetimes, τ 1 and τ 2 , respectively, of the energy levels, j1i and j2i, that are involved in the resonant transitions. The population in an energy level can relax through both radiative and nonradiative transitions to lower levels. Radiative relaxation is associated with population relaxation through spontaneous emission of radiation. The radiative relaxation rate of the transition from level j2i to level j1i is characterized by a rate constant A21 , known as the Einstein A coefﬁcient, which deﬁnes a time constant τ sp ¼ 1=A21 , known as the spontaneous radiative lifetime, between j2i and j1i. Both A21 and τ sp are discussed in further detail later. The total radiative relaxation rate, γrad all radiative 2 , of level j2i is the sum of the rates ofX rad spontaneous transitions from j2i to all levels of lower energies: γ2 ¼ A . The i 2i nonrad nonradiative relaxation rate, γ2 , accounts for all other population relaxation mechanisms that do not result in the emission of photons. The total relaxation rate, γ2 , of level j2i is the sum of its radiative and nonradiative relaxation rates, and the lifetime of the energy level has both radiative and nonradiative contributions: nonrad γ2 ¼ γrad , 2 þ γ2 1 1 1 ¼ rad þ nonrad , τ2 τ2 τ2 (7.6) rad nonrad ¼ 1=γnonrad . This concept can be applied to level j1i where τ 2 ¼ 1=γ2 , τ rad 2 ¼ 1=γ2 , and τ 2 2 to obtain similar relations for γ1 and τ 1 . Though τ 2 has contributions of both radiative and nonradiative relaxations, the ﬂuorescence due to spontaneous emission from level j2i decays in time at the total relaxation rate γ2 because its strength is proportional to the population in level j2i, which relaxes at the total relaxation Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.1 Optical Transitions 227 rate. Therefore, the decay time constant of the ﬂuorescent emission from level j2i is τ 2 , not τ rad 2 . For this reason, the total lifetimes τ 1 and τ 2 are known as the ﬂuorescence lifetimes of energy levels j1i and j2i, respectively. The contributions of various relaxation rates to the radiative and nonradiative lifetimes, and to the ﬂuorescence lifetimes, of the upper and lower energy levels are summarized in Fig. 7.2. The nonradiative relaxation rate of an energy level is a function of extrinsic factors, such as collisions and thermal vibrations. It can therefore be changed by varying the conditions of the surrounding environment. The minimum broadening is called natural broadening, which is caused only by radiative relaxation when all nonradiative processes are eliminated. The linewidth due to natural broadening is determined by the radiative phase relaxation rate caused by radiative decays of the two energy levels: 1 rad 1 1 1 natural rad rad γ21 (7.7) ¼ γ21 ¼ ðγ1 þ γ2 Þ ¼ þ rad : 2 2 τ rad τ2 1 The total phase relaxation rate that characterizes lifetime broadening of the linewidth accounts for the lifetimes of the two energy levels due to both radiative and nonradiative relaxation processes: 1 1 1 1 life þ : (7.8) γ21 ¼ ðγ1 þ γ2 Þ ¼ γnatural 21 2 2 τ1 τ2 and γlife The contributions to γnatural 21 21 are also summarized in Fig. 7.2. Note that the linewidth is determined by the lifetimes of both upper and lower levels. In the case when the lower level j1i is the ground level of an atomic system, we have γ1 ¼ 0 and τ 1 ¼ ∞. Then, the linewidth due to lifetime broadening is solely determined by the lifetime of the upper level, τ 2 . Other mechanisms that affect all atoms equally can further increase the homogeneous linewidth without changing the ﬂuorescence lifetime of either the upper or the lower level. One Figure 7.2 Contributions of various relaxation rates to the radiative and nonradiative lifetimes, and to the ﬂuorescence lifetimes, of the upper and lower energy levels. The homogeneous natural linewidth is determined by the radiative lifetimes, whereas the lifetime-broadened linewidth is determined by the ﬂuorescence lifetimes. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 228 Optical Absorption and Emission important mechanism is collision-induced phase randomization of the emitted radiation. Collisions among atoms in a gas or liquid and collisions between atoms and phonons in a solid normally have two possible effects. One effect is to reduce the ﬂuorescence lifetimes of the upper and lower levels by increasing the nonradiative relaxation rates. Such a process increases life nonrad lifetime broadening; its effect is included in γlife and 21 through the dependence of γ21 on γ1 nonrad contained in γ1 and γ2 , respectively. Collisions can also increase a homogeneous lineγ2 width without reducing the ﬂuorescence lifetimes by simply interrupting the phase of the radiation emitted through radiative relaxation. This dephasing process, quantiﬁed by a linewidth-broadening factor γdephase , is often more important than the lifetime-reduction pro21 cess, resulting in a homogeneous linewidth that is signiﬁcantly broader than the linewidth due to lifetime broadening. Therefore, the homogeneous linewidth can increase with both pressure and temperature in a gas medium, and with active-ion concentration and temperature in a liquid or solid medium. In general, the homogeneous linewidth including the contributions of such extrinsic mechanisms is a function of pressure, P, active-ion concentration, N, and temperature, T: dephase natural γ21 ðP, N, TÞ ¼ γlife γlife : 21 þ γ21 21 γ21 (7.9) EXAMPLE 7.1 The energy levels of Nd:YAG are shown in Fig. 7.3. The highest level 4 F3=2 of the active Nd3þ ion relaxes to four lower levels at different radiative relaxation rates characterized by the Einstein A coefﬁcients shown for different emission wavelengths. The lowest level 4 I9=2 is the ground level, which does not relax to any other level. The dominant transition of this system is that associated with the well-known Nd:YAG emission wavelength of λ ¼ 1:064 μm, which takes place between the upper level 4 F3=2 , labeled j2i, and the lower level 4 I11=2 , labeled j1i. The upper level 4 F3=2 has a lifetime of τ 2 ¼ 240 μs predominantly due to radiative relaxation; Figure 7.3 Energy levels of Nd:YAG. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.1 Optical Transitions 229 the lower level 4 I11=2 has a lifetime of τ 1 ¼ 200 ps purely from nonradiative relaxation. (a) Find the radiative, nonradiative, and total relaxation rates for the upper and lower levels, j2i and j1i, respectively. (b) Find the natural linewidth and the lifetime-broadened linewidth for the λ ¼ 1:064 μm emission line. If no other mechanisms further broaden this line, what is its lineshape and linewidth? (c) At room temperature, dephasing due to phonon collisions contributes a dephasing rate of γdephase ¼ 3:75 1011 s1 to the linewidth. What is the homogeneous line21 width of this emission line at room temperature? Solution: All of the processes considered here cause homogeneous broadening because they are common to all Nd3þ ions. Inhomogeneous broadening mechanisms are not considered in this example. (a) The upper level j2i relaxes both radiatively and nonradiatively to four lower levels, but the lower level j1i relaxes only nonradiatively to the ground level. The total relaxation rates of the two levels are, respectively, γ2 ¼ 1 1 ¼ s1 ¼ 4167 s1 , τ 2 240 106 γ1 ¼ 1 1 ¼ s1 ¼ 5 109 s1 : τ 1 200 1012 The radiative relaxation rates of the two levels are, respectively, X A2i ¼ 3868 s1 , γrad γrad 2 ¼ 1 ¼ 0: i The nonradiative relaxation rates of the two levels are, respectively, 1 γnonrad ¼ γ2 γrad 2 2 ¼ 299 s , 9 1 γnonrad ¼ γ1 γrad 1 1 ¼ 5 10 s : (b) Using the results from (a), we ﬁnd that 1 1 rad 1 ¼ ðγrad ¼ 1934 s1 , γnatural 21 1 þ γ2 Þ ¼ ð0 þ 3868Þ s 2 2 1 1 9 1 ¼ 2:5 109 s1 : γlife 21 ¼ ðγ1 þ γ2 Þ ¼ ð5 10 þ 4167Þ s 2 2 The natural linewidth and the lifetime-broadened linewidth are, respectively, Δνnatural ¼ γnatural 21 ¼ 616 Hz, π Δνlife ¼ γlife 21 ¼ 796 MHz: π If no other mechanisms further broaden this line, this emission line has a Lorentzian lineshape that has a homogeneously broadened linewidth of Δνh ¼ Δνlife ¼ 796 MHz: (c) With a dephasing rate of γdephase ¼ 3:75 1011 s1 , the total phase relaxation rate is 21 dephase ¼ 3:775 1011 s1 : γ21 ¼ γlife 21 þ γ21 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 230 Optical Absorption and Emission Thus, the homogeneous linewidth is broadened to Δνh ¼ γ21 ¼ 120 GHz: π The linewidth is further broadened by inhomogeneous mechanisms discussed below. For the λ ¼ 1:064 μm line of Nd:YAG, the total linewidth varies with temperature and with the quality of the YAG crystal. Increasing temperature increases the homogeneous linewidth, whereas a poorer crystal quality leads to a larger inhomogeneous linewidth. In any event, this emission line of Nd:YAG is predominantly homogeneously broadened at room temperature. 7.1.3 Inhomogeneous Broadening A resonant transition can be further broadened by inhomogeneous broadening if certain physical mechanisms exist that do not equally affect all atoms, causing energy levels j1i or j2i, or both, to shift differently among different groups of atoms. The resulting inhomogeneous shifts of the transition resonance frequency cause inhomogeneous broadening of the transition spectrum on top of the original homogeneous broadening. If we express the homogeneous lineshape function given in (7.4) as g^h ðν, ν21 Þ to explicitly indicate that its transition resonance frequency is ν21 , the homogeneously broadened spectrum of a group of atoms whose resonance frequency is shifted from ν21 to νk is g^h ðν, νk Þ. The distribution of atoms in an inhomogeneous system can be described by a probability density function pðνk Þ with ð∞ pðνk Þdνk ¼ 1: (7.10) 0 The probability that the resonance frequency of a given atom falls in the range between νk and νk þ dνk is pðνk Þdνk . Then, the overall spectral lineshape of the inhomogeneously broadened transition is ð∞ g^ðνÞ ¼ pðνk Þ^ g h ðν, νk Þdνk : (7.11) 0 The overall lineshape function obtained from (7.11) depends on the degree of inhomogeneous broadening in comparison to homogeneous broadening. Mathematically, it depends on the spread of the distribution function pðνk Þ in comparison to the homogeneous linewidth. One possibility for inhomogeneous broadening is the existence of different isotopes, which have slightly different resonance frequencies for a given resonant transition. In this situation, pðνk Þdνk represents the percentage of each isotope group among all atoms and (7.11) becomes simply the weighted sum of the isotope groups. Other mechanisms for inhomogeneous broadening include the Doppler effect in a gaseous medium at a low pressure and the random distribution of active impurity atoms doped in a solid Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.1 Optical Transitions 231 host. The inhomogeneous frequency shifts caused by these mechanisms are usually randomly distributed, resulting in a Gaussian functional distribution for pðνk Þ. In an extremely inhomogeneously broadened system, the spread of this distribution dominates the homogeneous linewidth. Then, the transition is characterized by a normalized Gaussian lineshape: " # 2ðln 2Þ1=2 ðν ν0 Þ2 g^ðνÞ ¼ 1=2 exp 4 ln 2 , (7.12) π Δνinh Δν2inh where ν0 is the center frequency and Δνinh is the FWHM of the inhomogeneously broadened spectral distribution. In terms of the angular frequency, the normalized Gaussian lineshape is " # 2ðln 2Þ1=2 ðω ω0 Þ2 g^ðωÞ ¼ 1=2 exp 4 ln 2 , (7.13) π Δωinh Δω2inh where ω0 ¼ 2πν0 and Δωinh ¼ 2πΔνinh . Whether a medium is homogeneously or inhomogeneously broadened is often a function of pressure and temperature. In a gas at a low pressure, the velocity distribution of the gas molecules in thermal equilibrium is characterized by the Maxwellian velocity distribution, which is a Gaussian function. This velocity distribution leads to a Gaussian distribution of Doppler frequency shifts with a linewidth of ΔνD given by 3=2 ΔνD ¼ 2 ðln 2Þ 1=2 k B T 1=2 23=2 ðln 2Þ1=2 k B T 1=2 ν ¼ , λ Mc2 M (7.14) where λ is the emission wavelength, kB is the Boltzmann constant, T is the temperature in kelvin, and M is the mass of the atom or molecule that emits the radiation. When this Doppler-broadening effect dominates, the Gaussian lineshape has an inhomogeneous linewidth of Δνinh ¼ ΔνD . Figure 7.4 Normalized Lorentzian (solid curves) and Gaussian (dashed curves) lineshape functions of the same FWHM with (a) a normalized area as g^ ðνÞ is deﬁned and (b) a normalized peak value. For the Lorentzian lineshape, ν0 ¼ ν21 and Δν ¼ Δνh . For the Gaussian lineshape, Δν ¼ Δνinh . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 232 Optical Absorption and Emission The normalized Lorentzian lineshape function and the normalized Gaussian lineshape function of the same FWHM are compared in Fig. 7.4. In Fig. 7.4(a), we show g^ðνÞ as expressed in (7.4) for the Lorentzian lineshape and in (7.12) for the Gaussian lineshape, both with a normalized area as deﬁned in (7.2). In Fig. 7.4(b), the lineshapes are normalized to have the same peak value. EXAMPLE 7.2 The transition for the well-known He–Ne emission wavelength of λ ¼ 632:8 nm takes place between the 3s2 level, which is the upper level j2i, and the 2p4 level, which is the lower level j1i, of the Ne atom. The upper and lower levels for this emission both relax 20 rad radiatively, with τ 2 ¼ τ rad and 2 ¼ 30 ns and τ 1 ¼ τ 1 ¼ 10 ns. Two Ne isotopes, Ne 22 20 Ne , contribute to this emission, with more than 90% due to Ne . For simplicity, we take the atomic mass number of Ne to be 20. The typical He–Ne laser medium operates at a temperature of T ¼ 400 K and a low gas pressure of P ¼ 2:5 torr. (a) Find the radiative, nonradiative, and total relaxation rates for the upper and lower levels, j2i and j1i, respectively. (b) Find the natural linewidth and the lifetime-broadened linewidth of the emission line. (c) Find the linewidth caused by Doppler broadening. (d) What is the lineshape and linewidth of this emission line? Solution: Natural broadening and lifetime broadening are homogeneous broadening mechanisms, whereas Doppler broadening is an inhomogeneous broadening mechanism. Pressure-induced broadening is a homogeneous mechanism, but it can be ignored in this problem because of the low gas pressure of P ¼ 2:5 torr. (a) Both the upper level j2i and the lower level j1i relax radiatively. For each level, the total relaxation rate is the same as the radiative relaxation rate: γ2 ¼ γrad 2 ¼ 1 1 ¼ s1 ¼ 3:3 107 s1 , τ 2 30 109 γ1 ¼ γrad 1 ¼ 1 1 ¼ s1 ¼ 1 108 s1 : τ 1 10 109 The nonradiative relaxation rates of the two levels are both zero: nonrad γnonrad ¼ γ2 γrad ¼ γ1 γrad 2 2 ¼ 0, γ1 1 ¼ 0: (b) Using the results from (a), we ﬁnd that 1 1 rad 8 7 1 ¼ ðγrad ¼ 6:7 107 s1 , γnatural 21 1 þ γ1 Þ ¼ ð1 10 þ 3:3 10 Þ s 2 2 1 1 8 7 1 γlife ¼ 6:7 107 s1 : 21 ¼ ðγ1 þ γ2 Þ ¼ ð1 10 þ 3:3 10 Þ s 2 2 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.1 Optical Transitions 233 The natural linewidth and the lifetime-broadened linewidth are the same: Δνlife ¼ γlife γnatural 21 ¼ Δνnatural ¼ 21 ¼ 21:2 MHz: π π If no other mechanisms further broaden this line, this emission line has a Lorentzian lineshape that has a homogeneously broadened linewidth of Δνh ¼ Δνlife ¼ 21:2 MHz. (c) The mass of a Ne atom is M ¼ 20 1:66 1027 kg ¼ 3:32 1026 kg for a mass number of 20. Therefore, the Doppler-broadened linewidth at T ¼ 400 K is 1=2 23=2 ðln 2Þ1=2 k B T 1=2 23=2 ðln 2Þ1=2 1:38 1023 400 ¼ Hz ¼ 1:5 GHz: ΔνD ¼ λ M 632:8 109 3:32 1026 (d) Because ΔνD Δνlife , the homogeneous lifetime broadening is completely dominated by the inhomogeneous Doppler broadening. Therefore, the lineshape of this emission line is Gaussian with a linewidth of Δνinh ΔνD ¼ 1:5 GHz: 7.1.4 Mixed Broadening When the pressure of a gaseous medium is increased, frequent collisions among the gas molecules shorten the lifetimes of the excited states of the molecules. This effect reduces τ 2 , and it can also reduce τ 1 if the lower level is not the ground level. The resulting pressureinduced lifetime broadening causes the homogeneous linewidth to increase. At a certain pressure, the homogeneous linewidth Δνh ﬁnally dominates the Doppler linewidth ΔνD . Then the medium becomes predominantly homogeneously broadened. Another good example is the linewidth associated with the impurity ions doped in a solid host, such as Nd:YAG or Nd:glass. At a low temperature, the homogeneous linewidth of the Nd3þ ions is narrow. The lineshape is dominated by inhomogeneous shifts of the resonance frequency due to variations in the local environment of individual Nd3þ ions. As a result, the lineshape function is inhomogeneously broadened. As the temperature increases, the homogeneous linewidth increases because of increased collisions of phonons with the ions. At room temperature, the spectral line of Nd:YAG at 1.064 μm has a total linewidth of Δν 120 to 180 GHz with an inhomogeneous component of only about 6 to 30 GHz. Therefore, as illustrated in Example 7.1, Nd:YAG is pretty much homogeneously broadened at room temperature. In comparison, Nd:glass has a much larger inhomogeneous linewidth than Nd:YAG because the glass host provides a larger range of local variations than the YAG crystal. At room temperature, the same spectral line of Nd: glass appears at 1.054 μm with a total linewidth of Δν 5 to 7 THz, which is almost all inhomogeneously broadened. Clearly a lineshape can be neither Lorentzian nor Gaussian when the homogeneously broadened linewidth Δνh and the inhomogeneously broadened linewidth Δνinh of an emission line are on the same order of magnitude. In this situation, the line proﬁle is a convolution of the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 234 Optical Absorption and Emission Lorentzian proﬁle of a width Δνh and the Gaussian proﬁle of a width Δνinh . The result is a Voigt lineshape that has a linewidth of Δν 0:5346Δνh þ ð0:2166Δν2h þ Δν2inh Þ1=2 : 7.2 (7.15) TRANSITION RATES .............................................................................................................. The probability per unit time for a resonant optical process to occur is measured by the transition rate of the process. Because of the resonant nature of the interaction, the transition rate of an induced process is a function of both the spectral distribution of the optical radiation and the spectral characteristics of the resonant transition. The spectral distribution of an optical ﬁeld is characterized by its spectral energy density, uðνÞ, which is the energy density of the optical radiation per unit frequency interval at the optical frequency ν. The total energy density of the radiation is ð∞ u ¼ uðνÞdν: (7.16) 0 The spectral intensity distribution, IðνÞ, of the radiation is related to uðνÞ by the relation c IðνÞ ¼ uðνÞ, n (7.17) where n is the refractive index of the medium, and the total intensity is simply ð∞ I ¼ IðνÞdν: (7.18) 0 Because an induced transition is stimulated by optical radiation, its transition rate is proportional to the energy density of the optical radiation within the spectral response range of the transition. The transition rate for the upward transition from j1i to j2i, associated with absorption, in the frequency range between ν and ν þ dν is W 12 ðνÞdν ¼ B12 uðνÞ^ g ðνÞdν, (7.19) whereas that for the induced downward transition from j2i to j1i, associated with stimulated emission, in the frequency range between ν and ν þ dν is W 21 ðνÞdν ¼ B21 uðνÞ^ g ðνÞdν: (7.20) Because the spontaneous emission rate is independent of the energy density of the radiation, the spontaneous emission spectrum is determined solely by the lineshape function of the transition: W sp ðνÞdν ¼ A21 g^ðνÞdν: (7.21) The A and B constants deﬁned above are known as the Einstein A and B coefﬁcients, respectively. The rates associated with the transitions between two atomic levels j1i and j2i Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.2 Transition Rates 235 Figure 7.5 Resonant transitions in the interaction of a radiation ﬁeld with two atomic levels j1i and j2i of population densities N 1 and N 2 , respectively. in the interaction with a radiation ﬁeld of an energy density uðνÞ are summarized in Fig. 7.5. The total induced transition rates are ð∞ ð∞ W 12 ¼ W 12 ðνÞdν ¼ B12 uðνÞ^ g ðνÞdν and 0 0 ð∞ ð∞ W 21 ¼ W 21 ðνÞdν ¼ B21 uðνÞ^ g ðνÞdν: 0 (7.22) (7.23) 0 The total spontaneous emission rate is ð∞ W sp ¼ W sp ðνÞdν ¼ A21 : (7.24) 0 The induced and spontaneous transition rates of a given system are not independent of each other but are directly proportional to each other. Their relationship was ﬁrst obtained by Einstein by considering the interaction of blackbody radiation with an ensemble of identical atomic systems in thermal equilibrium. The spectral energy density of blackbody radiation at a temperature T is given by Planck’s formula: uðνÞ ¼ 8πn3 hν3 1 , 3 hν=k T 1 B c e (7.25) where k B is the Boltzmann constant. As shown in Fig. 7.5, the population densities per unit volume of the atoms in levels j2i and j1i are N 2 and N 1 , respectively. The number of atoms per unit volume making the downward transition per unit time accompanied by the emission of radiation in a frequency range from ν to ν þ dν is N 2 ½W 21 ðνÞ þ W sp ðνÞdν, and the number of atoms per unit volume making the upward transition per unit time through the absorption of radiation in the same frequency range is N 1 W 12 ðνÞdν. In thermal equilibrium, both the spectral density of blackbody radiation and the atomic population density in each energy level reach a steady state, meaning that N 2 ½W 21 ðνÞ þ W sp ðνÞ ¼ N 1 W 12 ðνÞ: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 (7.26) 236 Optical Absorption and Emission This relation spells out the principle of detailed balance in thermal equilibrium. Therefore, the steady-state population distribution in thermal equilibrium satisﬁes N2 W 12 ðνÞ B12 uðνÞ ¼ ¼ : N 1 W 21 ðνÞ þ W sp ðνÞ B21 uðνÞ þ A21 (7.27) In thermal equilibrium at a temperature T, however, the population ratio of the atoms in the upper and the lower levels follows the Boltzmann distribution. Taking into account the degeneracy factors, g2 and g1 , of these energy levels, we have N 2 g2 hv=kB T ¼ e N 1 g1 (7.28) for the population densities associated with a transition energy of hν. Combining (7.27) and (7.28), we have uðνÞ ¼ A21 =B21 : ðg1 B12 =g2 B21 Þehv=kB T 1 (7.29) Identifying (7.29) with (7.25), we ﬁnd that A21 8πn3 hν3 ¼ B21 c3 (7.30) g1 B12 ¼ g2 B21 : (7.31) and The spontaneous radiative lifetime of the atoms in level j2i associated with the radiative spontaneous transition from j2i to j1i is τ sp ¼ 1 1 ¼ : W sp A21 (7.32) The spectral dependence of the spontaneous emission rate can be expressed as W sp ðνÞ ¼ 1 g^ ðνÞ: τ sp (7.33) According to the relations in (7.30) and (7.31), the transition rates of both of the induced processes of absorption and stimulated emission are directly proportional to the spontaneous emission rate. In terms of τ sp , the spectral dependence of the stimulated-emission transition from j2i to j1i can be generally expressed as W 21 ðνÞ ¼ c3 c2 uðνÞ^ g ðνÞ ¼ IðνÞ^ g ðνÞ, 8πn3 hv3 τ sp 8πn2 hv3 τ sp (7.34) and that for the absorption transition from j1i to j2i can be found as W 12 ðνÞ ¼ g2 W 21 ðνÞ: g1 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 (7.35) 7.2 Transition Rates 237 Because WðνÞ is the transition rate per unit frequency according to the deﬁnition in (7.19)– (7.21), we have WðνÞdν ¼ WðωÞdω. Therefore, W sp ðνÞ ¼ 2πW sp ðωÞ, W 21 ðνÞ ¼ 2πW 21 ðωÞ, and W 12 ðνÞ ¼ 2πW 12 ðωÞ. EXAMPLE 7.3 A cylindrical Nd:YAG rod has a length of l ¼ 5 cm and a diameter of d ¼ 6 mm. The Nd3þ ions are doped in the YAG host at 1.2% atomic concentration for a total concentration of N t ¼ 1:66 1020 cm3 . The rod is uniformly pumped such that 1% of the Nd3þ ions are excited to the 4 F3=2 level and then left to relax spontaneously. Use the parameters given in Fig. 7.3 for the energy levels of Nd:YAG to answer the following questions regarding the emission at the two lines of λ ¼ 1:064 μm and λ ¼ 1:34 μm. (a) Find the spontaneous radiative lifetimes for the transitions of the two emission lines, respectively. (b) What are the decay times of the spontaneous emission at the two emission lines, respectively? (c) What are the optical energies of the spontaneous emission at the two wavelengths, respectively? (d) What are the powers of the spontaneous emission at the two wavelengths, respectively? Solution: The Nd:YAG rod has a volume of V ¼ πðd=2Þ2 l ¼ πð6 103 =2Þ2 5 102 m3 ¼ 1:41 106 m3 : It is pumped to have a concentration in the upper level j2i of N 2 ¼ 1%N t ¼ 1:66 1018 cm3 ¼ 1:66 1024 m3 : (a) The spontaneous radiative lifetime of each transition is determined by the A coefﬁcient of the transition. From Fig. 7.3, we ﬁnd A1:064 ¼ 1940 s1 and A1:34 ¼ 493 s1 . Therefore, the spontaneous radiative lifetimes are, respectively, τ sp 1:064 ¼ 1 1 ¼ s ¼ 515 μs, A1:064 1940 τ sp 1:34 ¼ 1 1 ¼ s ¼ 2:03 ms: A1:34 493 (b) Because the spontaneous emission at both emission lines results from the population in level j2i, the number density S1:064 of the spontaneous photons that are emitted at λ ¼ 1:064 μm and the number density S1:34 of the spontaneous photons emitted at λ ¼ 1:34 μm are both proportional to N 2 . Therefore, the ﬂuorescence at both wavelengths decays at the same rate as that of N 2 . The ﬂuorescence time is the same for both wavelengths and is the lifetime τ 2 ¼ 240 μs of level j2i, given in Fig. 7.3. (c) Though the number densities S1:064 and S1:34 of the spontaneous photons emitted at λ ¼ 1:064 μm and λ ¼ 1:34 μm, respectively, are both proportional to N 2 and both decay at the same decay time, their magnitudes are respectively proportional to the spontaneous radiative relaxation rates, A1:064 and A1:34 , of their transitions: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 238 Optical Absorption and Emission S1:064 ¼ A1:064 N 2 ¼ A1:064 τ 2 N 2 ¼ 1940 240 106 1:66 1024 m3 ¼ 7:73 1023 m3 , γ2 S1:34 ¼ A1:34 N 2 ¼ A1:34 τ 2 N 2 ¼ 493 240 106 1:66 1024 m3 ¼ 1:96 1023 m3 : γ2 The photon energies at the two wavelengths are, respectively, hv1:064 ¼ 1:2398 1:2398 eV, hv1:34 ¼ eV: 1:064 1:34 The spontaneous optical energies emitted at the two wavelengths are, respectively, U 1:064 ¼ hv1:064 S1:064 V ¼ U 1:34 ¼ hv1:34 S1:34 V ¼ 1:2398 1:6 1019 7:73 1023 1:41 106 J ¼ 203 mJ; 1:064 1:2398 1:6 1019 1:96 1023 1:41 106 J ¼ 41 mJ: 1:34 Because these optical energies both decay at the ﬂuorescence time of τ 2 ¼ 240 μs, P1:064 ¼ P1:34 ¼ U 1:064 203 103 ¼ W ¼ 846 W, τ2 240 106 U 1:34 41 103 ¼ W ¼ 17 W: τ2 240 106 7.2.1 Transition Cross Section It is often useful to express the transition probability of an atom in its interaction with optical radiation at a frequency of ν in terms of the transition cross section, σðνÞ. For transitions between energy levels j1i and j2i, the transition cross sections σ 21 ðνÞ and σ 12 ðνÞ are deﬁned through the following relations to the transition rates, W 21 ðνÞ ¼ IðνÞ σ 21 ðνÞ hν (7.36) W 12 ðνÞ ¼ IðνÞ σ 12 ðνÞ: hν (7.37) and The transition cross section σ 21 ðνÞ, which is associated with stimulated emission, is also called the emission cross section, σ e ðνÞ, whereas σ 12 ðνÞ, which is associated with absorption, is also called the absorption cross section, σ a ðνÞ. From (7.34), we ﬁnd that σ e ðνÞ ¼ σ 21 ðνÞ ¼ c2 g^ ðνÞ: 8πn2 ν2 τ sp Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 (7.38) 7.2 Transition Rates 239 According to (7.35), we ﬁnd that g1 σ 12 ¼ g2 σ 21 . Therefore, σ a ðνÞ ¼ σ 12 ðνÞ ¼ g2 g σ 21 ðνÞ ¼ 2 σ e ðνÞ: g1 g1 (7.39) The transition cross sections have the unit of area in square meters but are often quoted in square centimeters. Note that σðνÞ ¼ σðωÞ because σðνÞ is simply deﬁned as the value of the transition cross section at the frequency ν rather than as that per unit frequency, but WðνÞ ¼ 2πWðωÞ and g^ ðνÞ ¼ 2π^ g ðωÞ. Therefore, in terms of ω, σ e ðωÞ ¼ σ 21 ðωÞ ¼ π 2 c2 g^ ðωÞ n2 ω2 τ sp and σ a ðωÞ ¼ g2 σ e ðωÞ: g1 (7.40) For the ideal Lorentzian and Gaussian lineshapes expressed in (7.4) and (7.12), respectively, the peak value of g^ ðνÞ occurs at the center of the spectrum and is a function of the linewidth Δν only. By applying this fact to (7.38), the peak value of the emission cross section at the center wavelength λ of the spectrum can be expressed as σ he ¼ λ2 4π 2 n2 Δνh τ sp (7.41) for a homogeneously broadened medium that has an ideal Lorentzian lineshape, and as σ inh e ¼ ðln 2Þ1=2 λ2 4π 3=2 n2 Δνinh τ sp (7.42) for an inhomogeneously broadened medium that has an ideal Gaussian lineshape. In practice, the experimentally measured peak emission cross section usually differs from that calculated using these formulas because the spectral lineshape of a realistic gain medium is generally determined by a combination of many different mechanisms and, consequently, is rarely ideal Lorentzian or ideal Gaussian. Nevertheless, these formulas provide a good estimate for the peak value of the emission cross section. They also clearly indicate that the emission cross section varies quadratically with the emission wavelength but is inversely proportional to both the emission linewidth and the spontaneous radiative lifetime of the transition. The characteristics of some representative laser materials are listed in Table 7.1. As seen in Table 7.1, the parameters vary over a wide range among different types of optical gain media. For example, the peak value of the emission cross section varies from 6 1025 m2 for Er:ﬁber to 2:5 1016 m2 for the Ar-ion laser, whereas the spontaneous emission linewidth varies from 60 MHz for CO2 to 100 THz for Ti:sapphire. The ﬂuorescence lifetime varies from the order of 1 ns for a semiconductor gain medium to the order of 10 ms for Er:ﬁber. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 240 Optical Absorption and Emission Table 7.1 Characteristics of some laser materials Gain medium Wavelength System λ (μm) a Cross section σe (m2) Spontaneous linewidthc b Lifetimesd Δν Δλ (nm) τ sp τ2 Index n He–Ne 0.6328 I,4 3.0 1017 1.5 GHz 0.002 300 ns 30 ns 1 Ar ion 0.488 I,4 2.5 1016 2.7 GHz 0.004 13 ns 10 ns 1 CO2 10.6 I,4 3.0 1022 60 MHz 0.02 4s 1 μs 1 Copper vapor 0.5105 I,3 8.6 1018 2.3 GHz 0.002 500 ns 500 ns 1 KrF excimer 0.248 H,3 2.6 1020 10 THz 2 10 ns 8 ns 1 R6G dye 0.57–0.65 H/I,Q2 2.3 1020 30 THz 33 6 ns 4 ns 1.4 Rubye 0.6943 H,3 1.25–2.5 1024 330 GHz 0.53 3 ms 3 ms 1.76 Nd:YAG 1.064 H,4 2–10 1023 150 GHz 0.56 515 μs 240 μs 1.82 Nd:glass 1.054 I,4 4.0 1024 6 THz 22 330 μs 330 μs 1.53 Er:ﬁber 1.53 H/I,3 6.0 1025 5 THz 40 10 ms 10 ms 1.46 0.66–1.1 H,Q2 3.4 1023 100 THz 180 3.9 μs 3.2 μs 1.76 0.78–1.01 H,Q2 4.8 1024 83 THz 200 67 μs 67 μs 1.4 H/I,Q2 1–5 1020 10–20 THz 20–100 1 ns 1 ns 3–4 Ti:sapphire Cr:LiSAF f Semiconductor 0.37–1.65 a H, homogeneously broadened; I, inhomogeneously broadened; Q2, quasi-two-level system; 3, three-level system; 4, four-level system. b Both the absorption and emission cross sections depend on the optical frequency. The absorption and emission cross sections generally have different peak values and different spectral dependences. Listed is the peak value of the emission cross section. c The spontaneous linewidth determines the gain bandwidth of a medium when population inversion is achieved. d The spontaneous lifetime τ sp is related to the transition rate, whereas the ﬂuorescence lifetime τ 2 is related to the upper-level population relaxation. The ﬂuorescence lifetime of a gaseous medium varies with temperature and pressure; that of a liquid or solid medium varies with temperature, the host material, and the concentration of the active ions or molecules. For example, τ 2 of CO2 varies from 100 ns to 1 ms depending on temperature and pressure. e Ruby is sapphire (Al2O3) doped with Cr3+ ions. The sapphire crystal is uniaxial. For ruby, the value of σe for emission with E⊥c, which is listed, is larger than that for Ekc. f For Ti:sapphire, the value of σ e for Ekc, which is listed, is larger than that for E⊥c. EXAMPLE 7.4 The λ ¼ 1:064 μm emission line of Nd:YAG considered in Example 7.1 has a predominantly homogeneously broadened total linewidth of 150 GHz and a spontaneous radiative relaxation rate of A ¼ 1940 s1 . The refractive index of the YAG crystal is n ¼ 1:82. The λ ¼ 632:8 nm Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 241 7.3 Attenuation and Ampliﬁcation of Optical Fields emission line of He–Ne considered in Example 7.2 has a predominantly inhomogeneously broadened total linewidth of 1.5 GHz and a spontaneous radiative lifetime of τ sp ¼ 300 ns. The refractive index of the low-pressure He–Ne gas is n ¼ 1. Find the peak emission cross sections for these two lines. Solution: For the λ ¼ 1:064 μm emission line of Nd:YAG, we take Δνh ¼ 150 GHz to be the homogeneous linewidth as an approximation because this line is predominantly homogeneously broadened. The spontaneous radiative lifetime is τ sp ¼ 1=A ¼ 515 μs. Then, using (7.41), the emission cross section is found to be σ he ¼ λ2 ð1:064 106 Þ2 ¼ m2 ¼ 1:12 1022 m2 , 4π 2 n2 Δνh τ sp 4π 2 1:822 150 109 515 106 which is slightly larger than, but consistent with, the value listed in Table 7.1. For the λ ¼ 632:8 nm emission line of He–Ne, we take Δνinh ¼ 1:5 GHz to be the inhomogeneous linewidth as an approximation because this line is predominantly inhomogeneously broadened. With a spontaneous radiative lifetime of τ sp ¼ 300 ns, the emission cross section is found using (7.42) to be σ inh e ¼ ðln 2Þ1=2 λ2 ðln 2Þ1=2 ð632:8 109 Þ2 ¼ m2 ¼ 3:33 1017 m2 , 4π 3=2 n2 Δνinh τ sp 4π 3=2 12 1:5 109 300 109 which is slightly larger than, but consistent with, the value listed in Table 7.1. 7.3 ATTENUATION AND AMPLIFICATION OF OPTICAL FIELDS .............................................................................................................. Optical absorption results in the attenuation of an optical ﬁeld, whereas stimulated emission leads to the ampliﬁcation of an optical ﬁeld. To quantify the net effect of a resonant transition on the attenuation or ampliﬁcation of an optical ﬁeld, we consider the interaction of a monochromatic plane optical ﬁeld at a frequency of ν with a material that consists of electronic or atomic systems with population densities N 1 and N 2 in energy levels j1i and j2i, respectively. Because the spectral intensity distribution of the monochromatic plane optical ﬁeld that has an intensity of I is simply IðνÞ ¼ Iδðν0 νÞ, the total induced transition rates between energy levels j1i and j2i in this interaction are I I (7.43) σ e ðνÞ and W 12 ¼ σ a ðνÞ: hν hν The net power that is transferred from the optical ﬁeld to the material is the difference between that absorbed by the material and that emitted due to stimulated emission: W 21 ¼ W p ¼ hνW 12 N 1 hνW 21 N 2 ¼ ½N 1 σ a ðνÞ N 2 σ e ðνÞI: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 (7.44) 242 Optical Absorption and Emission In the case when W p > 0, there is net power absorption by the medium from the optical ﬁeld due to resonant transitions between energy levels j1i and j2i. The absorption coefﬁcient, also called attenuation coefﬁcient, is g1 αðνÞ ¼ N 1 σ a ðνÞ N 2 σ e ðνÞ ¼ N 1 N 2 σ a ðνÞ: (7.45) g2 In the case when W p < 0, net power is transferred from the medium to the optical ﬁeld, resulting in the ampliﬁcation of the optical ﬁeld. The gain coefﬁcient, also called the ampliﬁcation coefﬁcient, is g2 gðνÞ ¼ N 2 σ e ðνÞ N 1 σ a ðνÞ ¼ N 2 N 1 σ e ðνÞ: (7.46) g1 The coefﬁcients α and g have the unit of per meter, also often quoted per centimeter. Note that αðνÞ ¼ αðωÞ and gðνÞ ¼ gðωÞ because σðνÞ ¼ σðωÞ. Note also that αðνÞ ¼ gðνÞ because a negative gain is a positive loss, and vice versa. According to (7.43), both σ e ðνÞ and σ a ðνÞ have positive values because W 21 0 and W 12 0 by deﬁnition. Therefore, αðνÞ > 0 and gðνÞ < 0 if N 1 > ðg1 =g2 ÞN 2 , whereas gðνÞ > 0 and αðνÞ < 0 if N 2 > ðg2 =g1 ÞN 1 . A material in its normal state in thermal equilibrium absorbs optical energy because the lower energy level is more populated than the upper energy level. In order to provide a net optical gain to the optical ﬁeld, a material has to be in a nonequilibrium state of population inversion for the upper level to be more populated than the lower level. EXAMPLE 7.5 The λ ¼ 1:064 μm emission line of Nd:YAG has τ 2 ¼ 240 μs for the upper level j2i and τ 1 ¼ 200 ps for the lower level j1i, as shown in Fig. 7.3. We consider here the Nd:YAG rod in Example 7.3, which is doped with Nd3þ ions at 1.2% atomic concentration for a total concentration of N t ¼ 1:66 1020 cm3 . If it is not pumped, what is its absorption coefﬁcient at λ ¼ 1:064 μm at T ¼ 300 K? If the rod is uniformly pumped such that 1% of the total Nd3þ ions are excited to level j2i, what is the absorption or gain coefﬁcient at λ ¼ 1:064 μm? Solution: The lower level j1i is not the ground level. From Fig. 7.3, we ﬁnd that its energy above the ground level is ΔE 10 ¼ 1:2398 1:2398 eV eV ¼ 0:21 eV: 0:9 1:064 At T ¼ 300 K, k B T ¼ 25:9 meV. Thus, the population density of Nd3þ ions in this level is approximately 0:21 ΔE 10 =kB T N 1 N te ¼ 3 104 N t ¼ N t exp 25:9 103 which is negligibly small because level j1i lies sufﬁciently high above the ground level. Therefore, the absorption coefﬁcient at λ ¼ 1:064 μm is negligibly small: α 0. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 7.3 Attenuation and Ampliﬁcation of Optical Fields 243 When 1% of the total Nd3þ ions are excited to level j2i, we have N 2 ¼ 1%N t ¼ 1:66 1018 cm3 ¼ 1:66 1024 m3 : In this situation, the excited ions can relax from level j2i to level j1i, but any ion reaching level j1i τ2. quickly relaxes to the ground level because level j1i has a short lifetime of τ 1 ¼ 200 ps Therefore, level j1i remains almost empty, N 1 0, as compared to level j2i. The emission cross section of the λ ¼ 1:064 μm line found in Example 7.4 is σ e ¼ 1:12 1022 m2 . Consequently, at λ ¼ 1:064 μm the Nd:YAG rod has a gain coefﬁcient of g ¼ N 2 σ e N 1 σ a N 2 σ e ¼ 1:66 1024 1:12 1022 m1 ¼ 186 m1 : This is a very large gain coefﬁcient even though only 1% of the total Nd3þ ions are excited. In practice, depending on the design of the laser cavity, only a smaller percentage of ions has to be excited for laser action. 7.3.1 Resonant Optical Susceptibility The macroscopic optical properties of a medium are characterized by its electric susceptibility. As seen in Section 2.3, resonances in an optical medium contribute to the dispersion in the susceptibility of the medium. Clearly, the optical properties of a medium are functions of the resonant optical transitions between the energy levels of the electrons in the medium. From the viewpoint of the macroscopic optical properties of a medium, the interaction between an optical ﬁeld and a medium is characterized by the polarization induced by the optical ﬁeld in the medium. The power exchange between the optical ﬁeld and the medium is given by (1.34). For the resonant interaction of an isotropic medium with a monochromatic plane optical ﬁeld at a frequency of ω ¼ 2πν, we have EðtÞ ¼ Eeiωt þ E∗ eiωt and ∗ iωt Pres ðtÞ ¼ ϵ 0 ½ χ res ðωÞEeiωt þ χ ∗ res ðωÞE e , where Pres is the polarization contributed by the resonant transitions and χ res is the resonant susceptibility. Using (1.34), we ﬁnd that the timeaveraged power density absorbed by the medium is ω W p ¼ 2ωϵ 0 χ 00res ðωÞjEj2 ¼ χ 00res ðωÞI: (7.47) nc By identifying (7.47) with (7.44), we ﬁnd that the imaginary part of the susceptibility contributed by the resonant transitions between energy levels j1i and j2i is χ 00res ðωÞ ¼ nc ½N 1 σ a ðωÞ N 2 σ e ðωÞ: ω (7.48) The real part χ 0res ðωÞ of the resonant susceptibility can be found through the Kramers–Kronig relations given in (2.53). As discussed in Sections 2.1 and 2.3, a medium causes an optical loss if χ 00 > 0, and it provides an optical gain if χ 00 < 0. It is also clear from (7.47) that there is a net power loss from the optical ﬁeld due to absorption by the medium if χ 00res > 0, but there is a net power gain for the optical ﬁeld if χ 00res < 0. By comparing (7.48) with (7.45) and (7.46), we ﬁnd that the medium has an absorption coefﬁcient given by Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 244 Optical Absorption and Emission αðωÞ ¼ ω 00 χ ðωÞ nc res (7.49) in the case of normal population distribution when χ 00res > 0, whereas it has a gain coefﬁcient given by gðωÞ ¼ ω 00 χ ðωÞ nc res (7.50) in the case of population inversion so that χ 00res < 0. Note that the material susceptibility characterizes the response of a material to the excitation of an electromagnetic ﬁeld. Therefore, the magnitude of the resonant susceptibility χ 00res only accounts for the contributions from the induced processes of absorption and stimulated emission, and not that from the process of spontaneous emission. Spontaneous emission causes natural broadening of the spectral width of χ 00res ðωÞ, as discussed in Section 7.1. The resonant susceptibility contributed by the induced transitions between two energy levels is proportional to the population difference between the two levels, but the power density of the optical radiation due to spontaneous emission is a function of the population density in the upper energy level alone. The coefﬁcients α and g respectively characterize the attenuation and growth of the optical intensity per unit length traveled by the optical wave in a medium. The intensity of a monochromatic plane wave at the resonance frequency varies with distance along its propagation direction, taken to be the z direction, as dI ¼ αI dz (7.51) in the case of optical attenuation when χ 00res > 0, and dI ¼ gI dz (7.52) in the case of optical ampliﬁcation when χ 00res < 0. EXAMPLE 7.6 What is the imaginary part χ 00res of the resonant susceptibility, at λ ¼ 1:064 μm, of the pumped Nd:YAG rod considered in Example 7.5? The refractive index of Nd:YAG is n ¼ 1:82. The rod has a length of l ¼ 5 cm. If a beam at λ ¼ 1:064 μm that has a power of Pin ¼ 1 mW is sent into one end of the Nd:YAG rod uniformly over the cross-sectional area of the rod, what is the optical power coming out at the other end? Solution: From Example 7.5, the gain coefﬁcient at λ ¼ 1:064 μm for the pumped Nd:YAG rod is g ¼ 186 m1 . Using (7.50), we ﬁnd the imaginary part of the resonant susceptibility at λ ¼ 1:064 μm: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 Problems χ 00res ¼ 245 nc nλ 1:82 1:064 106 186 ¼ 5:73 105 : g¼ g¼ 2π ω 2π For uniform illumination, (7.52) can be written in terms of the optical power to ﬁnd the output power as 2 dP ¼ gP ) Pout ¼ Pin egl ¼ 1 103 e186510 W ¼ 10:9 W: dz Problems 7.1.1 A ruby laser rod is a sapphire crystal doped with active Cr3þ ions. The upper level j2i of the transition for the ruby emission wavelength of λ ¼ 694:3 nm is the E level of the Cr3þ ion, and the lower level j1i is the 4 A2 ground level. The population in the E level relaxes only to the 4 A2 ground level, and the relaxation is purely radiative. The upper level lifetime is τ 2 ¼ 3 ms. At room temperature, this emission line has a predominantly homogeneous linewidth of Δν ¼ 330 GHz. (a) Find the radiative, nonradiative, and total relaxation rates for the upper and lower levels, j2i and j1i, respectively. (b) Find the natural linewidth and the lifetime-broadened linewidth for the λ ¼ 694:3 nm emission line. If no other mechanisms further broaden this line, what are its lineshape and linewidth? (c) The homogeneous broadening at room temperature is contributed by dephasing due to phonon collisions. What is the dephasing rate γdephase ? 21 7.1.2 Ti:sapphire and Cr:LiSAF are solid-state laser media. Ti:sapphire contains active Ti3þ ions doped in a sapphire crystal, and Cr:LiSAF contains active Cr3þ ions doped in a LiSAF crystal. The ﬂuorescence lifetime of Ti:sapphire is τ 2 ¼ 3:2 μs, and that of Cr: LiSAF is τ 2 ¼ 67 μs. For both systems, the lower level j1i is the ground level. Both media have very broad spontaneous linewidths that are predominantly homogeneously broadened, with Δν 100 THz for Ti:sapphire and Δν 83 THz for Cr:LiSAF. What are the expected lifetime-broadened homogeneous linewidths of these two media? Explain why these two media have such broad homogeneous linewidths. 7.1.3 The CO2 laser gain medium contains the gas mixture of CO2 , N2 , and He with about the same fractional ratio of CO2 and N2 , and somewhat more He. The λ ¼ 10:6 μm emission takes place between two vibrational levels of the CO2 molecule. The upper level j2i has a radiative lifetime of τ rad 2 ¼ 4 s, and the lower level j1i has a radiative lifetime of rad τ 1 ¼ 200 ms. The N2 molecules help to pump the CO2 molecules to the upper level j2i, while the He atoms help to de-excite the N2 and CO2 molecules back to their respective ground levels. The collisions of the CO2 molecules with the N2 molecules and the He atoms change the lifetimes τ 2 of the upper level and τ 1 of the lower level by inducing nonradiative relaxations from these levels. As a result, τ 2 and τ 1 depend on the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 246 Optical Absorption and Emission pressure and temperature of the gas mixture. The working temperature of a CO2 laser ranges from 400 K to 700 K. The working gas pressure varies from below 50 torr to 760 torr for different CO2 lasers. (a) Find the radiative relaxation rates for the upper and lower levels, j2i and j1i, respectively. What is the natural linewidth of the emission line? (b) The molecular mass number of CO2 is 44. Find the range of the Doppler-broadened linewidth for the CO2 lasers. (c) Consider a CO2 laser medium of a relatively low pressure working at T ¼ 400 K, which has τ 2 ¼ 10 μs and τ 1 ¼ 1 μs. Find the nonradiative and total relaxation rates for the upper and lower levels, j2i and j1i, respectively. What are the homogeneously and inhomogeneously broadened linewidths of the emission line? What are the lineshape and the total linewidth? Is it homogeneously or inhomogeneously broadened? (d) Consider a CO2 laser medium of a high pressure working at T ¼ 700 K, which has τ 2 ¼ 100 ns and τ 1 ¼ 1 ns. Find the nonradiative and total relaxation rates for the upper and lower levels, j2i and j1i, respectively. What are the homogeneously and inhomogeneously broadened linewidths of the emission line? What are the lineshape and the total linewidth? Is it homogeneously or inhomogeneously broadened? 7.1.4 The argon-ion laser has two emission lines at 488 nm and 514:5 nm. Both lines are almost entirely broadened by Doppler broadening at the typical operating temperature of T ¼ 1200 C. The Ar atom has an atomic mass number of 40. Find the linewidths and the lineshapes of the two emission lines, respectively. 7.2.1 A cylindrical ruby rod, which is a sapphire crystal doped with active Cr3þ ions, has a length of l ¼ 6 cm and a diameter of d ¼ 5 mm. The Cr3þ ions has a total concentration of N t ¼ 1:58 1019 cm3 . The upper level j2i of the transition for the ruby emission wavelength of λ ¼ 694:3 nm relaxes only radiatively through this emission line with a 3þ lifetime of τ 2 ¼ τ rad ions 2 ¼ 3 ms. The rod is uniformly pumped such that 50% of the Cr are excited to the upper level and then left to relax spontaneously. (a) Find the spontaneous radiative lifetime for the transition of this emission line. What is the decay time of the spontaneous emission? (b) What are the optical energy and the power of the spontaneous emission? 7.2.2 Two emission lines have exactly the same wavelength and the same linewidth, but one has a Lorentzian lineshape while the other has a Gaussian lineshape. If the optical transitions for both emission lines have the same spontaneous lifetime and the two media have the same refractive index, do they have the same peak emission cross section? If they do not have the same peak emission cross section, which one has a larger cross section? What is the difference? 7.2.3 Two emission lines have exactly the same center wavelength, the same linewidth, the same peak emission cross section, and they take place in two media that have the same refractive index, but one has a Lorentzian lineshape and the other has a Gaussian lineshape. What is the possible parameter that has different values for these two transitions? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 Problems 247 7.2.4 Are the emission cross section and the absorption cross section of the same spectral line associated with the transitions between the same pair of energy levels necessarily the same? Explain. 7.2.5 The upper level j2i of the transition for the ruby emission wavelength of λ ¼ 694:3 nm is the E level of the active Cr 3þ ions doped in the ruby crystal, which has a degeneracy of g2 ¼ 2, and the lower level j1i is the 4 A2 ground level, which has a degeneracy of g1 ¼ 4. The population in the E level relaxes radiatively only through this emission line to the 4 A2 ground level with τ 2 ¼ τ rad 2 ¼ 3 ms. At room temperature, this emission line has a homogeneous linewidth of Δν ¼ 330 GHz. The refractive index of the ruby crystal is n ¼ 1:76. Find the peak emission and absorption cross sections for this spectral line. 7.2.6 The λ ¼ 510:5 nm emission line of the copper vapor laser has a linewidth of 2:3 GHz, which is almost entirely caused by Doppler broadening, and a spontaneous radiative lifetime of τ sp ¼ 500 ns. The refractive index of the low-pressure gaseous medium is n 1. Find the peak emission cross section of this line. 7.3.1 A large absorption cross section of Ti:sapphire appears at the wavelength of λa ¼ 490 nm with σ a ðλa Þ ¼ 6:4 1024 m2 , while σ e ðλa Þ 3 1028 m2 . The peak emission cross section appears at the wavelength of λe ¼ 795 nm with σ e ðλe Þ ¼ 3:4 1023 m2 , while σ a ðλe Þ 8 1026 m2 . The lower level is the ground level. A Ti:sapphire rod that is not pumped is found to have an absorption coefﬁcient of αðλa Þ ¼ 200 m1 at λa ¼ 490 nm. (a) Find the total doping concentration N t of the active Ti3þ ions in this rod. (b) If a gain coefﬁcient of gðλe Þ ¼ 20 m1 is desired at λe ¼ 795 nm, what percent of the Ti3þ ions have to be excited to the upper level? 7.3.2 Ti:sapphire has a refractive index of n ¼ 1:76. A Ti:sapphire rod has a length of l ¼ 10 cm. (a) When it is not pumped, it has an absorption coefﬁcient of αðλa Þ ¼ 200 m1 at λa ¼ 490 nm. Find the imaginary part χ 00res of the resonant susceptibility at this wavelength. If a beam that has a power of Pin ðλa Þ ¼ 1 W at λa ¼ 490 nm is sent into the rod from one end, what is the output power at the other end? How much of the power is absorbed? (b) It is pumped so that it has a gain coefﬁcient of gðλe Þ ¼ 20 m1 at λe ¼ 795 nm. Find the imaginary part χ 00res of the resonant susceptibility at this wavelength. If a beam that has a power of Pin ðλe Þ ¼ 1 mW at λe ¼ 795 nm is sent into the rod from one end, what is the output power at the other end? How much of the power is emitted through stimulated emission? 7.3.3 Because the lower level of the He–Ne emission line at λ ¼ 632:8 nm is not the ground level, an unexcited Ne atom does not absorb light at this wavelength. The emission cross section of this emission line is σ e ¼ 3 1017 m2 . An optical beam at λ ¼ 632:8 nm is sent through a uniformly pumped He–Ne tube that has a length of l ¼ 1 m. If the output power is 120% of the input power, what is the population density of the excited Ne atoms in the upper level of the emission line? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 248 Optical Absorption and Emission 7.3.4 An Er:ﬁber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . It is found to have an absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼ 7:9 1025 m2 at the λ ¼ 1:53 μm wavelength. The lower level is the ground level. Assume uniform pumping throughout the ﬁber. Assume also that all Er3þ ions are distributed only between the two levels of the λ ¼ 1:53 μm transition. (a) What is its intrinsic absorption coefﬁcient α0 at this wavelength when the Er:ﬁber is not pumped? (b) What percent of the Er3þ ions have to be pumped to the upper level for the ﬁber to be transparent with α ¼ g ¼ 0? (c) What percent of the Er3þ ions have to be pumped to the upper level for a gain coefﬁcient of g ¼ 0:2 m1 ? (d) What percent of the Er3þ ions have to be pumped to the upper level for a gain coefﬁcient of g ¼ α0 ? (e) What is the maximum gain coefﬁcient g max when all Er3þ ions are pumped to the upper level? Compare it to the intrinsic absorption coefﬁcient α0 , which is the maximum value of the absorption coefﬁcient. Bibliography Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010. Rosencher, E. and Vinter, B., Optoelectronics. Cambridge: Cambridge University Press, 2002. Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991. Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986. Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996. Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010. Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995. Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University Press, 2007. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:30 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.008 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 8 - Optical Amplification pp. 249-273 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge University Press 8 8.1 Optical Ampliﬁcation POPULATION RATE EQUATIONS .............................................................................................................. From the discussion in the preceding chapter, it is clear that population inversion is the basic condition for an optical gain. For any system in its normal state in thermal equilibrium, a lowenergy level is always more populated than a high-energy level, hence there is no population inversion. Population inversion in a system can only be accomplished through a process called pumping by actively exciting the atoms in a low-energy level to a high-energy level. If left alone, the atoms in a system relax to thermal equilibrium. Therefore, population inversion is a nonequilibrium state that cannot be sustained without active pumping. To keep a constant optical gain, continuous pumping is required to maintain population inversion. This condition is clearly consistent with the law of conservation of energy: ampliﬁcation of an optical wave leads to an increase in optical energy, which is possible only if the required energy is supplied by a source. Pumping is the process that supplies energy to the gain medium for the ampliﬁcation of an optical wave. There are many different pumping techniques, including optical excitation, electric current injection, electric discharge, chemical reaction, and excitation with particle beams. The use of a speciﬁc pumping technique depends on the properties of the gain medium being pumped. The lasers and optical ampliﬁers of particular interest in photonic systems are made of either dielectric solid-state media doped with active ions, such as Nd:YAG and Er: glass ﬁber, or direct-gap semiconductors, such as GaAs and InP. For a dielectric gain medium, the most commonly used pumping technique is optical pumping using either an incoherent light source, such as a ﬂashlamp or a light-emitting diode, or a coherent light source from another laser. A semiconductor gain medium can also be optically pumped, but it is usually pumped by electric current injection. In this section, we consider the general conditions for pumping to achieve population inversion. Detailed pumping mechanisms and physical setups are not addressed here because they depend on the speciﬁc gain medium used in a particular application. The net rate of increase of the population density in a given energy level is described by a rate equation. As we shall see below, pumping for population inversion in any practical gain medium always requires the participation of more than two energy levels. In general, a rate equation has to be written for each energy level that is involved in the process. For simplicity but without loss of validity, however, we shall explicitly write down only the rate equations for the two energy levels, j2i and j1i, that are directly associated with the resonant transition of interest. We are not interested in the population densities of other energy levels but only in how they affect N 2 and N 1 . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 250 Optical Ampliﬁcation In the presence of a monochromatic optical wave that has an intensity of I at a frequency of v, the rate equations that govern the temporal evolution of N 2 and N 1 are dN 2 N2 I ðN 2 σ e N 1 σ a Þ, ¼ R2 dt τ 2 hv (8.1) dN 1 N1 N2 I ¼ R1 þ þ ðN 2 σ e N 1 σ a Þ, dt τ 1 τ 21 hv (8.2) where R2 and R1 are the total rates of pumping into energy levels j2i and j1i, respectively, and τ 2 and τ 1 are the ﬂuorescence lifetimes of levels j2i and j1i, respectively. The total rate of population relaxation, including radiative and nonradiative spontaneous relaxations, from level j2i to level j1i is τ 1 21 . Because it is possible for the population in level j2i to also relax to other 1 energy levels, the total population relaxation rate of level j2i is τ 1 2 τ 21 . Therefore, in general, we have τ 2 τ 21 τ sp : (8.3) 1 Note that τ 1 21 is not the same as γ21 deﬁned in (7.9): τ 21 is purely the rate of population relaxation from level j2i to level j1i, whereas γ21 is the rate of phase relaxation of the polarization associated with the transition between these two levels. For an optical gain medium, level j2i is known as the upper laser level, and level j1i is known as the lower laser level. The ﬂuorescence lifetime τ 2 of the upper laser level is an important parameter that determines the effectiveness of a gain medium. Generally speaking, for a gain medium to be useful, the upper laser level has to be a metastable state that has a relatively large τ 2 . To account for the difference between the emission cross section and the absorption cross section, the effective population inversion can be more accurately deﬁned as N ¼ N2 σa N 1: σe (8.4) With this deﬁnition for the effective population inversion, the gain coefﬁcient is simply g ¼ σ e N ¼ α: (8.5) This relation is also valid for ﬁnding the absorption coefﬁcient. A positive gain coefﬁcient g > 0 is found when the system reaches effective population inversion so that N > 0; it has a negative gain coefﬁcient, i.e., a positive absorption coefﬁcient, α ¼ g > 0 when effective population inversion is not accomplished so that N < 0. For the different systems discussed in the following section, the two rate equations given in (8.1) and (8.2) for N 2 and N 1 can be combined into one equation for the effective population inversion N: dN N I ¼ R β σ e N, dt τ2 hv (8.6) where R is the effective pumping rate for population inversion and β ¼1þ σa 1 σe Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 (8.7) 8.2 Population Inversion 251 is the bottleneck factor that characterizes the effectiveness of pumping a system for population inversion. It is more difﬁcult to reach population inversion in a system that has a larger value of β. Note that the detailed form of the effective pumping rate R depends on the pumping mechanism and the pumping scheme. It can be a function of the effective population inversion N, as in the situation when the gain medium contains a ﬁxed density of active atoms or molecules. In this case, the pumping rate R cannot be generally taken as an independent external parameter. However, it is possible in a different situation that the pumping rate can be taken as an independent external parameter, such as in the case of a semiconductor gain medium that is pumped by current injection where the pumping rate is determined by the injection current. In the following section, we consider the case when a gain medium contains a ﬁxed, ﬁnite concentration of active atoms or molecules so that the pumping rate R is a function of the effective population inversion N. EXAMPLE 8.1 A Nd:YAG crystal is doped with 1 at.% of Nd3þ ions for a concentration of N t ¼ 1:38 1026 m3 . For its λ ¼ 1:064 μm laser line, the emission cross section is found to be σ e ¼ 4:5 1023 m2 and the absorption cross section is σ a ¼ 0 because the lower laser level of this laser line is effectively empty all the time. A ruby crystal is doped with 0.05 wt.% of Cr3þ ions for a concentration of N t ¼ 1:58 1025 m3 . For its λ ¼ 694:3 nm laser line, the emission cross section is found to be σ e ¼ 1:34 1024 m2 and the absorption cross section is σ a ¼ 1:25 1024 m2 . The variations in the measured emission and absorption cross sections of these gain media are caused by the population ratios in the degenerate states of each laser level, which vary with doping and temperature. Find the bottleneck factors for these two laser media. Solution: The bottleneck factor of this Nd:YAG crystal at λ ¼ 1:064 μm is β ¼1þ σa 0 ¼1þ ¼ 1: σe 4:5 1023 The bottleneck factor of this ruby crystal at λ ¼ 694:3 nm is β ¼1þ σa 1:25 1024 ¼1þ ¼ 1:93: σe 1:34 1024 The λ ¼ 1:064 μm laser line of Nd:YAG has the smallest possible bottleneck factor of β ¼ 1 because σ a ¼ 0. The λ ¼ 694:3 nm laser line of ruby has a bottleneck factor of β ¼ 1:93, which is close to 2, because σ a is comparable to σ e . 8.2 POPULATION INVERSION .............................................................................................................. Population inversion between the upper laser level j2i of a degeneracy g2 and the lower laser level j1i of a degeneracy g1 in a medium is generally deﬁned as Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 252 Optical Ampliﬁcation N2 N1 g g > so that N 1 < 1 N 2 and N 2 > 2 N 1 : g2 g1 g2 g1 (8.8) According to (7.45) and (7.46), this condition makes αðvÞ < 0 and g ðvÞ > 0 so that the medium shows a positive optical gain. However, in many systems, the degenerate states in level j1i or j2i, or both, are split into closely spaced sublevels to form small energy bands. When the energy spread of the sublevels in a laser level is sufﬁciently large, the population in the level can be distributed unevenly so that (7.39) is not valid, i.e., σ a ðvÞ 6¼ ðg2 =g1 Þσ e ðvÞ. In this situation, the second equal sign in (7.45) and (7.46) is not valid though the ﬁrst equal sign is still valid: g1 (8.9) αðvÞ ¼ N 1 σ a ðvÞ N 2 σ e ðvÞ 6¼ N 1 N 2 σ a ðvÞ g2 and g ðvÞ ¼ N 2 σ e ðvÞ N 1 σ a ðvÞ 6¼ g2 N 2 N 1 σ e ðvÞ: g1 (8.10) For this reason, when the condition for population inversion given in (8.8) is achieved in a medium, we might ﬁnd σ a ðvÞ ðg2 =g1 Þσ e ðvÞ for an optical gain at an optical frequency v while at the same time we might ﬁnd σ a ðv0 Þ > ðg2 =g1 Þσ e ðv0 Þ for an optical loss at another frequency v0 . Therefore, the population inversion condition in (8.8) does not guarantee an optical gain at a particular optical frequency v in the case when the population in level j1i or j2i is distributed unevenly among its sublevels so that σ a ðvÞ 6¼ ðg2 =g1 Þσ e ðvÞ. What really matters to an optical wave at a given frequency is the optical gain at that speciﬁc frequency. For this reason, in the following discussion, we shall consider, instead of the condition in (8.8), the condition that guarantees an optical gain at the frequency v, g ðvÞ ¼ N 2 σ e ðvÞ N 1 σ a ðvÞ ¼ Nσ e ðvÞ > 0, (8.11) as the effective condition for population inversion as far as an optical signal at the frequency v is concerned. Clearly, by deﬁning the effective population inversion N as in (8.4), the effective condition for population inversion is simply N > 0. This population inversion condition can be reached even when N 2 < N 1 in the case when σ a < σ e . On the other hand, if σ a > σ e , it is possible that N 2 > N 1 but N 2 is not sufﬁciently large so that N < 0 and effective population inversion for an optical gain is not reached. The pumping requirement for the condition in (8.11) to be satisﬁed depends on the properties of a medium. For atomic and molecular media, there are three different basic systems. Each has a different pumping requirement to reach effective population inversion for an optical gain. The pumping requirement can be found by solving the coupled rate equations given in (8.1) and (8.2). EXAMPLE 8.2 Use the parameters given in Example 8.1 to ﬁnd the effective population inversion required to have a gain coefﬁcient of g ¼ 10 m1 for the λ ¼ 1:064 μm laser line of Nd:YAG and that required for the λ ¼ 694:3 nm laser line of ruby. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.2 Population Inversion 253 Solution: For the λ ¼ 1:064 μm laser line of Nd:YAG, σ e ¼ 4:5 1023 m2 . Therefore, the required effective population inversion is g 10 ¼ m3 ¼ 2:22 1023 m3 : 23 σ e 4:5 10 For the λ ¼ 694:3 nm laser line of ruby, σ e ¼ 1:34 1024 m2 . Therefore, the required effective population inversion is N¼ g 10 ¼ m3 ¼ 7:46 1024 m3 : σ e 1:34 1024 For the same gain coefﬁcient, the population inversion required for the ruby laser line is about 34 times that required for the Nd:YAG laser line because the emission cross section of the Nd:YAG laser line is about 34 times that of the ruby laser line. N¼ 8.2.1 Two-Level System When the only energy levels involved in the pumping and the relaxation processes are the upper and lower laser levels, j2i and j1i, the system can be considered as a two-level system, as shown in Fig. 8.1. In such a system, level j1i is the ground level, which has τ 1 ¼ ∞, and level j2i relaxes only to level j1i, so that τ 21 ¼ τ 2 . The total population density is N t ¼ N 1 þ N 2 . While a pumping mechanism excites atoms from the lower laser level to the upper laser level of a two-level system, the same pump also stimulates atoms in the upper laser level to relax to the lower laser level. Therefore, irrespective of the speciﬁc pumping technique used, it is always true that R2 ¼ R1 ¼ W p12 N 1 W p21 N 2 , where W p12 and W p21 are the pumping transition probability rates, or simply the pumping rates, from j1i to j2i and from j2i to j1i, respectively. Under these conditions, (8.1) and (8.2) are equivalent to each other. The upward and downward pumping rates are not independent of each other but are directly proportional to each other because both are associated with the interaction between the same pump source and a given pair of energy levels. We take the upward pumping rate to be W p12 ¼ W p and the downward Figure 8.1 (a) Pumping scheme of a true two-level system. (b) Pumping scheme of a quasi-two-level system. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 254 Optical Ampliﬁcation pumping rate to be W p21 ¼ pW p , where p is a constant that depends on the detailed characteristics of the two-level atomic system and the pump source. In the steady state when dN 2 =dt ¼ dN 1 =dt ¼ 0 , we then ﬁnd that g ¼ N 2σe N 1σa ¼ W p τ 2 ðσ e pσ a Þ σ a Nt: 1 þ ð1 þ pÞW p τ 2 þ ðIτ 2 =hvÞðσ e þ σ a Þ Using the relation in (7.43), we ﬁnd that, in the case of optical pumping, W p21 σ pe σ e λp p ¼ p ¼ p ¼ , W 12 σ a σ a λp (8.12) (8.13) where σ pa and σ pe are the absorption and emission cross sections, respectively, at the pump wavelength. In a true two-level system, shown in Fig. 8.1(a), the energy levels j2i and j1i can respectively be degenerate with degeneracies g2 and g1 , but the population density in each level is evenly distributed among the degenerate states in the level. In this situation, p ¼ σ pe =σ pa ¼ g1 =g2 ¼ σ e =σ a . Then, we ﬁnd from (8.12) that g ¼ N 2σe N 1σa ¼ σ a N t < 0: 1 þ Iτ 2 =hv þ W p τ 2 =σ a ðσ e þ σ a Þ (8.14) No matter how a true two-level system is pumped, it is clearly not possible to achieve population inversion for an optical gain in the steady state. This situation can be understood by considering the fact that the pump for a two-level system has to be in resonance with the transition between the two levels, thus simultaneously inducing downward and upward transitions. In the steady state, the two-level system reaches thermal equilibrium with the pump at a ﬁnite temperature, resulting in a Boltzmann population distribution of the form given in (7.28) without population inversion. As discussed above and illustrated in Fig. 8.1(b), however, in many systems an energy level is actually split into a band of closely spaced, but not exactly degenerate, sublevels with its population density unevenly distributed among these sublevels. This type of system is not a true two-level system, but is known as a quasi-two-level system, if either or both of the two levels are split in such a manner. By properly pumping a quasi-two-level system, it is possible to reach the needed population inversion in the steady state for an optical gain at a particular laser frequency v because the ratio p ¼ σ pe =σ pa at the pump frequency vp can now be made different from the ratio σ e =σ a at the laser frequency v due to the uneven population distribution among the sublevels within an energy level. From (8.12), we ﬁnd that the pumping requirements for a quasi-two-level system to have a steady-state optical gain are p¼ σ pe σ e σa and W p > : p < τ 2 ðσ e pσ a Þ σa σa (8.15) Because the absorption spectrum is generally shifted to the short-wavelength side of the emission spectrum, these conditions can be satisﬁed by pumping sufﬁciently strongly at a higher transition energy than the photon energy at the peak of the emission spectrum. In the case of optical pumping, this condition means that the pump wavelength has to be shorter than the emission wavelength. Figure 8.1(b) illustrates such a pumping scheme for a quasi-two-level Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.2 Population Inversion 255 system. Indeed, many laser gain media, including laser dyes, semiconductor gain media, and vibronic solid-state gain media, are often pumped as a quasi-two-level system. 8.2.2 Three-Level System Population inversion in the steady state is possible for a system that has three energy levels involved in the process. Figure 8.2 shows the energy-level diagram of an idealized three-level system. The lower laser level j1i is the ground level, E 1 ¼ E 0 , or is very close to the ground level, within an energy separation of ΔE10 ¼ E 1 E0 k B T from the ground level, so that it is initially populated. The atoms are pumped to an energy level j3i above the upper laser level j2i. An effective three-level system satisﬁes the following conditions. 1. Population relaxation from level j3i to level j2i is very fast and efﬁcient, ideally τ 2 τ 32 τ 3 , so that the atoms excited by the pump quickly end up in level j2i. 2. Level j3i lies sufﬁciently high above level j2i with ΔE32 ¼ E 3 E2 k B T so that the population in level j2i cannot be thermally excited back to level j3i. 3. The lower laser level j1i is the ground level, or its population relaxes very slowly if it is not the ground level, so that τ 1 ∞. Furthermore, level j2i relaxes mostly to level j1i so that τ 21 τ 2 . Under these conditions, R2 W p N 1 , R1 W p N 1 , and N 1 þ N 2 N t . The parameter W p is the effective pumping rate of exciting atoms in the ground level to eventually reach the upper laser level. It is proportional to the pump power. In the steady state under constant pumping, W p is a constant and dN 2 =dt ¼ dN 1 =dt ¼ 0. With these conditions, we ﬁnd that g ¼ N 2 σe N 1 σa ¼ W pτ2σe σa Nt: 1 þ W p τ 2 þ ðIτ 2 =hvÞðσ e þ σ a Þ (8.16) Therefore, the pumping requirement for a positive optical gain under steady-state population inversion is Wp > σa : τ2σe (8.17) This condition sets the minimum pumping requirement for a three-level system to have a positive optical gain. This requirement can be understood by considering the fact that almost Figure 8.2 Energy levels of a three-level system. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 256 Optical Ampliﬁcation all of the population initially resides in the lower laser level j1i. To achieve effective population inversion, the pump has to be strong enough to sufﬁciently depopulate level j1i while the system has to be able to keep the excited atoms in level j2i. In the case when σ a ¼ σ e , for a bottleneck factor of β ¼ 2, no population inversion occurs before at least half of the total population is transferred from level j1i to level j2i. This is the bottleneck effect that limits the energy conversion efﬁciency of a three-level laser system as compared to a quasi-two-level or four-level system. 8.2.3 Four-Level System A four-level system, shown schematically in Fig. 8.3, is more efﬁcient than a three-level system. A four-level system differs from a three-level system in that the lower laser level j1i lies sufﬁciently high above the ground level j0i with ΔE10 ¼ E 1 E0 k B T so that in thermal equilibrium the population in level j1i is negligibly small compared to that in level j0i. Pumping takes place from level j0i to level j3i. An effective four-level system also has to satisfy the conditions concerning levels j3i and j2i discussed above for an effective three-level system. In addition, it has to satisfy the condition that the population in level j1i relaxes very quickly to the ground level, ideally τ 1 τ 10 τ 2 , so that level j1i remains relatively unpopulated in comparison to level j2i when the system is pumped. Under these conditions, N 1 0 and R2 W p ðN t N 2 Þ, where the effective pumping rate W p is again proportional to the pump power. Because N 1 0, (8.2) can be ignored. For a four-level system, we can also take σ a ¼ 0, for a bottleneck factor of β ¼ 1, because its absorption coefﬁcient at the laser wavelength is zero even when it is not pumped. In the steady state with a constant W p , we ﬁnd by taking dN 2 =dt ¼ 0 for (8.1) and taking σ a ¼ 0 that g ¼ N 2σe ¼ W pτ2σe Nt: 1 þ W p τ 2 þ σ e Iτ=hv (8.18) This result indicates that there is no minimum pumping requirement for an ideal four-level system that satisﬁes the conditions discussed above. No bottleneck effect limits an ideal fourlevel system because level j1i is initially empty in such a system. Real systems are rarely ideal, but a practical four-level system is still much more efﬁcient than a three-level system. Figure 8.3 Energy levels of a four-level system. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.2 Population Inversion 257 8.2.4 Transparency When the gain coefﬁcient is zero, g ¼ 0, the medium becomes transparent, or bleached, to the optical signal, neither absorbing nor amplifying it. An ideal four-level system is transparent at no pumping. A quasi-two-level or three-level system reaches transparency, or the bleached condition, at the transparency pumping rate: W trp ¼ σa β1 ¼ , τ 2 ðσ e pσ a Þ τ 2 ½1 pðβ 1Þ (8.19) where β is the bottleneck factor deﬁned in (8.7). This relation is valid for all systems though it is obtained for a two-level or three-level system. For a four-level system, we simply ﬁnd from (8.19) that W trp ¼ 0 because σ a ¼ 0 and β ¼ 1 for the system. For a system to have an optical gain, the pumping rate has to be higher than the transparency pumping rate: W p > W trp . For a four-level system, any pumping leads to a gain because it is always true that W p > W trp ¼ 0 as long as the system is pumped. For a two-level or three-level system, which has σ a 6¼ 0 so that β > 1, it is possible for the system to have no optical gain but optical attenuation when it is not sufﬁciently pumped such that W trp > W p > 0. The relation in (8.19) gives the necessary pumping effort for a system to reach transparency and then an optical gain above it. Another useful measure is the population density N 2 that has to be pumped to the upper laser level in order for a system to have an optical gain. For a twolevel or three-level system, N 1 þ N 2 N t . By simultaneously solving N 1 þ N 2 N t and N 2 σ e N 1 σ a ¼ g, the population of the upper laser level is found: σaN t þ g 1 N N2 ¼ ¼ 1 Nt þ : (8.20) σe þ σa β β Though this relation is obtained by using N 1 þ N 2 N t , which is not valid for a four-level system, the relation is still valid for a four-level system because it reduces to N 2 ¼ g=σ e in the case of a four-level system, for which σ a ¼ 0. Therefore, this relation is valid for all systems. The relation given in (8.20) is valid for any valid value of g, which can be positive, zero, or negative. In the case of a four-level system, it is always true that g 0. In the case of a quasi-twolevel or three-level system, g ¼ α < 0 when the medium is not sufﬁciently pumped to reach transparency. Because the maximum value of the absorption coefﬁcient for a two-level or threelevel system is α0 ¼ σ a N t while α0 g α0 σ e =σ a , we ﬁnd from (8.20) that N 2 0 for any values of g, including g < 0 when the system has a positive absorption coefﬁcient of α ¼ g > 0 for optical attenuation, g ¼ 0 when the system neither attenuates nor ampliﬁes the optical signal, and g > 0 when the system has a positive gain coefﬁcient for optical ampliﬁcation. Because g ¼ 0 and N ¼ 0 at transparency, the transparency population density for the upper laser level is obtained from (8.20) as σa 1 tr N2 ¼ N t ¼ 1 Nt: (8.21) σe þ σa β Population inversion with N > 0 for a positive optical gain of g > 0 is reached when N 2 > N tr2 so that the system is above transparency. Clearly, the bottleneck factor gives a measure of the ease or difﬁculty in reaching the transparency point. For a four-level system, such as the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 258 Optical Ampliﬁcation Nd:YAG laser, β ¼ 1 because σ a ¼ 0; thus N tr2 ¼ 0. In this situation, any population density N 2 pumped to the upper laser level contributes to an optical gain even when most of the active atoms remain in the ground level, which is not the lower laser level of the system. For a twolevel or three-level system, β > 1; thus N tr2 > 0. In this situation, a population density of N 2 > N tr2 > 0 in the upper laser level is required for the system to have an optical gain, and it increases with the value of β. In many three-level systems, such as the ruby laser, the value of β is close to 2; in this situation, about half of all active atoms have to be pumped to the upper laser level before the system can have any optical gain. In some quasi-two-level systems, however, the value of β is close to 1 though larger than 1; then it is relatively easy, though not as easy as for a four-level system, for the system to reach population inversion for a positive optical gain. EXAMPLE 8.3 Consider the Nd:YAG and ruby crystals that have the parameters given in Example 8.1. Find the population density of the upper laser level required for the Nd:YAG crystal to reach transparency at its λ ¼ 1:064 μm laser line and that required for the ruby crystal to reach transparency at its λ ¼ 694:3 nm laser line. What percent of all active ions are excited in each case? Solution: For the Nd:YAG crystal, we have β ¼ 1 and N t ¼ 1:38 1026 m3 from Example 8.1. The population density of the upper laser level required for the Nd:YAG crystal to reach transparency at its λ ¼ 1:064 μm laser line is found using (8.21) to be 1 1 tr N2 ¼ 1 Nt ¼ 1 1:38 1026 m3 ¼ 0: β 1 The percentage of all active ions that are excited to the upper laser level is 0%. For the ruby crystal, we have β ¼ 1:93 and N t ¼ 1:58 1025 m3 from Example 8.1. The population density of the upper laser level required for the ruby crystal to reach transparency at its λ ¼ 694:3 nm laser line is found using (8.21) to be 1 1 tr N2 ¼ 1 Nt ¼ 1 1:58 1025 m3 ¼ 7:61 1024 m3 : β 1:93 The percentage of all active ions that are excited to the upper laser level is N tr2 7:61 1024 ¼ ¼ 48%: N t 1:58 1025 We ﬁnd that no active ions have to be excited for the Nd:YAG crystal to reach the transparency point because it is a four-level system that has a bottleneck factor of β ¼ 1. By comparison, as many as 48% of all active ions have to be excited to the upper laser level for the ruby crystal to reach transparency because it is a three-level system that has a large bottleneck factor of β ¼ 1:93. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.3 Optical Gain 8.3 259 OPTICAL GAIN .............................................................................................................. When the condition in (8.11) is satisﬁed for a system, an optical gain coefﬁcient at a speciﬁc optical frequency v can be found as g ðvÞ ¼ N 2 σ e ðvÞ N 1 σ a ðvÞ. The optical gain coefﬁcient is a function of the optical signal intensity, I, as a result of the dependence of N 2 and N 1 on I due to stimulated emission, which changes the population densities by causing downward transitions from level j2i to level j1i. This effect causes saturation of the optical gain coefﬁcient by the optical signal. For all three basic systems discussed above, the optical gain coefﬁcient can be expressed as a function of the optical signal intensity I: g¼ g0 , 1 þ I=I sat (8.22) where g 0 is the unsaturated gain coefﬁcient, which is independent of the optical signal intensity, and I sat is the saturation intensity of a medium, which can be generally expressed as I sat ¼ hv : τsσe (8.23) The time constant τ s is an effective saturation lifetime of the population inversion. It can be considered as an effective decay time constant for the optical gain coefﬁcient through the relaxation of the effective population inversion. Both g 0 and τ s are functions of the intrinsic properties of a gain medium, as well as of the pumping rate. They can be found from (8.12), (8.16), and (8.18) for the quasi-two-level, three-level, and four-level systems, respectively. The results are summarized below. Quasi-two-level system: g0 ¼ W pτsσe σa N t, τs ¼ τ2 Three-level system: 1 þ σ a =σ e : 1 þ ð1 þ pÞW p τ 2 g0 ¼ W pτsσe σa N t, τs ¼ τ2 1 þ σ a =σ e : 1 þ W pτ2 (8.24) (8.25) (8.26) (8.27) Four-level system: g0 ¼ W pτsσeN t, (8.28) τ2 : 1 þ W pτ2 (8.29) τs ¼ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 260 Optical Ampliﬁcation The minimum pumping requirement for a medium to have an optical gain is clearly g 0 > 0. This is the condition for reaching transparency discussed in Section 8.2. For a desired unsaturated gain coefﬁcient of g 0 , the required pumping rate can be found by solving (8.24) and (8.25) for a quasi-two-level system, (8.26) and (8.27) for a three-level system, and (8.28) and (8.29) for a four-level system. The results are summarized below. Quasi-two-level system: Wp ¼ 1 σaN t þ g0 : τ 2 ðσ e pσ a ÞN t ð1 þ pÞg 0 (8.30) Wp ¼ 1 σaN t þ g0 : τ2 σeN t g0 (8.31) Wp ¼ 1 g0 : τ2 σeN t g0 (8.32) Three-level system: Four-level system: The different forms of unsaturated gain coefﬁcient g 0 and saturation lifetime τ s found above for different systems can be expressed in a general form for all systems by using the parameter p and the bottleneck factor β to account for the differences among the systems. Meanwhile, the required pumping rate for an unsaturated gain coefﬁcient of g 0 can be found expressed in a general form for all systems. They are given below. General forms for all systems: g0 ¼ W pτs þ 1 β σeN t, (8.33) β , 1 þ ð1 þ pÞW p τ 2 (8.34) 1 ðβ 1Þσ e N t þ g 0 : τ 2 ½1 pðβ 1Þ σ e N t ð1 þ pÞg 0 (8.35) τs ¼ τ2 Wp ¼ For a quasi-two-level system, p 0 and β 1. When using a speciﬁc quasi-two-level system, it is desirable to make p as small as possible by properly choosing the pumping parameters and it is desirable to make β as close to unity as possible by properly choosing the laser emission wavelength. For a three-level system, p ¼ 0 and β > 1; the value of β is usually close to 2 for the typical three-level system, but it can be less than 2 or sometimes greater than 2. The large bottleneck factor makes a three-level system inefﬁcient, as discussed in Section 8.2. For a four-level system, p ¼ 0 and β ¼ 1, making the system most efﬁcient in pumping for an optical gain. In the limit when p ! 0, a quasi-two-level system is identical to a three-level system. In the limit when p ! 0 and σ a ! 0 ðβ ! 1Þ, a quasi-two-level system behaves like a four-level system. In the limit when σ a ! 0 ðβ ! 1Þ, a three-level system behaves like a four-level system. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.3 Optical Gain 261 EXAMPLE 8.4 The Nd:YAG laser crystal described in Example 8.1 has τ 2 ¼ 240 μs for its λ ¼ 1:064 μm laser line. The ruby laser crystal described in Example 8.1 has τ 2 ¼ 3 ms for its λ ¼ 694:3 nm laser line. (a) Find the pumping rates for the λ ¼ 1:064 μm Nd:YAG laser line to reach transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 , respectively. What are the saturation lifetime and the saturation intensity in each case? (b) Answer the same questions for the λ ¼ 694:3 nm ruby laser line. Solution: The two laser media belong to different systems and have different parameters. (a) The Nd:YAG at λ ¼ 1:064 μm is a four-level system with σ e ¼ 4:5 1023 m2 and σ a ¼ 0. The doping density is N t ¼ 1:38 1026 m3 . The photon energy is hv ¼ 1:2398 eV ¼ 1:165 eV: 1:064 Using (8.32), (8.29), and (8.23) for a four-level system, we ﬁnd the pumping rate, the saturation lifetime, and the saturation intensity for g 0 ¼ 0 at transparency to be W trp ¼ 0, τ trs ¼ τ 2 ¼ 240 μs, I trsat ¼ hv 1:165 1:6 1019 ¼ W m2 ¼ 17:3 MW m2 : 6 23 tr τ s σ e 240 10 4:5 10 The parameters for an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 are Wp ¼ 1 g0 1 10 s1 ¼ 6:72 s1 , ¼ τ 2 σ e N t g 0 240 106 4:5 1023 1:38 1026 10 τs ¼ I sat ¼ τ2 240 106 ¼ μs ¼ 239:6 μs, 1 þ W p τ 2 1 þ 6:72 240 106 hv 1:165 1:6 1019 ¼ W m2 ¼ 17:3 MW m2 : τ s σ e 239:6 106 4:5 1023 (b) The ruby at λ ¼ 694:3 nm is a three-level system with σ e ¼ 1:34 1024 m2 and σ a ¼ 1:25 1024 m2 . The doping density is N t ¼ 1:58 1025 m3 . The photon energy is hv ¼ 1239:8 eV ¼ 1:786 eV: 694:3 Using (8.31), (8.27), and (8.23) for a three-level system, we ﬁnd the pumping rate, the saturation lifetime, and the saturation intensity for g 0 ¼ 0 at transparency to be W trp ¼ 1 σa 1 1:25 1024 1 ¼ s ¼ 311 s1 , τ 2 σ e 3 103 1:34 1024 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 262 Optical Ampliﬁcation τ trs ¼ τ 2 I trsat ¼ 1 þ σ a =σ e ¼ τ 2 ¼ 3 ms, 1 þ W trp τ 2 hv 1:786 1:6 1019 ¼ W m2 ¼ 71:1 MW m2 : τ trs σ e 3 103 1:34 1024 The parameters for an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 are Wp ¼ 1 σaN t þ g0 1 1:25 1024 1:58 1025 þ 10 1 ¼ s ¼ 888 s1 , τ 2 σ e N t g 0 3 103 1:34 1024 1:58 1025 10 τs ¼ τ2 I sat ¼ 1 þ σ a =σ e 1 þ 1:25=1:34 ¼ 3 103 s ¼ 1:58 ms, 1 þ W pτ2 1 þ 888 3 103 hv 1:786 1:6 1019 ¼ W m2 ¼ 139:4 MW m2 : τ s σ e 1:58 103 1:34 1024 EXAMPLE 8.5 The Nd:YAG crystal considered in Example 8.4 can be optically pumped with an absorption cross section of σ pa ¼ 3:0 1024 m2 at the λp ¼ 808 nm pump wavelength, whereas the ruby crystal considered in Example 8.4 can be optically pumped with an absorption cross section of σ pa ¼ 2:0 1023 m2 at the λp ¼ 554 nm pump wavelength. Assume a 100% pump quantum efﬁciency for the following questions. (a) Find the required pump intensities at λp ¼ 808 nm to pump the λ ¼ 1:064 μm Nd:YAG laser line to transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 , respectively. (b) Find the required pump intensities at λp ¼ 554 nm to pump the λ ¼ 694:3 nm ruby laser line to transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 , respectively. Solution: The pumping transition probability rate W p determines the number per second of active atoms excited by the pump to the upper laser level. If the pump has a pump quantum efﬁciency of ηp when N p pump photons are absorbed, only ηp N p atoms are excited. Thus, the required pump intensity for a pumping transition probability rate of W p is Ip ¼ 1 hvp W p: ηp σ pa With ηp ¼ 1 assumed in this example, we have Ip ¼ hvp W p : σ pa (a) For the Nd:YAG crystal, λp ¼ 808 nm and σ pa ¼ 3:0 1024 m2 . The pump photon energy is hvp ¼ 1239:8 eV: 808 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.3 Optical Gain 263 From Example 8.4, the transparency pumping rate is W trp ¼ 0 and the pumping rate for g 0 ¼ 10 m1 is W p ¼ 6:72 s1 . Therefore, the required pump intensity for transparency is I trp ¼ hvp W trp σ pa ¼ 0, and that for g 0 ¼ 10 m1 is Ip ¼ hvp W p 1239:8 6:72 ¼ W m2 ¼ 550 kW m2 : 1:6 1019 p 24 808 σa 3:0 10 (b) For the ruby crystal, λp ¼ 554 nm and σ pa ¼ 2:0 1023 m2 . The pump photon energy is hvp ¼ 1239:8 eV: 554 From Example 8.4, the transparency pumping rate is W trp ¼ 311 s1 and the pumping rate for g 0 ¼ 10 m1 is W p ¼ 888 s1 . Therefore, the required pump intensity for transparency is I trp ¼ hvp W trp σ pa ¼ 1239:8 311 W m2 ¼ 5:57 MW m2 , 1:6 1019 554 2:0 1023 and that for g 0 ¼ 10 m1 is Ip ¼ hvp W p 1239:8 888 ¼ W m2 ¼ 15:9 MW m2 : 1:6 1019 p 554 σa 2:0 1023 8.3.1 Unsaturated Gain The unsaturated gain coefﬁcient g 0 is also known as the small-signal gain coefﬁcient because it is the gain coefﬁcient of a weak optical signal that does not saturate the gain medium. At transparency, g 0 ¼ 0 because g ¼ 0. For a four-level system, g 0 > 0 as long as the medium is pumped because there is no minimum pumping requirement for transparency. For a quasi-twolevel or three-level system, g 0 > 0 only when the pumping level exceeds its minimum pumping requirement for transparency; below that, the medium has absorption because g 0 < 0. It can be seen from (8.24)–(8.29) that for any system, g 0 increases with pump power less than linearly because τ s decreases with the pump power though W p is linearly proportional to the pump power. This dependence of τ s on the pump power is caused by the fact that as the pump excites atoms from the ground state to any excited state to eventually reach the upper laser level, it depletes the population in the ground level. Consequently, as the pump power increases, fewer atoms remain available for excitation in the ground level, thus reducing the differential increase of the effective population inversion with respect to the increase of the pump power. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 264 Optical Ampliﬁcation 8.3.2 Gain Saturation The optical gain coefﬁcient is a function of the intensity of the optical wave that propagates in the gain medium; it decreases as the optical signal intensity increases. According to (8.22), the optical gain coefﬁcient g is reduced to half of the unsaturated gain coefﬁcient g 0 when the optical signal intensity reaches the saturation intensity such that I ¼ I sat . The smaller the value of I sat , the easier it is for the gain to be saturated. For a quasi-two-level system, τ s ¼ τ 2 ð1 pσ a =σ e Þ at transparency. For a three-level or four-level system, τ s ¼ τ 2 at transparency. For all three systems, τ s < τ 2 when the gain medium is pumped above transparency for a positive gain coefﬁcient. Therefore, I sat increases as the gain medium is pumped harder for a larger unsaturated gain coefﬁcient. EXAMPLE 8.6 The Nd:YAG laser crystal considered in Example 8.4 has a saturation intensity of I sat ¼ 17:3 MW m2 when it is pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 at λ ¼ 1:064 μm. The ruby laser crystal also considered in Example 8.4 has a saturation intensity of I sat ¼ 139:4 MW m2 when it is pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 at λ ¼ 694:3 nm. Two Gaussian laser beams of the same power of P ¼ 1:5 W at these two wavelengths are both collimated to have the same spot size of w0 ¼ 300 μm in each crystal. Find the saturated gain coefﬁcient for each crystal when the beam at the respective wavelength is sent through each crystal. Solution: Each Gaussian beam has a cross-sectional area of 2 πw20 π 300 106 A¼ ¼ m2 ¼ 1:4 107 m2 : 2 2 The peak intensity of each beam is I¼ P 1:5 W m2 ¼ 10:7 MW m2 : ¼ A 1:4 107 For the Nd:YAG laser crystal, the saturated gain coefﬁcient is g¼ g0 ¼ 1 þ I=I sat 10 m1 ¼ 6:18 m1 : 10:7 1þ 17:3 For the ruby laser crystal, the saturated gain coefﬁcient is g¼ g0 ¼ 1 þ I=I sat 10 m1 ¼ 9:29 m1 : 10:7 1þ 139:4 The gain coefﬁcient of the Nd:YAG laser line is more saturated than that of the ruby laser line because the saturation intensity of the Nd:YAG laser line is lower than that of the ruby laser line. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.4 Optical Ampliﬁcation 8.4 265 OPTICAL AMPLIFICATION .............................................................................................................. Any medium that has an optical gain can be used to amplify an optical signal. Depending on the physical mechanism that is responsible for the optical gain, there are two different categories of optical ampliﬁers: nonlinear optical ampliﬁers and laser ampliﬁers. The optical gain of a nonlinear optical ampliﬁer originates from a nonlinear optical process in a nonlinear medium, whereas the gain of a laser ampliﬁer results from the population inversion in a gain medium as discussed in the preceding section. Ignoring the effect of noise, the ampliﬁcation of the intensity, I s , of an optical signal propagating in the z direction through a laser ampliﬁer can be described by dI s g 0 ðzÞ I s, ¼ gI s ¼ dz 1 þ I s =I sat (8.36) where g 0 ðzÞ is the unsaturated gain coefﬁcient and I sat is the saturation intensity of the gain medium, both deﬁned in the preceding section. Here we assume transverse uniformity but consider the possibility of longitudinal nonuniformity by taking the unsaturated gain coefﬁcient g 0 ðzÞ to be a function of z. Such a longitudinally nonuniform gain distribution is a common scenario for an ampliﬁer under longitudinal optical pumping because of pump absorption by the gain medium. In the following discussion, we assume for simplicity that the signal beam is collimated throughout the length of the ampliﬁer such that its divergence is negligible. This assumption allows us to express (8.36) in terms of the power, Ps , of the optical signal as dPs g 0 ðzÞ ¼ gPs ¼ Ps , dz 1 þ Ps =Psat (8.37) where Psat is the saturation power obtained by integrating I sat over the cross-sectional area of the signal beam. By integrating (8.37), the following relation is obtained: ðz Ps ðzÞ Ps ðzÞ Ps ð0Þ exp (8.38) ¼ exp g 0 ðzÞdz, Ps ð0Þ Psat 0 where Ps ð0Þ is the power of the signal beam at z ¼ 0. When Ps Psat , the power of the optical signal grows exponentially with distance. The growth slows down as Ps approaches the value of Psat . Eventually, the signal grows only linearly with distance when Ps Psat . The power gain of a signal is deﬁned as Pout s , (8.39) Pin s out where Pin s and Ps are the input and output powers of the signal, respectively. By using the relation in (8.38) while identifying Pout and Pin s s with Ps ðlÞ and Ps ð0Þ, respectively, for an ampliﬁer that has a length of l, an implicit relation is found for the power gain of the signal: Pin s G ¼ G0 exp ð1 GÞ , (8.40) Psat G¼ where G0 is the unsaturated power gain, or the small-signal power gain. For a single pass through the ampliﬁer, G0 is given by Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 266 Optical Ampliﬁcation Figure 8.4 Gain, normalized to the unsaturated gain as G=G0 , of a laser ampliﬁer as a function of the input signal power, normalized to the saturation power as Pin s =Psat , for different values of the unsaturated power gain G0 . ðl G0 ¼ exp g 0 ðzÞdz: (8.41) 0 Note that, according to (8.40), G0 G > 1 because g 0 > 0 for an ampliﬁer. For a small optical out signal such that Pin s < Ps Psat , the power gain is simply the small-signal power gain so that G ¼ G0 . If the signal power approaches or even exceeds the saturation power of the ampliﬁer, the relation in (8.40) clearly indicates that G < G0 because of gain saturation. In this situation, the overall gain G can be found by solving (8.40) when the values of Pin s and Psat , as well as that of G0 , are given. Figure 8.4 shows the ampliﬁer gain as a function of the input signal power for a few different values of the unsaturated power gain G0 . EXAMPLE 8.7 A Nd:YAG laser rod and a ruby laser rod with the properties described in the preceding examples both have a length of l ¼ 10 cm and a cross-sectional diameter of d ¼ 6 mm. The refractive index of Nd:YAG is 1.82, and that of ruby is 1.76. Each is uniformly pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 at its laser wavelength, λ ¼ 1:064 μm for Nd:YAG and λ ¼ 694:3 nm for ruby. The saturation intensities at g 0 ¼ 10 m1 are found in Example 8.4 to be I YAG ¼ 1:73 MW m2 for the Nd:YAG laser line and sat 2 I ruby for the ruby laser line. Two collimated Gaussian signal beams at the sat ¼ 139:4 MW m two laser wavelengths that have the same spot size of w0 ¼ 400 μm in the rod and the same power of Pin s ¼ 5 W are respectively sent through the Nd:YAG and ruby rods for ampliﬁcation. What are the output signal powers from the Nd:YAG and ruby ampliﬁers, respectively? Solution: The primary difference between the Nd:YAG ampliﬁer and the ruby ampliﬁer is their different saturation intensities. Because their signal wavelengths are different, the two Gaussian beams have different Rayleigh ranges when their spot sizes are the same. With w0 ¼ 400 μm, the Rayleigh ranges of the two beams are Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.5 Spontaneous Emission 267 2 πnw20 π 1:82 400 106 zR ¼ m ¼ 86 cm for λ ¼ 1:064 μm, ¼ λ 1:064 106 2 πnw20 π 1:76 400 106 ¼ m ¼ 1:27 m for λ ¼ 694:3 nm: zR ¼ λ 694:3 109 Both Rayleigh ranges are much larger than the l ¼ 10 cm length of each rod, and the spot size of each beam is much smaller than the cross-sectional diameter of each rod. Therefore, each Gaussian beam can be considered to be collimated throughout each rod with an approximate beam cross-sectional area of 2 πw20 π 400 106 A¼ ¼ m2 ¼ 2:51 107 m2 : 2 2 Then, the saturation powers are 6 7 ¼ I YAG W ¼ 4:34 W for the Nd:YAG amplifier, PYAG sat sat A ¼ 17:3 10 2:51 10 ruby 6 7 W ¼ 35 W for the ruby amplifier: Pruby sat ¼ I sat A ¼ 139:4 10 2:51 10 With l ¼ 10 cm and a uniform unsaturated gain coefﬁcient of g 0 ¼ 10 m1 for both rods, both ampliﬁers have the same unsaturated power gain of G0 ¼ exp ðg 0 lÞ ¼ e1:0 : Using (8.40), the power gain for an input signal power of Pin s ¼ 5 W can be found for each ampliﬁer: Pin 5 1:0 s GYAG ¼ G0 exp ð1 GYAG Þ YAG ¼ e exp ð1 GYAG Þ ) GYAG ¼ 1:51, 4:34 Psat " # Pin 5 1:0 s Gruby ¼ G0 exp 1 Gruby ruby ¼ e exp 1 Gruby ) Gruby ¼ 2:27: 35 Psat Thus, the output signal powers are in Pout s, YAG ¼ GYAG Ps ¼ 1:51 5 W ¼ 7:55 W for the Nd:YAG amplifier, in Pout s, ruby ¼ Gruby Ps ¼ 2:27 5 W ¼ 11:35 W for the ruby amplifier: 8.5 SPONTANEOUS EMISSION .............................................................................................................. Spontaneous emission occurs whenever the upper laser level of a system is populated, irrespective of the lower-level population. The population of the upper laser level for any system is given in (8.20): Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 268 Optical Ampliﬁcation N2 ¼ σaN t þ g , σe þ σa (8.42) where g > 0 when the system is above transparency with an optical gain, g ¼ 0 when the system is at transparency, and g < 0 when the system is below transparency with an optical attenuation coefﬁcient of α ¼ g. According to the discussion in Section 7.1, the spontaneous emission power is proportional to N 2 but is independent of N 1 . Therefore, regardless of whether the medium has a gain or a loss, the spontaneous emission power density, which is deﬁned as the spontaneous emission power per unit volume of the medium in watts per cubic meter, is ^ sp ¼ hv N 2 ¼ hv σ a N t þ g , P τ sp τ sp σ e þ σ a (8.43) where g can be positive for a medium pumped above transparency, zero for a system at transparency, or negative for a medium below transparency. For a gain volume of V, the spontaneous emission power is ^ sp V: Psp ¼ P (8.44) The spontaneous emission power density at transparency, which is known as the critical ﬂuorescence power density, is ^ trsp ¼ hv N 2 ¼ hv σ a N t : P τ sp τ sp σ e þ σa (8.45) The critical ﬂuorescence power for a gain volume of V is ^ trsp V: Ptrsp ¼ P (8.46) ^ trsp ¼ 0 and Ptrsp ¼ 0 because σ a ¼ 0 so that it is transparent For an ideal four-level system, P ^ trsp 6¼ 0 and Ptrsp 6¼ 0 without pumping. For a quasi-two-level system or a three-level system, P ^ trsp and Ptrsp are because σ a 6¼ 0. A practical quasi-two-level system usually has σ a σ e so that P ^ sp and Psp when the medium is pumped for a positive gain of respectively much smaller than P ^ sp and Psp ^ trsp and Ptrsp are often respectively comparable to P g > 0. For a three-level system, P when the medium is pumped for a positive gain of g > 0 because σ a and σ e are of the same order of magnitude. When an optical medium is pumped below transparency, it can still emit light through spontaneous emission as long as N 2 > 0 though N 2 < N tr2 in this situation. Even when an optical medium is pumped above transparency, spontaneous emission still occurs, and the power of spontaneous emission can still dominate that of stimulated emission before laser action takes place. Such spontaneous emission power is the basis of incoherent luminescent light sources. For example, light-emitting diodes are solid-state light sources that emit spontaneous emission generated by electroluminescence through radiative relaxation of electron–hole pairs that are injected by an electric current. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 8.5 Spontaneous Emission 269 In a laser ampliﬁer that ampliﬁes an optical signal through stimulated emission, the spontaneous emission is also ampliﬁed, resulting in ampliﬁed spontaneous emission. Ampliﬁed spontaneous emission is the major source of optical noise for a laser ampliﬁer. It is also the major source of optical noise for a laser oscillator. EXAMPLE 8.8 Consider the Nd:YAG and ruby crystals that have the characteristics described in the preceding examples. As found in Example 8.3, the population density of the upper laser level required for the λ ¼ 1:064 μm Nd:YAG laser line to reach transparency is N tr2 ¼ 0, whereas that required for the λ ¼ 694:3 nm ruby laser line to reach transparency is N tr2 ¼ 7:61 1024 m3 . The spontaneous lifetimes are τ sp ¼ 515 μs for the Nd:YAG laser line and τ sp ¼ 3 ms for the ruby laser line. A Nd:YAG laser rod and a ruby laser rod both have a length of l ¼ 10 cm and a cross-sectional diameter of d ¼ 6 mm. Find the critical ﬂuorescence power density and the critical ﬂuorescence power for each rod. Solution: The volume of each rod is 2 2 d 6 103 V¼π l¼π 10 102 m3 ¼ 2:83 106 m3 : 2 2 For the Nd:YAG rod, because N tr2 ¼ 0, both the critical ﬂuorescence power density and the critical ﬂuorescence power are zero: ^ trsp ¼ 0 and Ptrsp ¼ 0: P For the ruby rod, N tr2 ¼ 7:61 1024 m3 , τ sp ¼ 3 ms, and the photon energy is hv ¼ 1239:8 eV ¼ 1:786 eV: 694:3 Therefore, the critical ﬂuorescence power density and the critical ﬂuorescence power for the ruby rod are, respectively, hv tr 1:786 1:6 1019 tr ^ ¼ N ¼ 7:61 1024 W m3 ¼ 725 MW m3 P sp 3 τ sp 2 3 10 and ^ trsp V ¼ 725 106 2:83 106 W ¼ 2:05 kW: Ptrsp ¼ P Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 270 Optical Ampliﬁcation Problems 8.1.1 Show that the rate equation given in (8.6) for the effective population inversion is valid for all systems if the differences among the systems are accounted for by using the bottleneck factor deﬁned in (8.7). Show also that the effective pumping rate is R ¼ βR2 ðβ 1Þ Nt : τ2 (8.47) Hint: Use (8.20) directly for the relation between the population density of the upper laser level and the gain coefﬁcient deﬁned in (8.5). 8.1.2 A Ti:sapphire crystal is doped with 0.024 wt.% of Ti2 O3 for a Ti3þ ion concentration of N t ¼ 7:9 1024 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a ¼ 8 1026 m2 . Find its bottleneck factor at this laser wavelength. 8.1.3 An Er:ﬁber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . It has an absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼ 7:9 1025 m2 at the λ ¼ 1:53 μm wavelength. Find its bottleneck factor at this laser wavelength. What is the effective population inversion for a gain coefﬁcient of g ¼ 0:3 m1 at λ ¼ 1:53 μm? 8.2.1 Verify the relation given in (8.20) for the population density of the upper laser level for a gain coefﬁcient of g at an effective population inversion of N. 8.2.2 A Nd:YAG crystal is doped with 1 at.% of Nd3þ ions for a concentration of N t ¼ 1:38 1026 m3 . For its λ ¼ 1:064 μm laser line, the emission cross section is found to be σ e ¼ 4:5 1023 m2 and the absorption cross section is σ a ¼ 0 because the lower laser level of this laser line is effectively empty all the time. A ruby crystal is doped with 0.05 wt.% of Cr3þ ions for a concentration of N t ¼ 1:58 1025 m3 . For its λ ¼ 694:3 nm laser line, the emission cross section is found to be σ e ¼ 1:34 1024 m2 and the absorption cross section is σ a ¼ 1:25 1024 m2 . Find the effective population inversion and the population density of the upper laser level required for the λ ¼ 1:064 μm Nd:YAG laser line to have a gain coefﬁcient of g ¼ 6 m1 . Find those values required for the λ ¼ 694:3 nm ruby laser line to have a gain coefﬁcient of g ¼ 6 m1 . What percent of all active ions are excited in each case? Explain the difference between the two media. 8.2.3 A Ti:sapphire crystal is doped with 0.03 wt.% of Ti2 O3 for a Ti3þ ion concentration of N t ¼ 1:0 1025 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a ¼ 8 1026 m2 . (a) Find the population density of the upper laser level required for this Ti:sapphire crystal to reach transparency at λ ¼ 800 nm. What percent of all active ions are excited? (b) What is the effective population inversion for a gain coefﬁcient of g ¼ 15 m1 at λ ¼ 800 nm? What is the population density of the upper laser level for this effective population inversion? What percent of all active ions are excited? What percent of the excited ions effectively contribute to the population inversion? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 Problems 271 8.2.4 An Er:ﬁber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . It has an absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼ 7:9 1015 m2 at the λ ¼ 1:53 μm wavelength. (a) Find the population density of the upper laser level required for this Er:ﬁber to reach transparency at λ ¼ 1:53 μm. What percent of all active ions are excited? (b) What is the effective population inversion required for a gain coefﬁcient of g ¼ 0:3 m1 at λ ¼ 1:53 μm? What is the population density of the upper laser level for this effective population inversion? What percent of all active ions are excited? What percent of the excited ions effectively contribute to the population inversion? 8.3.1 With a constant upward pumping transition probability rate of W p into the upper laser level j2i by depleting the population in the lower laser level j1i, and a constant downward pumping transition probability rate of pW p that depletes the population in the upper level, the total pumping rate to the upper laser level is R2 ¼ W p ðN 1 pN 2 Þ. Show by using N 1 þ N 2 N t and (8.20) that the effective pumping rate found in Problem 8.1.1 can be expressed in terms of the total population N t and the effective population inversion N as β1 N t ð1 þ pÞW p N: (8.48) R ¼ ½1 ðβ 1Þp W p τ2 Use this pumping rate and the rate equation given in (8.6) for the effective population inversion to show that in the steady state the gain coefﬁcient can be expressed in the form of (8.22) with the saturation intensity I sat taking the form of (8.23), the unsaturated gain coefﬁcient g 0 having the form of (8.33), and the saturation lifetime τ s having the form of (8.34). 8.3.2 By using (8.33) and (8.34), show that the required pumping probability rate for an unsaturated gain coefﬁcient of g 0 is that given in (8.35). 8.3.3 By using the general expression in (8.34), ﬁnd the saturation lifetime at the transparency point for all systems. 8.3.4 A Ti:sapphire crystal is doped with 0.03 wt.% of Ti2 O3 for a Ti3þ ion concentration of N t ¼ 1:0 1025 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a 8 1026 m2 . It has an upper laser level lifetime of τ 2 ¼ 3:2 μs. It can be optically pumped at the pump wavelength of λp ¼ 532 nm, where the absorption cross section is σ pa ¼ 7:4 1024 m2 and the emission cross section is σ pe 3 1026 m2 . The pump quantum efﬁciency is ηp ¼ 0:9. (a) Find the pumping rates for this Ti:sapphire to reach transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 15 m1 at λ ¼ 800 nm, respectively. What are the saturation lifetime and the saturation intensity in each case? (b) Find the required pump intensities at λp ¼ 532 nm to pump this Ti:sapphire to transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 15 m1 , respectively. (c) When this Ti:sapphire is pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 15 m1 at λ ¼ 800 nm, a collimated Gaussian laser beam at this wavelength that has a power of P ¼ 1 W and a spot size of w0 ¼ 200 μm is sent through this crystal. Find the saturated gain coefﬁcient. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 272 Optical Ampliﬁcation 8.3.5 An Er:ﬁber is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 in its core. This ﬁber is a cylindrical waveguide that has a core radius of a ¼ 4:5 μm. At the λ ¼ 1:53 μm wavelength, the Er:ﬁber has an absorption cross section of σ a ¼ 5:7 1025 m2 , an emission cross section of σ e ¼ 7:9 1025 m2 , and an upper laser level lifetime of τ 2 ¼ 10 ms. It can be optically pumped as a three-level system at the pump wavelength of λp ¼ 980 nm, where the absorption cross section is σ pa ¼ 2:58 1025 m2 . At the signal wavelength of λ ¼ 1:53 μm and the pump wavelength of λp ¼ 980 nm, the guided signal and pump waves respectively have effective mode radii of ρ ¼ 4:1 μm and ρp ¼ 3:3 μm for their intensity proﬁles. The fractions of the signal and pump intensities that overlap with the core doped with active ions are determined by the conﬁnement factors, which are Γ ¼ 0:70 and Γp ¼ 0:72, respectively. The pump quantum efﬁciency is ηp ¼ 0:8. (a) Find the pumping rates for this Er:ﬁber to reach transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 0:3 m1 , respectively, at λ ¼ 1:53 μm. What are the saturation lifetime and the saturation intensity in each case? (b) Find the required pump intensities at λp ¼ 980 nm to pump this Er:ﬁber to transparency and to have an unsaturated gain coefﬁcient of g 0 ¼ 0:3 m1 , respectively. (c) Find the required pump powers for transparency and for g 0 ¼ 0:3 m1 by accounting for the overlap between the guided pump beam and the active core. (d) When this Er:ﬁber is pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 0:3 m1 at λ ¼ 1:53 μm, a guided laser beam at this wavelength that has a power of P ¼ 1 mW is sent through this ﬁber. Find the saturated gain coefﬁcient by accounting for the overlap between the guided signal beam and the active core. 8.4.1 If the spot sizes of both beams in Example 8.6 are increased to w0 ¼ 800 μm, what is the output power from each ampliﬁer? 8.4.2 A Ti:sapphire laser rod of the characteristics described in Problem 8.3.4 has a length of l ¼ 4 cm and a cross-sectional diameter of d ¼ 3 mm. The refractive index of sapphire is 1.76. The laser rod is uniformly pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 15 m1 at the wavelength of λ ¼ 800 nm. The saturation intensity at g 0 ¼ 15 m1 is I sat > 2 GW m2 . A collimated Gaussian signal beam at λ ¼ 800 nm that has a spot size of w0 ¼ 300 μm in the rod and a power of Pin s ¼ 1 W is sent through the Ti:sapphire ampliﬁer. What is the output signal power from this Ti:sapphire ampliﬁer? 8.4.3 An Er:ﬁber ampliﬁer of the characteristics described in Problem 8.3.5 has a length of l ¼ 10 m. It is uniformly pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 0:3 m1 at its laser wavelength of λ ¼ 1:53 μm. After accounting for the overlap between the guided signal beam and the active core, the saturation power at g 0 ¼ 0:3 m1 is Psat ¼ 1:49 mW. If a guided signal beam at λ ¼ 1:53 μm that has a power of Pin s ¼ 10 μW is sent through the Er:ﬁber ampliﬁer, what is the ampliﬁed output signal power? What is the output signal power if the input signal power is increased to Pin s ¼ 1 mW? 8.5.1 A Nd:YAG crystal is doped with a Nd3þ concentration of N t ¼ 1:38 1026 m3 . For its λ ¼ 1:064 μm laser line, the emission cross section is σ e ¼ 4:5 1023 m2 , the absorption cross section is σ a ¼ 0, and the spontaneous lifetime is τ sp ¼ 515 μs. A ruby crystal is doped with a Cr3þ concentration of N t ¼ 1:58 1025 m3 . For its λ ¼ 694:3 nm laser Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 Bibliography 273 line, the emission cross section is σ e ¼ 1:34 1024 m2 , the absorption cross section is σ a ¼ 1:25 1024 m2 , and the spontaneous lifetime is τ sp ¼ 3 ms. The refractive index of Nd:YAG is 1.82, and that of ruby is 1.76. A Nd:YAG laser rod and a ruby laser rod both have a length of l ¼ 10 cm and a cross-sectional diameter of d ¼ 6 mm. Find the spontaneous emission power density and the spontaneous emission power of each rod when each is uniformly pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 . 8.5.2 A Ti:sapphire laser rod has a length of l ¼ 4 cm and a cross-sectional diameter of d ¼ 3 mm. It is doped with a Ti3þ ion concentration of N t ¼ 1:0 1025 m3 . At the λ ¼ 800 nm wavelength, it has an emission cross section of σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a 8 1026 m2 . Its upper laser level for the λ ¼ 800 nm emission has a total lifetime of τ 2 ¼ 3:2 μs and a spontaneous lifetime of τ sp ¼ 3:9 μs. (a) Find the critical ﬂuorescence power density and the critical ﬂuorescence power of the rod. (b) Find the spontaneous emission power density and the spontaneous emission power of the rod when it is uniformly pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 15 m1 at λ ¼ 800 nm. 8.5.3 An Er:ﬁber that has a length of l ¼ 10 m is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 in its core, which has a radius of a ¼ 4:5 μm. It has an absorption cross section of σ a ¼ 5:7 1025 m2 and an emission cross section of σ e ¼ 7:9 1025 m2 at the λ ¼ 1:53 μm wavelength. Its upper laser level for the λ ¼ 1:53 μm emission has the same total lifetime and spontaneous lifetime of τ 2 ¼ τ sp ¼ 10 ms. (a) Find the critical ﬂuorescence power density and the critical ﬂuorescence power of the ﬁber. (b) Find the spontaneous emission power density and the spontaneous emission power of the ﬁber when it is uniformly pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 0:3 m1 at λ ¼ 1:53 μm. Bibliography Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Iizuka, K., Elements of Photonics for Fiber and Integrated Optics, Vol. II. New York: Wiley, 2002. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010. Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991. Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986. Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996. Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010. Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995. Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University Press, 2007. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:18:45 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.009 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 9 - Laser Oscillation pp. 274-296 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge University Press 9 9.1 Laser Oscillation CONDITIONS FOR LASER OSCILLATION .............................................................................................................. The word laser is the acronym of light ampliﬁcation by stimulated emission of radiation. A medium that is pumped to population inversion has an optical gain to amplify an optical ﬁeld through stimulated emission. Besides optical ampliﬁcation, however, positive optical feedback is normally required for laser oscillation. This requirement is fulﬁlled by placing the gain medium in an optical resonator. One major characteristic of laser light is that it is highly collimated and is spatially and temporally coherent. The directionality of laser light is a direct consequence of the fact that laser oscillation takes place only along a longitudinal axis deﬁned by the optical resonator. The spatial and temporal coherence results from the fact that a photon emitted by stimulated emission is coherent with the photon that induces the emission. The gain medium emits spontaneous photons in all directions, but only the radiation that propagates along the longitudinal axis within a small divergence angle deﬁned by the resonator obtains sufﬁcient regenerative ampliﬁcation through stimulated emission to reach the threshold for oscillation. In order for the oscillating laser ﬁeld to be most efﬁciently ampliﬁed in the longitudinal direction, any spontaneous photons emitted in a direction outside of that small angular range must not be allowed to compete for the gain. For this reason, a functional laser oscillator is necessarily an open cavity that provides optical feedback only along the longitudinal axis. Most of the randomly directed spontaneous photons quickly escape from the cavity through the open sides. Only a very small fraction of them that happen to be emitted within the divergence angle of the laser ﬁeld mix with the coherent oscillating laser ﬁeld to become the major incoherent noise source of the laser. A laser is basically a coherent optical oscillator, and the basic function of an oscillator is to generate a coherent signal through resonant oscillation without an input signal. No external optical ﬁeld is injected into the optical cavity for laser oscillation. The intracavity optical ﬁeld has to grow from the ﬁeld that is generated by spontaneous emission from the intracavity gain medium. When steady-state oscillation is reached, the coherent laser ﬁeld at any given location inside the cavity has to be a constant of time in both phase and magnitude. In the model shown in Fig. 9.1, the situation of steady-state laser oscillation requires that Ein ¼ 0 while Ec ðzÞ 6¼ 0 at any intracavity location z does not change with time. By applying this concept to (6.5) while using (6.4), we ﬁnd the condition for steady-state laser oscillation: a ¼ G exp ðiφRT Þ ¼ 1, (9.1) where a is the round-trip complex ampliﬁcation factor for the intracavity ﬁeld, G is the roundtrip gain factor for the intracavity ﬁeld amplitude, and φRT is the round-trip phase shift for the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.1 Conditions for Laser Oscillation 275 Figure 9.1 Fabry–Pérot laser. intracavity ﬁeld, as deﬁned in (6.4). This general condition for laser oscillation applies to lasers of various cavity structures that use different feedback mechanisms, including Fabry–Pérot lasers, ring lasers, and distributed-feedback lasers. To illustrate the implications of this condition, we consider in the following the simple Fabry–Pérot laser shown in Fig. 9.1 that contains an isotropic gain medium with a ﬁlling factor of Γ. The total permittivity of the gain medium, including the contribution of the resonant laser transition, is ϵ res ¼ ϵ þ ϵ 0 χ res , as given in (6.36). Therefore, the total complex propagation constant of the gain medium, including the contribution from the resonant transition, is g 1=2 (9.2) kg ¼ ωμ0 ðϵ þ ϵ 0 χ res Þ1=2 ¼ k þ Δkres i , 2 where χ 0res ω 0 ¼ (9.3) χ , 2 2n 2nc res χ 00 ω ¼ χ 00res : (9.4) g k res 2 n nc Here g is the gain coefﬁcient of the laser medium, which is identiﬁed in (7.50), and Δkres is the corresponding change in the propagation constant caused by the change in the refractive index of the gain medium due to the changes in the population densities of the laser levels. When population inversion is achieved, χ 00res < 0 so that the gain coefﬁcient g has a positive value. By replacing k for a cold medium with k g for a pumped gain medium, we ﬁnd that k given in (6.38) for a cold cavity has to be replaced with k þ ΓΔk res iΓg=2 when an actively pumped laser cavity is considered. We then ﬁnd for an active laser cavity the mode-dependent round-trip gain factor, Δkres k 1=2 1=2 Gmn ¼ R1 R2 exp ½ðΓmn g αmn Þl, (9.5) and the mode-dependent round-trip phase shift, RT φRT mn ¼ 2ðk þ ΓΔk res Þl þ ζ mn þ φ1 þ φ2 : (9.6) Because both Gmn and φRT mn are real parameters, the oscillation condition given in (9.1) can be satisﬁed for a given laser mode to oscillate only if the gain condition Gmn ¼ 1 (9.7) and the phase condition φRT mn ¼ 2qπ, q ¼ 1, 2, . . . are simultaneously fulﬁlled. Note that both Gmn and φRT mn are frequency dependent. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 (9.8) 276 Laser Oscillation 9.1.1 Laser Threshold The condition in (9.7) implies that there exist a threshold gain and a corresponding threshold pumping level for laser oscillation. For the Fabry–Pérot laser shown in Fig. 9.1, which has a length of l and contains a gain medium of a length lg for a ﬁlling factor of Γ ¼ lg =l, the threshold gain coefﬁcient, g th mn , of the TEMmn mode is given by 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ln R1 R2 , l (9.9) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ g th mn lg ¼ αmn l ln R1 R2 : (9.10) Γg th mn ¼ αmn or Because the distributed loss αmn is mode dependent, the threshold gain coefﬁcient g th mn varies from one transverse mode to another. In addition, the effective gain coefﬁcient can be different for different transverse modes because different transverse modes have different ﬁeld distribution patterns and thus overlap with the gain volume differently. The transverse mode that has the lowest loss and the largest effective gain at any given pumping level reaches threshold ﬁrst and starts oscillating at the lowest pumping level. In the typical laser, the transverse mode that reaches threshold ﬁrst is normally the fundamental TEM00 mode. Unless a frequency-selecting mechanism is placed in a laser to create a frequencydependent loss that varies from one longitudinal mode to another, the threshold gain coefﬁcient g th mn varies little among the mnq longitudinal modes of different q values that share the common mn transverse mode pattern. It is possible, however, to introduce a frequencyselecting device to a laser cavity to make αmn and, consequently, g th mn of a given mn transverse mode highly frequency dependent for the purpose of selecting or tuning the oscillating laser frequency. The power required to pump a laser to reach its threshold is called the threshold pump power, Pth p . Because the threshold gain coefﬁcient is mode dependent and frequency dependent, the threshold pump power is also mode dependent and frequency dependent. The threshold pump power of a laser mode can be found by calculating the power required for the gain medium to have an unsaturated gain coefﬁcient equal to the threshold gain coefﬁcient of the mode: g 0 ¼ g th mn ðωmnq Þ, assuming uniform pumping throughout the gain medium. For a quasi-two-level or three-level laser, there is also a transparency pump power, Ptrp , for g 0 ¼ 0, assuming uniform pumping. In the situation of nonuniform pumping, these conditions for reaching threshold and transparency have to be modiﬁed. Clearly, Ptrp < Pth p by deﬁnition. EXAMPLE 9.1 A Nd:YAG laser for the λ ¼ 1:064 μm laser wavelength consists of a Nd:YAG laser rod of a length lg ¼ 3 cm as a gain medium in a Fabry–Pérot cavity, which is formed by two mirrors of reﬂectivities R1 ¼ 90% and R2 ¼ 100% at a physical spacing of l ¼ 10 cm. The surfaces of the laser rod are antireﬂection coated to eliminate losses and undesirable effects. The crosssectional area of the laser rod is larger than that of the TEM00 Gaussian laser mode. This laser Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.2 Mode-Pulling Effect 277 mode has a distributed optical loss of α ¼ 0:1 m1 . Find the threshold gain coefﬁcient of this laser mode. Solution: Using (9.10), we ﬁnd with the given parameters that the threshold gain coefﬁcient of the TEM00 Gaussian laser mode is g th ¼ 9.2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 ðαl ln R1 R2 Þ ¼ ð0:1 0:1 ln 0:9 1Þ m1 ¼ 2:09 m1 : lg 0:03 MODE-PULLING EFFECT .............................................................................................................. Comparing (9.6) for an active Fabry–Pérot laser with (6.40) for its cold cavity, we ﬁnd that, through its dependence on Δkres , the round-trip phase shift of a ﬁeld in a laser cavity is a function of χ 0res . Consequently, the longitudinal mode frequencies ωmnq at which a laser oscillates are not exactly the same as the longitudinal mode frequencies ωcmnq given in (6.41) for the cold Fabry–Pérot cavity. Using (9.6) and (9.8), we ﬁnd that the longitudinal mode frequencies of a Fabry–Pérot laser are related to those of its cold cavity by the relation: ωmnq ¼ ωcmnq χ 0res 1 χ 0res c : 1þ ωmnq 1 2nn 2nn (9.11) Clearly, the laser mode frequencies ωmnq differ from the cold-cavity mode frequencies because they vary with the resonant susceptibility, which depends on the level of population inversion in the gain medium. This dependence of the laser mode frequencies on the population inversion in the gain medium is caused by the fact that the refractive index and the gain of the medium are directly connected to each other, as is dictated by the Kramers–Kronig relation. This effect causes a frequency shift of δωmnq ¼ ωmnq ωcmnq χ 0res c ω 2nn mnq (9.12) for the oscillation frequency of mode mnq. Because of the frequency dependence of χ 0res , the dependence of this frequency shift on χ 0res results in the mode-pulling effect demonstrated in Fig. 9.2. Near the transition resonance frequency, ω21 , of the gain medium, χ 0res is highly dispersive. When a medium is pumped to have population inversion for a transition that has a resonance frequency of ω21 , χ 00res ðωÞ < 0 for either ω < ω21 or ω > ω21 , but χ 0res ðωÞ < 0 for ω < ω21 and χ 0res ðωÞ > 0 for ω > ω21 . As a result, ωmnq > ωcmnq for ωcmnq < ω21 , whereas ωmnq < ωcmnq for ωcmnq > ω21 . Therefore, in comparison to the resonance frequencies of the cold cavity, the mode frequencies of a laser are pulled toward the transition resonance frequency of the gain medium. In Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 278 Laser Oscillation Figure 9.2 Frequency-pulling effect for laser modes. Compared to the resonance frequencies of the cold cavity shown as dotted lines, the mode frequencies of an active laser shown as solid lines are pulled toward the transition resonance frequency of the gain medium in the situation of population inversion. The real and imaginary parts of the gain susceptibility as a function of optical frequency are shown. addition, the longitudinal modes belonging to a common transverse mode are no longer equally spaced in frequency. In a laser of a relatively high gain and a large dispersion, such as a semiconductor laser, this effect can result in a large variation in the frequency spacing between neighboring laser modes. Because of the frequency dependence of the gain coefﬁcient g due to the frequency dependence of χ 00res , different longitudinal modes not only experience different values of refractive index but also see different values of gain coefﬁcient, as also illustrated in Fig. 9.2. A longitudinal mode that has a frequency close to the gain peak at the transition resonance frequency has a higher gain than one that has a frequency far away from the gain peak. EXAMPLE 9.2 A Nd:YAG laser contains a Nd:YAG rod described in Example 8.1 in a cavity described in Example 9.1. The refractive index of the Nd:YAG crystal is n ¼ 1:82. Find the largest frequency shift of the longitudinal mode frequencies of the Nd:YAG laser due to the modepulling effect. How large is this frequency shift compared to the longitudinal mode frequency spacing? Solution: From Example 9.1, we ﬁnd that the gain coefﬁcient is g ¼ g th ¼ 2:09 m1 when the TEM00 laser mode is pumped to its threshold. The overlap factor is Γ ¼ lg =l ¼ 0:3; thus, the weighted average refractive index seen by the laser mode is n ¼ 0:3 1:82 þ ð1 0:3Þ 1 ¼ 1:246: With λ ¼ 1:064 μm at the transition frequency ω21 , we ﬁnd that the maximum value of the imaginary part of the resonant susceptibility associated with this laser transition is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.3 Oscillating Laser Modes χ 00res ðω21 Þ ¼ 279 nc nλ 1:82 1:064 106 g¼ g¼ 2:09 ¼ 6:44 107 , 2π ω21 2π which appears at the line center. Because this laser transition is a discrete atomic transition, the real part χ 0res has the largest absolute value at two frequencies. With χ 00res ðω21 Þ < 0, χ 0res has the largest negative value of χ 0res ðω Þ ¼ χ 00res ðω21 Þ=2 at the frequency ω ¼ ω21 γ and the largest positive value of χ 0res ðωþ Þ ¼ χ 00res ðω21 Þ=2 at ωþ ¼ ω21 þ γ, as seen in Figs. 2.3 and 9.2. Thus, jχ 0res jmax ¼ jχ 00res ðω21 Þ=2j ¼ 3:22 107 : For a Nd:YAG laser at λ ¼ 1:064 μm, γ=ω21 2 104 because the gain linewidth is about Δvg ¼ γ=π 120 GHz, whereas the laser frequency is v21 ¼ ω21 =2π ¼ c=λ 283 THz. Therefore, we can take the approximation that ωc ¼ ω ¼ ω21 γ ω21 for (9.12) to ﬁnd the absolute value of the largest frequency shift caused by mode pulling: jδvjmax ¼ jδωjmax jχ 0res jmax 3:22 107 ν21 ¼ 283 1012 Hz ¼ 20:1 MHz: 2π 2nn 2 1:82 1:246 This is the largest amount of frequency shift, which occurs for a longitudinal mode that has a coldcavity mode frequency at either the positive or negative half-width points vc, ¼ v21 Δvg =2. As shown in Fig. 9.2, the mode that is closest to the lower frequency, vc, ¼ v21 Δvg =2, is pulled up by an amount of approximately jδvjmax , whereas the mode that is closest to the higher frequency, νc, þ ¼ v21 þ Δνg =2, is pulled down by an amount of approximately jδvjmax . The longitudinal mode frequency spacing is ΔνL ¼ c 3 108 Hz ¼ 1:204 GHz: ¼ 2nl 2 1:246 10 102 Thus, the percentage of the maximum mode-pulling frequency shift is jδνjmax 20:1 106 ¼ 1:67%: ΔνL 1:204 109 This frequency shift is appreciable though small. It is small because the dispersive effect of the optical gain is small in the Nd:YAG medium. It can be much larger in a highly dispersive gain medium, such as a semiconductor laser gain medium. 9.3 OSCILLATING LASER MODES .............................................................................................................. Because the gain coefﬁcient is a function of frequency, the net gain coefﬁcient, g g th mn , of a laser mode is always frequency dependent and varies among different transverse modes and among different longitudinal modes no matter whether the threshold gain coefﬁcient g th mn of a transverse mode is frequency dependent or not. At a low pumping level before the laser starts Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 280 Laser Oscillation oscillating, the net gain is negative for all laser modes. As the pumping level increases, the mode that ﬁrst reaches its threshold starts to oscillate. Once a laser starts oscillating in one mode, whether any other longitudinal or transverse modes have the opportunity to oscillate through further increase of the pumping level is a complicated issue of mode interaction and competition that depends on a variety of factors, including the properties of the gain medium, the structure of the laser, the pumping geometry, the nonlinearity in the system, and the operating condition of the laser. Here we only discuss some basic concepts in the situation of steady-state oscillation of a CW laser. Interaction and competition among laser modes are more complicated when a laser is pulsed than when it is in CW operation. Therefore, some of the conclusions obtained below may not be valid for a pulsed laser. The gain condition in (9.7) implies that once a given laser mode is oscillating in the steady state, the gain that is available to this mode does not increase with increased pumping above the threshold pumping level because Gmn has to be kept at unity for the steady-state oscillation of a laser mode. Thus the effective gain coefﬁcient of an oscillating mode is “clamped” at the threshold level of the mode as long as the pumping level is kept at or above threshold. The mechanism for holding down the gain coefﬁcient at the threshold level is the effect of gain saturation discussed in Section 8.3. An increase in the pumping level above threshold only increases the ﬁeld intensity of the oscillating mode in the cavity, but the gain coefﬁcient is saturated at the threshold value by the high intensity of the intracavity laser ﬁeld. The fact that the gain of a laser mode oscillating in the steady state is saturated at the threshold value has a signiﬁcant effect on the mode characteristics of a CW laser. 9.3.1 Homogeneously Broadened Lasers When the gain medium of a laser is homogeneously broadened, all modes that occupy the same spatial gain region compete for the gain from the population inversion in the same group of active atoms. As the mode that ﬁrst reaches threshold starts oscillating, the entire gain curve supported by this group of atoms saturates. Because this oscillating mode is normally the one that has a longitudinal mode frequency closest to the gain peak and a transverse mode pattern of the lowest loss, the gain curve is saturated in such a manner that its value at this longitudinal mode frequency is clamped at the threshold value of the transverse mode that has the lowest threshold gain coefﬁcient among all transverse modes. If the gain peak does not happen to coincide with this mode frequency, it still lies above the threshold when the gain curve is saturated, as shown in Fig. 9.3. Nevertheless, all other longitudinal modes belonging to this transverse mode have frequencies away from the gain peak. Therefore, even with increased pumping, they do not have sufﬁcient gain to reach threshold because the entire gain curve shared by these modes is saturated, as illustrated in Fig. 9.3. Other transverse modes that are supported solely by this group of saturated, homogeneously broadened atoms do not have the opportunity to oscillate either, because the gain curve is saturated below their respective threshold levels. Nevertheless, because different transverse modes have different spatial ﬁeld distributions, a high-order transverse mode may draw its gain from a gain region outside of the region that is saturated by a low-order transverse mode. Therefore, when the pumping level is increased, a high-order transverse mode may still reach its relatively high threshold for oscillation if a low-order transverse mode of a low threshold is already oscillating. Consequently, for a homogeneously broadened CW laser in steady-state oscillation, only one among all of the longitudinal modes belonging to a particular transverse mode will oscillate, but Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.3 Oscillating Laser Modes 281 Figure 9.3 Gain saturation in a homogeneously broadened laser. Only one longitudinal mode whose frequency is closest to the gain peak oscillates. The entire gain curve is saturated such that the gain at the single oscillating frequency remains at the loss level. it is possible for more than one transverse mode to oscillate simultaneously at a high pumping level. Note that this conclusion does not hold true for a pulsed laser. It is possible for multiple longitudinal modes belonging to the same transverse mode to oscillate simultaneously in a pulsed laser even when its gain medium is homogeneously broadened. EXAMPLE 9.3 The Nd:YAG laser described in Examples 9.1 and 9.2 has a Lorentzian gain lineshape that has a bandwidth of Δλg ¼ 0:45 nm for the laser line at λ ¼ 0:064 μm. It is pumped at a level such that the peak unsaturated gain coefﬁcient is twice the threshold gain coefﬁcient: g max ¼ 2g th . How 0 many longitudinal modes have their unsaturated gain coefﬁcients pumped above the threshold? How many longitudinal modes oscillate? Solution: The gain bandwidth in terms of frequency is Δν Δλ g g ¼ : ν λ With Δλg ¼ 0:45 nm and λ ¼ 1:064 μm, ν c 3 108 0:45 109 Hz ¼ 119:25 GHz: Δνg ¼ Δλg ¼ 2 Δλg ¼ λ λ ð1:064 106 Þ2 ¼ 2g th , the two frequencies at the two ends of the When the laser is pumped such that g max 0 FWHM of the gain bandwidth have an unsaturated gain coefﬁcient of g 0 ¼ g th . Therefore, every mode that has a frequency within the FWHM, Δνg ¼ 119:25 GHz, of the gain bandwidth has an unsaturated gain coefﬁcient above the threshold value. From Example 9.2, the longitudinal mode frequency spacing is ΔνL ¼ c 3 108 Hz ¼ 1:204 GHz: ¼ 2nl 2 1:246 10 102 Then, Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 282 Laser Oscillation Δνg 119:25 ¼ ¼ 99:04: ΔνL 1:204 Therefore, depending on where the longitudinal mode frequencies are located with respect to the gain peak, 99 or 100 longitudinal modes have unsaturated gain coefﬁcients that are above the threshold value. Because the gain spectrum has a Lorentzian lineshape, the laser is homogeneously broadened. Therefore, ideally only one longitudinal mode oscillates. Though 99 or 100 longitudinal modes are each pumped to have an unsaturated gain coefﬁcient above the threshold value, all of them except the oscillating mode are saturated below the threshold by the oscillating mode, which reaches the threshold ﬁrst. In practice, however, we often ﬁnd that a Nd:YAG laser oscillates steadily in more than one mode because it is not completely homogeneously broadened though it is predominantly so. The degree of inhomogeneous broadening determines the number of oscillating modes. 9.3.2 Inhomogeneously Broadened Lasers In a laser that has an inhomogeneously broadened gain medium, there are different groups of active atoms in the same spatial gain region. Each group saturates independently. Two modes occupying the same spatial gain region do not compete for the same group of atoms if the separation of their frequencies is larger than the homogeneous linewidth of each group of atoms. When one longitudinal mode reaches threshold and oscillates, the gain coefﬁcient is saturated only within the spectral range of a homogeneous linewidth around its frequency, while the gain coefﬁcient at frequencies outside this small range continues to increase with increased pumping. As the pumping level increases, other longitudinal modes can successively reach threshold and oscillate. As a result, at a sufﬁciently high pumping level, multiple longitudinal modes belonging to the same transverse mode can oscillate simultaneously. The saturation of the gain coefﬁcient in a small spectral range within a homogeneous linewidth around each of the frequencies of these oscillating modes, but not across the entire gain curve, creates the effect of spectral hole burning in the gain curve of an inhomogeneously broadened laser medium, as illustrated in Fig. 9.4. Different transverse modes Figure 9.4 Spectral hole burning effect in the gain saturation of an inhomogeneously broadened laser. Multiple longitudinal modes oscillate simultaneously at a sufﬁciently high pumping level. The gain at each oscillating frequency is saturated at the loss level. The mode-pulling effect is ignored in this illustration. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.3 Oscillating Laser Modes 283 also saturate independently in an inhomogeneously broadened medium if their frequencies are sufﬁciently separated. Therefore, an inhomogeneously broadened laser can also oscillate in multiple transverse modes. EXAMPLE 9.4 A He–Ne laser has a Doppler-broadened gain bandwidth of Δνg ¼ 1:5 GHz at its laser wavelength of λ ¼ 632:8 nm. The laser has a cavity length of l ¼ 32 cm. It is pumped at a level such that the peak unsaturated gain coefﬁcient is twice the threshold gain coefﬁcient: g max ¼ 2g th . How many longitudinal modes have their unsaturated gain coefﬁcients pumped 0 above the threshold? How many longitudinal modes oscillate? Solution: When the laser is pumped such that g max ¼ 2g th , the two frequencies at the two end of the 0 FWHM Δvg of the gain bandwidth have an unsaturated gain coefﬁcient of g 0 ¼ g th . Therefore, the laser has a bandwidth of Δv ¼ Δvg ¼ 1:5 GHz. Every mode that has a frequency within this bandwidth has an unsaturated gain coefﬁcient above the threshold value. With l ¼ 32 cm and n 1 for the gaseous He–Ne laser gain medium, the longitudinal mode frequency spacing is ΔνL ¼ c 3 108 ¼ Hz ¼ 468:75 MHz: 2nl 2 1 32 102 Then, Δν 1:5 109 ¼ ¼ 3:2: ΔνL 468:75 106 Therefore, three or four longitudinal modes have unsaturated gain coefﬁcients that are above the threshold value, depending on where the longitudinal mode frequencies are located with respect to the gain peak. Because the gain spectrum is Doppler broadened, the laser is inhomogeneously broadened. All longitudinal modes above threshold oscillate. 9.3.3 Laser Linewidth The linewidth of an oscillating laser mode is still described by (6.18): Δνmnq ¼ 1 Gmnq L Δνmn , πGmnq (9.13) where the longitudinal mode frequency spacing ΔνLmn might vary for different transverse modes. From this relation, we see that in practice the round-trip ﬁeld gain factor Gmnq of a laser mode in steady-state oscillation cannot be exactly equal to unity because the laser linewidth cannot be zero, due to the existence of spontaneous emission. In reality, in steady-state oscillation the value of Gmnq is slightly less than unity, with the small difference made up by spontaneous emission. Clearly, the linewidth of an oscillating laser mode is determined by the amount of Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 284 Laser Oscillation spontaneous emission that is channeled into the laser mode. Therefore, (9.13) is not very useful for calculating the linewidth of a laser mode in steady-state oscillation without knowing the exact value of Gmnq in the presence of spontaneous emission. A detailed analysis taking into account spontaneous emission yields the Schawlow–Townes relation for the linewidth of a laser mode in terms of the laser parameters: ΔνST mnq ¼ 2πhvðΔνcmnq Þ2 Pout mnq N sp ¼ hv 2πðτ cmnq Þ2 Pout mnq N sp , (9.14) where Δνcmnq and τ cmnq are respectively the cold-cavity linewidth and the photon lifetime of the oscillating mnq mode, Pout mnq is the output power of the oscillating laser mode, and N sp ¼ σeN 2 σeN 2 N 2 ¼ ¼ σeN 2 σaN 1 g N (9.15) is the spontaneous emission factor that measures the degree of the effective population inversion in the gain medium. The effective population inversion deﬁned as N ¼ g=σ e in (8.5) is the population density that is able to contribute to the coherent stimulate emission, which does not broaden the laser linewidth, whereas all of the upper level population N 2 contributes to the incoherent spontaneous emission, which broadens the laser linewidth. The effect of spontaneous emission on the linewidth of an oscillating laser mode enters the relation in (9.14) through the population densities of the laser levels in the form of the spontaneous emission factor. Because N sp 1, the ultimate lower limit of the laser linewidth, which is known as the Schawlow–Townes limit, is that given in (9.14) for N sp ¼ 1. It can also be seen that the linewidth of a laser mode decreases as the laser power increases. This phenomenon is easily understood. Because the gain of an oscillating laser mode is clamped at its threshold level, increased pumping above threshold does not increase the population inversion, and thus does not increase the spontaneous emission, which is proportional to the population of the upper laser level. When the power of an oscillating laser mode increases with increased pumping, the coherent stimulated emission increases proportionally but the incoherent spontaneous emission is clamped at its threshold level. As a result, the linewidth of the laser mode decreases with increasing laser power. EXAMPLE 9.5 Find the minimum possible linewidth that is set by the Schawlow–Townes limit for the oscillating laser mode of the Nd:YAG laser described in Examples 9.1 and 9.2 when the laser is pumped sufﬁciently above the threshold so that the output power of the mode at λ ¼ 1:064 μm is 100 mW. Solution: The Nd:YAG laser described in Examples 9.1 and 9.2 has a Fabry–Pérot cavity that has a length of l ¼ 10 cm, a weighted average index of n ¼ 1:246, a distributed loss of α ¼ 0:1 m1 , and mirror reﬂectivities of R1 ¼ 90% and R2 ¼ 100%. Therefore, from (6.45), the cold-cavity photon lifetime of the laser mode is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 285 9.4 Laser Power τc ¼ nl 1:246 10 102 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s ¼ 6:63 ns: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ cðαl ln R1 R2 Þ 3 108 ð0:1 10 102 ln 0:9 1Þ Because Nd:YAG is a four-level system which has σ a ¼ 0, it has N sp ¼ 1 as can be seen from (9.15). The photon energy at the λ ¼ 1:064 μm laser wavelength is hv ¼ 1:2398 eV ¼ 1:165 eV: 1:064 For an oscillating laser mode that has an output power of Pout ¼ 100 mW, the minimum possible linewidth set by the Schawlow–Townes limit is found using (9.14): ΔvST ¼ hv 1:165 1:6 1019 N ¼ 1 Hz ¼ 6:7 mHz: sp 2πτ 2c Pout 2π ð6:63 109 Þ2 100 103 This minimum possible oscillating laser mode linewidth is nine orders of magnitude smaller than the cold-cavity longitudinal linewidth of Δvc ¼ ð2πτ c Þ1 27:9 MHz. The signiﬁcant line narrowing is caused by the coherent stimulated emission. However, the Schawlow–Townes linewidth found above is only the fundamental lower bound limited by the spontaneous emission noise, which can be approached if all other noise sources are eliminated in the ideal condition. In practice, the linewidth of an oscillating laser mode is much larger than the Schawlow–Townes linewidth, though generally much smaller than the cold-cavity linewidth, because it is broadened by many mechanisms such as the noise from pump power ﬂuctuations, mechanical vibrations, and temperature ﬂuctuations of the laser. 9.4 LASER POWER .............................................................................................................. In this section, we consider the output power of a laser. Because the situation of a multimode laser can be quite complicated due to mode competition, we consider for simplicity only a CW laser that oscillates in a single longitudinal and transverse mode. The parameters mentioned in this section are not labeled with mode indices because all of them are clearly associated with the only oscillating mode. The simple case of a Fabry–Pérot cavity that contains an isotropic gain medium with a ﬁlling factor of Γ as shown in Fig. 9.1 is considered. To illustrate the general concepts, we consider the situation when the gain medium is uniformly pumped so that the entire gain medium has a spatially independent gain coefﬁcient. For the single oscillating mode of the Fabry–Pérot laser considered here, the round-trip gain factor G is that given by (9.5), and the cavity decay rate γc deﬁned by (6.23) is that given by (6.46). Therefore, G2 ¼ exp ð2Γgl γc TÞ: (9.16) Because G2 is the net ampliﬁcation factor of the intracavity ﬁeld energy, which is proportional to the intracavity photon number, in a round-trip time T of the laser cavity, we can deﬁne an Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 286 Laser Oscillation intracavity energy growth rate, or intracavity photon growth rate, Γg, for the oscillating laser mode through the relation G2 ¼ exp ½ðΓg γc ÞT: (9.17) We ﬁnd, by comparing (9.17) with (9.16), the gain parameter of the gain medium: g¼ 2gl cg ¼ : n T (9.18) By comparing (6.46) with (9.9), we ﬁnd that γc ¼ Γ 2g th l cg ¼ Γ th : T n (9.19) Note that while the unit of g and g th is per meter, the unit of g and γc is per second. The relation in (9.18) translates the gain coefﬁcient that characterizes spatially dependent ampliﬁcation through the gain medium of a propagating intracavity laser ﬁeld into an intracavity energy growth rate that characterizes the temporal growth of the energy in a laser mode. The relation in (9.19) clearly indicates that the threshold intracavity energy growth rate for laser oscillation is the cavity decay rate: Γgth ¼ γc : (9.20) This relation can also be obtained by applying the threshold condition of G ¼ 1 to the relation in (9.17). It is easy to understand because for a laser mode to oscillate, the growth of intracavity photons in that mode through ampliﬁcation by the gain medium has to completely compensate for the decay of photons caused by all the loss mechanisms. Therefore, we shall call the energy growth rate Γg and the cavity decay rate γc , both of which are speciﬁc to a laser mode, the gain parameter and the loss parameter, respectively, of the laser mode. Note that the gain parameter Γg of the laser mode is reduced by the ﬁlling factor Γ from the gain parameter g of the gain medium. By using temporal growth and decay rates instead of spatial gain and loss coefﬁcients to describe the characteristics of a laser, we are in effect moving from a spatially distributed description of the laser to a lumped-device description. In the lumped-device description, a laser mode is considered an integral entity with its spatial characteristics effectively integrated into the parameters Γg and γc . The detailed spatial characteristics of the mode are irrelevant and are lost in this description. Therefore, instead of the intensity of the oscillating laser ﬁeld, we have to consider the intracavity photon density, S, of the oscillating laser mode. For a Fabry–Pérot laser that contains a gain medium of a ﬁlling factor Γ so that the average refractive index inside the cavity is n ¼ Γn þ ð1 ΓÞn0 as deﬁned in (6.3), the average intracavity photon density of the laser mode is S¼ nI , chv (9.21) where I is the spatially averaged intracavity intensity and hv is the photon energy of the oscillating laser mode. Because the gain parameter g is directly proportional to the gain coefﬁcient g of the gain medium, the relation between the unsaturated gain parameter Γg0 and the saturated gain Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.4 Laser Power 287 parameter Γg of a laser mode in the lumped-device description can be obtained by converting the relation between g 0 and g discussed in Section 9.3 through the relation in (9.18). Therefore, for the gain parameter of a laser mode, we have g¼ g0 Γg0 and Γg ¼ 1 þ S=Ssat 1 þ S=Ssat (9.22) cg 0 n (9.23) where g0 ¼ is the unsaturated gain parameter of the gain medium and Ssat ¼ nI sat n ¼ chv cτ s σ e (9.24) is the saturation photon density of the laser mode. When a CW laser oscillates in the steady state, the value of Γg for the oscillating mode is clamped at its threshold value of γc , just as the value of g is clamped at g th . Therefore, by setting Γg to equal γc and using (9.22), we ﬁnd that the intracavity photon density of a CW laser mode in steady-state oscillation is Γg0 1 Ssat ¼ ðr 1ÞSsat , for r 1: (9.25) S¼ γc The dimensionless pumping ratio r represents that a laser is pumped at r times its threshold. It is deﬁned as r¼ Γg0 g 0 ¼ : γc g th (9.26) Assuming that the pumping efﬁciency is the same at transparency, at threshold, and at the operating point, the pumping ratio can be expressed in terms of the pump power as r¼ Pp Ptrp tr Pth p Pp , (9.27) where Ptrp is the pump power for the gain medium to reach transparency, Pth p is that for the laser to reach its threshold, and Pp is the pump power at the operating point. Note that (9.25) is valid only for r 1 when the laser oscillates because only then is the laser gain saturated. For r < 1, the laser does not reach threshold. The laser cavity is then ﬁlled with spontaneous photons at a density that is small in comparison to the high density of coherent photons when the laser oscillates at r 1. From the intracavity photon density of the oscillating laser mode, we can easily ﬁnd the total intracavity energy contained in this mode: U mode ¼ hvV mode S, (9.28) where V mode is the volume of the oscillating mode. The mode volume can be found by integrating the normalized intensity distribution of the mode over the three-dimensional Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 288 Laser Oscillation space deﬁned by the laser cavity; it is usually a fraction of the volume of the cavity. The output power of the laser is simply the coherent optical energy emitted from the laser per second. Therefore, it is simply the product of the mode energy and the output-coupling rate, γout , of the cavity: Pout ¼ γout U mode ¼ γout hvV mode S ¼ ðr 1Þγout hvV mode Ssat : (9.29) The output-coupling rate is also called the output-coupling loss parameter because it contributes to the total loss of a laser cavity; it is a fraction of the total loss parameter γc . One can indeed write γc ¼ γi þ γout , where γi is the internal loss of the laser that does not contribute to the output coupling of the laser power. As an example, for the Fabry–Pérot laser that has γc given by c 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (9.30) α ln R1 R2 γc ¼ n l as expressed in (6.46), we have the internal loss given by γi ¼ cα=n and the output-coupling loss given by γout ¼ c pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c pﬃﬃﬃﬃﬃ c pﬃﬃﬃﬃﬃ ln R1 R2 ¼ ln R1 ln R2 ¼ γout, 1 þ γout, 2 , nl nl nl where γout;1 ¼ c pﬃﬃﬃﬃﬃ ln R1 nl and γout, 2 ¼ c pﬃﬃﬃﬃﬃ ln R2 nl (9.31) (9.32) are the output-coupling losses of mirror 1 and mirror 2, respectively. In this case, γout is the total output-coupling loss through both mirrors. Therefore, Pout given in (9.29) is the total output power emitted through both mirrors. For the output power emitted through each mirror, we ﬁnd that Pout;1 ¼ U mode γout, 1 ¼ γout, 1 γ Pout and Pout, 2 ¼ U mode γout, 2 ¼ out, 2 Pout : γout γout (9.33) It is convenient to deﬁne the saturation output power as Psat out ¼ γout hvV mode Ssat : (9.34) Using the deﬁnition of Ssat in (9.24), it can be shown that pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Psat out ¼ Psat ln R1 R2 , (9.35) where Psat is the saturation power of the gain medium found by integrating I sat over the crosssectional area of the gain medium. Combining (9.29) with (9.34), we can express the output laser power in terms of Psat out as Pout ¼ ðr 1ÞPsat out : (9.36) Note that Psat out is not the level at which the output power of a laser saturates. Its physical meaning can be easily seen from (9.35) and (9.36). From (9.35), we ﬁnd that the output power of a laser is Psat out when the intracavity laser power is at the level Psat of the gain medium. From Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.4 Laser Power 289 sat (9.36), we ﬁnd that Pout ¼ Psat out when r ¼ 2; in other words, a laser has an output power of Pout when it is pumped at twice its threshold level. EXAMPLE 9.6 The Nd:YAG gain medium of the laser described in Examples 9.1 and 9.2 has a saturation intensity of I sat ¼ 17:3 MW m2 , which stays almost constant for an unsaturated gain coefﬁcient g 0 over the range from 0 to 10 m1. With a cavity length of l ¼ 10 cm, the two cavity mirrors are chosen such that at the λ ¼ 1:064 μm laser wavelength, the TEM00 Gaussian mode has a beam waist spot size of w0 ¼ 500 μm located at the center of the Nd:YAG rod, which has a length of lg ¼ 3 cm. (a) Find the pumping ratio r and the corresponding unsaturated gain coefﬁcient g 0 required for the laser mode to have an output power of 100 mW. (b) If the laser is pumped at a level for an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 , what is the pumping ratio and the output power of the laser mode? Solution: For the TEM00 Gaussian mode that has a beam waist spot size of w0 ¼ 500 μm in the Nd:YAG rod, the Rayleigh range, from (3.69), is zR ¼ πnw20 π 1:82 ð500 106 Þ2 ¼ m ¼ 1:34 m: λ 1:064 106 Because zR l > lg , the beam spot stays constant throughout the cavity. Therefore, the mode volume of the oscillating laser mode is V mode πw20 π ð500 106 Þ2 ¼ Al ¼ l¼ 10 102 m3 ¼ 3:93 108 m3 : 2 2 The weighted average refractive index of the laser mode is n ¼ 1:246, from Example 9.2. The photon energy for λ ¼ 1:064 μm is hv ¼ 1:165 eV, from Example 9.5. With a saturation intensity of I sat ¼ 17:3 MW m2 , the saturation photon density of the oscillating laser mode is Ssat ¼ nI sat 1:246 17:3 106 ¼ m3 ¼ 3:85 1017 m3 : chv 3 108 1:165 1:6 1019 The output coupling rate is γout ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 c pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 108 ln 0:9 1 s ¼ 1:27 108 s1 : ln R1 R2 ¼ nl 1:246 10 102 The saturation output power is found using (9.34): Psat out ¼ γout hvV mode Ssat ¼ 358 mW: (a) For an output power of Pout ¼ 100 mW, we ﬁnd by using (9.36) that the required pumping ratio is r ¼1þ Pout 100 ¼ 1:28: sat ¼ 1 þ Pout 358 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 290 Laser Oscillation From Example 9.1, the threshold gain coefﬁcient is g th ¼ 2:09 m1 . Therefore, by (9.26), the unsaturated gain coefﬁcient at this pumping ratio is g 0 ¼ rg th ¼ 1:28 2:09 m1 ¼ 2:68 m1 : (b) When the laser is pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 , by (9.26) the pumping ratio is r¼ g0 10 ¼ ¼ 4:78: g th 2:09 Therefore, from (9.36), the output laser power is 3 W ¼ 1:35 W: Pout ¼ ðr 1ÞPsat out ¼ ð4:78 1Þ 358 10 To explicitly express the output laser power as a function of the pump power, it is necessary to specify the pumping mechanism and the pumping geometry. Irrespective of the pumping details, it is generally true that a laser has zero coherent output power but only ﬂuorescence before it reaches threshold, whereas its coherent output power grows linearly with the pump power above threshold before nonlinearity occurs at a high pump power. Upon reaching the threshold, the output laser ﬁeld also shows dramatic spectral narrowing that accompanies the start of laser oscillation. According to (9.14) and the discussion following it, the linewidth of an oscillating laser mode continues to narrow with increasing laser power as the laser is pumped higher above threshold. The reason is that above threshold the coherent stimulated emission increases with the pumping ratio, whereas the spontaneous emission, which is proportional to the population of the upper laser level, is clamped at its threshold value. These are the unique characteristics that distinguish a laser from other types of light sources, such as ﬂuorescent light emitters and luminescent light sources. However, a real laser does not have such exact ideal characteristics, mainly because of the presence of spontaneous emission and nonlinearities in the gain medium. Figure 9.5 shows the typical characteristics of the output power Pout of a single-mode laser as a function of the pump power Pp . The linear relation between Pout and Pp is a consequence of applying the linear relation between g 0 and Pp to (9.26) for (9.27). As discussed in Section 8.3, the linear relation between g 0 and Pp is itself an approximation near the transparency point of a gain medium. As the pump power increases to a sufﬁciently high level, the unsaturated gain coefﬁcient of a medium cannot continue to increase linearly with the pump power because of the depletion of the ground-level population. Therefore, we should expect that the output power of a laser will not continue its linear increase with the pump power but will increase less than linearly with the pump power at high pumping levels. On the other hand, once the gain medium of a laser is pumped so that its upper laser level begins to be populated, it emits spontaneous photons regardless of whether the laser is oscillating or not. Clearly, the output power of a laser that is pumped below threshold is not exactly zero because ﬂuorescence from spontaneous emission is already emitted from the laser before the laser reaches threshold. Though this Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 9.4 Laser Power 291 Figure 9.5 Typical characteristics of the output power of a single-mode laser as a function of the pump power. ﬂuorescence is incoherent and its power is generally small for a practical laser, it is signiﬁcant for a laser below and right at threshold. Above threshold, it is the major source of incoherent noise for the coherent ﬁeld of the laser output. The overall efﬁciency of a laser, known as the power conversion efﬁciency, is ηc ¼ Pout : Pp (9.37) The approximately linear dependence of the laser output power on the pump power above threshold leads to the concept of the differential power conversion efﬁciency, also known as the slope efﬁciency, of a laser, deﬁned as ηs ¼ dPout : dPp (9.38) Referring to the laser power characteristics shown in Fig. 9.5, the threshold of a laser can usually be lowered by increasing the ﬁnesse of the laser cavity, thus lowering the values of γc and γout , but only at the expense of reducing the differential power conversion efﬁciency of the laser. In the linear region of the laser power characteristics, ηs is clearly a constant that is independent of the operating point of the laser. By contrast, ηc increases with the pump power, but ηc is always smaller than ηs in the linear region. At high pumping levels where the laser output power does not increase linearly with the pump power because of nonlinearity, ηs is no longer independent of the operating point. It can even become smaller than ηc in certain unfavorable situations. EXAMPLE 9.7 The Nd:YAG laser considered in Example 9.5 is optically pumped from two sides of the laser rod with two diode laser arrays at the 808 nm pump wavelength. Because the Nd:YAG laser is a four-level system, its transparency pump power is zero, Ptrp ¼ 0. Furthermore, the pumping ratio is approximately proportional to the pump power: r / Pp . It is found that the pump power Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 292 Laser Oscillation required to reach the pumping ratio for an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 is Pp ¼ 16:5 W. Use the data obtained in Example 9.6 to answer the following questions. (a) Find the threshold pump power. (b) Find the conversion efﬁciency and the slope efﬁciency when the laser has an output power of Pout ¼ 100 mW as in Example 9.6(a). (c) Find the conversion efﬁciency and the slope efﬁciency when the laser has an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 as in Example 9.6(b). Solution: From Example 9.6(b), r = 4.78 for g 0 ¼ 10 m1 . Therefore, r ¼ 4:78 for Pp ¼ 16:5 W. Because Nd:YAG is a four-level system, it is transparent without pumping. Therefore, Ptrp ¼ 0. From (9.27), we have r¼ Pp Ptrp Pth p Ptrp ¼ Pp , Pth p and dr r 4:78 1 ¼ ¼ W ¼ 0:29 W1 : dPp Pp 16:5 (a) The laser reaches its threshold when the pumping ratio is r th ¼ 1. Therefore, the threshold pump power is Pth p ¼ rth 1 W¼ W ¼ 3:45 W: 0:29 0:29 (b) From Example 9.6(a), we ﬁnd that r ¼ 1:28 for Pout ¼ 100 mW. At this pumping ratio, Pp ¼ rPth p ¼ 1:28 3:45 W ¼ 4:42 W: Therefore, from (9.37), the power conversion efﬁciency is ηc ¼ Pout 100 103 ¼ ¼ 2:26%: Pp 4:42 From Example 9.6, we have Psat out ¼ 358 mW. Using (9.38) and (9.36), we ﬁnd that the slope efﬁciency is ηs ¼ dPout dr sat ¼ P ¼ 0:29 358 103 ¼ 10:4%: dPp dPp out (c) When the laser is pumped with a pump power of Pp ¼ 16:5 W to give an unsaturated gain coefﬁcient of g 0 ¼ 10 m1 , we ﬁnd r = 4.78 and Pout ¼ 1:35 W from Example 9.6(b). Therefore, from (9.37), the power conversion efﬁciency is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 Problems ηc ¼ 293 Pout 1:35 ¼ ¼ 8:18%: Pp 16:5 The slope efﬁciency is the same as that found in (b): ηs ¼ dPout dr sat ¼ P ¼ 0:29 358 103 ¼ 10:4%: dPp dPp out Problems 9.1.1 A He–Ne laser has a Fabry–Pérot cavity formed by two mirrors of reﬂectivities R1 ¼ 95% and R2 ¼ 100% at its laser wavelength of λ ¼ 632:8 nm. The cavity length is l ¼ 32 cm. The effective refractive index of the He–Ne gas is n 1. The TEM00 Gaussian laser mode has a distributed optical loss of α ¼ 0:05 m1 . Find the threshold gain coefﬁcient of this laser mode. 9.1.2 An optical-ﬁber laser emitting at λ ¼ 1:53 μm has a ring cavity as shown in Fig. 6.1(d). It has one input–output coupler that has a coupling efﬁciency of η ¼ 10%. The ﬁber loop has a total length of l ¼ 10 m, which contains a gain section of a length lg ¼ 1 m. The effective index of the ﬁber laser mode is n ¼ 1:47 and the distributed loss is α ¼ 10 dB km1 . What is the threshold gain coefﬁcient of this laser mode? 9.1.3 A GaAs/AlGaAs semiconductor laser emitting at λ ¼ 860 nm has a Fabry–Pérot cavity formed by two ﬂat, cleaved surfaces of reﬂectivities R1 ¼ R2 ¼ 32% for the TE0 mode of the GaAs/AlGaAs waveguide. The gain region is the GaAs waveguide core, which is pumped uniformly throughout the cavity length such that the cavity and the gain medium have the same length of l ¼ lg ¼ 350 μm. The laser oscillates in the single transverse TE0 waveguide mode, which has a conﬁnement factor of Γ ¼ 0:3 deﬁned by the overlap factor of the TE0 mode intensity proﬁle with the waveguide core gain region. The distributed loss is α ¼ 25 cm1 . Find the threshold gain coefﬁcient of this laser mode. If one of the cleaved cavity surfaces is optically coated for 100% reﬂectivity, what is the threshold gain coefﬁcient? 9.2.1 The optical gain of a homogeneously broadened laser is contributed by a discrete optical transition between two atomic energy levels at a transition resonance frequency of ω21 . A longitudinal mode q of the laser has its cold-cavity frequency tuned to the transition resonance frequency such that ωcq ¼ ω21 . When the laser is pumped above the threshold for this mode to oscillate, what is the oscillating frequency of the laser? How much is the frequency shift due to mode pulling? 9.2.2 The optical gain in a semiconductor laser medium is contributed by excess electrons and holes in the conduction and valence bands, respectively, of the semiconductor. The gain is determined by the excess carrier concentration N, which is the density of the Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 294 Laser Oscillation electron–hole pairs in excess of the thermal-equilibrium concentrations of electrons and holes. As a result, the relationship between the real and imaginary parts of the resonant susceptibility is not simply the Lorentzian function that characterizes a discrete atomic transition. Nevertheless, an optical gain still causes a change in the refractive index of the medium. This effect is usually described by an experimentally measured antiguidance factor, also known as the linewidth enhancement factor, deﬁned as b¼ ∂n0 =∂N 2ω ∂n0 =∂N 4π ∂n0 =∂N ¼ ¼ , ∂n00 =∂N c ∂g=∂N λ ∂g=∂N (9.39) where n0 and n00 are, respectively, the real and imaginary parts of the refractive index of the medium, and g is the gain coefﬁcient. A GaAs/AlGaAs semiconductor laser emitting at λ ¼ 850 nm has a Fabry–Pérot cavity, which is pumped uniformly so that the cavity and the gain medium have the same length of l ¼ lg ¼ 300 μm. The gain medium has an antiguidance factor of b ¼ 3:5. The effective refractive index is n ¼ 3:65 when the laser medium is pumped to transparency at λ ¼ 850 nm. The laser is pumped to give a gain coefﬁcient of g ¼ 5 104 m1 . Besides shifting the frequency of each longitudinal mode, the mode-pulling effect caused by the antiguidance factor changes the longitudinal mode frequency spacing. Find the frequency shift of a longitudinal mode at the λ ¼ 850 nm laser wavelength. Find the change in the longitudinal mode frequency spacing. 9.3.1 A GaAs/AlGaAs vertical-cavity surface-emitting semiconductor laser emitting at λ ¼ 850 nm has a very short cavity. Its gain region is composed of a few thin quantum wells, and its reﬂective mirrors are distributed Bragg reﬂectors of periodic structures. For the longitudinal mode frequencies, the effective physical length of the cavity is leff ¼ 1:2 μm and the effective refractive index is neff ¼ 3:52. The laser is pumped to give a gain bandwidth of Δλg ¼ 48 nm above the laser threshold. How many longitudinal modes oscillate? 9.3.2 A He–Ne laser has a Doppler-broadened gain bandwidth of Δνg ¼ 1:5 GHz at its laser wavelength of λ ¼ 632:8 nm. The laser has a cavity length of l ¼ 32 cm. (a) It is pumped at a level such that the peak unsaturated gain coefﬁcient is four times the threshold gain coefﬁcient: g max ¼ 4g th . How many longitudinal modes have their 0 unsaturated gain coefﬁcients pumped above the threshold? How many longitudinal modes oscillate? (b) If a longitudinal mode frequency is tuned to the frequency of the gain peak, what is the value of g max for the laser to oscillate only in this mode? 0 9.3.3 An Er:ﬁber laser emitting at λ ¼ 1:53 μm has a cold-cavity linewidth of Δνc ¼ 520 kHz. It is doped with an Er3þ ion concentration of N t ¼ 2:2 1024 m3 . At λ ¼ 1:53 μm, the absorption cross section is σ a ¼ 5:7 1025 m2 , and the emission cross section is σ e ¼ 7:9 1025 m2 . The gain coefﬁcients of its oscillating modes are saturated at g ¼ 0:25 m1 . The population density of the upper laser level for this gain coefﬁcient can be found using (8.42). What is the minimum possible linewidth set by the Schawlow– Townes limit for an oscillating laser mode that has an output power of Pout ¼ 1 mW? Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 Problems 295 9.4.1 A Ti:sapphire laser consists of a Ti:sapphire crystal of a length lg ¼ 2 cm in a Fabry– Pérot cavity, which has a physical length of l ¼ 16 cm deﬁned by two mirrors of reﬂectivities R1 ¼ 100% and R2 ¼ 95% at the laser emission wavelength of λ ¼ 800 nm. The TEM00 Gaussian mode deﬁned by the cavity has a beam waist spot size of w0 ¼ 150 μm located at the center of the Ti:sapphire crystal, which has a refractive index of n ¼ 1:76. The end surfaces of the crystal are antireﬂection coated to eliminate undesirable losses. At the λ ¼ 800 nm laser wavelength, the Ti:sapphire crystal has an emission cross section of σ e ¼ 3:4 1023 m2 and an absorption cross section of σ a 8 1026 m2 . Over the range of laser operation considered here, the saturation lifetime can be taken as τ s τ 2 ¼ 3:2 μs. The distributed loss of the laser cavity, including the absorption of the Ti:sapphire crystal at λ ¼ 800 nm, is α ¼ 0:1 m1 . The laser is optically pumped at the pump wavelength of λp ¼ 532 nm. (a) Find the threshold gain coefﬁcient of this laser. (b) Find the saturation output power of this laser. (c) What are the pumping ratio and the unsaturated gain coefﬁcient required for the laser to have an output power of Pout ¼ 1 W? (d) The transparency pump power of the laser is Ptrp ¼ 1:4 W, and the threshold pump power is Pth p ¼ 5:0 W. What is the pump power that is required for Pout ¼ 1 W? (e) What are the power conversion efﬁciency and the slope efﬁciency when the laser has an output power of Pout = 1 W? 9.4.2 The Ti:sapphire laser described in Problem 9.4.1 is pumped to have an unsaturated gain coefﬁcient of g 0 ¼ 5 m1 . (a) What are the pumping ratio and the pump power? (b) Find the output laser power at this pumping level. (c) What are the power conversion efﬁciency and the slope efﬁciency at this operating point? 9.4.3 A semiconductor laser is pumped by current injection. The injected current generates excess electron–hole pairs in the active region of the laser. The excess electron–hole pairs act as the source of the optical gain. When the details of the laser structure and the parameters of the gain medium are known, the power and efﬁciency of a semiconductor laser can be analyzed as discussed in Section 9.4. Alternatively and equivalently, the output power of a semiconductor laser can be found by considering that one photon is emitted when an electron–hole pair recombines radiatively. Thus, for a semiconductor laser, Pout ¼ ηinj γout hv ðI I th Þ, γc e (9.40) where ηinj is the current injection efﬁciency, γout is the output coupling rate, γc is the cavity decay rate, hv is the laser photon energy, e is the electronic charge, I is the injection current, and I th is the threshold injection current for the laser to start oscillating. The injection efﬁciency ηinj is the fraction of the total injection current that actually Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 296 Laser Oscillation contributes to the generation of useful electron–hole pairs in the active region of the laser. If the bias voltage of the laser is V, the power conversion efﬁciency is Pout Pout γout hv I th ηc ¼ ¼ ηinj , (9.41) ¼ 1 Pp VI γc eV I and the slope efﬁciency is ηs ¼ dPout dPout γ hv ¼ ηinj out ¼ : dPp VdI γc eV (9.42) Now, consider the GaAs/AlGaAs laser described in Problem 9.1.3 but with R1 ¼ 1 and R2 ¼ 0:32. The effective refractive index of the laser mode is n ¼ 3:63. The injection efﬁciency is ηinj ¼ 0:7, the threshold current is I th ¼ 20 mA, and the bias voltage is V ¼ 2 V. (a) Find the output laser power for an injection current of I ¼ 40 mA. (b) What are the power conversion efﬁciency and the slope efﬁciency at this operating point? Bibliography Davis, C. C., Lasers and Electro-Optics: Fundamentals and Engineering, 2nd edn. Cambridge: Cambridge University Press, 2014. Iizuka, K., Elements of Photonics for Fiber and Integrated Optics, Vol. II. New York: Wiley, 2002. Liu, J. M., Photonic Devices. Cambridge: Cambridge University Press, 2005. Milonni, P. W. and Eberly, J. H., Laser Physics. New York: Wiley, 2010. Rosencher, E. and Vinter, B., Optoelectronics. Cambridge: Cambridge University Press, 2002. Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics. New York: Wiley, 1991. Siegman, A. E., Lasers. Mill Valley, CA: University Science Books, 1986. Silfvest, W. T., Laser Fundamentals. Cambridge: Cambridge University Press, 1996. Svelto, O., Principles of Lasers, 5th edn. New York: Springer, 2010. Verdeyen, J. T., Laser Electronics, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall, 1995. Yariv, A. and Yeh, P., Photonics: Optical Electronics in Modern Communications. Oxford: Oxford University Press, 2007. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:08 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.010 Cambridge Books Online © Cambridge University Press, 2016 Cambridge Books Online http://ebooks.cambridge.org/ Principles of Photonics Jia-Ming Liu Book DOI: http://dx.doi.org/10.1017/CBO9781316687109 Online ISBN: 9781316687109 Hardback ISBN: 9781107164284 Chapter 10 - Optical Modulation pp. 297-361 Chapter DOI: http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge University Press 10 Optical Modulation 10.1 TYPES OF OPTICAL MODULATION .............................................................................................................. Optical modulation allows one to control an optical wave or to encode information on a carrier optical wave. The inverse process that recovers the encoded information is demodulation. There are many types of optical modulation, which can be categorized in several different ways. 1. According to the particular optical-ﬁeld parameter being modulated, optical modulation can be categorized into different modulation schemes: phase modulation, frequency modulation, polarization modulation, amplitude modulation, spatial modulation, and diffraction modulation. 2. Depending on whether the information is encoded in the analog or digital form, optical modulation can be either analog modulation or digital modulation. 3. Optical modulation can be categorized as direct modulation or external modulation. Direct modulation is directly performed on an optical source, which is usually a light-emitting diode (LED) or a laser, without using a separate optical modulator. External modulation is performed on an optical wave using a separate optical modulator to change one or more characteristics of the wave. 4. Optical modulation is accomplished by varying the optical susceptibility of the modulator material. Depending on whether the real or imaginary part of the susceptibility is responsible for the functioning of the modulator, optical modulation can be categorized as refractive modulation or absorptive modulation. Refractive modulation is performed by varying the real part of the susceptibility, thus varying the refractive index of the material; absorptive modulation is performed by varying the imaginary part of the susceptibility, thus varying the absorption coefﬁcient of the material. 5. Optical modulation can be categorized according to the physical mechanism behind the change of the optical susceptibility, such as electro-optic modulation, acousto-optic modulation, magneto-optic modulation, all-optical modulation, and so forth. 6. Depending on the geometric relation between the modulating signal and the modulated optical wave, optical modulation can be transverse modulation or longitudinal modulation. In transverse modulation, the signal is applied in a direction perpendicular to the propagation direction of the optical wave. In longitudinal modulation, the signal is applied along the propagation direction of the optical wave. 7. Optical modulation can be performed on unguided or guided optical waves. Correspondingly, the structure of an optical modulator can take the form of a bulk or waveguide device. A bulk modulator is used to modulate an unguided optical wave. A waveguide modulator is used to modulate a guided optical wave. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 298 Optical Modulation Optical switching is a special form of optical modulation. Generally speaking, optical switching is large-signal digital optical modulation that results in the switching between two or more discrete values of an optical parameter or between two or more optical modes. It can be performed on any type of optical modulation. The characteristic of the optical wave being switched can be its phase, frequency, amplitude, polarization, propagation direction, or spatial pattern. Optical switching can also be performed between two or more normal modes in a waveguide structure. 10.2 MODULATION SCHEMES .............................................................................................................. As discussed in Section 1.7, an unguided optical ﬁeld is characterized by its polarization ^e , magnitude jE j, phase φE , wavevector k, and frequency ω: Eðr, tÞ ¼ Eðr, tÞ exp ðik r iωtÞ ¼ ^e jEðr, tÞjeiφE ðr, tÞ exp ðik r iωtÞ: (10.1) The total phase of this ﬁeld is that given in (1.83): φðr; t Þ ¼ k r ωt þ φE ðr; tÞ: (10.2) As described in (3.25), a guided optical ﬁeld propagating along the z direction can be expressed as a linear superposition of normal modes: X Eðr, tÞ ¼ Aν ðz, tÞE^ν ðx, yÞ exp ðiβν z iωtÞ ν X (10.3) ¼ E^ν ðx, yÞjAν ðz, tÞjeiφAν ðz, tÞ exp ðiβν z iωtÞ: ν The ﬁeld in a mode is also characterized by ﬁve ﬁeld parameters: the vectorial mode ﬁeld pattern E^v ðx; yÞ, the magnitude jAv ðz; tÞj of the complex mode amplitude Av ðz; tÞ, the phase φAv ðz; tÞ of the complex mode amplitude Av ðz; t Þ, the mode propagation constant βv , and the frequency ω. The total phase of the ﬁeld in mode v is φv ðz; t Þ ¼ βv z ωt þ φAv ðz; t Þ: (10.4) Optical modulation can be performed on any of the ﬁeld parameters. Therefore, there exist many modulation techniques based on different schemes. Each modulation scheme has been further developed into many advanced modulation formats. In general, the concept of a modulation scheme or format that is developed for an electromagnetic carrier wave at a low frequency, such as a radio frequency, can be adapted and applied to optical modulation. Also common to low-frequency carriers and optical carriers is that the modulation signal can be either analog or digital. The three basic modulation schemes for all carrier frequencies are phase modulation (PM), frequency modulation (FM), and amplitude modulation (AM) for analog modulation, which take the forms of phaseshift keying (PSK), frequency-shift keying (FSK), and amplitude-shift keying (ASK) for digital modulation. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.2 Modulation Schemes 299 Due to the differences between optical waves and low-frequency electromagnetic waves regarding the ﬁeld characteristics and the material properties in their respective spectral regions, some schemes and certain considerations are speciﬁc to optical modulation. In addition to the three basic modulation schemes of phase modulation, frequency modulation, and amplitude modulation, optical modulation can also be performed on the polarization ^e of the ﬁeld for polarization modulation, on the spatial distribution jE ðr; t Þj of the ﬁeld for spatial modulation, and on the direction k^ of wave propagation for diffraction modulation. Because of the dispersive nature and the intrinsic coupling between the real and imaginary parts of the optical susceptibility, as well as its tensorial nature in the case of an anisotropic crystal, a modulation signal often affects more than one parameter of the modulated optical ﬁeld. For example, amplitude modulation that is carried out by varying the absorption or ampliﬁcation coefﬁcient, through varying χ 00 , of the material in a modulator is usually accompanied by a variation in χ 0 , thus varying the refractive index and resulting in a modulation on the phase of the optical wave. This is the case for direct modulation discussed in Section 10.3. As another example, phase modulation using a modulator made of an anisotropic crystal can sometimes be accompanied by a polarization change of the optical ﬁeld. In any event, a modulation scheme is chosen based on the ﬁeld parameter on which we intend to code the information. The accompanying modulation on other ﬁeld parameters is a side effect that has to be avoided or suppressed as much as possible, if it is unavoidable. Phase modulation is the most fundamental of all modulation schemes. By controlling the optical phase while properly manipulating the optical wave, a desired modulation on any other ﬁeld parameter can be accomplished. On the other hand, certain ﬁeld parameters can be directly modulated without changing the optical phase. The concepts of basic optical modulation schemes are described in the following. The techniques and the physical mechanisms that can be used for these modulation schemes are discussed in later sections. 10.2.1 Phase Modulation A phase-modulated optical ﬁeld at a ﬁxed location, taken to be r ¼ 0 for simplicity of expression, is a function of time of the form: Eð0; tÞ ¼ ^e jE j exp ½iφE ðt Þ iωt, (10.5) where the time-varying phase φE ðt Þ carries the encoded information, whereas ^e , jE j, and ω do not vary with time. In analog phase modulation, φE ðt Þ is a continuous function of time; in digital phase modulation, i.e., PSK, φE ðt Þ changes stepwise with time. The temporal characteristics of the optical ﬁeld under analog and digital phase modulation are shown in Figs. 10.1(a) and (b), respectively. The magnitude and frequency of the carrier ﬁeld stay constant under phase modulation because only the phase varies with time. In phase modulation, the largest meaningful phase change is 2π because phase is periodic with a period of 2π; therefore, the range of phase modulation is usually chosen to be from 0 to 2π or from π to π. In PSK, the 2π phase range is equally divided into discrete levels representing different digital values. The phase shifts from one discrete level to another discrete Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 300 Optical Modulation Figure 10.1 (a) Analog phase modulation with an analog signal. (b) Digital phase modulation using two discrete phases separated by π for BPSK. The ﬁeld magnitude and the carrier frequency stay constant while the phase varies with time. level. In binary PSK (BPSK), two discrete phases separated by π, such as f0; π g or fπ=2; 3π=2g, are used to respectively represent the two binary bits of 0 and 1, as shown in Fig. 10.1(b). In quadrature PSK (QPSK), four discrete phases that are equally spaced at an interval of π=2, such as f0; π=2; π; 3π=2g or fπ=4; 3π=4; 5π=4; 7π=4g, are used to represent the four possible two-bit combinations of f00; 01; 10; 11g by encoding two bits with each phase. Optical phase modulation is normally accomplished through refractive modulation. By modulating the refractive index of a material through which an optical wave propagates, the phase of the wave can be modulated. The physical mechanisms that can be used for this purpose are discussed in Section 10.4. 10.2.2 Frequency Modulation A frequency-modulated optical ﬁeld has a time-varying frequency of ωðt Þ that carries the encoded information: Eð0; t Þ ¼ ^e jE j exp ½iφE iωðtÞt , (10.6) where ^e , jE j, and φE do not vary with time. In analog frequency modulation, ωðt Þ varies continuously with time; in digital frequency modulation, i.e., FSK, ωðt Þ shifts abruptly from one frequency to another. In binary FSK (BFSK), two different frequencies are used to represent the two binary bits of 0 and 1 for a digital signal. More than two frequencies can be used to digitize a signal in multiple symbols; for example, in quadrature FSK (QFSK), four frequencies are used to represent the four possible two-bit combinations of f00; 01; 10; 11g by encoding two bits with each frequency. Figures 10.2(a) and (b) show the temporal characteristics of the optical ﬁeld under analog frequency modulation and BFSK, respectively. The magnitude of the carrier ﬁeld stays constant Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 301 10.2 Modulation Schemes Figure 10.2 (a) Analog frequency modulation. (b) Digital frequency modulation using two different frequencies for BFSK. The ﬁeld magnitude stays constant while the carrier frequency varies with time. while the frequency varies with time. Note the ﬁne differences in the characteristics of the modulated waveforms between frequency modulation and phase modulation by comparing Fig. 10.2 to Fig. 10.1. Frequency modulation can be achieved by phase modulation over a large phase range because, from (1.87), ωðt Þ ¼ ∂φ ∂φ ¼ω E: ∂t ∂t (10.7) In contrast to the case for phase modulation discussed above, however, the modulated phase change for frequency modulation is not limited to a range of 2π. Instead, the range of phase change is a function of the magnitude and the duration of the frequency shift from the original, unshifted carrier frequency. For example, for BFSK that shifts the frequency between ω and ω0 , a time-varying phase of φE ðtÞ ¼ ðω0 ωÞðt t 0 Þ has to be maintained from the time t 0 when the frequency is shifted from ω to ω0 until the time when the frequency is shifted back to ω. EXAMPLE 10.1 The phase of a polarized plane optical ﬁeld is temporally modulated by a sinusoidal variation of a modulation amplitude φ0 and a modulation frequency Ω as φE ðt Þ ¼ φ0 sin Ωt. What happens to the polarization of this modulated optical ﬁeld? What happens to the magnitude and intensity of this optical ﬁeld? Does this phase modulation result in frequency modulation? What happens to the frequency of this optical ﬁeld in the time domain and in the frequency domain? Solution: The modulation is imposed only on the phase of the ﬁeld such that E ðt Þ ¼ ^e E exp½iφE ðt Þ ¼ ^e E exp ðiφ0 sin Ωt Þ: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 302 Optical Modulation Clearly, the polarization vector ^e is not affected by the phase modulation; thus, it remains a constant of time. The ﬁeld magnitude jE j is not affected by the phase modulation, either; therefore, both the ﬁeld magnitude and the intensity, which is I / jE j2 , remain constants of time. By contrast, this time-varying phase modulation does result in frequency modulation: ωðt Þ ¼ ∂φ ∂φ ¼ ω E ¼ ω φ0 Ω cos Ωt: ∂t ∂t In the time domain, we ﬁnd that the frequency of this optical ﬁeld varies sinusoidally with time around the center optical carrier frequency ω as ωðt Þ ¼ ω φ0 Ω cos Ωt. To ﬁnd the frequency components in the frequency domain, we use the identity: exp ðiφ0 sin ΩtÞ ¼ ∞ X J q ðφ0 Þ exp ðiqΩt Þ, q¼∞ where J q is the qth-order Bessel function of the ﬁrst kind, which has the property that J q ¼ ð1Þq J q . Therefore, we can express the phase-modulated optical ﬁeld as Eðt Þ ¼ ^e jE j( exp ðiφ0 sin Ωt iωt Þ ) ∞ h i X q iðωþqΩÞt iðωqΩÞt iωt þ J q ðφ0 Þ e þ ð1Þ e ¼ ^e jE j J 0 ðφ0 Þe : q¼1 It can be seen that in the frequency domain, the sinusoidal phase modulation generates a series of side bands at the harmonics of the modulation frequency Ω on both the low-frequency and high-frequency sides of the center optical carrier frequency ω. 10.2.3 Polarization Modulation Information can also be encoded on the polarization of an optical ﬁeld through polarization modulation so that the polarization vector is a time-varying function: Eð0; t Þ ¼ ^e ðt ÞjE j exp ðiφE iωt Þ, (10.8) where jE j, φE , and ω do not vary with time. In analog polarization modulation, ^e ðt Þ varies continuously with time; in digital polarization modulation, known as polarization-shift keying (PolSK), ^e ðt Þ changes abruptly from one polarization to another. In binary polarization-shift keying (BPolSK), two orthogonal polarization states are used to represent the two binary bits of 0 and 1 for a digital signal. Multiple polarization states can be used to represent multiple possible bit combinations; in this situation, the polarization states are not all mutually orthogonal because each polarization state has only one corresponding orthogonal polarization state. Polarization modulation can be achieved through differential phase modulation on two orthogonally polarized components of an optical ﬁeld by using, for example, the electro-optic Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.2 Modulation Schemes 303 Pockels effect or the magneto-optic Faraday effect. Any orthonormal set of unit polarization vectors f^e 1 ; ^e 2 g on the plane that is normal to the wave propagation direction k^ can be used to expand the unit polarization vector ^e on this plane as a linear superposition of two orthogonal polarizations: ^e ¼ c1 ^e 1 þ c2 ^e 2 , (10.9) where c1 and c2 are two complex constants subject to the normalization condition of ∗ c1 c∗ e 1 ; ^e 2 g basis, the unit polarization vector ^e ⊥ that is orthogonal 1 þ c2 c2 ¼ 1: On the f^ to the unit polarization vector ^e can be expressed as ^e ⊥ ¼ c∗ e 1 c∗ e2: 2^ 1^ (10.10) It is clear that f^e ; ^e ⊥ g is also an orthonormal basis because ^e ^e ∗ ¼ ^e ⊥ ^e ∗ ⊥ ¼ 1 and ∗ ∗ ^e ^e ⊥ ¼ ^e ⊥ ^e ¼ 0. Therefore, the two unit polarization vectors ^e 1 and ^e 2 can be expressed in terms of the f^e ; ^e ⊥ g basis as ^e 1 ¼ c∗ e þ c2 ^e ⊥ , 1^ ^e 2 ¼ c∗ e c1 ^e ⊥ : 2^ (10.11) As an example, any polarization state on the xy plane can be represented by the unit vector ^e ¼ ^x cos α þ ^y eiφ sin α given in (1.65), which is the linear superposition of the two orthonormal linear polarization unit vectors ^x and ^y with c1 ¼ cos α and c2 ¼ eiφ sin α. In this case, ^e 1 ¼ ^x , ^e 2 ¼ ^y , and ^e ⊥ ¼ ^x eiφ sin α ^y cos α. As another example, the linear polarization pﬃﬃﬃ unit vector ^x can be expressed as ^e ¼ ^x ¼ ð^e þ þ ^e Þ= 2 in terms of p the ﬃﬃﬃ linear superposition of the orthonormal circular polarization unit vectors with c1 ¼ c2 ¼ 1= 2. In this case, ^e 1 ¼ ^e þ , pﬃﬃﬃ ^e 2 ¼ ^e , and ^e ⊥ ¼ i^y ¼ ð^e þ ^e Þ= 2. When the phases of the two orthogonally polarized ﬁeld components are differentially modulated, the polarization vector of the modulated optical wave becomes a function of time: h i ^e m ðtÞ ¼ c1 eiφ1 ðtÞ ^e 1 þ c2 eiφ2 ðtÞ ^e 2 ¼ c1 ^e 1 þ c2 eiΔφðtÞ ^e 2 eiφ1 ðtÞ , (10.12) where ΔφðtÞ ¼ φ2 ðt Þ φ1 ðt Þ (10.13) is the time-varying phase difference due to differential phase modulation between the ^e 1 and ^e 2 components of the optical ﬁeld. By substituting ^e 1 and ^e 2 of (10.11) into (10.12), we can express the modulated time-varying unit polarization vector ^e m ðt Þ in terms of ^e and ^e ⊥ as iφ1 ðt Þ iφ2 ðt Þ ^e m ðtÞ ¼ c1 c∗ ^e þ c1 c2 eiφ1 ðtÞ c1 c2 eiφ2 ðtÞ ^e⊥ þ c2 c∗ 1e 2e (10.14) ∗ iΔφ1 ðt Þ iφ1 ðt Þ ^e þ c1 c2 1 eiΔφðtÞ eiφ1 ðtÞ ^e⊥ : ¼ c1 c∗ e 1 þ c2 c2 e It is clear from (10.14) that ^e m ðtÞ ^e ⊥ 6¼ 0 and ^e m ðt Þ 6¼ ^e when c1 c2 6¼ 0 and ΔφðtÞ 6¼ 2mπ, resulting in a polarization change caused by differential phase modulation. As discussed in Section 1.6, the polarization state of a wave depends only on the phase difference and the magnitude ratio of the two orthogonally polarized ﬁeld components. Therefore, the polarization state deﬁned by ^e m ðt Þ is determined by the phase difference ΔφðtÞ and the magnitude ratio jc1 =c2 j of the ^e 1 and ^e 2 components, and is independent of the common Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 304 Optical Modulation phase factor φ1 ðtÞ. Because the magnitude ratio jc1 =c2 j is not affected by phase modulation, thus remaining constant, the polarization state can be varied by varying only the phase difference Δφðt Þ. Consequently, polarization modulation of an optical ﬁeld can be accomplished through differential phase modulation on two orthogonally polarized components of the ﬁeld. EXAMPLE 10.2 An optical ﬁeld is initially linearly polarized in the x direction. Find two linearly polarized components of this polarization in the xy plane that are orthogonal to each other. How does the polarization of this ﬁeld change if the two orthogonally polarized components are differentially phase modulated by a phase difference of π=4, π=2, π, and 2π, respectively? Solution: In the xy plane, the two linearly polarized orthogonal components of the unit polarization vector ^e ¼ ^x can be chosen as ^x þ ^y ^x ^y ^e 1 ¼ pﬃﬃﬃ and ^e 2 ¼ pﬃﬃﬃ , 2 2 pﬃﬃﬃ ,which are arbitrarily chosen to be real vectors such that c1 ¼ c2 ¼ 1= 2 and arbitrarily assigned in the sequence of ^e 1 and ^e 2 . In the xy plane, the polarization that is orthogonal to ^e ¼ ^x is ^e ⊥ ¼ ^y . From (10.14), if the two orthogonally polarized components are differentially phase modulated such that φ2 ðt Þ φ1 ðtÞ ¼ Δφðt Þ, the polarization of the ﬁeld becomes 1 þ eiΔφðtÞ 1 eiΔφðtÞ ^e þ ^e ⊥ eiφ1 ðtÞ ^e m ðt Þ ¼ 2 2 1 þ eiΔφðtÞ 1 eiΔφðtÞ ^x þ ^y eiφ1 ðtÞ ¼ 2 2 Δφðt Þ Δφðt Þ ^x i sin ^y eiφ1 ðtÞþiΔφðtÞ=2 : ¼ cos 2 2 The common phase factor φ1 ðtÞ þ ΔφðtÞ=2 only changes the phase of the unit polarization vector ^e m ðtÞ and does not have an effect on the polarization state of the ﬁeld. Therefore, we can ignore this phase factor and consider only the polarization state vector of the differentially phase-modulated ﬁeld: Δφðt Þ ΔφðtÞ ^x i sin ^ ^e 0m ðtÞ ¼ cos y: 2 2 We ﬁnd different polarization states for different phase differences: π π π For Δφ ¼ , ^e 0m ¼ cos ^x i sin ^y ¼ 0:924^x i0:383^y , elliptically polarized; 4 8 8 ^x i^y π 0 π π For Δφ ¼ , ^e m ¼ cos ^x i sin ^y ¼ pﬃﬃﬃ , circularly polarized; 2 4 4 2 π π 0 For Δφ ¼ π, ^e m ¼ cos ^x i sin ^y ¼ i^y , linearly polarized parallel to ^e ⊥ ¼ ^y ; 2 2 0 For Δφ ¼ 2π, ^e m ¼ cos π^x i sin π^y ¼ ^x , linearly polarized parallel to ^e ¼ ^x . Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.2 Modulation Schemes 305 10.2.4 Amplitude Modulation One of the most common modulation schemes is amplitude modulation, which encodes information on the magnitude of an optical ﬁeld: Eð0; tÞ ¼ ^e jE ðtÞj exp ðiφE iωtÞ, (10.15) where ^e , φE , and ω do not vary with time. In analog amplitude modulation, jE ðt Þj varies continuously with time; in digital amplitude modulation, known as amplitude-shift keying (ASK), jE ðt Þj changes abruptly from one discrete value to another. In binary ASK, two ﬁeld magnitudes are used, with the binary bit 1 normally represented by a larger ﬁeld magnitude and the bit 0 represented by a smaller magnitude. A special case of binary ASK is on-off keying (OOK) where the optical ﬁeld is turned on at a ﬁxed magnitude level for bit 1 and turned off for bit 0. Multilevel ASK uses multiple discrete ﬁeld magnitudes to represent multiple possible bit combinations for each ﬁeld magnitude to encode one possible combination of an equal number of bits. Figures 10.3(a) and (b) show the temporal characteristics of the optical ﬁeld under analog modulation and binary ASK, respectively. The magnitude of the carrier ﬁeld varies with time while the frequency and phase stay constant. Amplitude modulation leads to intensity modulation (IM), in which the intensity and the power of an optical wave are modulated because the intensity and power of the wave are proportional to jE ðt Þj2 . Optical amplitude modulation can be accomplished in many different ways: by direct modulation on the optical source, as discussed in Section 10.3; by refractive modulation using any physical mechanism discussed in Section 10.4, followed by proper manipulation of the optical ﬁeld; or by absorptive modulation of a material through which the optical wave propagates, as discussed in Section 10.5. Figure 10.3 (a) Analog amplitude modulation. (b) Digital amplitude modulation using two different discrete ﬁeld magnitudes. Both the carrier frequency and phase of the ﬁeld stay constant while the magnitude varies with time. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 306 Optical Modulation Amplitude modulation on an optical ﬁeld can be achieved through polarization modulation by properly selecting a polarization component while ﬁltering out its orthogonal component after the ﬁeld is polarization modulated. As an example, consider the polarization-modulated optical ﬁeld characterized by the time-varying unit polarization vector ^e m ðt Þ expressed in (10.14). It is clear that by using a polarizer to select either only the ^e -polarized component or only the ^e ⊥ -polarized component, the resulting ﬁeld magnitude is modulated. For instance, by selecting only the ^e ⊥ -polarized component, the output ﬁeld has the time-varying magnitude: h i Δφðt Þ iΔφðt Þ E ¼ 2c1 c2 E sin , jE ⊥ ðtÞj ¼ c1 c2 1 e 2 (10.16) where E is the time-independent ﬁeld amplitude of the polarization-modulated optical ﬁeld. The intensity of this output ﬁeld is modulated as I ⊥ ðtÞ ¼ 4jc1 c2 j2 I sin2 ΔφðtÞ , 2 (10.17) where I / jE j2 is the time-independent intensity of the polarization-modulated optical wave. Though polarization modulation of the optical ﬁeld used in the above example is accomplished by differential phase modulation, the concept of obtaining amplitude modulation by selecting a polarization component while rejecting its orthogonal component is generally applicable to any polarization-modulated optical wave. Optical amplitude modulation can also be achieved through phase modulation to vary the coupling or interference between different components of an optical wave. 1. By varying the phase mismatch δ through differential phase modulation on two coupled modes in a coupler, the coupling efﬁciency η can be modulated, as discussed in Section 4.6. Thus, the ﬁeld amplitude of a mode is modulated. This general concept is applicable to any mode coupler. 2. By varying the interference of two or multiple waves through differential phase modulation, the superposition of the interfering waves can be amplitude modulated, as discussed in Section 5.1. This general concept is applicable to any interferometer discussed in Chapter 5. In analog amplitude modulation, the optical intensity varies continuously with time. To faithfully encode the analog information on the carrier optical wave, linearity of the modulation response is desired. However, as the example in (10.17) shows, the response of an amplitude modulator generally cannot be linear over the whole range of operation. For this reason, the linearity requirement for analog modulation often limits the modulation depth to a small linear range of the modulation response. In digital amplitude modulation, the optical intensity is switched between two or among multiple discrete levels. In this case, linearity is not required, but clear separation of the discrete levels is desired. In binary operation, where the switching takes place between a high-intensity level of I high and a low-intensity level of I low , it is desired that the ratio I low =I high is as small as possible while I high is sufﬁciently large. In digital amplitude modulation using an external modulator, the binary states are represented by a high transmittance T high and a low Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.2 Modulation Schemes 307 transmittance T low . The ratio of these two levels is deﬁned as the extinction ratio, which is usually measured in dB: ER ¼ 10 log I low T low ¼ 10 log : I high T high (10.18) A high extinction ratio allows clear separation of the two levels, thus clear identiﬁcation of the binary bits. Besides a high extinction ratio, the level of the high transmittance T high has to be sufﬁciently high for good performance. 10.2.5 Spatial Modulation Taking the propagation direction to be the z direction without loss of generality, a spatially modulated optical ﬁeld has a time-varying ﬁeld pattern of E ðx; y; 0; t Þ at the ﬁxed z ¼ 0 location on the plane that is perpendicular to the propagation direction: Eðx; y; 0; tÞ ¼ E ðx; y; 0; t Þexp ðiωt Þ ¼ ^e ðx; y; 0; tÞjE ðx; y; 0; t ÞjeiφE ðx;y;0;tÞ eiωt : (10.19) Spatial modulation can be on the ﬁeld polarization, with a space- and time-varying polarization vector ^e ðx; y; 0; tÞ; on the ﬁeld magnitude, with a space- and time-varying ﬁeld magnitude jE ðx; y; 0; t Þj; or on the phase, with a space- and time-varying ﬁeld phase φE ðx; y; 0; t Þ. The spatial variation can be either a continuous function of x and y, or a digitized function of x and y. If the spatial variation is expressed in terms of a linear superposition of transverse spatial normal modes, then X Eðx; y; 0; t Þ ¼ Av ðt ÞE^v ðx; yÞ exp ðiωt Þ (10.20) v according to (3.25). Thus, spatial modulation can be described as, and be accomplished through, the temporal variations of the mode expansion coefﬁcients Av ðt Þ. 10.2.6 Diffraction Modulation As discussed in Section 5.2, an optical grating diffracts an incident optical wave into multiple diffracted beams; the diffraction angle θq of the qth-order diffracted beam is determined by the phase-matching condition given in (5.32): k sin θq ¼ k sin θi þ qK (10.21) where k ¼ nω=c is the propagation constant of the optical wave, with n being the refractive index of the medium; θi is the incident angle of the incoming wave; and K ¼ 2π=Λ is the wavenumber of the grating, with Λ being the period of the grating. Clearly, the diffraction angle θq , and thus the diffraction pattern, can be varied by varying the refractive index n, the incident angle θi , the grating period Λ, or a combination of these parameters. Many refractive modulation mechanisms, as discussed in Section 10.4, can be used to modulate the refractive index of the grating material, thus accomplishing diffraction modulation. The grating period Λ Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 308 Optical Modulation can also be modulated if the grating is not a ﬁxed structure but is generated by an acoustic wave through an acousto-optic effect, by a low-frequency electric ﬁeld through an electro-optic effect, or by a periodic optical intensity pattern through optical interference. 10.3 DIRECT MODULATION .............................................................................................................. The most straightforward way to encode information on an optical wave is to directly modulate the optical source. This technique is often applied to an LED or a semiconductor laser, both of which are current-injection devices driven by current sources. Therefore, an LED or a semiconductor laser can be directly modulated by applying the modulation signal to the injection current, an approach known as direct current modulation. In this approach, the modulation signal takes the form of a modulating current, which is added to the DC bias current that supplies electrical power to the device. Figure 10.4 shows the schematic circuitry of direct current modulation. The LED or semiconductor laser is biased at a DC injection current level of I 0 and is modulated with a time-varying modulation current of I m ðt Þ that carries the modulation signal. Thus the total current injected into the device is I ðtÞ ¼ I 0 þ I m ðtÞ. The output optical power is Pout ðt Þ ¼ P0 þ Pm ðt Þ, where P0 is the constant output optical power at the bias current level of I 0 and Pm ðt Þ is the time-varying component of the modulated output optical power responding to the modulation current I m ðt Þ. Though the circuitry for direct modulation is the same for an LED and a semiconductor laser, the characteristics of their modulation responses are very different. LEDs and semiconductor lasers are both junction diodes that usually have sophisticated structures for improved performance. In operation as a light source, an LED or semiconductor laser is injected with a current of I to inject excess electrons and holes, i.e., excess charge carriers, into an active region of an area A and a thickness d. Taking into consideration the injection efﬁciency of the charge carriers, the current density J that actually contributes to carrier injection is related to the total current that is supplied to the device as J ¼ ηinj I , A (10.22) where ηinj is the carrier injection efﬁciency, which is determined by the device structure. The injected current creates an excess carrier density of N ¼ n n0 ¼ p p0 in the active region, where n0 and p0 are, respectively, the equilibrium electron and hole concentrations in the absence of current injection, and n and p are the electron and hole concentrations under current injection. Figure 10.4 Schematic circuitry of direct current modulation on an LED or semiconductor laser. A resistance in series with the device is normally used to protect the device. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.3 Direct Modulation 309 The excess carriers recombine through radiative and nonradiative mechanisms with a total spontaneous carrier recombination rate of γs and a corresponding spontaneous carrier recombination lifetime of τ s : γs ¼ 1 : τs (10.23) The output optical power of an LED is contributed by the spontaneous emission from spontaneous radiative recombination of the excess carriers. By contrast, the output optical power of a semiconductor laser comes from the resonant optical ﬁeld undergoing stimulated emission in the laser cavity. A semiconductor laser has a threshold for laser oscillation, but an LED does not have a turn-on threshold. These fundamental differences lead to very different modulation characteristics between an LED and a semiconductor laser, as discussed below. Direct current modulation on an LED or a semiconductor laser is a technique of amplitude modulation because its objective is the modulation of the output optical power. However, the time-varying current also causes the refractive index of the LED or laser material to vary with time; consequently, the phase and frequency of the output optical wave are also varied by the modulation current. The consequence is an accompanying phase and frequency modulation that is generally undesirable and difﬁcult to avoid because of the nonlinearity and dispersion in the variation of the refractive index in response to the modulation current. The temporal variation in the optical frequency results in frequency chirping in the modulated output optical wave. This effect is more signiﬁcant for direct current modulation on a semiconductor laser than on an LED. 10.3.1 Light-Emitting Diode An LED converts electrical energy to optical energy through the spontaneous emission resulting from spontaneous recombination of the excess carriers. Because spontaneous emission occurs whenever carriers are excited, an LED starts to emit light once current is injected, i.e., there is no threshold to turn an LED on. Therefore, the output optical power Pout is directly proportional to AdN=τ s , which is the total number of excess carriers recombining per second, and can be expressed as Pout ¼ ηe hvAd N, ηinj τ s (10.24) where ηe is the external quantum efﬁciency, ηinj is the carrier injection efﬁciency, both dependent on the structure of the LED, and hv is the photon energy. The temporal variation of the carrier density in response to the variation in the injection current I is described as ηinj dN J N N I , ¼ ¼ dt ed τ s eAd τs (10.25) where e is the electronic charge and J is the injection current density given in (10.22). The output optical power of an LED as a function of the injection current is known as the light–current characteristics, or simply the L–I characteristics, also called the power–current Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 310 Optical Modulation characteristics, or simply the P–I characteristics. The steady-state solution of (10.25) for N obtained by setting dN=dt ¼ 0 results in the ideal power–current relation for an LED in steadystate operation under DC current injection: Pout ¼ ηe hv I, e (10.26) which indicates that the output power of an LED increases linearly with the injection current. The L–I characteristics of a representative LED, shown in Fig. 10.5, are not exactly linear throughout the entire range of operation, however. These characteristics have several important features that distinguish an LED from a laser. First, there is no threshold in the L–I characteristics of an LED, indicating that an LED is turned on and starts emitting light once it is forward biased and injected with any amount of current. At moderate current levels, the L–I curve of an LED is indeed quite linear, as indicated by (10.26). This linearity is useful for analog modulation of an LED. Nonlinearities in the L–I relationship are usually found at very low and very high current levels. For high-speed applications, a large modulation bandwidth is desired. The intrinsic speed of an LED is primarily determined by the lifetime of the injected carriers in the active region. For an LED that is biased at a DC injection current level of I 0 and is modulated at a frequency of Ω ¼ 2πf with a modulation index of m, we can express the total time-dependent current that is injected to the LED as I ðtÞ ¼ I 0 þ I m ðtÞ ¼ I 0 ð1 þ m cos Ωt Þ ¼ I 0 þ mI 0 cos Ωt, Figure 10.5 Light–current characteristics and direct current modulation of a representative LED. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 (10.27) 10.3 Direct Modulation 311 where I m ðt Þ ¼ mI 0 cos Ωt is the modulation current signal, which has an amplitude of I m ¼ mI 0 . The time-varying output optical power Pout ðt Þ of an LED in response to this modulation can be found by using (10.24) after solving for N ðt Þ from (10.25). Note that the time-varying Pout ðt Þ cannot be found directly from (10.26) because (10.26) is valid only for the steady-state operation of an LED when it is only injected with a DC current. In the linear response regime under the condition that m 1, the output optical power can be expressed as Pout ðt Þ ¼ P0 þ Pm ðt Þ ¼ P0 ½1 þ jr j cos ðΩt φÞ, (10.28) where P0 is the constant output optical power found from (10.26) at the bias current level of I 0 , Pm ðt Þ ¼ jr jP0 cos ðΩt φÞ is the time-varying component of the modulated output power, jr j is the magnitude of the response to the modulation, and φ is the phase delay of the response to the modulation signal. The characteristics of direct current modulation on an LED are illustrated in Fig. 10.5. For an LED that is modulated in the linear response regime, the complex response as a function of the modulation frequency Ω is r ðΩÞ ¼ jr ðΩÞjeiφðΩÞ ¼ m : 1 iΩτ s (10.29) The frequency response and the modulation bandwidth of an LED are usually measured in terms of the electrical power spectrum using a broadband, high-speed photodetector that converts the output optical power of the LED into an output electrical current of the photodetector. In the linear operating regime of the detector, the detector current is linearly proportional to the optical power of the LED. Therefore, the electrical power spectrum of the detector output is proportional to jr j2 : Rðf Þ ¼ jr ðf Þj2 ¼ m2 m2 ¼ , 1 þ 4π 2 f 2 τ 2s 1 þ f 2 =f 23dB (10.30) which has a 3-dB modulation bandwidth of f 3dB ¼ 1 , 2πτ s (10.31) as shown in Fig. 10.6. The spontaneous carrier lifetime τ s is normally on the order of a few hundred nanoseconds to 1 ns for an LED. Therefore, the modulation bandwidth of an LED is typically in the range of a few megahertz to a few hundred megahertz. A modulation bandwidth up to 1 GHz can be obtained with a reduction in the internal quantum efﬁciency of an LED by reducing the carrier lifetime to the subnanosecond range. Aside from this intrinsic response speed determined by the carrier lifetime, the modulation bandwidth of an LED can be further limited by the parasitic effects from its electrical contacts and packaging, as well as from its driving circuitry. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 312 Optical Modulation Figure 10.6 Normalized current-modulation frequency response of an LED measured in terms of the electrical power spectrum using a photodetector. The spontaneous carrier lifetime is taken to be τ s ¼ 10 ns for this plot. EXAMPLE 10.3 An LED emitting at a center wavelength of λ ¼ 850 nm has an external quantum efﬁciency of ηe ¼ 21%. Its spontaneous carrier lifetime is τ s ¼ 10 ns. The LED is biased at a DC injection current of I 0 ¼ 20 mA and is modulated at a modulation frequency of f ¼ 10 MHz with a modulation current for a modulation index of m ¼ 10%. (a) Find the output power of the LED at the DC bias point. (b) What is the amplitude of the modulation current? (c) What are the amplitude of the modulated output power and the phase delay of the response to the current modulation? (d) Find the 3-dB modulation bandwidth of this LED in terms of its modulation response in the electrical power spectrum of the photodetector output. (e) At this modulation frequency, what is the modulation response in the electrical power spectrum of the photodetector used to measure the LED output? What is the normalized modulation response in dB? Solution: An LED has no threshold. Therefore, the DC output power is directly proportional to its DC bias current I 0 , and the modulation index is deﬁned as the ratio of the amplitude I m of the modulation current to I 0 . (a) The photon energy at λ ¼ 850 nm is 1239:8 eV ¼ 1:46 eV: 850 The DC output power of the LED is found using (10.26): hv ¼ hv I 0 ¼ 0:21 1:46 20 mW ¼ 6:13 mW: e (b) The amplitude of the modulation current for m ¼ 10% is P0 ¼ ηe I m ¼ mI 0 ¼ 10% 20 mA ¼ 2 mA: Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.3 Direct Modulation 313 (c) From (10.29), we ﬁnd m 0:1 ﬃ ¼ 8:47 102 , jr j ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 6 9 2 1 þ ð2πf τ s Þ 1 þ 2π 10 10 10 10 φ ¼ tan1 ð2πf τ s Þ ¼ tan1 2π 10 106 10 109 ¼ 0:56 rad: Note that it is always true that jrj < m for an LED at a nonzero modulation frequency. The amplitude of the modulated output power is Pm ¼ jrjP0 ¼ 8:47 102 6:13 mW ¼ 519 μW: and the phase delay of the modulation response is φ ¼ 0:56 rad. (d) The 3-dB modulation bandwidth of this LED is, from (10.31), f 3dB ¼ 1 1 ¼ Hz ¼ 15:9 MHz, 2πτ s 2π 10 109 as seen in Fig. 10.6. (e) At the modulation frequency of f ¼ 10 MHz, the modulation response in the electrical power spectrum of the photodetector output is, from (10.30), Rðf Þ ¼ m2 0:12 ¼ ¼ 7:2 103 : 1 þ f 2 =f 23dB 1 þ ð10=15:9Þ2 Because Rð0Þ ¼ m2 ¼ 1 102 , the normalized response is 10 log Rðf Þ 7:2 103 ¼ 1:43 dB: ¼ 10 log Rð0Þ 1 102 10.3.2 Semiconductor Laser For most applications, it is desired that a semiconductor laser oscillate in a single transverse mode and a single longitudinal mode. Many practical lasers indeed have such a desirable characteristic. For a single-mode semiconductor laser that is injected with a current of I, the temporal characteristics of its carrier density N and its intracavity photon density S can be described by the coupled rate equations: ηinj dN J N N I gS, ¼ gS ¼ eAd dt ed τ s τs (10.32) dS ¼ γc S þ ΓgS, dt (10.33) where e is the electronic charge, τ s is the spontaneous carrier lifetime, γc is the cavity decay rate, J is the injection current density deﬁned in (10.22), and g is the gain parameter of the gain region deﬁned in (9.18). The overlap factor Γ appears in the gain term of (10.33) because only Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 314 Optical Modulation that fraction of the laser mode volume overlaps with the gain region to receive stimulated ampliﬁcation. The threshold condition for a semiconductor laser is that in (9.20) for any laser: Γgth ¼ γc : (10.34) The gain parameter g is a function of the excess carrier density N, which in turn is determined by the injection current I. The threshold gain parameter gth determines a threshold carrier density N th at a threshold current density of J th that is supplied by a threshold injection current of I th . The characteristics of a semiconductor laser in steady-state oscillation above threshold can be obtained from the steady-state solutions of (10.32) and (10.33) by setting dN=dt ¼ dS=dt ¼ 0. It is found that in steady-state oscillation above threshold at an injection current of I > I th , the carrier density and the gain are clamped at their respective threshold values, N ¼ N th and g ¼ gth , while the intracavity photon density builds up for S 6¼ 0. Most of the concepts developed in Section 9.4 for laser power characteristics are directly applicable to a semiconductor laser. By directly applying the steady-state conditions of g ¼ gth ¼ γc =Γ and N ¼ N th ¼ J th τ s =ed ¼ ðηinj τ s =edAÞI th to (10.32) to obtain the steady-state solution of S for dS=dt ¼ 0, followed by using the relation J ¼ ðηinj =AÞI from (10.22) and the relation dA ¼ V gain ¼ ΓV mode , the CW output power of a semiconductor laser in steady-state oscillation under DC current injection can be found using (9.29) and can be expressed as a function of the injection current: Pout ¼ ηinj γout hv hv ðI I th Þ ¼ ηe ðI I th Þ, γc e e (10.35) where ηe ¼ ηinj γout =γc is the external quantum efﬁciency of the semiconductor laser. Figure 10.7 shows the power–current characteristics, i.e., the light–current characteristics, of a representative semiconductor laser. It can be seen from (10.35) that in an ideal situation, the output power of a semiconductor laser above threshold increases linearly with the injection current. This characteristic is indeed observed in most semiconductor lasers over a large range of operating conditions. This linearity is useful for analog modulation of a semiconductor laser over a large dynamic range. Nonlinearities in the L–I characteristics appear at high injection current levels. Like an LED, a semiconductor laser can be directly current modulated. Unlike an LED, however, the modulation speed of a semiconductor laser is not limited by the spontaneous carrier lifetime τ s in the active region of the laser. This difference is due to the fact that there is strong coupling between the carriers and the intracavity laser ﬁeld. The effective lifetime of the carriers in an oscillating laser is much shorter than the spontaneous lifetime because of the stimulated carrier recombination that takes place in a laser. The modulation speed of a semiconductor laser is primarily determined by the intracavity photon lifetime and the effective carrier lifetime. Because both the photon lifetime and the effective carrier lifetime of a semiconductor laser are generally much shorter than the spontaneous carrier lifetime, a semiconductor laser has a higher modulation speed than an LED. Because the stimulated recombination rate increases with the intracavity photon density, the modulation speed of a semiconductor laser increases with the laser power. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.3 Direct Modulation 315 Figure 10.7 Light–current characteristics and direct current modulation of a representative semiconductor laser. When a laser is in steady-state oscillation at a DC bias injection current of I 0 > I th in the absence of modulation, the laser gain and the carrier density are both clamped at their respective threshold values of gth and N th , but the photon density has a value of S0 corresponding to the laser output power P0 , which depends on the injection current at the bias point. Under the dynamical perturbation of a modulation signal, the gain can deviate from gth due to the variations in the carrier and photon densities caused by the external perturbation. To the ﬁrst order, the dependence of the gain parameter on the carrier and photon densities can be expressed as g ¼ gth þ gn ðN N th Þ þ gp ðS S0 Þ, (10.36) where gn is the differential gain parameter characterizing the dependence of the gain parameter on the carrier density and gp is the nonlinear gain parameter characterizing the effect of gain compression due to the saturation of the gain by intracavity photons. It has been found empirically that for a given laser, both gn and gp stay quite constant over large ranges of carrier density and photon density. For most practical purposes, they can be treated as constants over the operating range of a laser. These parameters are normally measured experimentally though they can also be calculated theoretically. Note that gn > 0 but gp < 0. It is convenient to deﬁne a differential carrier relaxation rate, γn , and a nonlinear carrier relaxation rate, γp , as γn ¼ gn S0 , γp ¼ Γgp S0 : (10.37) In addition, we have the cavity decay rate, γc ¼ 1=τ c , and the spontaneous carrier relaxation rate, γs ¼ 1=τ s . These four relaxation rates can be directly measured for a given semiconductor Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 316 Optical Modulation laser. They determine the current modulation characteristics of a laser. Note that, for a given laser, γc and γs are constants that are independent of the laser power, but γn and γp are linearly proportional to the laser power because they are linearly proportional to the photon density, as seen in (10.37). Because a semiconductor laser has a threshold, the modulation index m for a laser that is biased at a DC injection current of I 0 > I th and is modulated at a frequency of Ω ¼ 2πf is deﬁned as I ðt Þ ¼ I 0 þ I m ðt Þ ¼ I th þ ðI 0 I th Þð1 þ m cos Ωt Þ ¼ I 0 þ mðI 0 I th Þ cos Ωt, (10.38) where I m ðt Þ ¼ mðI 0 I th Þ cos Ωt is the modulation current signal, which has an amplitude of I m ðtÞ ¼ mðI 0 I th Þ. Note that the modulation index deﬁned in (10.38) for a semiconductor laser is different from that deﬁned in (10.27) for an LED because a laser has a threshold but an LED does not have a threshold. In the regime of linear response, the output power of the laser can be expressed in the same form as that in (10.28) of a directly modulated LED: Pout ðt Þ ¼ P0 þ Pm ðt Þ ¼ P0 ½1 þ jr j cos ðΩt φÞ: (10.39) The constant output power P0 corresponding to the DC bias current I 0 can be found from (10.35). However, the time-varying output power Pout ðt Þ cannot be found directly from (10.35) because the relation in (10.35) is valid only for the steady-state CW oscillation of a laser that is injected with a DC current. When the injection current is temporally modulated, the timevarying output optical power of the laser in response to the modulation can be found by using the relation Pout ðt Þ ¼ γout hvV mode Sðt Þ given in (9.29) after solving for the time-varying photon density SðtÞ from the coupled equations given in (10.32) and (10.33). For small-signal modulation of m 1, the complex response function of a laser is r ðΩÞ ¼ jrðΩÞjeiφðΩÞ ¼ mγc γn , Ω Ω2r þ iΩγr 2 (10.40) where Ωr is the relaxation resonance frequency and γr is the total carrier relaxation rate for the relaxation oscillation of the coupling between the carriers and the intracavity laser ﬁeld of the semiconductor laser. They are related to the intrinsic dynamical parameters of the laser as Ω2r ¼ 4π 2 f 2r ¼ γc γn þ γs γp (10.41) γr ¼ γs þ γn þ γp : (10.42) and Because γc and γs are constants while γn and γp are linearly proportional to the laser power, Ωr and f r are proportional to the square root of the laser power, whereas γr is a linear function of, but not proportional to, the laser power. The relation between the relaxation resonance frequency and the carrier relaxation rate is often characterized by a K factor that is independent of the laser power: K¼ γr γs : f 2r Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 (10.43) 10.3 Direct Modulation 317 Figure 10.8 Normalized current-modulation frequency response of a semiconductor laser measured in terms of the electrical power spectrum using a photodetector. The frequency response of a semiconductor laser depends on the output laser power, with its 3-dB bandwidth increasing approximately with the square root of the output pﬃﬃﬃﬃﬃﬃﬃﬃ power. These curves are generated with the relations: f r ðG H zÞ ¼ 5 Pout and γs ðns1 Þ ¼ 1:5 þ 11 Pout , where Pout is measured in mW. The modulation power spectrum of a semiconductor laser is Rðf Þ ¼ jr ðf Þj2 ¼ m2 γ2c γ2n 16π 4 f 2 f 2r 2 þ 4π 2 f 2 γ2r : (10.44) As shown in Fig. 10.8, this spectrum has a resonance peak at f pk ¼ f 2r γ2 r2 8π 1=2 (10.45) and a 3-dB modulation bandwidth of f 3dB ¼ 1 þ pﬃﬃﬃ 2 1=2 γ2 f 2r pﬃﬃrﬃ 8 2π 2 1=2 1:554 f pk : (10.46) 1=2 Because f r γr =2π for most lasers and because f r / P0 , the modulation bandwidth of a 1=2 semiconductor laser increases with the output laser power and scales roughly as f 3dB / P0 . An intrinsic modulation bandwidth on the order of a few gigahertz is common for a semiconductor laser. A high-speed semiconductor laser can have a bandwidth larger than 20 GHz. Because the intrinsic modulation bandwidth of a semiconductor laser is signiﬁcantly larger than that of an LED, it is very important to reduce the parasitic effects from electrical contacts and packaging for high-frequency modulation of a semiconductor laser. EXAMPLE 10.4 A semiconductor laser emitting at λ ¼ 850 nm has a current injection efﬁciency of ηinj ¼ 60% and an output coupling rate of γout ¼ 5:7 1010 s1 . Its spontaneous carrier lifetime is τ s ¼ 6:67 ns. It has a cavity decay rate of γc ¼ 2 1011 s1 , a differential carrier relaxation rate of γn ¼ 4:9P0 109 s1 , and a nonlinear carrier relaxation rate of γp ¼ 6:1P0 109 s1 , where Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 318 Optical Modulation P0 is the laser output power measured in mW. The laser has a threshold current of I th ¼ 12 mA. It is biased at a DC injection current of I 0 ¼ 28 mA and is modulated with a modulation current at a modulation frequency of f ¼ 10 GHz and a modulation index of m ¼ 10%. (a) Find the output power of the laser at the DC bias point. (b) What is the amplitude of the modulation current? (c) Find the relaxation resonance frequency f r and the total carrier relaxation rate γr of this laser at this operating point. What is the value of the K factor? (d) What are the amplitude of the modulated output power and the phase delay of the response to the current modulation? (e) Find the 3-dB modulation bandwidth of this laser at this operating point in terms of its modulation response in the electrical power spectrum of the photodetector output. (f) At this modulation frequency, what is the modulation response in the electrical power spectrum of the photodetector used to measure the laser output? What is the normalized modulation response in dB? Solution: A laser has a threshold. Therefore, the DC output power is not proportional to its DC bias current but is proportional to I 0 I th , and the modulation index is deﬁned as the ratio of the amplitude I m of the modulation current to I 0 I th . (a) The photon energy at λ ¼ 850 nm is hv ¼ 1239:8 eV ¼ 1:46 eV: 850 The DC output power of the laser is found using (10.35): P0 ¼ ηinj γout hv 5:7 1010 ðI 0 I th Þ ¼ 0:6 1:46 ð28 12Þ mW ¼ 4:0 mW: γc e 2 1011 (b) The amplitude of the modulation current for m ¼ 10% is I m ¼ mðI 0 I th Þ ¼ 10% ð28 12Þ mA ¼ 1:6 mA: (c) With τ s ¼ 6:67 ns, γc ¼ 2 1011 s1 , γn ¼ 4:9P0 109 s1 , and γp ¼ 6:1P0 109 s1 given, and P0 ¼ 4:0 mW found in (a), we have 9 1 11 1 10 1 10 1 γs ¼ τ 1 s ¼ 1:5 10 s , γc ¼ 2 10 s , γn ¼ 1:96 10 s , γp ¼ 2:44 10 s : Therefore, using (10.41) and (10.42), we ﬁnd fr ¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ γc γn þ γs γp ¼ 10 GHz, 2π γr ¼ γs þ γn þ γp ¼ 4:55 1010 s1 : The K factor is found using (10.43): K¼ γr γs 4:55 1010 1:5 109 ¼ s ¼ 440 ps: 2 f 2r 10 109 (d) For a modulation frequency of f ¼ 10 GHz, we ﬁnd that f ¼ f r , thus Ω ¼ Ωr , because f r ¼ 10 GHz as found in (c). Therefore, from (10.40), we ﬁnd Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.4 Refractive External Modulation jr j ¼ 319 mγc γn mγc γn 0:1 2 1011 1:96 1010 ¼ ¼ ¼ 1:37 101 , Ωγr 2πf γr 2π 10 109 4:55 1010 φ ¼ π tan1 Ωγr Ω Ω2r 2 Ωγr 0 ¼ π tan1 ¼ π rad: 2 Note that jr j > m at this modulation frequency because of the response enhancement from relaxation resonance in a semiconductor laser. The amplitude of the modulated output power is Pm ¼ jr jP0 ¼ 1:37 101 4 mW ¼ 548 μW, and the phase delay of the modulation response is φ ¼ π=2 rad: (e) The 3-dB modulation bandwidth of this laser is, from (10.46), f 3dB pﬃﬃﬃ ¼ 1þ 2 1=2 pﬃﬃﬃ ¼ 1þ 2 1=2 ¼ 14 GHz f 2r γ2 pﬃﬃrﬃ 8 2π 2 1=2 45:52 102 pﬃﬃﬃ 8 2π 2 1=2 GHz, as seen in Fig. 10.8 from the 4 mW curve. (f) At the modulation frequency of f ¼ 10 GHz ¼ f r , the modulation response in the electrical power spectrum of the photodetector output is, from (10.44), Rðf Þ ¼ m2 γ2c γ2n ¼ jr ðf Þj2 ¼ 1:37 101 2 2 2 4π f γr 2 ¼ 1:88 102 : From (10.46), we have Rð0Þ ¼ m2 γ2c γ2n , 16π 4 f 4r Therefore, for f ¼ 10 GHz ¼ f r , we ﬁnd that Rðf Þ 4π 2 f 2r 4π 2 f 2r 4π 2 10 109 ¼ 2 2 ¼ 2 ¼ 2 γr Rð0Þ f γr 4:55 1010 2 ¼ 1:91, and the normalized response is 10 log Rðf Þ ¼ 10 log1:91 ¼ 2:8 dB: Rð0Þ 10.4 REFRACTIVE EXTERNAL MODULATION .............................................................................................................. The basic principle of refractive modulation is to modulate the real part of a principal dielectric constant, thus modulating the corresponding principal refractive index, of an optical medium. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 320 Optical Modulation The direct effect is phase modulation on an optical wave that propagates through the medium. Modulating the real part of a dielectric constant also changes the imaginary part because the real and imaginary parts are intrinsically related through the Kramers–Kronig relations. This effect leads to undesirable amplitude modulation that appears as a side effect, which can be minimized by operating the modulator at an optical carrier frequency that is far away from the transition resonance frequencies of the material. For this reason, refractive modulation is generally performed using a material that has little absorption in the spectral region of the modulated optical wave. As discussed in Section 10.2, any other form of optical modulation can be accomplished through phase modulation followed by properly manipulating the phasemodulated optical wave. Refractive modulation through varying the principal refractive indices usually causes differential changes in the principal normal modes of polarization, resulting in induced linear or circular birefringence, which can be applied to polarization modulation. The induced birefringence that is desired for a speciﬁc polarization modulation can usually be achieved by properly choosing the parameters of the optical wave and the material. Therefore, polarization modulation can often be directly accomplished through proper refractive modulation without indirectly manipulating a phase-modulated wave. In principle, any physical mechanism that can cause a change in the refractive index of an optical medium can be used for refractive modulation. Refractive modulation is most often implemented through electro-optic modulation using the Pockels effect. It is also implemented through magneto-optic modulation using the Faraday effect, through acousto-optic modulation using Bragg diffraction, or through all-optical modulation using the optical-ﬁeld-induced birefringence caused by the third-order nonlinear optical susceptibility. The concepts of these physical mechanisms are discussed in Sections 2.6 and 2.7. The principles of refractive modulation based on these physical mechanisms are discussed in the following. 10.4.1 Electro-optic Modulation Practical electro-optic modulators are based on the Pockels effect, which is the ﬁrst-order electrooptic effect, though it exists only in noncentrosymmetric crystals, as discussed in Section 2.6. The electro-optic Kerr effect, being a second-order effect, is relatively weak, and thus not practically useful, though it exists in all materials. As seen in Section 2.6, depending on the direction and the magnitude of the applied electric ﬁeld with respect to the principal axes of the crystal, the linear birefringence induced by the Pockels effect results in changes in the principal indices that might or might not be accompanied by a rotation of the principal axes. An electro-optically induced rotation of the principal axes is not required for the functioning of an electro-optic modulator though it often accompanies the index changes. However, the directions of the principal axes in the presence of an applied electric ﬁeld, whether rotated or not, have to be taken into consideration in the design and operation of an electro-optic modulator. For simplicity without loss of the general concept, we consider in the following a special case where the electro-optically induced linear birefringence causes only index changes without rotating the principal axes. We consider the LiNbO3 crystal, which is the most well-known electro-optic crystal. LiNbO3 is a negative uniaxial crystal of principal indices nx ¼ ny ¼ no > nz ¼ ne . Because of its 3m symmetry, the r αk matrix deﬁned in (2.60) for its Pockels coefﬁcients has only eight Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.4 Refractive External Modulation 321 nonvanishing elements with four independent values: r13 ¼ r 23 , r 12 ¼ r 61 ¼ r 22 , r 33 , and r42 ¼ r 51 . In order that the electro-optically induced linear birefringence changes only the values of the principal refractive indices without rotating the principal axes, the electric ﬁeld is applied along the optical axis such that E0x ¼ E 0y ¼ 0 but E 0z 6¼ 0. In this case, the changes caused by the Pockels effect are found from (2.60) to be Δη1 ¼ Δη2 ¼ r13 E 0z , Δη3 ¼ r33 E 0z , and Δη4 ¼ Δη5 ¼ Δη6 ¼ 0, which can be expressed as Δηxx ¼ Δηyy ¼ r 13 E 0z , Δηzz ¼ r33 E 0z , and Δηij ¼ 0 for i 6¼ j by applying the index contraction rule given in (2.59). By using (2.62) and (2.63), the ﬁeld-dependent dielectric permittivity tensor can be found: 0 2 1 no n4o r 13 E 0z 0 0 A: (10.47) ϵ ðE 0 Þ ¼ ϵ 0 @ 0 0 n2o n4o r13 E 0z 2 4 0 0 ne ne r 33 E 0z ^ ¼ ^x , Y^ ¼ ^y , and Z^ ¼ ^z . The crystal remains Clearly, the principal axes are not rotated: X uniaxial with the same optical axis, but the indices of refraction are changed. Since the induced 2 changes are generally so small that jr 13 E 0z j n2 o and jr 33 E 0z j ne , the new principal indices of refraction can be expressed as nX ¼ nY no n3o r 13 E 0z , 2 nZ ne n3e r33 E 0z : 2 (10.48) The phase of an optical wave can be electro-optically modulated. For this type of application, the optical wave is linearly polarized in a direction that is parallel to one of the principal axes, ^ Y^ , or Z^ , of the crystal that is subjected to a modulation ﬁeld. The preferred choice is a X, principal axis that has a large electro-optically induced index change but remains in a ﬁxed direction as the magnitude of the modulation electric ﬁeld varies. In LiNbO3, this can be accomplished by applying the electric ﬁeld along the z axis, as discussed above and shown in Figure 10.9. There are two possible arrangements: transverse modulation, which has the modulation ﬁeld perpendicular to the direction of optical wave propagation, as shown in Fig. 10.9(a), and longitudinal modulation, which has the modulation ﬁeld parallel to the direction of optical wave propagation, as shown in Fig. 10.9(b). Figure 10.9 (a) LiNbO3 transverse electro-optic phase modulator for an optical wave propagating in the X direction. (b) LiNbO3 longitudinal electro-optic phase modulator for an optical wave propagating in the Z direction. In both cases, the modulation ﬁeld is applied in the Z direction. The ^x , ^y , and ^z unit vectors represent the original principal axes of the crystal, and X^ , Y^ , and Z^ represent its new principal axes in the presence of the modulation voltage. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 322 Optical Modulation Transverse Phase Modulation We ﬁrst consider the situation of the transverse phase modulator shown in Fig. 10.9(a), where the optical wave propagates in the X direction. In this case, the optical wave can be polarized in either the Y or Z direction. If it is linearly polarized in the Z direction, its space and time dependence can be written as EðX; tÞ ¼ Z^ E exp ik Z X iωt ¼ Z^ E exp ðiφz iωtÞ: (10.49) For propagation through a crystal that has a length of l, the total phase shift is φZ ¼ kZ l ¼ ω ω n3 nZ l ¼ ne l e r 33 E 0z l 2 c c ¼ ω n3 l ne l e r 33 V , 2 c d (10.50) where V ¼ E 0z d is the voltage applied to the modulator shown in Fig. 10.9(a). For sinusoidal modulation of a modulation frequency f ¼ Ω=2π, the modulation voltage can be written as V ðt Þ ¼ V m sin Ωt, (10.51) which has a modulation amplitude of V m . The Z-polarized optical ﬁeld at the output plane, X ¼ l, of the crystal is phase modulated: Eðl; tÞ ¼ Z^ Eeiωne l=c exp½iðωt þ φm sin ΩtÞ, (10.52) ω n3e l πn3 l Vm π r 33 V m ¼ e r 33 V m ¼ λ c 2 d d Vπ (10.53) where φm ¼ is the peak modulated phase shift, known as the phase modulation depth, and Vπ ¼ λ d l (10.54) n3e r33 is the modulation voltage that is required for a phase shift of π, known as the half-wave voltage, also denoted as V λ=2 . If the optical ﬁeld is instead linearly polarized in the Y direction, the phase shift after propagation through the crystal is φY ¼ kY l ¼ ω ω n3 nY l ¼ no l o r13 E 0z l 2 c c ¼ ω n3 l no l o r 13 V : 2 c d (10.55) The phase modulation depth for the modulation voltage given in (10.51) is then φm ¼ ω n3o l πn3 l Vm r 13 V m ¼ o r13 V m ¼ π, λ c 2 d d Vπ (10.56) where the half-wave voltage for this arrangement is Vπ ¼ λ d : l n3o r 13 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 (10.57) 10.4 Refractive External Modulation 323 Because no ne but r 33 3:6r13 , it can be seen by comparing (10.57) with (10.54) that for a desired modulation depth, the modulation voltage required for a Y-polarized optical wave is about 3.6 times that for a Z-polarized wave. Longitudinal Phase Modulation For the longitudinal phase modulator shown in Fig. 10.9(b), an optical wave of any polarization in the XY plane experiences the same amount of phase shift because nX ¼ nY ¼ no . For a crystal of a length l as shown in Fig. 10.9(b), we have φX ¼ φ Y ¼ ω n3 no l o r 13 E 0z l 2 c ¼ ω n3 no l o r 13 V 2 c (10.58) where V ¼ E 0z l for the longitudinal modulator. For a sinusoidal modulation voltage as given in (10.51), the modulation depth of the longitudinal phase modulator is φm ¼ ω n3o πn3 Vm π, r 13 V m ¼ o r 13 V m ¼ λ Vπ c 2 (10.59) where Vπ ¼ λ n3o r 13 : (10.60) Both φm and V π for longitudinal modulation are independent of the crystal length l. It is seen that the voltage required for a given modulation depth is independent of the physical dimensions of the modulator in the case of longitudinal modulation, whereas it is proportional to d=l in the case of transverse modulation. One advantage of transverse modulation is that the required modulation voltage can be substantially lowered by reducing the d=l dimensional ratio of a transverse modulator. Another advantage is that the electrodes of a transverse modulator can be made using standard techniques and can be patterned if desired, while those of a longitudinal modulator have to be made of transparent conductors that can be very difﬁcult, if not impossible, to fabricate in the dimensions of the typical optical waveguide. However, if a large input and output aperture is desired such that d=l > 1, it becomes advantageous to use longitudinal modulation rather than transverse modulation. EXAMPLE 10.5 LiNbO3 is a negative uniaxial crystal, which has nx ¼ ny ¼ no ¼ 2:251 and nz ¼ ne ¼ 2:170 at the λ ¼ 850 nm wavelength. It has eight nonvanishing Pockels coefﬁcients, which are r13 ¼ r23 ¼ 8:6 pm V1 , r 12 ¼ r61 ¼ r 22 ¼ 3:4 pm V1 , r 33 ¼ 30:8 pm V1 , and r 42 ¼ r51 ¼ 28 pm V1 . Consider transverse and longitudinal modulation of an optical wave at λ ¼ 850 nm using a LiNbo3 electro-optic modulator in the conﬁgurations shown in Figs. 10.9 (a) and (b), respectively. The LiNbo3 modulator has the dimensions of l ¼ 3 cm and d ¼ 3 mm. (a) Find the values of the half-wave voltage V π for transverse and longitudinal modulation, respectively, in the case when the optical wave is polarized along the y principal axis. (b) The Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 324 Optical Modulation largest Pockels coefﬁcient is r 33 . If this coefﬁcient can be used, what are the values of V π for transverse and longitudinal modulation, respectively? Solution: In both conﬁgurations shown in Figs. 10.9(a) and (b), the voltage is applied in the direction along the z principal axis. Therefore, the Pockels coefﬁcients that are useful for the modulation are r 13 for x-polarized wave, r23 for y-polarized wave, and r 33 for z-polarized wave. Note that r 13 ¼ r 23 ¼ 8:6 pm V1 and r33 ¼ 30:8 pm V1 . (a) For a y-polarized wave, we use r 23 , which is the same as r 13 . For transverse modulation in this case, the half-wave voltage is that given in (10.57). With l ¼ 3 cm and d ¼ 3 mm, we ﬁnd Vπ ¼ λ d 850 109 3 103 V ¼ 867 V: ¼ n3o r 13 l 2:2513 8:6 1012 3 102 For longitudinal modulation, the half-wave voltage is that given in (10.60): Vπ ¼ λ 850 109 ¼ V ¼ 8:67 kV: n3o r 13 2:2513 8:6 1012 (b) To use r 33 , the optical wave has to be polarized along the z principal axis while the applied voltage has to be in this direction as well. This is possible for transverse modulation but is not possible for longitudinal modulation, as can be seen by examining Figs. 10.9(a) and (b). For transverse modulation on a z-polarized optical wave in this case, the half-wave voltage is that given in (10.54). With l ¼ 3 cm and d ¼ 3 mm, we ﬁnd Vπ ¼ λ d 850 109 3 103 V ¼ 270 V: ¼ n3e r 33 l 2:1703 30:8 1012 3 102 This half-wave voltage is less than one third of that found in (a) for transverse modulation on a y-polarized optical wave because r 33 is more than three times larger than r 23 . Polarization Modulation As discussed in Section 10.2, polarization modulation can be accomplished by differential phase modulation between two orthogonally polarized ﬁeld components. For electro-optic polarization modulation, the optical wave is not linearly polarized in a direction that is parallel to any of the principal axes in the presence of the modulation ﬁeld. The optical ﬁeld can be decomposed into two linearly polarized normal modes. If the two normal modes see different ﬁeld-induced refractive indices, an electric-ﬁeld-dependent phase retardation between the two modes occurs, resulting in a change of the polarization of the optical wave at the output of the crystal. The LiNbO3 transverse modulator discussed above becomes a polarization modulator if the polarization of the input optical ﬁeld is not parallel to Y^ or Z^ so that (10.61) Eð0; t Þ ¼ Y^ E Y þ Z^ E Z eiωt , Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 325 10.4 Refractive External Modulation Figure 10.10 LiNbO3 transverse electro-optic polarization modulator. The ^x , ^y , and ^z unit vectors represent the original principal axes of the crystal, and X^ , Y^ , and Z^ represent its new principal axes in the presence of the modulation voltage. where E Y 6¼ 0 and E Z 6¼ 0, as shown in Fig. 10.10. At the output, we ﬁnd Y Z Y Eðl; tÞ ¼ Y^ E Y eik l þ Z^ E Z eik l eiωt ¼ Y^ E Y þ Z^ E Z eiΔφ eik l eiωt , where 3 π l 3 2ðne no Þl þ no r 13 ne r 33 V Δφ ¼ k k l ¼ λ d Z Y (10.62) (10.63) is the phase retardation between the Y and Z components. The polarization of the output optical ﬁeld can be electro-optically modulated by a modulation electric ﬁeld of E 0z ðt Þ ¼ V ðt Þ=d that causes a time-varying phase retardation of Δφðt Þ following the time-varying voltage V ðtÞ. EXAMPLE 10.6 The phase retardation given in (10.63) between the Y and Z components of the optical ﬁeld for the transverse polarization modulator shown in Fig. 10.10 has a background value that is independent of the applied voltage V because ne 6¼ no . This voltage-independent background phase retardation can be compensated by using a DC bias voltage of V b such that Δφ ¼ Δφb ¼ 2mπ when V ¼ V b . Then (10.63) can be expressed as Δφ ¼ Δφb þ V Vb V Vb π ¼ 2mπ þ π: Vπ Vπ In practice, V b can be adjusted to make sure that Δφb ¼ 2mπ. Find the expression for V π in the above relation. Use the parameters of LiNbO3 given in Example 10.5 to ﬁnd the value of V π at λ ¼ 850 nm for a LiNbO3 polarization modulator of the dimensions of l ¼ 3 cm and d ¼ 3 mm. Solution: The expression for V π can be found by taking Δφ ¼ π while ignoring the voltage-independent background term in (10.63). Thus, we ﬁnd that Vπ ¼ n3o r13 λ d : 3 ne r 33 l Using the parameters given in Example 10.5, we ﬁnd that Vπ ¼ 850 109 3 103 V ¼ 392 V: 2:2513 8:6 1012 2:1703 30:8 1012 3 102 Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 326 Optical Modulation Amplitude Modulation As discussed in Section 10.2, amplitude modulation can be achieved through polarization modulation by properly selecting a polarization component of the polarization-modulated ﬁeld while ﬁltering out its orthogonal component. This can be done by simply placing a polarization modulator between a polarizer at the input end and another polarizer, often referred to as an analyzer, at the output end. The axes of the polarizer and the analyzer are often orthogonally crossed, though other arrangements are possible. Figure 10.11 shows such an arrangement using the LiNbO3 polarization modulator discussed above and shown in Fig. 10.10. Following the discussion in Section 10.2 on polarization modulation and amplitude modulation, here we take ^e þ ^e ⊥ ^e ^e ⊥ Y^ þ Z^ Y^ Z^ ^e ¼ pﬃﬃﬃ , ^e⊥ ¼ pﬃﬃﬃ , ^e 1 ¼ Y^ ¼ pﬃﬃﬃ , ^e 2 ¼ Z^ ¼ pﬃﬃﬃ , (10.64) 2 2 2 2 pﬃﬃﬃ with c1 ¼ c2 ¼ 1= 2. The axis of the input polarizer is along ^e , and that of the output analyzer is along ^e ⊥ , as shown in Fig. 10.11. The polarizer ensures that the input optical wave is linearly polarized in the ^e direction, whereas the analyzer passes only the ^e ⊥ component of the optical wave at the output end. Thus, the input ﬁeld is E Eð0; t Þ ¼ ^e Eeiωt ¼ pﬃﬃﬃ Y^ þ Z^ eiωt : 2 (10.65) Then, from (10.62), the ﬁeld at the end of the crystal is Y Y E E 1 þ eiΔφ ^e þ 1 eiΔφ ^e⊥ eik liωt , Eðl; tÞ ¼ pﬃﬃﬃ Y^ þ Z^ eiΔφ eik liωt ¼ 2 2 (10.66) where Δφ is that given in (10.63). Because the analyzer passes only the ^e ⊥ component of the optical ﬁeld, the transmittance of the amplitude modulator is I out I ⊥ Δφ 1 ¼ sin2 ¼ ¼ ð1 cos ΔφÞ, (10.67) I in I 2 2 pﬃﬃﬃ which agrees with (10.17) for c1 ¼ c2 ¼ 1= 2. Electro-optic amplitude modulation can also be accomplished by varying the coupling or interference between two ﬁelds that have differential phase modulation, as discussed in Section 10.2. This concept can be implemented with many different structures, both in free space and in waveguides. Here we illustrate the concept using a guided-wave electro-optic modulator in the T¼ Figure 10.11 Electro-optic amplitude modulator using two cross polarizers at the input and the output of the LiNbO3 transverse electro-optic polarization modulator shown in Fig. 10.10. Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books Online © Cambridge University Press, 2016 10.4 Refractive External Modulation 327 Figure 10.12 Mach–Zehnder waveguide interferometric modulator using Y junctions fabricated on an x-cut, ypropagating LiNbO3 substrate. form of the Mach–Zehnder waveguide interferometer, shown in Fig. 10.12. This structure uses Y-junction couplers as input and output couplers. It is fabricated in an x-cut, y-propagating LiNbO3 crystal. In the electrode conﬁguration shown in Fig. 10.12, the modulation voltage is applied to the central electrode while the outer electrodes are grounded so that the upper arm sees a modulation ﬁeld of E0z ¼ V=se but the lower arms sees E0z ¼ V=se , where se is the separation between two neighboring electrodes. The modulation electric ﬁelds appearing in the two arms point in opposite directions, resulting in a push–pull operation with equal but opposite phase shifts in the optical waves propagating through the two arms. For an interferometer that has identical arms, any other background phase shifts are exactly canceled. Thus the total phase difference is twice the electro-optically induced phase shift in each arm. If the two arms are identical single-mode waveguides, the phase difference induced by a modulation voltage V is Δφ ¼ π V , Vπ (10.68) where V π is the half-wave voltage for a phase difference of π between the two arms. For a TElike mode, the transverse optical ﬁeld component is primarily the E z component so that Vπ ¼ λ se , 2n3e r 33 ΓTE l (10.69) where ΓTE is the overlap factor that accounts for the overlap between the spatial distributions of the modulation ﬁeld and the TE-like mode ﬁeld. For a TM-like mode, the transverse optical ﬁeld component is primarily the E x component so that Vπ ¼ λ 2n3o r 13 ΓTM se , l (10.70) where ΓTM is the overlap factor that accounts for the overlap between the spatial distributions of the modulation ﬁeld and the TM-like mode ﬁeld. If both input and output Y junctions of the Mach–Zehnder waveguide interferometer are ideal 3-dB couplers, i.e., the input power is split equally between the two arms and the ﬁelds from the two arms are combined equally for the output, the power transmittance due to interference at the output between the ﬁelds coming from the two arms is Downloaded from Cambridge Books Online by IP 131.111.164.128 on Sat Aug 20 20:19:33 BST 2016. http://dx.doi.org/10.1017/CBO9781316687109.011 Cambridge Books