Digital image plane holography as a three-dimensional flow
velocimetry technique
J. Lobera*, N. Andrés, M.P. Arroyo
Dpto. Física Aplicada. Facultad de Ciencias. Universidad de Zaragoza
C/ Pedro Cerbuna 12, 50.009-Zaragoza, SPAIN
ABSTRACT
This paper shows the feasibility of applying digital image plane holography (DIPH) as a fluid velocimetry technique for
simultaneous measurements of all three components of the velocity field. As a first approach DIPH has been set up to
measure a single fluid plane. The recording apparatus is a digital speckle interferometer (DSPI) with spatial phase
shifting (SPS). The speckle interferometer has an out-of-plane sensitivity and the off-axis reference beam produces a
spatial modulation in the pattern (hologram) recorded by the CCD camera. From the interferometric and photographic
analysis of the reconstructed object wave, the three velocity components in the fluid plane are obtained. The complex
amplitude of the object wave is calculated by the application of a Fourier-transform method to the hologram. The phase
change between two subsequent frames yields an out-of-plane component of the velocity field. The two in-plane
components are obtained, as in digital speckle photography, by cross correlation of the reconstructed object wave’s
intensity. Some quantitative results in a Rayleigh-Bénard convective flow are presented. In the final setup, angular
multiplexing with coherence length control has been introduced in order to simultaneously measure the velocity fields
in two fluid planes. Some preliminary results from the convective flow are presented.
Keywords: Digital holography, fluid velocimetry, holographic interferometry, spatial phase shifting.
1
INTRODUCTION
Holographic techniques have been widely used for non-destructive measurement of displacements, deformations and
even vibrations in experimental solid mechanics. All of them are based on recording a hologram, i.e. the interference
between an object beam, that carries the information on the solid object, and its corresponding reference beam. The
reconstruction of the object wave field from the hologram gives full information about the solid object.
In optical holography, the recording is done on photographic films and the object wave is reconstructed by illuminating
the developed film with a beam similar to the reference beam. For many years, optical holography was the only choice
for any application. However, in the last few years, digital holography1 has become very popular. In digital holography,
both the hologram recording and the object wave reconstruction are digital processes. Temporal and spatial phase
shifting speckle pattern interferometry2 can also be included in digital holography.
Many different arrangements for digital hologram recording have already been reported: in-line3, off-axis[4-7], lensless[3, image plane[6-9]. Different numerical reconstruction processes are used depending on the recording arrangement: a
Fourier method7 for the image plane recording, a propagation/diffraction method4 for the lensless recording and an
intensity based algorithm2 for the speckle pattern interferometry arrangement. Interferometric techniques[3-5],7,8 that only
make use of the object wave phase information have been used more often than photographic/imaging techniques6,9
where the important information is carried on the object wave amplitude. Static deformation problems have been
analysed more often than dynamic events.
5]
In the digital holography applications in solid mechanics, the object wave is formed by the light scattered by the solid
surface after being illuminated by an extended laser beam, whose propagation direction has been appropriately chosen.
*jlobera@unizar.es;
phone: 34 976 762605;
fax: 34 976 761233
Speckle Metrology 2003, Kay Gastinger, Ole Johan Løkberg, Svein Winther,
Editors, SPIE Vol. 4933 (2003) © 2003 SPIE . 0277-786X/03/$15.00
279
The object wave phase depends on the observation direction and on the illumination direction while the object wave
intensity depends more on the surface properties.
Applications of laser speckle metrology techniques in fluid mechanics are much less common. Although any solid
deformation/displacement measurement technique could be adapted as a fluid velocity measurement technique, the
practical implementation is more difficult due to the intrinsic dynamic nature of the fluid mechanics problems. Particle
Image Velocimetry (PIV)10, which is now a widely used fluid velocimetry technique, is an example of a very successful
adaptation of a solid deformation measurement technique (speckle photography). In PIV, a thin laser sheet is used to
illuminate the fluid, previously seeded with small tracer particles. The object wave is formed by the light scattered by
the tracers inside the illuminated fluid plane; this fluid plane plays the same role as the solid object surface. Some work
has also been done on adapting some interferometric techniques11 for fluid velocimetry.
This paper will focus in showing the feasibility of digital holography as a fluid velocimetry technique. In this
application, a light sheet illumination for marking a fluid plane and an image plane hologram recording arrangement
will be used. We will start describing the recording setup and the numerical reconstruction for this digital image plane
holography technique (DIPH). Then, the interferometric and photographic analysis used to measure the 3-C velocity
field will be presented together with some results in a convective flow. Finally, the extension of DIPH for measurement
in a 3-D region by means of an angular multiplexed holographic recording will be described.
2
HOLOGRAM RECORDING
The setup for the hologram recording (Fig.1) is typical of a spatial phase shifting – digital speckle pattern interferometer
(SPS-DSPI). As is usual in DSPI, a lens is used to image the object onto the CCD sensor. Specific of this fluid
velocimetry application is the sheet like illumination beam that marks a fluid plane playing the same role as a rough
surface. Specific of this SPS-DSPI setup is the divergent beam that has to be placed in a very precise position respect to
the object beam (see Fig. 1b). Because the divergent reference beam has to originate in the same plane and at a suitable
distance from the lens aperture,
Flow
this SPS-DSPI set-up can be also
Object
beam
u
viewed as an off-axis digital
holography set-up. Because of the
Laser sheet
u
K
small angles allowed by the low
Beam
spatial resolution of CCD sensors,
Reference
splitter
Object beam
beam
the reference and the object beams
are combined by means of a beam
Lens
splitter placed between the lens
Optic fiber
and the CCD camera sensor.
Beam splitter
CCD
b)
Reference beam
i
o
a)
CCD
camera
Fig 1. a) Setup for hologram recording in DIPH ;b) Relative position of the reference and object beams.
The hologram recorded by the CCD sensor has been also referred as specklegram because it is always a speckle field. It
contains information on the amplitude and phase of both the object and the reference beams. As in any interference
pattern between two beams, the specklegram/hologram intensity can be expressed as
I = I o + I r + 2 I o I r cos(φo − φ r ) ,
(1)
where I and φ are the intensity and phase, with o and r referring to the object and the reference waves. Let us note that
φo is a spatially random phase because it changes from one speckle/particle to the next. This φo needs to be locally
constant in a 2-3 pixels area in order for the CCD sensor to be able of recording the spatial frequencies associated to the
hologram recording. The lens aperture is thus chosen to ensure a minimum mean speckle size of 2.4 pixels. This
requirement is common to digital PIV for an appropriate accuracy. The spatial dependence of φr is approximately linear
with the rate of change depending on the angle α subtended by the object and reference beams. Thus, the hologram
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intensity will locally show fringes whose spacing depends on this angle α. If α is too small or
too big, the hologram will look like a random speckle field with no modulation; if α is
appropriate the fringe modulation will be seen (Fig. 2).
Fig 2.Specklegram with spatial phase shifting modulation.
3
NUMERICAL RECONSTRUCTION
In digital holography, the most common reconstruction process is similar to optical holography: the hologram intensity
is multiplied by the reference or its conjugated beam and the output wave is propagated back to either the real or the
virtual image plane. With our recording setup, the hologram behaves as a lensless Fourier transform hologram of the
lens aperture12. Thus the most appropriate numerical reconstruction is based on a Fourier method. The Fourier
transform of the digital hologram (Fig. 3) contains three distinct parts: a dc term and a virtual and a real image of the
lens aperture. The dc term correspond to the Io+Ir terms of eq. 1; the decaying of the intensity of this term is typical of
the speckle statistics due to the Io; the Ir contribution is a delta function in the origin. The two heptagons are the images
of the lens aperture. The other features in this Fourier plane are due to some spurious interference patterns on the
recording and care should be taken not to have any of them over the heptagons. By selecting any of the heptagons and
calculating the inverse Fourier transform we obtain a complex wave whose amplitude and phase approximately
correspond to the amplitude and phase of the original object wave in the image plane (CCD sensor). Only the sign of
the reconstructed phase depends on the selected heptagon. For an appropriate object reconstruction, the heptagons
should neither overlap with each other nor with the dc term. The position of the two heptagons is directly related to the
relative position of the reference and object beams on the hologram recording. In our case, these two beams have been
displaced along a diagonal direction in order to have the lens aperture as big as possible.
Let us note that the amplitude of the reconstructed wave correspond to the object wave
amplitude multiplied by the reference beam amplitude, which do not have any effect if this Ir is
approximately constant. The reconstructed wave phase is the object wave phase plus a small
contribution from the reference due to its spherical geometry. This small phase contribution has
to be taken into account if propagation back to the fluid plane or to any other plane is going to
be done. For fluid velocimetry, we do not have taken it into account.
Fig 3.Fourier transform of fig. 2.
4
4.1
3-C FLUID VELOCIMETRY
Basic principles
For fluid velocimetry, as for solid deformation measurements, two holograms are recorded corresponding to different
object states. In fluid velocimetry, where we always deal with dynamic events, the recordings are taken in different time
instants. The time interval ∆T between the two recordings is an important parameter that is determined for each flow
taking into account the velocity field and the recording geometry. Following eq. 1, the specklegram intensities can be
written as
I1 = I o,1 + I r + 2 I o,1 I r cos(φo ,1 − φ r ) ,
(2)
I 2 = I o,2 + I r + 2 I o,2 I r cos(φo ,1 + ∆φo − φ r ) ,
(3)
with 1 and 2 referring to the first and the second specklegrams and where the (x,y) dependence of all the magnitudes
has been omitted for clarity. Let us note that not only φo but also Io are spatially random magnitudes because the object
is a speckle (or particle image) field. However, for each (x,y) position the change in φo, ∆φo, is not random but related
to the local fluid displacement. This relationship can be expressed as
r r
∆ φo = K⋅ V ∆T ,
(4)
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281
r
r
r r
r
r
where V is the local fluid velocity, and K =(2π/λ)( u o- u i) is the sensitivity vector, u o and u i being unity vectors in the
observation and the illumination directions respectively (see Fig. 1) and λ being the laser wavelength in the fluid.
The changes from Io,1 to Io,2 are negligible when the particle image moves less
r than itsr diameter during ∆T. In this case,
∆φo is the useful quantity that provides information about the projection of V along K , Vk. In fluid velocimetry,
because of the illumination geometry, Vk is always an out of plane velocity component. ∆φo is obtained by subtracting
the reconstructed object wave phases corresponding to the two recorded holograms. When the particle image moves
more than its diameter, Io,1 and Io,2 are the useful quantities.
r By cross correlating the Io,1 and Io,2 reconstructed
amplitudes, as in DSP or PIV, the in-plane displacement d is obtained. The velocity is then calculated as
r r
V = d/M ∆T ,
(5)
being M the imaging system magnification.
Let us note that the three velocity components (3-C velocity field) can be measured by using both the phase and the
amplitude of the numerically reconstructed object wave. For 3-C fluid velocimetry to work with only two hologram
recordings, Vk needs to be much smaller than the in-plane velocity projection due to the interferometric sensitivity for
the measurements of Vk and the photographic sensitivity for the measurement of the in-plane velocity. However, in
general, three holograms with two different ∆T need to be recorded.
4.2
Experimental results
To demonstrate the performance of DIPH in a real flow, we applied it to a Rayleigh-Bénard convective flow13 setup in a
cell filled with an aqueous glycerol solution that was seeded with 5µm diameter latex particles. The dimensions of the
cell were Lx=25 mm, Ly= 25 mm and Lz=12.5 mm. A 17mW He-Ne laser beam was shaped into a 15mm high 1mm
wide light sheet that illuminates an XZ fluid plane. An electromechanical shutter was used to control the exposure time
for the hologram recording and the time interval between holograms. While reference beam energy was never a
problem, a minimum exposure time of about 5 ms was needed for adequate energy on the object wave. The convective
flow velocity was appropriately chosen in the range of 10-20 µm/s so that no time averaging holography was produced.
Pairs of holograms with time interval ∆T1 suitable for the interferometric analysis were recorded at time intervals ∆T2
suitable for the photographic analysis. Figure 4 shows some typical results obtained from a y= 2 mm plane, with
∆T1=80 ms and ∆T2= 8s. Figure 4a presents the phase map obtained from the interferometric analysis; it shows the
spatial pattern of the projection of the velocity vector on a direction at 45º with the X and Y axis.
a)
b)
c)
12
-12
9
18
6
-6
0
6
12
0
6
12
0
-6
-6
-12
e)
18
24
12
0
d)
6
6
f)
Fig 4. 3-C velocity measurements in a convective flow with DIPH. a) phase map from the interferometric analysis; b) reconstructed
intensity field; c) in-plane vector map from the photographic analysis; d) isoline map for Vx; e) isoline map for Vy; f) isoline map for
Vz
Figure 4b presents the intensity field obtained from one hologram; it shows a good image of the cell. Figure 4c presents
the in-plane vector map obtained from the photographic/PIV analysis of two images like the one in fig. 4b. By solving a
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three equation system with the in-plane velocity data of figure 4c and the velocity projection of figure 4a, the three
components of the vector field can be calculated. Figures 4d to 4f presents the results as isoline maps for Vx, Vy and
Vz. Let us note that, as a first approximation, the Vx and the Vz data come directly from the PIV data while Vy data
comes from eliminating from the 45º projection the Vx component. The fact that the spatial pattern of the Vy field is
completely different than Vx and than the phase map demonstrates the accuracy of this interferometric/photographic
analysis for 3-C velocimetry.
5
5.1
3-D FLUID VELOCIMETRY
Basic principles
For 3-D velocimetry, the whole fluid instead of just one plane should be recorded. Although illuminating a whole
volume is possible, recording it properly presents a lot of problems in practice such as noise from out of focus particles.
Furthermore, digital holographic interferometry will not work because the reconstructed wave phase will always contain
information from the whole volume. The approach we have chosen is to record only a limited number of fluid planes,
by keeping the sheet like illumination. This should reduce the noise on the photographic analysis. However, in order
solve the interferometric analysis problems, each fluid plane has to be recorded independently of each other. As shown
by Schedin et al8, several holographic recordings can be angularly multiplexed by using reference beams with point
sources in different position for each recording, and exploiting the beam temporal coherence length properties. For 3-D
fluid velocimetry, each reference beam has to be made coherent with only one fluid plane (fig. 5).
Flow
Laser sheet 2
Laser sheet 1
b)
Lens
Beam splitter
Reference beam 1
Reference beam 2
CCD
camera
c)
a)
Fig 5. a) Setup for a two-plane recording with DIPH. b) Digital hologram. c) Fourier transform of the digital hologram
It can be done using one laser and adjusting the optical path lengths of each reference beam and each light sheet.
Another possibility would be to use an array of lasers so that each light sheet and its corresponding reference beam
comes from a different laser. For demonstration purposes, we show an only two-plane hologram recording setup
(fig.5a). The fluid is illuminated with two laser sheets whose scattered waves will not interfere with each other. Two
reference beams are used. Reference beam 1 interferes only with the object wave 1 while reference beam 2 interferes
only with object wave 2. In a two-plane system, laser polarization properties can also be used to make the two recording
systems non coherent with each other. A typical specklegram (fig.5b) and its corresponding Fourier transform (fig.5c)
are shown. The two reference beams have been displaced along the two diagonals (for maximum lens aperture in the
recording). Let us note that the specklegram and its Fourier transform will look the same no matter if beams 1 and 2 are
coherent with each other or not. It will be on the reconstruction where it will be apparent whether the two object waves
have been recorded independently or not.
5.2
Results
The two-plane hologram recording setup has also been tested on the Rayleigh-Bénard convective flow using the 17 mw
He-Ne laser as the only light source. The laser coherence properties were checked with DIPH. Some holograms with
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one laser sheet and one reference beam were taken for increasingly bigger optical path lengths on the reference beam.
By looking at the intensity of the heptagons (the interference terms) on the Fourier plane, we measured the coherence
length to be minimum for an optical path length of 35 cm. Then, this optical path length difference was introduced
between the recording systems (laser sheet and reference beam) 1 and 2. By taking holograms with the two reference
beams and only one laser sheet the perfect matching of all beams could be checked by looking at the intensity of the
heptagons obtained for the laser sheets 1 and 2.
Figure 6 presents some results obtained from recording two planes at y=6mm and y=19 mm. The phase maps obtained
for each plane are so different that nothing would have been seen without the coherence control. The phase maps are in
agreement with the expected pattern for the toroidal
3-D flow normally appearing in this convective flow.
The intensity field also allowed a PIV analysis.
However, no 3-C quantitative analysis in the twoplane DIPH has been done because both the
interferometric and the PIV data were less accurate
than in the previous setup due to of a too low energy
a)
b)
per fluid plane in the present setup.
Fig 6.Results with DIPH as a 3-D velocimetry technique. Wrapped phase
maps for: a) plane 1 (y= 19 mm); b) plane 2 (y= 6 mm).
6
CONCLUSIONS
An off-axis DIPH setup for fluid velocimetry has been described. Preliminary experiments in a convective flow with a
continuous He-Ne laser have shown the feasibility of measuring 3-C velocity fields in one plane. For 3-D, 3-C velocity
measurements a multiple light sheet illumination with controlled optical path length is proposed. A preliminary setup
with two light sheets has shown the feasibility of independently obtaining information form each fluid plane.
Quantitative measurements are expected to be possible when using a pulsed laser.
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