From ESPI to Digital Image Plane Holography
(DIPH): Requirements, Possibilities and
Limitations for Velocity Measurements in a 3-D
Region.
J. Lobera, N. Andrés and M. P. Arroyo
Dpto. Física Aplicada. Facultad de Ciencias. Universidad de Zaragoza.
C/ Pedro Cerbuna, 12, 50009- Zaragoza. SPAIN.
Abstract
The present work shows how the SPS-ESPI recordings can alternatively be analysed as Digital Image Plane Holographic (DIPH) recordings. When the SPS-ESPI
recordings are analysed using a DIPH analysis, both the phase and intensity can be
independently retrieved at each point of the fluid plane. The three velocity components in a fluid plane could be measured, since the phase maps detect an out-ofplane velocity component while the intensity data can be analysed with standard
PIV methods to detect the two in-plane components. Finally, more than one fluid
plane have been simultaneously recorded but independently reconstructed using
DIPH. Some preliminary results from a convective flow with a He-Ne laser illustrate these features. A discussion of the requirements, possibilities and limitations
of multiple plane DIPH is also presented.
1 Introduction
Several optical metrology techniques have been developed recently because of the
need of the non-intrusive measurements in solids and in fluids mechanics. One
such velocimetry technique is particle image velocimetry (PIV), which is a well
developed and widely used technique to measure the two in-plane velocity components from a single plane in the fluid.
Some efforts have been made to extend PIV to the measurement of full velocity
vectors in a volume of the fluid. The most straightforward approach to obtain 3-C
measurements is stereoscopic PIV, based on recording two views of the fluid
plane from different directions. But only a plane can be measured if a photographic recording is used.
A holographic recording is the only option to extend PIV to a whole volume. In
Holographic PIV (HPIV), a fluid volume is illuminated. Although HPIV is very
promising, its main drawback is the painful developing process. Digital holography can overcome this drawback but suffers from smaller spatial resolution.
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In this work, a multiple plane digital holographic technique is proposed to make
a more efficient use of the limited spatial resolution. We will start by describing a
digital speckle pattern interferometry [1] with spatial phase shifting technique
(SPS-DSPI), and its evolution into digital image plane holography (DIPH). We
will also show the ability of DIPH for 3-C velocity measurements. We will present
some preliminary experiments to demonstrate the feasibility of DIPH as a 3-D
velocimetry technique. Finally we will give some limits about the promising future of DIPH as a 3-D velocimetry technique.
2 From SPS-DSPI to DIPH
In DSPI the interference between an object wave and a reference wave is recorded. The interference pattern is known as specklegram because of its speckled
appearance. The object wave is obtained, as in PIV, by illuminating the fluid with
a sheetlike beam and focusing the light scattered by the small particles inside the
fluid onto a CCD detector using a convergent lens (Fig. 1). The reference beam is
obtained by diverting a small amount of the main laser beam. In this work, the reference beam is guided through an optical fibre, whose end is at the same distance
from the CCD sensor than the lens aperture, and the object and reference beam are
made to overlap by means of a non-polarizing cube beam splitter.
x
Flow
y
ui
uo
K
Laser sheet
Object
beam
Object beam
Beam
splitter
Lens
Beam splitter
Reference
beam
Reference beam
a)
CCD
camera
Specklegram
b)
CCD
Fig. 1. Optical setup of SPS-DSPI technique: a) general layout; b) reference and object
wave overlapping.
Fig. 2a shows a typical specklegram for a general position of the optical fibre
end, and a typical lens aperture of f/16. When the fibre end is precisely positioned
so that there is a small angle between the local propagation directions of the two
overlapping beams, a spatial phase shift [2,3] is introduced into the speckle field.
Fig. 2b shows a typical SPS specklegram where the spatial modulation is apparent.
In order for the CCD sensor to be able of recording this modulation, the phase
shift β from one pixel to the next must be smaller than π [4].
Holography and ESPI 365
As in digital PIV, two camera frames record two fluid states at a time interval
∆T. The specklegram intensities can be expressed as
I1 = I o,1 + I r + 2 I o,1 I r cos(φo ,1 − φ r )
(1)
I 2 = I o,2 + I r + 2 I o,2 I r cos(φo ,1 + ∆φo − φr )
(2)
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
(3)
r
r
r r
where V is the local fluid velocity, and K =(2π/λ)( u o- u i) is the sensitivity vector,
r
r
u o and u i being unity vectors in the observation and the illumination directions
respectively and λ being the laser wavelength in the fluid.
For a phase-shifted interferogram, Eq. 1 can be rewritten as
(
I n = I dc + mcos φ n + n β
)
(4)
where In is the intensity at pixel n, Idc=Io+Ir is the average intensity, m=2 I o I r
is the modulation of the interference term, and φn is related to the object phase at
pixel n. The intensities of 3 consecutive pixels are used as a set of equations to resolve the three unknowns[5]: Idc, m, φn.
The phase, which is the only magnitude of interest in DSPI can be calculated as
 1 - cos β
I n -1 − I n +1 

sin
β
2
I
I
I
−
−
n
n -1
n +1 

φn = tan -1 
(5)
The phase difference, ∆φo, can be obtained as
∆φ o = φ 2,n - φ 1,n
(6)
Fig. 4 shows a typical phase difference map obtained with the 3-step algorithm.
Let us note that the detected phase is not the true phase ∆φo but its modulo 2π. In
this type of wrapped phase maps null phase is mapped into black, while 2π is
mapped into white.
The SPS-DSPI optical setup can also be viewed as a digital image plane offaxis holographic setup. Now specklegrams can be called digital holograms. The
complex object wave can be reconstructed from the digital holograms using a
number of different numerical methods [6]. The most appropriate method for our
holographic recording is a global and frequency based method called Fourier
transform method (FTM) [2]. In this method, the first step is to calculate the Fourier transform of the digital hologram. This Fourier transform have three terms
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(Fig. 3). The dc term corresponding to Idc is at the centre (null frequency). The
other two terms come from the interference effects and are shifted symmetrically
from the centre by an amount that depends on the carrier frequency introduced
with the off-axis setup. Because of the specific recording geometry used in this
work the two off-centred terms are the real and the virtual image of the lens aperture, and are well separated from the dc term. In a second step, one of the aperture
images is windowed off from the rest. By calculating its inverse Fourier transform,
a complex wave with an intensity of I o I r and a phase of φo- φr is reconstructed.
The intensity field is a particle image field, like the ones recorded in PIV,
modified by the reference beam intensity. To reduce the noise due to an inhomogeneous reference beam and its influence on the particle image field, the reference
beam is recorded separately and subtracted from the digital hologram before
starting the FT calculations. Fig 3b shows the efficient removal of the dc term and
some spurious signals in the FT of the holograms. Besides, by dividing the reconstructed intensity field (squared) by the recorded reference beam intensity, the influence of the reference beam on the particle field is removed [3].
a)
b)
a)
b)
Fig. 2. 128x128 region of a typical speckle- Fig. 3. Fourier Transform of a SPS specklegram obtained without (a) or with (b) SPS gram before (a) and after (b) subtracting the
reference beam intensity.
modulation.
3 DIPH as a 3-C Velocimetry Technique
In DIPH as a 3-C velocimetry technique, not only the phase field but also the
intensity field are used [3,7]. As in DSPI, the subtraction of two phase maps corresponding to two recorded fluid states gives a wrapped phase difference map, being
∆φo the detected phase (Fig. 5a). However DIPH produces less noisy wrapped
phase difference maps than DSPI. As already stated ∆φo is related to the projection
r
r
of the velocity vector V along the sensitivity vector K direction and the phase
maps are always sensitive to an out-of-plane velocity component.
The wrapped phase difference maps need some further processing to produce
data ready to be used for velocity calculation. The essential part of this processing
is to unwrap the phase difference map so that the true ∆φo phase is known. It is
also convenient to remove the speckle noise from the phase difference map. In this
work both process are done simultaneously by using a regularized phase tracking
method [8] which is very robust. Fig. 6a shows the unwrapped phase difference
map obtained from the phase map shown in Fig. 5a.
Holography and ESPI 367
In the image, the full range of phases is mapped into the full range of grey levels. When the unwrapped phase is shown rewrapped (Fig. 6b) the quality of the
unwrapping process is more obvious.
The phase difference is only the important magnitude when the particle image
displacement between the two holograms is smaller than the particle image diameter. However when the displacement is bigger, the important magnitudes are
Io1 and Io2. In this case the cross-correlation of the intensity fields, Io1 and Io2, as in
r
PIV, gives information about the projection of the velocity vector V on a plane
r
perpendicular to the observation direction u o . Fig. 5b shows the intensity field Io1
reconstructed from one digital hologram and Fig. 6c presents the 2-C velocity map
obtained from the PIV analysis of two maps like the one in Fig. 5b.
The 3-C velocity measurement in a Cartesian coordinate system can be obtained by solving a three-equation system with the in-plane velocity data provided
by the PIV analysis of the reconstructed intensity field and the out-of-plane velocity data provided by the DSPI analysis of the reconstructed phase field.
To demonstrate the performance of DIPH as a 3-C velocimetry technique some
experiments in a Rayleigh-Bénard convective flow [9] have been carried out.
A cell filled with an aqueous glycerol solution that was seeded with 5µm diameter latex particles was used. The dimensions of the cell were Lx=25 mm, Ly=
25 mm and Lz=12.5 mm. An XZ fluid plane is illuminated by a 15mm high 1mm
wide light sheet beam from a 17mW He-Ne laser. The camera is a PCO Pixel-Fly
camera with 12-bit dynamic resolution, 1280x1024 pixels, and a pixel size of
6.7µm. An aperture of f/16 and a magnification of 0.3 are used, giving a field of
view of 28.5mm x 22.9mm and a speckle size of 67µm (in object space). 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 just
enough energy on the object wave.
a)
b)
Fig. 4. Wrapped phase dif- Fig. 5. Results from a FTM analysis: a) wrapped phase difference map obtained with ference map corresponding to ∆φ ; b) particle field correo
a 3-step algorithm.
sponding to Io1.
The convective flow velocity was appropriately chosen in the range of 15-25
µm/s so that no time averaging holography was produced. Pairs of holograms with
time interval ∆T1 suitable for the DSPI analysis were recorded at time intervals
∆T2 suitable for the PIV analysis. The typical results shown so far in Fig. 2 to 6
were obtained from the y=5mm plane recorded with ∆T1=40 ms, ∆T2= 4s and f/16.
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Fig. 7 presents the 3-C velocity measurements as isoline maps. 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 the Vx component from the 45º projection.
The fact that the measured Vy field is completely different than the Vx field and
its spatial pattern is also different from the wrapped phase difference map demonstrates the feasibility of using DIPH as a 3-C velocimetry.
The wriggling isolines show the non-optimum accuracy of the measurements.
The main source of error was the noise in both the phase and the intensity fields,
due to a very low energy on the object wave because of the low power He-Ne laser and the short exposure time required by the holographic recording. Much more
accurate results are expected with the use of pulsed lasers, even with the lowest
pulse energy models.
b)
a)
c)
Fig. 6. Raw velocity data from the DIPH analysis: a) unwrapped phase difference map; b)
rewrapping of the unwrapped phase map (Fig. 6a); c) in-plane vector map from PIV analysis.
24
-16
16
0
8
16
a)
24
16
16
8
-8
0
0
0
-8
-8
-8
-16
-16
b)
-16
c)
Fig. 7. 3C measurements presented as isoline maps labeled with the velocity values in
µm/s: a) Vx field; b) Vy field; c) Vz field
4 DIPH as a 3-D Velocimetry Technique
A consequence of the DIPH approach and the frequency analysis is the feasibility
of a multiplexed recording in one frame. Schedin et al. [10] have already shown in
a solid mechanics application that three holograms can be angularly multiplexed
by using three reference beams with point sources in different positions for each
hologram recording and exploiting the beam temporal coherence length properties.
For 3-D fluid velocimetry a multiple light sheet illumination is proposed. Each
reference beam has to be kept coherent with only one light sheet in the measurement region.
Holography and ESPI 369
As a first approach, a DIPH arrangement for recording two planes has been
setup (Fig. 8). Two beams, coming from the same laser and with an appropriate
optical path length difference, are used to form two pairs of object and reference
beams. The object beam 1 interferes only with the reference wave 1 while object
beam 2 interferes only with reference wave 2. The two object beams are shaped
into two sheetlike beams, whose scattered waves will not interfere with each other.
The Fourier transform of a typical hologram is shown in Fig. 9.
The two reference beams have been adjusted so that the corresponding aperture
images stay along the two diagonals to allow the maximum lens aperture in the recording without overlapping.
Flow
Laser sheet 2
Laser sheet 1
Lens
Beam splitter
Reference beam 1
Reference beam 2
CCD
camera
Fig. 9. Fourier Transform of a two multiFig. 8. DIPH setup for multiplexing two
plexed plane digital image plane hologram.
plane recordings.
a)
b)
c)
Fig. 10. Results with DIPH as a 3-D velocimetry technique for two planes (y=4mm in the
upper row and y=1mm in the lower row): a) wrapped phase difference maps; b) reconstructed particle field; c) in plane vector maps from a PIV analysis of two particle fields.
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By applying the DIPH analysis to each aperture, information from only one
fluid plane is obtained. The two-plane hologram recording setup has also been
tested on the Rayleigh-Bénard convective flow using the 17mw He-Ne laser as the
only light source. Fig. 10 presents some results obtained from recording two
planes at y=4mm and y=1 mm, with ∆T1=83ms, ∆T2=8s, f/16 and 7ms of exposure time. The phase maps obtained for each plane are so different that nothing
would have been seen without the coherence control.
However, no 3-C quantitative analysis in the two-plane DIPH has been done
because both the DSPI and the PIV data were less accurate than in the one-plane
setup due to an even lower energy per fluid plane, than before.
5 Requirements, possibilities and limitations of DIPH
DIPH is a powerful technique that should allow measuring 3-C velocity fields in
3-D regions.
In a first stage, DIPH can be used to record two planes very close together.
Measuring the 3-C velocity fields in these two planes should allow calculating
other magnitudes with spatial derivatives of the velocity field such as the vorticity
vector.
For separating the two plane information not only coherence control, but also
polarization control could be used. Even a setup with two independent lasers can
be envisaged. In any case, the laser coherence length has to be as long as the
length of the measurement region. Pulsed lasers with a pulse width as small as
possible are a must. For a pulse width of 10ns DIPH is feasible in flows with ver
r
locities up to 25m/s in the u o and u i directions. The velocity component in the
other direction can be an order of magnitude bigger.
In flows with wide enough optical access, DIPH can be used just to record the
two planes simultaneously. The 3-C velocity field could be obtained using a stereo-PIV arrangement for the DIPH recording and a stereo-PIV analysis, using only
the reconstructed intensity field. Two laser pulses will be enough for this configuration.
In flows with limited optical access the full DIPH analysis would have to be
used. In this case, the temporal resolution of the DSPI analysis is different from
the resolution of the PIV analysis, which might be a problem in unsteady flows.
Regarding the accuracy of the 3-C measurements, a 3-cavity laser but only one
camera will be needed.
For DIPH to record a 3-D region, a multiple plane illumination setup is envisaged. Up to ten holograms can be multiplexed in a 12-bits and 1000x1000 pixel
sensor. The multiple planes can be arranged to be equally spaced or to be in
groups of two very close planes. In the second case not only 3-C velocity field, but
3-C velocity derivative fields can be obtained.
A stereoscopic recording setup and a stereo-PIV analysis can be used to calculate the 3-C velocity field in each fluid plane. For this multiplane stereo DIPH,
two cameras and a wide enough optical access are needed. However, full DIPH
analysis will be the only possibility in flows with limited optical access. Two or
Holography and ESPI 371
three cavity lasers are necessary depending on the type of analysis (stereo or full
DIPH). Coherence control or independent lasers are the only choices to separate
the information recorded for each plane.
6 Conclusions
In this work the feasibility of DIPH technique as a fluid velocimetry technique has
been shown. Both phase maps and particle fields are obtained from each recording. The three velocity components are obtained from three DIPH recordings separated by different temporal intervals, in order to be able to use both, the PIV algorithms on the reconstructed object intensity, and the DSPI analysis on the
reconstructed object phase.
Two planes have been simultaneously recorded but independently reconstructed, showing the way for DIPH to be extended to a quasi 3-D technique.
Acknowledgements:
This research was supported by a Spanish Research Agency Grant (DPI20001578-C02-02) and by the EUROPIV 2 project. EUROPIV 2 (A joint program to
improve PIV performance for industry and research) is a collaboration between
LML URA CNRS 1441, DASSAULT AVIATION, DASA, ITAP, CIRA, DLR,
ISL, NLR, ONERA and the Universities of Deft, Madrid, Oldenburg, Rome,
Rouen (COIRA URA CNRS 230), St Etienne (TSI URA CNRS 842), Zaragoza.
The project is managed by LML URA CNRS 1441 and is funded by the European
Union within the 5th framework (Contract nº G4RD-CT-2000-00190).
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