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Digital holographic interferometry in flow research
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OPTICS COMMUNICATIONS
1 November 1990
Full length article
Digital holographic interferometry in flow research T.A.W.M. Lanen Delft University qf Technology. Deparlment of Aerospace Engineering, Kluyverweg 1. P.O. Box 5058. 2629 HS Delft, The Netherlands Received 25 June 1990
A plane-waveholographic interferometric set-up for the quantitative determination of the density distribution existing in the flow field around arbitrarily shaped objects using digital image-processingis presented. By utilizing two reference/reconstruction beams, generated by a slightly misaligned Twyman-Green (T-G) interferometer in the reference/reconstruction beam path, the wavelength used to reconstruct the holographic plate is allowed to differ from the recording wavelength. To obtain again an infinite fringe pattern after such wavelengthchange, one of the two mirrors constituting the T-G interferometer is slightlyrotated. Phase-stepped interferograms are generated successivelyby piezo-eleetricallytranslating the other mirror of the T-G interferometer. Overlapping cross reconstructions, which are inherently induced in such a set-up, are eliminated by a spatial filter in the focus generated by the imaging lens. By applying plane traversing waves instead of diffuse light, automatic distinction between the foreground pixels (i.e., the pixels not lying in the shadow of the object) and the background pixels is made, starting from the modulation intensity as a discriminating parameter. The above procedure is demonstrated by measuring the density distribution in an axisymmetric supersonicjet of air.
1. Introduction With the development of high resolution CCD cameras and the availability of image-processing packages r u n n i n g on powerful computers, digital holographic interferometry ( D H I ) has become an important diagnostic tool in the quantitative experimental investigation of flow fields [ 1,2]. DHI combines holographic interferometric imaging with image-processing techniques. The images are digitized and are represented in the computer by 2-D blocks of data. Typical block sizes are 512 X 512 pixels large with 256 discernable grey levels in each pixel. By storing at least three phase-stepped interferograms in the computer memory, a pixelwise d e t e r m i n a t i o n of the wave front deformation with an accuracy up to 2 / 1 0 0 is obtainable [3]. In a case of general interest, the compressible unsteady flow field around an airfoil in a wind tunnel is dealt with. To determine the projected density distribution around the airfoil, this requires the ability of the image-processing routines to automatically distinguish between the foreground pixels (i.e., the t~ixels not lvina in the shadow of the airfoil) and the
background pixels. Furthermore, because of the unsteady character of the flow field an analysis limited to the realization of the field at a unique m o m e n t of time must be carried out. However, thus far most of the applications of DHI were limited to relatively simple test fields, either due to (i) shortcomings of the image-processing routines or due to (ii) limitations of the interferometric set-up. To illustrate the present requirements imposed on the image-processing routines, again consider the flow field around a non-transparent airfoil. A successful analysis can only be carried out if it is possible to automatically distinguish between the foreground pixcls and the background pixels. Furthermore, all subsequent image-processing routines must have the ability to circumvent the background pixels when performing their specific tasks. A detailed description of the routines fulfilling these requirements, illustrated by the analysis of the free convection flow field a r o u n d a heated, horizontal, cylindrical bar using plane-wave DHI was recently given by Lanen et al. [4]. As the main limitation of most interferometric setuos. the inability of studvin~ raoidlv-varvinz un-
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steady flow fields can be mentioned. Such fields require the intermediate freezing of the flow field on a holographic plate for subsequent analysis. Limited power of CW lasers together with the known sensitivity of the holographic plates determines the minimum exposure time, yielding typical values of about 8 ms. The growing commercial availability of pulsed laser systems has solved this problem by enabling exposure times as short as 30 ns. However, with the application of pulsed lasers a new problem was introduced. The utilization of a pulsed laser implies that the wavelength used during the recording of the holographic plate differs from the wavelength used during the reconstruction of the holographic plate. Several investigators tried to solve this problem by switching from plane waves traversing the flow field to diffuse illumination [ 1,2 ]. As will be shown, there are several important disadvantages to diffuse illumination when compared to plane-wave illumination. In this paper the attention will be focused on new developments in holographic interferometric imaging and not on software developments. The final aim can be formulated as to combine plane-wave DHI with the allowance of a wavelength change between the recording and the reconstruction of the holographic plate.
2. Digital holographic interferometry (DHI) In interferometry the wave fronts of two light beams, the disturbed scene beam and the undisturbed scene beam (both originating from one laser source), are made to interfere. The disturbed scene beam has passed through the flow field in study while the undisturbed scene beam has passed through a region of uniform density. In real-time holographic interferometry one of the two beams is reconstructed from a holographic plate. It is even possible to reconstruct both beams from the same holographic plate. This situation is usually indicated as two-reference-beam holographic interferometry. Because of their importance to the investigation of flow fields both types of holographic interferometry and their specific properties will be briefly discussed below.
1 November 1990
2.1. Real-time, plane-wave DHI In real-time holographic interferometry usually the wave fronts of the undisturbed scene beam are recorded on the holographic plate [5]. For this purpose, the holographic plate is exposed at the same time by both a reference beam and the undisturbed scene beam. During reconstruction the holographic plate is illuminated by the reconstruction beam (i.e., the beam which serves as the reference beam when exposing the holographic plate) and the undisturbed scene beam will be reproduced. If at the same time the disturbed scene beam is generated, an interference pattern is obtained. In the case of an unsteady flow field this interference pattern will show a time-dependent character. This enables one to real-time monitor the evolution of the flow field by a continuously varying interference pattern. However, modern interferogram evaluation procedures require at least three so-called phase-stepped interferograms as input data [3]. These interferograms are generated in time by piezoelectrically translating one of the mirrors in the reconstruction beam path, resulting in a path-length variation and in a corresponding shift of the fringes in the interference pattern [4,6]. To allow a pixelwise determination of the projected density distribution in the flow field, all interferograms must be related to identical flow conditions, i.e., to the realization of the flow field at a unique moment of time. Hence, the time dependency of the interference pattern due to the unsteadiness of the flow must be eliminated. This would be realized if it were possible to record the disturbed scene beam on the holographic plate instead of the undisturbed scene beam. However, as briefly remarked in the introduction, the limited power of CW lasers prevents one from doing so. Hence, it can be concluded that real-time DHI is limited to the investigation of steady flow fields. In real-time DHI plane-wave illumination is generally preferable over diffuse illumination especially if the flow field around a non-transparent object is investigated. As in a series of phase-stepped interferograms the background pixels will not show any difference in intensity, it follows that these pixels (as opposed to foreground pixels) are characterized by relalivelv ~rnall vahle~ c~flhelr rncwhdatic~n intensity
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This enables one to make use of the modulation intensity (which is computed from the digitized interFerograms) as the parameter discriminating between foreground pixels and background pixels. This distinctive information is an essential input to subsequent image-processing routines [4]. However, following the above strategy in the case of diffuse illumination will lead to erroneous decisions as the probability distribution of the modulation intensity o f the foreground pixels (speckles) shows m a x i m u m occurrence of 'zero-modulation-intensity' [ 7 ].
2.2. T~o-reJ~,rence-beam, diJ'/use DHI In this type of D H I diffuse light traverses the flow field. Furthermore, both the disturbed scene beam and the undisturbed scene beam are recorded on one and the same holographic plate by two reference beams which have different propagation directions. One of the reference beams serves to record the undisturbed scene beam and the other serves to record the disturbed scene beam. During reconstruction both beams illuminate the holographic plate at the same time and a steady interference pattern will result. Phase-stepped interferograms are induced by a piezoelectric length variation of one of the reconstruction beam paths. This type of D H I originated in efforts to determine the deformation of diffusely reflecting objects under load [8] and was later applied to flow fields [ 1,2 ]. One of the major problems of two-reference-beam holographic interferometry is the generation of cross reconstructions. The reference/reconstruction beam which served to record the undisturbed scene beam on the holographic plate will not only reconstruct the undisturbed scene beam but also the disturbed scene beam and vice versa. In all, eight reconstructions will be generated: four normal reconstructions and four conjugate reconstructions. Only the four normal reconstructions are of interest and these can be classified into two interfering reconstructions and two undesirable cross reconstructions. The situation in which the cross reconstructions overlap the interference pattern can be avoided by having the reference beams well separated. However, in this case the repositioning sensitivity of the holographic plate, after being subjected to the photographic process, is far lar~er than with the two reference beams close to-
1 N o v e m b e r 1990
gether [8]. The occurrence of overlapping cross reconstructions is not an insolvable problem if diffuse light is applied. The two cross reconstructions, which have a small angular separation, will superimpose a decorrelated speckle pattern over the interference pattern in the image plane. In this situation the conventional data-acquisition procedure and the imageprocessing routines apply again. With the above set-up it is possible to make use of a pulsed laser to investigate rapidly-varying unsteady flow fields. In this case the holographic plate is exposed by a pulsed laser serving as light source and is reconstructed by a CW (alignment) laser. The inherent wavelength change must be corrected for by a small angular rotation of one of the reconstruction beams. This process was clearly demonstrated by Breuckmann and Thieme [ 1 ] and by Watt and Vest [2]. Despite the merits of the above described set-up in the area of compressible unsteady flow fields (and of diffusely reflecting deformed objects), some important disadvantages of the use of diffuse light have to be mentioned. In the first place, the superposition of a decorrelated speckle pattern over the interference pattern in the image plane will reduce the contrast of the interference pattern to at least half the maximum value obtainable with plane-wave illumination [8]. Secondly, the statistical predominance of zero-modulated speckles in the foreground space of the interference pattern hampers the investigation of flow fields existing around non-transparent objects (see earlier remarks on this topic). Furthermore, fringe localization in a diffuse illumination interferogram strongly depends on the structure of the flow field and the viewing direction [ 7 ]. Finally, diffuse illumination is disadvantageous compared to plane-wave illumination because of the noisy character of the speckle interference patterns, resulting in less accuracy. Therefore, an attempt was made to combine the advantages of real-time plane-wave DHI (i.e., investigating flow fields around non-transparent objects) with the advantages of two-reference-beam, diffuse DHI (i.e., the introduction of a wavelength change and hence the possible investigation of unsteady flow fields by applying a pulsed laser), resulting in two-
re~'rence-beam, plane-wave DHI.
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3. Two-reference-beam, plane-wave DHI
3. I. Experimental set-up Figure 1 gives a schematic overview of the holographic interferometer used in the experiments. The solid lines indicate the outer dimensions of the mobile, closed construction containing the interferometer. For ease of experimentation, instead of a pulsed laser/ CW alignment laser-combination the light source of the interferometer is a CW Ar ion laser with a maximum output of 1 W at a wavelength of 514.5 nm. By adjusting the rear high reflector of the laser a number of different wavelengths in the interval [454.5 nm, 514.5 nm] can be selected, allowing to simulate the wavelength change. The main laser beam is folded by the mirrors M1 and M2 and split by the beam splitter cube BSC (fig. 1 ) into the reference/reconstruction beam and the scene beam. First the path traversed by the scene beam will be followed. The scene beam reaches via the mirrors M3 and M4 the objective-pinhole combination PH~ which is exactly positioned via the
Ht
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mirror M5 in the focal point of the lens L,. Hence, the flow field in study will be traversed by a scene beam which is collimated. Finally, the scene beam is projected by the lens L2 on the holographic plate H. Starting from the beam splitter cube BSC the reference/reconstruction beam path will now be described. The reference/reconstruction beam enters via the mirror M6 the Twyman-Green interferometer section (outlined by dashed lines in fig. 1). In this interferometer the beam is expanded to a collimated beam by the objective-pinhole combination PH2 and by the lens L3. Subsequently, this beam is partially reflected/transmitted by the beam splitter plate BSP. The reflected beam is again reflected in the mirror MT, mounted on the piezo-electric translator PZT, and illuminates through the beam splitter plate the holographic plate. In the sequel of this paper this beam will be called the first reference/reconstruction beam. The transmitted beam is reflected in the mirror Ms, which can be rotated about two axes, and is reflected again on the beam splitter plate to reach the holographic plate. This beam will be called the second reference/reconstruction beam.
I $1 CW Ar LASER
,,
Y
RI
BSC
V
R2
Sz
R3
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I
-1
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FLOW FELD
Fig. 1. Experimental set-up. BSC:beam splitter cube, BSP:beam splitter plate, H: holographicplate, L~,..., L3: lens, M~, ..., Ma: mirror, PHi, PH2: objective-pinholecombination, PZT: piezo-electrictranslator, R~,..., R3: 2/2-retardation plate, S~, $2: shutter, --: TwymanGreen interfernmeter
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For reasons to be explained later, the Twyman-Green interferometer is slightly misaligned so that two collimated reference/reconstruction beams with slightly different propagation directions are generated.
3.2. Exposing the holographic' plate "\
The holographic plate is exposed in two successive stages. In the first stage the flow field is absent and one of the arms of the T-G interferometer is blocked. By opening the shutter S~ (fig. 1 ) for a short time (8 ms) the undisturbed scene beam is recorded on the holographic plate. In the second stage the flow field is present and the other arm of the T-G interferometer is blocked. Again the shutter S~ in the main laser beam is opened for a short time and the disturbed scene beam is recorded on the holographic plate. The intensity rate of the scene beam and the reference beam in the hologram plane is controlled by the 2/2-retardation plate R~ and the polarizing beam splitter cube BSC. Two other 2/2-retardation plates R2 and R3 serve to choose identical polarization directions for the scene beam and the reference beam. After exposure of the holographic plate it is taken out of the interferometer for photographic processing.
3.3. Reconstructing the holographic plate The reconstruction process is performed in the setup of fig. 1 with the holographic plate back in its original position. In this stage the shutter S~ remains permanently opened and the scene beam path is blocked by closing the shutter S> Furthermore, none of the two arms of the T-G interferometer is blocked so that the holographic plate is illuminated by both reconstruction beams. As remarked before, eight reconstructions will be produced: four converging normal reconstructions and four diverging conjugate reconstructions. However, in the present case of oil:axis holography (i.e., the situation in which the reference/reconstruction beam and the scene beam have different propagation directions), only the normal reconstructions play a role. To compute the propagation directions of these reconstructions the sign conventions of fig. 2 must be k e p t in mind. In this figure the angle between the nnrmnl
nftho
halnornnhic
nlnto
nncl lh~ e.eono heam
/ec
t
Fig. 2. Sign conventions.
is positive and equal to cg and the angle between the normal of the holographic plate and the reference beam is negative and equal to Or. The angles as and Or are related to the situation when the holographic plate is exposed. The angle between the reconstruction beam and the normal of the holographic plate is negative and equal to 0c. Then the angle between the reconstructed scene beam and the normal of the holographic plate, c~'s, is given by the following formula [ 9 ] o!'s=sin ~(~.'. ( s i n % - s i n O~)+sin Oc),
(l)
where ~ is the recording wavelength and 2' is the reconstructing wavelength. In the present situation the following set of equations determines the propagation directions of the four normal reconstructions O~'s,li----sin- '(@ (sin Ces--sin 0~ ) + sin 0 c , ) ,
(2)
ot~,~=sin 2 ' , _ ~( ~-(sin c ~ - s i n 0,.~)+sin 0~2) ,
(3)
a's,2, =sin
,(2~ ( s i n o ~ - s i n 0 r 2 ) + s i n 0 ~ l ) ,
(4)
cQ22=sin
~
(sincg-sin0~2)+sin0~2
.
(5)
Here 0,1, 0r2 and % indicate the respective orientations of the two reference beams and the scene beam when exposing the holographic plate. 0c~ and 0~2 indicate the orientations of the two reconstruction beams. Unequal subscripts to c~'~ refer to cross reconstructions and equal subscripts refer to legal reenn~lruetion~
Obviously
toohtainnninfinilefrin~e
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pattern the propagation directions of the two legal reconstructions must be identical: t ! O(s,l I = OL s,22
~r2 ----~r,
As the angles, as, 0r~ and 0r2 a r e constant once the holographic plate is exposed, the only free parameters in eqs. (2) and (5) are the reconstruction beam angles 0c~ and 0c2. However, taking a look at the setup of fig. 1 and bearing in mind that the mirror M7 is rigidly mounted on the PZT, allowing only translation and no rotation, it will be clear that 0cl = 0r~. This implies that a's.~ is strictly determined. Only the mirror Ms can be rotated to affect the parameter 0~2 and hence a's.22. Suppose that 0c2 deviates from 0r2 by the angle A 0 r 2 , ( 7
)
To satisfy eq. (6), A0r2must be chosen according to the following equation zXO~2~
~- zX0r,
(9)
"~- A 0 r .
If the second reconstruction beam is given the propagation direction prescribed by eqs. (7) and (8), eqs. ( 3 ) and (4) determine the propagation directions of the two cross reconstructions. In fig. 3 the propagation directions of the two cross reconstructions, relative to the propagation direction of the two coinciding legal reconstructions, are given as a function of the angle A0~ between the two reference beams. Furthermore, the angle over which the second reference/reconstruction beam must be rotated, A0r2, has been drawn. Fig. 3 has been computed for the intended use of a pulsed laser for the exposure of the holographic plate and of a CW laser in the reconstruction stage. Accordingly, in the computation the recording wavelength 2 is taken equal to 694.3 nm, being the wavelength of a pulsed ruby laser, and the reconstructing wavelength )~' is taken equal to 632.8 nm, being the wavelength of a CW HeNe laser. The angles as and 0r~(=0c,) are taken from the set-up of fig. 1 and are equal to 7 ° and - 4 5 ° respectively. From fig. 3 it follows that the angular separation between each of the cross reconstructions and the legal reconstructions is almost linearly related with the angle between the reference beams. Also the
(6)
0c2 ~-~ 0r2 + A0r2 .
1 November 1990
(8)
where it is supposed that m 0 r / 0 r l < < 1, m 0 r 2 / 0 r 2 < < 1 and that A0r is the angle between the two reference beams at the moment of exposure
16
(~s. 12 ~ a s . II
14 12
(degree)
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-8 -10 -12 -14 -16
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Fi2. 3. Prona2atinn directions hi'cross r@(:NnsWll(~tlnn~ ~ n d ~Nt'r~tiNn ~nol~ v~r~u~ ~nole between ruf~ronr~ henrnr
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I November 1990
i /
H7 BSP
H
____
Fz FI I SF
~
[L,
~/ ,,I
~ ' ~ - ~ - ~ ,
J CROSS LEGAL
i il~cR°ss
Fig. 4. Experimental set-up at reconstruction. See also fig. 1. Ft: focal plane of holographic plate H, F2: focal plane of lens L2, SF: spatial filter.
angle over which the second reference/reconstruction beam must be rotated to obtain an infinite fringe pattern shows an almost linear relation. Figure 4 shows a detail of the set-up of fig. 1 as it will look like during reconstruction. In the focal plane of the holographic plate Ft (which does not coincide with the focal plane F2 of the lens L2 because of the wavelength change) the two cross reconstructions are spatially separated from the legal reconstructions. An analogous situation will exist in the focus of the imaging lens which projects the interference pattern on a CCD camera. In this focal plane the cross reconstructions are filtered out by a spatial filter, transmitting only the two (coinciding) legal reconstructions. In this paper no attention is paid to the holographic aberrations caused by the wavelength change. As described by Meier [10], under some circumstances one or more of these aberrations can be made to vanish.
computer memory. In the reconstruction set-up depicted in fig. 4 these are generated in time by a length variation of one of the arms of the T-G interferometer. This length variation is achieved by a computer-controlled voltage on the piezo-electric crystal present in the PZT, translating the mirror M7. The resulting interferogram is stored in the computer memory by digitizing the video signal from the CCD camera. Depending on the number of phase-stepped interferograms needed by the image-processing routines, this process is repeated for different positions of the mirror M 7. Before starting the data-acquisition procedure, it is possible to superimpose arbitrarily oriented reference fringes over the interference pattern. This is achieved by rotating the mirror M8 of the T-G interferometer about two axes by adjusting two micrometers. The superposition of reference fringes is essential when the fringe spacing of the infinite fringe pattern locally approximates the pixel resolution.
3.4. Data-acquisition 4. Experimental results As remarked in the introduction, pixelwise determination of the wave front deformation induced by the flow field requires at least three phase-stepped
To demonstrate the above interferometric set-up, the flow field of an axisymmetric supersonic jet of
i n t e r f e r n ~ r a m ~ In b e d i ~ i t i T e d a n d to h e ~ t n r e d in t h e
a i r is e x a m i n e d
Theieli~urranndedhvamhienlair
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Fig. 5. Interferogram showing reference fringes parallel to jet axis due to wavelength change (2 = 514.5 nm--,2' = 472.7 n m ).
I November 1990
Fig. 6. Interferogram with reference fringes normal to jet axis.
at rest. The main flow direction of the jet is oriented normal to the scene beam axis. The holographic plate is recorded at the wavelength 2=514.5 nm (visible green). To simulate the wavelength change as will necessarily take place in pulsed laser interferometry (pulsed ruby laser: 2=694.3 n m - . c w HeNe laser: 2' = 632.8 nm) the reconstructing wavelength is chosen in the visible blue equal to 2' =472.7 nm. This makes the quotient 2 ' / 2 appearing in eqs. (2) to (5) almost identical for both situations. The CCD camera viewing the flow field is oriented at 45 ° to the flow direction. Starting from the infinite fringe pattern in the visible green, fig. 5 shows the finite fringe pattern which is obtained after the wavelength change. By adjusting the mirror M8 of fig. 1 the orientation of the reference fringes can be arbitrarily chosen as well as their relative spacing. Figure 6 shows the finite fringe pattern with the reference fringes normal to the jet axis. The mirror Ms is ultimately adjusted so that the infinite fringe pattern of fig. 7 results. In all three figs. 5 to 7 the cross reconstructions have been filtered out by the spatial filter in the focal plane of the imaging lens. Figure 8 shows the distorted interference pattern which re-
distorting presence of the two overlapping cross reconstructions. The interferogram of fig. 7 serves as
~lllt~ i f t h ~ ~ n n l i n l f i l l e r i~ r ~ r n n v o c l
n n e nHt n f t h ~ ~erie~ n f n h a ~ e - ~ l e n n e d
1~ i l l l ~ l r n t o ~ l h o
Fig. 7. Infinite interferogram of an axisymmetric jet of air.
inlerferc~rams
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Fig. 8, Interferogram without spatial filtering,
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Due to the axial symmetry of the jet several characteristics o f the structure of the flow can be derived from this result. The jet is bounded by a shear layer which grows in a downstream direction. Close to the axis of the jet, a series of cells resulting from alternate regions of expansion and compression can be discerned. Discharged by the nozzle at sonic speed and at higher than ambient pressure, the air is expanded by expansion waves to supersonic speed and the density decreases. By reflection of the expansion waves at the free surface of the jet as compression waves, the density regains approximately its sonic value at the axis of the jet, etc. Further downstream the cell structure finally becomes invisible by viscous effects. From fig. 9 artificial Schlieren images can be computed with the knife-edge in an arbitrarily chosen direction. For this purpose, the partial derivatives of the projected density distribution p(x, y) (which is linearly related to the wave front deformation of fig. 9) in the x-direction and in the y-direction, p.~ and p,., are computed using the Sobel difference operator [11]. Combining these results according to p,. cos O+pv sin 0, the Schlieren image with the knifeedge in the 0-direction is obtained. Figure 10 depicts
Fig. 9. Wave front deformation. from which the wave front deformation is computed. Figure 9 shows the result of this computation, yielding a m a x i m u m wave front deformation of 1.05X2'.
Fig. 10. Computed Schlieren image (knife-edge normal to jet axis).
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1 November 1990
ti
,$ -, D Fig. 11. Computed Schlieren image (knife-edge parallel to jet axis)• the computed Schlieren image with the knife-edge oriented normal to the axis of the jet, whereas in fig. 1 1 the knife-edge orientation is parallel to the jet axis. In fig. 11 the shear layer indicating the outer dimensions of the jet is visible and fig. 10 clearly demonstrates the diamond cell structure characteristic for a supersonic jet. As a final result, fig. 12 depicts the axial density distribution in the jet. This density distribution is computed from the projected density distribution of fig. 9 by the inverse Abel transformation, following the numerical scheme suggested by Nestor and Olsen [ 12 ]. The transformation is executed for the first four cells, starting at the nozzle's outlet (top of fig. 12). The deviation of the axial density from the ambient density varies between - 0 . 1 6 k g / m ~ and 1.16kg/m 3. It is interesting to notice that two successive local areas of higher density are separated from each other by a local area of lower density, a phenomenon which is not directly clear from fig. 9 alone. Furthermore, the density of the first m i n i m u m lies below the ambient density.
Fig. 12. Axial density distribution with curves of constant density (curve spacing 0.082 kg/m3). 5. Conclusions and future prospects A successful demonstration of two-reference-beam, plane-wave DHI in the investigation of flow fields was given. Overlapping cross reconstructions are prevented by spatial filtering. Non-transparent objects intersecting the interference patterns are easily detected and successively circumvented by the imageprocessing routines as plane-wave illumination is used instead of diffuse light. Since the reconstructing wavelength is allowed to differ from the recording wavelength in the presented set-up, a pulsed laser can be applied instead of the CW Ar ion laser to make the investigation of unsteady flow fields nossible, The only necessary
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m o d i f i c a t i o n to t h e s e t - u p o f fig. 1 is t h e t r a n s i t i o n to H E L ( high energy level ) o p t i c s for t h e m i r r o r s M j, M > M3, M4 a n d M6. F u r t h e r m o r e , in all p r o b a b i l i t y t h e o b j e c t i v e - p i n h o l e c o m b i n a t i o n s P H i a n d PH2 h a v e to b e r e p l a c e d b y n e g a t i v e lenses to a v o i d foci.
References [ 1 ] B. Breuckmann and W. Thieme, Appl. Optics 24 ( 1985 ) 2145. [2 ] D.W. Watt and C.M. Vest, Exp. Fluids 5 ( 1987 ) 401. [3] J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White and D.J. Brangaccio, Appl. Optics 13 (1974) 2693.
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[4] T.A.W.M. Lanen, C. Nebbeling and J.L. van Ingen, Optics Comm. 76 (1990) 268. [5] P. Hariharan, B.F. Oreb and N. Brown, Optics Comm. 41 (1982) 393. [6] W. Jfiptner, T.M. Kreis and H. Kreitlow, SPIE Proceedings 398 (1983) 22. [ 7] C.M. Vest, Holographic interferometry (Wiley, New York, 1979). [8] R. D~indliker, R. Thalmann and J.F. Willemin, Optics Comm. 42 (1982) 301. [9] M. Franqon, Holography (Academic Press, New York, 1974). [10] R.W. Meier, J. Opt. Soc. Am. 55 (t965) 987. [11 ] A. Rosenfeld and A.C. Kak, Digital picture processing (Academic Press, New York, 1982). [ 12] O.H. Nestor and H.N. Olsen, SIAM Rev. 2 (1960) 200.
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Full length article
Digital holographic interferometry in flow research T.A.W.M. Lanen Delft University qf Technology. Deparlment of Aerospace Engineering, Kluyverweg 1. P.O. Box 5058. 2629 HS Delft, The Netherlands Received 25 June 1990
A plane-waveholographic interferometric set-up for the quantitative determination of the density distribution existing in the flow field around arbitrarily shaped objects using digital image-processingis presented. By utilizing two reference/reconstruction beams, generated by a slightly misaligned Twyman-Green (T-G) interferometer in the reference/reconstruction beam path, the wavelength used to reconstruct the holographic plate is allowed to differ from the recording wavelength. To obtain again an infinite fringe pattern after such wavelengthchange, one of the two mirrors constituting the T-G interferometer is slightlyrotated. Phase-stepped interferograms are generated successivelyby piezo-eleetricallytranslating the other mirror of the T-G interferometer. Overlapping cross reconstructions, which are inherently induced in such a set-up, are eliminated by a spatial filter in the focus generated by the imaging lens. By applying plane traversing waves instead of diffuse light, automatic distinction between the foreground pixels (i.e., the pixels not lying in the shadow of the object) and the background pixels is made, starting from the modulation intensity as a discriminating parameter. The above procedure is demonstrated by measuring the density distribution in an axisymmetric supersonicjet of air.
1. Introduction With the development of high resolution CCD cameras and the availability of image-processing packages r u n n i n g on powerful computers, digital holographic interferometry ( D H I ) has become an important diagnostic tool in the quantitative experimental investigation of flow fields [ 1,2]. DHI combines holographic interferometric imaging with image-processing techniques. The images are digitized and are represented in the computer by 2-D blocks of data. Typical block sizes are 512 X 512 pixels large with 256 discernable grey levels in each pixel. By storing at least three phase-stepped interferograms in the computer memory, a pixelwise d e t e r m i n a t i o n of the wave front deformation with an accuracy up to 2 / 1 0 0 is obtainable [3]. In a case of general interest, the compressible unsteady flow field around an airfoil in a wind tunnel is dealt with. To determine the projected density distribution around the airfoil, this requires the ability of the image-processing routines to automatically distinguish between the foreground pixels (i.e., the t~ixels not lvina in the shadow of the airfoil) and the
background pixels. Furthermore, because of the unsteady character of the flow field an analysis limited to the realization of the field at a unique m o m e n t of time must be carried out. However, thus far most of the applications of DHI were limited to relatively simple test fields, either due to (i) shortcomings of the image-processing routines or due to (ii) limitations of the interferometric set-up. To illustrate the present requirements imposed on the image-processing routines, again consider the flow field around a non-transparent airfoil. A successful analysis can only be carried out if it is possible to automatically distinguish between the foreground pixcls and the background pixels. Furthermore, all subsequent image-processing routines must have the ability to circumvent the background pixels when performing their specific tasks. A detailed description of the routines fulfilling these requirements, illustrated by the analysis of the free convection flow field a r o u n d a heated, horizontal, cylindrical bar using plane-wave DHI was recently given by Lanen et al. [4]. As the main limitation of most interferometric setuos. the inability of studvin~ raoidlv-varvinz un-
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steady flow fields can be mentioned. Such fields require the intermediate freezing of the flow field on a holographic plate for subsequent analysis. Limited power of CW lasers together with the known sensitivity of the holographic plates determines the minimum exposure time, yielding typical values of about 8 ms. The growing commercial availability of pulsed laser systems has solved this problem by enabling exposure times as short as 30 ns. However, with the application of pulsed lasers a new problem was introduced. The utilization of a pulsed laser implies that the wavelength used during the recording of the holographic plate differs from the wavelength used during the reconstruction of the holographic plate. Several investigators tried to solve this problem by switching from plane waves traversing the flow field to diffuse illumination [ 1,2 ]. As will be shown, there are several important disadvantages to diffuse illumination when compared to plane-wave illumination. In this paper the attention will be focused on new developments in holographic interferometric imaging and not on software developments. The final aim can be formulated as to combine plane-wave DHI with the allowance of a wavelength change between the recording and the reconstruction of the holographic plate.
2. Digital holographic interferometry (DHI) In interferometry the wave fronts of two light beams, the disturbed scene beam and the undisturbed scene beam (both originating from one laser source), are made to interfere. The disturbed scene beam has passed through the flow field in study while the undisturbed scene beam has passed through a region of uniform density. In real-time holographic interferometry one of the two beams is reconstructed from a holographic plate. It is even possible to reconstruct both beams from the same holographic plate. This situation is usually indicated as two-reference-beam holographic interferometry. Because of their importance to the investigation of flow fields both types of holographic interferometry and their specific properties will be briefly discussed below.
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2.1. Real-time, plane-wave DHI In real-time holographic interferometry usually the wave fronts of the undisturbed scene beam are recorded on the holographic plate [5]. For this purpose, the holographic plate is exposed at the same time by both a reference beam and the undisturbed scene beam. During reconstruction the holographic plate is illuminated by the reconstruction beam (i.e., the beam which serves as the reference beam when exposing the holographic plate) and the undisturbed scene beam will be reproduced. If at the same time the disturbed scene beam is generated, an interference pattern is obtained. In the case of an unsteady flow field this interference pattern will show a time-dependent character. This enables one to real-time monitor the evolution of the flow field by a continuously varying interference pattern. However, modern interferogram evaluation procedures require at least three so-called phase-stepped interferograms as input data [3]. These interferograms are generated in time by piezoelectrically translating one of the mirrors in the reconstruction beam path, resulting in a path-length variation and in a corresponding shift of the fringes in the interference pattern [4,6]. To allow a pixelwise determination of the projected density distribution in the flow field, all interferograms must be related to identical flow conditions, i.e., to the realization of the flow field at a unique moment of time. Hence, the time dependency of the interference pattern due to the unsteadiness of the flow must be eliminated. This would be realized if it were possible to record the disturbed scene beam on the holographic plate instead of the undisturbed scene beam. However, as briefly remarked in the introduction, the limited power of CW lasers prevents one from doing so. Hence, it can be concluded that real-time DHI is limited to the investigation of steady flow fields. In real-time DHI plane-wave illumination is generally preferable over diffuse illumination especially if the flow field around a non-transparent object is investigated. As in a series of phase-stepped interferograms the background pixels will not show any difference in intensity, it follows that these pixels (as opposed to foreground pixels) are characterized by relalivelv ~rnall vahle~ c~flhelr rncwhdatic~n intensity
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This enables one to make use of the modulation intensity (which is computed from the digitized interFerograms) as the parameter discriminating between foreground pixels and background pixels. This distinctive information is an essential input to subsequent image-processing routines [4]. However, following the above strategy in the case of diffuse illumination will lead to erroneous decisions as the probability distribution of the modulation intensity o f the foreground pixels (speckles) shows m a x i m u m occurrence of 'zero-modulation-intensity' [ 7 ].
2.2. T~o-reJ~,rence-beam, diJ'/use DHI In this type of D H I diffuse light traverses the flow field. Furthermore, both the disturbed scene beam and the undisturbed scene beam are recorded on one and the same holographic plate by two reference beams which have different propagation directions. One of the reference beams serves to record the undisturbed scene beam and the other serves to record the disturbed scene beam. During reconstruction both beams illuminate the holographic plate at the same time and a steady interference pattern will result. Phase-stepped interferograms are induced by a piezoelectric length variation of one of the reconstruction beam paths. This type of D H I originated in efforts to determine the deformation of diffusely reflecting objects under load [8] and was later applied to flow fields [ 1,2 ]. One of the major problems of two-reference-beam holographic interferometry is the generation of cross reconstructions. The reference/reconstruction beam which served to record the undisturbed scene beam on the holographic plate will not only reconstruct the undisturbed scene beam but also the disturbed scene beam and vice versa. In all, eight reconstructions will be generated: four normal reconstructions and four conjugate reconstructions. Only the four normal reconstructions are of interest and these can be classified into two interfering reconstructions and two undesirable cross reconstructions. The situation in which the cross reconstructions overlap the interference pattern can be avoided by having the reference beams well separated. However, in this case the repositioning sensitivity of the holographic plate, after being subjected to the photographic process, is far lar~er than with the two reference beams close to-
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gether [8]. The occurrence of overlapping cross reconstructions is not an insolvable problem if diffuse light is applied. The two cross reconstructions, which have a small angular separation, will superimpose a decorrelated speckle pattern over the interference pattern in the image plane. In this situation the conventional data-acquisition procedure and the imageprocessing routines apply again. With the above set-up it is possible to make use of a pulsed laser to investigate rapidly-varying unsteady flow fields. In this case the holographic plate is exposed by a pulsed laser serving as light source and is reconstructed by a CW (alignment) laser. The inherent wavelength change must be corrected for by a small angular rotation of one of the reconstruction beams. This process was clearly demonstrated by Breuckmann and Thieme [ 1 ] and by Watt and Vest [2]. Despite the merits of the above described set-up in the area of compressible unsteady flow fields (and of diffusely reflecting deformed objects), some important disadvantages of the use of diffuse light have to be mentioned. In the first place, the superposition of a decorrelated speckle pattern over the interference pattern in the image plane will reduce the contrast of the interference pattern to at least half the maximum value obtainable with plane-wave illumination [8]. Secondly, the statistical predominance of zero-modulated speckles in the foreground space of the interference pattern hampers the investigation of flow fields existing around non-transparent objects (see earlier remarks on this topic). Furthermore, fringe localization in a diffuse illumination interferogram strongly depends on the structure of the flow field and the viewing direction [ 7 ]. Finally, diffuse illumination is disadvantageous compared to plane-wave illumination because of the noisy character of the speckle interference patterns, resulting in less accuracy. Therefore, an attempt was made to combine the advantages of real-time plane-wave DHI (i.e., investigating flow fields around non-transparent objects) with the advantages of two-reference-beam, diffuse DHI (i.e., the introduction of a wavelength change and hence the possible investigation of unsteady flow fields by applying a pulsed laser), resulting in two-
re~'rence-beam, plane-wave DHI.
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3. Two-reference-beam, plane-wave DHI
3. I. Experimental set-up Figure 1 gives a schematic overview of the holographic interferometer used in the experiments. The solid lines indicate the outer dimensions of the mobile, closed construction containing the interferometer. For ease of experimentation, instead of a pulsed laser/ CW alignment laser-combination the light source of the interferometer is a CW Ar ion laser with a maximum output of 1 W at a wavelength of 514.5 nm. By adjusting the rear high reflector of the laser a number of different wavelengths in the interval [454.5 nm, 514.5 nm] can be selected, allowing to simulate the wavelength change. The main laser beam is folded by the mirrors M1 and M2 and split by the beam splitter cube BSC (fig. 1 ) into the reference/reconstruction beam and the scene beam. First the path traversed by the scene beam will be followed. The scene beam reaches via the mirrors M3 and M4 the objective-pinhole combination PH~ which is exactly positioned via the
Ht
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mirror M5 in the focal point of the lens L,. Hence, the flow field in study will be traversed by a scene beam which is collimated. Finally, the scene beam is projected by the lens L2 on the holographic plate H. Starting from the beam splitter cube BSC the reference/reconstruction beam path will now be described. The reference/reconstruction beam enters via the mirror M6 the Twyman-Green interferometer section (outlined by dashed lines in fig. 1). In this interferometer the beam is expanded to a collimated beam by the objective-pinhole combination PH2 and by the lens L3. Subsequently, this beam is partially reflected/transmitted by the beam splitter plate BSP. The reflected beam is again reflected in the mirror MT, mounted on the piezo-electric translator PZT, and illuminates through the beam splitter plate the holographic plate. In the sequel of this paper this beam will be called the first reference/reconstruction beam. The transmitted beam is reflected in the mirror Ms, which can be rotated about two axes, and is reflected again on the beam splitter plate to reach the holographic plate. This beam will be called the second reference/reconstruction beam.
I $1 CW Ar LASER
,,
Y
RI
BSC
V
R2
Sz
R3
":2/-.U
I
-1
L' l
FLOW FELD
Fig. 1. Experimental set-up. BSC:beam splitter cube, BSP:beam splitter plate, H: holographicplate, L~,..., L3: lens, M~, ..., Ma: mirror, PHi, PH2: objective-pinholecombination, PZT: piezo-electrictranslator, R~,..., R3: 2/2-retardation plate, S~, $2: shutter, --: TwymanGreen interfernmeter
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For reasons to be explained later, the Twyman-Green interferometer is slightly misaligned so that two collimated reference/reconstruction beams with slightly different propagation directions are generated.
3.2. Exposing the holographic' plate "\
The holographic plate is exposed in two successive stages. In the first stage the flow field is absent and one of the arms of the T-G interferometer is blocked. By opening the shutter S~ (fig. 1 ) for a short time (8 ms) the undisturbed scene beam is recorded on the holographic plate. In the second stage the flow field is present and the other arm of the T-G interferometer is blocked. Again the shutter S~ in the main laser beam is opened for a short time and the disturbed scene beam is recorded on the holographic plate. The intensity rate of the scene beam and the reference beam in the hologram plane is controlled by the 2/2-retardation plate R~ and the polarizing beam splitter cube BSC. Two other 2/2-retardation plates R2 and R3 serve to choose identical polarization directions for the scene beam and the reference beam. After exposure of the holographic plate it is taken out of the interferometer for photographic processing.
3.3. Reconstructing the holographic plate The reconstruction process is performed in the setup of fig. 1 with the holographic plate back in its original position. In this stage the shutter S~ remains permanently opened and the scene beam path is blocked by closing the shutter S> Furthermore, none of the two arms of the T-G interferometer is blocked so that the holographic plate is illuminated by both reconstruction beams. As remarked before, eight reconstructions will be produced: four converging normal reconstructions and four diverging conjugate reconstructions. However, in the present case of oil:axis holography (i.e., the situation in which the reference/reconstruction beam and the scene beam have different propagation directions), only the normal reconstructions play a role. To compute the propagation directions of these reconstructions the sign conventions of fig. 2 must be k e p t in mind. In this figure the angle between the nnrmnl
nftho
halnornnhic
nlnto
nncl lh~ e.eono heam
/ec
t
Fig. 2. Sign conventions.
is positive and equal to cg and the angle between the normal of the holographic plate and the reference beam is negative and equal to Or. The angles as and Or are related to the situation when the holographic plate is exposed. The angle between the reconstruction beam and the normal of the holographic plate is negative and equal to 0c. Then the angle between the reconstructed scene beam and the normal of the holographic plate, c~'s, is given by the following formula [ 9 ] o!'s=sin ~(~.'. ( s i n % - s i n O~)+sin Oc),
(l)
where ~ is the recording wavelength and 2' is the reconstructing wavelength. In the present situation the following set of equations determines the propagation directions of the four normal reconstructions O~'s,li----sin- '(@ (sin Ces--sin 0~ ) + sin 0 c , ) ,
(2)
ot~,~=sin 2 ' , _ ~( ~-(sin c ~ - s i n 0,.~)+sin 0~2) ,
(3)
a's,2, =sin
,(2~ ( s i n o ~ - s i n 0 r 2 ) + s i n 0 ~ l ) ,
(4)
cQ22=sin
~
(sincg-sin0~2)+sin0~2
.
(5)
Here 0,1, 0r2 and % indicate the respective orientations of the two reference beams and the scene beam when exposing the holographic plate. 0c~ and 0~2 indicate the orientations of the two reconstruction beams. Unequal subscripts to c~'~ refer to cross reconstructions and equal subscripts refer to legal reenn~lruetion~
Obviously
toohtainnninfinilefrin~e
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pattern the propagation directions of the two legal reconstructions must be identical: t ! O(s,l I = OL s,22
~r2 ----~r,
As the angles, as, 0r~ and 0r2 a r e constant once the holographic plate is exposed, the only free parameters in eqs. (2) and (5) are the reconstruction beam angles 0c~ and 0c2. However, taking a look at the setup of fig. 1 and bearing in mind that the mirror M7 is rigidly mounted on the PZT, allowing only translation and no rotation, it will be clear that 0cl = 0r~. This implies that a's.~ is strictly determined. Only the mirror Ms can be rotated to affect the parameter 0~2 and hence a's.22. Suppose that 0c2 deviates from 0r2 by the angle A 0 r 2 , ( 7
)
To satisfy eq. (6), A0r2must be chosen according to the following equation zXO~2~
~- zX0r,
(9)
"~- A 0 r .
If the second reconstruction beam is given the propagation direction prescribed by eqs. (7) and (8), eqs. ( 3 ) and (4) determine the propagation directions of the two cross reconstructions. In fig. 3 the propagation directions of the two cross reconstructions, relative to the propagation direction of the two coinciding legal reconstructions, are given as a function of the angle A0~ between the two reference beams. Furthermore, the angle over which the second reference/reconstruction beam must be rotated, A0r2, has been drawn. Fig. 3 has been computed for the intended use of a pulsed laser for the exposure of the holographic plate and of a CW laser in the reconstruction stage. Accordingly, in the computation the recording wavelength 2 is taken equal to 694.3 nm, being the wavelength of a pulsed ruby laser, and the reconstructing wavelength )~' is taken equal to 632.8 nm, being the wavelength of a CW HeNe laser. The angles as and 0r~(=0c,) are taken from the set-up of fig. 1 and are equal to 7 ° and - 4 5 ° respectively. From fig. 3 it follows that the angular separation between each of the cross reconstructions and the legal reconstructions is almost linearly related with the angle between the reference beams. Also the
(6)
0c2 ~-~ 0r2 + A0r2 .
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(8)
where it is supposed that m 0 r / 0 r l < < 1, m 0 r 2 / 0 r 2 < < 1 and that A0r is the angle between the two reference beams at the moment of exposure
16
(~s. 12 ~ a s . II
14 12
(degree)
10 8 6 4. 2 0
&0,z
-2 -4 -6
-8 -10 -12 -14 -16
I
-20
I
-I0
I
I
0
1 10
(XI,21
-- CI[I, i i
20 ---)
aO, (degree)
Fi2. 3. Prona2atinn directions hi'cross r@(:NnsWll(~tlnn~ ~ n d ~Nt'r~tiNn ~nol~ v~r~u~ ~nole between ruf~ronr~ henrnr
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i /
H7 BSP
H
____
Fz FI I SF
~
[L,
~/ ,,I
~ ' ~ - ~ - ~ ,
J CROSS LEGAL
i il~cR°ss
Fig. 4. Experimental set-up at reconstruction. See also fig. 1. Ft: focal plane of holographic plate H, F2: focal plane of lens L2, SF: spatial filter.
angle over which the second reference/reconstruction beam must be rotated to obtain an infinite fringe pattern shows an almost linear relation. Figure 4 shows a detail of the set-up of fig. 1 as it will look like during reconstruction. In the focal plane of the holographic plate Ft (which does not coincide with the focal plane F2 of the lens L2 because of the wavelength change) the two cross reconstructions are spatially separated from the legal reconstructions. An analogous situation will exist in the focus of the imaging lens which projects the interference pattern on a CCD camera. In this focal plane the cross reconstructions are filtered out by a spatial filter, transmitting only the two (coinciding) legal reconstructions. In this paper no attention is paid to the holographic aberrations caused by the wavelength change. As described by Meier [10], under some circumstances one or more of these aberrations can be made to vanish.
computer memory. In the reconstruction set-up depicted in fig. 4 these are generated in time by a length variation of one of the arms of the T-G interferometer. This length variation is achieved by a computer-controlled voltage on the piezo-electric crystal present in the PZT, translating the mirror M7. The resulting interferogram is stored in the computer memory by digitizing the video signal from the CCD camera. Depending on the number of phase-stepped interferograms needed by the image-processing routines, this process is repeated for different positions of the mirror M 7. Before starting the data-acquisition procedure, it is possible to superimpose arbitrarily oriented reference fringes over the interference pattern. This is achieved by rotating the mirror M8 of the T-G interferometer about two axes by adjusting two micrometers. The superposition of reference fringes is essential when the fringe spacing of the infinite fringe pattern locally approximates the pixel resolution.
3.4. Data-acquisition 4. Experimental results As remarked in the introduction, pixelwise determination of the wave front deformation induced by the flow field requires at least three phase-stepped
To demonstrate the above interferometric set-up, the flow field of an axisymmetric supersonic jet of
i n t e r f e r n ~ r a m ~ In b e d i ~ i t i T e d a n d to h e ~ t n r e d in t h e
a i r is e x a m i n e d
Theieli~urranndedhvamhienlair
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Fig. 5. Interferogram showing reference fringes parallel to jet axis due to wavelength change (2 = 514.5 nm--,2' = 472.7 n m ).
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Fig. 6. Interferogram with reference fringes normal to jet axis.
at rest. The main flow direction of the jet is oriented normal to the scene beam axis. The holographic plate is recorded at the wavelength 2=514.5 nm (visible green). To simulate the wavelength change as will necessarily take place in pulsed laser interferometry (pulsed ruby laser: 2=694.3 n m - . c w HeNe laser: 2' = 632.8 nm) the reconstructing wavelength is chosen in the visible blue equal to 2' =472.7 nm. This makes the quotient 2 ' / 2 appearing in eqs. (2) to (5) almost identical for both situations. The CCD camera viewing the flow field is oriented at 45 ° to the flow direction. Starting from the infinite fringe pattern in the visible green, fig. 5 shows the finite fringe pattern which is obtained after the wavelength change. By adjusting the mirror M8 of fig. 1 the orientation of the reference fringes can be arbitrarily chosen as well as their relative spacing. Figure 6 shows the finite fringe pattern with the reference fringes normal to the jet axis. The mirror Ms is ultimately adjusted so that the infinite fringe pattern of fig. 7 results. In all three figs. 5 to 7 the cross reconstructions have been filtered out by the spatial filter in the focal plane of the imaging lens. Figure 8 shows the distorted interference pattern which re-
distorting presence of the two overlapping cross reconstructions. The interferogram of fig. 7 serves as
~lllt~ i f t h ~ ~ n n l i n l f i l l e r i~ r ~ r n n v o c l
n n e nHt n f t h ~ ~erie~ n f n h a ~ e - ~ l e n n e d
1~ i l l l ~ l r n t o ~ l h o
Fig. 7. Infinite interferogram of an axisymmetric jet of air.
inlerferc~rams
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Fig. 8, Interferogram without spatial filtering,
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Due to the axial symmetry of the jet several characteristics o f the structure of the flow can be derived from this result. The jet is bounded by a shear layer which grows in a downstream direction. Close to the axis of the jet, a series of cells resulting from alternate regions of expansion and compression can be discerned. Discharged by the nozzle at sonic speed and at higher than ambient pressure, the air is expanded by expansion waves to supersonic speed and the density decreases. By reflection of the expansion waves at the free surface of the jet as compression waves, the density regains approximately its sonic value at the axis of the jet, etc. Further downstream the cell structure finally becomes invisible by viscous effects. From fig. 9 artificial Schlieren images can be computed with the knife-edge in an arbitrarily chosen direction. For this purpose, the partial derivatives of the projected density distribution p(x, y) (which is linearly related to the wave front deformation of fig. 9) in the x-direction and in the y-direction, p.~ and p,., are computed using the Sobel difference operator [11]. Combining these results according to p,. cos O+pv sin 0, the Schlieren image with the knifeedge in the 0-direction is obtained. Figure 10 depicts
Fig. 9. Wave front deformation. from which the wave front deformation is computed. Figure 9 shows the result of this computation, yielding a m a x i m u m wave front deformation of 1.05X2'.
Fig. 10. Computed Schlieren image (knife-edge normal to jet axis).
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1 November 1990
ti
,$ -, D Fig. 11. Computed Schlieren image (knife-edge parallel to jet axis)• the computed Schlieren image with the knife-edge oriented normal to the axis of the jet, whereas in fig. 1 1 the knife-edge orientation is parallel to the jet axis. In fig. 11 the shear layer indicating the outer dimensions of the jet is visible and fig. 10 clearly demonstrates the diamond cell structure characteristic for a supersonic jet. As a final result, fig. 12 depicts the axial density distribution in the jet. This density distribution is computed from the projected density distribution of fig. 9 by the inverse Abel transformation, following the numerical scheme suggested by Nestor and Olsen [ 12 ]. The transformation is executed for the first four cells, starting at the nozzle's outlet (top of fig. 12). The deviation of the axial density from the ambient density varies between - 0 . 1 6 k g / m ~ and 1.16kg/m 3. It is interesting to notice that two successive local areas of higher density are separated from each other by a local area of lower density, a phenomenon which is not directly clear from fig. 9 alone. Furthermore, the density of the first m i n i m u m lies below the ambient density.
Fig. 12. Axial density distribution with curves of constant density (curve spacing 0.082 kg/m3). 5. Conclusions and future prospects A successful demonstration of two-reference-beam, plane-wave DHI in the investigation of flow fields was given. Overlapping cross reconstructions are prevented by spatial filtering. Non-transparent objects intersecting the interference patterns are easily detected and successively circumvented by the imageprocessing routines as plane-wave illumination is used instead of diffuse light. Since the reconstructing wavelength is allowed to differ from the recording wavelength in the presented set-up, a pulsed laser can be applied instead of the CW Ar ion laser to make the investigation of unsteady flow fields nossible, The only necessary
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m o d i f i c a t i o n to t h e s e t - u p o f fig. 1 is t h e t r a n s i t i o n to H E L ( high energy level ) o p t i c s for t h e m i r r o r s M j, M > M3, M4 a n d M6. F u r t h e r m o r e , in all p r o b a b i l i t y t h e o b j e c t i v e - p i n h o l e c o m b i n a t i o n s P H i a n d PH2 h a v e to b e r e p l a c e d b y n e g a t i v e lenses to a v o i d foci.
References [ 1 ] B. Breuckmann and W. Thieme, Appl. Optics 24 ( 1985 ) 2145. [2 ] D.W. Watt and C.M. Vest, Exp. Fluids 5 ( 1987 ) 401. [3] J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White and D.J. Brangaccio, Appl. Optics 13 (1974) 2693.
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[4] T.A.W.M. Lanen, C. Nebbeling and J.L. van Ingen, Optics Comm. 76 (1990) 268. [5] P. Hariharan, B.F. Oreb and N. Brown, Optics Comm. 41 (1982) 393. [6] W. Jfiptner, T.M. Kreis and H. Kreitlow, SPIE Proceedings 398 (1983) 22. [ 7] C.M. Vest, Holographic interferometry (Wiley, New York, 1979). [8] R. D~indliker, R. Thalmann and J.F. Willemin, Optics Comm. 42 (1982) 301. [9] M. Franqon, Holography (Academic Press, New York, 1974). [10] R.W. Meier, J. Opt. Soc. Am. 55 (t965) 987. [11 ] A. Rosenfeld and A.C. Kak, Digital picture processing (Academic Press, New York, 1982). [ 12] O.H. Nestor and H.N. Olsen, SIAM Rev. 2 (1960) 200.