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Additive-subtractive phase-modulated speckle interferometry: Fringe visibility under partial decorrelation
Opdcs and Lasers in Engineering
26 (1997) 179-197
Copyright Q 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0143-8166/97/$1500
ELSEVIER
0143-8166(%)001l2-3
Additive-Subtractive Phase-Modulated Speckle Interferometry: Fringe Visibility Under Partial Decorrelation Sridhar
Krishnaswamy,
Bruno
F. Pouet
& Tom C. Chatters*
Center for Quality Engineering and Failure Prevention, Northwestern Evanston, IL 60208-3020, USA (Received 9 September
University,
1994; revised version received and accepted 9 December
1995)
ABSTRACT Additive-subtractive phase modulated speckle interferometry (ASPMSI) is a technique that minimizes the susceptibility of speckle methods to environmental noise while providing fringes of good visibility. The method requires the acquisition of two consecutive video frames of additive-speckle images of the same two deformed states of an object at a rapid enough rate such that ambient noise is not a problem. The additive-speckle images as expected are of very poor visibility due to the presence of the self -interference term. An interframe phase-modulation is introduced and the two additive-speckle images are digitally subtracted to improve the fringe visibility by removing the self -interference term. The ASPM-SI method works with in-plane and out-of-plane deformation sensitive ESPI as well as with displacement-gradient sensitive speckle-shearing interferometry. It is shown that the ASPM-SI scheme has higher visibility than conventional additive-SI and performs consistently better than subtractive-H schemes in the presence of partial interframe speckle decorrelating optical noise. Furthermore, it is shown that the fringe visibility of the out-of-plane displacement sensitive interferometer which uses a protected reference beam separate from the object beam can be made to be essentially unity even at complete interframe decorrelation. Copyright 0 1996 Elsevier Science Ltd.
1 INTRODUCTION
Video-based electronic speckle correlation interferometry has proved to be a very useful laboratory tool for the measurement of full-field surface deformation of diffuse objects. Several different optical configurations are * Present address: Idaho National Engineering USA. 179
Laboratory,
Idaho Falls, Idaho 834152209,
180
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
possible to provide interferometers that are sensitive to in-plane or out-of-plane object displacements (electronic speckle pattern interferometry or ESPI) or displacement gradients (speckle shearing interferometry).‘*’ Typically, speckle patterns from the object are acquired for two deformation states of the object, and these patterns are correlated interferometry using additive or subtractive schemes.1-3 Additive-speckle can be done in an analog manner by double exposure of the camera sensor and therefore can be done rapidly enough that ambient noise does not degrade the signal. Unfortunately, the fringe visibility of pure additive schemes is very poor due to the added presence of self-interference terms that do not contain information about the object deformation.4 Subtractive-S1 schemes have fringe visibility of close to unity under noise-free conditions, but since the subtraction process has to be digital, these schemes are substantially slower and are very difficult to use outside of a noise-free laboratory. Noise in an industrial environment can range from large amplitude but typically low frequency motion due to object drift and low frequency building vibrations, to higher frequency but typically smaller amplitude building vibrations and thermal currents. Large amplitude low frequency noise can cause speckle decorrelation and hence complete loss of fringes in the case of interferometers operating in the subtractive mode. The higher frequency noise sources generally do not lead to decorrelation but can cause significant optical phase variation leading to fringe distortion5-’ The low frequency noise susceptibility of subtractive41 may be substantially reduced by synchronizing the optical interferometer and the object stressing system with the image acquisition and processing system, and by performing what amounts to a repetitive sequence of rapid SI tests;’ By this scheme a stable fringe pattern can be obtained as long as the ambient noise is of substantially lower frequency than the video acquisition rate (typically 30Hz). Using this approach the possibility of noiseinduced speckle decorrelation is essentially eliminated. More recently, we have described a hybrid additive-subtractive phase-modulation (ASPM) technique that can freeze out the fringe distortions caused by unwanted environmental noise of frequencies much higher than video rates.- The basic principle of the ASPM-SI technique can be summed up as follows: (i) acquire additive speckle images containing information about the same two deformed object states in every frame of the video image acquisition sequence, (ii) suitably scramble the random speckle phases between consecutive frames, and (iii) subtract every consecutive pair of speckle images to extract visible fringes corresponding to the relative deformation between the two deformed states of the test object. The advantage of this scheme is that the additive speckle images can be obtained very rapidly
ASPM-SI
181
by double exposure such that noise-induced decorrelation or phase shifts do not occur during this process. The resulting additive speckle images however have the poor visibility typical of additive speckle correlation techniques. To mitigate this, the phase modulation and digital subtraction steps are taken. Under noise-free conditions, the resulting fringes are of unit visibility. It will be shown in this paper that even under conditions where noise is important upto video acquisition rates or higher, the resulting fringe visibility can be substantially increased in the case of the out-of-plane displacement sensitive interferometer.
2 ADDITIVE-SUBTRACTIVE PHASE-MODULATED INTERFEROMETRY
SPECKLE
The steps involved in ASPM-SI are described below in detail. The test object is subject to repetitive deformation through acoustic (vibrational) stressing or other similar meanss4 A continuous sequence of speckle patterns is recorded at video framing rates (30 frames/second). The primary speckle image that is recorded in each frame of the sequence is due to the interference of two coherent beams one of which is always speckled and the other can be either speckled or uniform. In the case of ESPI with out-of-plane deformation sensitivity (Fig. l(a)) (hereafter referred to as ASPM-ESPI-out), one of these beams is the scatter from the object and is therefore speckled, whereas the reference beam is uniform. In the case of ESPI with in-plane sensitivity (Fig. l(b)) (hereafter referred to as ASPM-ESPI-in) or in the case of out-of-plane speckleshearing interferometry (Fig. l(c)) (hereafter referred to as ASPMshearography), both the beams that form the primary image are speckled since both arise from the object. Step 1 Additive-S1 images of the same two deformed object states are acquired in frame No of the video sequence. The resulting intensity recorded on the camera is: INo= %&I + W&, + t@b, + u,,u: + u&J,*,+ ub&* + ~~% + u,,&
(1)
where u is the complex wavefront amplitude for the ZV,,thframe; subscripts ‘a’ and ‘b’ refer to the two legs that interfere; subscripts ‘1’ and ‘2’ refer to the two deformed states of the object; and the asterisk represents complex conjugation. The above addition of the two primary speckle images corresponding to the object in its two deformed states can be done very
Sridhar Krishnaswamy,
182 (4
Bruno F. Pouet, Tom C. Chatters Imaging Lsn¶
Obiect
‘a’_ /
d
/
/
D
/’ Lag‘b‘
0))
Imaging LMIS
Obiect
/ / / / / *
F
\
\
k \
a
Phase-tiulator
CCDArray
Fig. I. (a) ASPM-ESPI-out for out-of-plane deformation sensitivity. (b) ASPM-ESPI-in for in-plane deformation sensitivity. (c) ASPM-shearography for out-of-plane deformation gradient sensitivity.
ASPM-Si
183
rapidly in an analog fashion by double exposing the camera sensor array. It can therefore be assumed that the resulting additive speckle image is acquired without noise-induced speckle decoirelation or even spurious phase shift. This additive-S1 image is digitized and stored. Step 2 An electro-optic modulator is synchronized with the image processor so as to introduce a spatially uniform inter-frame phase shift of Q in leg ‘b’ of the interferometer. Once again, additive speckle interference images of the same two deformed object states (as in step 1) are acquired in frame N1 of the video sequence (typically frames N1 and No are consecutive). The resulting intensity recorded on the camera is then: &I*= %,U,*,+ &&
+ +bl
+ &&I + %,U,*,+ &&* + u,*,ub2+ u,,ux,
(2)
where the various wavefronts are denoted by u for frame ZV1.This second additive-S1 image is also digitized and stored. Step 3 The two acquired additive speckle images are digitally subtracted in the image processor and the square of the result is output on the display monitor to obtain the ASPM-SI correlation fringe pattern given by: (4) = WN, - LJ2), where ( ) represents to yield:
(3)
ensemble averaging. Expression
(3) can be expanded
(4) = (A2) + 03’) + X4@, where the intensity-type terms wavefront) are grouped as:
(involving
A = u,,u,*, - u,,u,*, + u,,u,*, - IJ&
and the phase-type grouped as:
+ ub&
combinations
+ u,&
of the same
- u,&~ + ubu& - I+,&
terms (involving combinations
B = u$,u,,~- u,*,u,, + u,*2ub2- u:&
(4)
(5)
of the different legs) are
- u,,u:~ + u,,ut - u,u&
(6)
The above process is repeated continuously in real time using a dedicated image processor system. Following Goodman,9 we assume that the speckle phase obeys uniform statistics of zero mean, and that the speckle amplitude obeys circular Gaussian statistics (and therefore the speckle intensities obey negative exponential statistics). Furthermore the
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Sridhar Krishnaswamy,
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speckle amplitude and phase are stochastically independent. Under these conditions, the term (AB) identically vanishes in expression (4). The remaining terms can be evaluated only if we prescribe the amount of noise-induced decor-relation between the eight primary wavefronts that form the ASPM-SI interferogram. For simplicity, we will neglect any deformation-induced speckle decorrelation as this has been studied extensively by Slettemoen4 for additive-S1 and by Owner-Petersen” for subtractive-SI. Furthermore, as argued earlier, we will not allow for the possibility of noise-induced decorrelation or spurious speckle phase-shift in the acquisition of the primary additive-speckle interferograms since this can be done in a very rapid analog manner. We will, however, allow for the possibility of noise-induced speckle decorrelation between steps ‘1’ and ‘2’ since this has to be done digitally and is consequently a slower process. In this paper, the term speckle decorrelation is therefore used only in this limited sense of decorrelation that is noise-induced and that occurs over a time scale that is significant only for the process of digital subtraction of the two primary additive-S1 images. Perfect correlation in the total absence of noise In the total absence of ambient optical phase-shifting or decorrelating noise, the u and LJ wavefronts are identical except for the ‘b’ leg wavefronts which are intentionally modified by the phase-modulator by a spatially uniform constant phase. Therefore, under total absence of noise it is apparent that the intensity-type term (A’) identically vanishes as well. In this case we can relate the various wavefronts as follows: W32=
u,le’M-;
ub2 =
ubleiMb
(w
where M, and Mb are the deformation-induced phase shifts between states ‘1’ and ‘2’ in legs ‘a’ and ‘b’, respectively (note that Mb = 0 for ASPM-ESPI-out). u al
=
&I,;
uti
=
uti
m
=
u,,e ih4..,
vbl = ublein
where R is the phase-modulation
that electronic
(74
induced in leg ‘b’ only;
ub2 = &e” Assuming
(74
= ubletieiMb.
(7e)
noise (in the camera sensor and the digitization
ASPM-SI
process) is insignificant, given by:
the resulting
(4) = 32(~&)
185
ASPM-SI
correlation
fringes are
sin 2[;]cos2[“;“],
are the average intensities of the leg ‘a’ and leg ‘b’ beams, respectively. Clearly it would be most advantageous to have the induced phase modulation G = n. Defining the fringe visibility as: V = (qrnax)(4max)
+
(4min) (4min)
(IO)
'
it is clear that though the resulting cosine-squared fringes of ASPM-SI are of the additive-S1 type, the fringe visibility is unity, a level which is not obtainable with additive-S1 even in a noise-free environment.4 This is true for all three configurations shown in Fig. 1. A typical ASPM-ESPI-out fringe pattern obtained in a noise-free situation is shown in Fig. 2(A). In this and in all the other experimental results cited in this paper, the test object used was an aluminum plate (2-5 X 25 X 30-5 cm) containing a centrally located flat-bottomed hole of 7.6 cm diameter leaving a O-08 cm thick membrane that acted as a circular plate which was acoustically excited by a broadband piezoelectric transducer to activate one of its resonant modes of vibration. A custom-built electronic synchronization system allowed us to control the excitation of the transducer and the object illumination (using an acousto-optic
Fig. 2.
(A) ASPM-ESPI-out.
(B) Conventional
subtractive
ESPI.
186
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
modulator to shutter the laser beam) such that only the relative deformation corresponding to the maximum and minimum of the acoustic cycle were obtained. The synchronization electronics also controlled the electro-optic phase modulator to provide the interframe phase shifting required in step 2. As seen from the figure, the fringes obtained using the ASPM method (Fig. 2(A)) are of squared-cosine nature which is the complement of that obtained using conventional subtractive-ESPI (Fig. 2(B)) where the fringes are of squared-sine nature. Complete noise-induced decorrelation If there is ambient noise that causes complete speckle decorrelation between steps ‘1’ and ‘2’, then the u and u wavefronts are no longer the same in all cases, and we have to analyze the different configurations in Fig. 1 separately. For the ASPM-ESPI-out setup (Fig. l(a)), we can ensure that the reference leg ‘b’ wavefronts are protected (by piping through a fiber-optic system for instance) such that only the object leg ‘a’ wavefronts are affected by ambient noise from the object or between the object and the camera. In this case, we have: KG?= ualeW*;
is& ubl
=
uble
,
Ub2
vb2
=
=
Ubl;
ub’&‘”
W)
=
uble’“;
WC)
ual are assumed to be uncorrelated due to complete interframe Q, decorrelation of the object leg. Note however that for the average intensity of the ‘a’ wavefront eqn (9a) still holds. The ASPM-ESPI-out correlation fringes in this case are given by: and
(4) = f3V,J2 + 16(&)(1,) COs2
(12)
,
where we have set the induced phase modulation Q = x as before, and we find that the fringe visibility in this case becomes: V =-
1 1 +x’
@a) is the object-to-reference
where x For the case of ASPM-ESPI-in =-
03) beam-intensity
ratio.
(Ib>
(Fig. l(c)) where both wavefronts
(Fig. l(b)) or for ASPM-shearography ‘a’ and ‘b’ forming the primary speckle
ASPM-SI
187
image arise from the object, both legs will be subject decorrelation. In this case, the wavefronts are given by:
to interframe
uQ = u,,eW*;
ub2 = ubleiMb;
Wa)
ua2 = ualeiM’;
ub2 = q,le’Mb;
Wb)
where ual, ual are uncorrelated as are r&l, &,l due to interframe decorrelation of both legs. (It may be guessed that the inter-frame phase-modulation is unnecessary in this case since the induced phase modulation will be swamped by the noise-induced phase decorrelation.) The correlation fringes in this case are given by:
[“‘1Mb],
(4) = 8W2 + (Ibj2) + 16(za&)COS2
(15)
and so the fringe visibility reduces to: v= X=;;;B
x 1+x+x2;
is the leg ‘a’ to leg ‘b’ beam-intensity
(16) ratio. Figure 3 shows a plot
b
of ASPM-ESPI-out and ASPM-shearography fringe visibilities as a function of beam-intensity ratio X for the case of complete interframe decorrelation of the wavefronts arising from the object. For ASPM-ESPIout it is clear from Fig. 3 that by weakening the intensity of the object leg
t>
0.8-
0.6v - ASPM-ESPI-out (theoretical) - -- ASPM-shearography (theoretical) 0 ASPM-ESPI-out (experimental) 0.2
r
Fig. 3.
Visibility
as a function
of beam intensity ratio X for the case of complete interframe decorrelation.
188
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
with respect to the reference leg, the fringe visibility can be enhanced to close to unity! This is because the noise-induced decorrelation is only in the object leg, whereas the reference leg is protected so as to be noise-free. This increase in visibility is clearly illustrated experimentally in the ASPM-EPSI-out fringe patterns shown in Fig. 4(a)-(c). To get a quantitative comparison between theory and experiment, the experimental fringe visibility was measured from these sequences as follows. The average brightness along lines of constant maximum and also minimum displacement-induced phase (i.e. along centers of bright and dark vibrational fringes) were computed from the digitized images. The resulting fringe visibility was then computed for each case and these are plotted in Fig. 3. The agreement between theory and experiment is quite good though the experimental values are somewhat lower than predicted due to other sources of error arising from electronic noise and the limited dynamic range of the camera used in this particular experiment which are not included in the analysis. In fact, it must be pointed out that from eqn (13) it appears that unit visibility will be obtained at X = 0, that is by turning off the object beam! This is obviously incorrect and the reason for that is as the signal term becomes weaker and weaker the effect of electronic noise, which should appear as an additional term in eqn (12), becomes significant and is no longer negligible as we assumed. The issue of electronic noise is explained in greater detail in Ref. 7. For the case of ASPM-ESPI-in and ASPM-shearography, the maximum fringe visibility at total decorrelation is only V = l/3 which occurs when both legs are of equal intensity. This is not surprising since both legs are now subject to decorrelation since they both arise from the object. Thus it appears that it would be best to use ASPM-ESPI-out (or in any case ASPM-ESPI with only one object leg and one protected reference leg even if the sensitivity vector is not aligned so as to give only the out of plane displacement of the object) with a weak object leg if noise induced visibility degradation is of concern. Partial decorrelation The intermediate case of partial noise-induced interframe decorrelation is considerably more complex to analyze. To get a quantitative measure, the degree of correlation between the u and u wavefronts needs to be established. Noise-induced decorrelation can be due to a variety of reasons (object drift, ambient vibrations, thermal currents in the air between the object and the camera, etc.), and it is in general impossible to relate the distortion in the u and u wavefronts unless the details of the noise are given. Therefore, we will restrict attention to ASPM-ESPI-out
ASPM-SI
189
Fig. 4. ASPM-.ES ?PI cJut images at complete interframe decorrek xtion. Note 6-inge visibi llity increas ies as cjbject beam intensity is reduced: (a) X = 1; (b) x = l/25; ; Cc) x = l/112.
190
Sridhar Krishnaswamy, Bruno F. Pouet, Tom C. Chatters
(Fig. l(a)) and ASPM-shearography (Fig. l(c)) configurations only. Furthermore we will consider only the case of decorrelation arising from interframe in-plane translation of the object. In this case, the various wavefronts are again given by eqns (lla)-(llc) for ASPM-ESPI-out and by eqns (14a)-(14b) for ASPM-shearography, but now ual, ual and t&l, ub,, etc., are no longer totally uncorrelated because they arise from the same resolution element on the object except for a possible in-plane translation of the resolution element itself. Therefore, the u and corresponding u wavefronts at the image plane arising from the same resolution element on the object can be related through a correlation coefficient in a manner described by Owner-Petersen. lo The process involves tracking the scattered wavefronts from a resolution element on the object plane as it propagates in space to the imaging lens and then diffracts through the lens pupil to the image plane. The process is repeated for the scatter from the same resolution element after it is displaced in the plane of the object. The details are sketched in the Appendix. For the ASPM-ESPI-out set-up, the resulting correlation fringes can be shown to be given by?’ (q) = 8(z,)*(l - c’) + 16(Z&)(l+ and for ASPM-shearography
the correlation
[ 1,
c) cos* !$
(17)
fringes are given by:”
(4) = WJ2 + (zb)*Nl- c*) + 16(1,)(&)(1+ c*) cos* [“, where the correlation coefficient due to in-plane interframe real-valued number given by:
“1, translation
(IS) is a
which depends on the amount of in-plane interframe object translation distance IdIaccording to the Bessel function of the first order; and D is the aperture size and LO is the distance between the object and the pupil plane of the imaging lens (see Fig. Al). The value of the correlation coefficient indicates the level of in-plane translation-induced decorrelation: i.e. full-correlation c = 1 occurs when the interframe in-plane translation IdI= 0 and total-decorrelation c = 0 occurs when the interframe in-plane translation Jdl= 1.22AL0/D = IdmaxI.A plot of the correlation coefficient
191
ASPM-SI
0
0.4
0.2
0.6
0.8
1
Normalized Translation Fig. 5.
Variation
of the correlation
coefficient c with normalized distance ldl/ld,,l.
as a function of the normalized in-plane translation Fig. 5. The associated fringe visibility for ASPM-ESPI-out V=
1
translation
distance is shown in becomes:
x=i!?L)
1+(1+x;
and for ASPM-shearography
in-plane
UtJ ’
the visibility becomes:
(1 + c’)X v = (1 - c2) + (1 + 3)X + (1 - cqx*
*
;
(21)
Clearly, the above visibility expressions reduce to the values obtained previously for perfect correlation (c = 1) and for complete decor-relation (c = 0) as they must. The fringe visibilities for conventional subtractive speckle interferometry have been developed by Owner-Petersen” and are reproduced here for subtractive-ESPI: V= and for subtractive
2c
x
2+(1-c*)x;
-
(ZJ (lb)
(22)
’
shearography: v=
2c*x 2c*x + (1 - c’)(l + X2) *
(23)
192
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
0.8
0.6
V 0.4
0 0
0.4 0.6 0.8 0.2 NonmlizedTranslation Distance
1
(a) ESPI-out
V
\ 0.20
. -.
\ '\ l. , *-.*-___ , I 0.6 0.8 1
. . ., . . . ,. . 1 0
0.4
0.2
Normalized Translation Distance
(b) Shearography Fig. 6. Fringe visibility as a function
of normalized interframe ratio X = 1).
translation
distance
ldl/ld,,l (for beam intensity
The fringe visibilities for the ASPM techniques are plotted in Fig. 6(a) and (b) for different levels of decorrelation and for unit beam-intensity ratio. Also plotted for comparison are the fringe visibilities for the conventional subtractive-ESPI and speckle-shearing interferometers. Note that for the same levels of decorrelation the ASPM method produces fringes with
ASPM-SI
193
consistently higher visibility than conventional subtractive methods. This is of course due to the fact that in the ASPM method additive correlation fringes are obtained within each video frame without any decorrelation. The interframe decorrelation only contributes to degrading the cancellation of the self-interference intensities in the ASPM method, whereas in the subtractive schemes it affects the very formation of the subtractive correlation fringes. The above theoretical development is borne out by the experimental results shown in Fig. 7(a) and (b) where the acoustically vibrating object has been intentionally translated between frames by various amounts. The beam-intensity ratio was kept at unity. A very interesting feature of the fringe patterns in Fig. 7(a) is the presence of two sets of fringes as the vibrating object is translated: one corresponding to the object vibration which is acquired in the additive-EPSI sense, and the other corresponding to the inter-frame phase shifts due to object translation between frames and which is acquired in the subtractive sense. These fringes appear because the experimental configuration was such that the object illumination was not normal and so the interferometer was somewhat sensitive to in-plane motion. These translational fringes appear multiplicatively superposed on each other. As expected translation fringes do not appear in the displacement-gradient sensitive shearography sequence shown in Fig. 7(b). Note however that as the object is translated in its own plane inter-frame speckle decorrelation occurs. An important point to note is that whereas inter-frame total decorrelation completely kills the formation of correlation fringes in the conventional scheme, it only affects the visibility for the ASPM scheme.
3 CONCLUSION The salient points of this paper are: (i) additive-subtractive phasemodulated speckle interferometry provides both the good noise immunity of conventional additive techniques as well as the good fringe visibility of conventional subtractive methods; (ii) experimental data as well as fringe visibility analyses for noise-free and partial to complete interframe speckle decorrelation conditions show that the technique provides consistently better visibility than conventional schemes; and (iii) the fringe visibility of the out-of-plane displacement sensitive interferometer (with one object leg and a separate protected reference leg) can be made to be essentially unity even under conditions of total interframe speckle decorrelation.
194
Sridhar Krishnaswamy,
Bruno F. Pouel, Tom C. Chatrers
(a) ASPM-ESPI
(b) ASPM-shearography Fig. 7.
(a) ASPM-ESPI and (b) ASPM-shearography images of defect vibrating at 1.2 kHz as object undergoes interframe translation of ldl/&,,l = 0, l/6, l/3, l/2,2/3 and 1 (from top left to bottom right).
ASPM-SI
195
REFERENCES 1. Jones, R. & Wykes, C., Holographic and Speckle Interferometry. Cambridge University Press, Cambridge, 1989. 2. Sirohi, R. S., Speckle Metrology. Marcel Dekker, New York, 1993. 3. Spooren, R., Double-pulse subtraction TV holography. Opt. Engng, 31 (1992), looo-1007. 4. Slettemoen, G. A., General analysis of fringe contrast in electronic speckle pattern interferometry. Optica Acta, 26 (1979), 313-327. 5. Pouet, B. F., Chatters, T. C. & Krishnaswamy, S., Synchronized reference updating technique for electronic speckle interferometry. J. Nondestructiue Evaluation, 12 (1993), 133-138. 6. Pouet, B. F. & Krishnaswamy, S., Additive-subtractive decorrelated ESPI. Opt. Engng, 32 (1993), 1360-1369. 7. Pouet, B. F. & Krishnaswamy, S., Additive-subtractive phase-modulated electronic speckle interferometry: Analysis of fringe visibility. Appf. Optics, 33 (1994), 6609-6616. 8. Chatters, T. C., Pouet, B. F. & Krishnaswamy, S., Additive-subtractive phase modulated shearography with synchronized acoustic stressing. Exper. Mech., 35 (1995). 159-165. 9. Goodman, J. W., Statistical properties of laser speckle patterns. In Laser Speckle and Related Phenomena, ed. J. C. Dainty. Springer-Verlag, 1984. 10. Owner-Petersen, M., Decorrelation and fringe visibility: on the limiting behavior of various electronic speckle-pattern correlation interferometers. J. Opt. Sot. Am. A8 (1991), 1082-1089. 11. Chatters, T. C., Noise reduction techniques in optical nondestructive evaluation. PhD dissertation, Northwestern University, 1994.
APPENDIX The u and corresponding u wavefronts that come from the object can be related through a correlation coefficient as described by Owner-Petersen.” The process involves tracking the scattered wavefronts from a resolution element on the object plane as it propagates in space to the imaging lens and then diffracts through the lens pupil to the image plane (see Fig. Al). The process is repeated for the scatter from the same resolution element after it is displaced in the plane of the object. In this analysis, we make the following assumptions about the wavefronts scattered from a diffuse object: (i) plane wavefront illumination normal to the object provides a scattered object wavefront; (ii) the object wavefront is imaged by a thin lens of aperture D and focal length fL onto the image plane, and therefore the field at any point on the image plane comes only from a small resolution element on the object; (iii) speckle decorrelation due to in-plane inter-frame translation d of the object is significant, but decorrelation effects due to object deformation itself are negligible; (iv) load-induced phase-
Sridhar Krkhnaswamy,
196
Bruno F. Pouet, Tom C. Chatters
c
imaging Object
Lens
I
t I0
----------
T---------7
-J+
L-
Fig. Al.
image
-__li----,
Diffraction of scattered wavefront from object to image plane.
shift on the object is constant across the resolution element which is true if the fringe spacing is much larger than the resolution element size. Under the above assumptions, it can be shown”‘-” that the wavefront Uim(cm) at position q, = (Xi,,yh) on the image plane is related to the scattered wavefront u&c,) from a resolution element at position r,, = (x,, y,) on the object plane through: Uim(lm) = F(ri,; Lo, L&
la P(fp)k,(fp>e2”f~~fP df,,
(AlI
--m
where fi, = Q,,,LJL~ FL;
is related to the image plane coordinates
L, L)
e
= AZ; z.
rim; and
-i(21~/A)(&,+L&~-i(n/AL&x&+y2,) ,
WI
Cl IIn
is a phase and amplitude scaling term; &,(f,) is the Fourier transform of the object beam u,(r,) and is given by: u,(ro)e2”if~.‘odr,;
643)
A is the wavelength of light used; L, is the object to lens distance and Li, is the lens to image plane distance. The aperture function is given by: I DI2AL, for IfpI
> DI2AL,
’
(A4)
where f, = r,/AL, is related to the pupil plane coordinate rp = (x,, yp). If now the object is translated in its own plane by an amount d, then the scattered wavefront at the object plane from the same resolution element as above is given by: u,(r,) = u,(r,, - d)e*‘“, (As)
ASPM-SI
where &R is an intentionally image plane wavefront is then:
induced
Uim(rim)= F(ri,; Lo, Li,)e*in In computing the average evaluate terms such as:
=e Tin
where an unimportant simplifies to:
brightness
197
phase modulation.
The corresponding
I-m P(fp)ii,(fp)eZrid*f~eznie*f~df,. of the correlation
fringes,
we need to
P(fp)P(f#&,(fp)ii,(f~))e2tid’f~e2niL~(f~-~) df, df& scaling (magnification) Gh&&h(bJ>
where (I,) = (u,u$ is the average coefficient is given by: c=i
(A7)
factor has been ignored. The above
= e”“4J,
object
646)
beam intensity,
J-1 P2(fp)e2”idmfpdfp,
VW
and the correlation
(A9)
where A, = 7rD2/4 is the area of the circular pupil aperture. The correlation coefficient evaluates to the expression in eqn (19). Expressions similar to eqn (A8) that are needed in the evaluation of eqns (8), (17) and (18) are obtained in like manner by using the appropriate wavefronts in legs ‘a’ and ‘b’ and states ‘1’ and ‘2’ as the object is deformed and/or translated.
26 (1997) 179-197
Copyright Q 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0143-8166/97/$1500
ELSEVIER
0143-8166(%)001l2-3
Additive-Subtractive Phase-Modulated Speckle Interferometry: Fringe Visibility Under Partial Decorrelation Sridhar
Krishnaswamy,
Bruno
F. Pouet
& Tom C. Chatters*
Center for Quality Engineering and Failure Prevention, Northwestern Evanston, IL 60208-3020, USA (Received 9 September
University,
1994; revised version received and accepted 9 December
1995)
ABSTRACT Additive-subtractive phase modulated speckle interferometry (ASPMSI) is a technique that minimizes the susceptibility of speckle methods to environmental noise while providing fringes of good visibility. The method requires the acquisition of two consecutive video frames of additive-speckle images of the same two deformed states of an object at a rapid enough rate such that ambient noise is not a problem. The additive-speckle images as expected are of very poor visibility due to the presence of the self -interference term. An interframe phase-modulation is introduced and the two additive-speckle images are digitally subtracted to improve the fringe visibility by removing the self -interference term. The ASPM-SI method works with in-plane and out-of-plane deformation sensitive ESPI as well as with displacement-gradient sensitive speckle-shearing interferometry. It is shown that the ASPM-SI scheme has higher visibility than conventional additive-SI and performs consistently better than subtractive-H schemes in the presence of partial interframe speckle decorrelating optical noise. Furthermore, it is shown that the fringe visibility of the out-of-plane displacement sensitive interferometer which uses a protected reference beam separate from the object beam can be made to be essentially unity even at complete interframe decorrelation. Copyright 0 1996 Elsevier Science Ltd.
1 INTRODUCTION
Video-based electronic speckle correlation interferometry has proved to be a very useful laboratory tool for the measurement of full-field surface deformation of diffuse objects. Several different optical configurations are * Present address: Idaho National Engineering USA. 179
Laboratory,
Idaho Falls, Idaho 834152209,
180
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
possible to provide interferometers that are sensitive to in-plane or out-of-plane object displacements (electronic speckle pattern interferometry or ESPI) or displacement gradients (speckle shearing interferometry).‘*’ Typically, speckle patterns from the object are acquired for two deformation states of the object, and these patterns are correlated interferometry using additive or subtractive schemes.1-3 Additive-speckle can be done in an analog manner by double exposure of the camera sensor and therefore can be done rapidly enough that ambient noise does not degrade the signal. Unfortunately, the fringe visibility of pure additive schemes is very poor due to the added presence of self-interference terms that do not contain information about the object deformation.4 Subtractive-S1 schemes have fringe visibility of close to unity under noise-free conditions, but since the subtraction process has to be digital, these schemes are substantially slower and are very difficult to use outside of a noise-free laboratory. Noise in an industrial environment can range from large amplitude but typically low frequency motion due to object drift and low frequency building vibrations, to higher frequency but typically smaller amplitude building vibrations and thermal currents. Large amplitude low frequency noise can cause speckle decorrelation and hence complete loss of fringes in the case of interferometers operating in the subtractive mode. The higher frequency noise sources generally do not lead to decorrelation but can cause significant optical phase variation leading to fringe distortion5-’ The low frequency noise susceptibility of subtractive41 may be substantially reduced by synchronizing the optical interferometer and the object stressing system with the image acquisition and processing system, and by performing what amounts to a repetitive sequence of rapid SI tests;’ By this scheme a stable fringe pattern can be obtained as long as the ambient noise is of substantially lower frequency than the video acquisition rate (typically 30Hz). Using this approach the possibility of noiseinduced speckle decorrelation is essentially eliminated. More recently, we have described a hybrid additive-subtractive phase-modulation (ASPM) technique that can freeze out the fringe distortions caused by unwanted environmental noise of frequencies much higher than video rates.- The basic principle of the ASPM-SI technique can be summed up as follows: (i) acquire additive speckle images containing information about the same two deformed object states in every frame of the video image acquisition sequence, (ii) suitably scramble the random speckle phases between consecutive frames, and (iii) subtract every consecutive pair of speckle images to extract visible fringes corresponding to the relative deformation between the two deformed states of the test object. The advantage of this scheme is that the additive speckle images can be obtained very rapidly
ASPM-SI
181
by double exposure such that noise-induced decorrelation or phase shifts do not occur during this process. The resulting additive speckle images however have the poor visibility typical of additive speckle correlation techniques. To mitigate this, the phase modulation and digital subtraction steps are taken. Under noise-free conditions, the resulting fringes are of unit visibility. It will be shown in this paper that even under conditions where noise is important upto video acquisition rates or higher, the resulting fringe visibility can be substantially increased in the case of the out-of-plane displacement sensitive interferometer.
2 ADDITIVE-SUBTRACTIVE PHASE-MODULATED INTERFEROMETRY
SPECKLE
The steps involved in ASPM-SI are described below in detail. The test object is subject to repetitive deformation through acoustic (vibrational) stressing or other similar meanss4 A continuous sequence of speckle patterns is recorded at video framing rates (30 frames/second). The primary speckle image that is recorded in each frame of the sequence is due to the interference of two coherent beams one of which is always speckled and the other can be either speckled or uniform. In the case of ESPI with out-of-plane deformation sensitivity (Fig. l(a)) (hereafter referred to as ASPM-ESPI-out), one of these beams is the scatter from the object and is therefore speckled, whereas the reference beam is uniform. In the case of ESPI with in-plane sensitivity (Fig. l(b)) (hereafter referred to as ASPM-ESPI-in) or in the case of out-of-plane speckleshearing interferometry (Fig. l(c)) (hereafter referred to as ASPMshearography), both the beams that form the primary image are speckled since both arise from the object. Step 1 Additive-S1 images of the same two deformed object states are acquired in frame No of the video sequence. The resulting intensity recorded on the camera is: INo= %&I + W&, + t@b, + u,,u: + u&J,*,+ ub&* + ~~% + u,,&
(1)
where u is the complex wavefront amplitude for the ZV,,thframe; subscripts ‘a’ and ‘b’ refer to the two legs that interfere; subscripts ‘1’ and ‘2’ refer to the two deformed states of the object; and the asterisk represents complex conjugation. The above addition of the two primary speckle images corresponding to the object in its two deformed states can be done very
Sridhar Krishnaswamy,
182 (4
Bruno F. Pouet, Tom C. Chatters Imaging Lsn¶
Obiect
‘a’_ /
d
/
/
D
/’ Lag‘b‘
0))
Imaging LMIS
Obiect
/ / / / / *
F
\
\
k \
a
Phase-tiulator
CCDArray
Fig. I. (a) ASPM-ESPI-out for out-of-plane deformation sensitivity. (b) ASPM-ESPI-in for in-plane deformation sensitivity. (c) ASPM-shearography for out-of-plane deformation gradient sensitivity.
ASPM-Si
183
rapidly in an analog fashion by double exposing the camera sensor array. It can therefore be assumed that the resulting additive speckle image is acquired without noise-induced speckle decoirelation or even spurious phase shift. This additive-S1 image is digitized and stored. Step 2 An electro-optic modulator is synchronized with the image processor so as to introduce a spatially uniform inter-frame phase shift of Q in leg ‘b’ of the interferometer. Once again, additive speckle interference images of the same two deformed object states (as in step 1) are acquired in frame N1 of the video sequence (typically frames N1 and No are consecutive). The resulting intensity recorded on the camera is then: &I*= %,U,*,+ &&
+ +bl
+ &&I + %,U,*,+ &&* + u,*,ub2+ u,,ux,
(2)
where the various wavefronts are denoted by u for frame ZV1.This second additive-S1 image is also digitized and stored. Step 3 The two acquired additive speckle images are digitally subtracted in the image processor and the square of the result is output on the display monitor to obtain the ASPM-SI correlation fringe pattern given by: (4) = WN, - LJ2), where ( ) represents to yield:
(3)
ensemble averaging. Expression
(3) can be expanded
(4) = (A2) + 03’) + X4@, where the intensity-type terms wavefront) are grouped as:
(involving
A = u,,u,*, - u,,u,*, + u,,u,*, - IJ&
and the phase-type grouped as:
+ ub&
combinations
+ u,&
of the same
- u,&~ + ubu& - I+,&
terms (involving combinations
B = u$,u,,~- u,*,u,, + u,*2ub2- u:&
(4)
(5)
of the different legs) are
- u,,u:~ + u,,ut - u,u&
(6)
The above process is repeated continuously in real time using a dedicated image processor system. Following Goodman,9 we assume that the speckle phase obeys uniform statistics of zero mean, and that the speckle amplitude obeys circular Gaussian statistics (and therefore the speckle intensities obey negative exponential statistics). Furthermore the
184
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
speckle amplitude and phase are stochastically independent. Under these conditions, the term (AB) identically vanishes in expression (4). The remaining terms can be evaluated only if we prescribe the amount of noise-induced decor-relation between the eight primary wavefronts that form the ASPM-SI interferogram. For simplicity, we will neglect any deformation-induced speckle decorrelation as this has been studied extensively by Slettemoen4 for additive-S1 and by Owner-Petersen” for subtractive-SI. Furthermore, as argued earlier, we will not allow for the possibility of noise-induced decorrelation or spurious speckle phase-shift in the acquisition of the primary additive-speckle interferograms since this can be done in a very rapid analog manner. We will, however, allow for the possibility of noise-induced speckle decorrelation between steps ‘1’ and ‘2’ since this has to be done digitally and is consequently a slower process. In this paper, the term speckle decorrelation is therefore used only in this limited sense of decorrelation that is noise-induced and that occurs over a time scale that is significant only for the process of digital subtraction of the two primary additive-S1 images. Perfect correlation in the total absence of noise In the total absence of ambient optical phase-shifting or decorrelating noise, the u and LJ wavefronts are identical except for the ‘b’ leg wavefronts which are intentionally modified by the phase-modulator by a spatially uniform constant phase. Therefore, under total absence of noise it is apparent that the intensity-type term (A’) identically vanishes as well. In this case we can relate the various wavefronts as follows: W32=
u,le’M-;
ub2 =
ubleiMb
(w
where M, and Mb are the deformation-induced phase shifts between states ‘1’ and ‘2’ in legs ‘a’ and ‘b’, respectively (note that Mb = 0 for ASPM-ESPI-out). u al
=
&I,;
uti
=
uti
m
=
u,,e ih4..,
vbl = ublein
where R is the phase-modulation
that electronic
(74
induced in leg ‘b’ only;
ub2 = &e” Assuming
(74
= ubletieiMb.
(7e)
noise (in the camera sensor and the digitization
ASPM-SI
process) is insignificant, given by:
the resulting
(4) = 32(~&)
185
ASPM-SI
correlation
fringes are
sin 2[;]cos2[“;“],
are the average intensities of the leg ‘a’ and leg ‘b’ beams, respectively. Clearly it would be most advantageous to have the induced phase modulation G = n. Defining the fringe visibility as: V = (qrnax)(4max)
+
(4min) (4min)
(IO)
'
it is clear that though the resulting cosine-squared fringes of ASPM-SI are of the additive-S1 type, the fringe visibility is unity, a level which is not obtainable with additive-S1 even in a noise-free environment.4 This is true for all three configurations shown in Fig. 1. A typical ASPM-ESPI-out fringe pattern obtained in a noise-free situation is shown in Fig. 2(A). In this and in all the other experimental results cited in this paper, the test object used was an aluminum plate (2-5 X 25 X 30-5 cm) containing a centrally located flat-bottomed hole of 7.6 cm diameter leaving a O-08 cm thick membrane that acted as a circular plate which was acoustically excited by a broadband piezoelectric transducer to activate one of its resonant modes of vibration. A custom-built electronic synchronization system allowed us to control the excitation of the transducer and the object illumination (using an acousto-optic
Fig. 2.
(A) ASPM-ESPI-out.
(B) Conventional
subtractive
ESPI.
186
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
modulator to shutter the laser beam) such that only the relative deformation corresponding to the maximum and minimum of the acoustic cycle were obtained. The synchronization electronics also controlled the electro-optic phase modulator to provide the interframe phase shifting required in step 2. As seen from the figure, the fringes obtained using the ASPM method (Fig. 2(A)) are of squared-cosine nature which is the complement of that obtained using conventional subtractive-ESPI (Fig. 2(B)) where the fringes are of squared-sine nature. Complete noise-induced decorrelation If there is ambient noise that causes complete speckle decorrelation between steps ‘1’ and ‘2’, then the u and u wavefronts are no longer the same in all cases, and we have to analyze the different configurations in Fig. 1 separately. For the ASPM-ESPI-out setup (Fig. l(a)), we can ensure that the reference leg ‘b’ wavefronts are protected (by piping through a fiber-optic system for instance) such that only the object leg ‘a’ wavefronts are affected by ambient noise from the object or between the object and the camera. In this case, we have: KG?= ualeW*;
is& ubl
=
uble
,
Ub2
vb2
=
=
Ubl;
ub’&‘”
W)
=
uble’“;
WC)
ual are assumed to be uncorrelated due to complete interframe Q, decorrelation of the object leg. Note however that for the average intensity of the ‘a’ wavefront eqn (9a) still holds. The ASPM-ESPI-out correlation fringes in this case are given by: and
(4) = f3V,J2 + 16(&)(1,) COs2
(12)
,
where we have set the induced phase modulation Q = x as before, and we find that the fringe visibility in this case becomes: V =-
1 1 +x’
@a) is the object-to-reference
where x For the case of ASPM-ESPI-in =-
03) beam-intensity
ratio.
(Ib>
(Fig. l(c)) where both wavefronts
(Fig. l(b)) or for ASPM-shearography ‘a’ and ‘b’ forming the primary speckle
ASPM-SI
187
image arise from the object, both legs will be subject decorrelation. In this case, the wavefronts are given by:
to interframe
uQ = u,,eW*;
ub2 = ubleiMb;
Wa)
ua2 = ualeiM’;
ub2 = q,le’Mb;
Wb)
where ual, ual are uncorrelated as are r&l, &,l due to interframe decorrelation of both legs. (It may be guessed that the inter-frame phase-modulation is unnecessary in this case since the induced phase modulation will be swamped by the noise-induced phase decorrelation.) The correlation fringes in this case are given by:
[“‘1Mb],
(4) = 8W2 + (Ibj2) + 16(za&)COS2
(15)
and so the fringe visibility reduces to: v= X=;;;B
x 1+x+x2;
is the leg ‘a’ to leg ‘b’ beam-intensity
(16) ratio. Figure 3 shows a plot
b
of ASPM-ESPI-out and ASPM-shearography fringe visibilities as a function of beam-intensity ratio X for the case of complete interframe decorrelation of the wavefronts arising from the object. For ASPM-ESPIout it is clear from Fig. 3 that by weakening the intensity of the object leg
t>
0.8-
0.6v - ASPM-ESPI-out (theoretical) - -- ASPM-shearography (theoretical) 0 ASPM-ESPI-out (experimental) 0.2
r
Fig. 3.
Visibility
as a function
of beam intensity ratio X for the case of complete interframe decorrelation.
188
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
with respect to the reference leg, the fringe visibility can be enhanced to close to unity! This is because the noise-induced decorrelation is only in the object leg, whereas the reference leg is protected so as to be noise-free. This increase in visibility is clearly illustrated experimentally in the ASPM-EPSI-out fringe patterns shown in Fig. 4(a)-(c). To get a quantitative comparison between theory and experiment, the experimental fringe visibility was measured from these sequences as follows. The average brightness along lines of constant maximum and also minimum displacement-induced phase (i.e. along centers of bright and dark vibrational fringes) were computed from the digitized images. The resulting fringe visibility was then computed for each case and these are plotted in Fig. 3. The agreement between theory and experiment is quite good though the experimental values are somewhat lower than predicted due to other sources of error arising from electronic noise and the limited dynamic range of the camera used in this particular experiment which are not included in the analysis. In fact, it must be pointed out that from eqn (13) it appears that unit visibility will be obtained at X = 0, that is by turning off the object beam! This is obviously incorrect and the reason for that is as the signal term becomes weaker and weaker the effect of electronic noise, which should appear as an additional term in eqn (12), becomes significant and is no longer negligible as we assumed. The issue of electronic noise is explained in greater detail in Ref. 7. For the case of ASPM-ESPI-in and ASPM-shearography, the maximum fringe visibility at total decorrelation is only V = l/3 which occurs when both legs are of equal intensity. This is not surprising since both legs are now subject to decorrelation since they both arise from the object. Thus it appears that it would be best to use ASPM-ESPI-out (or in any case ASPM-ESPI with only one object leg and one protected reference leg even if the sensitivity vector is not aligned so as to give only the out of plane displacement of the object) with a weak object leg if noise induced visibility degradation is of concern. Partial decorrelation The intermediate case of partial noise-induced interframe decorrelation is considerably more complex to analyze. To get a quantitative measure, the degree of correlation between the u and u wavefronts needs to be established. Noise-induced decorrelation can be due to a variety of reasons (object drift, ambient vibrations, thermal currents in the air between the object and the camera, etc.), and it is in general impossible to relate the distortion in the u and u wavefronts unless the details of the noise are given. Therefore, we will restrict attention to ASPM-ESPI-out
ASPM-SI
189
Fig. 4. ASPM-.ES ?PI cJut images at complete interframe decorrek xtion. Note 6-inge visibi llity increas ies as cjbject beam intensity is reduced: (a) X = 1; (b) x = l/25; ; Cc) x = l/112.
190
Sridhar Krishnaswamy, Bruno F. Pouet, Tom C. Chatters
(Fig. l(a)) and ASPM-shearography (Fig. l(c)) configurations only. Furthermore we will consider only the case of decorrelation arising from interframe in-plane translation of the object. In this case, the various wavefronts are again given by eqns (lla)-(llc) for ASPM-ESPI-out and by eqns (14a)-(14b) for ASPM-shearography, but now ual, ual and t&l, ub,, etc., are no longer totally uncorrelated because they arise from the same resolution element on the object except for a possible in-plane translation of the resolution element itself. Therefore, the u and corresponding u wavefronts at the image plane arising from the same resolution element on the object can be related through a correlation coefficient in a manner described by Owner-Petersen. lo The process involves tracking the scattered wavefronts from a resolution element on the object plane as it propagates in space to the imaging lens and then diffracts through the lens pupil to the image plane. The process is repeated for the scatter from the same resolution element after it is displaced in the plane of the object. The details are sketched in the Appendix. For the ASPM-ESPI-out set-up, the resulting correlation fringes can be shown to be given by?’ (q) = 8(z,)*(l - c’) + 16(Z&)(l+ and for ASPM-shearography
the correlation
[ 1,
c) cos* !$
(17)
fringes are given by:”
(4) = WJ2 + (zb)*Nl- c*) + 16(1,)(&)(1+ c*) cos* [“, where the correlation coefficient due to in-plane interframe real-valued number given by:
“1, translation
(IS) is a
which depends on the amount of in-plane interframe object translation distance IdIaccording to the Bessel function of the first order; and D is the aperture size and LO is the distance between the object and the pupil plane of the imaging lens (see Fig. Al). The value of the correlation coefficient indicates the level of in-plane translation-induced decorrelation: i.e. full-correlation c = 1 occurs when the interframe in-plane translation IdI= 0 and total-decorrelation c = 0 occurs when the interframe in-plane translation Jdl= 1.22AL0/D = IdmaxI.A plot of the correlation coefficient
191
ASPM-SI
0
0.4
0.2
0.6
0.8
1
Normalized Translation Fig. 5.
Variation
of the correlation
coefficient c with normalized distance ldl/ld,,l.
as a function of the normalized in-plane translation Fig. 5. The associated fringe visibility for ASPM-ESPI-out V=
1
translation
distance is shown in becomes:
x=i!?L)
1+(1+x;
and for ASPM-shearography
in-plane
UtJ ’
the visibility becomes:
(1 + c’)X v = (1 - c2) + (1 + 3)X + (1 - cqx*
*
;
(21)
Clearly, the above visibility expressions reduce to the values obtained previously for perfect correlation (c = 1) and for complete decor-relation (c = 0) as they must. The fringe visibilities for conventional subtractive speckle interferometry have been developed by Owner-Petersen” and are reproduced here for subtractive-ESPI: V= and for subtractive
2c
x
2+(1-c*)x;
-
(ZJ (lb)
(22)
’
shearography: v=
2c*x 2c*x + (1 - c’)(l + X2) *
(23)
192
Sridhar Krishnaswamy,
Bruno F. Pouet, Tom C. Chatters
0.8
0.6
V 0.4
0 0
0.4 0.6 0.8 0.2 NonmlizedTranslation Distance
1
(a) ESPI-out
V
\ 0.20
. -.
\ '\ l. , *-.*-___ , I 0.6 0.8 1
. . ., . . . ,. . 1 0
0.4
0.2
Normalized Translation Distance
(b) Shearography Fig. 6. Fringe visibility as a function
of normalized interframe ratio X = 1).
translation
distance
ldl/ld,,l (for beam intensity
The fringe visibilities for the ASPM techniques are plotted in Fig. 6(a) and (b) for different levels of decorrelation and for unit beam-intensity ratio. Also plotted for comparison are the fringe visibilities for the conventional subtractive-ESPI and speckle-shearing interferometers. Note that for the same levels of decorrelation the ASPM method produces fringes with
ASPM-SI
193
consistently higher visibility than conventional subtractive methods. This is of course due to the fact that in the ASPM method additive correlation fringes are obtained within each video frame without any decorrelation. The interframe decorrelation only contributes to degrading the cancellation of the self-interference intensities in the ASPM method, whereas in the subtractive schemes it affects the very formation of the subtractive correlation fringes. The above theoretical development is borne out by the experimental results shown in Fig. 7(a) and (b) where the acoustically vibrating object has been intentionally translated between frames by various amounts. The beam-intensity ratio was kept at unity. A very interesting feature of the fringe patterns in Fig. 7(a) is the presence of two sets of fringes as the vibrating object is translated: one corresponding to the object vibration which is acquired in the additive-EPSI sense, and the other corresponding to the inter-frame phase shifts due to object translation between frames and which is acquired in the subtractive sense. These fringes appear because the experimental configuration was such that the object illumination was not normal and so the interferometer was somewhat sensitive to in-plane motion. These translational fringes appear multiplicatively superposed on each other. As expected translation fringes do not appear in the displacement-gradient sensitive shearography sequence shown in Fig. 7(b). Note however that as the object is translated in its own plane inter-frame speckle decorrelation occurs. An important point to note is that whereas inter-frame total decorrelation completely kills the formation of correlation fringes in the conventional scheme, it only affects the visibility for the ASPM scheme.
3 CONCLUSION The salient points of this paper are: (i) additive-subtractive phasemodulated speckle interferometry provides both the good noise immunity of conventional additive techniques as well as the good fringe visibility of conventional subtractive methods; (ii) experimental data as well as fringe visibility analyses for noise-free and partial to complete interframe speckle decorrelation conditions show that the technique provides consistently better visibility than conventional schemes; and (iii) the fringe visibility of the out-of-plane displacement sensitive interferometer (with one object leg and a separate protected reference leg) can be made to be essentially unity even under conditions of total interframe speckle decorrelation.
194
Sridhar Krishnaswamy,
Bruno F. Pouel, Tom C. Chatrers
(a) ASPM-ESPI
(b) ASPM-shearography Fig. 7.
(a) ASPM-ESPI and (b) ASPM-shearography images of defect vibrating at 1.2 kHz as object undergoes interframe translation of ldl/&,,l = 0, l/6, l/3, l/2,2/3 and 1 (from top left to bottom right).
ASPM-SI
195
REFERENCES 1. Jones, R. & Wykes, C., Holographic and Speckle Interferometry. Cambridge University Press, Cambridge, 1989. 2. Sirohi, R. S., Speckle Metrology. Marcel Dekker, New York, 1993. 3. Spooren, R., Double-pulse subtraction TV holography. Opt. Engng, 31 (1992), looo-1007. 4. Slettemoen, G. A., General analysis of fringe contrast in electronic speckle pattern interferometry. Optica Acta, 26 (1979), 313-327. 5. Pouet, B. F., Chatters, T. C. & Krishnaswamy, S., Synchronized reference updating technique for electronic speckle interferometry. J. Nondestructiue Evaluation, 12 (1993), 133-138. 6. Pouet, B. F. & Krishnaswamy, S., Additive-subtractive decorrelated ESPI. Opt. Engng, 32 (1993), 1360-1369. 7. Pouet, B. F. & Krishnaswamy, S., Additive-subtractive phase-modulated electronic speckle interferometry: Analysis of fringe visibility. Appf. Optics, 33 (1994), 6609-6616. 8. Chatters, T. C., Pouet, B. F. & Krishnaswamy, S., Additive-subtractive phase modulated shearography with synchronized acoustic stressing. Exper. Mech., 35 (1995). 159-165. 9. Goodman, J. W., Statistical properties of laser speckle patterns. In Laser Speckle and Related Phenomena, ed. J. C. Dainty. Springer-Verlag, 1984. 10. Owner-Petersen, M., Decorrelation and fringe visibility: on the limiting behavior of various electronic speckle-pattern correlation interferometers. J. Opt. Sot. Am. A8 (1991), 1082-1089. 11. Chatters, T. C., Noise reduction techniques in optical nondestructive evaluation. PhD dissertation, Northwestern University, 1994.
APPENDIX The u and corresponding u wavefronts that come from the object can be related through a correlation coefficient as described by Owner-Petersen.” The process involves tracking the scattered wavefronts from a resolution element on the object plane as it propagates in space to the imaging lens and then diffracts through the lens pupil to the image plane (see Fig. Al). The process is repeated for the scatter from the same resolution element after it is displaced in the plane of the object. In this analysis, we make the following assumptions about the wavefronts scattered from a diffuse object: (i) plane wavefront illumination normal to the object provides a scattered object wavefront; (ii) the object wavefront is imaged by a thin lens of aperture D and focal length fL onto the image plane, and therefore the field at any point on the image plane comes only from a small resolution element on the object; (iii) speckle decorrelation due to in-plane inter-frame translation d of the object is significant, but decorrelation effects due to object deformation itself are negligible; (iv) load-induced phase-
Sridhar Krkhnaswamy,
196
Bruno F. Pouet, Tom C. Chatters
c
imaging Object
Lens
I
t I0
----------
T---------7
-J+
L-
Fig. Al.
image
-__li----,
Diffraction of scattered wavefront from object to image plane.
shift on the object is constant across the resolution element which is true if the fringe spacing is much larger than the resolution element size. Under the above assumptions, it can be shown”‘-” that the wavefront Uim(cm) at position q, = (Xi,,yh) on the image plane is related to the scattered wavefront u&c,) from a resolution element at position r,, = (x,, y,) on the object plane through: Uim(lm) = F(ri,; Lo, L&
la P(fp)k,(fp>e2”f~~fP df,,
(AlI
--m
where fi, = Q,,,LJL~ FL;
is related to the image plane coordinates
L, L)
e
= AZ; z.
rim; and
-i(21~/A)(&,+L&~-i(n/AL&x&+y2,) ,
WI
Cl IIn
is a phase and amplitude scaling term; &,(f,) is the Fourier transform of the object beam u,(r,) and is given by: u,(ro)e2”if~.‘odr,;
643)
A is the wavelength of light used; L, is the object to lens distance and Li, is the lens to image plane distance. The aperture function is given by: I DI2AL, for IfpI
> DI2AL,
’
(A4)
where f, = r,/AL, is related to the pupil plane coordinate rp = (x,, yp). If now the object is translated in its own plane by an amount d, then the scattered wavefront at the object plane from the same resolution element as above is given by: u,(r,) = u,(r,, - d)e*‘“, (As)
ASPM-SI
where &R is an intentionally image plane wavefront is then:
induced
Uim(rim)= F(ri,; Lo, Li,)e*in In computing the average evaluate terms such as:
=e Tin
where an unimportant simplifies to:
brightness
197
phase modulation.
The corresponding
I-m P(fp)ii,(fp)eZrid*f~eznie*f~df,. of the correlation
fringes,
we need to
P(fp)P(f#&,(fp)ii,(f~))e2tid’f~e2niL~(f~-~) df, df& scaling (magnification) Gh&&h(bJ>
where (I,) = (u,u$ is the average coefficient is given by: c=i
(A7)
factor has been ignored. The above
= e”“4J,
object
646)
beam intensity,
J-1 P2(fp)e2”idmfpdfp,
VW
and the correlation
(A9)
where A, = 7rD2/4 is the area of the circular pupil aperture. The correlation coefficient evaluates to the expression in eqn (19). Expressions similar to eqn (A8) that are needed in the evaluation of eqns (8), (17) and (18) are obtained in like manner by using the appropriate wavefronts in legs ‘a’ and ‘b’ and states ‘1’ and ‘2’ as the object is deformed and/or translated.