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Phase-shifting speckle interferometry

Katherine Creath

Speckle patterns have high frequency phase data, which make it difficult to find the absolute phase of a sin-gle speckle pattern; however, the phase of the difference between two correlated speckle patterns can be de-termined. This is done by applying phase-shifting techniques to speckle interferometry, which will quanti-tatively determine the phase of double-exposure speckle measurements. The technique uses computer con-trol to take data and calculate phase without an intermediate recording step. The randomness of the speck-le causes noisy data points which are removed by data processing routines. One application of this tech-nique is finding the phase of deformations, where up to ten waves of wavefront deformation can easily bemeasured. Results of deformations caused by tilt of a metal plate and a disbond in a honeycomb structurebrazed to an aluminum plate are shown.

1. Introduction

Speckle interferometry techniques for double-exposuremeasurements such as electronic speckle pattern in-terferometry (ESPI) which measure deformations andcontours have been in use for about 15 years.1-6 Thesetechniques produce correlation fringes which corre-spond to the object movement between exposures or theobject's shape. To obtain fringes corresponding to anobject deformation, primary interferograms of theobject are recorded using a TV camera before and afterthe deformation. These interferograms are then elec-tronically processed to produce correlation fringes ora secondary interferogram, which corresponds exactlyto the object movement between exposures. However,the correlation fringes are not clearly defined becausethey are composed of variations in speckle contrast [forexample, see Fig. 5(a)]. Speckle interferometry tech-niques have been good for qualitative measurements buthave not produced good quantitative results, since it ishard to determine the fringe centers.

Quantitative data can be obtained using phase-shifting interferometry (PSI).7 PSI has been used tonondestructively test specular objects by shifting thephase of one beam in the interferometer with respectto the other. The phase of the test wavefront relativeto the reference wavefront can be simply calculated

The author is with University of Arizona, Optical Sciences Center,Tucson, Arizona 85721.

Received 22 March 1985.0003-6935/85/183053-06$02.0010.© 1985 Optical Society of America.

from the interferogram intensities measured for mul-tiple phase shifts. Double-exposure phase measure-ments (i.e., of deformations) have been made by com-bining double-exposure holography and phase-shiftinginterferometry in digital holographic interferometry(DHI).8,9 DHI requires the making of an intermediatehologram (using something such as a thermoplasticcamera) with the test object in place. Then the objectis deformed, and the phase of one beam in the inter-ferometer is changed, and the interference between thewavefront recorded in the hologram and the deformedwavefront produces secondary interference fringescorresponding to the deformation. The intensities ofthese secondary fringes are recorded at different relativephase shifts so that the phase can be calculated. DHIhas been used to quantitatively test both opticallysmooth and diffuse surfaces.

Phase-shifting can be applied to speckle interfer-ometry to produce phase maps for double exposures ofdiffuse surfaces without an intermediate recording step.The phase maps are calculated from sets of phase-shifted intensity data taken before and after the de-formation and then combined to produce the phase ofthe difference between exposures. Each set of phase-shifted intensity data is obtained in the same way asPSI. To find the phase of the object deformation, thephases of the speckle patterns before and after defor-mation are simply subtracted. This paper presents thetheory for this technique along with experimental re-sults and a discussion of its limitations.

II. Technique and Limitations

Figure 1 shows a schematic of the speckle interfer-ometer used in this work. This digital sepckle patterninterferometer (DSPI) is a variation on electronicspeckle pattern interferometers which processes speckle

15 September 1985 / Vol. 24, No. 18 / APPLIED OPTICS 3053

two intensities by subtracting and taking the modulussquared. When an ensemble average is taken overmany realizations of these secondary fringes, the sec-ondary fringe function becomes

( III2) = 8(I1) (12) sin 2(AO/2). (3)

Fig. 1. Schematic of digital speckle pattern interferometer with asingle-mode optical fiber used to produce a clean spherical reference

wave.

data inside a computer rather than in hardware.10 Toget the information needed to obtain fringes or phaseinformation, the detector array must resolve the speckle.In this case, the aperture limits the bundle of rays in-cident on the image plane to f/40 to produce 60-pumspeckles. A spherical reference wave is produced by asingle-mode optical fiber mounted in the center of theaperture, and the object and reference beams are col-linear at the detector array. A diffuse object is illumi-nated by a nearly collimated beam to conserve light andis imaged onto the face of the detector array. The de-tector array is a 100 X 100-element Reticon diode array,which is electronically interfaced to an HP-9836Cdesk-top computer. To measure a deformation usingESPI or DSPI, a speckle pattern exposure is taken ofthe object in one position. Then the object is deformed,and another exposure is taken. These two specklepatterns are subtracted, and their difference is squaredto obtain speckle correlation fringes corresponding tothe object's deformation.

This process can be shown mathematically. Theintensity in a speckle interferometer is written as

Ibefore = II(x,y) + I2(xy) + 2VIl(x,y)I 2(x,y) cos[(x,y)], (1)

where I, and I2 are the intensities of the reference andobject beams, and (x,y) is the phase of the specklepattern. The first two terms are known as self-inter-ference terms because they relate to the interference ofeach beam in the interferometer with itself, and thethird term is the cross-interference between the objectand reference beams. The modulation, contrast, orvisibility of the speckle pattern is given by the modulusof the cross-interference term. The modulation is animportant factor for determining if data points are goodor bad when using phase-shifting techniques. After theobject has been deformed, the intensity becomes

'after = Ii(x,y) + I2(XY)

+ 2 VII(xy)I 2 (xy) cos[o(x,y) - Ap(xy)], (2)

where AO is the phase change due to the object defor-mation. Secondary fringes are calculated from these

This fringe function is similar to fringes in a classicalinterferometer where the fringes are sinusoidally de-pendent on a relative phase difference; however, Eq. (3)depends on the phase difference between exposuresrather than the phase difference between object andreference beams.

Phase-shifting interferometry (PSI) is a techniquewhich can determine the shape of a surface or wavefrontby calculating a phase map from the measured inten-sities. The technique used here was introduced byCarr6 to calculate the phase modulo 27r.1 For eachdata frame, the intensity is integrated over the time ittakes to move a reference mirror linearly pushed by apiezoelectric transducer (PZT) through a 2a (usually900) change in phase. Four frames of intensity data arerecorded in this manner:

A(x,y) = loll + y cos[(x,y) -3a]},

B(x,y) = loll + y cos[i(x,y) - all,

C(x,y) = boil + y cos[0(x,y) + all, (4)

D(xy) = Ioll + y cos[4,(xsy) + 3a]l,

where Io is the average intensity, and y is the modula-tion of the interference term. The phase calculated ateach detected point (x,y) in the interferogram is

= arctan[N1 [(A D) + (B - C)][3(B -C) - (A ID)]ljI l(B+C)-(A+D)l

(5)

where the phase falls between 0 and r/2 (quadrant 1).The signs of the quantities

(B - C) = [2 Ioy sina] sinq/,

(B + C) - (A + D) = [2 Ioy cosa sin2a] cos,

(6a)

(6b)

determine in which quadrant to place the phase, be-cause they are proportional to sinq/ and cos4. WhenEq. (6a) is negative and Eq. (6b) positive, 4 is subtractedfrom 27r to place it in quadrant 2. If both Eqs. (6a) and(6b) are negative, 7r is added to 4 to put the phase inquadrant 3. Finally, if Eq. (6a) is positive and Eq. (6b)is negative, 4' is subtracted from 7r to place it in quad-rant 3. If the denominator of Eq. (5) equals zero, 4equals either r/4 or 3-7r/4 depending on the sign of Eq.(6a), and when the numerator equals zero, 4' = 7r. 27rambiguities are removed by adding or subtracting 27rfrom individual pixels until the phase difference be-tween adjacent pixels is <7r. Equation (5) calculatesthe phase independent of the phase shift 2a betweenframes of intensity data eliminating the need for PZTcalibration. Since the phase shift may vary from pointto point in the image, owing to tilt of the mirror as it istranslated by the PZT, this method can more accuratelycalculate the phase than methods which depend on howmuch the phase is shifted. The optical path difference

3054 APPLIED OPTICS / Vol. 24, No. 18 / 15 September 1985

(OPD) for the object wavefront relative to the referencewavefront can be determined from the phase:

OPD(x,y) = 2 Y)* (7)

The OPD is related to surface heights by a multiplica-tive factor and accounts for angles of illumination andviewing which may differ from the surface normal.This factor is one-half for a double-pass inteferometerlike a Twyman-Green. For phase-shifting techniquesto work, the phase difference between adjacent pixelsin the measured wavefront must be <7r (one-half waveof OPD). This restriction limits the total number ofwaves departure measurable across the test surface andwill limit the test sensitivity.

Speckle patterns have high frequency phase datawhich cause the phase difference between adjacentpixels to be >7r. Thus the phase of a single specklepattern can only be determined modulo 27r. However,the phase of the difference between two speckle patternscan be completely determined as long as the absolutephase change between exposures does not vary by morethan 7r between adjacent detector elements. To findthe phase of a deformation using speckle, four framesof intensity data are taken while shifting the phase ofone beam in the interferometer. The object is thendeformed, and four more frames of intensity data aretaken while shifting the phase the same amount as forthe first set of data. Once these data are taken, thephase of the deformation can be calculated from

Ark(x,y) = (x,y) - [(xy) - A(xy)]= 1'before - 1after, (8)

where 4' is given by Eq. (5). This calculation yieldsphases which are modulo 27r. The 27r ambiguities canbe removed as described for PSI.

When phase-shifting techniques are applied tospeckle interferometry, the randomness of the specklewill cause many noisy data points. Other than pointswhere the intensity saturates the detector element,there are two main contributors to these noisy pointswhich are fundamental limitations to this technique.They are (1) decorrelation of the speckle between theexposures before and after deformation, and (2) lowmodulation of the measured intensity at a given pixelas the phase is shifted.

Speckle decorrelation in this case is caused bychanges in the collected scattering angles from theobject which contribute to a single image point. Eachpoint on the object imaged onto the detector arrayscatters light into a large solid angle. The f/40 opticalsystem will only collect a small portion of this light. Asthe object is deformed, points on it will tilt, and differentscattering contributions will be collected by the opticalsystem than before the deformation. These tiltedpoints will have a displaced collecting cone comparedto the undeformed points (see Fig. 2). Thus the in-tensity and phase of the imaged speckle will changeslightly. For the system shown in Fig. 1, five waves ofsurface tilt will displace the collecting angle by 3%. Astatistical analysis of the speckle decorrelation for this

amount of tilt shows a 330 rms phase error.1 6 One wayto reduce this problem is by using a larger aperture, butthis produces smaller speckles, so that a detector arraywith smaller pixels would be needed. Simply changingthe geometry of the system would not solve this prob-lem. Because the phases may be slightly off, this willcause problems in removing 27r ambiguities.

Low modulation is a fundamental problem in allphase-shifting techniques.12 Because speckle is sam-pled with a finite-sized detector element, the intensitymeasured for that detector element is going to be anaverage over its area. To resolve the speckle in theimage plane, the f/No. of the system is set to make thespeckle size equal to the pixel size. With this specklesize, two speckles will influence one pixel on the aver-age.13 For only one speckle to influence each detector,the detector size would have to be about a tenth of thespeckle size. If the aperture were made this small, verylittle light would get through. When the phase of onebeam in the interferometer is shifted, the intensity ofthe speckle will change; but, with a detector element thesize of the speckle, the intensity will not modulate asmuch as it would for point sampling. This is the effectof averaging over a pixel with more than one speckleinfluencing it.

Both problems could be corrected by using pointdetector elements. A point detector would enable alarger f/No. to be used, so that the problem with thespeckle correlation could be reduced, and it would in-crease the modulation by reducing the averaging effects.The spacing between detectors is not critical, becauseit is assumed that a speckle is incident on the same de-tector both before and after deformation. The onlyconstraint on spacing is that from one point detector tothe next, the change in phase due to deformation cannotbe larger than 7r. This is a consequence of the samplingtheorem, and it influences the overall sensitivity (howmany waves of deformation can be measured) of thetest.

Data points which are saturated or have low modu-lation can be removed from the data sets. Points wherethe phase is slightly off due to a decorrelation betweenspeckles will be modulated sufficiently and not re-moved; but they still cause problems and will be dis-cussed in more detail later. To remove most noisy datapoints, the intensity measured in the interferometer[Eq. (1)] is tested to determine if the data points aregood. There are three contributions to bad pixels: (1)the data point is saturated; (2) its modulation as thephase is shifted is not greater than some Imin; and (3) agiven pixel is bad in either data set. The modulation

DIFFUSE SURFACE IMAGE PLANE

Fig. 2. Speckle decorrelation due to the collection of differentscattering contributions.


of the speckle is determined by calculating the magni-tude of the cross-interference term and checking itagainst an intensity minimum:

2V\-II < mm. (9)

This quantity can be calculated from the four intensityframes given in Eq. (4) by

277i~ = [(B - C) +(A- D)]2 + [(B + C) - (A + D)]2

(10)

where this equation assumes that 2ac is .90°. It shouldbe noted that Imin can be related to the amount ofelectronic noise present if point detectors are used.14

The integration routine first passes along the rowsand then the columns. It adds or subtracts 2r to asingle pixel until the difference between adjacent pixelsis <7r. If there are bad pixels (those that have beenthrown out), they are skipped, and the phase differenceis check between nearest good pixels. These missingbad points sometimes cause problems, because thephase may change a lot between the good pixels. Datapoints which are slightly off in phase will not be flaggedby the bad pixel tests. The result of having points thatslightly err in phase is that from one point to the nextthe phase may jump by 2r and then jump back. Theintegration routine can handle these problems; but, ifthe phase jumps by 27r and then returns only partiallyat the next point, the integration will add 2r and thennot subtract it again afterward as it should. Thisproblem increases as the amount of phase deformationis increased. To reduce the effect of these points, somedata processing must be done. The integration prob-lems encountered for slight phase errors are not presentwhen this technique is used with a Twyman-Green in-teferometer to test specular surfaces.

To improve the results of integrating the phase, twoprocessing routines have been implemented. The firstis a median window,15 which takes all the data pointscontained within the window, sorts them until the me-dian is found, and then places the median value into alocation in another array corresponding to the centerof the window. The window is then moved over by onepixel, and the procedure is repeated until the entirearray has been processed. Features which are smallerthan (N + 1)/2 in extent are removed, where N is the1-D size of the window. Features greater than 2N inextent are retained by the median window to preserveedges. This is important when processing raw phasedata, because boundaries where the phase jumps by 27rneed to be preserved.

The median window does a good job filling in baddata points but does not change the pixel values wherethe phase errs. To smooth data points where the phaseis slightly off and preserve edges as well, another pro-cessing routine was developed. This routine compareseach data point to eight nearly pixels in the form of aplus sign. If most of the nearby points are close inphase value to the center pixel, it is left alone. But, ifmost of them are different, the center pixel is replacedwith the average of the good nearby pixels. By com-

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C) D)Fig. 3. Results of calculating the phase for a small object tilt: (A)raw calculated phase of deformation; (B) integration of 27r ambigui-ties; (C) removed streaks and filled in bad points with a 5 X 5 median

window; (D) contour map of the tilt.

A) B)

C) D)Fig. 4. Results of calculating phase for a larger object tilt: (A) rawcalculated phase; (B) integration of 27r ambiguities with phase errorspresent; (C) smoothed raw phase data followed by a 3 X 3 medianwindow; (D) contour map of smoothed and integrated data for object

tilt.


I,' By_ aI' A i

. ~ f. A, .F.H, rV moo -1 C o ' :

bining this routine with the median window, the effectof 27r jumps by the integration routine which do notreturn to the desired phase can be reduced or elimi-nated. Hence good phase contour maps may be gen-erated.

Ill. Experimental Results

A computer program was written to calculate phasefrom double-exposure measurements. This programcontrols the PZT, transfers data from the detector array,and calculates the phase at each data point as describedearlier. It also utilizes the bad data point tests for re-moving pixels with low modulation and has the abilityto smooth the data and apply median windows of dif-ferent sizes.

The optical layout of the interferometer, which wasdescribed earlier, is shown in Fig. 1. A 10-mW He-Nelaser provides the source, while a variable beam splittercontrols the ratio of intensities between the object andreference beams to optimize the cross-interference term.The PZT is linearly ramped by the computer while thedetector array integrates the intensity for 20 msec/frame. The object is both illuminated and viewed at45°. A 3- X 4-cm illuminated area on the object is im-aged onto the detector array. A wedge is placed on thewindow over the detector array to reduce the effect ofinterference fringes between its front and rear surfaces.Because the single-mode optical fiber preserves polar-ization and has a preferred coupling axis, halfwaveplates are placed in each beam to align the polarizationsat the image plane. Finally, the difference in pathlength between the beams must be within the coherencelength of the laser to ensure good interference.

To determine the precision of this technique, sets ofdata are taken without deforming the object. Thus thephases of two identical speckle patterns (except for airturbulence) are subtracted and then integrated. Theresulting phase distribution has a peak-to-valley (P-V)of X/20 and an rms of X/250. When the data are pro-cessed using smoothing and a 3 X 3 median windowbefore integration, the P-V is X/40, and the rms is un-changed. This test accounts for electronic noise andlow modulation but does not include speckle decorre-lation.

Figure 3(A) shows the raw phase obtained from tiltinga rough metal plate between the two sets of phase data.The modulation threshold Imin was set at 5, which re-moved 4% of the data pixels. These data have a verysalt-and-peppery appearance due to the randomnessof the speckle. When these raw phase data are inte-grated [Fig. 3(B)] to remove 27r ambiguities, streakingis caused by points where the phase errs. These streakscan be removed along with filling in bad data points byusing a 5 X 5 median window [Fig. 3(C)]. The finalcontour map of the deformation is in Fig. 3(D), and ithas a P-V of 3.5 waves OPD.

When a deformation of less than 5 waves OPD is in-troduced, the raw phase data can be directly integratedand bad data points filled in without any problem; but,for larger deformations, the raw phase data must besmoothed and median windowed before integration.

A) B)

C)Fig. 5. (A) DSPI fringes of disbond in braze between a honeycombstructure and aluminum plate where the object was heated to deformit; (B) raw calculated phase of same structure; (C) contour map after

smoothing, median window, and integration of raw phase.

Figure 4(A) shows raw phase data for a tilt deformationof about eight waves. With Imin = 8, 9.6% of the datawere flagged as bad. When integrated, large areas ofthe image are incorrectly integrated owing to phaseerrors and bad pixels [Fig. 4(b)]. To help this problem,the data are smoothed and followed by a 3 X 3 medianwindow which fills in bad points [Fig. 4(c)]. After thesesmoothed data are integrated, a few streaks are presentcaused by the points which still err in phase. Theseresidual streaks can be removed by a 5 X 5 medianwindow. The contour map of this tilt [Fig. 4(D)] showsa P-V wavefront deformation of 7.8 waves tilt. Thesmoothing of the data enables larger deformations tobe processed. The ultimate sensitivity limit is thespeckle noise due to decorrelation and low modulation.For larger deformations, higher Imin values need to beused. This causes more data points to be flagged as badand not used. Data points have to be smoothed andfilled in to obtain good results.

A final example is a disbonded area in a honeycombstructure brazed to an aluminum plate. Speckle cor-relation fringes using DSPI are shown in Fig. 5(A). Theobject was deformed between exposures by heating thealuminum plate, where the number of fringes seen re-lates to how much the disbonded area was heated.Figure 5(B) shows the raw calculated phase with Imin= 2. After processing these data in the same manneras the last example, the resulting contour map of thedisbonded area has aP-V of 2.25 waves OPD [Fig. 5(C)].


This technique enables the location, size, and shape ofthe defect to be determined quantitatively.

V. Conclusions

The phase contour of an object deformation can bedetermined quantitatively by applying phase-shiftingtechniques to speckle interferometry. This allows thelocation, size, and shape of a defect to be measured.Because the phase is calculated directly, no interme-diate recording step is needed, and there is no need toprocess speckle data to form speckle correlation fringesand locate fringe centers. The ultimate limitations tothis technique are pixels with low intensity modulationas the phase is changed, and pixels having decorrelationof the speckles between data recorded before and afterdeformation. These limitations cause phase errors butcan be smoothed or filled in by processing the data toenable measurement of larger deformations. Over a100-element wide detector array, ten waves of wavefrontdeformation are measurable to within /10. Themeasurement range could be extended with the use ofpoint detectors, because they would reduce the effectsof speckle noise and low modulation.

This same algorithm can be applied to both specularand diffuse surfaces by changing the optical setup.When this technique is used with a Twyman-Greeninterferometer and a specular object, the speckle de-correlation effects are not seen; however, there still islow modulation of some data points. Another appli-cation of this technique utilizing speckle is contouringusing either two wavelengths or two angles of illumi-nation.

The author wishes to thank Gudmunn Slettemoenand Jim Wyant for their helpful discussions.

References1. J. N. Butters and J. A. Leendertz, "Speckle Pattern and Holo-

graphic Techniques in Engineering Metrology," Opt. LaserTechnol. 3, 26 (1971).

2. R. Jones and C. Wykes, "General Parameters for the Design andOptimization of Electronic Speckle Pattern Interferometers,"Opt. Acta 28, 949 (1981).

3. C. Wykes, "Use of Electronic Speckle Pattern Interferometry(ESPI) in the Measurement of Static and Dynamic Surface Dis-placements," Opt. Eng. 21, 400 (1982).

4. R. Jones, and C. Wykes, Holographic and Speckle Interferometry(Cambridge U. P., London, 1983).

5. G. A. Slettemoen, "General Analysis of Fringe Contrast in Elec-tronic Speckle Pattern Interferometry," Opt. Acta 26, 313(1979).

6. 0. J. Lokberg, "Advances and Applications of Electronic SpecklePattern Interferometery (ESPI)," Proc. Soc. Photo-Opt. Instrum.Eng. 215, 92 (1980).

7. J. C. Wyant et al., "An Optical Profilometer for Surface Char-acterization of Magnetic Media," ASLE Trans. 27, 101 (1984).

8. P. Hariharan, Optical Holography (Cambridge U. P., London,1984).

9. P. Hariharan, B. F. Oreb, and N. Brown, "A Digital Phase-Mea-surement System for Real-Time Holographic Interferometry,"Opt. Commun. 41, 393 (1982).

10. K. Creath, "Digital Speckle Pattern Interferometry (DSPI) Usinga 100 X 100 Imaging Array," Proc. Soc. Photo-Opt. Instrum. Eng.501, 292 (1984).

11. P. Carr6, "Installation et utilisation du compateur photoelectriqueet interferential du Bureau International des Poids et Mesures,"Metrologia 2, 13 (1966).

12. K. Creath, Y.-Y. Cheng, and J. C. Wyant, "Contouring AsphericSurfaces Using Two-Wavelength Phase-Shifting Interferometry,"to be published in Opt. Acta. 32, (1985).

13. J. W. Goodman, "Statistical Properties of Laser Speckle Pat-terns," in Laser Speckle and Related Phenomena, J. C. Dainty,Ed. (Springer-Verlag, Berlin, 1975).

14. G. A. Slettemoen and J. C. Wyant, "Maximal Fraction of Ac-cepted Measurements in Phase-Shifting Speckle Interferometry:a Theoretical Study," to be published in J. Opt. Soc. Am. A 2(1985).

15. B. R. Frieden, "Some Statistical Properties of the Median Win-dow," Proc. Soc. Photo-Opt. Instrum. Eng. 373, 219 (1981).

16. K. Creath, "Phase-Shifting Speckle Interferometry," Proc. Soc.Photo-Opt. Instrum. Eng. 556, in press (1985).

The Thirty-Third Annual onference of the WesternSpectroscopy Association will be held at the AsilomarConference Center, Pacific Grove, California,January 29-31, 1986. The program consists ofinvited lectures and a poster session of contributedpapers. Interested persons should contact DavidSaperstein, EM, E42i13, 5600 Cottle Road, San Jose,CA 9193.

The Thirty-Fourth Annual Conference of the WesternSpectroscopy Association will be held at the AsilomarConference Center, Pacific Grove, California,January 28-30, 1987. The program consists ofinvited lectures and a poster session of contributedpapers. Interested persons should contact DavidSaperstein, IBM, E42/13, 600 Cottle Road, San Jose,CA 95193


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