Fringe scanning speckle-pattern interferometry

Fringe scanning speckle-pattern interferometry

Suezou Nakadate and Hiroyoshi Saito

Digital speckle-pattern interferometry systems for automatic measurement of deformations of a diffuseobject are presented, which are based on a fringe scanning method with phase-shifted speckle interfero-

grams. A digital speckle pattern before deformation of an object is recorded in the mass storage device of

a computer facility. After deformation, four digital speckle patterns are recorded as changing the phase of

reference light such as 0, r/2, 7r, and 37r/2, respectively. Four speckle interferograms, whose phases are

shifted by 0, 7r/2, 7r, and 37r/2, are generated by calculating the square of the differences between speckle pat-

terns before and after deformation. These interferograms are low-pass filtered to reduce speckle noise.The calculation of the arctangent with four phase-shifted speckle interferograms gives the optical path dif-ference which is proportional to the deformation. A correction of the discontinuity of the calculated phasegives the numerical data of the deformation in the whole object area. Some experimental results for themeasurement of out-of-plane, in-plane, and 3-D deformations are presented.

1. Introduction

Speckle interferometry allows us to measure surfacedeformations of a diffuse object.' Speckle interfer-ometry provides not only the measurement of out-of-plane deformation but also gives a fringe pattern de-picting directly in-plane displacement or spatial de-rivatives of 3-D deformation of an object. Low-reso-lution devices, such as instant Polaroid film and a TVcamera, can be used in speckle interferometry. TVdetection of speckle patterns makes it possible to per-form the measurements in real time.2 This method iscalled electronic speckle-pattern interferometry (ESPI),and it has been developed using analog3 and digital4

signal processing techniques. Speckles, however, yieldto noise in a speckle interferogram and deteriorate thefringe quality. To reduce speckle noise and evaluatea speckle interferogram, digital image processingtechniques using a computer facility were developed. 5

A few problems to be solved remain, which are (1) de-termination of fringe order numbers and (2) interpo-lation of fractional fringe order numbers. High-accu-racy measurement systems for evaluating interfero-grams have recently been developed by several authorsusing heterodyne detection,6 fringe scanning,7 andFourier transform.8 Holographic interferometry sys-

The authors are with Institute of Physical & Chemical Research,2-1 Hirosawa, Wako-shi, Saitama 351-01, Japan.

Received 28 January 1985.0003-6935/85/142172-09$02.00/0.( 1985 Optical Society of America.

tems using fringe scanning have been reported bySommargren,9 Hariharan et al.,10 and Nakadate andSaito." In these interferometers, experimental resultswith high accuracy were obtained over the whole surfaceof the object measured.

In this paper digital speckle-pattern interferometrysystems for measuring deformations using fringescanning are described. Spatial filtering for reducingspeckle noise, correction of discontinuities of calculatedphase differences, and smoothing of the numerical dataafter the correction are also described. Some experi-mental results with interferometers are presented forthe measurement of out-of-plane, in-plane, and 3-Ddeformations.

11. Out-of-Plane Deformation Measurement

A. Method and System Description

A schematic diagram of a fringe scanning speckleinterferometer for measurement of out-of-plane de-formation is shown in Fig. 1. The light from a He-Nelaser (50 mW) is expanded by an objective lens and splitinto object and reference surface illumination beams bya beam splitter. Diffusely scattered light from objectand reference surfaces is collected by an imaging lens(105-mm focal length) and focused on the Chalnicontarget of a TV camera. The optical path differencebetween reference and object light can be changed byapplying a high voltage to the PZT device attached tothe reference surface. The light intensity of the speckleimage is sampled to yield a digital picture made up of512 X 512 sample points. Each sample point is quan-tized to 256 discrete gray levels. The digital picture isstored in a digital frame memory in 1/30 sec. Thestored picture is transferred to a minicomputer

2172 APPLIED OPTICS / Vol. 24, No. 14 / 15 July 1985

Fig. 1. Schematic diagram of the fringe scanning speckle-patterninterferometer for measuring out-of-plane deformation.

(PDP-11/34) by direct memory access and recorded ona mass storage device such as a disk or a magnetic tape(MT).

For fast digital image processing, a large scale com-puter (FACOM M380) was used in our experiments.The speckle pattern before deformation of the object,denoted by So, is stored on the disk. After deformation,the reference surface is displaced by amounts of 0, X/8,X/4, and 3X/8 when applying the high voltage to thePZT device; here X is the wavelength of light. Thecorresponding four phase-shifted speckle patterns,S1-S4, are stored on the disk.

In our experiments, the high voltages to the PZT aredetermined manually, but it can be controlled by thecomputer, then the signal from the computer is shownby the dotted line in Fig. 1. Four phase-shifted speckleinterferograms, F1-F4, are generated by calculating thesquare of the difference between speckle patterns beforeand after deformation, that is, Fi = (S, - So)2 (i = 1-4).To reduce speckle noise in the speckle interferogram,spatial low-pass filtering is performed by five iterationsof averaging over 3 X 3 sample points. The resultantspeckle interferograms are represented by I1-14. Thephases of the speckle interferograms, I2-14, with respectto the interferogram I,, are changed by 7r/2, 7r, and 3ir/2,respectively. Then the optical phase difference beforeand after the deformation of the object, which is de-noted by 4, can be calculated from

tan [I42 (1)

Because the phase is the principal value of thearctangent calculation, it ranges from -7r to 7 rad,therefore, the discontinuities of the phase occur near -7ror -7r rad. To obtain the numerical value of the defor-mation over the whole area of the object, it is necessaryto correct the discontinuity of the calculated phase ?by the following algorithm. The calculated phase at theith sample point lying on a vertical or a horizontal linein the phase image is expressed by Pi (i = 1,2, .. ., N),where N is the number of sample points in the line.The bias component of the phase, which is denoted byBIAS, is set to zero when i = 1. The difference betweenthe ith phase Pi and the i-lth phase Pi-, is calculated

xi X2

Fig. 2. Schematic illustration for the determination of an absolutefringe order number.

to yield 3P, that is, 6P = Pi-pi_,. If 3P is greater thana positive threshold value T, BIAS is replaced by BIAS-27r. If bP is less than a negative threshold value-T,BIAS is replaced by BIAS + 27r. Otherwise, BIAS re-mains. The output phase 0, where the discontinuity iscorrected, is set to Pi + BIAS. These operations aresuccessively repeated for i = 2,3, . . , N.

Because speckle noise cannot be perfectly filtered inspeckle interferograms, the calculated phase 4 also hasnoise and behaves in a complicated way near -r or 7r

rad. Then the correction procedure described aboveis performed twice as T = 7r in each horizontal andvertical line. In this correction procedure it is assumedthat the phase difference between two adjacent samplepoints is not greater than 7r. It is confirmed by the ex-periment that double iteration of the correction pro-cedure corrects the discontinuity of the phases moresmoothly.

Next, the absolute phase at the point (ij) is deter-mined by the following procedure. The correctedphases in the horizontal and vertical lines are repre-sented by H(i,]) and V(ij)(ij = 1, 2, . . , N), re-spectively. Two base lines in the horizontal and verticaldirections are determined in the phase image as shownin Fig. 2, which are denoted by Y1,Y2 and X1,X2, re-spectively. It is necessary to confirm that the phasedata on each base line are connected smoothly; if theyare not another base line must be selected. If the ab-solute phase at a point on a base line is determined byan operator, all the absolute phase on the base lines isdetermined automatically. In general, the phase at thepoint where the object is not deformed is set to zero bythe operator. Let 0H1,0V1,... , 0V2 shown in Fig. 2denote the differences of the phases, H(ij) -0H(X1,),0V(ij) - V(i,Y1),.. , V(ij) - OV(i,Y2),respectively. Then four phases at the point (ij) areestimated as 01 = 0(Xlj) + H1,02 = (i,Y1) +WV,..., 04 = 0(i,Y2) + 0V2, where0(X1j), .. . , 0(i,Y2) represent the absolute phases onthe base lines. The most possible value of the absolutephase 0(ij) is either of two phases whose absolute dif-ference is the smallest of the six values of 101 - 021,101-031,..., 103-041. In our experiments, the absolute

15 July 1985 / Vol. 24, No. 14 / APPLIED OPTICS 2173

phase 0 at the point (ii) is determined as the meanvalue of two phases whose absolute difference is thesmallest of the six values. Subsequently, eight itera-tions of a median filtering within an 11 X 11 sample areaare performed to reduce salt-pepper noise caused byspeckle noise and to remove a small area of phasediscontinuities caused by the correction procedure. Forthe measurement of out-of-plane deformation, thenumerical value of the deformation, dz, can be calcu-lated from

2 2(2)

Therefore, numerical data of the deformation at anysample point on the surface of the object can be ob-tained by the procedure described above.

B. Experimental Results

To demonstrate the method described above, themeasurement of out-of-plane deformation was per-formed with a 20-mm diam aluminum disk 0.1 mmthick. The circular edge of the object except for a 3-mmlength is fixed to an acrylic holder by an adhesive. Thesurface of the object is sprayed white. To deform theobject, a 5-mm diam PZT device is attached in thecentral part of the object. The object is deformedstepwise to the nine states by increasing the high voltageto the PZT. In each state of the object, four specklepatterns are recorded on the MT by displacing the ref-erence surface by amounts of 0, X/8, X/4, and 3X/8.Then, thirty-seven speckle images are recorded on theMT including the image of the undeformed object.Low-pass filtered speckle interferograms are shown inFig. 3, generated by taking the square of the differencebetween the speckle images of the undeformed objectand the first speckle image in each state of the object.Figures 3(a)-(i) show the deformations in each state.Fringes appearing near the edge of the attached PZTcannot be resolved in Figs. 3(e)-(i). However, the nu-merical data of the deformation can be obtained withoutupper limit by the following procedure:

Four speckle interferograms in the third state of theobject are shown in Figs. 4(a)-(d), whose phases areshifted by 0, 7r/2, 7r, and 37r/2, respectively. Fourspeckle interferograms in another state of the object canbe generated with the first speckle pattern in the pre-vious state of the object. Then nine phase imagesproportional to the deformations in the adjacent statesof the object can be obtained by using Eq. (1) as shownin Figs. 5(a)-(i). The phases ranging from -r to ir radare displayed as 0-255 levels. The discontinuities ofthe phases in each phase image in Fig. 5 appear in thecentral part of the object. The discontinuities of thephase can be corrected by the procedure described. Ifthe discontinuities are corrected and summed succes-sively from Fig. 5(a) to Fig. 5(i), the deformation in theninth state of the object can be obtained.

The resultant numerical value of the deformation isshown in Figs. 6(a) and (b), which represent contour andperspective maps, respectively. The interval betweencontours in Fig. 6(a) is 0.3 Mim. The noise in Fig. 6 is

Fig. 3. Speckle interferograms in nine deformation states of the

20-mm circular object which are obtained by five iterations of aver-

aging over 3 X 3 sample points. The out-of-plane deformation in-creases gradually from (a) to (i).

Fig. 4. Four phase-shifted speckle interferograms, (a)-(d), in thethird state of deformation, whose phases are 0, 7r/2, 7r, and 37r/2,

respectively.

Fig. 5. Phase distribution proportional to the deformation betweenthe adjacent deformation states shown in Fig. 3. Phases ranging from

-7r to 7r rad are displayed as 0-255 levels on the monitor.

caused by speckles. The phase image resulting fromthe median filtering is shown in Figs. 7(a) and (b), whichare contour and perspective representations, respec-tively. The noise, similar to shot noise, in Fig. 6 is re-moved very well. The interval between contours in Fig.7(a) is 0.3 Atm. The perspective map in Fig. 7(b) showsclearly how the object deformed. The maximum valueof the deformation is -1.4 ,tm, the minimum value isabout -0.8 ttm. The numerical value of the deforma-tion in another state of the object can also be obtained


(a)

1.0

0.5(b)

Fig. 6. Numerical values of the out-of-plane deformation in the ninthstate of the object, which is obtained by correcting discontinuities ofthe phases shown in Figs. 5(a)-(i), and the results are summed: (a)contour and (b) perspective representations. The interval between

contours is 0.3 Atm.

(a) (b)

Fig. 7. Results from eight iterations of the median filtering withinan 11 X 11 sample area in Fig. 6: (a) contour and (b) perspective

representations. The interval between contours is 0.3 ,im.

Z. _ _

a -**U 0 V 2

(a) (b)

Fig. 8. Second spatial derivative proportional to the bending mo-ment of the deformation, which is obtained by polynomial fitting tothe data in each horizontal line in Fig. 7; the polynomials are differ-entiated: (a) contour and (b) perspective representations. The in-

terval between contours is 5 X 10-3 (1/m).

in the same manner. The second spatial derivative ofout-of-plane deformation, which is proportional to thebending moment, can be obtained with the data shownin Fig. 7. By a least-squares approximation, a twenti-eth-order polynomial is fitted to the data in each hori-zontal line in Fig. 7. The second derivative of the de-formation is obtained to differentiate the polynomialapproximated. The results are shown in Figs. 8(a) and(b), which are contour and perspective representations.The interval between contours is 5 X 10- 3(1/m).

The maps in Fig. 8 show that the bending momentconcentrates at the edge of the PZT. The bendingmoment in the vertical direction can also be obtainedin the same manner, and the result shows that the mo-ment also concentrates at the edge of the PZT device.

-1.0 -0.5 0 0.5 1.0DEVIATION(x A)

Fig. 9. Normalized histograms of the deviations between the ap-proximated first-order polynomials and the measured value of thedeformation, where the object is tilted. Lines (a) and (b) result fromthe correction of phase discontinuities and median filtering, respec-tively. The standard deviations in lines (a) and (b) are 0.0639 X 27r

and 0.0489 X 2r rad, respectively.

C. Error Estimation

The measurement error is estimated experimentally.An acrylic rectangular object 50 X 50 mm2 and 10 mmthick is used, whose surface is sprayed white. Theobject is tilted horizontally and the displacement ismeasured in the same manner. The measured valuesof each horizontal line are fitted to first-order polyno-mials by a least-squares approximation. The valuecalculated from the approximated polynomials is re-garded as the true value of the deformation, and thedeviation between the true and measured values is ob-tained at each sample point. The normalized histo-grams of the deviations are shown in Fig. 9, where linesa and b show the histograms after correction for phasediscontinuity and median filtering, respectively. Theselines show that the histograms obey the Gaussian dis-tribution. The standard deviations, Oc and o-m, re-sulting from the correction of the phase discontinuityand the median filtering, are 0.0639 X 2 r and 0.0489 X27r rad, respectively.

These standard deviations are regarded as the casualmean errors in fringe scanning speckle-pattern inter-ferometry (FSSPI). In fringe scanning inteferometry,systematic and casual errors caused by mechanical andelectronic instruments can be reduced by increasing thenumber of divisions within the period of a fringe.7 Thephase noise caused by speckles, which cannot be com-pletely removed, is dominant in FSSPI. Therefore, themeasurement accuracy with FSSPI does not becomemuch higher even if the number of divisions of the pe-riod of the fringe is increased. The measurement ac-curacies of out-of-plane deformation, which are re-garded as 3c and 3 m, are equal to 9.59/100 X and7.34/100 X, respectively.


Ill. In-Plane Deformation Measurement

The schematic diagram of FSSPI for measurementof in-plane deformation is shown in Fig. 10. The objectmeasured is illuminated by two symmetrical beams withrespect to the normal direction of the surface of theobject. The mirror attached to the PZT is inserted inthe path of one illuminating beam to change the pathdifference between two illuminating beams. A specklepattern So is recorded on the MT before deformationof an object. After deformation, four phase-shiftedspeckle patterns, S1 -S4 , are also recorded on the MTwhere the high voltage to the PZT is changed stepwiseas the phase difference between two illuminating beamsbecomes 0, 7r/2, 7r, and 371r/2. Speckle interferogramsFi (i = 1-4) are generated by calculating (Si - So)2 (i =

1-4), and subsequently the interferogram Fi is low-passfiltered by five iterations of averaging over a 3 X 3sample area. The low-pass filtered interferogram isrepresented by It (i = 1-4). The phase proportional tothe in-plane deformation can be obtained from Eq. (1),which is denoted by -1. Assuming that (p denotes thephase after correction of the phase discontinuity, thenumerical value of in-plane deformation dx can becalculated by

2 sinO 27r

Fig. 10. Schematic diagram of the fringe scanning speckle-patterninteferometer for measuring in-plane deformation.

(3)

where a is the angle between the illuminating beams andthe normal direction of the surface of the object.

The object measured is a rectangular brass plate 20mm high, 50 mm wide and 0.5 mm thick; a 3-mm diamhole is made in its center. The white metal strip isclamped at its left and right ends and is subject to ninesuccessive uniform tensions at its right end. The ob-served area is 35 X 35 mm2 at the center of the object.Speckle interferograms in each state of the object withrespect to the undeformed object are shown in Figs.11(a)-(i), where the images are enlarged longitudinallyto the full scale of the TV monitor. The number offringes on the whole surface of the object increasessuccessively from Figs. 11(a) to 11(i), but fringes shownin Figs. 11(g)-(i) are not well resolved because of thedecorrelation of the corresponding speckles between thetwo images. In FSSPI, however, the numerical valueof the deformation in the ninth state of the object canbe obtained by the procedure described in Sec. II.B.

Phase images proportional to the deformation be-tween adjacent states of the object are shown in Figs.12(a)-(i), where the phases ranging from-7r to 7r rad aredisplayed as 0-255 levels on the monitor. Because theangle 0 is 21°, the sensitivity of one fringe is 0.88 m.The discontinuity of phases is corrected and subse-quently summed from Figs. 12(a) to 12(i). Deformationnumerical data after the correction of phase disconti-nuity are shown in Figs. 13(a) and (b), which are contourand perspective representations. The interval betweencontours is 1 gum. The numerical data resulting fromeight iterations of the median filtering within an 11 X11 sample area are shown in Figs. 14(a) and (b). Theinterval between contours corresponds to a 1-,m in-plane displacement. These figures show that the

Fig. 11. Low-pass filtered speckle interferograms in nine states ofin-plane deformation of a brass plate with a hole. The deformations

increase gradually from (a) to (i).

Fig. 12. Phase distribution proportional to the in-plane deformationbetween adjacent states of the deformations shown in Fig. 11. Phasesranging from -7r to 7r rad are displayed as 0-255 levels on the

monitor.

maximum value of the deformation is -14 m, and theobject deformed uniformly in the horizontal directionexcept for the region near the hole. The data on eachhorizontal line in Fig. 14 are fitted to a twentieth-orderpolynomial, and the results are spatially differentiatedto show strain and stress. The distribution of the strainis shown in Fig. 15, where the interval between contoursis equal to 1 X 10-4 strains. These figures show that thestress concentrates near the edge of the hole.


pNRW(a)

B.S.

Fig. 13. Numerical value of the in-plane deformation in the ninthstate of the object, which is obtained by correcting the phase discon-tinuities in Figs. 12(a)-(i); the results are summed: (a) contour and(b) perspective representations. The inteval between contours is 1

Jim.

(a)

ShearingCamera,

|MemoryI | L

tor I PDP-11/34 |Tape l

Fig. 16. Schematic diagram of fringe scanning speckle-patternshearing interferometry for 3-D deformation measurement.

(b)Fig. 14. Numerical values of the deformation result from eight it-erations of the median filtering within an 11 X 11 sample area: (a)contour and (b) perspective representations. The interval between

contours is 1 /Am.

(b)

Fig. 15. First spatial derivative of the deformation in the horizontaldirection, which is proportional to the strain or stress. A twenti-eth-order polynomial is fitted to the data on each horizontal line inFig. 14 and the resultant polynomial is spatially differentiated: (a)contour and (b) perspective representations. The interval between

contours is equal to 1 X 10-4 strains.

IV. Three-Dimensional Deformation Measurementwith a Shearing Camera

A. Method and System Description

Hybrid systems for measuring 3-D deformation withholographic interferometry have been developed byseveral authors.101 2 In this section a method for the3-D deformation measurement with fringe scanningspeckle-pattern shearing interferometry (FSSPSI) ispresented.

A speckle-pattern shearing interferogram is a func-tion of the spatial derivatives of 3-D deformation of anobject.13 Therefore measurement of 3-D deformationwith speckle-pattern shearing interferometry can beperformed using three illuminating beams. A sche-

Fig. 17. Shearing camera for fringe scanning speckle-patternshearing interferometry.

matic FSSPSI diagram for 3-D deformation measure-ment is shown in Fig. 16. The light from a He-Ne laseris split into two illuminating beams, I, and I2. The lightfrom an Ar-ion laser is the other illuminating beam I3.Either the same or separate lasers can be used for threeilluminating beams. Two lasers in this experiment areused for obtaining sufficient intensities of the illumi-nating light. In the shearing camera, diffusely scatteredlight from the object is split into two wave fronts, andtwo sheared images of the object are formed on thetarget of the TV camera. The sampled and digitizedimage is stored in the digital frame memory and trans-ferred to the mass storage device of the computer sys-tem. A schematic diagram of the shearing camera forfringe scanning is shown in Fig. 17; a Twyman-Greeninterferometer is used. A cubic beam splitter 50 mmin width is used and the surfaces are coated for antire-flection. The tilt of two mirrors 50 mm in diametergives a variable amount of shearing of the two images.


MVUhMi

A PZT device is attached to the mirror for changing thepath difference between the two wave fronts.

Before deforamtion of the object, speckle images Sio(i= 1-3) are stored on the MT as successively changingilluminating beams denoted by the subscript i. Afterdeformation, speckle images S' (i = 1-3,j = 1-4) arestored on the MT as changes in the phase differencebetween two sheared lights by the amounts of 0, 7r/2, r,

and 3ir/2, which are denoted in order by the subscriptof j. Calculating (S - S.o)2 gives a speckle interfero-gram Fij, which is subsequently low-pass filtered by fiveiterations of averaging over 3 X 3 sample points. Thecalculation from Eq. (1) with low-pass filtered speckleinterferograms, Iij (j = 1-4), gives the phase difference,Oi(i = 1-3). Assuming that the vectors d = (u,v,w) andIi = (l,m,ni) denote the deformation of the object andthe unit vectors of the illuminating directions, respec-tively, the phase change Oi can be written as' 3

2r= [i - u+ mi - v + ( + ni) - w], (4)iv

where bu, 5v, and bw represent the differences of com-ponents of 3-D deformation between two points sepa-rated by an amount of shearing in the X direction. Thewavelength of the ith illuminating beam is denoted byXi. Here, it is assumed that the surface is observed fromthe Z direction. If the phase Oi is integrated in the Xdirection, phase Ai due to 3-D deformation d can bewritten as

Ai = 2iw [li u + mi v + (1 + n) w].

The matrix representation becomes

A = Ad,

where

1 1 'X11A= A2 X2

LA3. X3

d= v '[w]

11 ml + n;

A= 12 M2 1+n2

13 M3 + n3

Then the 3-D deformation vector d canfrom

d = A-1A,

be obtained

where A-' denotes the inverse matrix of A. The firstor second spatial derivative of a polynomial fit to acomponent of vector d gives strain or bending momentof the object deformation.

B. Experimental Results

The object to be measured is an aluminum disk 80mm in diameter and 0.1 mm thick covered with whitemagnesium oxide powder. The area observed by theTV camera is -56 X 56 mm2 . The vector diagram ofthree illumination directions with direction cosines is

1 J-370

I =(-0.036, 0.59,0.80)

12=(-0.55,-0.42, 0.73)

I3=(0.54,-0.41, 0.74)Fig. 18. Vector diagram of three illuminating directions with di-rection cosines. II and I2 represent the unit vectors of illuminatingbeams from the He-Ne laser; 13 represents that from the Ar-ion laser

shown in Fig. 16.

(5)

(6)

Fig. 19. Four phase-shifted low-pass filtered speckle interferogramsin the direction of 13 in Fig. 18.

shown in Fig. 18. I1 and 12 represent the unit vectorsin the illuminating light from the He-Ne laser and 13 isthat from the Ar-ion laser. The wavelengths of theHe-Ne and Ar-ion lasers are 0.6328 and 0.5145 ,m, re-spectively. Four phase-shifted and low-pass filteredspeckle interferograms in the I3 direction are shown inFig. 19, which are obtained by five iterations of aver-aging over 3 X 3 sample points. Phase images due tothe deformation are shown in Figs. 20(a)-(c), whichcorrespond to the directions I, I2, and 13, respectively.In these figures, the phases ranging from -7r to 7r radare displayed as gray levels from 0 to 255.

The phase difference due to the illuminating beam12 is shown in Fig. 21, which results from the correctionof the phase discontinuity and eight iterations of themedian filtering within an 11 X 11 sample area. Theinterval between contours in Fig. 21(a) is 21 rad. Theshearing is in the horizontal direction, which is equal tothe X direction. The amount of shearing is 3.6 mm onthe object surface which corresponds to thirty-twosample points in the digital picture. Therefore suc-


(b)

Fig. 23. The Z component w of 3-D deformation: (a) contour and(b) perspective representations. The interval between contours is

0.5 gm.

Fig. 20. Phase distribution proportional to 3-D deformation of thecircular object in the direction of (a) I1, (b) 12, and (c) I. The amountof shearing is 3.6 mm in the horizontal direction. Phases ranging from

-'ir to ir rad are displayed as 0-255 levels on the monitor.

(a)

Fig. 21. Phase difference proportional to the spatial derivative ofthe deformation, which results from the correction of phase discon-tinuities and eight iterations of the median filtering within an 11 X11 sample area: (a) contour and (b) perspective representations. The

interval between contours is 27r rad.

(b) (c)

Fig. 22. Phase distributions proportional to 3-D deformation usingthe illuminating beams of (a) I,, (b) 12, and (c) 13, which are obtainedby successive summation in the horizontal direction using data shown

in Fig. 21.

cessive summation of the phase data Oi at everythirty-two sample points gives the phase difference Aiin Eq. (5). Resultant phase distributions due to thethree illuminating beams, I1, 12, and I3, are shown inFigs. 22(a)-(c), respectively. The interval betweencontours is 27r rad. The calculation of Eq. (9) with thephase data shown in Fig. 22 gives each component of

(b)

Fig. 24. Second spatial derivative in the horizontal direction,02 w/x 2 , using the data shown in Fig. 23, which is proportional to thebending moment: (a) contour and (b) perspective representations.

The interval between contours is 1 X 10-5 (1/m).

(a) (NFig. 25. First spatial derivative of the in-plane deformation com-ponent in the horizontal direction, du/Ox: (a) contour and (b) per-spective representations. The interval between contours is 1 X 10-5

strains.

3-D deformation. The Z component of the deformationis shown in Figs. 23(a) and (b), where the interval be-tween contours is 0.5 gm. The Z component of thedeformation is fitted to a twentieth-order polynomialby the least-squares method, and the second spatialderivative of the polynomial gives the data proportionalto the bending moment. The contour and perspectiverepresentations of the second derivative, a2wIax2, areshown in Figs. 24(a) and (b), respectively. The intervalbetween contours is 1 X 10-5 (1/m).

Other data, such as the first spatial derivative of in-plane deformation au/ax, are shown in Figs. 25(a) and(b), where the interval between contours is 1 X 10-5strains. These figures show that bending moment andstress concentrate about the central area between theleft edge of the object and the center point. In thisexperiment the Z component of the deformation isdominant; other components calculated are not so ac-

15 Julv 1985 / Vol. 24, No. 14 / APPLIED OPTICS 2179

?-I"~~~~~~~~~~~~

I iGQ0

UU\~~~ 7

I2

Me

curate. However, other components of 3-D deforma-tion can be known qualitatively.

Speckle-pattern shearing interferometry has severalfeatures: (1) simple optical setup; (2) considerabletolerance of environmental disturbances; (3) no needfor path length adjustment. Therefore, the presentmethod is well suited for 3-D deformation measure-ment.

V. Conclusion

Computer-aided speckle-pattern interferometry withfringe scanning has been discussed, which makes itpossible to evaluate automatically speckle interfero-grams to give numerical data of the deformation. Be-cause a speckle noise in a speckle inteferogram cannotbe completely removed, FSSPI does not present themeasurement with very high accuracy compared withfringe scanning holographic interferometry. However,FSSPI has several attractive features: (1) use of a lowresolution device; (2) short exposure time; (3) no needfor photographic processing; (4) automatic determi-nation of fringe order numbers; and (5) automatic in-terpolation of fractional fringe order numbers. Themost attractive feature of FSSPI is that, if a largenumber of images can be recorded in every state of theobject, it is possible to perform the measurementwithout an upper limit.

The authors would like to thank Toyohiko Yatagaiof University of Tsukuba for his helpful suggestions.

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Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975),Vol. 9, pp. 203-253.

2. J. N. Butters and J. A. Leendertz, "Speckle Pattern and Holo-graphic Techniques in Engineering Metrology," Opt. LaserTechnol. 3, 26 (1971).

3. 0. J. Lokbrg, 0. M. Holje, and H. M. Pedersen, "Scan ConverterMemory Used in TV-Speckle Interferometry," Opt. LaserTechnol. 8, 17 (1976).

4. S. Nakadate, T. Yatagai, and H. Saito, "Electronic Speckle Pat-tern Interferometry Using Digital Image Processing Techniques,"Appl. Opt. 19, 1879 (1980).

5. S. Nakadate, T. Yatagai, and H. Saito, "Computer-Aided SpecklePattern Interferometry," Appl. Opt. 22, 237 (1983).

6. R. Daindliker, B. Ineichen, and F. M. Mottier, "High ResolutionHologram Interferometry by Electronic Phase Measurement,"Opt. Commun. 9,412 (1973).

7. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld,A. D. White, and D. J. Brangaccio, "Digital Wavefront MeasuringInterferometer for Testing Optical Surfaces and Lenses," Appl.Opt. 13, 2693 (1974).

8. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-TransformMethod of Fringe-Pattern Analysis for Computer-Based To-pography and Interferometry," J. Opt. Soc. Am. 72, 156 (1982).

9. G. E. Sommargren, "Double-Exposure Holographic Interfer-ometry Using Commonpath Reference Waves," Appl. Opt. 16,1736 (1977).

10. P. Hariharan, B. F. Oreb, and N. Brown, "Real-Time HolographicInterferometry: a Microcomputer System for the Measurementof Vector Displacements," Appl. Opt. 22, 876 (1983).

11. S. Nakadate and H. Saito, "Fringe Scanning Holographic andSpeckle Interferometers for Deformation Measurements," Ko-gaku (Jpn. J. Opt.) 13, 299 (1984), in Japanese.

12. S. Nakadate, N. Magome, T. Honda, and J. Tsujiuchi, "HybridHolographic Interferometer for Measuring Three-DimensionalDeformations," Opt. Eng. 20, 246 (1981).

13. S. Nakadate, T. Yatagai, and H. Saito, "Digital Speckle-PatternShearing Interferometry," Appl. Opt. 19, 4241 (1980).

1985August

Fourth Australian Laser Conference, Sydney.,P.G. Browne, School of Maths and Physics,Macquarie University, North Ryde NSW 2113,Australia.


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Documents

Fringe scanning speckle-pattern interferometry