Interferometric comparison of displacements by electronicspeckle pattern interferometry

Ole J. L0kberg and Gudmunn A. Slettemoen

Using the image waves from two similar objects as, respectively, object wave and reference wave in an elec-tronic speckle pattern interferometer (ESPI) setup, it is possible to obtain interference patterns that indi-cate the amount of relative displacement between the two objects. Experimental results demonstrating di-rect interferometric subtraction and addition of vibration patterns are presented. Possible applications ofthe technique are also discussed.

1. Introduction

Electronic speckle pattern interferometry (ESPI) isa hologram interferometry technique in which videoregistration and electronic processing have replaced theconventional film recording and optical reconstructionsteps (for example, see Ref. 1). ESPI is used mainly tostudy and measure the deformations or vibrations of awide variety of objects.

In ordinary ESPI the reference wave is a sphericalwave which serves as a reference to changes in the objectwave. The resulting fringe patterns indicate the ab-solute displacement of the object's surface, which isusually the measurement desired. For some applica-tions, however, it would be useful to obtain fringe pat-terns that indicate the relative movement between twosimilar objects, as, for example, a test object and itsprototype.

In this paper we show how an ESPI with specklereference beam can be used to subtract (and add) in-terferometrically the displacements of two objects.

11. Speckle Reference ESPI

A. Basic Principle

The theory and design of a speckle reference ESPIwere presented in an earlier paper,2 and the reader

When this work was done both authors were with Norwegian In-stitute of Technology, Physics Department, N-7034 Trondheim-NTH, Norway; G. A. Slettemoen is now with University of Arizona,Optical Sciences Center, Tucson, Arizona 85721.

Received 30 March 1981.0003-7935/81/152630-05$00.50/0.

) Optical Society of America.

should consult that reference for a complete description.Only a rough idea is given here of the working principleof this technique. Referring to Fig. 1, O1 and 02 rep-resent two similar objects that reflect the illuminatinglight diffusely. (The illumination is not shown in thisfigure.) The object waves are directed toward lens Lby means of a beam combiner BC, which is the essentialcomponent in the interferometer. It consists of anarray of reflecting r, transmitting t, and opaque o slitsarranged in the order. . . t, o, r, o, t ... as indicated inthe upper part of Fig. 1. The slits are equal in widthand are orthogonal to the TV scan direction. The 01wave is reflected by the slit mirrors, and the 02 wave istransmitted through the open slits. The waves arecombined by the lens into two overlapping pictures ofthe objects on the target T of the video camera. Thesewaves are coherent and interfere. Due to action of theopaque slits, the cross-interference term, 01-02* (theinformation), and the self-interference terms, 01-01*and 02-02* (the noise), occupy separate spatial fre-quency bands. The intensity distribution in the in-terferogram is converted into a corresponding videosignal by the photoelectric action of the TV camera.This also converts the spatial frequencies into temporalfrequencies. Therefore, by adjusting the widths of theslits and passing the video signal through an appropriatebandpass filter, the self-interference terms are removedand only the useful cross-interference term is passed.The filtered video signal is demodulated by full-waverectification. It is subsequently converted into animage on the video monitor.

B. Interferometric ComparisonWe designate 01 the test object and write the complex

amplitude of its image wave as

Ao(r,t) = Ao(r) -exp[iko * ro(t)], (1)

where r is a position vector in the image and the move-

2630 APPLIED OPTICS / Vol. 20, No. 15 / 1 August 1981

0,

0 ,

Fig. 1. The optical part of a speckle reference ESPI. Upper partshows a head-on view (enlarged) of the beam combiner BC.

ment of the corresponding object points is given by thedisplacement vector ro(t); ko is the holographic sen-sitivity vector at those same points, and the effect of theobject movement is factored out as an exponential fac-tor. That is, we suppose that the object movement issmall such that the image speckles do not decorrelateand Ao(r,t)J stays constant during measurement.

In the same manner we represent the complex am-plitude of the image wave due to the reference objectas

AR(r,t) = AR(r) exp[ikR rR(t)I (2)

The waves are combined on the TV target to give anintensity distribution

IT(r,t) = Ao(r,t) + AR(r,t)12

= Ao(r)Ao*(r)1 2 + AR(r)AR*(r)J 2

(Iw (r)() 2 (Io ) (IR ) I M(r) 1 2, (5)

where (I ) is the average intensity of the object image,and (IR) is the average intensity of the reference objectimage. For our purpose the fringe function dependenceon the relative object movements ro(t) - rR(t) is ofmajor concern. In ordinary hologram interferometrythe fringe function is given by the magnitude and thetime dependence of the object movement during theeffective recording time. But in our case the fringefunction depends on the relative movement betweencorresponding object points. By far the most importantcases are the two-step change and the out-of-plane si-nusoidal vibration.

The two-step change may be produced by a staticdeformation recorded by double exposures in a videomemory or two stages of a dynamic deformation frozenby short exposures within a TV frame. We assume theobjects's motion during the exposures to be negligible,and the resulting fringes will then be described by

IM12 = cos2[k(ro - rR)]. (6)

If r = r for all object positions we have identical de-formations, and a zero order fringe would cover theentire image. Note that this fluffing out of the fringeswill work however complex the objects and their dis-placements are. Any differences in the two deforma-tions will show up as darkened areas or local fringepatterns. Therefore, this measurement of relativedisplacement works provided the sensitivity vector isthe same for corresponding object positions and pro-vided the movements are so small that IAo (r,t) I andI AR(r,t)I do not completely change during measure-ment.

For vibrating objects we consider the out-of-planemovement of the vibrating test object given by

(3a)

+ 2 Re1Ao(r)AR*(r) exp[i(ko ro(t)-kR rR(t)I. (3b)

The first two terms on the right-hand side of Eq. (3b)represent noise, and, as explained in the previous sec-tion, they are removed by electronic filtering of thevideo signal. We now introduce a normalized exposurefunction f(t), which describes the effective exposure ofthe camera target. The maximum exposure time isequal to 1/25 sec (European standard), and f (t) can takeany form inside this time interval (as, for example, singleor double exposure). After a square law rectificationof the video signal, we are then left with an image in-tensity on the TV monitor given by

Im(r) 1 12 Re[Ao(r)AR*(r)ff(t) explik[ro(t) - rR(t)IdtI}w=12 Re[Ao(r)AR*(r)M(r)]1 2 , (4)

where, or corresponding object points, a commonsensitivity vector k = ko = kR has been assumed. Themultiplicative fringe function M(r) will modulate themonitor intensity, and it is given by the integral in Eq.(4). The random nature of AO (r) and AR (r) causes theimages to have the speckled appearance. If it is as-sumed that the complex amplitudes Ao(r) and AR(r)obey circular Gaussian statistics, the expectation valueof the monitor intensity at position r will be equal to2

Iro(t)l = ao(x,y) cos[27rft + Oo(X,y)], (7a)

where ao(x,y) and ko(x,y) are, respectively, the am-plitude distribution and phase distribution across theobject, and f is the vibration frequency.

Similarly, for the reference object,

IrR(t) = aR(x,y) cos[2irft + (PR(X,y)]. (7b)

For illumination and viewing directions normal to thesurfaces we can now use the result from, for example,Ref. 3 to express the squared fringe function as

I~2 = J202 klao(xy) + a (x,y) -2ao(x,y) aR(x,y)* os[0ko(x,y) - OR(X,y)] I"2I. (8)

From Eq. (8) it is seen that the zero order fringe, or 0will cover the entire image only if ao(x ,y) = aR (x ,y), and

o(X,Y) = R(x,y). Any difference caused by eitheramplitude or phase will produce higher order fringes.To determine the cause of the difference it is possibleto vary the amplitude and/or the phase of the test objectto obtain a zero order fringe at the area of interest. Thistechnique as applied to vibrating objects might beconsidered as a phase modulation technique, 4 where thereference object acts like an optical element producing

1 August 1981 / Vol. 20, No. 15 / APPLIED OPTICS 2631

I

spatially varying phase modulation. The compensationprinciple of phase modulation is effective for all kindsof periodic movements. 5

Generally it is noted that for both static and dynamicdeformation the ordinary fringe pattern representingthe behavior of either object individually is obtained bykeeping the other object at its rest position during therecording.

The interferometric sensitivity can be doubled byadding optical path lengths instead of subtracting asdescribed above. This is particularly easy in vibrationwork. Here, the amplitude and phase of the test objectare first adjusted to make the area of the zero orderfringe as large as possible. Then, by shifting the phaseof either object by 1800, the object waves are caused tointerfere when corresponding object points move in theopposite direction. This doubles the number of fringesas compared to using a static reference object. Forstatic deformations the increased sensitivity can beobtained whenever the objects are deformable in bothdirections. An example is nondestructive testing byESPI using under- or over-pressure to load theobject.

M

0/L BC

BANDPASSTVFILTER. MONITOR

RECTIFIER.

0 I

02:

I '

I I

r- ---- - - JSIGNAL J I PHASE

GENERATOR - SHIFTER _J

Fig. 2. Schematic drawing of the speckle reference ESPI setup usedto compare the vibrational displacement of object 01 with that of

object 02-

Finally, it is also possible to compare the test objectwith a reference object of a difference scale by intro-ducing an extra imaging lens in one of the branches ofthe interferometer.

Ill. Experimental Setup and Results

The layout of the ESPI setup is shown in Fig. 2. Thelight from the argon laser was expanded from a distantpoint to give nearly plane wave illumination normal tothe objects 01 and 02. Both objects were rectangularsteel plates 10 X 13 cm. They were made, mounted, andexcited as identically as possible. The 02 wave wastransmitted directly through the open slits of beamcombiner BC. The 01 wave was directed by mirror Monto the reflecting slits on BC. (Note that this mirroris necessary to get correct overlap of the objects. Figure1 gives overlap between the image of object 02 and themirror image of O1.) As indicated in Fig. 2 object 02

had to be moved farther back to equalize the pathlengths 02- BC and 01 - M - BC. The objects wereimaged by lens L onto the 2.5-cm (1-in.) Chalnicon tubeof the video camera. (The active area is -1 cm2 .) Thelens was used at f/14 aperture with 150-mm focal length.The video signal was passed through an active bandpassfilter (2-4 MHz) followed by a full-wave rectifier. Theprocessed signal was finally displayed as an image on a22.8-cm (9-in.) video monitor. To improve the picturequality of the fringe patterns speckle averaging wasused, as described in Ref. 2. Recording static defor-mations by means of ESPI required a video store of highquality, which was unavailable. Therefore, we wereable to demonstrate only how the vibration patterns ofobjects 01 and 02 could be subtracted and added.

The objects were excited by piezoelectric transducersthat were connected to separate outlets on a signalgenerator. A continuous phase shifter was used tocontrol the phase of the signal that excited object 02-

A typical example of direct subtraction and additionof vibration patterns is shown in Fig. 3. Figures 3(a)and (b) show the individual vibration patterns of objects

Fig. 3. Objects vibrating at 1575 Hz:(a)-(b) individual vibration patterns ofobjects 01 and 02; (c) subtraction pattern,objects 01 and 02 vibrating in phase; (d)sum pattern, O1 and 02 vibrating

in antiphase.

2632 APPLIED OPTICS / Vol. 20, No. 15 / 1 August 1981

Fig. 4. Objects vibrating at 2069 Hz:(a)-(b) individual vibration patterns ofobjects 01 and 02; (c) optimized sub-traction pattern; (d) sum pattern, relative

phase changed 1800 from (c).

Fig. 5. Objects vibrating at 10,150 Hz:(a)-(b) individual vibration patterns ofobjects 01 and 02; (C) subtraction pattern; L(d) sum pattern, relative phase changed

180° from (c).

01 and 02 at 1575 Hz. These patterns were recordedin the setup using one object at rest as a reference to theother vibrating object. Figure 3(c) shows the vibrationpattern recorded when both objects vibrated in phase,whereby the movements were effectively subtracted.The zero order fringe covered most of the monitorimage, indicating that at this frequency the objects'vibrations were almost identical. In Fig. 3(d) the phasehad been shifted by 180° to make the objects vibrate inantiphase. As expected the vibration amplitudes wereeffectively added, which doubled the number of fringesin relation to the pattern of Figs. 3(a) and 3(b).

In Fig. 4 the objects vibrated at a higher frequency:2069 Hz. The individual vibration patterns shown inFigs. 4(a) and 4(b) were quite different. Object 02 Vi-brated in a combination of two modes, which was indi-

cated by the nodal lines degenerating into nodal points. 6

In this case the phase setting that gave a minimumnumber of fringes resulted in an apparently new modepattern [Fig. 4(c)]. This pattern looked as if the con-tribution from one of the 02 modes had been removed.The summation pattern on Fig. 4(d) behaved as ex-pected.

The last example, in Fig. 5, shows the technique usedat higher frequencies, where the vibration patterns weremore complex. Here we also observe some minor dif-ferences between the two object displacements.

Before closing this section a comment should be madeon an experimental problem caused by the proximityof the objects. At certain frequencies, especially in thelower range, a strong acoustical coupling was observedbetween the objects. This coupling would either gve

1 August 1981 / Vol. 20, No. 15 / APPLIED OPTICS 2633

unstable patterns or make the vibration amplitude ofone object dependent on the vibration level of the otherobject. The latter effect meant that individual patternsthat should have subtracted to a zero order fringe gaveinstead a quite different value in the actual experiment.However, by adjusting the excitation of one of theobjects the expected fringe pattern was obtained. Thisproblem could have been reduced by use of acousticaldamping, for example, by simply moving the objectsfarther apart, but in these experiments the troublesomefrequencies were simply avoided.

IV. Concluding Discussion

A simple optical method has been described for directmeasurement of the relative displacement between twoobjects' surfaces. Although the method has beendemonstrated for vibrating objects only, the most usefulapplication will probably be for static deformation ofobjects. Here the most obvious application would befor testing the behavior of mass-produced objects inrelation to that of a prototype. It should be straight-forward to automate inspection of whether an objectdeforms within the prototype specification. For ex-ample, small loads could be exerted on the objects andspecify that the acceptable difference pattern shouldbe within the first dark fringe. The whole inspectionwould then be reduced to an either/or test based onsimple measurement in the video image.

Direct interferometric subtraction of the staticmovement of two similar objects would also be usefulin nondestructive testing by ESPI. Here the loadingof the test object might produce complex fringe patternswhere the contribution due to local deformations indi-

cating material failure can be easily overlooked. Pro-vided the prototype and the test object can be equallyloaded these gross movements should be fully canceled,however, complex they might be. But decorrelation ofthe speckle patterns, influencing the absolute value ofthe complex image amplitudes and caused by, for ex-ample, large object tilts, limits the maximum displace-ments that can be measured by this technique.

In vibration analysis this method has the advantageof comparing not only the amplitude distribution butalso the phase distribution. This additional informa-tion is very valuable especially when complex vibrationsare likely, as in a loudspeaker, for example. Automatedinspection of vibrating objects is also possible, but theproblem is that similar objects can have their resonancefrequencies several hertz apart. Therefore they haveto be separately excited and the net effect of amplitude,phase, and frequency difference measured. The re-sulting fringe pattern will vary cyclically at the differ-ence frequency between the sum and difference patternsof the two movements.

We thank Martha Stockton of the Optical SciencesCenter and Randi Bolso Moen of the Department ofPhysics, NTH, for help in preparing the manuscript.

References1. 0. J. L0kberg, Phys. Technol. 11, 16 (1980).2. G. A. Slettemoen, Appl. Opt. 19, 616 (1980).3. C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).4. 0. J. Lokberg and K. Hogmoen, Proc. Soc. Photo-Opt. Instrum.

Eng. 136, 222 (1977).5. K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).6. N. E. Molin and K. A. Stetson, J. Phys. E: 2, 609 (1969).

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2634 APPLIED OPTICS / Vol. 20, No. 15 / 1 August 1981