strain measurement by heterodyne holographic interferometry

Strain measurement by heterodyne holographic interferometry

Ruedi Thalmann and Rene Dandliker

Heterodyne holographic interferometry provides automated interference phase measurement with highresolution and allows the quantitative evaluation of 3-D displacement and strain on solid objects. Thefundamentals of heterodyning in double-exposure holography are reviewed and its possibilities and limita-tions are discussed in detail. Experimental results of strain measurement on a curved object surface arereported. Three double-exposure hologram recordings with different illuminations are used to get the vectordisplacement. Based on locally calculated derivatives of the displacement field, a sensitivity for the straincomponents of 1 gm/m with a spatial resolution of a few millimeters has been achieved. The features ofheterodyne holographic interferometry are compared with those of quasiheterodyne (phase stepping) fringeinterpolation.

1. Introduction

The automated and quantitative evaluation of 3-Ddisplacement and strain from holographic interfero-grams requires accurate interference phase measure-ment independent of fringe position and intensityvariations in the reconstructed image. In many inter-ferometric arrangements, the heterodyne techniquehas become a powerful tool for high resolution interfer-ence fringe interpolation. In holographic interferom-etry, heterodyning has been successfully applied tosurface deformation measurements of solid objects,1 2

measurements of phase objects,3 and surface contour-ing. 4

The aim of this paper is to demonstrate the perfor-mance and the theoretical and experimental limits ofsurface strain measurement by heterodyne holograph-ic interferometry. After a brief introduction into theprinciple of heterodyning, the experimental arrange-ment is described. Then, the numerical evaluation of3-D displacement and strain from the holographic in-terferogram is explained. The limitations for the ac-curacy of the interference phase measurement are in-vestigated in detail. Experimental results arereported for the determination of surface rotation andstrain with double-exposure holography using a setupwith three different illumination beams. Finally, the

The authors are with University of Neuchatel, Institute of Micro-technology, CH-2000 Neuchatel, Switzerland.

Received 3 December 1986.0003-6935/87/101964-08$02.00/0.© 1987 Optical Society of America.

different methods of electronic fringe interpolation inholographic interferometry are compared and theirfields of application are outlined.

II. Principle of Heterodyne Holographic Interferometry

The application of the heterodyne technique in dou-ble-exposure holographic interferometry requires asetup with two reference beams5 to have independentaccess to the relative phase between the two interferingreconstructions, which allows shifting the fringes inthe interferogram. The two states of the mechanicalobject (before and after deformation), which shall beinvestigated by double-exposure holography, are re-corded consecutively, each with another referencewave on the same hologram plate. The simultaneousreconstruction with, both reference beams yields fourprimary reconstructed images, the two desired inter-fering self-reconstructions and the two undesired crossreconstructions.5 To avoid disturbing overlapping ofthe different reconstructions, the two referencesources must be chosen on the same side of the object,with a mutual separation larger than the angular size ofthe object in the corresponding direction.

During reconstruction, the relative phase betweenthe two references is shifted linearly in time by intro-ducing an offset Aco to the optical frequency in one ofthe two reference beams. Assuming that the two wavefields to be compared are given by

Vk(x) = ak(X) CoS[Wkt + lk(X)], (1)

the local intensity of their superposition becomes

I(x) = V, + V212 = a(x){1 + m(x) cos[Awt + 0(x)], (2)

where a(x) is the local mean intensity, m(x) is thefringe contrast, and 0(x) = 01(x) - 02(x) is the phase

1964 APPLIED OPTICS / Vol. 26, No. 10 / 15 May 1987

difference of the two wave fields. Due to the opticalfrequency offset Aco = xl- 2 during reconstruction,the superposition intensity is time dependent and theinterference phase 0(x) can be detected as the phase ofthe beat frequency signal. Since the beat frequencyAw/27r is chosen low enough (<100 MHz) to be resolvedby the optoelectronic detector employed, the interfer-ence phase can be measured with high accuracy, inde-pendent of variations of the mean intensity and thefringe contrast, using an electronic phasemeter. Thisway, both the interpolation problem and the sign am-biguity of classical interferometry are solved.

Fig. 1. Arrangement for heterodyne holographic interferometryusing two acoustooptic modulators to generate the frequency offset.

Ill. Experimental Arrangement

The experimental arrangement for double-exposureheterodyne holographic interferometry, as it has beenused in the experiments reported here, is sketched inFig. 1. The setup is essentially based on the one de-scribed in Ref. 1. The frequency difference Aco/27r of100 kHz between the two reference beams is realizedby two commercially available acoustooptic modula-tors (AOM) in cascade to give opposite frequencyshifts. During recording, both modulators are drivenwith 40 MHz, so that the net shift is zero. Duringreconstruction, one modulator is driven with 40 MHzand the other with 40.1 MHz, so that the net shift is thedesired beat frequency of 100 kHz.

The interferogram is observed in the 1:1 imge of thevirtual object obtained with an f/5.6, f = 300-mm Ro-denstock repro lens. The interference phase is deter-mined by scanning an array of three detectors whichmeasures the phase differences in two orthogonal di-rections in the image plane. The total phase 0(x) canbe obtained afterward by appropriate integration.The detector array is made up of the ends of three fiberbundles, which feed the light to photomultiplier tubes.The beat frequency signals at 100 kHz are filtered witha bandwidth of 10 kHz, and the amplitudes are keptconstant, independent of the intensity variationsacross the image, by a feedback control of the photo-multiplier supply voltage. The phase differences be-tween the detected signals are measured with zero-crossing phasemeters, which interpolate the phaseangles to 0.10 and also count the multiples of 360°(fringe number). Note that all electronic amplifiersand filters in the signal path should be carefully de-signed to avoid phase distortion that could reduce theaccuracy of the phase measurement.6 The detectorarray is mounted on a stepper motor driven stage toscan the image. Scanning and data acquisition areautomated and computer controlled. The measuringtime for one position, including the displacement ofthe detector head, is -1 s.

For the determination of vector displacement andstrain, the interference phase caused by the objectdeformation must be measured for different sensitiv-ity vectors, by changing either the observation, or theillumination directions, as explained in Sec. IV. Tohave a fixed imaging system and changes of the sensi-tivity vector which are large enough, we used a setupwith three illumination sources. The deformation is

j;-hheadImage plane

Illuminationbeam \

M

Object

Fig. 2. Schematic representation of the optical arrangement forhologram recording and observation. With the help of removablemirrors (RM), the illumination beam can be switched from source S1

to S2 and S3 (diverging lenses).

recorded with three double-exposure holograms, eachwith another object illumination. For reasons of sim-plicity in the recording procedure and sensitivity tomisalignment errors of the holograms, only one holo-graphic plate is used. The different holograms arespatially multiplexed by rotating a symmetric aper-ture in front of the holographic plate. The three illu-mination sources (diverging lenses with f = -6 mm)are arranged around the optical axis as shown in Fig. 2.The angle between the illumination directions and theoptical axis is typically 26. Switching from oneillumination source to the other is managed by remov-able mirrors mounted on precision translation stages.7First, the three holograms of the undeformed objectstate are recorded with the first reference wave on theappropriate part of the hologram plate, changing theillumination sources in the order 1, 2, and 3 (Fig. 2).Then the corresponding holograms of the deformedstate are recorded with the second reference wave andthe illumination beams 3, 2, and 1, respectively. Fol-lowing this procedure, no repositioning of the remov-able mirrors between the hologram exposures is re-quired. It is extremely important that the opticalsetup and the object in its undeformed and deformedstates are stable during the whole recording procedure.

IV. Evaluation of Displacement and Strain

A. Determination of Vector Displacement

The interference phase 0 and the displacement vec-tor u at the object point x are related by8

15 May 1987 / Vol. 26, No. 10 / APPLIED OPTICS 1965

V

kMf

In

w I

~-P ( )

Fig. 3. (,) is a plane tangential to the object surface at point Pn.The derivatives of the displacement at Pn are obtained by fitting a 2-D linear function through Pn and its (4,8, or more) nearest neighbors

Pi~ti,ni)-

, = (g u), (3)

where k = 27r/X is the wavenumber and g denotes thesensitivity vector

g = k-h. (4)

k and h are the unit vectors of observation and illumi-nation direction, respectively given by

k = (XA - X0)/IXA - xJ and h = (x0 - x,)/Ix0 - xJI. (5)

X,, XA, and x, are the coordinates of the object point,the observation point, and the illumination source,respectively. It can be seen from Eqs. (4) and (5) thatthe sensitivity vector g is not constant along the objectsurface, but it depends on the coordinate x,. There-fore, g has to be calculated for each object point x,independently.

To determine all three components of the displace-ment vector, three interference phase measurementsfrom three hologram recordings with different illumi-nation sources and thus different sensitivity vectors g,are carried out. Equation (3) can be written in matrixform as

0 = kGu, (6)

with the 3-D vector 4 = (01,02,03) and the 3 X 3 matrixG with Gnk = (n)k. Equation (6) can now be solved bytrigonalization of the matrix G. If the interferencephase is measured for more than three independentsensitivity directions, the system of linear Eqs. (6) isoverdetermined. Its best solution is found by least-squares fitting.9

As mentioned in the preceding section, heterodyneholographic interferometry provides the measurementof phase differences Aq5, and Aq5y in the orthogonaldirections x and y rather than the total interferencephase j(x,y) itself. For the evaluation of the displace-ment field, O(x,y) can easily be calculated by summa-tion of the measured phase differences along a givenpath. Obviously, with respect to the propagation ofmeasurement errors, a simple summation procedure isnot ideal. More sophisticated methods with least-squares fitting for the summation of the interference

phase are described in the literature (see, e.g., Ref. 10),but they all need extensive computations. However,for strain analysis this is not necessary, since the phasedifferences between the object points rather than thetotal phase are important.

B. Determination of Surface Strain and Rotation

In-plane strain and rotation of the object surface canbe determined by numerical differentiation of the dis-placement field in the tangential plane at each point.The grid of sample points on the surface of nonplaneobjects is generally distorted by perspective. There-fore standard finite difference methods cannot be ap-plied. However, the required derivatives can be calcu-lated by 2-D least-squares fitting of the displacementvector components through the desired point and itsnearest neighbors, as described hereafter.

The location of the sample points on the objectsurface are obtained by a central projection of theregular grid, which is defined by the interference phasemeasurement in the image, from the observation pointonto the object surface. The derivatives shall be cal-culated in an arbitrarily chosen sample point, denotedby P,. The object point P, is taken as the origin of anew coordinate system (Q,-), tangential to the surfaceat this point, with the t axis in the x,z plane. A certainnumber N of points Pi around Pn are selected, e.g., 4, 8,or even more, depending on the desired spatial resolu-tion and the accuracy of the displacement measure-ment. Points Pi are projected onto the (,i7 plane,where they form in general an irregular grid of points(, ) (see Fig. 3). The components of the displace-ment vector u are also expressed in the local coordinatesystem Qxq). For each component Uk of the displace-ment vector, a 2-D linear function

UkQn) = Uk(0,0) + O8Uk/8 + OUk/an (7)

is established, which fits the values uk(ttij1 i) at the Nsample points best. Least-squares fitting yields thefirst derivatives by solving the system of equations

Bv = b, (8)

where B is the symmetrical 3 X 3 matrix B = ATA and Ais the 3 X N matrix:

/1 n 1

A= . . . . (9)1 (N 77N

The vector v of the three unknownsvector b are defined by

and the constant

V = [Uk(0,0),9Uk/d8U0k/8OI, (10)

b = ATa, with ai = uki,1i), i = 1,...,N. (11)

Now, the in-plane strain E and the rotation wc are givenby


l In

-e

0 ml so

PI/ En

4)

aut 1 ut du \ea r at , =,, ' 2 __ a )

dun 2 /du ant (12)

The remaining components of strain and rotation,which are related to the out-of-plane derivatives of thedisplacement, can be calculated by using the mechani-cal boundary conditions for an object surface free fromexternal forces."1

C. Accuracy of Surface Strain Determination

The accuracy for the determination of surface strainby holographic interferometry depends essentially onthe error in the interference phase measurement.This has been investigated for heterodyne holographicinterferometry in Ref. 11.

To study the influence of interference phase errorson the evaluation of displacement and strain, one hasto distinguish between relative phase errors (subse-quently denoted by 6) and errors in the absolutephase (AO). The former ones determine the accuracyfor the measurement of the phase difference betweentwo neighboring points in the interferogram (sensitiv-ity of phase measurement). This relative phase erroris mainly of statistical nature. It is the essential limi-tation for surface strain determination. The absolutephase (absolute fringe order), on the other hand, isdirectly related to the change of the optical pathlength. Its uncertainty is the same all over the inter-ferogram and limits mainly the determination of thedisplacement.

Although the accuracy for the different componentsof displacement and strain depends on the opticalarrangement, it is appropriate to carry out an analyti-cal estimation for a simple geometry. It shall be as-sumed that three holographic interferograms with dif-ferent illumination directions are evaluated. Theillumination sources are arranged symmetricallyaround the optical axis, at a distance from the originon the object surface. The illumination directionsintersect the optical axis with an angle a and the obser-vation direction is parallel to the optical axis (see Fig.2).

If the strain components are calculated by numericaldifferentiation of the displacement field, one obtainsfor the standard deviation of the surface strain as afunction of the relative phase errors12

be C ad 0'(13)Ax sina 2 13

where X is the wavelength and the constant C has avalue near 1, depending on the different componentsand the choice of the coordinate system. Ax denotesthe base for the numerical differentiation of the dis-placement components and determines thus the spa-tial resolution of strain determination. Figure 4 showsthe different be vs the angle a for an interference phasemeasurement accuracy of 1, i.e., 6b0/27r = 1/360, aspatial resolution Ax = 3 mm, and X = 514 nm. Thiscorresponds very well with the results given in Ref. 11.

3 ar , . A11[

=10

2

0 -

0 300 600 900

Fig. 4. Accuracy of surface strain determination vs angle a betweenillumination and observation directions in the case of three symmet-

rically arranged illumination sources.

The absolute phase (fringe order) is not known apriori from the measurement in the interferogram,therefore the measured interference phase contains anunknown additive constant. The problem can beovercome by referring to an object point without dis-placement (fixation, vibration node), or by interfero-metric observation of the deformation in real time andcounting the fringes during displacement. Whereasthe knowledge of the absolute fringe order is indis-pensable for the determination of displacement, itsinfluence on the surface strain components is verysmall. For the particular geometry described above,one obtains for the error of as a function of theuncertainty AN = AOk/27r of the absolute fringe order'2

A 1 cosa 1 XJiS (+ cosa) 2 1S (14)

where is the distance of the illumination sourcesfrom the object surface. For example, with X = 514 nmand I = 1 m, one gets AEii n AN Am/m. Thus the errorin strain determination due to an absolute phase erroris in general negligible.

V. Sources of Error in Interference Phase Measurement

A. Two-Reference-Beam Holography

Two-reference-beam holography requires specialattention to the multiplicity of the reconstructed im-ages and the influence of misalignment of the holo-gram with respect to the reference beams.5 To avoiddisturbing overlapping of the different reconstruc-tions, the reference sources must be well separated.However, the consequence of large separation of thereferences is high sensitivity to repositioning errors.Small rotations of the hologram plate result in anadditional mutual shift of the two reconstructed im-ages (which causes speckle decorrelation and thus areduced fringe contrast, see Sec. C) and a linear phase


deviation in the interferogram.5"13 The experimentalpractice shows'2 that, in a typical setup with well-separated reference sources and with a well-designedhologram plate holder, it is possible to realign thehologram to achieve a fringe contrast of better than80% (for an /15.6 imaging lens aperture) and a phasegradient in the interferogram of <27r X 10-3 mm-.

The use of an optical arrangement with referencesources close together, which would greatly reduce thesensitivity to misalignment,'3 is not adequate in het-erodyne holographic interferometry, because the re-sulting overlapping of the cross reconstructions intro-duces a statistical interference phase error,14 which issignificantly larger than the phase measurement accu-racy which can be obtained by heterodyning.

B. Phase Errors Due to the Electronics

In heterodyne holographic interferometry, the in-terference phase is measured electronically as the rela-tive phase of two detector signals with a zero-crossingphasemeter. Fluctuations in the detected signals in-troduce a statistical phase error which depends on thesignal-to-noise ratio of the detector signal, the hetero-dyne frequency, and the integration time of the phase-meter.' Using photomultipliers with shot noise-limit-ed detection, a detector area larger than 0.1 mm2 andan argon laser for hologram reconstruction, a SNR of40 dB, and a phase error below 0.1° can easily beachieved. Thus the electronic detector noise does notusually introduce any significant error to the phasemeasurement.

Attention must be paid to the crosstalk in the twoacoustooptic frequency shifters. The rms value of therandom phase error produced by the frequency cross-talk is given by & = fl/j2,12 where fi is the opticalamplitude of the crosstalk component of the frequencyX in the frequency shifted beam at X + Aw. Using twoindependent quartz oscillators (with HF amplifiers) todrive the AOMs, we achieved f 2

= -52 dB and thus 60= 0.1° rms.

C. Speckle Decorrelation

The presence of speckles in the image of objects withdiffusely scattering surfaces gives rise to a statisticalerror for the measured interference phase. The inter-ference of noncorrelated speckles (or noncorrelatedparts of speckles) introduces a random contribution tothe phase of the mutual intensity of the interferingwave fields. Partial decorrelation of the speckle pat-terns may be caused by the transverse component ofthe object displacement.

The influence of speckle decorrelation on the inter-ference phase measurement has been rigorously treat-ed in Ref. 14. The resulting mean-square phase errordue to the speckle noise as a function of the in-planedisplacement component is given by

1 -Ch(2u) (15)(6p)=2N C2(u1) (5

60 is inversely proportional to the number N of speck-les within the detector area. Ch is the autocorrelation

0. 5' 1 "

u -2 p

0.2' 2101 02 103 104

Fig. 5. Statistical phase error hi due to speckle decorrelation vsnumber N of detected speckles for different in-plane displacements

u, (detector area AD = 1 mm2).

of the point spread function h of the imaging system inthe holographic reconstruction. For a circular pupil ofthe lens, Ch is given by the Airy function."l4 Thephase error can be adequately reduced by increasingthe detector surface and thus averaging over manyspeckles, at the expense of spatial resolution, of course.Figure 5 shows, for a given detector area of 1 mm2 , thestatistical phase error At vs the number N of detectedspeckles for different in-plane displacements u1 as pa-rameters. The number of speckles is related to the f/No. of the imaging lens. The figure shows that over awide range the phase error for constant u, does notdepend on the lens aperture. The reason is that, whenthe lens aperture is closed, the reduction of the numberof speckles which increases the phase error [Eq. (15)] iscompensated by the increase of the average specklesize, and thus the fringe contrast, which reduces thephase error. For a given maximum in-plane displace-ment uJ, the optimum choice for the f/No. (minimumphase error and brightest possible image) is deter-mined by the bend of the curves (dashed line in Fig. 5),which corresponds to a reduction of the optimumfringe contrast to -70%. The minimum phase errordue to the speckle decorrelation is given by the simplerelation

lim h = +5A4 UI = UIIRD, (16)

where AD and RD are the area and the radius of the(circular) detector aperture, respectively.

VI. Experimental Results

A. Undeformed Object and Rigid Body Motion

The sensitivity and reliability of experimental straindetermination can best be tested by measuring anundeformed object or an object undergoing only a rigidbody motion. An aluminum plate, which can be rotat-ed about its normal axis, has been investigated by acomplete 3-D evaluation of three holograms with dif-ferent object illuminations, as described in Sec. III.The interference phase has been measured on a grid of15 X 15 sample points, 3 mm apart. After evaluationof the displacement vector field, the three surfacestrain components and the three rotation components


Table 1. Experimentally Determined Mean Values and Standard Deviations of Strain and Rotation Components from a Holographic Interferogram of anUndeformed Object

EXX 1EYY Exy Wx WY Wz

Mean value (gm/m) -0.25 0.23 -0.22 -0.26 0.01 -0.08Standard Deviation (um/in) 0.22 0.3 0.14 0.04 0.04 0.17

Table II. Experimentally Determined Mean Values and Standard Deviations of Strain and Rotation Components from a Holographic Interferogram of aRotated Plate

(xx syy Exy Cox Wy co

Mean value (Um/m) -2.61 -1.43 -0.91 0.03 -0.92 38.0Standard Deviation (m/m) 0.71 1.35 0.71 0.3 0.5 0.68

were calculated by numerical differentiation of thedisplacement field (see Sec. IV).

Table I shows the mean values and the standarddeviations of strain and local rotation for the evaluatedsample points of the undeformed object. The stan-dard deviations (rms values) for surface strain and in-plane rotation are all - 0.2 ,gm/m, the mean values areall zero within the standard deviation error interval.The out-of-plane rotation is significantly differentfrom zero due to a systematic deviation of the u, com-ponent.

Table II shows the mean values and the standarddeviations of strain and rotation evaluated on a platewith a small in-plane rotation of w = 38 X 10-6. Therotation has been chosen so small in order to have smallenough in-plane displacement component (<1.5 ,gm)at the border of the evaluated surface area, which donot yet introduce a detectable statistical phase errordue to speckle decorrelation (see Fig. 5). Theoretical-ly, all the components, except the rotation wu, shouldbe equal to zero. The measured rms errors are largerthan for the undeformed object, but still of the order ofa fraction of one microstrain for a spatial resolution of6 mm. Only the standard deviation for the Eyy compo-nent is somewhat larger. Furthermore, the mean val-ue of eXx is significantly different from zero. From ananalysis of the 2-D phase and displacement fields, itcan be seen that the deviations of the two normal straincomponents are of a systematic nature and not due tostatistical errors in the interference phase measure-ment.

The systematic deviations observed in the aboveexperiments are mainly caused by repositioning errorsof the hologram plate and atmospheric turbulencesduring hologram recording. The statistical interfer-ence phase measurement accuracy has been shown tobe much better, namely, of the order of 1/1000 fringe or0.30 rms. This shows clearly that limitations to straindetermination in heterodyne holographic interferome-try do not arise from limitations in the interferencephase measurement technique, but rather from theinterferometry itself, that is to say from the stability ofthe optical setup, air turbulences, zero-order fringeerrors, and inaccurately known geometrical data of thesetup.

fZFig. 6. Cross section through the cylindrical tube loaded by inter-nal pressure. The displacement u and the strain along the curved

surface will be represented in the local coordinate sytem Qn,).

B. Cylindrical Vessel with Internal Pressure

Another investigated test object was a cylindricalvessel loaded by internal pressure." In this case, thestrain has to be evaluated on a curved surface. Thecylindrical vessel was an aluminum tube with a radiusof R = 10 cm and a wall thickness of b = 5 mm (Fig. 6).In the experiment described hereafter, the internalpressure difference was 2.6 bar.

For a curved object surface, it is adequate to intro-duce a surface oriented local coordinate system (Q,rn,)at each observed object point. For the cylindricalobject, the t axis is chosen tangential to the surface ateach point and perpendicular to the axis of the cylinder(iq axis) (see Fig. 6). The displacement components,which are calculated in the laboratory coordinate sys-tem (x,y,z) are then transformed into tangential (us),axial (u,), and radial (u) components. The surfacestrain components Ee, %E0, and eV, are calculated bynumerical differentiation of the displacement field inthe surface oriented local coordinate system.

Figure 7 shows the interference fringe patterns ofthe three holograms recorded with different illumina-tion directions. The indicated rectangular area corre-sponds to the field of 31 X 5 sample points for which


70

60

Fig. 7. Interference fringe patterns on a cylindrical vessel de-formed by internal pressure for three illumination directions. Therectangular field indicates the area where the interferogram has

been evaluated.

the interferogram has been analyzed. In Fig. 8 themeasured phase functions are plotted along a horizon-tal line through the three interferograms. The abso-lute fringe order for each illumination direction hasbeen determined from three corresponding real-timeholograms of the cylinder deformation.

Figure 9 shows the experimentally determined dis-placement components along a radial section on thecylinder surface. The corresponding strain compo-nents are displayed in Fig. 10. The reliability of themeasurements has been checked by the evaluation ofthe interferogram for the same holographic recordingon two interlaced grids of sample points ( and *,respectively). The standard deviation of the differ-ence between the two measurements is of the order of 1gum/m. These results impressively illustrate the pow-

er of heterodyne holographic interferometry. It ispossible to determine all the components of surfacestrain and rotation on a curved surface with an accura-cy of about one microstrain for a spatial resolution of afew millimeters.

VIl. Conclusions

It is interesting to compare the features and theperformance of heterodyne holographic interferome-try and quasiheterodyne'3 "15 "16 (stepwise phase shift-ing) techniques with video-electronic detection in viewof their appropriate applications. Both heterodyneand quasiheterodyne techniques offer automated dataacquisition for a large number of sample points withhigh spatial resolution. The phase measurement ishighly accurate, independent of variations of the meanintensity and the fringe contrast, and there is no signambiguity.

Heterodyne holographic interferometry offers veryhigh accuracy for the interference phase measurement(better than 1/1000 of a fringe). The principle ofheterodyning requires sophisticated electronic equip-ment, such as AOMs with HF drivers and phasemeters.The mechanical scanning of the image by photodetec-tors makes it slow (-1 s/point). Since the phase mea-surement accuracy can be very high, an arrangementwith reference sources close together, which wouldintroduce additional errors due to the overlapping ofdecorrelated speckles, is not adequate. Well-separat-ed references sources, on the other hand, impose strongrequirements to the stability of the optical setup and to

10

100

90

80-4 -2 0 2 4

Fig. 8. Measured phase functions along a horizontal line throughthe interferograms of Fig. 7.

6

4

2

C

-4 -2 0 2 4

Fig. 9. Experimentally determined radial, axial, and tangentialdisplacement components on the (curved) cylinder surface.

60

40

20

0-10

-4 -2 0 2 4

Fig. 10. Experimentally determined surface strain components eat,c,,, and Et, of the cylindrical vessel evaluated from interferencephases measured on two interlaced grids of sample points (0 and *).

the repositioning of the hologram. For recording andreconstruction, one and the same experimental setuphas to be used.

The described arrangement, using three double-ex-posure hologram recordings with different illumina-tion beams, provides surface strain measurement ofextremely high resolution. For the strain compo-nents, obtained from the local derivatives of the ex-perimentally determined displacement field, an accu-racy of about one microstrain with a spatial resolutionof a few millimeters has been achieved. The determi-nation of the displacement itself requires however, inaddition, knowledge of the absolute fringe order for


¢1 3 / 2 ¢2

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000000000000000000000000000 00u NMuI=u

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each holographic interferogram, because the informa-tion on the mutual phase between the three interfero-grams is not directly available from the separatelyrecorded holograms. The experimental results showclearly that it is not the heterodyne fringe interpola-tion technique that limits the measurement accuracy,but rather air turbulence, hologram repositioning, andspeckle decorrelation due to large in-plane displace-ments.

Quasiheterodyne holographic interferometry withTV detection offers moderate accuracy for fringe in-terpolation (about 1/100 of a fringe), with all the ad-vantages of phase shifting interferometry. An opticalarrangement with two reference sources close togetheris nearly as simple and uncritical for hologram record-ing and reconstruction as standard double-exposureholography. It requires no special instrumentationapart from a microcomputer with a video frame grab-ber, which is implemented in a standard image pro-cessing system. Data acquisition from the interfero-gram is very fast, even for a large number of samplepoints. With the experimental arrangement de-scribed in Ref. 16, good results for 3-D displacementanalysis have been achieved. For the evaluation ofsurface strain, a typical sensitivity of -10 ,gm/m with aspatial resolution of a few millimeters can be expected.

To summarize this comparison: Heterodyne holo-graphic interferometry is well suited for scientific ap-plications in laboratory environments, where the ulti-mate limits for strain determination are required. Onthe other hand, quasiheterodyne holographic interfer-ometry with TV detection is suited for applications inindustrial environments, for deformation and vibra-tion analysis, where moderate accuracy is required.Note however that this means moderate comparedwith heterodyne systems and that the capabilities ofphase shifting techniques for displacement and strainmeasurement are anyway at least 1 order of magnitudebetter than those of conventional holographic interfer-ometry.

References1. R. Dandliker, "Heterodyne Holographic Interferometry," Prog.

Opt. 17, 1 (1980).2. R. J. Pryputniewicz, "Heterodyne Holography Applications in

Studies of Small Components," Opt. Eng. 24, 849 (1985).3. P.V. Farrell, G. S. Springer, and C. M. Vest, "Heterodyne Holo-

graphic Interferometry: Concentration and Temperature Mea-surements in Gas Mixtures," Appl. Opt. 21, 1624 (1982).

4. R. Thalmann and R. Dandliker, "Holographic Contouring UsingElectronic Phase Measurement," Opt. Eng. 24, 930 (1985).

5. R. Dandliker, E. Marom, and F. M. Mottier, "Two-Reference'-Beam Holographic Interferometry," J. Opt. Soc. Am. 66, 23(1976).

6. J. Mastner and V. Masek, "Electronic Instrumentation for Het-erodyne Holographic Interferometry," Rev. Sci. Instrum. 51,926 (1980).

7. P. Hariharan, B. F. Oreb, and N. Brown, "Real-Time Holo-graphic Interferometry: a Microcomputer System for the Mea-surement of Vector Displacements," Appl. Opt. 22, 876 (1983).

8. See, e. g., C. M. Vest, Holographic Interferometry (Wiley, NewYork, 1979), Chap. 2.

9. K. A. Stetson, "Matrix Methods in Hologram Interferometryand Speckle Metrology," Nato Advanced Study Institute onOptical Metrology, July 1984, Viana de Castelo, Portugal(NATO ASI Series, Martinus Nijhoff Publishers, The Hague,1986).

10. B. R. Hunt, "Matrix Formulation of the Reconstruction ofPhase Values from Phase Differences," J. Opt. Soc. Am. 69,393(1979).

11. R. Dandliker and B. Eliasson, "Accuracy of Heterodyne Holo-graphic Strain and Stress Determination," Exp. Mech. 19, 93(1979).

12. R. Thalmann, "Electronic Fringe Interpolation in HolographicInterferometry, Applied to Deformation Measurement of SolidObjects," Ph.D. Thesis, U. Neuchatel, Switzerland (1986).

13. R. Dandliker, R. Thalmann, and J.-F. Willemin, "Fringe Inter-polation by Two-Reference-Beam Holographic Interferometry:Reducing Sensitivity to Hologram Misalignment," Opt. Com-mun. 42, 301 (1982).

14. R. Thalmann and R. Dandliker, "Statistical Properties of Inter-ference Phase Detection in Speckle Fields Applied to Holo-graphic Interferometry," J. Opt. Soc. Am. A 3, 972 (1986).

15. R. Dandliker and R. Thalmann, "Heterodyne and Quasi-Het-erodyne Holographic Interferometry," Opt. Eng. 24,824 (1985).

16. R. Thalmann and R. Dandliker, "Automated Evaluation of 3-DDisplacement and Strain by Quasi-Heterodyne Holographic In-terferometry," Proc. Soc. Photo-Opt. Instrum. Eng. 599, 141(1986).

Meetings continued from page 1934

Pulse Compression), and R. C. McPhedran, U. Sydney (Long-Wave-length Diffraction Optics).

Current office bearers of the Australian Optical Society includeM. Waterworth (President), A. G. Klein (Past President), P. Hari-haran (Vice-President), R. C. McPhedran (Secretary), R. Carter(Treasurer). The next meeting of the Australian Optical Society isscheduled to be held at Sydney 24-29 Jan. 1988. The 26th ofJanuary is also the city's bicentenary, and those attending the con-ference will have a unique opportunity to participate in the celebra-tions.

Jean M. Bennett and W. H. (Beattie) Steel at the FourthAnnual Meeting of the Australian Optical Society, Melbourne.


strain measurement by heterodyne holographic interferometry

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