Two-way gratings for strain analysis using moire and moireinterferometry techniques

Krzysztof Patorski and M. Kujawinska

A specimen grating design with two linear rulings with axes 600 apart is proposed for strain analysis usingmoire and moire interferometry techniques. The grating facilitates derivation of three displacement pat-terns using one analyzer grating or a single pair of illuminating beams. Advantages relating to the gratingtechnology and experimental setup simplification are discussed. The results of experiments are presented.

1. Introduction

The moire fringe method is well known as one of themost powerful techniques in scientific and industrialmetrology. It is one of a group of metrological tech-niques performing measurement by comparison. In aconventional test configuration proper superimposi-tion of two periodic structures is essential in which oneof them represents the object under test and the oth-er-the reference-yields moire fringes giving infor-mation about the object (in the form of relative fringedisplacement). The information extraction from themoire pattern is equivalent to classical interferogramevaluation based on fringe detection.

The moire fringe method finds one of its most im-portant applications in strain analysis.1 2 Two differ-ent approaches have been used for obtaining completestrain information. In the first one a crossline gratingof the same spatial period in two mutually perpendicu-lar directions is attached to the specimen under load.It is analyzed by one linear master grating alignedparallel to each of the specimen component rulings.Because of the problem of accuracy with this approachrelated to precise analyzer grating orientation and theneed to calculate displacement cross derivatives, thesecond approach is preferred. Three separate dis-placement (strain) fields must be generated. 3 4 Subse-quent use of the rosette equations provides full straininformation. This approach is less time-consumingand more accurate.

The authors are with Warsaw Technical University, Institute ofDesign of Precise & Optical Instruments, 8 Chodkiewicza Street,02-525 Warsaw, Poland.

Received 6 June 1985.0003-6935/86/071105-06$02.00/0.©P 1986 Optical Society of America.

Two designs of the specimen grating have been usedto obtain three displacement (strain) fields. The firstis the above-mentioned x-y crossline grating. It isanalyzed by two linear master rulings4 of spatial periodequal to that of the specimen grating and is \/2 timessmaller than the specimen grid period. The first ana-lyzer is aligned successively with the x and y axes; thesecond one is aligned at 450 to the x-y axes. Thesecond alternative design proposed uses an equilateraltriangular specimen grating formed by the superimpo-sition of three linear gratings.3 5 A single linear ana-lyzer grating is aligned with the x, x + 60, and x + 1200axes.

Both designs of the specimen grating mentionedabove have their merits and demerits. In the cross-line-type grid the contrast of moire fringes is betterthan in the triangular (three-way) grating, but twodifferent period analyzer rulings are required. In thetriangular grating only one analyzer is used but thevisibility of moire bands is worse. This is why the useof a crossline grid is preferred.

The problem of optimum design of the specimengrating is also faced in the moire interferometry tech-nique.6 7 It evolved as the result of research on thesensitivity enhancement of the classical moire fringemethod. The reflection-type phase grating cast on thespecimen is symmetrically illuminated by two mutual-ly coherent plane beams. Providing proper incidenceangles the plus first diffraction order of one illuminat-ing beam and the minus first order of the other beamcoincide in space along the specimen normal. As aresult, fringes representing a contour map of in-planedisplacements along the axis perpendicular to thespecimen grating lines are obtained. Since, as men-tioned above, it is necessary to determine more thanone displacement (or strain) field a crossline-typespecimen grating is used. Each linear grating is ana-lyzed separately by a pair of mutually coherent planebeams. This is achieved by rotating the specimen

1 April 1986 / Vol. 25, No. 7 / APPLIED OPTICS 1105

under load6 by 90° or by providing a three-beam illu-mination system.7 The latter method is more versatilesince the specimen can remain in place and becausethree (not two) displacement fields are readily ob-tained by sequentially cutting one of the three illumi-nating beams. The setup with the rotated specimenunder load6 can be supplemented by two additionalilluminating beams to obtain a third displacement pat-tern8 (exploiting the crossline grating functioning as agrating in the +450 directions; in fact, the three-beamillumination interferometer6 is based on the sameprinciple). Recently, two designs of three-way (three-axis) linear gratings for obtaining three displacementfields with a single pair of illuminating beams havebeen proposed.9"10 However, because three exposuresare required for the specimen grating formation, thelight intensity in the observation plane is reduced.

In this paper we present a simple new design of thespecimen grating providing three displacement(strain) fields using a single analyzer grating (classicalmoire fringe technique) or a single pair of illuminatingbeams (moire interferometry technique). The gratingis composed of two identical linear rulings with axes600 apart. Such angular orientation of componentgratings results in the spatial beat formation of thethird linear grating of the same spatial period as that ofthe two rulings physically present. The moire pat-terns formed by subsequent overlapping of the threerulings with an analyzer grating are very bright and ofgood visibility. In addition to simpler technology, theuse of a two-way reflection-type specimen grating inmoire interferometry leads to much better diffractionefficiency compared with that with triple exposuregratings. 9 "10

The principles of the novel specimen grating designare theoretically explained and the results of experi-ments using two-way transmission-type binary grat-ings are given.

11. Principle

The purpose of the novel grating design is to providethe surface of the specimen under test with three ap-proximately oriented linear diffraction gratings of thesame spatial period. This can be achieved by a three-way specimen grating 9"10 or, in a much simpler way, bya specimen grating with two linear rulings having axes60° apart. An enlarged part of the latter configurationis shown in Fig. 1.

Let us consider the spatial frequency spectrum ofsuch a two-way grating. Since its amplitude (or inten-sity) transmittance is equal to the product of the am-plitude (intensity) transmittances of the componentlinear gratings, beat frequencies are formed. Figure 2is a schematic representation of the central part of thespatial frequency plane. The localization of the spec-tral point representing the beat frequency formed byspatial frequencies +1A and - 1

B of component grat-ings A and B, respectively, is given by

R = 2r2 - 2r2 cos6O0 = r, (1)

where r = lid, and d is the spatial period of the compo-

............. ............ .............. .........I. . .. . .. . .. . . .. . .. . .. . ............................ ___..................... ____................................ I...

.. . . . . . . . . . . . . . . . . . .____.................._ _ _ _ __................ ..........I. . . . .. . . . .. . . . .. . . . .. .. .

._................._ _ __.....................Fi.. Enlarementof pat of te proosed wo-waygratigwit

two linear rulings.mutually.rotated.by60°...

\ (A+B)Fig. 2. Schematic representation of the spectrum of the gratingwith two linear rulings with axes 60° apart. A and B designatecomponent gratings, r is the fundamental frequency of the compo-

nent gratings, R is the beat frequency (+1A, -B).

nent rulings. It follows that the resulting beat fre-quency grating is of the same spatial frequency as thatof the component gratings. In this way the two-waygrating with two linear rulings having axes 600 apartcan serve as the three-way grating of the same spatialperiod.

It can be seen from Fig. 2 that there is no danger ofoverlap of the beat and fundamental frequencies whenthe grating becomes deformed (due to the specimenload). A spread of spatial frequencies'0 occurs aroundthe spots shown in Fig. 2. However, no overlap will beencountered because of the large enough separation ofthe spots (much larger than for the three-way gratingdesign9" 0; a detailed discussion of the frequencyspread in moire strain analysis is given in Ref. 10).

Let us compare the diffraction efficiency obtainedfor the two-way grating proposed above and the three-way9" 0 gratings. In both cases component binarygratings of the same operating ratio : (defined as theratio of the transparent linewidth to the grating peri-od) is assumed. The diffraction efficiency is deter-mined by the value of light intensity in the first diffrac-tion order of the structure under investigation (undercoherent light illumination). It also characterizes thevisibility of the moire pattern formed when overlap-

1106 APPLIED OPTICS / Vol. 25, No. 7 / 1 April 1986

ping this structure with the analyzer grating underincoherent or coherent illumination (classical moirefringe technique). 1 1, 2

The intensity values in the spectral point diffractionorders' 3 are as follows:

(a) in the first diffraction order of component rulingsforming the two-way grating

I2(uv) = O2 sinC2(3), (2)

where I2 relates to the two-way grating, and u and v arethe coordinates in the spectral plane;

(b) in the first diffraction order of component rul-ings forming the three-way grating

I 3(U,V) = 33 SinC2 (3), (3)

where 13 relates to the three-way grating; and(c) in the beat frequency spectral point formed by

+1A and -1B frequencies of the two-way grating1 2b(U,V) = 2 sinc4 (0).

(A B)

" BA r

Fig. 3. Influence of the error in angular orientation of componentgratings in the two-way grating on the beat-frequency R formed by

component frequencies +1A and - 1 B-

(4)

In the most frequent case of the grating opening ratio a= 0.5, we obtain, by comparison,

12(u,v) I2b(U,V)= 2; _ 0.8

13(u,v) ' 3(U,v)(5)

It follows that use of the two-way grating leads tobetter diffraction efficiency (or visibility) of all threecomponent gratings being used (two of them are pho-tographically exposed, the third one is the spatial beatproduct) compared with that of the three-way grating(three gratings photographically exposed).

The diffraction efficiency superiority of the pro-posed grating design proved for binary transmission-type gratings (classical moire fringe technique) is alsoshown for the reflection-type phase gratings in themoire interferometry technique.

Now the problem of the accuracy of the 600 separa-tion angle between the two linear rulings should bediscussed. The angular misalignment 60 + Aa inproducing the grating results in a different period ofthe resultant beat frequency grating. The explana-tion of this effect is shown in Fig. 3. If the angularseparation of the gratings is 60° + Aa, the direction ofthe vector R changes by + Aca/2 and is accompanied bya change in value, i.e., R' = R ± AR = r AR. Thechange of length of the beat frequency vector (or of thelocalization of the beat frequency spot) corresponds tothe change of beat grating period. Therefore, whenusing the same analyzer grating for overlapping thethree component rulings a different period and orien-tation of moire fringes is obtained when studying thebeat-formed grating (as compared to the two-compo-nent rulings). This should be avoided by providingsufficiently high accuracy of the angular orientation oftwo linear rulings. The period of mismatching moirefringes resulting from the angular misalignment Aashould be larger than the lateral dimensions of thespecimen grating.

Since the required accuracy of the separation angle600 is higher the finer is the spatial period of thecomponent specimen rulings, the problem is more pro-

(41BL-1AL) L+AL-lk)s (+1AR 1BR)0 0 0

(1AR +BL)

iBL AP 1BR

AL BL AR BR

Fig. 4. Spectrum of the 60° two-way grating under double-beamillumination (moire interferometry method). The axial beat fre-quencies +lAL,-BL and- 1 AR, +1BL are filtered. R andL designate

the two conjugate beams illuminating the specimen grating.

nounced in the moire interferometry method. Spatialfrequency of the component gratings is 500 lines/mmand higher.6 7 For example, when using the 500 lines/mm component gratings the period of mismatchingmoire fringes equals 1.6 mm for Aa = 5 min and 15.9mm for Aa= 30 sec. Since for the grating productionprocess the photographic plate can be placed on theprecise optical rotation table (providing 5-sec of arc orbetter rotation setting accuracy), the identical periodproblem for three component rulings can be readilysolved.

Figure 4 is a schematic representation of the centralpart of the spatial frequency plane with the moireinterferometry technique. When the angular separa-tion between the two illuminating beams L and R(Refs. 6-10) is approximately tuned to the componentgratings frequency, the spatial coincidence of the beatdiffraction orders (+1A, -1B)L and (-lA, +lB)R is en-countered. Otherwise, for angular misalignment ofthe component rulings the beat-frequency orders(+1A, -1B)L and (-1A, +lB)R become spatially separat-ed and mismatching interference fringes are observedat the output of the interferometer.

1 April 1986 / Vol. 25, No. 7 / APPLIED OPTICS 1107

..................... ~ ~ ,* *

.............~ ~ ~~~~* G................ ~~.., ,, ,

Fig. 5. Enlarged central part of the two-way synthetic interfero-gram grating.

a o

Fig. 7. Moire fringes resulting from overlapping the analyzer grat-ing with (a) one of the two-component synthetic interferograms and(b) the beat-frequency formed component grating. In both casesthe analyzer lines are set parallel to the lines of the specimen grating

under study.

posing analyzer and specimen gratings we obtain theproduct TA(xy)TS(xy). The moire fringes are de-scribed by the lowest spatial frequency term, i.e.,

Im(xy) 4 + 2 co{27r - X + A(X2 + y2)] - (8)

In the special case of dA = ds we see that points of thesame intensity value in the moire pattern satisfy theequality

Fig. 6. Spectra of the two-way gratings used in the experimentscomposed of (a) linear component gratings and (b) synthetic inter-

ferograms.

Ill. Experimental Work

Experiments have been conducted to verify the pro-posed two-way specimen grating principle. Two typesof grating have been used: a two-way linear binaryamplitude grating, see Fig. 1, and a two-way binarysynthetic interferogram with coded spherical wave-fronts,' 0 see Fig. 5. The Fourier spectra of these twogratings representing model undeformed and de-formed specimen gratings are shown in Fig. 6.

A. Moire Fringe Technique

As mentioned above by the classical moire fringetechnique we can understand the overlapping processof the specimen and analyzer gratings under incoher-ent or coherent light illumination. For our experi-ments we have used a linear Ronchi type / = 0.5 binaryamplitude grating of 10-lines/mm spatial frequency.

For a quantitative description of the experiment weassume cosinusoidal amplitude transmittance of bothspecimen and analyzer gratings. This is done for thesake of simplicity of the description only; this assump-tion does not restrict the generality of analysis.Therefore, the analyzer grating can be represented as

TA(x,y) = 2[1 + cos(27rx/dA)I, (6)

where dA designates the period of the analyzer grating.Each component synthetic interferogram of the two-way specimen grating can be described by the form14

TS(xy) = 2-1 + co 27rd+ A(X2 + y2) (7)

where A designates the coefficients of a spherical wavefront encoded in the interferogram. When superim-

A(x 2 + y2 ) = D, (9)

where D is a constant. Therefore, the moire fringesare in the form of circular bands centered at the originof the coordinate system.

When dA # ds (but when a small difference betweendA and ds is required for moire fringe formation), themoire fringes satisfy the equality

Bx + A(x2 + y 2) = D, (10)

where B = 27r[(1/d) - (1/d')]. After simple rearrange-ment, Eq. (10) reads

(x + E)2 +y 2= (DIA) -E 2, (11)

where E = B/2A. Equation (11) describes circularmoire fringes with the center shifted along the x direc-tion by -E.

In a similar manner, we can easily prove that in thecase of mutual rotation of the specimen and analyzergratings of equal spatial period the circular moirefringes shift along the y direction.

Figure 7(a) shows the moire fringes obtained for aparallel setting of the lines of the analyzer and one ofthe component specimen gratings when 1/dA = 10 lines/mm and 1l/ds = 11 lines/mm. Figure 7(b) shows moirefringes generated when the analyzer grating is set par-allel to lines of the beat-frequency component in thespecimen grating. As mentioned before, the funda-mental frequency of the beat grating results from theinteraction between the diffraction orders +1A and-1B. Since these diffraction orders carry the samespherical wave front distortion but of opposite sign,the resulting spatial beat frequency has uniform phasedistribution. Because of the frequency mismatch be-tween the beat specimen grating and the analyzer grat-ing, equidistant straight-line fringes are obtained. InSec. II we mentioned that the exact moire fringe periodand its orientation is influenced by the accuracy of

1108 APPLIED OPTICS / Vol. 25, No. 7 / 1 April 1986

adjusting the 600 angle when producing the two-wayspecimen grating. The number of fringes in Fig. 7(b)agrees well with the predicted number under the esti-mated sensitivity of setting and angular separationbetween the component rulings and the sensitivity ofangular orientation between the analyzer and the beat-frequency formed component specimen grating.

B. Moire Interferometry Technique

The experimental setup is shown in Fig. 8. A two-way synthetic interferogram (serving as a model speci-men grating) with coded spherical wave fronts is sym-metrically illuminated by two plane beams. They areformed using two first diffraction orders of a binarymaster grating of spatial frequency equal to the carrierfrequency of a synthetic interferogram.

Figure 9 shows the Fourier spectrum plane of a two-way synthetic grating under double-beam illumina-tion. The plane of incidence of the illuminatingbeams was set perpendicular to the lines of the beat-frequency formed component grating. Therefore, theintensity distribution shown in Fig. 9 corresponds tothe spectral distribution schematically presented inFig. 4 (with the spatial frequency spreads due to grat-ing line deformations).

The interferograms corresponding to the case of an-alyzing one of the two component synthetic interfero-grams (i.e., the incidence plane of the illuminatingbeams is perpendicular to the component interfero-gram fringes) are shown in Fig. 10. The null fieldfringe detection mode corresponds to Fig. 10(a) (thetwo interfering diffraction orders propagate colinear-ly). Since in this case we have interference of twoconjugate first diffraction orders of the componentgrating, the number of fringes is doubled6-'0 comparedto the case shown in Fig. 7(a). The latter correspondsto the interferogram formed by one of the first-orderdiffracted beams and a plane reference wave front.Again centered circular fringes are encountered.When the angle between the illuminating beams is notequal to the double value of the first-order diffractionangle of the specimen grating, the finite fringe detec-tion mode results. A tilt between the interfering dif-fraction orders is introduced, as shown in Fig. 10(b).As explained by Eq. (11) [although Eqs. (8)-(11) havebeen derived for the classical moire fringe techniquethey can be easily transformed to describe the moireinterferometry method; this is a very straightforwardoperation and it will not be presented here], the intro-duction of a carrier pattern along the x direction re-sults in a lateral shift of circular fringes in this direc-tion, see Fig. 10(b). On the other hand, the rotation ofthe model specimen grating (synthetic interferogram)in its plane leads to a lateral shift of circular fringesalong the y direction.

Figure 11 shows the interference pattern obtainedby setting the lines of the beat-frequency formed com-ponent grating perpendicular to the incidence plane oftwo illuminating beams. Similarly in view of the ex-planation given when discussing the corresponding

G LI SF1 L2 IS L3 SF2 L4 I

-I-'a 1- I-XL * F F F' F F FI F

Fig. 8. Experimental setup for optical simulation of the moireinterferometry techniques using a two-way grating: G linear grat-ing; SF1 and SF2, spatial filters; L1, L2, L3, and L4, lenses of the

coherent processor; IS, specimen grating; I, image plane.

Fig. 9. Spatial frequency of the two-way synthetic interferogram(model specimen grating) under double-beam illumination. Theintensity distribution corresponds to the schematic presentation of

Fig. 4 with the specimen line deformations introduced.

Fig. 10. Results of the analysis of one of the component syntheticinterferograms by the moire interferometry method: (a) collinearpropagation of interfering diffraction orders, (b) tilt angle between

the interfering orders in the incidence plane.

Fig. 11. Interferogram obtained in the finite fringe detection modewhen analyzing the beat-frequency formed grating by the moireinterferometry method. The straight-line fringes are a result of the

interference of the two wave front diffraction orders.

1 April 1986 / Vol. 25, No. 7 / APPLIED OPTICS 1109

moire pattern obtained in the classical moire fringetechnique, see Fig. 7(b), we should expect a uniforminterference field when the two interfering diffractionorders coincide in space. Otherwise, straight-linefringes are encountered when a tilt between the ordersis introduced. The latter is shown in Fig. 11. The twointerfering diffraction orders (+1AL, -1BL) and (-AR,+lBR), see Figs. 4 and 9, are plane wave front beamsresulting from the beat phenomenon between the con-jugate diffraction orders +1AL and -1 BL or - 1 BL and+1BL of two identical component synthetic interfero-grams. In the moire interferometry method, like theclassical moire fringe technique, reference fringes canbe introduced either by period or angular mismatchingbetween the reference grating (formed by two illumi-nating wave fronts) and the specimen grating understudy.

It follows from the experimental results presented inFigs. 9-11 that the principle of a two-way linear gratingwith axes 600 apart functioning as a three-way gratinghas been fully verified.

IV. Conclusions

A simple two-way specimen grating design has beenproposed for strain analysis using the well-known moi-re method or the more recent moire interferometrytechnique. The grating consists of two linear gratingswith lines mutually tilted by 600. These two compo-nent rulings form, due to the spatial beat effect be-tween their frequencies, a third specimen grating ofthe same period as that of the two components. Thelines of all three gratings form an equilateral triangle.In this way, only two component gratings are requiredto obtain a three-way specimen ruling. The latterprovides three displacement (or strain) patterns bysuperimposing a single linear analyzer grating overeach component linear specimen ruling. Subsequentuse of the rosette equations provides full strain infor-mation.

The grating design proposed has several advantages.Only one master grating is required compared to twoanalyzers when using the cross line-type specimengrating. Much better brightness and visibility of moi-re fringes is encountered than when employing anequilateral triangular specimen grating formed bythree superimposed linear gratings. The two advan-tages just mentioned met in the classical moire fringemethod (contact or image superimposition of ampli-tude-type specimen and analyzer gratings) have theircounterparts in the moire interferometry technique.In a configuration using a rotatable specimen underload, the need for two separate pairs of illuminatingbeams encountered in using the cross line-type speci-men grating is obviated. On the other hand, the lightintensity in the interferogram plane is considerablyhigher when using our grating design compared to athree-exposure type specimen grating (three linearrulings superimposed), because of the higher diffrac-tion efficiency of all three component rulings of thetwo-way grating proposed.

All theoretical predictions together with a discus-sion of the influence of the accuracy of setting the angle600 between the axes of the two linear gratings havebeen experimentally verified. The synthetic interfer-ogram (fringe-type computer-generated hologram)with an encoded spherical wave front) has been used asa deformed model grating. It provided a very illustra-tive interpretation of the beat-frequency formationand the characteristics of the third-component speci-men grating.

It is hoped that because of its advantages the two-way grating proposed will interest experimentalistsdealing with moire strain analysis. Moreover, it isbelieved that the moire grid-analyzer technique pro-posed by Post,' 5 using the crossline-type specimen andanalyzer gratings, can also be extended to the noveltwo-way grating design.

The authors wish to thank P. Szwaykowski for help-ful suggestions and discussions.

References1. P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon,

Oxford, 1969).2. A. J. Durelli and V. J. Parks, Moire Analysis of Strain (Prentice-

Hall, Englewood Cliffs, NJ, 1970).3. M. Rogozinski, "An Attempt to Establish the Theoretical Foun-

dations of the Moire Method of Strain and Stress Analysis,"Arch. Mech. Stosow. 9, 191 (1957).

4. P. Dantu, Exp. Mech. 4, 64 (1964).5. R. Fidler, I. Law, and P. Nurse, Strain 6,111 (1970).6. D. Post, "Developments in Moire Interferometry," Opt. Eng. 21,

458 (1982), and references therein.7. A. McDonach, J. McKelvie, P. MacKenzie, and C. A. Walker,

"Improved Moire Interferometry and Applications in FractureMechanics, Residual Stress and Damaged Composites," Exp.Tech. 7, 20 (1983), and references therein.

8. E. M. Weissman and D. Post, "Full-Field Displacement andStrain Rosettes by Moire Interferometry," Exp. Mech. 22, 342(1982).

9. E. M. Weissman, D. Post, and A. Asundi, "Whole-Field StrainDetermination by Moire Shearing Interferometry," J. StrainAnal. 19, 77 (1984).

10. K. Patorski and M. Kujawinska, "Three-way Gratings for MoireInterferometry," Opt. Commun. 53, 285 and 428 (1985).

11. 0. Bryngdahl, "Moire and Higher Grating Harmonics," J. Opt.Soc. Am. 65, 685 (1975).

12. K. Patorski, S. Yokozeki, and T. Suzuki, "Moire Profile Predic-tion by Using Fourier Series Formalism," Jpn. J. Appl. Phys. 15,443 (1976).

13. M. Kujawinska, "The Analysis of the Product- and Sum-TypeMulti-Exposure Synthetic Holograms," Opt. Commun. 44, 85(1982).

14. M. Kujawinska, "Coding of Many Phase Objects Using Comput-er-Generated Binary Holograms," Opt. Acta 28, 843 (1981).

15. D. Post, "The Moire Grid-analyzer Method for Strain Analysis,"Exp. Mech. 5, 368 (1965).

1110 APPLIED OPTICS / Vol. 25, No. 7 / 1 April 1986