Talbot interferometry with increased shear: further considerations

Talbot interferometry with increased shear: furtherconsiderations

Krzysztof Patorski

Further modification is proposed of the Talbot interferometer configuration for obtaining moire fringeshearing interferograms with increased shear. The configuration is described with the beam splitter gratingrotated about the axis parallel to its lines. The second grating detects only one two-beam interference patternat a particular angular setting of the beam splitter grating from its multiple beam interference field. Thissignificantly reduces the low quality of the Talbot interferograms at large shear values.

1. Introduction

The so-called Talbot interferometer' 3 utilizes theproperties of the Fresnel diffraction field of periodicobjects. When the phase object under study is placedin front of or just behind an amplitude-type lineardiffraction grating (the type of grating most frequentlyused for this purpose), the lines of its diffraction im-ages depart from straightness by an amount propor-tional to the first derivative of the object phase distri-bution. The derivative information, in a directionperpendicular to the grating lines, is obtained whenthe shear amount, i.e., the lateral separation betweenthe diffraction orders in the observation plane, issmall. A second identical diffraction grating is placedin the observation plane coinciding with one of the self-image planes of the self-imaging (beam splitter) grat-ing. As a result, the shear-type interferogram is dis-played in the form of moire fringes. Depending onwhether the lines of both gratings are mutually parallelor slightly inclined, infinite and finite fringe detectionmodes are achieved.

Since the Talbot interferometer consists of only twodiffraction gratings, it is easy to align and to work with.It was originally proposed for an investigation of phaseobjects and optical elements.1 -3 Recently, the Talbotinterferometry principle was extended to optical dif-ferentiation of quasi-periodic patterns, i.e., distorted

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

Received 5 October 1985.0003-6935/86/071111-06$02.00/0.C 1986 Optical Society of America.

specimen gratings used in the moire method of strainanalysis. 4

From the experimental point of view it is useful to beable to change continuously the shear amount fromzero to any desired value. In a Talbot interferometerthe shear amount is proportional to the product of thegratings' separation distance and the first-order dif-fraction angle. Therefore, it can be easily varied bychanging the gratings' separation gap. However, onlyrelatively small shears can be allowed due to w3rseningof the contrast of self-images of the beam splitter grat-ing (and, consequently, of the moire fringes) as theshear increases,5 because of the multiple beam inter-ference of laterally separated diffraction orders of thefirst grating. For example, even at the recommendedshear amount equal to 0.1 of the diameter of the exitpupil of the lens under test6 7 useful moire fringescannot be obtained in the Talbot interferometer.

This drawback of the Talbot interferometer hasbeen treated in a recent publication.5 It was shownthat by tilting the beam splitter grating (or alternative-ly, the detecting grating) about the axis perpendicularto the grating lines the useful moire fringes corre-sponding to a lateral shear interferogram can be ob-tained. In the same interferometer setup with an un-tilted grating no clear moire fringes could be discerned.In the configuration proposed two identical gratingswere used. The above-mentioned rotation of one ofthe gratings about an axis perpendicular to its linesresults in the angular separation of two two-beam in-terference patterns formed by the dominating diffrac-tion order pairs (0,+1) and (0,-1). The appropriatetilt of the other grating about the optical axis helps todetect, independently, one of two patterns using auniform field detection mode, whereas the moirefringes due to the other two-beam field have differentspacing and orientation. This is why the two-product-type moire fringe patterns corresponding to the in-

1 April 1986 / Vol. 25, No. 7 / APPLIED OPTICS 1111

creased lateral shear interferograms can be easily dis-cerned from each other and analyzed. In the usualTalbot interferometer setup with the beam splitterand detecting grating lying in mutually parallel planes,the two moire fringe patterns interact. This results ina considerable deterioration of the final interferogram.

In this paper, for the sake of completeness of theproblem analysis, the further modification of the Tal-bot interferometer configuration for obtaining goodinterferogram quality at large shear values is present-ed. It is shown that, by rotating the beam splittergrating about the axis parallel to its lines, useful moirefringes representing the interferogram with increasedshear value can be obtained. The rotation of the self-imaging grating causes a change in the spatial periodsof the two-beam interference patterns formed by thediffraction order pairs (0,+1) and (0,-1) (although thefringes are mutually parallel). Therefore, using thedetecting grating of the spatial period equal to thefringe spacing of one of the two-beam interferencepatterns (or in practice, adjusting the fringe period tothe period of the detecting grating), the moire fringesdue to the interaction of these two structures can beeasily discerned from the other moire pattern of higherfringe density. A slight rotation of the beam splittergrating about the same axis (parallel to grating lines)permits one to observe, independently, the moirefringes formed by the detecting grating and by theother two-beam pattern. By introducing a slight tiltof one of the gratings about the optical axis, the uni-form field or finite fringe detection modes can be cho-sen. The theoretical analysis is given, the merits anddemerits are discussed and illustrated by the experi-ments.

II. Analysis

Figure 1 shows a schematic representation of themodified Talbot interferometer under discussion.Let us assume that lens L is the lens under test. Sincethe point source S is assumed to be at the front focalplane of L, a quasi-plane aberrated wave front illumi-nates beam splitter grating GI. The grating is rotatedabout the axis parallel to the grating lines. A detectinggrating G2 lies at a finite distance from GI in the planeperpendicular to the optical axis (illumination direc-tion). The lines G1 and G2 are initially assumed to beparallel.

For the sake of simplicity and clearness of the analy-sis, let us assume the cosinusoidal amplitude transmit-tance of diffraction gratings G1 and G2. As men-tioned in previous papers,1 45 this assumption does notrestrict the generality of the approach; the results ob-tained are directly applicable to binary gratings, forexample. The amplitude transmittance of untiltedgrating GI is described by

T(x,y) = AO + 2A1 cos(2irx/d), (1)

where d designates the grating period, x is the directionperpendicular to the grating lines, and AO and Al de-scribe the amplitude modulation parameters. Whenthe plane of G1 is normal to the illuminating beam the

GI X

s

G2

Fig. 1. Schematic representation of the modified Talbot interfer-ometer under investigation: S, point source; L, lens under test; G1and G2, linear amplitude-type diffraction gratings axially separatedby a distance z0 . In the usual configuration G1 lies in the planeperpendicular to the optical axis z; in the configuration proposed it is

rotated about the y axis.

three diffraction orders 0, +1, and -1 propagate in onecommon plane and are equally angularly separatedfrom one another. However, when the grating is rotat-ed about the axis parallel to the grating lines the angu-lar distance between adjacent orders is no longer thesame. This follows the well-known exact diffractiongrating equation8:

sina'- sina = n/d, (2)

where a'n designates the nth order diffraction angleand a is the incidence angle with respect to the gratingnormal, respectively. For further analysis it is conve-nient to introduce the notation

P = a n - a, (3)

where p designates the diffraction angle with respectto the illumination direction (coinciding with the zero-order diffraction direction). The fact of nonequidis-tant angular separation between diffraction orders fol-lowing from Eqs. (2) and (3) has been derived in anapproximate manner elsewhere.9"10 It is expressed bythe approximation

/ ~2nX \p cota 1 1 d cs tana) (4)d cosa

The nonequidistant angular separation between dif-fraction orders has a certain influence on the proper-ties of the Fresnel diffraction field of obliquely illumi-nated diffraction gratings.11""2

Let us investigate the complex amplitude distribu-tion E(x,y,z) in the observation plane just in front ofthe detecting grating G2. The interference pattern isformed by three diffraction orders of the rotated beamsplitter grating G1. Grating G1 is illuminated by anaberrated wave front described by exp [ikg(x,y)J, whereg(x,y) is the OPD departure from the plane wave frontand k = 27r/X. Assuming, as is usual in the Talbotinterferometry theoretical model,13 5 slow changes ofg(x,y) with the propagation distance z and using theapproach of an angular spectrum of plane waves,1-5 wehave

1112 APPLIED OPTICS / Vol. 25, No. 7 / 1 April 1986

E(x,y,z) = AO exp[ikg(xy)]

+ A1 explik[p+lx + g(x - A+1 ,y) -ZP+2/2]

+ A1 exp-ik[px - g(x + A-1,y) + zp.. 2/2]}, (5)

where p+1 and p-1 designate the propagation angles ofdiffraction orders +1 and -1 in the x-z plane, respec-tively, and A+, and A-, describe the lateral shifts of I1diffraction orders with respect to the zero-order beamin the x direction, respectively. We have A+,,-, = ztanp+,,-,. It is seen from Fig. 1 that z = zo - x tanaimplies that the parameters A+, and A-, are not con-stant in the observation field, they vary along the xdirection. The importance of this fact will be dis-cussed in the following.

The intensity distribution in the observation planeis calculated from Eq. (5) in the form

I(x,y,z) = AO + 2A12 + 2AoA1 cosk[p+ix- ax 2 P+1]

2A0 A1 - og(x,y) z 2+ 2AOA1 cosk[P 1 x - 1 ax 2 P 1

+ 2A12 coskk(p+i + p-1 )x -(A+ 1 + A1 ) Ox

-2 (Po1 - ) . (6)

Equation (6) has been derived approximating theslowly varying function g(x,y) by its Taylor series ex-pansion up to the first term. This is usually done inTalbot interferometry analyses.

It is seen from Eq. (6) that the two dominating two-beam interference patterns formed by the diffractionorder pairs (0,+1) and (0,-1) have slightly differentspatial periods X/p+1 and X/p-1, respectively. As men-tioned before, this fact results from the non-normalillumination of beam splitter grating G1. In conse-quence, the spatial frequency beat phenomenon be-tween these two patterns occurs and a complicatedintensity pattern is encountered. It is expressed bythe following form of Eq. (6):

I(x,y,z) =AO2 + 2A 12 + 4AoA cosk +121

A+ 1 - A+1 g(xy) 2]2 Ox 2 2

X cosk P+1 + P-1 -A+, + A-, Og(x,y)L 2 2 Ax

+ 2 p+2)] 2A cosk[(p+l + p-1 )x

- (A+1 + A-1 ) Ox - (P+ 12

- P-1 2)] (7)

Some characteristics of the Fresnel diffraction fieldfollowing from Eq. (7) have been discussed in previouspapers.1"12 The fundamental term of the intensitydistribution of spatial period 2X/(p+l + p-1), carryingthe information proportional to [(A+, +

A_1)/2]9g(x,y)/ax, is strongly amplitude (contrast)modulated by the first cosine term. As a result, noreadable moire fringes can be observed just behind thedetecting grating G2 of the same period as G1.

To obtain a useful moire fringe pattern we proposeselecting only one of the two-beam interference pat-terns, see Eq. (6), using the detecting grating G2 of anappropriate spatial period. In other words, we rotatebeam splitter grating G1 by the angle a (depending onthe spatial frequency of this grating) to differentiatethe periods X/p+1 and X/p_1 of the two two-beam inter-ference patterns. Then the detecting grating G2 ofspatial period equal to, for example, X/p+1 is used.Writing the amplitude transmission function of G2 inthe form A'0 + 2A', coskp+lx, we obtain the moirefringe pattern expressed by the terms with the lowestspatial frequency, i.e.,

IM(x,y,z) = DC + 4AoA1A'1 cosk[A+l Ogx +2P+] (8)

where DC designates the bias term. It is seen that themoire-fringe intensity distribution is proportional tothe first derivative of the phase object g(x,y). Againapproximation of g(x,y) by Taylor expansion up to thefirst term has been used. For larger shear amountsA+,, the second term of the series expansion should betaken into account. Equation (8) corresponds to theuniform field detection mode. It follows that thechange of the axial separation distance z between G1and G2 does not influence the contrast of the shearinginterferogram fringes; only a phase shift is encoun-tered. On the other hand, the contrast deterioration ispertinent to the classical Talbot interferometry config-uration (multiple, at least three, beam interference) atincreased shear values. The improvement attained inthe proposed setup is an obvious consequence of de-tecting a two-beam interference pattern instead of athree-beam one.

When using detecting grating 2 of spatial periodX/p+,, in addition to the moire pattern described byEq. (8), other fringes due to the interaction between G2and the second two-beam interference pattern 2A0A1cosk[plx - A_1ag(x,y)/Ox + zp_12/2] of Eq. (6) are alsogenerated. The intensity distribution of the set offringes with lowest spatial frequency is now propor-tional to

IM'(x,y,z) = DC + 4AOA1A1 cosk[(P+ - P-dx

+ A_ Og(xy) Z p1 2Ox 2 (9)

It is seen that the period of moire fringes expressed byEq. (9) is equal to X/(p+ - p-1). These carrier fringesrun perpendicular to shear direction x and are de-formed proportionally to Aldg(x,y)/6x. Due to aproper choice of rotation angle a of beam splitter grat-ing G1, the period value X/(p+ - p-1) can be set smallenough. As a consequence, the fringes of Eq. (9) con-stitute an undisturbing background to the fringes ex-


pressed by Eq. (8). Therefore, the latter can be easilydiscerned by the observer.

As explained above, the period of detecting gratingG2 should match the period of the two-beam interfer-ence fringes formed by the diffraction orders 0 and +1of the beam splitter grating at the particular choice ofits rotation angle a. In practice, we proceed in theopposite way. Having the diffraction gratings ofknown spatial frequencies we can adjust by rotatingGi, the period of interference fringes formed by thepair (0,+1) of diffracted waves to the period of thedetecting grating G2. This is a much more usefulapproach and is suggested for experimental practice.

At this point the additional property of the proposedconfiguration should be mentioned. For example, forthe rotation direction of G1 shown in Fig. 1, the angularseparation between diffraction orders increases for thepositive (upward-bent) orders and decreases for thenegative orders. It means thatp+, > p- [according tothe notation introduced in Eq. (4) the signs of p+1 andP-1 are the same]. Therefore, when we rotate G1 toobtain the equality X/p+1 = d2 (d2 designates the spa-tial period of G2) the infinite- and finite-period moirefringes are simultaneously encountered as explainedby Eqs. (8) and (9). However, if we slightly increasethe rotation angle a of G1, both values p+1 and p-1 willincrease and we can reach the condition X/p_1 = d2.Now the interaction of the interference fringes formedby the pair (0,+1) of diffracted beams and detectinggrating G2 results in finite period moire fringes, where-as the interaction of the pair (0,-1) of diffraction or-ders with the same G2 gives uniform field moirefringes. Therefore, the situation is reversed with re-spect to that described by Eqs. (8) and (9). We con-clude that in the proposed configuration the same twoshearing moiregrams can be generated by a slight an-gular adjustment of the rotation angle a of G1. Cer-tainly there is the intermediate angular position of GIin which the identical period in both two-beam inter-ference patterns (0,+1) and (0,-1) is encountered.This case, however, is of no practical use since the twointerference fringe sets mutually interact. This re-sults in the essential quality decrease of the observedpattern.

All analytical and heuristic explanations givenabove assumed a mutual parallel setting of the lines ofgratings G1 and G2 (lines of both gratings lie in planesparallel to the y-z plane). Therefore, uniform field orfinite period carrier moire fringes parallel to gratinglines could be obtained. However, if we introduce anadditional slight tilt of, for example, the detectinggrating G2 about the optical axis z, the other directionof reference (carrier) fringes can be achieved. Forexample, when observing an infinite fringe moire pat-tern and then slightly tilting grating G2, moire fringesparallel with the shear direction are generated.

In our configuration, contrary to previously de-scribed systems,' 7 shear amount A is not uniform overthe observation plane because of the rotation of beamsplitter grating G1. This is a disadvantage but theprofits gained from the rotation of G1 are more signifi-

cant. The linear change of the shear value along the xdirection can be taken into account when quantitative-ly analyzing the moire interferogram. Alternatively,when using the configuration proposed for qualitativetesting and providing z0 is at least a few times largerthan D tana (D is the diameter of the illuminatingbeam), the influence of the nonuniform shear amountA on the coordinate x can be neglected (see Sec. III).

Ill. Experimental Work

In the theoretical analysis presented above we haveshown that by tilting the beam splitter grating in theTalbot interferometer about the axis parallel to thegrating lines and using the detecting grating of anappropriately chosen spatial period, the range of shearvalues can be significantly increased. For the experi-mental verification the setup shown schematically inFig. 1 was used. Ronchi-type binary amplitude grat-ings of 25- and 40-lines/mm spatial frequencies wereused, respectively, as beam splitter grating G1 anddetecting grating G2. For demonstration purposes, acollimator objective of 45-mm diameter and 200-mmfocal length, set opposite its correction direction (theobjective rotated by 180° about the vertical axis), wasused as the lens under test L. In the classical Talbotinterferometer configuration- 5 with beam splittergrating G1 perpendicular to the illuminating beam, therecommended lateral shear value of 0.1 of the testedbeam diameter6 7 requires a gratings' separation dis-tance z0 = 285 mm. However, as has been generallyshown in an earlier paper, 5 no useful moire fringeshearing interferogram can be observed in the plane ofthe second grating G2 at such a shear value (see Fig. 2in Ref. 5). This fact was noted again in our prelimi-nary experiments using a pair of gratings of 25-lines/mm spatial frequency separated by a distance of285 mm.

According to the theoretical analysis in Sec. II, beamsplitter grating G1 was rotated by the angle a about theaxis parallel to the grating lines. It is easy to show thatto match the period of two-beam interference fringesformed by the diffraction orders 0 and +1 of G1 withthe period of the detecting grating G2 of 40-lines/mmfrequency, it is necessary to set the angle a equal to50.6°. In this case we have p+1 = 0.0253147 and thefringe period is equal to 25 Am for X = 0.6328 Am.Therefore, when G1 is illuminated by a plane wavefront beam, i.e., g(x,y) = 0, infinite-period moirefringes are encountered in the detecting grating planeG2, see Eq. (8). This was verified experimentally. Atthe same time we have p-1 = 0.0245574. The two-beam interference fringes formed by diffraction orders0 and -1 generate the moire pattern in plane G2, seeEq. (9), of spatial period X/(p+l - p-1) = 0.84 mm.This value has also been verified experimentally.

A similar intensity distribution of the moire consist-ing of infinite- and finite-period fringes is encounteredfor the rotation angle a = 52.050. Then the period ofthe two-beam interference fringes formed by the or-ders 0 and -1 of Gi is equal to the period of thedetecting grating G2. Moire fringes of 0.75-mm peri-


b

Fig. 2. Intensity distributions observed in the plane of the detect-ing grating G2 when G1 is rotated to detect its two-beam interferencepatterns formed by the diffraction orders: (a) 0 and +1, (b) 0 and-1. Some carrier fringes parallel to the lines of both gratings are

introduced.

od formed by the interaction of G2 with the patterngenerated by orders 0 and +1 of G1 are encountered, aswell as a uniform field intensity distribution. Theseanalytical figures have been verified by the experi-ment.

Figure 2 shows the intensity distributions in theplane of G2 when detecting the two-beam interferencepatterns generated by the diffraction order pairs(0,+1) and (0,-i) of GI by the grid G2, respectively.Now, when the angle between the interfering orders isequal to 0.0253147 rad, the shear value of 0.1 of thetested 45-mm beam diam is obtained at a distance of z0= 177.7 mm. The photographs in Fig. 2 were taken atthis axial separation. However, to emphasize the ef-fectiveness of the technique, an even larger shear val-ue, i.e., 0.15 of the tested beam diameter, is presentedin Fig. 2. This was achieved by reducing the beamdiameter. In both cases the carrier fringes parallel tothe grating lines were introduced for better represen-tation of the fringe pattern, i.e., the angular setting ofGI was slightly different from the values of a equal to50.6° and 52.05°. The characteristic shearing inter-ferogram fringes of the beam with spherical aberrationare easily recognized. At the same time other, moredense moire fringes are visible in the photographs.Their presence is explained by Eq. (9) and the discus-sion following it.

When setting the rotation angle a of G equal to50.6° and 52.05° (the uniform field detection mode),the shear amount A changes from 3.8 to 5.2 mm andfrom 3.76 to 5.24 mm, respectively, in the 45-mm diam

Fig. 3. Finite fringe detection mode; the carrier fringes run parallelwith the shear direction. The detecting grating G2 is slightly rotat-

ed about the z axis, a = 50.6°.

observation field along the x direction. The averageshear value for the cross section passing through the y-z plane is 4.5 mm in both cases so the change of shearamounts to 15.5% of the average. The linear shearchange can be neglected in qualitative testing (forexample, for fast lens selection). However, for quanti-tative control the linear shear change over the observa-tion field must be taken into account.

The photographs shown in Fig. 2 correspond to thefinite fringe detection mode with a few fringes parallelto the grating lines of both gratings (i.e., perpendicularto the shear direction). They were introduced, asexplained above, by a proper angular setting of theangle a. However, it is useful to be able to introducecarrier fringes running parallel to the shear direction,i.e., in a direction perpendicular to the grating lines.Such a detection mode is easily achieved, as in a con-ventional Talbot interferometer configuration,' byslightly rotating one of the gratings about the z axis.Figure 3 shows the moire fringes obtained by a slighttilt of G2. These fringes evolve from the pattern corre-sponding to the uniform field detection mode whensetting the angle a = 50.6°. As is well known, thefringes in Fig. 3 represent a map of the transversespherical aberration.7

It follows from Figs. 1-3 that, when increasing therotation angle a, the lateral dimension of the beamsplitter grating influences the diameter of the observa-tion field. Therefore, the method requires a gratingGi of dimensions appropriately larger than the diame-ter of the beam under test.

IV. Conclusions

We have extended the investigations presented in aprevious paper5 aiming at the development of Talbotinterferometer configurations giving interferogramswith increased amounts of lateral shear. In a conven-tional Talbot interferometer setup very small shearscan be achieved due to the multiple-beam interferenceof diffraction orders of the beam splitter grating. Thistype of interference leads to considerable deteriorationof the interferogram quality at increased shear values.For example, interferograms with shears of the order of0.1 or larger of the diameter of the beam under testcannot be obtained.


The common idea behind the previous5 and presentmethods is to mutually separate the dominant two-beam interference patterns formed by the diffractionorder pairs (0,+1) and (0,-1) of the beam splitter (self-imaging) grating. This could be done by appropriatedifferentiation of their angular localization5 or, as pro-posed in this paper, by proper differentiation of theirspatial period. Then, contrary to the case of a classicalTalbot interferometer (where the two two-beam inter-ference patterns having the same spatial period andfringe angular orientation mutually interact), the twofringe patterns do not disturb each other and can beeasily discerned. This results in sufficient quality ofthe shearing interferogram with increased shear value.

In this paper a method of rotating the beam splittergrating about the axis parallel to its lines has beendescribed. By adjusting the grating rotation angle tomatch the period of one of the two-beam interferencepatterns with the period of the detecting grating, theperiod of one of two simultaneously present product-type moire patterns can be chosen for clear observa-tion. The other moire pattern represents much denserbut non disturbing background fringes. The shearvalue varies linearly in the observation field in a direc-tion perpendicular to the grating lines. This effect canbe taken into account during the quantitative tests orneglected in the qualitative ones. The shear amountcan be changed continuously by varying the gratings'separation distance without impairing the moire inter-ferogram quality. This property, as well as the in-crease in the shear value, represents the advantage ofthe proposed configuration over a conventional Talbotinterferometer setup. The experiments conductedfully verified the characteristics of the proposed con-figuration.

The methods described in the previous5 and presentpaper can be used complementarily depending on thelaboratory facilities, i.e., the type of gratings and rota-tional mounts available. Both methods enlarge theapplication field of Talbot interferometry and, in gen-eral, of the self-imaging phenomenon.

References1. S. Yokozeki and T. Suzuki, "Shearing Interferometer Using the

Grating as the Beam Splitter," Appl. Opt. 10, 1575 (1971).2. A. W. Lohmann and D. E. Silva, "An Interferometer Based on

the Talbot Effect," Opt. Commun. 2, 413 (1971).3. D. E. Silva, "Talbot Interferometer for Radial and Lateral Deri-

vatives," Appl. Opt. 11, 2613 (1972).4. K. Patorski and P. Szwaykowski, "Optical Differentiation of

Quasi-Periodic Patterns Using Talbot Interferometry," Opt.Acta 31, 23 (1984).

5. K. Patorski, "Talbot Interferometry with Increased Shear,"Appl. Opt. 24, 4448 (1985).

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

7. J. Schwider, "Single Sideband Ronchi Test," Appl. Opt. 20,2635(1981).

8. M. Born and E. Wolf, Principles of Optics (Pergamon, NewYork, 1975), Chap. 8.

9. Y. V. Chugui, V. P. Koronkevitch, B. E. Krivenkov, and S. V.Mikhlyaev, "Quasi-Geometrical Method for Fraunhofer Dif-fraction Calculations for Three-Dimensional Bodies," J. Opt.Soc. Am. 71, 483 (1981).

10. K. Patorski, "Fraunhofer Diffraction Patterns of Tilted PlanarObjects," Opt. Acta 30, 673 (1983); "Errata," Opt. Acta 31, 147(1984).

11. K. Patorski, "Fresnel Diffraction Field (Self-Imaging) ofObliquely Illuminated Linear Diffraction Gratings," Optik 69,30 (1984).

12. P. Bialobrzeski and K. Patorski, "Self-Imaging Phenomenon ofTilted Linear Periodic Objects," Opt. Appl. 15, 000 (1985).

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Talbot interferometry with increased shear: further considerations