Moiré methods in interferometry

Optics and Lasers in Engineering 8 (1988) 147-170

Moir6 Methods in Interferometry

Krzysztof Patorski

Institute of Design of Precise and Optical Instruments, Warsaw University of Technology, 8 Chodkiewicza Street, 02-525 Warsaw, Poland

(Received 14 November 1987; accepted 9 December 1987)

A B S T R A C T

The applications o f moir~ methods to optical interferometry are reviewed. They are used for instrumental error subtraction, reduction of spatial frequency of interferogram fringes, increase of sensitivity and the range o f applications of interferometric methods, and various information processing operations. The cases of the reference beam and shear type interferometry are treated separately.

INTRODUCTION

Moir6 fringe techniques represent a powerful tool in scientific and workshop metrology. Several review works have been published on this subject, and one of them appeared recently in this journal. 1

Interferometry relates to optical metrology and it has been dramati- cally expanded since the development of the laser. This statement is valid even if we do not take into consideration such techniques as holographic or speckle interferometry. The progress in various fields in interferometry has been mutually stimulating. The application of the moir6 technique is one of the examples.

The present paper reviews moir6 fringe applications in optical interferometry used for studying phase objects in transmission and reflection. Holographic techniques are excluded. The two beam configurations using the reference wave and the shear type interference are

147 Optics and Lasers in Engineering 0143-8166/88/$03.50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Northern Ireland

148 Krzysztof Patorski

described. The common purpose of applying moir6 to interferometry is to display and process information about the object. The subtraction of instrumental errors, control of the sensitivity of the method and phase information processing (i.e. whole-field differentiation) can be quoted as examples.

The moir6 method can be realized using coherent or incoherent superposition of interferograms or diffraction gratings. Either of these two processes can be called moir6 interferometry. However, this name will be restricted, as is widely accepted, to the special technique employed for strain analysis 2 and discussed in a separate article by C. A. Walker in this issue pp. 213-62. It is to be noted that the name moir6 interferometry was proposed previously by Yokozeki and Mihara 3 for two-beam interferometry using incoherent superposition techniques.

In cases of both coherent and incoherent superposition, the contrast of moir6 fringes can be improved by spatial filtering. In the first case an optical system for filtering is placed directly at the output of an interferometer. In the second case the two interferograms recorded on separate plates or by a double exposure are placed in the input plane of the processor. The operation of spatial filtering results in a two-beam interference and fringes of high contrast. In the present paper, however, the main interest will be focused on systems in which the moir6 pattern can be observed directly. The spatial filtering will be treated as an accompanying although secondary operation.

MOIRI~ METHODS IN TWO-BEAM I N T E R F E R O M E T R Y WITH A R E F E R E N C E WAVE FRONT

Subtraction of instrumental errors

An advantage of the interferometer with a reference beam is that the contour map of the object phase distribution can be obtained directly from the interferogram. However, an interferometer with quality optics is required for this purpose to avoid the influence of instrumental errors.

On the other hand it has been shown 4-14 that certain interferometer errors can be subtracted when using the moir6 fringe method. Two interferograms with carrier fringes of appropriate spatial frequency are superimposed. The first interferogram is produced with an empty interferometer and the second one with the object under study present.

MoirF methods in interferometry 149

Since the moir6 fringe effect corresponds to the subtraction of the two interferograms, the common part in both of them, i.e. the information about instrumental errors (for example, defects of mirrors and beam splitters), is eliminated. Mathematical description of this principle is given by Yokozeki and Suzuki. 9-11 To observe a phase object using the finite fringe detection mode, the angle between the reference and object beams should be slightly changed between the two exposures.

Certainly, the superposition of two interferograms can be effected in real time or after recording them. In the first case the photographic plate with an empty interferometer fringe pattern is replaced back into the system. Then the moir6 produced by the interferogram recorded and the one formed in the interferometer with the object inserted can be viewed in real time. 8 Higher spatial frequencies can also be eliminated in real time by placing the optical coherent processor at the output of the interferometer. The accuracy of repositioning of the photographic plate is not as high as in holographic interferometry. If the interferometer errors are known, the first (reference) interferogram can be synthesized, i.e. computer generated.

When the two interferograms superimposed are registered by photographic or video techniques, the superposition can be of product or sum type. The first one corresponds to recording the two patterns on separate plates and their subsequent overlap. The second superposition is realized by the double exposure technique. However, since the additive superposition can be readily converted to the product type one (using a nonlinear recording or spatial filtering of the doubly exposed plate) the mathematical model presented in the papers mentioned above 3'9-n applies to both techniques.

Figures 1, 2 and 3 show experimental illustration of the instrumental error subtraction by the moir6 technique (courtesy of S. Yokozeki). Figures I(A) and I(B) show interference fringes obtained in an empty Mach-Zehnder interferometer and with a glass plate inserted in one arm of the interferometer, respectively. The density of reference fringes is the same in both photographs and set arbitrarily by tilting one of the mirrors of the interferome'ter. Figure I(C) shows double-exposure recording of the two patterns. Additive-type moir6 fringes correspond to an infinite fringe map of the optical path difference introduced by a glass plate. They are of low contrast, mainly because of a too coarse fringe spacing of the interferograms being superimposed. Figure 2 corresponds to the case of overlapping higher fringe density interferograms; moir6 bands can be seen here much more easily. Moir6grams of Figs I(C) and 2 are equivalent to an infinite fringe interferogram of a glass plate generated in a good Mach-Zehnder system (Fig. 3).


(A) (B)

(c)

Fig. 1. Interferograms obtained in a non-perfect Mach-Zehnder interferometer. (A) Fringes for an empty interferometer (reference fringes are introduced arbitrarily). (B) Fringes for a glass plate inserted. (C) Double-exposure recording of (A) and (B).

(Courtesy of S. Yokozeki.)

Reduction of the spatial frequency of interferogram fringes

The next application of the moir6 technique to two-beam interferometry relates to the reduction of the spatial frequency of dense interferogram fringes 15 or to the conversion of finite fringe interferograms into infinite fringe ones. ~6 As an example, the configuration of a Fizeau multipass moir6 interferometer, as proposed by Langenbeck, 15 will be described. Figure 4 shows schematically an optical system. Mirror under test, M, and beam-splitter, B, are mutually inclined to achieve lateral separation of light source images at the back focal plane of objective O. Only two beams are allowed to pass through the double slit filter, SF. These are the 0th (reflected from the beam-splitter surface) and the n th beams. The latter carries n-times multiplied wave

Moird methods in interferometry 151

Fig. 2. Double-exposure recording of interference patterns without and with a glass plate with higher frequency of reference (carrier) fringes. (Courtesy of S. Yokozeki.)

Fig. 3. Interferogram of a glass plate under the uniform field detection mode. (Courtesy of S'. Yokozeki.)

front deformation introduced by M or by M and B. This approach gives an interferogram with n-times increased sensitivity as compared to the usual Fizeau pattern. The contrast of the two-beam interference can be controlled by polarization techniques.

Because of a finite angular distance between the interfering beams the fringes are dense and their frequency must be reduced for visual inspection. This is done by inserting a linear diffraction grating, G, in the observation plane. The grating period should be close to an average spacing of interference fringes. Infinite and finite fringe detection modes are realized by changing the difference between periods or mutual orientation of the two overlapped patterns.


i, ol ', / \ ' , / / /

o_ ~_ SF /\ /~

Fig. 4. Multipass Fizeau

- - G

interferometer using the moir6 technique for fringe detection.

Interferogram analysis with increased sensitivity

Langenbeck's paper 15 contains further suggestions for using the moir6 technique for increasing the sensitivity of interferometry. Again, the interferogram is inspected by moir6 effect but now the frequency of detecting grating is higher. It is appropriately chosen to visualize the nth harmonic of the fringes. For this purpose the spatial frequency of the detecting grid should be n times higher than the frequency of the analyzed interferogram. This technique requires strongly nonsinusoidal intensity distribution of the fringes to assure the presence of higher spatial harmonics. At the same time the transmission profile of the detecting grid can be properly chosen to provide acceptable contrast of moir6 fringes. This problem was studied in detail in the papers of Post, 17 Bryngdahl TM and Patorski e t a l . ~9 Although the advantage of the higher-order moir6 fringe method just mentioned is that it can be used with incoherent light (coherent techniques based on spatial filtering are sensitive to various noise sources--their description can be found in the papers of Post 2° and Matsumoto and Takashima21), its use has been limited by low light level when detecting higher harmonics.

Moir6 methods in interferometry 153

The configuration with a grating detecting an interference pattern can be used for further high-precision interferogram analysis by a phase- stepping method. Three interferograms with different phase shifts between the object and reference beams are required in this method and are readily produced in the form of moir6grams by laterally displacing the detecting grating with respect to the interferogram being analyzed.

Whole-field differentiation of fringe patterns

In some cases, besides information about the object phase distribution, its spatial derivative is required. This corresponds to lateral aberration, surface slope and principal stresses when studying optical systems, reflective surfaces and structures under load, respectively. Numerical or graphical differentiation methods of interferogram fringes can be time consuming and introduce substantial errors. The application of moir6 techniques for the purpose of generating the derivative can be very helpful because of its simplicity.

The dupligram method using the moir6 effect was introduced by Lau. 2a The works of Lau 23 and others 24'2s concerned its application in the field of lens testing. Two copies of an interferogram are overlapped and laterally displaced along a selected direction. If the interferogram fringes are dense enough a moir6 pattern is formed. This depicts the spatial derivative of the interferogram along the shift direction. The derivative is averaged over the shift distance.

Figure 5 shows the result of the dupligram method using a Fresnel zone pattern. As is well known, the Fresnel pattern can be treated as a binarized interferogram of a parabolic wave front interfering with a plane one. Straight moir6 fringes (known as Schuster fringes) are generated by two identical, laterally displaced plates. They are equivalent to two-beam lateral shear interference fringes obtained when testing spherical wave fronts.

The principle of dupligrametry has been widely used in other fields of metrology in which the information is displayed in the form of interference or moir6 fringes. For example, in the application of moir6 to strain analysis the dupligram method is known as mechanical differentiation. 26

Addition of phase functions

An interferogram obtained in the reference beam arrangement corresponds to a difference between the object phase distribution and the


Fig. 5. Fresnel zone plate and moir6 fringes obtained by the dupligram method.

reference wave front. In other words, the interferometer realizes the operation of subtraction. However, in some cases, not the difference but the sum of two functions is required. Addition can be achieved by overlapping two interferograms with conjugate carriers, 2~ i.e. two fringe patterns with fringes distorted in mutually opposite directions.

Their formation is shown schematically in Fig. 6 corresponding to an interference arrangement with the object and plane reference beams mutually inclined. For clarity of explanation let the object beam propagate along the normal to the observation plane. The conjugate interferogram is formed by making the reference beam impinge on the opposite side of the object beam with respect to the first interferogram


p, IP

Fig. 6. Recording of conjugate interferograms. O, object beam; R, reference beam; IP, interferogram plane.

recording configuration. In the moir6 approach, two conjugate moir6grams can be obtained by superimposing the interferogram (deformed grating) with the master grating. The lines of overlapped structures are mutually inclined at the same angle as in the case of the first moir6gram formation, but of opposite sign. 27'28 Another method is to use two master gratings with spatial frequency larger and smaller, by the same amount, with respect to the frequency of the distorted grating.

The concept of addition of two phase functions by moir6 of two conjugate, reference beam interferograms has been employed to determine the out-of-plane displacements of a deformed body by holographic 27 and moir6 interferometry 29'3° techniques. In the latter case (see the paper by C. A. Walker in this issue) conjugate interferograms of the two diffraction orders of the specimen grating are recorded separately. In the two interferograms the information about in-plane and out-of-plane displacements are recorded with the same and opposite signs, respectively. Therefore, when the interferograms are overlapped, the moir6 fringes represent a map of out-of-plane displacements.

Two-wavelength interferometry

Until now overlapping of two interferograms formed by the same light source, i.e. of the same wavelength, has been discussed. However, we can imagine the use of different wavelengths in the same interferometer system. The interferograms corresponding to each wavelength are mutually incoherent and have different fringe spacing. Therefore, when overlapped, they will produce a moir6 pattern. Wyant showed that two-wavelength holographic interferometry enables study of aspheric mirrors at effectively less sensitive, longer wavelength. 31 The method was generalized and extended to non-holographic interferometry by Polhemus. 32 Two methods of interferogram superposition can be distinguished. The first can be called static interferometry and is similar


to coherent superposition of identical wavelength interferograms discussed above. An interferogram with carrier fringes is recorded at wavelength ~,l, processed and replaced at its original position. It serves the role of the reference pattern. Next, the interferometer is operated at wavelength 3,2. The product of intensity distributions of the reference and the second interferogram contains a low spatial frequency moir6 pattern. It is equivalent to the interferogram one would obtain by using an equivalent wavelength /~eq ( '~eq=~1'~2/1'~1"1--/~'21) • The technique provides a wide range of sensitivities when using an ion argon laser source operating at several wavelengths. However, the approach of static interferometry using a reference mask cannot be considered, in a strict sense, as a real-time method. Any change of the object or the reference interferogram position during the test, results in the change of moir6 fringes.

A modification of the above technique allows real-time evaluation (dynamic interferometry). 32 Now the wavelengths ,~1 and )~2 operate simultaneously in the interferometer. Intensity distributions of the two interferograms are added, and the moir6 pattern appears in the form of high-frequency fringe contrast modulation (additive-type moir6). A nonlinear detecting device allows us to monitor the contrast modulation in real time. Subsequent processing renders the desired signal suitable for further processing.

Polhemus provides mathematical descriptions of both the static and dynamic two-wavelength interferometry. 32

MOIRI~ METHODS IN SHEARING INTERFEROMETRY

Shearing interferometry has a remarkable advantage over interferometry with a reference beam. Simply, the reference beam is not required. The object beam interferes with its replica and under the conditions of small shear value and slow variations of the object phase distribution, the interferogram fringes carry information about the derivative of the phase function. Several applications of the moir6 fringe technique in interferometry reviewed in the previous section (i.e. subtraction of instrumental errors, change from the finite to infinite fringe detection mode, and interferogram processing) are directly transferable to shearing interferometry. Therefore, instead of reviewing them here in full, we will limit our considerations to two cases.

In the first case the moire technique plays a fundamental part in the interferometer performance. Shearing interferometers using diffraction


gratings as beam-splitters and recombiners represent such optical configurations. The second case relates to a superimposition of shearing interferograms to select the information sought. Such processing of shear-type interferograms by moir6 has recently been found very useful in moir6 interferometry (see the paper by C. A. Walker in this issue).

Talbot interferometry

Grating shearing interferometers can be of multiple- or two-beam type. The first group relates to so-called Talbot interferometry developed independently by Yokozeki and Suzuki 33'34 and Lohmann and Silva. 35-37 Ten years later a similar technique was described by Kafri under the name of moir6 deflectometry. 38 The diffraction approach was used by the first authors to explain the system performance, whereas Kafri's description was based on ray optics. The first approach is more universal, taking into consideration various effects involved. The two treatments give equivalent results as signaled by Patorski 39 although Keren and Kafri 4° and their collegues 41 first mentioned the difference between the cases of placing the phase object in front of and behind the first grating. The origin of Talbot interferometry and moir6 deflectometry can be traced to the paper of Oster et al.42

Figure 7 shows a schematic representation of the Talbot interferometer. A spatially coherent plane wave front illuminates binary amplitude diffraction grating G1 of low spatial frequency. The self- imaging phenomenon takes place behind this grating. Self-images are detected using the moir6 fringe method, i.e. the second identical grating Gz is placed in one of the self-image planes of G1. An aberration of the illuminating beam or phase changes in a phase object O cause distortions of the self-images. The departure of self-image lines from straightness is proportional to the first derivative of the phase distribution under study in the direction perpendicular to grating lines. It is visualized in the form of moir6 fringes that are equivalent to

Fig. 7.

) ,,

Optical setup of the Talbot interferometer. S, light source; L, collimation optics; G1 and G2, diffraction gratings; O, phase object under study.


interference fringes obtained in a two-beam lateral shear interferometer. The derivative is averaged over the shear distance equal to the product of the grating first-order diffraction angle and the distance of an object (or the wave front under study) from the second grating. The uniform field display mode is obtained when the lines of both gratings are set mutually parallel. The finite fringe interferogram with reference fringes parallel to the shear direction is realized by slightly rotating the detecting grating about its normal. The reference fringes perpendicular to the shear direction (i.e. parallel to grating lines) can be obtained by using a detecting grating of a slightly different period.

The remarkable simplicity of the Talbot interferometer is clearly seen from Fig. 7. It only requires a well collimated spatially coherent light beam from a quasi-monochromatic light source. The sensitivity of the method is easily controlled by changing the shear amount, i.e. by changing the axial separation distance between the object and observation planes. The system is confined to rather small shear values and to the slow gradient phase functions because of the disturbing diffraction effects. The range of applications of Talbot interferometry can be extended by shifting from the multiple-beam interference of diffraction orders of G1 to the two-beam interference. The solutions were described by Silva, 37 Keren e t a l . , 43 and Patorski. 44'45 In the first two papers two-beam interference was realized by using a spatial filtering technique (the moir6 method is not used for fringe detection) and a specially prepared beam-splitter grating, respectively. The solutions of Patorski 44'45 use the basic system of the Talbot interferometer shown in Fig. 7. Two-beam interference patterns formed by the diffraction order pairs 0, + 1 and 0, - 1 were made different from each other by rotating the first grating about the axis parallel or perpendicular to its lines. In this way spatial frequency or orientation of the two interferograms differ and proper tuning of the detecting grating allows monitoring of one of the two interference patterns. Figure 8 shows schematically one of these configurations. Note that when the object is placed in front of the first grating a change in shear value over the observation plane is encountered. This disadvantage is absent when the object is between the gratings.

Talbot interferometry using linear diffraction gratings provides derivative information along the direction perpendicular to the lines of the beam-splitter grating. The radial derivative is obtained by using circular gratings. 36'37 Evolvent gratings, as proposed by Szwaykowski, 46 provide the radial as well as the azimuthal derivative.

Figure 9 shows the experimental results obtained in a lateral shear Talbot interferometer with an acousto-optic cell serving as the object

G2

Fig. 8. Modified configuration of the Talbot interferometer with increased shear. ~ Beam-splitter grating G~ is rotated about the axis perpendicular to its lines to differentiate the spatial orientation of two-beam interference patterns formed by

adjacent grating orders. Grating G2 is tuned to one of these interferograms.

(A)

(B) Fig. 9. Experimental results obtained in the Talbot interferometer when testing temperature gradients in an acousto-optic cell. (A) and (B) show the infinite and finite

detection modes, respectively.


under test. The temperature gradients near the piezoelectric transducer are easily detected.

Among the modifications to the Talbot interferometer that are aimed at improvements in its performance and utilize various concepts of the moir6 fringe technique, it is necessary to mention an electronic superimposition video technique, 47 the use of computer-generated detection grating, 48 modern methods of moir6gram analysis, 49-51 and the development and implementation of the concept of conjugate lateral shear interferometry. 52 An incoherent version of the Talbot interferometer using a periodic extended source was proposed by Patorski. 53"54

The following applications of Talbot interferometry have been reported: beam collimation test, 55-58 testing and focal length measurement of optical elements, 33'37"44'45'59-64 analysis of phase ob j ec t s , 33-37'65-69

optical differentiation of quasi-periodic s t ruc tures ,70,71 optical alignment, 72'73 analysis of vibrating objects, TM depth encoding, 75 produc- tion and testing of diffraction gratings, 76-79 and measurement of small tilts. 8°

Applications of the moir6 deflectometry technique were reviewed by Kafri and Glatt. 81 This paper contains an extensive list of references and photographic documentation of the experimental results.

Both Talbot interferometry and moir6 deflectometry represent multiple-beam interference systems. All diffraction orders of the beam-splitter grating are allowed to interfere. In a special case of a grating with a sinusoidal amplitude transmittance only three diffraction orders are present. On the other hand, the three lowest order diffraction beams are also predominant when using square-wave type binary gratings. The final interference can be treated as the sum-type m o i r 6 44'45 due to the additive overlap of the two-beam interference patterns formed by diffraction order pairs 0, + 1 and 0, - 1. When the shear amount is not small and when the phase variations in the object under test are considerable, the contrast modulation effect characteris- tic to additive moir6 displays the second derivative of the phase distribution. 33 Application of this additive moir6 effect in three-beam interferometry for aberration analysis was described by Komissaruk 82 and Lin and Cowley. 83

Two-beam lateral shear interferometers

Interferometers of this type are widely used because of very good contrast of interference fringes. Various configurations have been reported but the most elegant ones provide variable shear and fringe


Fig. 10.

0 L1 G1 F1G2 L2 0P1 L3 F-'2 L.4 OP2 I ) I

f f f f f f f f 4

Double grating lateral shear interferometer using moir6 fringe detection technique. 87

orientation. Moreover, these two parameters should be controlled independently. The double grating interferometer developed independently by Spornik and Yanichkin 84 and Wyant et al. 85'86 represents a very elegant solution to this problem. It requires, however, slightly compli- cated opto-mechanical design.

An alternative system that possesses the same features (arbitrary shear amount and arbitrary selection of the period and orientation of reference fringes) was proposed by Patorski. 87 It uses relatively low frequency binary gratings, spatial filtering and moir6 technique. Figure 10 shows a schematic drawing of the optical configuration.

A phase object O under investigation is placed in the front focal plane of the first imaging system Ll-La. Two gratings of different spatial frequency are located on opposite sides of the focal plane F1 where the spatial filter is introduced. First diffraction orders of G1 are allowed to pass and illuminate the second grating. The doubly diffracted beams (+ 1 , - 1) and ( - 1, + 1) (the numbers in parentheses indicate diffraction order numbers at G1 and G2, respectively) are utilized for the formation of a two-beam lateral shear interferogram. They are mutually inclined behind Ga because of the different spatial frequencies of the gratings. This results in lateral displacement between the interfering beams in the observation plane. The interference pattern frequency should be reduced by the moir6 fringe technique using the third grating. How- ever, it cannot be placed directly in the observation plane OP1. Because of the multiple interferences of doubly diffracted beams generated by G2 and the interactions of these interferences with the third grating and its harmonics, the contrast of the desired pattern is considerably reduced. Therefore, grating G3 is placed in the image plane of the second optical processor L3-L4 used for isolation of the two beams desired. By changing the separation distance between G1 and G2 and rotating the detection grating G3 about the optical axis, arbitrary orientation and number of moir6 fringes can be selected. The shear amount is varied stepwise by substituting one of the beam-splitting gratings of different spatial frequency.


Superposition of shear-type intederograms

The operations of optical subtraction or addition realized by the moir6 fringe technique can be performed on phase functions and their derivatives. The latter case is of importance in strain analysis where not only the displacement maps are required but also their partial derivatives.

For example, shearing interferometry and moir6 are readily ap- plicable to the moir6 interferometry technique for producing the derivatives of in-plane and out-of-plane displacements. It is worth recalling at this moment (see the paper by C. A. Walker in this issue) that the two orders of the specimen grating carry mutually conjugate information about the in-plane displacements (corresponding to the departure of grating lines from straightness) and the same wave front deformation caused by the out-of-plane displacements. Therefore, if lateral shear interferograms with dense carrier fringes of each specimen grating diffraction order are recorded separately and overlapped, the moir6 fringe pattern displays the in-plane displacement derivative information. This approach was reported by Weissman et al. 88 It represents an optical equivalent of the mechanical shearing (dupligram) technique. 89-9°

Real-time differentiation by moir6 requires real-time optical shearing of displacement patterns or real-time overlap of the lateral shear intefferograms. These two approaches were developed by Patorski et

al. 91 Since the interference patterns overlapped are present simultaneously they add in intensity. The derivative information is then given by additive-type moir6 fringes. Figure 11 shows the experimental results presented in the cited paper.

The superposition of shear-type interferograms discussed up to now has concerned the subtraction of derivatives encoded in the component patterns. The operation of addition can be performed by overlapping two mutually conjugate shear-type interferograms, i.e. the interferograms with carrier fringes distorted in opposite directions? 2 In this case moir6 fringes display information proportional to the sum of the derivatives of the functions encoded in the component interferograms. A simple method for generating conjugate interferograms using a modified Ronchi test configuration and the self-imaging phenomenon was proposed by Patorski? 2 It can be briefly explained using Fig. 12. The object under study is placed in the front focal plane of an afocal imaging system El-L2. The lateral shear interferograms are recorded in the back focal plane of lens L2. A low-frequency amplitude grating G is placed at a distance z from the common focal plane of L1 and L2. To


Fig. 11. Experimental results of real-time differentiation of in-plane displacement patterns of a cantilever beam. 91 Top, au/Ox; bottom, au/Oy, where u is the x

(horizontal) component of specimen displacement. Ax = 1-2 mm, Ay = 0.9 mm.

obtain high contrast of the interferogram (which corresponds to a distorted self-image of G) , the distance z must be equal to a multiple of the so-called Talbot distance d2/A, where d designates the grating per iod and ~. is the light wavelength. Two mutually conjugate interferograms are recorded for two symmetrical localizations of the grating with respect to the beam focus.


Fig. 12. Schematic representation of the optical system for recording conjugate lateral shear interferograms. The method is based on the modified Ronchi test and self-imaging phenomenon. OP, object plane; IP, image plane; L1 and L2, afocal

imaging optics; G, diffraction grating.

Figure 13 shows moir6 fringes formed by superposition of the conjugate shearing interferograms with a master straight-line grating, (A) and (B), and by overlapping the two interferograms (C). Finite fringe patterns are presented. The opposite direction of fringe defor- mations in (A) and (B) proves the conjugate relationship between the patterns. The addition operation in Fig. 13(C) results in a double fringe deformation as compared to Figs 13(A) and 13(B).

The concept of addition of lateral shear interferograms was developed to determine the first derivative of out-of-plane displacement in the moir6 interferometry configuration. 92 The conjugate carrier fringe lateral shear interferograms of each of the two specimen grating diffraction orders are taken separately. When overlapped the subtractive moir6 fringes display the sum of derivatives of the wave front deformation in two diffraction orders. As a result, information about the derivative of out-of-plane displacements is obtained.

In the method discussed above, the carrier fringes were perpendicular to the shear direction. In some applications of the conjugate lateral shear interferometry to experimental mechanics, the other direction of the carrier fringes, i.e. parallel to the shear, is required as well. The method providing such conjugate interferograms, described by Patorski and Ulinowicz, 93 uses the double-grating interferometer system with independent control of shear and tilt. 84-86 It has several practical advantages following from the two-beam interference and the lack of axial movement of the shearing element, i.e. a diffraction grating.

CONCLUSIONS

Moir6 methods are well suited to various types of interferometry. By overlapping the interference patterns of appropriately high fringe


(A)

(B)

(c) Fig. 13. Moir6 fringe patterns formed by overlapping a master grating with two intefferograms obtained in the system of Fig. 12 for two symmetrical positions of G with respect to beam focus, (A) and (B), and by overlapping the two interferograms

themselves (C).

f requency, moi r rg rams can provide information about the object without instrumental errors, or give it in a processed form. All these operat ions are realized optically without the necessity of computer calculations. In the first case high demands on the in te r fe rometer quality are reduced and the cost of experiments becomes considerably lower. On the o ther hand, the use of the moir6 me thod in its simplest


subtractive version, supplemented by the technique of overlapping the patterns with conjugate carriers, results in an attractive approach producing mathematical operations such as subtraction, addition and differentiation of phase functions. Because of this possibility the potential applications of a specific interferometer system are considerably broadened.

Al though many of the merits of using moir6 described in this paper can be readily obtained by computer processing, which has been developing so fast in the recent years, the author believes in further development of applications of the moir6 fringe method. Inter- ferometry is one of these fields, among others. Its simplicity, low cost and high accuracy remain undisputable factors that attract scientists and engineers.

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Moir~ methods in interferometry 167

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Moiré methods in interferometry