Three-way gratings for moiré interferometry

Volume 53, number 5 OPTICS COMMUNICATIONS 1 April 1985

THREE-WAY GRATINGS FOR M O I R E I N T E R F E R O M E T R Y

K. PATORSKI and M. KUJAWIlqSKA

Warsaw Technical University, Institute of Design of Precise and Optical Instruments, 8 Chodkiewicza Str., 02-525 Warsaw, Poland

Received 2 January 1985

A special grating with three linear rulings angularly separated by 30 and 60 degrees is proposed for use in the Moir6 interferometry technique. The principles and performance of the novel type special grating as well as the advantages relating to the simplification of the experimental setup design are discussed. The results of a preliminary model experiments are presented.

1. I n t r o d u c t i o n G

The so-called Moir6 interferometry technique has been developed for high sensitivity, whole field analysis of in-plane and out-of-plane displacements of deformed bodies. It represents an improved version of Moir6 fringe multiplication technique proposed by Post [1 ]. The first practical implementations have been presented by Matsumoto and Takashima [2], Marchant and Bishop [3], and Walker and McKelvie [4]. Moir6 interferometry has been greatly developed during the last five years, the main innovations and applications having been introduced at the Virginia Polytechnic Institute and State University [5] and at the University of Strathclyde [6].

In Moir6 interferometry high frequency (500 £/mm and more) reflective type phase grating is formed on the specimen under investigation. Since usually it is necessary to determine more than one displacement or strain field the specimen grating is of the crossed-line type. Each linear grating (with rulings parallel to the x and y axes, respectively) is analysed separately by two mutually coherent plane beams, fig. 1. The illumination configuration is symmetrical, i.e., the beams im- pinge on opposite sides of the specimen grating normal at angles equal to the first order diffraction angle. In this way appropriate first diffraction orders of the illuminating beams, i.e., the wave +1L and - 1 R , coin- cide in space and propagate along the grating normal (other diffraction orders of illuminating beams do not

0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

• • IL

P

Fig. 1. Schematic representation of the moir~ interferometry principle. G, one of the component specimen gratings. L and R, plane wave illuminating beams. IL, imaging optics. P, observation plane.

take part in the interferogram formation and are not indicated in fig. 1).

Because of the specimen loading the grating lines become deformed. The wavefronts of the +1L and - 1 R diffraction orders are no longer plane, the wavefront deformations of the +1L and - 1 R beams are equal but longitudinally reversed. This is why the final interferogram obtained in the observation plane (conjugate to the specimen grating) has double number of fringes as compared to the case of interference of one of the diffracted beams with a plane reference wave.

The u and v displacement fields, i.e., the displacements in the x andy directions, are obtained by analysing linear rulings (with lines perpendicular to x and y axis, respectively) of the crossed-line specimen grat-

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ing. This is realized by rotating the specimen under load by 90 degree [5] with respect to the incidence plane defined by the two illuminating beams or by providing a three beam system [6]. The latter method is more versatile since the specimen can remain in place and because three (not two) displacement fields are readily obtained by sequentially cutting one of the three illuminating beams. Three displacement components transformed to three normal strain components enable to obtain full field strain information by the rosette method [7]. In this way the need of calculating the derivatives ~u/ay and ~o/~x, very sensitive to accidental specimen rotations, is obviated. The two beam illumination setup of Post [5] supplemented by two additional illuminating beams serves the purpose of obtaining the third displacement pattern. This is due to the fact that cross-line gratings can also be used as gratings functioning in the -+45 degree directions to the x andy axes [8], in fact the same principle is utilized in the three-beam illumination interferometer [6].

In this paper we would like to present a new design of specimen grating that can be used with simple two beam illuminating system [5]. The grating is composed of three appropriately oriented linear gratings of the same spatial period. The angular orientation of the rulings follows from the beat-frequency analysis of multiple-exposure gratings [9-11] . Since the periods of component gratings are equal the rulings can be sequentially analyzed by single pair of illuminating beams. This leads to significant simplification of the interferometer construction as compared with the Post's solution [7].

The principles of the novel specimen grating design will be theoretically explained and the results of preliminary experiments using three-way transmission type gratings will be given.

2. Principle

The purpose of the novel grating design is to provide the surface of the specimen under test with three appropriately oriented linear diffraction gratings of the same spatial period. Since the composite three-way grating can be treated as the product of component linear rulings, many beat frequencies (resulting in the multiple diffraction orders) appear in the frequency plane of such a composite grating. The should be spa-

tially separated in order to have only two diffraction orders + 1L and -1R propagating along the specimen surface normal when analysing the component gratings.

A single one-way distorted specimen grating can be expressed by a Fourier series expansion

n--+oo

g(x,y) = ~ A n exp{i[2nnfox+nu(x,y)]}, (1) n = m ~

wheref 0 = 1/d, d is the grating period, u(x,y) the local displacement of the grating lines due to specimen loading. In general, the grating cannot be treated as a sinusoidal one because of technological factors in- fluencing the groove profile (for example, the non- linearity of photographic emulsion).

Considering a single component grid of the three- way grating we can treat it as a linear periodic structure being represented in the Fourier space by the vector

rk/= k/r/exp(i0/), (2)

where j = 1,2,3 the number of the linear grating, rj = fo, Oj the angle between the normal to grating lines and the x axis, k/the number of diffraction order of the j th structure. Using the vector summation approach [9] the spectrum of the product of three linear gratings of the same spatial period is represented by vec- tors described by the following general formula

R = klr exp(i01)

+ k2r exp(i02) + k3r exp(i03). (3)

Taking into consideration the symmetrical illumination configuration of the component gratings we must assure that:

- The beat frequencies in the spectrum of composite specimen grating do not overlap with the first diffraction order of each of the component gratings under single beam illumination.

- In the case of symmetrical illumination, fig. 1, the beat frequencies due to beams L and R do not appear within the common region of the diffraction orders +ljL and --l jR. This condition has to be fulfilled for three different angular orientations of the specimen.

The diffraction efficiency of higher orders of quasi- sinusoidal grating is low so it seems quite resonable to limit the number of diffraction orders under consideration to k/E ( -2 ,2) .

286

Volume 53, number 5 OPTICS COMMUNICATIONS

C

1 April 1985

.2*J*c

8 I,A*J,C .~+8.¢

*2A "f,4 f,a',J-c "f,,,-c f-~

f,,-J..b

2,A-C

~2c

J-A .~lt

" 2-~,c

• ~A÷C .2-~,c

"fA -ZA

",c'2-'~':," ' - A - c ~ . ~ .2.~-,

"f-J-c "l- , .s-c~ 2c "2-A-c

"2+c

Fig. 2. The spectrum of a three-way grating under single beam illumination. A, B, and C designate the component gratings.

From the detailed analysis (carried according to the methods given in refs. [9,11]) of different angle con- figurations of equal-period component gratings we have found out that the most suitable angular orientation of component gratings is 0, 30 and 90 degree. Fig. 2 shows the spectrum of a composite grating under single beam illumination. A, B and C designate the component gratings and correspond t o / = 1,2 and 3 in eq. (2), respectively. Accordingly, the spectra of three-way structure under two beam illumination for three different angular orientations of the specimen are found, fig. 3. In fig. 3 beat frequencies are marked by points. The first diffraction orders and the beat frequencies which are closest to analysed orders 1/L and 1/R are indexed in the figure. In such a configuration the diffraction orders + 1 (A,B,C)L and - 1 (A,B,C)R propagating in the common direction are spatially separated from the beat frequencies characterized by considerable values o f intensity.

The above analysis has been given for linear gratings. When the component structures become deformed the points in the diffraction grating spectrum become ex- tended regions. The frequency bandwidth of a component grating is given by

Af] x =/r -1 maXx,y ~ u / ( x , y ) / ~ x ,

Af/y = 7r -1 maXx,y ~)u](x ,y) /ay. (4)

Let us assume that the frequency band width Af of the function u ( x , y ) is not larger than 1% of the carrier frequency value f0. That means

o.o110, (Sa) and the beat frequencies bandwidth is given by

Aflq. x , Afk/y <~ 0.01 k f o . (5b)

This assumption seems quite reasonable since it im- poses that the strain should be smaller than 10 -2 ,

287


B~

As

(a)

C~

+tA

CL

- h

• BL

(b)

& CL

a,

8L

(c)

AR B e

-IA - fs " "

A L

/ - t ¢ •

Fig. 3. The spectrum of a three-way grating under two-beam illumination for different angular orientations of the grating A: (a) a = 0 °, (b) c~ = 30 °, and (c) c~ = 90 °, where c~ designates the angle between the component grating A and the x-axis.

which is mostly the case in Moir6 interferometry in- vestigations [5,6]. As the radius r/. o f the free area around the utilized orders +IlL and --I/R obtained for rulings separated by 30 and 60 degree equals, see fig. 3,

r n = 0.5 f0 , rB = 0.5 f0 , and r C = 0.25 f0 , this values

assure the possibility of spatial filtering of first diffrac- t ion orders of the above analysed component gratings.

3. Experiments

The validity of the proposed gratings configuration has been examined by experiments using the transmission type binary synthetic interferogram with coded spherical wavefronts of radius R = 10 m (for X = 488 nm). The experimental setup is shown in fig. 4. Two- beam symmetrical i l lumination is realized by using two first diffraction orders of a binary grating with frequency f0 equal to the carrier frequency of the synthetic interferogram [9].

First, a simple one-way synthetic interferogram has been illuminated by two beams L and R. This arrange- ment results in the spectra shown in fig. 5. The over- lapping first diffraction orders +1L and - 1 R are filtered and the result of their interference (two spherical wavefronts with conjugate phase) is shown in fig. 6.

For further experiments a composite synthetic interferogram with three identical components separated by 30 and 60 degree has been used, see fig. 7. The spectra obtained for three angular positions of synthetic interferogram are shown in fig. 8. In each position the

overlapped + 1 (A,B,C)L and - 1 (A,B,C)R diffraction orders are filtered and the interferograms are recorded in the observation plane, fig. 9.

The information content of the employed component interferogram has been A f ~ 0.25 f0 , so the qua- l i ty of interferogram obtained from the three-way grating in comparison with the one-way grating, fig. 6, is worse. Herein, fig. 8, the noisy fringes appear at the borders of the interferogram. However, it is obvious

G LI ~K'f L2 J5

I I

f f f f

L3 5F3 L4 J

IT ( f f f

Fig. 4. Experimental setup for optica simulation of Moird interferometry using three-way grating. G, linear grating, SFI and SF2, spatial filters; L1, L2, L3 and L4, lenses of the coherent optical processor, IS, synthetic interferogram composed of three quasi-linear gratings, I, image plane.

288


Fig. 5. The spectrum of a single synthetic interferogram under two beam illumination.

that in the case of smaller frequency bandwidth of the component gratings the spectra do not overlap and the interferogram contain the information about +1 L and - 1 R orders only.

Fig. 6. The interferomgram obtained as a result of interference of +_ 1 diffraction orders of a single synthetic interferogram with coded spherical wavefront.

Fig. 7. The central portion of a three-way grating with angular separation 30 and 60 deg between the component rulings A-B and B-C, respectively.

4. Conclusions

The design of a composite specimen grating for use in Moir~ interferometry employing rotable specimen and a two-beam illumination system has been proposed. It has been shown that proper angular orientation of composite linear gratings enables separate analysis of in-plane deformations in three directions. As conse- quence, complete strain information can be obtained without calculating the cross-derivatives which are very sensitive to accidental specimen rotations. Only one pair of illuminating beams is required which results in a considerable simplification of the interferometer.

The addition of the 30 degree grating to the cross- line grating employed in the experimental setups described in the literature results in the decrease of light intensity in the observed and recorded interference pat- terns. However, this problem seems to be a not very serious one when using the advanced technology of casting the relief type diffraction gratings [5,6]. The detailed analysis is deferred to subsequent paper.

The principles of the composite grating design have been verified using optical simulation of a three-way ruling in the form of transmission-type superposition of three binary synthetic interferograms.

289


~ O 0 ~ • 6 I

b

Fig. 8. The spectra obtained for three angular positions of the composite synthetic interferogram: (a) c~ = 0 °, (b) a = 30 °, and (c) a = 90 °.

Fig. 9. Interferograms obtained in the image plane I, fig. 4, for the following combinations of diffraction orders: (a) +IAL and - 1 A R , (b) +IBL and - 1 B R , (c) + I c L and - 1 C R .

2 9 0

Volume 5 3, number 5 OPTICS COMMUNICATIONS 1 April 1985

References

[I] D. Post, Appl. Optics6 (1967) 1938. [2] K. Matsumoto and M. Takashima, Appl. Optics 12 (1973)

858.

[3] M. Marchant and J.M. Bishop, J. Strain Anal. 9 (1974) 36.

[ 41 C.A. Walker and J. McKelvie, Exper. Mechanics 18 (1978)

316.

[5] D. Post, Opt. Engineering 21 (1982) 458, and the referen-

ces therein.

[6] A. McDonach, .I. McKelvie, P. MacKenzie and C.A.

Walker, Exper. Technique 7 (1983) 20, and the referen-

ces therein.

[7] E.M. Weissman and D. Post, Exper. Mechanics 22 (1982)

324.

[ 81 P. Dantu, Exper. Mechanics 4 (1964) 64.

[9] M. Kujawinska, Optica Acta 28 (1981) 843.

[IO] M. Kujawmska, Optics Comm. 44 (1982) 85.

[ 111 M. Kujawmska, Optica Acta 30 (1983) 1319.

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Three-way gratings for moiré interferometry