Abstract

The fringe profile of the moiré pattern resulting from the interaction between a regular and a distorted binary grating is studied. The moiré fringe profile prediction method in a generalized form is deduced and summarized in a table illustrating the relationship between the profile parameters of superposed gratings and those of moiré fringes. The method is confirmed by computer simulation and experiment. The conditions for dark and bright fringe sharpening are obtained by using the derived prediction method.

© 1978 Optical Society of America

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References

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  1. R. J. Pegis, U.S. Patent3,162,711 (1964).
  2. L. O. Vargady, U.S. Patent3,166,624 (1965).
  3. D. Post, Exp. Mech. 7, 154 (1967).
    [CrossRef]
  4. R. H. Katyl, Appl. Opt. 11, 2278 (1972).
    [CrossRef] [PubMed]
  5. S. Yokozeki, Y. Kusaka, K. Patorski, Appl. Opt. 15, 2223 (1976).
    [CrossRef] [PubMed]
  6. K. Patorski, S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 15, 443 (1976).
    [CrossRef]
  7. O. Bryngdahl, J. Opt. Soc. Am. 66, 87 (1976).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.
  9. H. H. Hsu, Fourier Analysis (Simon and Schuster, New York, 1970), p. 47.

1976 (3)

1972 (1)

1967 (1)

D. Post, Exp. Mech. 7, 154 (1967).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.

Bryngdahl, O.

Hsu, H. H.

H. H. Hsu, Fourier Analysis (Simon and Schuster, New York, 1970), p. 47.

Katyl, R. H.

Kusaka, Y.

Patorski, K.

K. Patorski, S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

S. Yokozeki, Y. Kusaka, K. Patorski, Appl. Opt. 15, 2223 (1976).
[CrossRef] [PubMed]

Pegis, R. J.

R. J. Pegis, U.S. Patent3,162,711 (1964).

Post, D.

D. Post, Exp. Mech. 7, 154 (1967).
[CrossRef]

Suzuki, T.

K. Patorski, S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

Vargady, L. O.

L. O. Vargady, U.S. Patent3,166,624 (1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.

Yokozeki, S.

K. Patorski, S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

S. Yokozeki, Y. Kusaka, K. Patorski, Appl. Opt. 15, 2223 (1976).
[CrossRef] [PubMed]

Appl. Opt. (2)

Exp. Mech. (1)

D. Post, Exp. Mech. 7, 154 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

K. Patorski, S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

Other (4)

R. J. Pegis, U.S. Patent3,162,711 (1964).

L. O. Vargady, U.S. Patent3,166,624 (1965).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.

H. H. Hsu, Fourier Analysis (Simon and Schuster, New York, 1970), p. 47.

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Figures (5)

Fig. 1
Fig. 1

Definition of function f(X).

Fig. 2
Fig. 2

Function F(X).

Fig. 3
Fig. 3

Parameters of the moiré fringe profile.

Fig. 4
Fig. 4

Computed and experimental moiré fringe profiles together with moiré fringe profile parameters obtained by the prediction method. The moiré fringes were formed by superimposing two binary grating with an opening ratio of 0.8. Fringe multiplication numbers are: (a) M = 1, (b) M = 2, and (c) M = 3. The solid lines illustrate the computed profiles, and the dashed lines show the changes of the computed profiles due to the opening ratio error. The cross marks are for experimental results.

Fig. 5
Fig. 5

Examples of moiré fringe sharpening. The parameters of two binary gratings: (a) O1 = O2 = 0.9 and M = 1; (b) O1 = 0.9, O2 = 0.1, and M = 1; (c) O1 = O2 = 0.1 and M = 1.

Tables (1)

Tables Icon

Table I Relation between profile parameters.

Equations (29)

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T P ( X ) = a 0 e 0 + k = 1 [ ( b M k f k 2 + c M k g k 2 ) cos 2 π k X + ( c M k f k 2 b M k g k 2 ) sin 2 π k X ] ,
O 1 = h 1 / d 1 , O 2 = h 2 / d 2 ,
a 0 = O 1 , b M k = 2 O 1 sin c ( k M O 1 ) , c M k = 0 , e 0 = O 2 , f k = 2 O 2 sin c ( k O 2 ) , g k = 0 ,
T p ( X ) = O 1 O 2 + 2 k = 1 [ O 1 O 2 sin c ( k M O 1 ) sin c ( k O 2 ) ] cos 2 π k X .
M O 1 = N + O 3 ,
T p ( X ) = ( N + O 3 ) O 2 / M + ( 2 / M ) k = 1 [ O 3 O 2 cos ( π k N ) sin c ( k O 3 ) sin c ( k O 2 ) ] cos ( 2 π k X ) .
T p ( X ) = T p 1 ( X ) + T p 2 ( X ) ,
T p 1 ( X ) = N O 2 / M ,
T p 2 ( X ) = ( 1 / M ) { O 3 O 2 + 2 O 3 O 2 k = 1 [ sin c ( k O 3 ) sin c ( k O 2 ) ] cos ( 2 π k X ) } .
F ( X ) = m = f ( X m ) ,
S + P < 1 ,
F ( X ) = ( A / S ) { S ( S + P ) + 2 S ( S + P ) n = 1 sin c ( n S ) sin c [ n ( S + P ) ] cos ( 2 π n X ) } .
T p 2 ( X ) F ( X ) ,
S = O 3 , P = O 2 O 3 , A = O 3 / M for O 3 O 2 , S = O 2 , P = O 3 O 2 , A = O 2 / M for O 3 > O 2 .
V = ( T max T min ) / ( T max + T min ) .
T min = N O 2 / M for O 3 + O 2 1 , T min = ( O 3 + O 2 1 + N O 2 ) / M for O 3 + O 2 > 1 ,
O 3 + O 2 1 and N = 0 ,
W B = O 2 and W D = 1 O 2 for O 3 O 2 , W B = O 3 and W D = 1 O 3 for O 3 > O 2 .
O 2 1 for O 3 O 2 and O 3 1 for O 3 > O 2 .
T max = O 3 / M ( 1 O 2 ) / M = W D / M for O 3 O 2 , T max = O 2 / M ( 1 O 3 ) / M = W D / M for O 3 > O 2 .
O 3 + O 2 = 1 and N = 0
O 2 0 for O 3 O 2 and O 3 0 for O 3 > O 2 .
T max = O 3 / M O 2 / M = W B / M for O 3 O 2 , T max = O 2 / M < O 3 / M = W B / M for O 3 > O 2 .
O 2 = O 3 0.
N = 0
O 1 = O 2 = 1 / ( M + 1 ) .
F ( X ) = ( P + 2 S 1 ) A / S + ( A / S ) × { [ 1 ( S + P ) ] ( 1 S ) + 2 [ 1 ( S + P ) ] ( 1 S ) × n = 1 sinc n [ 1 ( S + P ) ] sinc n ( 1 S ) cos ( 2 π n X ) } .
F ( X ) = ( A / S ) × { S ( S + P ) + 2 [ 1 ( S + P ) ] ( 1 S ) × n = 1 { ( 1 ) n + 1 ( S + P ) / [ 1 ( S + P ) ] } } sinc [ n ( S + P ) ] × [ ( 1 ) n + 1 S / ( 1 S ) ] sinc ( n S ) cos ( 2 π n X ) } .
F ( X ) = ( A / S ) { S ( S + P ) + 2 S ( S + P ) n = 1 sinc [ n ( S + P ) ] sinc ( n S ) cos ( 2 π n X ) } .

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