Abstract

The objective of this study is twofold: to design reticle patterns with desirable alignment properties; to build an automatic alignment system using these patterns. We design such reticle patterns via a synthetic approach; the resultant patterns, so-called pseudonoise arrays, are binary and their autocorrelation functions are bilevel. Both properties are desirable in optical alignment. Besides, these arrays have attractive signal-to-noise ratio performance when employed in alignment. We implement the pseudonoise array as a 2-D cross-grating structure of which the grating period is much less than the wavelength of impinging light used for alignment. The short grating period feature, together with the use of polarized light, enables us to perform essentially 2-D optical alignment in one dimension. This alignment separability allows us to build a system that performs alignment automatically according to a simple 1-D algorithm.

© 1988 Optical Society of America

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References

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  1. M. C. Hutley, Diffraction Gratings (Academic, New York, 1982), pp. 309–310.
  2. N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328 (1973).
    [Crossref] [PubMed]
  3. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,169,002 (9Nov.1971).
  4. R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).
  5. J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).
  6. Henry Stark, Ed. Image Recovery: Theory and Application (Academic, New York, 1987), Chap. 7.
  7. M. R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information, and Computing (Springer-Verlag, New York, 1984).
  8. D. Calobro, J. K. Wolf, “On the Synthesis of Two-Dimensional Array with Desirable Correlation Properties,” Inf. Control 2, 537 (1968).
  9. E. E. Fenimore, T. M. Cannon, “Coded Aperture Imaging with Uniformly Redundant Arrays,” Appl. Opt. 17, 337 (1978).
    [Crossref] [PubMed]
  10. R. Petit, Ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980), Chap. 6.
    [Crossref]

1981 (1)

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).

1978 (1)

1973 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

1968 (1)

D. Calobro, J. K. Wolf, “On the Synthesis of Two-Dimensional Array with Desirable Correlation Properties,” Inf. Control 2, 537 (1968).

Calobro, D.

D. Calobro, J. K. Wolf, “On the Synthesis of Two-Dimensional Array with Desirable Correlation Properties,” Inf. Control 2, 537 (1968).

Cannon, T. M.

Fenimore, E. E.

Fienup, J. R.

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Hirsch, P. M.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,169,002 (9Nov.1971).

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982), pp. 309–310.

Jordan, J. A.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,169,002 (9Nov.1971).

Lesem, L. B.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,169,002 (9Nov.1971).

Liu, B.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Schroeder, M. R.

M. R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information, and Computing (Springer-Verlag, New York, 1984).

Wolf, J. K.

D. Calobro, J. K. Wolf, “On the Synthesis of Two-Dimensional Array with Desirable Correlation Properties,” Inf. Control 2, 537 (1968).

Appl. Opt. (2)

Inf. Control (1)

D. Calobro, J. K. Wolf, “On the Synthesis of Two-Dimensional Array with Desirable Correlation Properties,” Inf. Control 2, 537 (1968).

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).

Other (5)

Henry Stark, Ed. Image Recovery: Theory and Application (Academic, New York, 1987), Chap. 7.

M. R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information, and Computing (Springer-Verlag, New York, 1984).

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982), pp. 309–310.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,169,002 (9Nov.1971).

R. Petit, Ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980), Chap. 6.
[Crossref]

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Figures (15)

Fig. 1
Fig. 1

Alignment of two sets of cross hairs in the boresight region.

Fig. 2
Fig. 2

Array 1 is imaged onto array 2 to form autocorrelation on the photodetector. Array 2 is adjusted in position to determine the maximum value of the autocorrelation.

Fig. 3
Fig. 3

Iterative scheme.

Fig. 4
Fig. 4

Root mean square error and desirable autocorrelation function.

Fig. 5
Fig. 5

Real Part of G(f).

Fig. 6
Fig. 6

Real Part of g(x) after threshold binarization.

Fig. 7
Fig. 7

One-dimensional pseudonoise array.

Fig. 8
Fig. 8

Grating polarizer.

Fig. 9
Fig. 9

Grid grating response.

Fig. 10
Fig. 10

One-dimensional grating strip.

Fig. 11
Fig. 11

One-dimensional alignment reticle pattern.

Fig. 12
Fig. 12

One-dimensional reticle pattern response.

Fig. 13
Fig. 13

Two-dimensional reticle pattern.

Fig. 14
Fig. 14

Image of the 2-D pattern under different polarized light.

Fig. 15
Fig. 15

System configuration.

Equations (19)

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g ˜ b k ( x ) = { 1 g ˜ k ( x ) > threshold , 0 otherwise .
error rms ( k ) = g k + 1 ( x ) - g k ( x ) g k ( x ) = G k + 1 ( f ) - G k ( f ) G k ( f ) ,
error rms b ( k ) = g k b ( x ) - g k ( x ) g k ( x ) ,
error rms M ( k ) = G k + 1 ( f ) - G k b ( f ) G k b ( f ) .
A x 2 + B x + C = 0 mod ( p )
y 2 = b mod ( p ) ,
y = x + A - 1 B 2 ,
b = ( A - 1 B 2 ) 2 - A - 1 C ,
A - 1 A 1 mod ( p ) .
( x / p ) { 0 if x 0 mod ( p ) , 1 if x square mod ( p ) , - 1 if x nonsquare mod ( p ) .
{ a n , p }
P 1. { a n , p } is a periodic sequence with period p , P 2. a n , p = ( 1 / p ) = 1 for any p , P 3. a m n , p = a m , p × a n , p , P 4. a n , p = n ( p - 1 ) 2 mod ( p ) , P 5. of n p - 1 = 1 a n , p = 0 , P 6. n = 1 p - 1 a n + i , p = { 0 if i 0 mod ( p ) , - 1 otherwise , P 7. n = 1 p - 1 ( a n , p ) 2 = p - 1 , p 8. n = 1 p - 1 ( a n , p ) × ( a n + i , p ) = - 1 for any i .
b n , p = ( a n , p + 1 ) 2 .
n = 1 p - 1 ( b n , p ) 2 = n = 1 p - 1 ( a n , p + 1 2 ) 2 = 1 4 × n = 0 p - 1 [ a n , p 2 + 2 × a n , p + 1 ] = 1 4 × [ n = 0 p - 1 a n , p 2 + 2 × n = 0 p - 1 a n , p + ( p - 1 ) ] = 1 4 × [ ( p - 1 ) + 2 × 0 + ( p - 1 ) ] = ( p - 1 ) 2 ,
n = 1 p - 1 ( b n , p ) × ( b n + i , p ) = n = 1 p - 1 [ a n , p + 1 2 ] × ( a n + i , p + 1 2 ) = 1 4 × { n = 1 p - 1 [ ( a n , p ) × ( a n + i , p ) + a n , p + a n + i , p + 1 ] } = 1 4 × [ n = 1 p - 1 ( a n , p ) × ( a n + i , p ) + 0 + ( - 1 ) + ( p - 1 ) ] = 1 4 [ ( - 1 ) + ( - 1 ) + ( p - 1 ) ] = p - 3 4 .
p 4 + 1 4 and 2 ( p - 3 p - 1 ) ,
A n , p = ( n p ) = n ( p - 1 ) 2 mod ( p ) ,
n ( p - 1 ) 2 = ( 1 ) mod ( p ) when n is quadratic residue , n ( p - 1 ) 2 = ( - 1 ) mod ( p ) when n is not a quadratic residue .
b = y 2 - | y 2 p | p ,

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