Abstract

Symbolic substitution is a method for manipulating binary data that depends on both the value of the data and its spatial location to realize logical operations. A substitution system requires only a pattern recognizer, a nonlinear device, and a pattern substituter. Using classical optical elements and phase-only holographical elements, it is possible to construct optical systems for both recognition and substitution. Systems using one or two holographical elements are presented, and procedures for designing the elements are also discussed.

© 1988 Optical Society of America

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References

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  1. K.-H. Brenner, A. Huang, N. Streibl, “Digital Optical Computing with Symbolic Substitution,” Appl. Opt. 25, 3054 (1986).
    [CrossRef] [PubMed]
  2. K.-H. Brenner, “New Implementation of Symbolic Substitution Logic,” Appl. Opt. 25, 3061 (1986).
    [CrossRef] [PubMed]
  3. H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, Eds., Optical Bistability III, Springer Proceedings in Physics 8 (Springer-Verlag, New York, 1986).
    [CrossRef]
  4. M. J. Murdocca, “Digital Optical Computing with One-Rule Cellular Automata,” Appl. Opt. 26, 682 (1987).
    [CrossRef] [PubMed]
  5. J. N. Mait, K.-H. Brenner, “Dual-Phase holograms: Improved Design,” Appl. Opt. 26, 4883 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
    [CrossRef]
  8. H. Dammann, E. Klotz, “Coherent Optical Generation and Inspection of Two-Dimensional Periodic Structures,” Opt. Acta 24, 505 (1977).
    [CrossRef]
  9. J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann Gratings as Array Generators,” J. Opt. Soc. Am. A 4(13), 69 (1987).
  10. W. B. Veldkamp, J. R. Leger, G. J. Swanson, “Coherent Summation of Laser Beams Using Binary Phase Gratings,” Opt. Lett. 11, 303(1986).
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    [CrossRef] [PubMed]

1987 (3)

1986 (3)

1980 (1)

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
[CrossRef]

1978 (1)

1977 (1)

H. Dammann, E. Klotz, “Coherent Optical Generation and Inspection of Two-Dimensional Periodic Structures,” Opt. Acta 24, 505 (1977).
[CrossRef]

1973 (1)

1972 (1)

Brenner, K.-H.

Chu, D. C.

Dammann, H.

H. Dammann, E. Klotz, “Coherent Optical Generation and Inspection of Two-Dimensional Periodic Structures,” Opt. Acta 24, 505 (1977).
[CrossRef]

Downs, M. M.

Fienup, J. R.

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
[CrossRef]

Gallagher, N. C.

Goodman, J. W.

Hsueh, C.-K.

Huang, A.

Jahns, J.

Klotz, E.

H. Dammann, E. Klotz, “Coherent Optical Generation and Inspection of Two-Dimensional Periodic Structures,” Opt. Acta 24, 505 (1977).
[CrossRef]

Leger, J. R.

Liu, B.

Mait, J. N.

Murdocca, M. J.

Prise, M. E.

Sawchuk, A. A.

Streibl, N.

Swanson, G. J.

Veldkamp, W. B.

Walker, S. J.

Appl. Opt. (7)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent Optical Generation and Inspection of Two-Dimensional Periodic Structures,” Opt. Acta 24, 505 (1977).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
[CrossRef]

Opt. Lett. (1)

Other (1)

H. M. Gibbs, P. Mandel, N. Peyghambarian, S. D. Smith, Eds., Optical Bistability III, Springer Proceedings in Physics 8 (Springer-Verlag, New York, 1986).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Examples of pattern transformation rules: (a) dual-rail implementation of logical AND; (b) single-rule due to Murdocca (from Ref. 4). A darkened pixel indicates high intensity.

Fig. 2
Fig. 2

Recognition and substitution for logical and assuming dual-rail logic and NOR-type optical nonlinearity: (a) binary input data; (b) the eight shifted replicas necessary for recognition of the four transformation rules shown in Fig. 1(a) (the arrowhead indicates the row to be masked off); (c) output from optical NOR gate; (d) shifted replicas necessary for scribing operation; (e) binary output.

Fig. 3
Fig. 3

Fixed bit-plane array for one-bit intensity-coded implementation of logical AND using Murdocca’s single rule. The high-intensity bits indicated are fixed. The dashed boxes on the left indicate the locations of the inputs, and the dashed box on the right indicates the output. Four iterations are necessary for the input to propagate to the output (after Ref. 4).

Fig. 4
Fig. 4

Single-channel optical systems for recognition using (a) dual-rail logic and (b) polarization-based logic.

Fig. 5
Fig. 5

Single-channel optical systems for substitution using (a) dual-rail logic and (b) polarization-based logic.

Fig. 6
Fig. 6

(a) Even and (b) odd representations of pupil functions.

Fig. 7
Fig. 7

Dual-channel optical system for symbolic substitution.

Fig. 8
Fig. 8

Imaging and Fourier transform properties of an optical system using cylindrical lenses: (a) optical system; (b) x-cross-sectional view of system; (c) y-cross-sectional view of system.

Fig. 9
Fig. 9

Summation of two phase-only quantities for realizing a complex number.

Equations (26)

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N T = n = 1 N R n 0 ,
N T = n = 1 N R n 1 ,
p r ( x , y ) = i = 1 N T δ ( x x i , y y i ) ,
p s , n ( x , y ) = i = 1 S n 1 δ ( x x i , y y i ) ,
t ( x , y ) = s ( x ξ , y η ) p ( ξ , η ) d ξ d η = s ( x , y ) * * p ( x , y ) ,
f ( x , y ) = | s ( x , y ) | 2 * * | p ( x , y ) | 2 .
P e ( u ) = P ˜ e ( u ) * comb ( u ) ,
p e ( x ) = p ˜ e ( x ) comb ( x ) = n = p ˜ e ( n ) δ ( x n ) .
p ˜ e ( x ) = { 1 π x k = 0 N ( 1 ) k ( sin 2 π x z e , k + 1 sin 2 π x z e , k ) , x 0 , 2 k = 0 N ( 1 ) k ( z e , k + 1 z e , k ) , x = 0 .
p ˜ e ( n ) = { 1 π n k = 0 N ( 1 ) k ( sin 2 π n z e , k + 1 sin 2 π n z e , k ) , n 0 , 2 k = 0 N ( 1 ) k ( z e , k + 1 z e , k ) , n = 0 .
p ( x , y ) = i = 1 N T δ ( x x i , y y i ) = i = 1 N x j = 1 N i y δ ( x x i , y y j ) ,
p ( x ) = p e ( x ) + p o ( x ) ,
p e ( x ) = p ( x ) + p ( x ) 2 ,
p o ( x ) = p ( x ) p ( x ) 2 .
P ( u ) = P e ( u ) + j P o ( u ) .
P 1 ( u ) = P o ( u ) ,
P 2 ( u ) = P e ( u ) ,
ϕ = π / 2 .
P o ( u ) = P ˜ o ( u ) * comb ( u ) ;
p o ( x ) = p ˜ o ( x ) comb ( x ) = n = p ˜ o ( n ) δ ( x n ) ;
p ˜ o ( n ) = { 1 j π n k = 0 N ( 1 ) k ( cos 2 π n z o , k + 1 cos 2 π n z o , k ) , n 0 , 0 , n = 0 .
P 1 ( u , υ ) = exp [ j Ө + ( u , υ ) / 2 ] ,
P 2 ( u , υ ) = exp [ j Ө ( u , υ ) / 2 ] ,
P ( u , υ ) = | P ( u , υ ) | exp [ j Ө ( u , υ ) ] = ( 1 / 2 ) { exp [ j Ө + ( u , υ ) ] exp ( j ϕ ) + exp [ j Ө ( u , υ ) ] } ,
Ө + ( u , υ ) = Ө ( u , υ ) + cos 1 | P ( u , υ ) | ϕ ,
Ө ( u , υ ) = Ө ( u , υ ) cos 1 | P ( u , υ ) | .

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