Abstract

The capabilities of optical computers are extended to perform the class of bilinear transformations (of nonzero spread) on 1-D signals. Use is made of the additional degree of freedom in 2-D linear processing. The technique is applied to the study of partially coherent optical systems and to systems in which coherent optical processing is followed by postdetection linear spatial filtering.

© 1978 Optical Society of America

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References

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  1. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
    [CrossRef]
  2. G. W. Stroke, IEEE Spectrum 9, 24 (1972).
    [CrossRef]
  3. K. Preston, Coherent Optical Computers (McGraw-Hill, New York, 1972).
  4. A. Vander Lugt, Proc. IEEE 62, 1300 (1974).
    [CrossRef]
  5. J. W. Goodman, Proc. IEEE 65, 29 (1977).
    [CrossRef]
  6. G. M. Robbins, T. S. Huang, Proc. IEEE 60, 862 (1972).
    [CrossRef]
  7. A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
    [CrossRef]
  8. J. W. Goodman, P. Kellman, E. W. Hansen, Appl. Opt. 16, 733 (1977).
    [CrossRef] [PubMed]
  9. R. J. Marks, J. F. Walkup, M. O. Hagler, T. F. Krile, Appl. Opt. 16, 739 (1977).
    [CrossRef]
  10. K. T. Stalker, S. H. Lee, J. Opt. Soc. Am. 64, 545 (1974).
  11. T. C. Strand, Opt. Commun. 15, 60 (1975).
    [CrossRef]
  12. H. Kato, J. W. Goodman, Appl. Opt. 14, 1818 (1975).
  13. S. R. Dashiell, A. A. Sawchuk, Opt. Commun. 15, 66 (1975).
    [CrossRef]
  14. S. R. Dashiell, A. A. Sawchuk, Appl. Opt. 16, 1009 (1977).
    [CrossRef] [PubMed]
  15. S. R. Dashiell, A. A. Sawchuk, Appl. Opt. 16, 2279 (1977).
    [CrossRef] [PubMed]
  16. R. J. Marks, J. F. Walkup, T. F. Krile, Appl. Opt. 16, 746 (1977).
    [CrossRef]
  17. W. T. Rhodes, J. M. Florence, J. Opt. Soc. Am. 65, 1178 (1975).
  18. W. T. Rhodes, J. M. Florence, Appl. Opt. 15, 3073 (1977).
    [CrossRef]
  19. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1974).
  20. T. N. Cornsweet, Visual Perception (Academic, New York, 1970).

1977 (7)

1975 (4)

T. C. Strand, Opt. Commun. 15, 60 (1975).
[CrossRef]

H. Kato, J. W. Goodman, Appl. Opt. 14, 1818 (1975).

S. R. Dashiell, A. A. Sawchuk, Opt. Commun. 15, 66 (1975).
[CrossRef]

W. T. Rhodes, J. M. Florence, J. Opt. Soc. Am. 65, 1178 (1975).

1974 (2)

A. Vander Lugt, Proc. IEEE 62, 1300 (1974).
[CrossRef]

K. T. Stalker, S. H. Lee, J. Opt. Soc. Am. 64, 545 (1974).

1972 (3)

G. M. Robbins, T. S. Huang, Proc. IEEE 60, 862 (1972).
[CrossRef]

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

G. W. Stroke, IEEE Spectrum 9, 24 (1972).
[CrossRef]

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1974).

Cornsweet, T. N.

T. N. Cornsweet, Visual Perception (Academic, New York, 1970).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Dashiell, S. R.

Florence, J. M.

W. T. Rhodes, J. M. Florence, Appl. Opt. 15, 3073 (1977).
[CrossRef]

W. T. Rhodes, J. M. Florence, J. Opt. Soc. Am. 65, 1178 (1975).

Goodman, J. W.

J. W. Goodman, P. Kellman, E. W. Hansen, Appl. Opt. 16, 733 (1977).
[CrossRef] [PubMed]

J. W. Goodman, Proc. IEEE 65, 29 (1977).
[CrossRef]

H. Kato, J. W. Goodman, Appl. Opt. 14, 1818 (1975).

Hagler, M. O.

Hansen, E. W.

Huang, T. S.

G. M. Robbins, T. S. Huang, Proc. IEEE 60, 862 (1972).
[CrossRef]

Kato, H.

H. Kato, J. W. Goodman, Appl. Opt. 14, 1818 (1975).

Kellman, P.

Krile, T. F.

Lee, S. H.

K. T. Stalker, S. H. Lee, J. Opt. Soc. Am. 64, 545 (1974).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Marks, R. J.

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1974).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Preston, K.

K. Preston, Coherent Optical Computers (McGraw-Hill, New York, 1972).

Rhodes, W. T.

W. T. Rhodes, J. M. Florence, Appl. Opt. 15, 3073 (1977).
[CrossRef]

W. T. Rhodes, J. M. Florence, J. Opt. Soc. Am. 65, 1178 (1975).

Robbins, G. M.

G. M. Robbins, T. S. Huang, Proc. IEEE 60, 862 (1972).
[CrossRef]

Sawchuk, A. A.

Stalker, K. T.

K. T. Stalker, S. H. Lee, J. Opt. Soc. Am. 64, 545 (1974).

Strand, T. C.

T. C. Strand, Opt. Commun. 15, 60 (1975).
[CrossRef]

Stroke, G. W.

G. W. Stroke, IEEE Spectrum 9, 24 (1972).
[CrossRef]

Vander Lugt, A.

A. Vander Lugt, Proc. IEEE 62, 1300 (1974).
[CrossRef]

Walkup, J. F.

Appl. Opt. (7)

IEEE Spectrum (1)

G. W. Stroke, IEEE Spectrum 9, 24 (1972).
[CrossRef]

IRE Trans. Inf. Theory (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

J. Opt. Soc. Am. (2)

K. T. Stalker, S. H. Lee, J. Opt. Soc. Am. 64, 545 (1974).

W. T. Rhodes, J. M. Florence, J. Opt. Soc. Am. 65, 1178 (1975).

Opt. Commun. (2)

S. R. Dashiell, A. A. Sawchuk, Opt. Commun. 15, 66 (1975).
[CrossRef]

T. C. Strand, Opt. Commun. 15, 60 (1975).
[CrossRef]

Proc. IEEE (4)

A. Vander Lugt, Proc. IEEE 62, 1300 (1974).
[CrossRef]

J. W. Goodman, Proc. IEEE 65, 29 (1977).
[CrossRef]

G. M. Robbins, T. S. Huang, Proc. IEEE 60, 862 (1972).
[CrossRef]

A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
[CrossRef]

Other (3)

K. Preston, Coherent Optical Computers (McGraw-Hill, New York, 1972).

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Redondo Beach, Calif., 1974).

T. N. Cornsweet, Visual Perception (Academic, New York, 1970).

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Figures (7)

Fig. 1
Fig. 1

Optical simulation of a 1-D bilinear space-invariant transformation. ζ ¯ is the 2-D Fourier transform of ζ.

Fig. 2
Fig. 2

Factorization of a 1-D space-variant bilinear transformation into a 2-D space-invariant linear transforamtion followed by a 1-D space-variant linear transformation.

Fig. 3
Fig. 3

Simulation of a 1-D optical system illuminated by partially coherent light of complex degree of coherence γ. The intensities of the object and image are Io and Ii. The system has a coherent transfer function H.

Fig. 4
Fig. 4

A system equivalent to that of Fig. 3. γ ¯ is the Fourier transform of γ.

Fig. 5
Fig. 5

Coherent system followed by a detector and a postdetection linear system.

Fig. 6
Fig. 6

Optical simulation of the system in Fig. 5.

Fig. 7
Fig. 7

Filters corresponding to the systems H*(x)H(y) and X(x + y).

Equations (16)

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g ( x ) = d x 1 d x 2 f ( x 1 ) f ( x 2 ) ζ ( x ; x 1 , x 2 ) ,
ζ ( x ; x 1 , x 2 ) = μ ( x , x 1 ) μ ( x , x 2 ) .
r ( x , y ) = d x d y f ( x ) f ( y ) h ( x , y ; x , y ) .
ζ ( x ; x 1 , x 2 ) = h ( x , x ; x 1 , x 2 ) .
g ( x ) = r ( x , x ) .
ζ ( x , x 1 , x 2 ) = γ ( x 1 , x 2 ) μ ( x x 1 , x x 2 ) s ( x , x ) d x ,
g ( x ) = s ( x , x ) g 1 ( x ) d x ,
g 1 ( x ) = d x 1 d x 2 μ ( x x 1 , x x 2 ) f ( x 1 ) f ( x 2 ) γ ( x 1 , x 2 ) .
g ( x ) = f ( x 1 ) f ( x 1 x ) d x 1 .
g ( 2 x ) = f ( x 1 ) f ( x 2 ) ζ ( x x 1 , x x 2 ) d x 1 d x 2 , ζ ( x , y ) = δ ( x + y ) .
g ( x ) = f ( x 1 ) f ( x x 1 ) d x 1
g Ω ( x ) = f ( x 1 ) f ( x 1 x ) exp ( j Ω x 1 ) d x 1
ζ ( x ; x 1 , x 2 ) = δ ( x x 1 + x 2 ) exp ( j Ω x 1 ) .
g Ω ( 2 x ) exp ( j Ω x ) = f ( x 1 ) f ( x 2 ) ζ ( x x 1 , x x 2 ) d x 1 d x 2 , ζ ( x , y ) = δ ( x + y ) exp ( j Ω y ) , ζ ¯ ( x , y ) = δ ( x y Ω ) .
g ( x ) = d x 1 d x 2 f ( x 1 ) f ( x 2 ) γ ( x 1 , x 2 ) h * ( x , x 1 ) h ( x , x 2 ) ,
ζ ¯ ( ω x , ω y ) = H * ( ω x ) H ( ω y ) S ( ω x + ω y ) ,

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