Abstract

In the past, most optical data processing systems have been restricted to performing linear space-invariant operations. However, a wide class of interesting data processing operations require linear space-variant filtering. Three methods for performing linear space-variant processing of 1-D inputs are described. Experimental results obtained with all three systems are presented, and their relative advantages and disadvantages are discussed.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1968), Sec. 2-2.
  2. L. J. Cutrona, “Recent Developments in Coherent Optical Technology,” in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C.J. Koester, A. Vanderburgh, Eds. (MIT Press, Cambridge, 1965), Chap. 6.
  3. J. F. Walkup, M. O. Hagler, “Volume Hologram Representations of Space-Variant Optical Systems,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.
  4. L. M. Deen, J. F. Walkup, M. O. Hagler, Appl. Opt. 14, 2438 (1975).
    [CrossRef] [PubMed]
  5. G. M. Robbins, T. S. Huang, Proc. IEEE 60, 862 (1972).
    [CrossRef]
  6. A. A. Sawchuk, Proc. IEEE 60, 854 (1972).
    [CrossRef]
  7. J. W. Goodman, “Operations Achievable with Coherent Optical Data Processing,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 1–8.
  8. J. W. Goodman, Proc. IEEE 65, 29 (1977).
    [CrossRef]
  9. D. Casasent, D. Psaltis, “Mellen Transforms in Optical DataProcessing,” in Proceedings 1975 Electro-Optical System, Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.
  10. D. Casasent, D. Psaltis, Opt. Eng. 15, 258 (1976).
    [CrossRef]
  11. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  12. The quadratic phase factor existing in the y direction in plane P3 is of no consequence, since L3 images in this direction.
  13. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 27.
  14. W. T. Rhodes, J. M. Florence, Appl. Opt. 15, 3073 (1976).
    [CrossRef] [PubMed]
  15. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
    [CrossRef]

1977 (1)

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

1976 (2)

1975 (1)

1974 (1)

1972 (2)

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

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

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Bryngdahl, O.

Casasent, D.

D. Casasent, D. Psaltis, Opt. Eng. 15, 258 (1976).
[CrossRef]

D. Casasent, D. Psaltis, “Mellen Transforms in Optical DataProcessing,” in Proceedings 1975 Electro-Optical System, Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.

Cutrona, L. J.

L. J. Cutrona, “Recent Developments in Coherent Optical Technology,” in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C.J. Koester, A. Vanderburgh, Eds. (MIT Press, Cambridge, 1965), Chap. 6.

Deen, L. M.

Florence, J. M.

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1968), Sec. 2-2.

J. W. Goodman, “Operations Achievable with Coherent Optical Data Processing,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 1–8.

Hagler, M. O.

L. M. Deen, J. F. Walkup, M. O. Hagler, Appl. Opt. 14, 2438 (1975).
[CrossRef] [PubMed]

J. F. Walkup, M. O. Hagler, “Volume Hologram Representations of Space-Variant Optical Systems,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.

Huang, T. S.

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

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 27.

Psaltis, D.

D. Casasent, D. Psaltis, Opt. Eng. 15, 258 (1976).
[CrossRef]

D. Casasent, D. Psaltis, “Mellen Transforms in Optical DataProcessing,” in Proceedings 1975 Electro-Optical System, Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.

Rhodes, W. T.

Robbins, G. M.

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

Sawchuk, A. A.

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

Walkup, J. F.

L. M. Deen, J. F. Walkup, M. O. Hagler, Appl. Opt. 14, 2438 (1975).
[CrossRef] [PubMed]

J. F. Walkup, M. O. Hagler, “Volume Hologram Representations of Space-Variant Optical Systems,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

D. Casasent, D. Psaltis, Opt. Eng. 15, 258 (1976).
[CrossRef]

Proc. IEEE (3)

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

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

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

Other (8)

D. Casasent, D. Psaltis, “Mellen Transforms in Optical DataProcessing,” in Proceedings 1975 Electro-Optical System, Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1968), Sec. 2-2.

L. J. Cutrona, “Recent Developments in Coherent Optical Technology,” in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C.J. Koester, A. Vanderburgh, Eds. (MIT Press, Cambridge, 1965), Chap. 6.

J. F. Walkup, M. O. Hagler, “Volume Hologram Representations of Space-Variant Optical Systems,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 38–41.

J. W. Goodman, “Operations Achievable with Coherent Optical Data Processing,” in Proceedings 1975 Electro-Optical System Design Conference (Industrial and Scientific Conference Management, Inc., Chicago, 1975), pp. 1–8.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

The quadratic phase factor existing in the y direction in plane P3 is of no consequence, since L3 images in this direction.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962), p. 27.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Filtering system which uses two spherical–cylindrical lens combinations.

Fig. 2
Fig. 2

Filtering system which uses one spherical–cylindrical lens combination and one purely spherical lens.

Fig. 3
Fig. 3

Filtering system which uses two purely spherical lenses.

Fig. 4
Fig. 4

Variable magnifier realized with the system of Fig. 1: (a) slit angle 10°; (b) slit angle 30°; (c) slit angle 45° (output slit removed).

Fig. 5
Fig. 5

Variable blur realized with the system of Fig. 1: (a) input function; (b) mask in plane P2; (c) output (output slit removed).

Fig. 6
Fig. 6

Recording of the required filter: (a) transparent slits; (b) recording geometry.

Fig. 7
Fig. 7

Output of the system of Fig. 2, with output slit removed, and using the holographic filter generated as in Fig. 6.

Fig. 8
Fig. 8

Output of the system of Fig. 3 operated as a variable magnifier (output slit removed): (a) slit angle 20°; (b) slit angle 40°; (c) slit angle 60°.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

g ( y ) = - h ( y , x ) f ( x ) d x ,
g ( y ) = f [ z ( y ) ] = - δ [ z ( y ) ] - x ] f ( x ) d x , h ( y , x ) = δ [ z ( y ) - x ] ;
g ( y ) = 0 x y - 1 f ( x ) d x , h ( y , x ) = x y - 1 U ( x ) ,
U ( x ) = { 1 x > 0 ½ x = 0 0 x < 0 ;
g ( y ) = - 1 π y f ( x ) d x ( x 2 - y 2 ) 1 / 2 , h ( y , x ) = - 1 π ( x 2 - y 2 ) 1 / 2 U ( x - y ) d d x [ ] ,
f ^ ( ν X ) = - f ( x ) exp ( - i 2 π ν X x ) d x , h ^ ( y , ν X ) = - h ( y , x ) exp ( - i 2 π ν X x ) d x ,
g ( y ) = - h ^ ( y , - ν X ) f ^ ( ν X ) d ν X .
g ^ ( ν Y ) = - h ^ ( ν Y , - ν X ) f ^ ( ν X ) d ν X ,
g ^ ( ν Y ) = - g ( y ) exp ( - i 2 π ν Y y ) d y , h ^ ^ ( ν Y , - ν X ) = - h ( y , x ) exp [ - i 2 π ( ν Y y - ν X x ) ] d y d x .
g ( y ) = - h ^ ^ ( ν Y , - ν X ) f ^ ( ν X ) exp ( i 2 π ν Y y ) d ν X d ν Y .
h ( y , x ) = δ ( y - M x ) .
f ( x ) = δ ( x - x 1 ) + δ ( x - x 2 )
h ( y , x ) = rect ( y - x b x ) ,
t ( x , y ) = δ [ x - x O 2 - ( y + y 0 ) 1 / 2 ] + δ [ x + x 0 2 + ( y + y 0 ) 1 / 2 ]
t ( ν X , y ) = k 1 + k 2 cos [ π ν X x 0 + 2 π ν X ( y + y 0 ) 1 / 2 ] ,
h ^ ^ ( ν Y , - ν X ) = δ ( ν X - M ν Y ) .

Metrics