Abstract

The analogy between optical one-to-one point transformations and optical one-to-one interconnections is discussed. Methods for performing both operations are reviewed and compared. The multifacet and multistage architectures have the flexibility to implement any arbitrary one-to-one transformation or interconnection pattern. The former would be preferred for low-cost and low-resolution applications, whereas the latter would be preferred for high-cost and high-performance applications.

© 1993 Optical Society of America

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  1. J. Duvernoy, “Modele synthese non-lineaire pour le traitment optique des ecritures,” Opt. Commun. 11, 373–377 (1974).
    [CrossRef]
  2. A. Sawchuk, “Space-variant image restoration by coordinate transformation,” J. Opt. Soc. Am. 64, 138–144 (1974).
    [CrossRef]
  3. D. Casasent, D. Psaltis, “Deformation invariant, space variant optical pattern recognition,” Prog. Opt. 16, 289–356 (1978).
    [CrossRef]
  4. Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
    [CrossRef]
  5. D. Casasent, S.-F. Xia, A. J. Lee, J.-Z Shong, “Real-time deformation invariant optical pattern recognition using coordinate transformations,” Appl. Opt. 26, 938–942 (1987).
    [CrossRef] [PubMed]
  6. D. Mendlovic, N. Konforti, E. Marom, “Scale and projection invariant pattern recognition,” Appl. Opt. 28, 4982–4986 (1989).
    [CrossRef] [PubMed]
  7. G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
    [CrossRef]
  8. A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
    [CrossRef] [PubMed]
  9. O. Bryngdahl, “Geometrical transformation in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
  10. W. H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [CrossRef] [PubMed]
  11. M. A. Stuff, J. N. Cederquist, “Coordinate transformations realizable with multiple holographic optical elements,” J. Opt. Soc. Am. A 7, 977–981 (1990).
    [CrossRef]
  12. N. Davidson, A. A. Friesem, E. Hasman, “Optical coordinate transformations,” Appl. Opt. 31, 1067–1073 (1992).
    [CrossRef] [PubMed]
  13. S. K. Case, P. R. Haugen, O. J. Løberge, “Multifacet holographic optical elements for wave front transformations,” Appl. Opt. 20, 2670–2675 (1981).
    [CrossRef] [PubMed]
  14. H. M. Ozaktas, Y. Amitai, J. W. Goodman, “Comparison of system size for some optical interconnection architectures and the folded multi-facet architecture,” Opt. Commun. 82, 225–228 (1991).
    [CrossRef]
  15. H. M. Ozaktas, Y. Amitai, J. W. Goodman, “A three dimensional optical inter-connection architecture with minimal growth rate of system size,” Opt. Commun. 85, 1–4 (1991).
    [CrossRef]
  16. J. W. Goodman, A. R. Dias, L. M. Woody, “Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms,” Opt. Lett. 2, 1–3 (1978).
    [CrossRef] [PubMed]
  17. H. M. Ozaktas, J. W. Goodman, “Lower bound for the communication volume required for an optically interconnected array of points,” J. Opt. Soc. Am., A 7, 2100–2106 (1990).
    [CrossRef]
  18. G. E. Lohman, A. W. Lohmann, “Optical interconnection network utilizing diffraction gratings,” Opt. Eng. 27, 893–900 (1988).
  19. G. E. Lohman, K.-H. Brenner, “Space-variance in optical computing systems,” Optik 89, 123–124 (1992).
  20. R. K. Kostuk, J. W. Goodman, L. Hesselink, “Optical imaging applied to microelectronic chip-to-chip interconnections,” Appl. Opt. 24, 2851–2858 (1985).
    [CrossRef] [PubMed]
  21. R. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
    [CrossRef] [PubMed]
  22. M. R. Feldman, C. C. Guest, T. J. Drabik, S. C. Esener, “Comparison between electrical and free space optical interconnects for fine grain processor arrays based on interconnect density capabilities,” Appl. Opt. 28, 3820–3829 (1989).
    [CrossRef] [PubMed]
  23. M. R. Feldman, C. C. Guest, “Interconnect density capabilities of computer generated holograms for optical interconnection of very large scale integrated circuits,” Appl. Opt. 28, 3134–3137 (1989).
    [CrossRef] [PubMed]
  24. A. W. Lohmann, “What classical optics can do for the digital optical computer,” Appl. Opt. 25, 1543–1549 (1986).
    [CrossRef] [PubMed]
  25. A. W. Lohmann, W. Stork, G. Stucke, “Optical perfect shuffle,” Appl. Opt. 25, 1530–1531 (1986).
    [CrossRef] [PubMed]
  26. K.-H. Brenner, A. Huang, “Optical implementations of the perfect shuffle interconnection,” Appl. Opt. 27, 135–137 (1988).
    [CrossRef] [PubMed]
  27. M. J. Murdocca, A. Huang, J. Jahns, N. Streibl, “Optical design of programmable logic arrays,” Appl. Opt. 27, 1651–1660(1988).
    [CrossRef] [PubMed]
  28. A. W. Lohmann, F. Sauer, “Holographic telescope array,” Appl. Opt. 27, 3003–3007 (1988).
    [CrossRef] [PubMed]

1992 (2)

N. Davidson, A. A. Friesem, E. Hasman, “Optical coordinate transformations,” Appl. Opt. 31, 1067–1073 (1992).
[CrossRef] [PubMed]

G. E. Lohman, K.-H. Brenner, “Space-variance in optical computing systems,” Optik 89, 123–124 (1992).

1991 (2)

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “Comparison of system size for some optical interconnection architectures and the folded multi-facet architecture,” Opt. Commun. 82, 225–228 (1991).
[CrossRef]

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “A three dimensional optical inter-connection architecture with minimal growth rate of system size,” Opt. Commun. 85, 1–4 (1991).
[CrossRef]

1990 (2)

H. M. Ozaktas, J. W. Goodman, “Lower bound for the communication volume required for an optically interconnected array of points,” J. Opt. Soc. Am., A 7, 2100–2106 (1990).
[CrossRef]

M. A. Stuff, J. N. Cederquist, “Coordinate transformations realizable with multiple holographic optical elements,” J. Opt. Soc. Am. A 7, 977–981 (1990).
[CrossRef]

1989 (3)

1988 (4)

1987 (2)

1986 (2)

1985 (1)

1983 (2)

A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
[CrossRef] [PubMed]

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

1982 (1)

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

1981 (1)

1978 (2)

1974 (4)

Amitai, Y.

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “A three dimensional optical inter-connection architecture with minimal growth rate of system size,” Opt. Commun. 85, 1–4 (1991).
[CrossRef]

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “Comparison of system size for some optical interconnection architectures and the folded multi-facet architecture,” Opt. Commun. 82, 225–228 (1991).
[CrossRef]

Brenner, K.-H.

G. E. Lohman, K.-H. Brenner, “Space-variance in optical computing systems,” Optik 89, 123–124 (1992).

K.-H. Brenner, A. Huang, “Optical implementations of the perfect shuffle interconnection,” Appl. Opt. 27, 135–137 (1988).
[CrossRef] [PubMed]

Bryngdahl, O.

Casasent, D.

Case, S. K.

Cederquist, J. N.

Davidson, N.

Dias, A. R.

Drabik, T. J.

Duvernoy, J.

J. Duvernoy, “Modele synthese non-lineaire pour le traitment optique des ecritures,” Opt. Commun. 11, 373–377 (1974).
[CrossRef]

Esener, S. C.

Feldman, M. R.

Friesem, A. A.

Goodman, J. W.

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “Comparison of system size for some optical interconnection architectures and the folded multi-facet architecture,” Opt. Commun. 82, 225–228 (1991).
[CrossRef]

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “A three dimensional optical inter-connection architecture with minimal growth rate of system size,” Opt. Commun. 85, 1–4 (1991).
[CrossRef]

H. M. Ozaktas, J. W. Goodman, “Lower bound for the communication volume required for an optically interconnected array of points,” J. Opt. Soc. Am., A 7, 2100–2106 (1990).
[CrossRef]

R. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
[CrossRef] [PubMed]

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Optical imaging applied to microelectronic chip-to-chip interconnections,” Appl. Opt. 24, 2851–2858 (1985).
[CrossRef] [PubMed]

J. W. Goodman, A. R. Dias, L. M. Woody, “Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms,” Opt. Lett. 2, 1–3 (1978).
[CrossRef] [PubMed]

Guest, C. C.

Hasman, E.

Haugen, P. R.

Häusler, G.

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Hesselink, L.

Huang, A.

Jahns, J.

Komatsu, S.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Konforti, N.

Kostuk, R.

Kostuk, R. K.

Lee, A. J.

Lee, W. H.

Løberge, O. J.

Lohman, G. E.

G. E. Lohman, K.-H. Brenner, “Space-variance in optical computing systems,” Optik 89, 123–124 (1992).

G. E. Lohman, A. W. Lohmann, “Optical interconnection network utilizing diffraction gratings,” Opt. Eng. 27, 893–900 (1988).

Lohmann, A. W.

Marom, E.

Mendlovic, D.

Murdocca, M. J.

Ohzu, H.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “Comparison of system size for some optical interconnection architectures and the folded multi-facet architecture,” Opt. Commun. 82, 225–228 (1991).
[CrossRef]

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “A three dimensional optical inter-connection architecture with minimal growth rate of system size,” Opt. Commun. 85, 1–4 (1991).
[CrossRef]

H. M. Ozaktas, J. W. Goodman, “Lower bound for the communication volume required for an optically interconnected array of points,” J. Opt. Soc. Am., A 7, 2100–2106 (1990).
[CrossRef]

Psaltis, D.

D. Casasent, D. Psaltis, “Deformation invariant, space variant optical pattern recognition,” Prog. Opt. 16, 289–356 (1978).
[CrossRef]

Saito, Y.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Sauer, F.

Sawchuk, A.

Shong, J.-Z

Stork, W.

Streibl, N.

Stucke, G.

Stuff, M. A.

Woody, L. M.

Xia, S.-F.

Appl. Opt. (15)

D. Casasent, S.-F. Xia, A. J. Lee, J.-Z Shong, “Real-time deformation invariant optical pattern recognition using coordinate transformations,” Appl. Opt. 26, 938–942 (1987).
[CrossRef] [PubMed]

D. Mendlovic, N. Konforti, E. Marom, “Scale and projection invariant pattern recognition,” Appl. Opt. 28, 4982–4986 (1989).
[CrossRef] [PubMed]

A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983).
[CrossRef] [PubMed]

N. Davidson, A. A. Friesem, E. Hasman, “Optical coordinate transformations,” Appl. Opt. 31, 1067–1073 (1992).
[CrossRef] [PubMed]

S. K. Case, P. R. Haugen, O. J. Løberge, “Multifacet holographic optical elements for wave front transformations,” Appl. Opt. 20, 2670–2675 (1981).
[CrossRef] [PubMed]

W. H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
[CrossRef] [PubMed]

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Optical imaging applied to microelectronic chip-to-chip interconnections,” Appl. Opt. 24, 2851–2858 (1985).
[CrossRef] [PubMed]

R. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, T. J. Drabik, S. C. Esener, “Comparison between electrical and free space optical interconnects for fine grain processor arrays based on interconnect density capabilities,” Appl. Opt. 28, 3820–3829 (1989).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Interconnect density capabilities of computer generated holograms for optical interconnection of very large scale integrated circuits,” Appl. Opt. 28, 3134–3137 (1989).
[CrossRef] [PubMed]

A. W. Lohmann, “What classical optics can do for the digital optical computer,” Appl. Opt. 25, 1543–1549 (1986).
[CrossRef] [PubMed]

A. W. Lohmann, W. Stork, G. Stucke, “Optical perfect shuffle,” Appl. Opt. 25, 1530–1531 (1986).
[CrossRef] [PubMed]

K.-H. Brenner, A. Huang, “Optical implementations of the perfect shuffle interconnection,” Appl. Opt. 27, 135–137 (1988).
[CrossRef] [PubMed]

M. J. Murdocca, A. Huang, J. Jahns, N. Streibl, “Optical design of programmable logic arrays,” Appl. Opt. 27, 1651–1660(1988).
[CrossRef] [PubMed]

A. W. Lohmann, F. Sauer, “Holographic telescope array,” Appl. Opt. 27, 3003–3007 (1988).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

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

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

H. M. Ozaktas, J. W. Goodman, “Lower bound for the communication volume required for an optically interconnected array of points,” J. Opt. Soc. Am., A 7, 2100–2106 (1990).
[CrossRef]

Opt. Commun. (5)

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “Comparison of system size for some optical interconnection architectures and the folded multi-facet architecture,” Opt. Commun. 82, 225–228 (1991).
[CrossRef]

H. M. Ozaktas, Y. Amitai, J. W. Goodman, “A three dimensional optical inter-connection architecture with minimal growth rate of system size,” Opt. Commun. 85, 1–4 (1991).
[CrossRef]

J. Duvernoy, “Modele synthese non-lineaire pour le traitment optique des ecritures,” Opt. Commun. 11, 373–377 (1974).
[CrossRef]

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

G. Häusler, N. Streibl, “Optical compensation of geometrical distortion by deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Opt. Eng. (1)

G. E. Lohman, A. W. Lohmann, “Optical interconnection network utilizing diffraction gratings,” Opt. Eng. 27, 893–900 (1988).

Opt. Lett. (1)

Optik (1)

G. E. Lohman, K.-H. Brenner, “Space-variance in optical computing systems,” Optik 89, 123–124 (1992).

Prog. Opt. (1)

D. Casasent, D. Psaltis, “Deformation invariant, space variant optical pattern recognition,” Prog. Opt. 16, 289–356 (1978).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

General optical system for coordinate transformation. The dashed lens appears only in some of the mapping methods. In these cases, the focal length of the lens is f = z/2.

Fig. 2
Fig. 2

Optical perfect-shuffle stage.

Fig. 3
Fig. 3

Cost versus number of pixels for the multifacet architecture (solid curve), and the multistage architecture (the dashed, dotted, and dashed–dotted curves correspond to C2/C1, = 1, 5, and 25, respectively).

Tables (2)

Tables Icon

Table 1 Comparison of the Main Optical Coordinate Transformation Methods

Tables Icon

Table 2 Comparison of Optical Interconnection Architectures

Equations (9)

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

N = D λ f # .
S W = ( D λ f # ) 2 N ,
z = 2 D ,
N = D λ 2 D λ .
O i = j M i j I j
12 λ f # 2 N 1 / 2 ,
36 λ f # 2 N 1 / 2 log 2 N .
C 1 S W = C 1 N 2 ,
C 2 S W = C 2 4 N .

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