Abstract

A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs. This can be accomplished with passive components, such as 2 × 2 couplers, and propagation delays. The Hadamard transform can be performed similarly.

© 1987 Optical Society of America

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References

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  1. M. E. Marhic, Opt. Lett. 9, 368 (1984).
    [CrossRef] [PubMed]
  2. M. E. Marhic, in Proceedings of FOC/LAN ’84 (Information Gatekeepers, Boston, Mass., 1984).
  3. T. Hermes, J. Saniter, F. Schmidt, Nachrichten-techn. Z. 37, 636 (1984).
  4. H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
    [CrossRef]
  5. C. C. Wang, W. K. Burns, C. A. Villaruel, Opt. Lett. 10, 49 (1985).
    [CrossRef] [PubMed]
  6. D. B. Mortimore, Electron. Lett. 21, 502 (1985).
    [CrossRef]
  7. H. Stone, IEEE Trans. Comput. C-20, 153 (1971).
    [CrossRef]
  8. M. C. Pease, J. ACM 15, 252 (1968).
    [CrossRef]
  9. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), Chap. 6.
  10. R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1977), p. 96.
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1985 (2)

1984 (3)

M. E. Marhic, Opt. Lett. 9, 368 (1984).
[CrossRef] [PubMed]

T. Hermes, J. Saniter, F. Schmidt, Nachrichten-techn. Z. 37, 636 (1984).

H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
[CrossRef]

1971 (1)

H. Stone, IEEE Trans. Comput. C-20, 153 (1971).
[CrossRef]

1968 (1)

M. C. Pease, J. ACM 15, 252 (1968).
[CrossRef]

Burns, W. K.

Döldissen, W.

H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
[CrossRef]

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), Chap. 6.

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1977), p. 96.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Heidrich, H.

H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
[CrossRef]

Hermes, T.

T. Hermes, J. Saniter, F. Schmidt, Nachrichten-techn. Z. 37, 636 (1984).

Hoffman, D.

H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
[CrossRef]

Klug, M.

H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
[CrossRef]

Marhic, M. E.

M. E. Marhic, Opt. Lett. 9, 368 (1984).
[CrossRef] [PubMed]

M. E. Marhic, in Proceedings of FOC/LAN ’84 (Information Gatekeepers, Boston, Mass., 1984).

Mortimore, D. B.

D. B. Mortimore, Electron. Lett. 21, 502 (1985).
[CrossRef]

Pease, M. C.

M. C. Pease, J. ACM 15, 252 (1968).
[CrossRef]

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), Chap. 6.

Saniter, J.

T. Hermes, J. Saniter, F. Schmidt, Nachrichten-techn. Z. 37, 636 (1984).

Schmidt, F.

T. Hermes, J. Saniter, F. Schmidt, Nachrichten-techn. Z. 37, 636 (1984).

Stone, H.

H. Stone, IEEE Trans. Comput. C-20, 153 (1971).
[CrossRef]

Villaruel, C. A.

Wang, C. C.

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1977), p. 96.

Electron. Lett. (2)

H. Heidrich, D. Hoffman, W. Döldissen, M. Klug, Electron. Lett. 20, 1058 (1984); W. Döldissen, H. Heidrich, D. Hoffmann, H. Dbhler, M. Klug, in Proceedings of ECIO ’85 (Springer-Verlag, Berlin, 1985), p. 225.
[CrossRef]

D. B. Mortimore, Electron. Lett. 21, 502 (1985).
[CrossRef]

IEEE Trans. Comput. (1)

H. Stone, IEEE Trans. Comput. C-20, 153 (1971).
[CrossRef]

J. ACM (1)

M. C. Pease, J. ACM 15, 252 (1968).
[CrossRef]

Nachrichten-techn. Z. (1)

T. Hermes, J. Saniter, F. Schmidt, Nachrichten-techn. Z. 37, 636 (1984).

Opt. Lett. (2)

Other (4)

M. E. Marhic, in Proceedings of FOC/LAN ’84 (Information Gatekeepers, Boston, Mass., 1984).

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1975), Chap. 6.

R. C. Gonzalez, P. Wintz, Digital Image Processing (Addison-Wesley, Reading, Mass., 1977), p. 96.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (3)

Fig. 1
Fig. 1

An N × N star coupler.

Fig. 2
Fig. 2

A 2N × 2N star coupler.

Fig. 3
Fig. 3

Detailed diagram of a 4 × 4 Fourier-transform network, showing location of required phase shifters.

Equations (13)

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b j = 1 N i = 1 N j i a i ,
j i = 1.
j = 1 N b j b j * = i = 1 N a i a i * ,
1 N k = 1 N l = 1 N a k a l * j = 1 N j k j l * = i = 1 N a i a i * .
j = 1 N j k j l * = N δ k , l ,
j k = exp ( i 2 π j k / N ) .
b j = 1 N k = 1 N exp ( i 2 π j k / N ) a k ,
b k = 1 N j = 1 N u j k a j ,
b N + k = 1 N j = 1 N u j k a N + j ,
β 1 = 1 2 ( - α 1 + α 2 ) , β 2 = 1 2 ( - α 1 + α 2 ) .
c k = 1 v k b k + b N + k , c N + k = 1 v k b k + b N + k ,
c l = m = 1 2 N v l m a m ,
a p = a 2 p - 1 for p N , a p = a 2 ( p - N ) for N + 1 p 2 N ,

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