Abstract

To accommodate the available spatial light modulators and to achieve the maximum possible storage capacity of a phase-code-multiplexed holographic memory, it is often necessary to generate an orthogonal phase code whose length is not a power of 2. We describe an algorithm for the generation of basic Hadamard matrices of order M = 4p (with p an odd number) and show that Hadamard matrices of order N = 4t (with t a positive integer) can be constructed by combination of the algorithms presented with the tensor-product extension method.

© 1996 Optical Society of America

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References

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  1. J. F. Heanue, M. C. Bashaw, L. Hesselink, Science 265, 749 (1994).
    [Crossref] [PubMed]
  2. D. Psaltis, F. Mok, Sci. Am. 273, 70 (1995).
    [Crossref]
  3. J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
    [Crossref]
  4. G. A. Rakuljic, V. Leyva, A. Yariv, Opt. Lett. 17, 1471 (1992).
    [Crossref] [PubMed]
  5. F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  8. X. Yang, Z.-H. Gu, Opt. Eng. 35, 408 (1996).
  9. K. Curtis, D. Psaltis, J. Opt, Soc. Am. A 10, 2547 (1993).
    [Crossref]
  10. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), pp. 136–143.
  11. R. E. A. C. Paley, J. Math, Phys. (Cambridge, Mass.) 12, 311 (1933).
  12. J. Williamson, Duke Math. J. 11, 65 (1944).
    [Crossref]
  13. S. W. Golomb, L. D. Baumert, Am. Math. Mon. 70, 27 (1963).
  14. C. V. Eynden, Elementary Number Theory (Random House, New York, 1987), pp. 238–240.

1996 (1)

X. Yang, Z.-H. Gu, Opt. Eng. 35, 408 (1996).

1995 (2)

D. Psaltis, F. Mok, Sci. Am. 273, 70 (1995).
[Crossref]

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

1994 (2)

J. F. Heanue, M. C. Bashaw, L. Hesselink, Science 265, 749 (1994).
[Crossref] [PubMed]

C. Alves, G. Pauliat, G. Roosen, Opt. Lett. 19, 1894 (1994).
[Crossref] [PubMed]

1993 (1)

K. Curtis, D. Psaltis, J. Opt, Soc. Am. A 10, 2547 (1993).
[Crossref]

1992 (2)

1991 (1)

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[Crossref]

1963 (1)

S. W. Golomb, L. D. Baumert, Am. Math. Mon. 70, 27 (1963).

1944 (1)

J. Williamson, Duke Math. J. 11, 65 (1944).
[Crossref]

1933 (1)

R. E. A. C. Paley, J. Math, Phys. (Cambridge, Mass.) 12, 311 (1933).

Alves, C.

Bashaw, M. C.

J. F. Heanue, M. C. Bashaw, L. Hesselink, Science 265, 749 (1994).
[Crossref] [PubMed]

Baumert, L. D.

S. W. Golomb, L. D. Baumert, Am. Math. Mon. 70, 27 (1963).

Chang, T. Y.

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

Christian, W.

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

Curtis, K.

K. Curtis, D. Psaltis, J. Opt, Soc. Am. A 10, 2547 (1993).
[Crossref]

Denz, C.

Eynden, C. V.

C. V. Eynden, Elementary Number Theory (Random House, New York, 1987), pp. 238–240.

Golomb, S. W.

S. W. Golomb, L. D. Baumert, Am. Math. Mon. 70, 27 (1963).

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), pp. 136–143.

Gu, Z.-H.

X. Yang, Z.-H. Gu, Opt. Eng. 35, 408 (1996).

Heanue, J. F.

J. F. Heanue, M. C. Bashaw, L. Hesselink, Science 265, 749 (1994).
[Crossref] [PubMed]

Hesselink, L.

J. F. Heanue, M. C. Bashaw, L. Hesselink, Science 265, 749 (1994).
[Crossref] [PubMed]

Hong, J.

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

Leyva, V.

Math, J.

R. E. A. C. Paley, J. Math, Phys. (Cambridge, Mass.) 12, 311 (1933).

Mayers, A. W.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[Crossref]

McMichael, I.

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

Mok, F.

D. Psaltis, F. Mok, Sci. Am. 273, 70 (1995).
[Crossref]

Opt, J.

K. Curtis, D. Psaltis, J. Opt, Soc. Am. A 10, 2547 (1993).
[Crossref]

Paley, R. E. A. C.

R. E. A. C. Paley, J. Math, Phys. (Cambridge, Mass.) 12, 311 (1933).

Pauliat, G.

Peak, E. G.

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

Psaltis, D.

D. Psaltis, F. Mok, Sci. Am. 273, 70 (1995).
[Crossref]

K. Curtis, D. Psaltis, J. Opt, Soc. Am. A 10, 2547 (1993).
[Crossref]

Rajan, S.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[Crossref]

Rakuljic, G. A.

Roosen, G.

Tschudi, T.

Williamson, J.

J. Williamson, Duke Math. J. 11, 65 (1944).
[Crossref]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), pp. 136–143.

Wu, S.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[Crossref]

Yang, X.

X. Yang, Z.-H. Gu, Opt. Eng. 35, 408 (1996).

Yariv, A.

Yu, F. T. S.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[Crossref]

Am. Math. Mon. (1)

S. W. Golomb, L. D. Baumert, Am. Math. Mon. 70, 27 (1963).

Appl. Opt. (1)

Duke Math. J. (1)

J. Williamson, Duke Math. J. 11, 65 (1944).
[Crossref]

Opt. Commun. (1)

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[Crossref]

Opt. Eng. (2)

X. Yang, Z.-H. Gu, Opt. Eng. 35, 408 (1996).

J. Hong, I. McMichael, T. Y. Chang, W. Christian, E. G. Peak, Opt. Eng. 34, 2193 (1995).
[Crossref]

Opt. Lett. (2)

Phys. (1)

R. E. A. C. Paley, J. Math, Phys. (Cambridge, Mass.) 12, 311 (1933).

Sci. Am. (1)

D. Psaltis, F. Mok, Sci. Am. 273, 70 (1995).
[Crossref]

Science (1)

J. F. Heanue, M. C. Bashaw, L. Hesselink, Science 265, 749 (1994).
[Crossref] [PubMed]

Soc. Am. A (1)

K. Curtis, D. Psaltis, J. Opt, Soc. Am. A 10, 2547 (1993).
[Crossref]

Other (2)

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993), pp. 136–143.

C. V. Eynden, Elementary Number Theory (Random House, New York, 1987), pp. 238–240.

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Figures (2)

Fig. 1
Fig. 1

Flow chart for the computer search of H-matrices of order M = 4p. Init., initial; Pro., product; Decom., decomposition.

Fig. 2
Fig. 2

Four matrices A, B, C, and D that generate a H-matrix of order 116.

Tables (2)

Tables Icon

Table 1 Signs of the Eigenelements of a H-Matrix of Order 116

Tables Icon

Table 2 Signs of the Eigenelements of Two H-Matrices of Order 92

Equations (11)

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

[ 3 4 5 6 ] * [ 2 2 3 0 3 0 1 2 0 ] = [ 6 8 6 8 9 12 10 12 10 12 15 18 0 0 9 12 0 0 0 0 15 18 0 0 3 4 6 8 0 0 5 6 10 12 0 0 ] .
M = 4 p ,
H = [ A B C D B A D C C D A B D C B A ] .
α i j = ± 1 α i j = α ( i + 1 ) ( j + 1 ) ( α = a , b , c , or d ) ,
α 0 = 1 , α i = α p i ( α = a , b , c , or d ; i = 1 , 2 , 3 , , p 1 2 ) .
A 2 + B 2 + C 2 + D 2 = 4 p I ,
i = 0 p 1 ( a i a i + r + b i b i + r + c i c i + r + d i d i + r ) = 0 , ( r = 1 , 2 , , p 1 2 ) ,
( i = 0 p 1 a i ) 2 + ( i = 0 p 1 b i ) 2 + ( i = 0 p 1 c i ) 2 + ( i = 0 p 1 d i ) 2 = 4 p .
[ 1 + 2 i = 1 ( p 1 ) / 2 a i ] 2 + [ 1 + i = 1 ( p 1 ) / 2 b i ] + [ 1 + i = 1 ( p 1 ) / 2 c i ] 2 + [ 1 + i = 1 ( p 1 ) / 2 d i ] 2 = q a 2 + q b 2 + q c 2 + q d 2 .
α 1 + α 2 + + α ( p 1 ) / 2 = 1 ± q α 2 , ( α = a , b , c , or d ) .
α 1 + α 2 + + α 14 = 1 + 9 2 = 4 .

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