Abstract

We present here a new family of diffusers suitable for use in holography. We first exhibit a method for obtaining mathematical descriptions of diffusers that give nearly uniform amplitude at the hologram before insertion of an object. From these diffusers we select a few that redundantly map information into the hologram. Finally, we present a method of comparing the performance of different diffusers using a computer simulation technique that has a simple experimental analog.

© 1973 Optical Society of America

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  1. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 224.
  3. B. Julesz, K. Pennington, J. Opt. Soc. Am. 55, 604 (1965).
  4. B. M. Oliver, Proc. IEEE 51, 220 (1963).
    [CrossRef]
  5. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [CrossRef]
  6. L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
  7. S. Lowenthal, H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
    [CrossRef]
  8. L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  9. D. Gabor, SPIE Seminar Proc. 25, 129 (1971).
    [CrossRef]
  10. D. Gabor, in Second U.S.-Japan Seminar on Information Processing by Holography, 1969. Applications of Holography, E. S. Barrakette et al., Eds. (Plenum, New York, 1971).
  11. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [CrossRef] [PubMed]
  12. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1139 (1967).
    [CrossRef] [PubMed]
  13. H. Dammann, Phys. Lett. 29A, 301 (1969).
  14. B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
    [CrossRef]
  15. E. N. Leith, J. Upatnieks, Appl. Opt. 7, 2085 (1968).
    [CrossRef] [PubMed]
  16. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
    [CrossRef] [PubMed]
  17. D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
    [CrossRef]
  18. R. H. Katyl, Appl. Opt. 11, 198 (1972).
    [CrossRef] [PubMed]
  19. M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
    [CrossRef]
  20. J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
    [CrossRef]
  21. R. C. Heimiller, IRE Trans. Inform. Theory IT-7, 254 (1961).
    [CrossRef]
  22. R. L. Frank, S. A. Zadoff, IRE Trans. Inform. Theory IT-8, 381 (1962).
    [CrossRef]
  23. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 405.
  24. D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
    [CrossRef] [PubMed]
  25. D. Gabor, Proc. Roy. Soc. A197, 454 (1949).
  26. L. Mertz, N. O. Young, in Proc. Conf. Optical Instruments and TechniquesK. J. Habell, Ed. (Wiley, New York, 1963), p. 305.

1972

1971

D. Gabor, SPIE Seminar Proc. 25, 129 (1971).
[CrossRef]

1970

M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
[CrossRef]

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

S. Lowenthal, H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
[CrossRef]

1969

H. Dammann, Phys. Lett. 29A, 301 (1969).

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1968

1967

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1139 (1967).
[CrossRef] [PubMed]

1966

1965

B. Julesz, K. Pennington, J. Opt. Soc. Am. 55, 604 (1965).

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
[CrossRef]

1964

1963

B. M. Oliver, Proc. IEEE 51, 220 (1963).
[CrossRef]

1962

R. L. Frank, S. A. Zadoff, IRE Trans. Inform. Theory IT-8, 381 (1962).
[CrossRef]

1961

R. C. Heimiller, IRE Trans. Inform. Theory IT-7, 254 (1961).
[CrossRef]

1949

D. Gabor, Proc. Roy. Soc. A197, 454 (1949).

Arsenault, H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 405.

Brown, B. R.

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[CrossRef] [PubMed]

Chu, D. C.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Dammann, H.

H. Dammann, Phys. Lett. 29A, 301 (1969).

Enloe, L. H.

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

Frank, R. L.

R. L. Frank, S. A. Zadoff, IRE Trans. Inform. Theory IT-8, 381 (1962).
[CrossRef]

Gabor, D.

D. Gabor, SPIE Seminar Proc. 25, 129 (1971).
[CrossRef]

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

D. Gabor, Proc. Roy. Soc. A197, 454 (1949).

D. Gabor, in Second U.S.-Japan Seminar on Information Processing by Holography, 1969. Applications of Holography, E. S. Barrakette et al., Eds. (Plenum, New York, 1971).

Gerritsen, H. J.

Goldfischer, L. I.

Goodman, J. W.

D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
[CrossRef] [PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 224.

Hannan, W. J.

Heimiller, R. C.

R. C. Heimiller, IRE Trans. Inform. Theory IT-7, 254 (1961).
[CrossRef]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Julesz, B.

B. Julesz, K. Pennington, J. Opt. Soc. Am. 55, 604 (1965).

Katyl, R. H.

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lohmann, A. W.

Lowenthal, S.

Mertz, L.

L. Mertz, N. O. Young, in Proc. Conf. Optical Instruments and TechniquesK. J. Habell, Ed. (Wiley, New York, 1963), p. 305.

Oliver, B. M.

B. M. Oliver, Proc. IEEE 51, 220 (1963).
[CrossRef]

Paris, D. P.

Pennington, K.

B. Julesz, K. Pennington, J. Opt. Soc. Am. 55, 604 (1965).

Ramberg, E. G.

Schroeder, M. R.

M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Upatnieks, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 405.

Young, N. O.

L. Mertz, N. O. Young, in Proc. Conf. Optical Instruments and TechniquesK. J. Habell, Ed. (Wiley, New York, 1963), p. 305.

Zadoff, S. A.

R. L. Frank, S. A. Zadoff, IRE Trans. Inform. Theory IT-8, 381 (1962).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

IBM J. Res. Dev.

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

IEEE Trans. Inform. Theory

M. R. Schroeder, IEEE Trans. Inform. Theory IT-16, 85 (1970).
[CrossRef]

IRE Trans. Inform. Theory

R. C. Heimiller, IRE Trans. Inform. Theory IT-7, 254 (1961).
[CrossRef]

R. L. Frank, S. A. Zadoff, IRE Trans. Inform. Theory IT-8, 381 (1962).
[CrossRef]

J. Opt. Soc. Am.

Math. Comput.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Phys. Lett.

H. Dammann, Phys. Lett. 29A, 301 (1969).

Proc. IEEE

B. M. Oliver, Proc. IEEE 51, 220 (1963).
[CrossRef]

Proc. Roy. Soc.

D. Gabor, Proc. Roy. Soc. A197, 454 (1949).

SPIE Seminar Proc.

D. Gabor, SPIE Seminar Proc. 25, 129 (1971).
[CrossRef]

Other

D. Gabor, in Second U.S.-Japan Seminar on Information Processing by Holography, 1969. Applications of Holography, E. S. Barrakette et al., Eds. (Plenum, New York, 1971).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 224.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 405.

L. Mertz, N. O. Young, in Proc. Conf. Optical Instruments and TechniquesK. J. Habell, Ed. (Wiley, New York, 1963), p. 305.

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Figures (8)

Fig. 1
Fig. 1

Hologram recording and reconstruction geometries.

Fig. 2
Fig. 2

Phase functions of three diffusers

Fig. 3
Fig. 3

(a). Computer generated binary image hologram of nonredundant ideal diffuser; (b). diffuser wave at Fourier hologram; (c). diffuser, partially blocked; (d). wave from partially blocked diffuser (c) at hologram.

Fig. 4
Fig. 4

Merit curves for three unblocked sixteen-element diffusers.

Fig. 5
Fig. 5

Merit curves for sixteen-element diffusers with four elements blocked.

Fig. 6
Fig. 6

Merit curves for sixteen-element diffusers with four elements blocked.

Fig. 7
Fig. 7

Merit curves for sixteen-element diffusers with four elements blocked.

Fig. 8
Fig. 8

Merit curves for sixteen-element diffusers with four elements blocked.

Equations (26)

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H j k = M - 1 / 2 N - 1 / 2 m = 0 M - 1 n = 0 N - 1 G m n exp [ - 2 π i ( j m / M + k n / N ) ] .
G m n = 1             and             H j k = 1.
H j k = A - 1 / 2 B - 1 / 2 p = 0 A - 1 q = 0 B - 1 T j k p q exp [ - 2 π i ( j p / M + k q / N ) ] ,
T j k p q = a - 1 / 2 b - 1 / 2 α = 0 a - 1 β = 0 b - 1 G ( α A + p ; β B + q ) exp [ - 2 π i ( j α / a + k β / b ) ] .
T p q ( γ a + σ ; δ b + τ ) = T σ τ p q ,
T σ τ p q = a - 1 / 2 b - 1 / 2 α = 0 a - 1 β = 0 b - 1 G ( α A + p ; β B + q ) exp [ - 2 π i ( σ α / a + τ β / b ) ] ;
H ( γ a + σ ; δ b + τ ) = A - 1 / 2 B - 1 / 2 p = 0 A - 1 q = 0 B - 1 T σ τ p q exp { - 2 π i [ ( γ a + σ ) p / M + ( δ b + τ ) q / N ] } ,
H ( γ a + σ ; δ b + τ ) = A - 1 / 2 B - 1 / 2 p = 0 A - 1 q = 0 B - 1 S σ τ p q exp { - 2 π i [ ( γ p / A + δ q / B ) ] .
H ( γ M 1 / 2 + σ ; δ N 1 / 2 + τ ) = M - 1 / 4 N - 1 / 4 p = 0 M 1 / 2 - 1 q = 0 N 1 / 2 - 1 S σ τ p q exp [ - 2 π i ( γ p / M 1 / 2 + δ q / N 1 / 2 ) ]
G ( α M 1 / 2 + p ; β N 1 / 2 + q ) = M - 1 / 4 N - 1 / 4 σ = 0 M 1 / 2 - 1 τ = 0 N 1 / 2 - 1 T σ τ p q exp [ + 2 π i ( α σ / M 1 / 2 + β τ / N 1 / 2 ) ]
S σ τ p q = T σ τ p q exp [ - 2 π i ( σ p / M + τ q / N ) ] .
T σ τ p q = ( M N ) - 1 / 4 δ σ τ η ( p , q ) e i ϕ p q ,
δ σ τ p q = { 1 if ( σ , τ ) = ( p , q ) 0 otherwise ,
T σ p = M - 1 / 4 δ σ p ; note that η ( p ) = p .
H ( γ M 1 / 2 + σ ) = exp ( - 2 π i σ 2 / M ) exp ( - 2 π i γ σ / M 1 / 2 ) ,
G ( α M 1 / 2 + p ) = exp ( + 2 π i α p / M 1 / 2 ) .
G m = exp ( - 2 π i m 2 / M )
G ˇ m = G m , 0 m M - 1 , G ˇ m = 0 , m m k M - 1 ,
H ˇ j = ( k M ) - 1 / 2 m = 0 k M - 1 G ˇ m exp ( - 2 π i j m / k M ) .
Z j ( W c ) = ( 1 / W ) p = 0 W - 1 H ˇ j + p 2 ,
Z ( W c ) ¯ = ( 1 / k M ) p = 0 k M - 1 Z j ( W c ) 2 .
σ 2 ( W c ) = ( 1 / k M ) j = 0 k M - 1 [ Z j ( W c ) - Z ( W c ) ¯ ] 2 ,
σ R ( W c ) = [ σ ( W c ) ] / [ Z ( W c ) ¯ ] ,
M R ( W c ) = { Maximum [ Z j ( W c ) - Z ( W c ) ¯ ] } / [ Z ( W c ) ¯ ] ,
H j = m = 0 M - 1 G m exp [ - 2 π i ( m / a - j / b ) 2 ] ,
exp ( - i π j 2 / b 2 ) H j = m = 0 M - 1 G m exp ( - 2 π i m j / M ) = H j .

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