Abstract

A technique to determine a real nonnegative function representing the transmittance of a synthesized hologram is described. The technique uses the positions of the samples in the synthesized hologram to record the phase information of a complex wavefront. Synthesized holograms are displayed on a flying spot scanner and recorded on film. The transmittance of the synthesized hologram is quantized into 256 levels because of a hardware limitation of the scanner.

© 1970 Optical Society of America

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References

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  1. A. Kozma, D. L. Kelly, Appl. Opt. 4, 387 (1965); B. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966); J. P. Waters, Appl. Phys. Lett. 9, 405 (1966); J. J. Burch, Proc. IEEE 55, 5591 (1967); A. J. Meyer, R. Hickling, J. Opt. Soc. Amer. 57, 1388 (1967); P. M. Hirsch, L. B. Lesem, J. A. Jordan, J. Opt. Soc. Amer. 57, 1406A (1967); J. P. Waters, J. Opt. Soc. Amer. 57, 563A (1967); B. Brown, J. Opt. Soc. Amer. 58, 729A (1968); R. Hickling, J. Opt. Soc. Amer. 58, 455 (1968); P. M. Hirsch, J. A. Jordan, J. Opt. Soc. Amer. 58, 729A (1968); S. C. Keeton, Proc. IEEE 56, 325 (1968); A. W. Lohmann, D. P. Paris, Appl. Opt. 7, 651 (1968); J. P. Waters, J. Opt. Soc. Amer. 58, 1284 (1968); L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Develop. 13, 150 (1969); B. R. Brown, A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).
  2. E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).
  3. T. S. Huang, B. Prasada, Quarterly Progress Rept. No. 81, Research Laboratory of Electronics, MIT, Cambridge, Mass., 15April, 1966, pp. 199–205.
  4. J. W. Cooley, J. W. Tukey, Math. Comput. 11, 297 (1965).
  5. C. B. Burckhart, E. T. Doherty, Appl. Opt. 7, 651 (1968).

1968

1965

1962

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).

Burckhart, C. B.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 11, 297 (1965).

Doherty, E. T.

Huang, T. S.

T. S. Huang, B. Prasada, Quarterly Progress Rept. No. 81, Research Laboratory of Electronics, MIT, Cambridge, Mass., 15April, 1966, pp. 199–205.

Kelly, D. L.

Kozma, A.

Leith, E. N.

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).

Prasada, B.

T. S. Huang, B. Prasada, Quarterly Progress Rept. No. 81, Research Laboratory of Electronics, MIT, Cambridge, Mass., 15April, 1966, pp. 199–205.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 11, 297 (1965).

Upatnieks, J.

E. N. Leith, J. Upatnieks, J. Opt. Soc. Amer. 52, 1123 (1962).

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Figures (5)

Fig. 1
Fig. 1

Optical system for reconstructing Fourier transform holograms.

Fig. 2
Fig. 2

Computer-generated hologram of the letters MIT.

Fig. 3
Fig. 3

The letters MIT reconstructed from the hologram in Fig. 2.

Fig. 4
Fig. 4

Original continuous-tone picture.

Fig. 5
Fig. 5

Images of the picture shown in Fig. 4 reconstructed from a synthesized hologram.

Equations (16)

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H ( u , υ ) = | A exp ( j 2 π a u ) + F ( u , υ ) | 2 = A 2 + | F ( u , υ ) | 2 + A F ( u , υ ) exp ( j 2 π a u ) + A F * ( u , υ ) exp ( j 2 a u ) ,
H ( u , υ ) = B + F ( u , υ ) exp ( + j 2 π a u ) + F * ( u , υ ) exp ( j 2 π a u )
H ( u , υ ) = 2 | F ( u , υ ) | + F ( u , υ ) exp ( + j 2 π a u ) + F * ( u , υ ) exp ( j 2 π a u ) .
h ( x , y ) = H ( u , υ ) exp [ j 2 π ( x u + y υ ) / λ f ] d u d υ ,
H ( u , υ ) = H * ( u , υ ) H ( u , υ ) > 0 f ( x , y ) = h ( x W , y ) × M ( x , y ) .
| F ( u , υ ) | exp [ j ϕ ( u , υ ) ] = F 1 ( u , υ ) F 3 ( u , υ ) + j F 2 ( u , υ ) j F 4 ( u , υ ) ,
F 1 ( u , υ ) = { | F ( u , υ ) | cos ϕ ( u , υ ) if cos ϕ ( u , υ ) > 0 0 otherwise , F 3 ( u , υ ) = F 1 ( u , υ ) | F ( u , υ ) | cos ϕ ( u , υ ) , F 2 ( u , υ ) = { | F ( u , υ ) | sin ϕ ( u , υ ) if sin ϕ ( u , υ ) > 0 0 otherwise , F 4 ( u , υ ) = F 2 ( u , υ ) | F ( u , υ ) | sin ϕ ( u , υ ) .
G s ( u , υ ) = G ( u , υ ) × m = n = δ ( u n / W + b , υ m / W ) ,
g s ( x , y ) = n = m = t n ( x n W , y m W ) ,
H ( u , υ ) = m = n = k = 1 4 F k ( u , υ ) × δ [ u ( 4 n k + 1 ) / 4 W , υ m / W ] .
h ( x , y ) = m = n = k = 1 4 f k ( x n W , y m W ) exp [ j n ( k 1 ) π / 4 ] = m = n = T n ( x n W , u m W ) .
T n ( x , y ) = i = 1 4 f i ( x , y ) exp [ j n ( i 1 ) π / 4 ] .
F ( k Δ u / 4 , k Δ υ ) = m = ( N / 2 ) ( N / 2 ) 1 n = ( N / 2 ) ( N / 2 ) 1 f ( m Δ x , n Δ y ) × exp [ j 2 π ( m k 4 + n k ) / N ]
Δ x × Δ u = Δ y × Δ υ = 1 / N , k = 2 N , 2 N + 1 , , 2 N 1
k = N / 2 , N / 2 + 1 , , N / 2 1.
H { ( 4 m + k 1 2 N ) ( Δ u / 4 ) , [ n ( N / 2 ) ] Δ υ } = { R e ( ( j ) k 1 F { ( 4 m + k 1 2 N ) ( Δ u / 4 ) , [ n ( N / 2 ) ] Δ υ } ) if H { ( 4 m + k 1 2 N ) ( Δ u / 4 ) , [ n ( N / 2 ) ] Δ υ } > 0 0 otherwise

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