Abstract

The positions and width of the fringes in a binary synthetic hologram are determined by the points at (x,y) that satisfy −q/2 ≤ x/T + ϕ (x,y)/2π + nq/2, where ϕ(x,y) is the phase variation of the wave front T is the grating period, n is an integer, and q is a constant determining the fringe width. A methed for finding an exact solution to this inequality is presented. The feasibility of the procedure has been tested by making binary synthetic holograms. Experimental results are presented, and extensions of the method are discussed.

© 1974 Optical Society of America

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References

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  1. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [CrossRef] [PubMed]
  2. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 739 (1967).
  3. A. Kozma, D. L. Kelly, Appl. Opt. 4, 3871 (1965).
    [CrossRef]
  4. A. W. Lohmann, D. P. Paris, Appl. Opt. 9, 1567 (1967).
    [CrossRef]
  5. J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1969).
    [CrossRef]
  6. N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Application to Computer Technology (McGraw-Hill, New York, 1967), p. 16.
  7. O. Bryngdahl, J. Opt. Soc. Am. 63, 1098 (1973).
    [CrossRef]
  8. Y. Ichioka, A. W. Lohmann, Appl. Opt. 11, 2597 (1972).
    [CrossRef] [PubMed]

1973 (1)

1972 (1)

1969 (1)

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1969).
[CrossRef]

1967 (2)

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 739 (1967).

A. W. Lohmann, D. P. Paris, Appl. Opt. 9, 1567 (1967).
[CrossRef]

1966 (1)

1965 (1)

A. Kozma, D. L. Kelly, Appl. Opt. 4, 3871 (1965).
[CrossRef]

Brown, B. R.

Bryngdahl, O.

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1969).
[CrossRef]

Ichioka, Y.

Kelly, D. L.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 3871 (1965).
[CrossRef]

Kozma, A.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 3871 (1965).
[CrossRef]

Lohmann, A. W.

Y. Ichioka, A. W. Lohmann, Appl. Opt. 11, 2597 (1972).
[CrossRef] [PubMed]

A. W. Lohmann, D. P. Paris, Appl. Opt. 9, 1567 (1967).
[CrossRef]

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 739 (1967).

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

Paris, D. P.

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 739 (1967).

A. W. Lohmann, D. P. Paris, Appl. Opt. 9, 1567 (1967).
[CrossRef]

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1969).
[CrossRef]

Szabo, N. S.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Application to Computer Technology (McGraw-Hill, New York, 1967), p. 16.

Tanaka, R. I.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Application to Computer Technology (McGraw-Hill, New York, 1967), p. 16.

Appl. Opt. (5)

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

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 739 (1967).

A. Kozma, D. L. Kelly, Appl. Opt. 4, 3871 (1965).
[CrossRef]

A. W. Lohmann, D. P. Paris, Appl. Opt. 9, 1567 (1967).
[CrossRef]

Y. Ichioka, A. W. Lohmann, Appl. Opt. 11, 2597 (1972).
[CrossRef] [PubMed]

IBM J. Res. Dev. (1)

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (1)

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Application to Computer Technology (McGraw-Hill, New York, 1967), p. 16.

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Figures (5)

Fig. 1
Fig. 1

Ideal limiter for generating a binary hologram. The parameter C is equal to cosπq, where 0 ≤ q ≤ 1/2. For a wavefront with amplitude A(x,y), q is space-variant and is given by [sin−1A(x,y)]/π. The maximum value of A(x,y) is normalized to 1.

Fig. 2
Fig. 2

Spherical wavefront. (a) Center portion of the synthetic binary hologram. (b) Interferogram of the spherical wavefront recorded using the set up shown in Fig. 3. (c) Fraunhofer diffraction pattern of the hologram.

Fig. 3
Fig. 3

Optical setup for obtaining an interferogram of the wavefront produced by the hologram. H, hologram; L, lens having focal length f; F, spatial filter; I, interferogram.

Fig. 4
Fig. 4

Conical wavefront. (a) Center portion of the synthetic binary hologram. (b) Interferogram of the conical wavefront recorded using the set up shown in Fig. 3. (c) Fraunhofer diffraction pattern of the hologram.

Fig. 5
Fig. 5

Helical wavefront. (a) Center portion of the synthetic binary hologram. (b) Interferogram of the helical wavefront recorded using the set up shown in Fig. 3. (c) Fraunhofer diffraction pattern of the hologram.

Equations (39)

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( x / Δ x ) - [ ϕ ( x , y ) / 2 π ] = n ,
x = n Δ x + { [ Δ x ϕ ( n Δ x , m Δ y ) ] / 2 π } .
h ( x , y ) = | m = 0 N - 1 1 N     exp [ j m ϕ ( x , y ) ] | 2 .
h ( x , y ) = ( sin N ϕ ( x , y ) / 2 N     sin ϕ ( x , y ) / 2 ) 2 .
ϕ ( x , y ) = 2 π n ,
h ( x , y ) = m = - sin ( m π q ) m π             exp [ j m ϕ ( x , y ) ] .
cos ϕ ( x , y ) cos π q ,
- ( q / 2 ) [ ϕ ( x , y ) / 2 π ] + n q / 2 ,
- ( q / 2 ) x T + [ ϕ ( x , y ) / 2 π ] + n q 2 .
ν x ( x , y ) = 1 T + 1 2 π ϕ ( x , y ) x .
B x = 1 π Max x , y | ϕ ( x , y ) x | ,
B y = 1 π Max x , y | ϕ ( x , y ) y | .
1 / T 1.5 B x ,
- M q 2 k x + M ϕ ( T k x / M , 2 T k y ) 2 π + M n M q 2 .
Mod ( - M q 2 ) Mod [ k x + M ϕ ( T k x / M , 2 T k y ) 2 π + M n ] Mod ( M q 2 ) .
( 1 ) Mod ( a + M n ) = Mod ( a ) , n = integer ; and ( 2 ) Mod ( - a ) = Mod ( M - a ) , for a < M .
Mod [ k x + M ϕ ( T k x / M , 2 T k y ) 2 π ] Mod [ M q 2 ] ,
Mod [ k x + M ϕ ( T k x / M , 2 T k y ) 2 π ] Mod [ M - M q 2 ] .
ϕ ( x , y ) = π ( x 2 + y 2 ) / λ F .
1 2 π ϕ ( x , y ) x = x λ F ,
T = λ F / 2 D .
ϕ ( T k x / M , 2 T k y ) = ( π T 2 / λ F ) ( k x 2 / M 2 + 4 k y 2 ) .
T 2 / λ F = T / 2 D = 1 / 2 N ,
ϕ ( T k x M , 2 T k y ) = π ( k x 2 + 4 M 2 k y 2 ) 2 M 2 N .
Mod [ k x + ( k x 2 + 4 M 2 k y 2 ) 4 M N ] = 0 ,
ϕ ( x , y ) = 2 π r / r 0 ,
Mod { k x + [ ( k x 2 + 4 M 2 k y 2 ) 1 / 2 4 ] } = 0.
ν x ( r , θ ) = - sin θ / r θ 0 .
Mod [ k x + 3 M tan - 1 ( k y / k x ) ] = 0.
( k x 2 + 4 M 2 k y 2 ) 1 / 2 12 M .
- q ( x , y ) 2 x T + ϕ ( x , y ) 2 π + n q ( x , y ) 2 ,
q ( x , y ) = [ sin - 1 A ( x , y ) ] / π .
h ( x ) = n = 0 N δ ( x - n Δ x + S n ) ,
f ( x ) = g 0 ( x ) exp [ j 2 π x / Δ x ] ,
g 0 ( x ) = sin π B x / π x .
w ( x ) = n = 0 N g 0 ( x - n Δ x + S n )             exp [ j 2 π ( x - n Δ x + S n ) / Δ x ] = exp [ j 2 π x / Δ x ] n = 0 N exp [ j 2 π S n / Δ x ] g 0 ( x - n Δ x + S n ) .
g 0 ( x - n Δ x + S n ) = k = 0 ( Δ x B ) k [ ϕ ( n Δ x ) / 2 π ] k × g k ( x - n Δ x ) .
g k ( x ) = 1 k ! B k d k d x k g 0 ( x ) .
w ( x ) = exp [ j 2 π x / Δ x ] k = 0 ( Δ x B ) k n = 1 N [ ϕ ( n Δ x ) / 2 π ] k × exp [ j ϕ ( n Δ x ) ] g k ( x - n Δ x ) .

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