Abstract

Generation of holograms by computer allows the possibility of better controlling the hologram formation process and of displaying a synthesized image in the case where the object does not exist physically. However, limitations of equipment used to plot the hologram can cause degradation of the reconstructed image. We examine these degradations for the binary Fourier transform hologram and present a method by which the plotting procedure may be designed so as to yield a most faithful reconstructed image. Experimental results which support the analysis are included.

© 1970 Optical Society of America

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References

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  1. J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1967).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Co., New York, 1968).
  3. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).
  4. L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, New York, 1967), p. 681.
  5. L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Fall Joint Computer Conference (AFIPS, Philadelphia, 1967), p. 41.
  6. A. J. Meyer, R. Hickling, J. Opt. Soc. Amer. 57, 1388 (1967).
    [CrossRef]
  7. L. B. Lesem, P. Hirsch, J. A. Jordan, IBM J. Res. Devel. 13, 150 (1969).
    [CrossRef]
  8. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  9. B. R. Brown, A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).
    [CrossRef]
  10. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill Book Co., New York, 1968), Chap. 4.
  11. J. Katzenelson, IRE Trans. Automatic Control, 7, 58 (1962). See also, D. J. Watts, J. Katzenelson, IEEE Trans. Automatic Control 8, 187 (1963).
    [CrossRef]
  12. A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill Book Co., New York, 1965), Sec. 10-5.

1969 (2)

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

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

1967 (2)

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
[CrossRef] [PubMed]

A. J. Meyer, R. Hickling, J. Opt. Soc. Amer. 57, 1388 (1967).
[CrossRef]

1962 (1)

J. Katzenelson, IRE Trans. Automatic Control, 7, 58 (1962). See also, D. J. Watts, J. Katzenelson, IEEE Trans. Automatic Control 8, 187 (1963).
[CrossRef]

Brown, B. R.

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

DeVelis, J. B.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1967).

Goodman, J. W.

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

Hickling, R.

A. J. Meyer, R. Hickling, J. Opt. Soc. Amer. 57, 1388 (1967).
[CrossRef]

Hirsch, P.

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

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, New York, 1967), p. 681.

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Fall Joint Computer Conference (AFIPS, Philadelphia, 1967), p. 41.

Jordan, J. A.

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

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Fall Joint Computer Conference (AFIPS, Philadelphia, 1967), p. 41.

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, New York, 1967), p. 681.

Katzenelson, J.

J. Katzenelson, IRE Trans. Automatic Control, 7, 58 (1962). See also, D. J. Watts, J. Katzenelson, IEEE Trans. Automatic Control 8, 187 (1963).
[CrossRef]

Lesem, L. B.

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

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Fall Joint Computer Conference (AFIPS, Philadelphia, 1967), p. 41.

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, New York, 1967), p. 681.

Lohmann, A. W.

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

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
[CrossRef] [PubMed]

Meyer, A. J.

A. J. Meyer, R. Hickling, J. Opt. Soc. Amer. 57, 1388 (1967).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill Book Co., New York, 1968), Chap. 4.

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill Book Co., New York, 1965), Sec. 10-5.

Paris, D. P.

Reynolds, G. O.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1967).

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).

Appl. Opt. (1)

IBM J. Res. Devel. (1)

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

IBM J. Res. Develop. (1)

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

IRE Trans. Automatic Control (1)

J. Katzenelson, IRE Trans. Automatic Control, 7, 58 (1962). See also, D. J. Watts, J. Katzenelson, IEEE Trans. Automatic Control 8, 187 (1963).
[CrossRef]

J. Opt. Soc. Amer. (1)

A. J. Meyer, R. Hickling, J. Opt. Soc. Amer. 57, 1388 (1967).
[CrossRef]

Other (7)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill Book Co., New York, 1968), Chap. 4.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1967).

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

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Symposium on Modern Optics (Polytechnic Press, Brooklyn, New York, 1967), p. 681.

L. B. Lesem, P. Hirsch, J. A. Jordan, in Proceedings of the Fall Joint Computer Conference (AFIPS, Philadelphia, 1967), p. 41.

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill Book Co., New York, 1965), Sec. 10-5.

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Figures (13)

Fig. 1
Fig. 1

Cell dimensions.

Fig. 2
Fig. 2

Amplitude and phase quantizing functions.

Fig. 3
Fig. 3

Image error at the point (x, y).

Fig. 4
Fig. 4

Amplitude quantizing error as a function of power spectral density.

Fig. 5
Fig. 5

Simulated image reconstruction: (a) original image; (b) simulation for N = 11; (c) simulation for N = 41; (d) simulation for N = 65.

Fig. 6
Fig. 6

Flow chart for simulating image reconstruction.

Fig. 7
Fig. 7

Optical image reconstructions. (a) Typical binary hologram (N = 17); (b) optical reconstruction for N = 17 (enlarged 3 × approx.); (c) optical reconstruction for N = 47 (enlarged 1.3 × approx.); (d) optical reconstruction for N = 65.

Fig. 8
Fig. 8

Error measurements for deterministic image.

Fig. 9
Fig. 9

Simulation for deterministic pattern.

Fig. 10
Fig. 10

Error measurements for stochastic image.

Fig. 11
Fig. 11

Simulation for stochastic pattern.

Fig. 12
Fig. 12

Sequence of apertures with increasing areas.

Fig. 13
Fig. 13

Error reduction with proper phase design.

Equations (62)

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S = { ( x , y ) : | x | X / 2 , | y | X / 2 } ,
g ( x , y ) = m , n = G m n e j ( 2 π / X ) ( m x + n y ) = m , n = A m n e j ϕ m n e j ( 2 π / X ) ( m x + n y ) ,
G m n = G ( m Δ , n Δ ) , A m n = | G ( m Δ , n Δ ) | ,
ϕ m n = arg [ G ( m Δ , n Δ ) ] .
H m n = H ( m Δ , n Δ ) = { 0 for m , n I ( A m n + N m n ) e j ( ϕ m n + θ m n ) for m , n I ,
m , n I means | m | ( N 1 ) / 2 and | n | ( N 1 ) / 2 , m , n I means | m | > ( N 1 ) / 2 or | n | > ( N 1 ) / 2 .
Q = max m n I A m n / M ,
h ( x , y ) = m , n = H m n e j ( 2 π / X ) ( m x + n y ) = m , n , I ( A m n + N m n ) e j ( ϕ m n + θ m n ) e j ( 2 π / X ) ( m x + n y ) .
g ( x , y ) = m , n , = A m n e j ϕ m n e j ( 2 π / X ) ( m x + n y )
h ( x , y ) = m , n I ( A m n + N m n ) e j ( ϕ m n + θ m n ) e j ( 2 π / X ) ( m x + n y )
Ψ = { 1 X 2 S [ | g ( x , y ) | | h ( x , y ) | ] 2 d x d y } .
e ( x , y ) = g ( x , y ) h ( x , y ) .
( | g | | h | ) 2 = ( g h ) 2 = | e | 2 .
Ψ = ε { 1 X 2 S | e ( x , y ) | 2 d x d y } .
e ( x , y ) = m , n = G m n e j ( 2 π / X ) ( m x + n y ) m , n , = H m n × e j ( 2 π / X ) ( m x + n y ) = m , n = E m n e j ( 2 π / X ) ( m x + n y ) ,
E m n = { G m n for m , n I G m n ( 1 e j θ m n ) + N m n e j ( ϕ m n + θ m n ) for m , n I .
1 X 2 S | e ( x , y ) | 2 d x d y = 1 X 2 S m , n , p , q = E m n E p q * e j ( 2 π / X ) [ ( m p ) x + ( n q ) y ] × d x d y = m , n = | E m n | 2
1 X X / 2 X / 2 e j ( 2 π / X ) ( m p ) x d x = { 1 for m = p 0 for m p ,
ε ( | E m n | 2 ) = { | G m n | 2 for m , n , I | G m n | 2 ε { | 1 e j θ m n | 2 } + G m n ε { N m n e j ( ϕ m n + θ m n ) ( 1 e j θ m n ) } + G m n * ε { N m n e j ( ϕ m n + θ m n ) ( 1 e j θ m n ) } + ε { N m n 2 } for m , n I .
ε { N m n } = 0 , ε { N m n 2 } = σ 2 = Q 2 / 12 ,
ε { | 1 e j θ m n | 2 } = 2 ( 1 sin Ө / 2 Ө / 2 ) ,
Ψ = ε { 1 X 2 S | e ( x , y ) | 2 d x d y } = m , n I | G m n | 2 + 2 ( 1 sin Ө / 2 Ө / 2 ) m , n , I | G m n | 2 + N 2 σ 2 .
ε { 1 X 2 S | e ( x , y ) | 2 d x d y } = m , n I | G m n | 2 + 2 ( 1 sin Ө / 2 Ө / 2 ) m , n , I | G m n | Q / 2 | G m n | 2 + N 0 σ 2 + m , n , I | G m n | Q / 2 | G m n | 2 ,
| h | = | g | [ 1 + ( | e | 2 / | g | 2 ) 2 ( | e | / | g | ) cos γ ] 1 2 ,
( | g | | h | ) 2 | e | 2 cos 2 γ,
Ψ = ε { 1 X 2 S | e ( x , y ) | 2 cos 2 γ ( x , y ) d x d y } .
Ψ = ε { cos 2 γ } ε { 1 X 2 | e ( x , y ) | 2 d x d y } = ε { cos 2 γ } [ m , n I | G m n | 2 + 2 ( 1 sin Ө / 2 Ө / 2 ) m , n I | G m n | Q / 2 | G m n | 2 + N 0 σ 2 + m , n I | G m n | < Q / 2 | G m n | 2 ] .
Ψ = 1 2 ε { 1 X 2 S | e ( x , y ) | 2 d x d y } .
Λ = 1 X 2 S [ | g ( x , y ) | | h ( x , y ) | ] 2 d x d y .
g ( x , y ) = k , i = g 0 ( x + k X , y + i X ) ,
R g g ( Δ x , Δ y ) = ε { g ( x + Δ x , y + Δ y ) g * ( x , y ) } = R g g ( Δ x + X , Δ y + X ) ;
g ( x , y ) = m , n = G m n e j ( 2 π / X ) ( m x + n y ) ,
ε ( G m n ) = { ε [ g ( x , y ) ] for m = n = 0 0 otherwise
ε ( G m n G p q * ) = { β m n if m = p and n = q 0 otherwise .
β m n = 1 X 2 S R g g ( Δ x , Δ y ) e j ( 2 π / X ) ( m Δ x + n Δ y ) d Δ x d Δ y
R g g ( Δ x , Δ y ) = m , n = β m n e j ( 2 π / X ) ( m Δ x + n Δ y ) ,
S g g ( f x , f y ) = R g g ( Δ x , Δ y ) e j 2 π ( f x Δ x + f y Δ y ) d Δ x d Δ y = m , n = β m n δ ( f x m X , f y n X ) ,
Ψ = 1 X 2 S ε [ | e ( x , y ) | 2 ] ε [ cos 2 γ ( x , y ) ] d x d y .
ε { | e ( x , y ) | 2 } = m , n I β m n + 2 ( 1 sin Ө / 2 Ө / 2 ) m , n I β m n + N 2 σ 2 .
Ψ = ε { | e | 2 } 1 X 2 ε { cos 2 γ ( x , y ) } d x d y for | g | | e |
= 1 2 ε { | e | 2 } for | g | | e | and γ uniformly distributed on ( π , π )
= ε { | e | 2 } for | g | > | e | and g real .
ε ( N m n 2 ) σ 0 2 for β m n σ 0 2 β m n for β m n < σ 0 2 .
Ψ = 1 X 2 ε { cos 2 ( x , y ) } d x d y · [ m , n I β m n + 2 ( 1 sin Ө / 2 Ө / 2 ) m , n ε I β m n + N 0 σ 0 2 + m , n I β m n < σ 0 2 β m n ] ,
W { G ( u , υ ) } = k , i = G ( u + k Ω , υ + i Ω ) ,
G m n = k , i = G ( m Δ + k Ω , n Δ + i Ω ) = k , i = G m + k N , n + i N ;
A m n = | G m n | ; and ϕ m n = arg ( G m n ) .
H m n = ( A m n + N m n ) e j ( ϕ m n + θ m n ) ,
H m n = { G m n e j θ m n + k 0 or i 0 G m + k N , n + i N e j θ m n + N m n e j ( ϕ m n + θ m n ) for m , n I 0 for m , n I
e ( x , y ) = m , n = ( G m n H m n ) e j ( 2 π / X ) ( m x + n y ) = m , n I G m n e j ( 2 π / X ) ( m x + n y ) + m , n I G m n ( 1 e j θ m n ) e j ( 2 π / X ) ( m x + n y ) + m , n I k 0 or i 0 G m + k N , n + i N , e j θ m n e j ( 2 π / X ) ( m x + n y ) + m , n I N m n e j ( ϕ m n + θ m n ) e j ( 2 π / X ) ( m x + n y ) .
ε { p , q Im , n I k 0 or i 0 G m G p q * × e j ( 2 π / X ) [ ( m p ) x + ( n q ) y ] }
m , n I k 0 or i 0 β m n e j ( 2 π / X ) ( k N x + i N y ) .
ε { | e ( x , y ) | 2 } = m , n I β m n + 2 ( 1 sin Ө / 2 Ө / 2 ) m n I β m n + m , n I β m n + m , n I β m n < σ 0 2 β m n + N 0 σ 0 2 + 2 m , n I k 0 or i 0 β m n e j ( 2 π / X ) ( k N x + i N y ) .
Ψ = ε { cos 2 γ } 1 X S ε { | e ( x , y ) | 2 } d x d y = ε { cos 2 γ } [ 2 m , n I β m n + 2 ( 1 sin Ө / 2 Ө / 2 ) × m , n I β m n + m , n I β m n < σ 0 2 β m n + N 0 σ 0 2 ] .
S g g ( f x , f y ) = β m n δ ( f x m X ) δ ( f y n X ) ,
β m n = 27 / [ 27 + ( m 2 + n 2 ) 3 2 ] .
G m n = G ( m Δ u , n Δ υ ) .
G ( u , υ ) = G ( u , υ ) e j π ( u 2 + υ 2 ) ( f z ) / λ f z = G ( u , υ ) e j α ( u 2 + υ 2 ) ,
g ( x , y ) = F 1 { G ( u , υ ) e j α ( u 2 + υ 2 ) }
h ( x , y ) = F 1 { H ( u , υ ) e j α ( u 2 + υ 2 ) } ,
g ˜ ( x , y ) = g ( x , y ) e j ξ ( x , y )
G ˜ ( u , υ ) = G ( u , υ ) * W ( u , υ ) ,

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