Abstract

A study of the effects of photographic nonlinearities in recording coherent wavefronts is reported. Particular emphasis is placed on holography using the two-beam interferometry technique. A phenomenological model which provides the mathematical formulation for describing the effects of nonlinearities is described. The model includes a zero-memory nonlinearity which represents experimentally derived transmittance–exposure curves for various photographic emulsions. An analysis of this model reveals many interesting phenomena which are supported experimentally. In particular, the nonlinearity of the film generates false targets, causes weak signal suppression, and can introduce additional noise as a result of spectral folding. Experimental results, verifying the analysis, are presented.

© 1967 Optical Society of America

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References

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  1. D. H. Kelly, J. Opt. Soc. Am. 50, 269 (1966).
    [CrossRef]
  2. R. F. Van Ligten, J. Opt. Soc. Am. 56, 1 (1966).
    [CrossRef]
  3. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [CrossRef]
  4. A. B. Vander Lugt, IEEE Trans. IT-10, 139 (1964).
  5. R. O. Harger, Optimization of Some Noisy Nonlinear Systems, Rept. No. 2900–290–T, Institute of Science and Technology, The University of Michigan, October1961.
  6. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [CrossRef]
  7. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill Book Co. Inc., New York, 1958), pp. 277–311.
  8. W. Magnus, F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea, New York, 1949).
  9. J. J. Jones, IEEE Trans. IT-9, 34 (1963).
  10. A. A. Friesem, J. S. Zelenka, J. Opt. Soc. Am. 56, 542A (1966).

1966

1964

A. B. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

1963

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill Book Co. Inc., New York, 1958), pp. 277–311.

Friesem, A. A.

A. A. Friesem, J. S. Zelenka, J. Opt. Soc. Am. 56, 542A (1966).

Harger, R. O.

R. O. Harger, Optimization of Some Noisy Nonlinear Systems, Rept. No. 2900–290–T, Institute of Science and Technology, The University of Michigan, October1961.

Jones, J. J.

J. J. Jones, IEEE Trans. IT-9, 34 (1963).

Kelly, D. H.

Kozma, A.

Leith, E. N.

Magnus, W.

W. Magnus, F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea, New York, 1949).

Oberhettinger, F.

W. Magnus, F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea, New York, 1949).

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill Book Co. Inc., New York, 1958), pp. 277–311.

Upatnieks, J.

Van Ligten, R. F.

Vander Lugt, A. B.

A. B. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

Zelenka, J. S.

A. A. Friesem, J. S. Zelenka, J. Opt. Soc. Am. 56, 542A (1966).

IEEE Trans.

A. B. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

J. J. Jones, IEEE Trans. IT-9, 34 (1963).

J. Opt. Soc. Am.

Other

R. O. Harger, Optimization of Some Noisy Nonlinear Systems, Rept. No. 2900–290–T, Institute of Science and Technology, The University of Michigan, October1961.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signal and Noise (McGraw-Hill Book Co. Inc., New York, 1958), pp. 277–311.

W. Magnus, F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea, New York, 1949).

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Figures (10)

Fig. 1
Fig. 1

Transmission vs exposure characteristics. 649F film. intensity = constant = 0.44 mW/cm2.

Fig. 2
Fig. 2

Transmission vs exposure characteristics. Panatomic-X film. intensity = constant = 0.175 × 10−4 mW/cm2.

Fig. 3
Fig. 3

Transmission vs exposure characteristics. HCC film. intensity = constant = 0.175 × 10−2 mW/cm2.

Fig. 4
Fig. 4

Model for representing nonlinear hologram recording.

Fig. 5
Fig. 5

Reconstruction of hologram of two-point target, linear recording.

Fig. 6
Fig. 6

Reconstruction of hologram of two-point target, soft nonlinear recording.

Fig. 7
Fig. 7

Reconstruction of hologram of two-point target, strong nonlinear recording.

Fig. 8
Fig. 8

Reconstruction of hologram of two-point target, low offset frequency, nonlinear recording.

Fig. 9
Fig. 9

Reconstruction of hologram made with low off-set frequency, linear recording.

Fig. 10
Fig. 10

Reconstruction of hologram made with low offset frequency, nonlinear recording.

Equations (12)

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T = T 0 - k I ,
r ( x ) = A r exp [ j θ r ( x ) exp ( j ω 0 t ) ,
s ( x ) = A s ( x ) exp [ j θ s ( x ) ] exp ( j ω 0 t ) ,
I ( x ) = A r 2 + A s 2 + A r A s × { exp [ j ( θ r - θ s ) ] + exp [ - j ( θ r - θ s ) ] } ,
I ( x ) = A r 2 + A s 2 + 2 A r A s cos ( θ r - θ s ) .
v ( x ) = 1 2 π j σ - j σ + j G ( ξ ) ( exp { A r [ 2 A s cos ( θ r - θ s ) + α ] ξ } - exp { - A r [ 2 A s cos ( θ s - θ r ) + α ] ξ } ) d ξ
G ( ξ ) = 0 g ( x ) e - ξ x d x
exp ( z cos θ ) = m = 0 m I m ( z ) cos m θ ,
v ( x ) = m = 0 n = 0 m n cos [ m ( θ r - θ s ) ] × 1 2 π j σ - j σ + j G ( ξ ) [ I m ( 2 A r A s ξ ) I n ( A r α ξ ) - I m ( - 2 A r A s ξ ) I n ( - A r α ξ ) ] d ξ .
s ( x ) = A s ( x ) exp [ j θ s ( x ) ] = A 1 exp [ j θ 1 ( x ) ] + A 2 exp [ j θ 2 ( x ) ] ,
w ( x ) = K { A 1 cos ( θ r - θ 1 ) + [ ( 1 + ν ) / 2 ] A 2 cos ( θ r - θ 2 ) + m = 1 p m ( A 2 / A 1 ) m - 1 A 2 cos [ θ r - ( m + 1 ) θ 1 + m θ 2 ] + n = 1 q n ( A 2 / A 1 ) n A 2 cos [ θ r - ( n + 1 ) θ 2 + n θ 1 ] + } ,
K = 4 ( A r ) v Γ ( ν + 1 ) ( 1 + ν ) Γ 2 [ ( ν + 1 ) / 2 ] A 1 v - 1 ,

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