Abstract

An expression derived for hologram exposures made along the straight-line portion of an H–D curve of a photographic plate gives the first-order transmittance of a hologram made of several object points exposed simultaneously (conventional holograms). This expression is compared with a similar expression derived previously for holograms made of several object points exposed sequentially (synthetic holograms). Theory and experiments show the effect of the nonlinearity of the photographic process on the contrast of the reconstruction of conventional holograms. Synthetic and conventional holograms are studied theoretically and experimentally to determine how the total amount of light in the reconstruction image depends upon the number of object points when the total amount of light in the object is constant. It is shown that the reconstructed image formed by a conventional hologram contains more light than the image formed by a synthetic hologram of the same number of object points. Both synthetic and conventional holograms are also studied to determine the ratio of reference-beam illuminance to object-beam illuminance that maximizes the amount of light in the reconstructed image.

© 1969 Optical Society of America

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References

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  1. J. C. Wyant and M. P. Givens, J. Opt. Soc. Am. 58, 357 (1968).
    [Crossref]
  2. F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 56, 1414 (1966).
  3. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [Crossref]
  4. A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1960), p. 167.

1968 (1)

1966 (2)

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 56, 1414 (1966).

A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
[Crossref]

Geffner, J.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1960), p. 167.

Givens, M. P.

Kaspar, F. G.

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 56, 1414 (1966).

Kozma, A.

Lamberts, R. L.

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 56, 1414 (1966).

Worthing, A. G.

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1960), p. 167.

Wyant, J. C.

J. Opt. Soc. Am. (3)

Other (1)

A. G. Worthing and J. Geffner, Treatment of Experimental Data (John Wiley & Sons, Inc., New York, 1960), p. 167.

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Figures (9)

F. 1
F. 1

Apparatus for making the holograms. L, He–Ne laser; MO, 20× microscope objective; PH, 15-μ pinhole; PMT, photomultiplier tube; CL, collimating lens; L, two −3 diopter negative lenses; P, Polaroid film; F1 and F2, neutral-density filters; and H, hologram plane.

F. 2
F. 2

Ratio of total amount of light in the hologram reconstruction of N object points, each of illuminance I/N, to amount of light in hologram reconstruction of one object point of illuminance I, as function of number of object points, N. For N equal ∞, the luminance ratio equals 0.643. × are experimentally measured data points. Gamma = 4.7, pre-exposed density = 0.18, and average hologram density is approximately 0.75.

F. 3
F. 3

Relative luminance of hologram reconstruction of single object point as a function of beam-balance ratio. Experimentally measured data points are given by Δ for Curve A and × for Curve B. Gamma = 5.8. For beam-balance ratio = 1, average hologram density equals 0.73 for Curve A and 0.86 for Curve B.

F. 4
F. 4

Relative luminance of synthetic hologram reconstruction of three object points as a function of beam-balance ratio. Experimentally measured data points are given by Δ for Curve A and × for Curve B. Gamma = 4.8. For beam-balance ratio = 1, average hologram density equals 0.72 for Curve A and 0.92 for Curve B.

F. 5
F. 5

Relative luminance of conventional hologram reconstruction of three object points as a function of beam-balance ratio. Experimentally measured data points are given by Δ for Curve A and × for Curve B. Gamma = 3.85. For beam balance ratio = 1, average hologram density equals 0.75 for Curve A and 0.9 for Curve B.

F. 6
F. 6

Relative luminance of hologram reconstruction of one object point as a function of beam-balance ratio. × are experimentally measured data points.

F. 7
F. 7

Relative luminance of conventional hologram reconstruction of four object points as a function of beam-balance ratio, × are experimentally measured data points.

F. 8
F. 8

Relative luminance of conventional hologram reconstruction of eight object points as a function of beam-balance ratio, × are experimentally measured data points.

F. 9
F. 9

Relative luminance of conventional hologram reconstruction of twelve object points and an infinite number of object points, as a function of beam-balance ratio. × are experimentally measured data points.

Tables (6)

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Table I Ratio of reconstructed-image luminances of conventional holograms of two object points.

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Table II Ratio of reconstructed-image luminance for different numbers of superimposed holograms.

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Table III Ratio of reconstructed-image luminances of conventional holograms of different number of object points.

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Table IV Ratio of total amount of light in reconstruction of hologram of N superimposed exposures, each of exposure E/N, to amount of light reconstruction of hologram of single exposure E.

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Table V Ratio of total amount of light in reconstruction of holograms of N object points, each of illuminance I/N, to amount of light in reconstruction of hologram of one object point of illuminance I.

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Table VI Relative luminance of hologram reconstruction as a function of beam-balance ratio. Note: Relative luminance of reconstruction equals 1 for a beam-balance ratio of 1.

Equations (10)

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T A = ( E / E ) γ / 2 ,
A = R + n = 1 N A n ,
E H = t ( A A * ) = t ( R + n = 1 N A n ) ( R + n = 1 N A n ) * = t ( I R + I + R * n = 1 N A n + R n = 1 N A n * + n = 1 N m = 1 m n N A n A m * ) ,
| T A | = ( E C ) γ / 2 [ 1 + C ( a + b + d ) ] γ / 2 ,
C = [ E 0 + t ( I R + I ) ] 1 , a = t R * n = 1 N A n , b = t R n = 1 N A n * ,
d = t i = 1 N j = 1 j i N A i A j * .
I K = | D | 2 ( E C ) γ ( 1 2 γ ) 2 C 2 t 2 I R I K ( 1 U + V S + higher order terms ) 2 ,
U = ( 1 2 γ + 1 ) C t n = 1 n k N I n and I n = | A n | 2 ,
V = ( 1 2 γ + 1 ) ( 1 2 γ + 2 ) C 2 t 2 [ I R ( 1 2 I K + n = 1 n k N I n ) + I K n = 1 n k N I n + 3 2 n = 1 n k N j = 1 j k j n N I n I j ] ,
S = ( 1 2 γ + 1 ) ( 1 2 γ + 2 ) ( 1 2 γ + 3 ) C 3 ( 1 6 ) t 3 ( I R ( 3 n = 1 N i = 1 i n N I n I i + 6 I K j = 1 j k N I j + 9 n = 1 n k N i = 1 i n i k N I n I i ) + n = 1 N i = 1 i n N j = 1 j k N I n I i I j + 2 n = 1 N i = 1 i n N j = 1 j i j n N I n I i I j + 3 i = i i k N j = 1 j k j i N n = 1 n j n k N I i I j I n + n = 1 N i = 1 i n i k N j = 1 j k j n N I n I i I j + i = 1 N n = 1 n i n k N j = 1 j n j i j k N I i I n I j + n = 1 n k N i = 1 i n N j = 1 j i j n N I n I i I j + 2 n = 1 n k N i = 1 i n i k N j = 1 j k j i j n N I n I i I j ) ,