Abstract

The contrast and diffraction efficiency of a dielectric hologram made from a diffuse signal beam is calculated. Diffraction efficiency of 64% with acceptable image contrast is possible. Experimental results with holograms made by bleaching photographic emulsions are presented.

© 1970 Optical Society of America

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References

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  1. Diffraction efficiency of a hologram is defined as the ratio of diffracted light flux to the incident light flux.
  2. H. Kogelnik, in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, N. Y., 1967).
  3. C. B. Burckhardt, J. Opt. Soc. Am. 57, 601 (1967).
    [Crossref]
  4. T. A. Shankoff, Appl. Opt. 7, 2101 (1968).
    [Crossref] [PubMed]
  5. L. H. Lin and H. L. Beauchamp, J. Opt. Soc. Am. 58, 1551A (1968).
    [Crossref]
  6. J. N. Latta, Appl. Opt. 7, 2409 (1968).
    [Crossref] [PubMed]
  7. J. Upatnieks and C. Leonard, Appl. Opt. 8, 85 (1969).
    [Crossref] [PubMed]
  8. K. S. Pennington and J. S. Harper, J. Opt. Soc. Am. 59, 481A (1969).
  9. J. Upatnieks and C. Leonard, J. Opt. Soc. Am. 59, 481A (1969).
  10. Rayleigh, Phil. Mag. 10, 73 (1880).
  11. E. Jahnke and F. Emde, Tables of Functions (Dover Publications, Inc., New York, 1945).
  12. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).
    [Crossref] [PubMed]
  13. R. W. James, The Optical Principles of the Diffraction of X-Rays (Cornell University Press, Ithaca, N. Y., 1965).
  14. Pennington and Harper8 demonstrated at the 1969 Spring Meeting of the Optical Society a bleached hologram having 28% diffraction efficiency and a contrast ratio of 50. This contrast ratio corresponds to (S/N)c= 25 in our experiments, when adjustments are made for differences of test objects.
  15. W. H. Barkas, Nuclear Research Emulsions I (Academic Press Inc., New York, 1963). Numerous other processing techniques to minimize emulsion distortion are also described in this book.
  16. Yu N. Deninsyuk and I. R. Protas, Opt. Spectrosc. 14, 381 (1963).
  17. L. H. Lin and C. V. LoBianco, Appl. Opt. 7, 1255 (1967).
    [Crossref]
  18. H. M. Smith, J. Opt. Soc. Am. 58, 533 (1968).
    [Crossref]

1969 (3)

J. Upatnieks and C. Leonard, Appl. Opt. 8, 85 (1969).
[Crossref] [PubMed]

K. S. Pennington and J. S. Harper, J. Opt. Soc. Am. 59, 481A (1969).

J. Upatnieks and C. Leonard, J. Opt. Soc. Am. 59, 481A (1969).

1968 (4)

1967 (2)

C. B. Burckhardt, J. Opt. Soc. Am. 57, 601 (1967).
[Crossref]

L. H. Lin and C. V. LoBianco, Appl. Opt. 7, 1255 (1967).
[Crossref]

1966 (1)

1963 (1)

Yu N. Deninsyuk and I. R. Protas, Opt. Spectrosc. 14, 381 (1963).

1880 (1)

Rayleigh, Phil. Mag. 10, 73 (1880).

Barkas, W. H.

W. H. Barkas, Nuclear Research Emulsions I (Academic Press Inc., New York, 1963). Numerous other processing techniques to minimize emulsion distortion are also described in this book.

Beauchamp, H. L.

L. H. Lin and H. L. Beauchamp, J. Opt. Soc. Am. 58, 1551A (1968).
[Crossref]

Burckhardt, C. B.

Deninsyuk, Yu N.

Yu N. Deninsyuk and I. R. Protas, Opt. Spectrosc. 14, 381 (1963).

Emde, F.

E. Jahnke and F. Emde, Tables of Functions (Dover Publications, Inc., New York, 1945).

Harper, J. S.

K. S. Pennington and J. S. Harper, J. Opt. Soc. Am. 59, 481A (1969).

Jahnke, E.

E. Jahnke and F. Emde, Tables of Functions (Dover Publications, Inc., New York, 1945).

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-Rays (Cornell University Press, Ithaca, N. Y., 1965).

Kogelnik, H.

H. Kogelnik, in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, N. Y., 1967).

Kozma, A.

Latta, J. N.

Leith, E. N.

Leonard, C.

J. Upatnieks and C. Leonard, J. Opt. Soc. Am. 59, 481A (1969).

J. Upatnieks and C. Leonard, Appl. Opt. 8, 85 (1969).
[Crossref] [PubMed]

Lin, L. H.

L. H. Lin and H. L. Beauchamp, J. Opt. Soc. Am. 58, 1551A (1968).
[Crossref]

L. H. Lin and C. V. LoBianco, Appl. Opt. 7, 1255 (1967).
[Crossref]

LoBianco, C. V.

L. H. Lin and C. V. LoBianco, Appl. Opt. 7, 1255 (1967).
[Crossref]

Marks, J.

Massey, N.

Pennington, K. S.

K. S. Pennington and J. S. Harper, J. Opt. Soc. Am. 59, 481A (1969).

Protas, I. R.

Yu N. Deninsyuk and I. R. Protas, Opt. Spectrosc. 14, 381 (1963).

Rayleigh,

Rayleigh, Phil. Mag. 10, 73 (1880).

Shankoff, T. A.

Smith, H. M.

Upatnieks, J.

Appl. Opt. (5)

J. Opt. Soc. Am. (5)

H. M. Smith, J. Opt. Soc. Am. 58, 533 (1968).
[Crossref]

C. B. Burckhardt, J. Opt. Soc. Am. 57, 601 (1967).
[Crossref]

K. S. Pennington and J. S. Harper, J. Opt. Soc. Am. 59, 481A (1969).

J. Upatnieks and C. Leonard, J. Opt. Soc. Am. 59, 481A (1969).

L. H. Lin and H. L. Beauchamp, J. Opt. Soc. Am. 58, 1551A (1968).
[Crossref]

Opt. Spectrosc. (1)

Yu N. Deninsyuk and I. R. Protas, Opt. Spectrosc. 14, 381 (1963).

Phil. Mag. (1)

Rayleigh, Phil. Mag. 10, 73 (1880).

Other (6)

E. Jahnke and F. Emde, Tables of Functions (Dover Publications, Inc., New York, 1945).

Diffraction efficiency of a hologram is defined as the ratio of diffracted light flux to the incident light flux.

H. Kogelnik, in Proceedings of the Symposium on Modern Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, N. Y., 1967).

R. W. James, The Optical Principles of the Diffraction of X-Rays (Cornell University Press, Ithaca, N. Y., 1965).

Pennington and Harper8 demonstrated at the 1969 Spring Meeting of the Optical Society a bleached hologram having 28% diffraction efficiency and a contrast ratio of 50. This contrast ratio corresponds to (S/N)c= 25 in our experiments, when adjustments are made for differences of test objects.

W. H. Barkas, Nuclear Research Emulsions I (Academic Press Inc., New York, 1963). Numerous other processing techniques to minimize emulsion distortion are also described in this book.

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Figures (9)

Fig. 1
Fig. 1

Calculated diffraction efficiency and signal-to-noise ratio (S/N)T vs the ratio of reference beam to average signal-beam irradiances K for several values of m and B: —, decreasing with K, diffraction efficiency using approximation Δns = 〈Δn〉; …., diffraction efficiency using statistical distribution of Δns; —, increasing with K, signal-to-noise ratio (S/N)T.

Fig. 2
Fig. 2

Vector diagrams for calculation of spatial-frequency vectors.

Fig. 3
Fig. 3

Contrast of reconstructed image (S/N)c vs density of exposed plate before bleaching: ▲- - - -▲ dichromate bleach, modified Kodak R-10, K = 7.5; ●....● potassium ferricyanide bleach, K = 8; ■——■ dichromate bleach, modified Kodak R-10, xylene in liquid gate, K = 7.5; ●- - - -● potassium ferricyanide bleach, xylene in liquid gate, K = 8.

Fig. 4
Fig. 4

Diffraction efficiency and image contrast (S/N)c vs the ratio of reference-beam to average signal-beam irradiances K: — diffraction efficiency; - - - - image contrast (S/N)c for unexpanded emulsion; — - — image contrast (S/N)c for expanded emulsion. Test parameters: carrier frequency 820 lines/mm, signal beam width β =2.3°, emulsion in xylene liquid gate, minimum density before bleaching 1.9, potassium ferricyanide bleach.

Fig. 5
Fig. 5

Diffraction efficiency and signal-to-noise ratio vs the ratio of reference-beam to average signal-beam irradiances K: — diffraction efficiency; ●....● image contrast (S/N)c for emulsion in air; △....△ signal-to-noise ratio (S/N)T for emulsion in air; ● - - - - ● image contrast (S/N)c for emulsion in liquid gate, n = 1.497; △- - - -△ signal-to-noise ratio (S/N)T for emulsion in liquid gate, n = 1.497; —●— image contrast (S/N)c for emulsion in Kodak Aroclor 1232, liquid gate, n =1.618. Test parameters: carrier frequency 820 lines/mm, signal beam width 2.3°, expanded emulsion, minimum density before bleaching 1.9, potassium ferricyanide bleach.

Fig. 6
Fig. 6

Diffraction efficiency and image contrast (S/N)c vs the ratio of reference-beam to average signal-beam irradiances K, for several signal beam widths β. Test parameters: carrier frequency 1330 lines/mm, minimum density before bleaching 2.7, expanded emulsion, potassium ferricyanide bleach.

Fig. 7
Fig. 7

Diffraction efficiency and image contrast (S/N)c vs the ratio of reference-beam to average signal-beam irradiances K. Spatial-carrier frequencies, corresponding to the indicated angles between reference and signal beams are, for 30°, 820 lines/mm; 50°, 1330 lines/mm; 90°, 2230 lines/mm. Test parameters: signal beam width β = 2.3°, expanded emulsion, minimum density before bleaching 2.7, potassium ferricyanide bleach.

Fig. 8
Fig. 8

Diffraction efficiency and image contrast (S/N)c vs the ratio of reference-beam to average signal-beam irradiances K: — potassium ferricyanide bleach; - - - - dichromate bleach, modified Kodak R-10 with potassium bromide in solution B. The lower (S/N)c curves are for unpolarized and the upper curves are for polarized signal beams. Diffraction-efficiency curves are shown for polarized signal beams and the points below these (at low K) indicate efficiencies for unpolarized signal beams. Test parameters: carrier frequency 1330 lines/mm, signal beam width β = 15°, emulsions bleached with potassium ferricyanide bleach are expanded, while those bleached in modified R-10 bleach are not.

Fig. 9
Fig. 9

Diffraction efficiency and image contrast (S/N)c vs the ratio of reference-beam to average signal-beam irradiances K for several signal beam widths β and polarized and unpolarized signal beams: ▲ —— ▲ unpolarized signal beam, β = 15°; polarized signal beams: ○- - -○ β = 15°, ■ — · — ■ β = 6°, ●....● β = 2.3°. The same symbols indicate experimental points of diffraction efficiencies for the corresponding beam widths, with the curve drawn only for β = 2.3°. Test parameters: emulsion not expanded, minimum density 2.7, modified R-10 bleach.

Equations (26)

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s 0 + s 2 = a 0 2 + 2 a 0 i = 1 M a i cos ϕ 0 i + 2 l = i M i = 1 M a i a l cos ϕ i l ,
T = exp ( j m τ s 0 + s 2 ) .
T = exp ( j m τ a 0 2 ) exp [ j ( 2 a 0 m τ i = 1 M a i cos ϕ 0 i ) ] × exp [ j ( 2 m τ l = i M i = 1 M a i a l cos ϕ i l ) ] .
T = exp ( j m τ a 0 2 ) exp [ j ( 2 m τ a 0 2 ( K M ) 1 2 i = 1 M cos ϕ 0 i ) ] × exp [ j ( 2 m τ a 0 2 ( K M ) l = i M i = 1 M cos ϕ i l ) ] .
A = exp [ - j μ I ( x , y ) ] exp [ j ( p x + q y ) ] d x d y / d x d y | p = q = 0 = exp [ j μ I ( x , y ) ] d x d y / d x d y .
A = 0 p ( I ) exp ( - j μ I ) d I ,
A = 1 / ( 1 + j 2 μ σ 2 ) .
( S / N ) T = 1 / ( 2 μ σ 2 ) 2 .
σ 2 = a 0 2 / 2 K .
Δ ψ = 2 π d Δ n N / λ cos θ ,
( S / N ) T = K 2 / ( m 2 B 2 ) ,
E ( Δ n s ) = sin 2 ( π d Δ n s / λ cos θ ) ,
2 m τ a 0 2 ( K M ) 1 2 i = 1 M cos ϕ 0 i .
Δ n = 2 m τ a 0 2 / K
E ( Δ n ) sin 2 ( 2 π m d τ a 0 2 K λ cos θ ) sin 2 ( m B K ) ,
p ( a s ) = a s / σ 2 exp ( - a s 2 / 2 σ 2 ) .
E = 0 a s σ 2 exp ( - a s 2 2 σ 2 ) sin 2 b a s d a s ,
E = 2 b 2 σ 2 exp ( - 2 b 2 σ 2 ) F 1 1 ( 1 2 ; 3 2 ; 2 b 2 σ 2 ) ,
F 1 1 ( α ; v ; x ) = 1 + α v x + α v α + 1 v + 1 x 2 2 ! + α v α + 1 v + 1 α + 2 v + 2 x 3 3 ! + .
a s = [ a 0 / ( K M ) 1 2 ] i = 1 M cos ϕ 0 i .
E ( α s ) = sin 2 [ ( 2 π m τ d a 0 / λ cos θ ) a s ] = sin 2 b a s .
E = m 2 B 2 K exp ( - m 2 B 2 K ) F 1 1 ( 1 2 ; 3 2 ; m 2 B 2 K ) .
ω 12 = ω 12 r ˆ 12 = ( r ˆ 1 - r ˆ 2 ) / λ 0 .
ω 14 = p ω 12 ± q ω 23
r ˆ 4 = ( p - q ) r ˆ 2 + q r ˆ 3 - ( p - 1 ) r ˆ 1
r ˆ 4 = - q r ˆ 3 - ( p - 1 ) r ˆ 1 + ( p + q ) r ˆ 2 ,