Abstract

Optimum linearity, and maximum efficiency, for thin-amplitude and thin-phase holograms of small modulation depth are realized at different values of average exposure (bias). We utilize a threshold model for the photographic detection process to predict the bias values. These values are shown to be in agreement with available experimental results. For large modulation, the biasing point depends upon both modulation depth and average exposure.

© 1976 Optical Society of America

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References

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  1. C. T. Chang and J. L. Bjorkstam, J. Opt. Soc. Am. 65, 1495 (1975).
    [Crossref]
  2. K. Biedermann, Appl. Opt. 10, 584 (1971).
    [Crossref] [PubMed]
  3. C. E. Thomas, Appl. Opt. 11, 1756 (1972).
    [Crossref] [PubMed]
  4. A. Kozma, G. W. Jull, and K. O. Hill, Appl. Opt. 9, 721 (1970).
    [Crossref] [PubMed]
  5. W. H. Lee and M. O. Greer, J. Opt. Soc. Am. 61, 402 (1971).
    [Crossref]
  6. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), p. 17.
  7. G. R. Bird, R. C. Jones, and A. E. Ames, Appl. Opt. 8, 2389 (1969).
    [Crossref] [PubMed]
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 376.
  9. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1973), pp. 171 and 376.
  10. K. Biedermann, Optik 28, 160 (1968/69).
  11. D. Gabor, Proc. R. Soc. Ser. A 197, 454 (1949).
    [Crossref]
  12. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 160.
  13. D. Falconer, Phot. Sci. Eng. 10, 133 (1966).
  14. J. C. Urbach and R. W. Meier, Appl. Opt. 8, 2269 (1969).
    [Crossref] [PubMed]
  15. K. Biedermann and K. A. Stetson, Phot. Sci. Eng. 13, 361 (1969).

1975 (1)

1972 (1)

1971 (2)

1970 (1)

1969 (3)

1966 (1)

D. Falconer, Phot. Sci. Eng. 10, 133 (1966).

1949 (1)

D. Gabor, Proc. R. Soc. Ser. A 197, 454 (1949).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 376.

Ames, A. E.

Biedermann, K.

K. Biedermann, Appl. Opt. 10, 584 (1971).
[Crossref] [PubMed]

K. Biedermann and K. A. Stetson, Phot. Sci. Eng. 13, 361 (1969).

K. Biedermann, Optik 28, 160 (1968/69).

Bird, G. R.

Bjorkstam, J. L.

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 160.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1973), pp. 171 and 376.

Chang, C. T.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 160.

Falconer, D.

D. Falconer, Phot. Sci. Eng. 10, 133 (1966).

Gabor, D.

D. Gabor, Proc. R. Soc. Ser. A 197, 454 (1949).
[Crossref]

Greer, M. O.

Hill, K. O.

Jones, R. C.

Jull, G. W.

Klauder, J. R.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), p. 17.

Kozma, A.

Lee, W. H.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 160.

Meier, R. W.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 376.

Stetson, K. A.

K. Biedermann and K. A. Stetson, Phot. Sci. Eng. 13, 361 (1969).

Sudarshan, E. C. G.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), p. 17.

Thomas, C. E.

Urbach, J. C.

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

Optik (1)

K. Biedermann, Optik 28, 160 (1968/69).

Phot. Sci. Eng. (2)

D. Falconer, Phot. Sci. Eng. 10, 133 (1966).

K. Biedermann and K. A. Stetson, Phot. Sci. Eng. 13, 361 (1969).

Proc. R. Soc. Ser. A (1)

D. Gabor, Proc. R. Soc. Ser. A 197, 454 (1949).
[Crossref]

Other (4)

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 160.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 376.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1973), pp. 171 and 376.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), p. 17.

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Figures (4)

FIG. 1
FIG. 1

Diagram of the experimental arrangement for recording a two-beam interference grating. The symbols have the following meaning: BS, variable beam splitter; F, photographic plate; Gp, 4 × 5 in. ground glass plate with absorbing black paint on its rough side; L, index matching liquid; Lo, microscopic objective lens; Lc, collimating lens; M, mirror; pi, 25 μm pinhole; S, electronic shutter.

FIG. 2
FIG. 2

Normalized theoretical average density versus log exposure curves (f vs log N ¯) for a grating formed with modulation depths, m = 0.0, 0.6, 0.8, and 1.0.

FIG. 3
FIG. 3

Theoretical average amplitude transmittance versus average exposure curves ( t ¯ vs N ¯) for the grating with modulation depths, m = 0.0, 0.6, 0.8, and 1.0. Different modulation depths give different static characteristic curves (m ≠ 0 curve). Modulation depth m, together with average exposure N ¯, determines the average amplitude transmittance. For example, with m = 0.6 this is point A. The corresponding biasing point B (tB, NB), on the dynamic characteristic curve (m = 0 curve), is chosen to have the same transmittances as the spatial average transmittance t ¯ (i.e., tB = t ¯). However, the exposure is different (i.e., NB N ¯).

FIG. 4
FIG. 4

Experimental average density versus log average exposure curves ( D ¯ vs logĒ) for the grating formed with modulation depths, m = 0.0, 0.6, 0.8, 1.0.

Equations (28)

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I ( x ) = ( I o + I r ) ( 1 + m cos θ ) .
θ ( 4 π x / λ ) sin ( ϕ / 2 ) ,
m 2 I o I r / ( I o + I r ) .
d = λ [ 2 sin ( ϕ / 2 ) ] - 1 .
P n ¯ ( n ) = e - n ¯ n ¯ n / n ! .
n ¯ ( I o + I r ) ( 1 + m cos θ ) T A N ¯ ( 1 + m cos θ ) .
f 1 ( n ¯ ) = 1 - n = o 3 e - n ¯ n ¯ n n ! .
f ( N ¯ , m ) = 1 π 0 π ( 1 - n = o 3 e - N ¯ ( 1 + m cos θ ) [ N ¯ ( 1 + m cos θ ) ] n n ! ) d θ = 1 - e - N ¯ [ ( 1 + N ¯ + N ¯ 2 2 + N ¯ 2 m 2 4 + N ¯ 3 6 + N ¯ 3 m 2 4 ) I o ( - N ¯ m ) + ( N ¯ m + N ¯ 2 m + N ¯ 3 m 2 + N ¯ 3 m 3 8 ) I 1 ( - N ¯ m ) + N ¯ 2 m 2 4 ( 1 + N ¯ ) I 2 ( - N ¯ m ) + N ¯ 3 m 2 24 I 3 ( - N ¯ m ) ] .
I ν ( z ) 1 π 0 π e z cos θ cos ( ν θ ) d θ
t = 10 - 0.5 D = e - 1.15 D .
t ¯ = 1 π 0 π exp [ - 1.15 D o × ( 1 - n = o 3 e - N ¯ ( 1 + m cos θ ) [ N ¯ ( 1 + m cos θ ) ] n n ! ) ] d θ .
t ¯ = exp { - 1.15 D o [ 1 - e - N ¯ ( 1 + N ¯ + N ¯ 2 / 2 + N ¯ 3 / 6 ) ] } .
t = exp { - 1.15 D o [ 1 - e - n ¯ ( 1 + n ¯ + n ¯ 2 / 2 + n ¯ 3 / 6 ) ] } ,
D = D o [ 1 - e - n ¯ ( 1 + n ¯ + n ¯ 2 / 2 + n ¯ 3 / 6 ) ] .
n ¯ - N ¯ = N ¯ m cos θ .
t ( n ¯ ) = t | N ¯ + d t d n ¯ | N ¯ ( N ¯ m cos θ ) + 1 2 d 2 t d n ¯ 2 | N ¯ ( N ¯ m cos θ ) 2 + .
d 2 t d n ¯ 2 | N ¯ = 0.
- 1.15 D o e - N ¯ ( N ¯ 3 / 6 ) = 1 - ( 3 / N ¯ ) .
t ( n ¯ ) = e i p D ( n ¯ ) .
t ( n ¯ ) = e i p D ( N ¯ ) + ( e i p D ( n ¯ ) i p d D d n ¯ ) | N ¯ N ¯ m cos θ + 1 2 [ e i p D ( n ¯ ) i p d 2 D d n ¯ 2 - e i p D ( n ¯ ) p 2 ( d D d n ¯ ) 2 ] | N ¯ ( N ¯ m cos θ ) 2 + ,
d 2 D d n ¯ 2 | N ¯ = 0.
1.15 D o e - N ¯ ( N ¯ 2 2 - N ¯ 3 6 ) = 0.
η = ( m / 2 ) 2 [ n ¯ ( d t / d n ¯ ) ] N ¯ 2 = [ 1 4 ( t γ ) N ¯ m ] 2 .
d d n ¯ ( d t d n ¯ n ¯ ) | N ¯ = 0.
t ( N ¯ ) [ ( - 1.15 D o e - N ¯ N ¯ 3 / 6 ) 2 N ¯ + ( 1.15 D o e - N ¯ ) ( N ¯ 4 / 6 - 2 N ¯ 3 / 3 ) ] = 0.
η = [ 1 2 ( d D d n ¯ ) n ¯ | N ¯ p m ] 2 = ( 1 4.6 γ ( N ¯ ) p m ) 2 .
d d n ( d D d n ¯ n ¯ ) | N ¯ = 0.
D o e - N ¯ ( 2 N ¯ 3 / 3 - N ¯ 4 / 6 ) = 0.