Abstract

A statistical analysis of the effects of film nonlinearities on wavefront-reconstruction images of diffuse objects is presented. Regardless of the particular nature of the nonlinearity, the image distortions are shown to consist of additive irradiance contributions which may be found as multiple autoconvolutions of the ideal image irradiance distribution produced by a linear film. The predictions of the general nature of the image distortions are supported by experimental evidence.

© 1968 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
    [Crossref]
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); J. Opt. Soc. Am. 53, 1377 (1963); J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  3. E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
    [Crossref]
  4. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [Crossref]
  5. R. F. van Ligten, J. Opt. Soc. Am. 56, 1009 (1966).
    [Crossref]
  6. J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
    [Crossref] [PubMed]
  7. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [Crossref]
  8. A. A. Friesem and J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
    [Crossref] [PubMed]
  9. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 116.
  10. G. R. Knight, Ph.D. dissertation, Stanford University, Stanford, California, 1967 (University Microfilms, Ann Arbor, Mich.).
  11. M. Born and E. Wolf, Principles of Optics, 2nd (rev.) ed. (Pergamon Press, New York, 1964), p. 508.
  12. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 288.
  13. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Sec. 7.7.
  14. Reference 13, Sec. 9.1.
  15. Reference 12, p. 289.
  16. Here, as throughout this paper, we neglect the finite aperture of the hologram, concentrating our attention on nonlinear effects rather than on diffraction and its limitations of resolution.
  17. Reference 12, p. 281.
  18. J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
    [Crossref]
  19. J. H. Van Vleck and D. Middleton, Proc. IEEE 54, 2 (1967).
    [Crossref]
  20. Reference 12, p. 163.

1967 (4)

J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
[Crossref] [PubMed]

A. A. Friesem and J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
[Crossref] [PubMed]

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
[Crossref]

J. H. Van Vleck and D. Middleton, Proc. IEEE 54, 2 (1967).
[Crossref]

1966 (2)

1965 (2)

1962 (1)

1948 (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd (rev.) ed. (Pergamon Press, New York, 1964), p. 508.

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 288.

Friesem, A. A.

Gabor, D.

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
[Crossref]

J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
[Crossref] [PubMed]

Haines, K. A.

Knight, G. R.

G. R. Knight, Ph.D. dissertation, Stanford University, Stanford, California, 1967 (University Microfilms, Ann Arbor, Mich.).

Kozma, A.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
[Crossref]

Leith, E. N.

Meier, R. W.

Middleton, D.

J. H. Van Vleck and D. Middleton, Proc. IEEE 54, 2 (1967).
[Crossref]

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Sec. 7.7.

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 288.

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 116.

Upatnieks, J.

van Ligten, R. F.

Van Vleck, J. H.

J. H. Van Vleck and D. Middleton, Proc. IEEE 54, 2 (1967).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd (rev.) ed. (Pergamon Press, New York, 1964), p. 508.

Zelenka, J. S.

Appl. Opt. (1)

Appl. Phys. Letters (1)

J. W. Goodman and R. W. Lawrence, Appl. Phys. Letters 11, 77 (1967).
[Crossref]

J. Opt. Soc. Am. (6)

Nature (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
[Crossref]

Proc. IEEE (1)

J. H. Van Vleck and D. Middleton, Proc. IEEE 54, 2 (1967).
[Crossref]

Other (10)

Reference 12, p. 163.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 116.

G. R. Knight, Ph.D. dissertation, Stanford University, Stanford, California, 1967 (University Microfilms, Ann Arbor, Mich.).

M. Born and E. Wolf, Principles of Optics, 2nd (rev.) ed. (Pergamon Press, New York, 1964), p. 508.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 288.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Sec. 7.7.

Reference 13, Sec. 9.1.

Reference 12, p. 289.

Here, as throughout this paper, we neglect the finite aperture of the hologram, concentrating our attention on nonlinear effects rather than on diffraction and its limitations of resolution.

Reference 12, p. 281.

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Figures (12)

Fig. 1
Fig. 1

Hologram-recording geometry.

Fig. 2
Fig. 2

Film nonlinearities: typical transmittance–vs–exposure curve.

Fig. 3
Fig. 3

The Kozma model of film nonlinearity.

Fig. 4
Fig. 4

Image-reconstruction geometry.

Fig. 5
Fig. 5

Pictorial representation of the effects of film nonlinearities. The nonlinear contributions may be found as multiple autoconvolutions of the ideal image produced by a linear film characteristic.

Fig. 6
Fig. 6

Transmittance–exposure characteristic of an ideal hard-limiting film.

Fig. 7
Fig. 7

Predicted first-order image-irradiance distribution for an object consisting of a uniformly bright diffuse patch, and a hard-limiting film. The reference irradiance is assumed to be much greater than the object irradiance.

Fig. 8
Fig. 8

Measured transmittance–vs–exposure curve for Kodak 649F spectroscopic plate, exposed to 6328-Å light, developed 5 min in HRP developer. (Courtesy of M. Lehmann.)

Fig. 9
Fig. 9

Predicted images for the nonlinearities of the Kodak 649F spectroscopic plate. The object is assumed to be a uniformly bright diffuse patch. (a) Object-radiance distribution; (b) Irradiance distribution in the first-order image, reference–to–object irradiance ratio 2.5 to 1; (c) Irradiance distribution in the first-order image, reference–to–object irradiance ratio 1.25 to 1.

Fig. 10
Fig. 10

Experimentally obtained twin images of a rectangular aperture backlighted through a diffuser. The recording medium is Kodak 649F spectroscopic plate, the reference–object irradiance ratio is 1.4 to 1, and the average amplitude transmittance is 0.5.

Fig. 11
Fig. 11

Images for conditions stated in Fig. 10, except the average amplitude transmittance is 0.24, which places the operating point in a region of greater nonlinearities.

Fig. 12
Fig. 12

Images of a bar-chart transparency illuminated through a diffuser. (a) Conventional photographs of the object and coplanar reference; (b) Holographic images showing nonlinear effects.

Equations (34)

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U ( x ) = k exp [ j π λ d 0 ( x - ξ r ) 2 ] + a ( x ) exp [ j ϕ ( x ) ] exp ( j π x 2 λ d 0 ) ,
E ( x ) = τ U ( x ) 2 = τ { k 2 + a 2 ( x ) + 2 k a ( x ) cos [ 2 π a x + ϕ ( x ) + Ω ] } ,
α sin θ / λ .
E ( x ) = τ a 2 ( x ) + 2 τ k a ( x ) cos [ 2 π α x + ϕ ( x ) + Ω ] .
R t ( Δ x ) = i ( x ) t ( x + Δ x ) ,
I i ( p ) = K 1 - R t ( Δ x ) exp ( - j 2 π λ f p Δ x ) d Δ x ,
R t ( Δ x ) = 1 ( 2 π j ) 2 C C f ( w 1 ) f ( w 2 ) M E ( w 1 , w 2 ) d w 1 d w 2 ,
f ( w ) = - t ( E ) e - w E d E ,
M E ( w 1 , w 2 ) = exp ( w 1 E 1 + w 2 E 2 ) ,
U 0 ( x ) = a ( x ) exp [ j ϕ ( x ) ] ,
R 0 ( Δ x ) = U 0 ( x ) U 0 * ( x + Δ x )
R 0 ( Δ x ) = K 2 - I 0 ( ξ ) exp ( - j 2 π λ f Δ x ξ ) d ξ ,
E ( x ) τ k 2 + 2 τ k a ( x ) cos [ 2 π α x + ϕ ( x ) + Ω ] .
E ( x ) = 2 τ k a ( x ) cos [ 2 π α x + ϕ ( x ) + Ω ] .
R E ( Δ x ) = E ( x ) E * ( x + Δ x ) = τ 2 k 2 [ R 0 ( Δ x ) e - j 2 π α Δ x + R 0 * ( Δ x ) e j 2 π α Δ x ] ,
R E ( Δ x ) = R c ( Δ x ) cos 2 π α Δ x + R c s ( Δ x ) sin 2 π α Δ x ,
R c ( Δ x ) = 2 τ 2 k 2 Re { R 0 ( Δ x ) } R c s ( Δ x ) = 2 τ 2 k 2 Im { R 0 ( Δ x ) } .
M E ( w 1 , w 2 ) = exp { - 1 2 [ σ 2 w 1 2 + σ 2 w 2 2 + 2 R E ( Δ x ) w 1 w 2 ] } ,
σ 2 = 2 τ 2 k 2 a 2 .
R t ( Δ x ) = m = 0 h m 2 m ! R E m ( Δ x ) ,
h m = 1 2 π j C f ( w ) w m exp ( - 1 2 σ 2 w 2 ) d w .
I i ( p ) = K 1 m = 0 h m 2 m ! × [ - R E m ( Δ x ) exp ( - j 2 π λ f p Δ x ) d Δ x ] .
- R E ( Δ x ) exp ( - j 2 π λ f p Δ x ) d Δ x = K 3 [ I 0 ( p - ξ r ) + I 0 ( - p - ξ r ) ] ,
t ( E ) = K ν ( E ) ν ,
f ( w ) = K ν Γ ( ν + 1 ) / w ν + 1 .
R t ( Δ x ) = K ν 2 Γ 2 ( ν + 1 ) 4 l = 0 ν ( σ 2 / 2 ) ν - l Γ 2 [ ( ν - l + 2 ) / 2 ] R E l ( Δ x )
ρ t ( Δ x ) = ( 2 / π ) sin - 1 ρ E ( Δ x ) ,
M E ( j w 1 , j w 2 ) = 0 0 0 2 π 0 2 π p ( a 1 , a 2 , ϕ 1 , ϕ 2 ) × exp [ j ( w 1 E 1 + w 2 E 2 ) ] d ϕ 1 d ϕ 2 d a 1 d a 2 ,
p ( a 1 , a 2 , ϕ 1 , ϕ 2 ) = a 1 a 2 4 π Λ 1 2 exp { - 1 2 Λ 1 2 [ σ 2 ( a 1 2 + a 2 2 ) - 2 R c ( Δ x ) a 1 a 2 cos ( ϕ 2 - ϕ 1 ) - 2 R c s ( Δ x ) a 1 a 2 sin ( ϕ 2 - ϕ 1 ) ] }
Λ = [ σ 4 - R c 2 ( Δ x ) - R c s 2 ( Δ x ) ] 2 .
M E ( w 1 , w 2 ) = [ 1 - w 1 w 2 Λ 1 2 k 4 - j σ 2 k 2 ( w 1 + w 2 ) ] - 1 × exp { - 1 2 w 1 2 ( σ 2 - j w 2 Λ 1 2 k 2 ) + 2 w 1 w 2 ( R c cos 2 π α Δ x + R c s sin 2 π α Δ x ) + w 2 2 ( σ 2 - j w 1 Λ 1 2 k 2 ) 1 - w 1 w 2 Λ 1 2 k 4 - j σ 2 k 2 ( w 1 + w 2 ) } .
R t ( Δ x ) = K ν 2 Γ 2 ( ν + 1 ) ( k 2 τ ) 2 ν Λ 1 2 2 σ 4 m = 0 ν m σ 2 m [ e - j 2 π m α Δ x ( R c + j R c s ) m + e j 2 π m α Δ x ( R c - j R c s ) m ] l = 0 [ ( ν - m ) / 2 ] n = 0 [ ( ν - m ) / 2 ] [ 2 Λ 1 2 σ 2 k 2 τ ] 2 ν - m - l - n × ( ν - l ) ! ( ν - n ) ! m ! l ! n ! ( m + l ) ! ( m + n ) ! ( ν - 2 l - m ) ! ( ν - 2 n - m ) ! F 2 1 ( ν - n + 1 , ν - l + 1 ; m + 1 ; R c 2 + R c s 2 σ 4 ) ,
t ˆ ( E ) - t 0 = K 1 E + K 2 ( E ) 2 + + K N ( E ) N ,
t ˆ ( E ) - 0.5 = - 1.37 × 10 - 2 E + 1.044 × 10 - 4 ( E ) 2 - 1.358 × 10 - 7 ( E ) 3 ,