Abstract

When an object and collecting pupil are separated by a medium that has random spatial and temporal variations of refractive index, a conventionally formed image may be severely degraded over the entire field of view of the imaging system. By using lensless Fourier-transform holography, considerable improvement of image resolution can be obtained within a limited field of view. The nature of the degradation of the reconstructed images is analyzed for both long and short exposures under the assumption that the random log amplitude and phase fluctuations across the collecting pupil are locally stationary processes with gaussian statistics. Experimental results support the analysis in a qualitative manner.

© 1968 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  2. G. O. Reynolds and T. J. Skinner, J. Opt. Soc. Am. 54, 1302 (1964).
    [Crossref]
  3. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  4. D. L. Fried, J. Opt. Soc. Am. 56, 1381 (1966).
  5. J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
    [Crossref]
  6. M. J. Beran and G. B. Parrent, Theory of Partial Coherence, (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 175.
  7. Here we are not concerned with the practical aspects of how the object illumination and reference are provided. However, we do assume that both are linearly polarized in the same direction and that the state of polarization is not changed by the perturbing medium.
  8. See Ref. 3, p. 1374.
  9. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [Crossref]
  10. Here we neglect a constant as well as a quadratic phase factor that would vanish later.
  11. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]
  12. The analysis of this section becomes invalid if the irradiance fluctuations across the reference beam are sufficiently large to drive the hologram exposure out of the linear region of the transmittance-exposure curve. However, this problem can be eliminated by pre-exposing the photographic plate.
  13. Ref. 3, p. 1375.
  14. Ref. 11, Eq. (3.3). This equation cannot be strictly correct because of its behavior for large r2, but its use is justified for the purpose of computing image resolution.
  15. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Company, New York, 1965), p. 226, Eq. (7–14).

1967 (1)

1966 (4)

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
[Crossref]

D. L. Fried, J. Opt. Soc. Am. 56, 1381 (1966).

J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
[Crossref]

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

1964 (2)

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence, (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 175.

Fried, D. L.

Goodman, J. W.

J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
[Crossref]

Hufnagel, R. E.

Huntley, W. H.

J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
[Crossref]

Jackson, D. W.

J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
[Crossref]

Kozma, A.

Lehmann, M.

J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Company, New York, 1965), p. 226, Eq. (7–14).

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence, (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 175.

Reynolds, G. O.

Skinner, T. J.

Stanley, N. R.

Appl. Phys. Letters (1)

J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, Appl. Phys. Letters 8, 311 (1966).
[Crossref]

J. Opt. Soc. Am. (6)

Other (8)

Here we neglect a constant as well as a quadratic phase factor that would vanish later.

The analysis of this section becomes invalid if the irradiance fluctuations across the reference beam are sufficiently large to drive the hologram exposure out of the linear region of the transmittance-exposure curve. However, this problem can be eliminated by pre-exposing the photographic plate.

Ref. 3, p. 1375.

Ref. 11, Eq. (3.3). This equation cannot be strictly correct because of its behavior for large r2, but its use is justified for the purpose of computing image resolution.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Company, New York, 1965), p. 226, Eq. (7–14).

M. J. Beran and G. B. Parrent, Theory of Partial Coherence, (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 175.

Here we are not concerned with the practical aspects of how the object illumination and reference are provided. However, we do assume that both are linearly polarized in the same direction and that the state of polarization is not changed by the perturbing medium.

See Ref. 3, p. 1374.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Hologram-recording geometry.

Fig. 2
Fig. 2

Geometry for reconstructing the images.

Fig. 3
Fig. 3

Sketch of m ˜ ( r 2 ; 0 ) showing how the maximum value depends on the strength of the turbulence.

Fig. 4
Fig. 4

Sketch of m ˜ ( r 2 ; ) showing how the width depends on the strength of the turbulence.

Fig. 5
Fig. 5

Conventional photograph of the object used in the experiment with no perturbing medium.

Fig. 6
Fig. 6

One of the reconstructed images of the object with no perturbing medium. (The bright rings around the reference position result from film halation.)

Fig. 7
Fig. 7

Long-exposure conventional photograph with electric heater under the propagation path, showing blurring of entire image.

Fig. 8
Fig. 8

Reconstructed images for long-exposure case with electric heater under propagation path. Note how the image irradiance falls off with distance from the reference position, but that the resolution over the entire visible part of the image is quite good.

Fig. 9
Fig. 9

Conventional photograph for simulated short-exposure case with perturbing plate directly in the propagation path, displaying degradation over the entire image.

Fig. 10
Fig. 10

Reconstructed images for simulated short-exposure case. (The bright rings are a result of interference arising from reflections between the surfaces of the glass plate.) Notice how the resolution near the reference position is better than at any point in Fig. 9, and at the points farthest from the reference position the resolution is about the same in each figure.

Equations (71)

Equations on this page are rendered with MathJax. Learn more.

u 1 ( x 1 , y 1 ) = r ( x 1 , y 1 ) + s ( x 1 , y 1 ) .
u 2 ( x 2 ) = u 1 ( x 1 ) h ( x 2 ; x 1 ) d x 1 ,
h ( x 2 ; x 1 ) = h 0 ( x 2 ; x 1 ) exp { j [ k z + ( π / λ z ) h ( x 2 x 1 ) 2 ] } ,
h ( x 2 ; x 1 ) = h 0 ( x 2 ; x 1 ) exp ψ ( x 2 ; x 1 ) ,
ψ ( x 2 ; x 1 ) = L ( x 2 ; x 1 ) + j Φ ( x 2 ; x 1 ) .
u 2 ( x 2 ) = exp { j [ k z + π x 2 2 λ z ] } u 1 ( x 1 ) × exp { j π x 1 2 λ z + ψ ( x 2 ; x 1 ) } exp { j 2 π ( x 2 λ z ) x 1 } d x 1 .
I 2 ( x 2 ) = u 2 ( x 2 ) u 2 * ( x 2 ) .
υ ( x 1 ) u 1 ( x 1 ) exp { j ( π x 1 2 / λ z ) + ψ ( x 2 ; x 1 ) } ,
I 2 ( x 2 ) = υ ( x 1 ) υ * ( x 1 ) × exp { j 2 π ξ 2 ( x 1 x 1 ) } d x 1 d x 1 .
I 2 ( x 2 ) = υ ( x 1 ) υ * ( x 1 r 1 ) exp { j 2 π ξ 2 r 1 } d x 1 d r 1 = F { υ ( x 1 ) υ * ( x 1 r 1 ) d x 1 } ,
υ 0 ( x 1 ) u 1 ( x 1 ) exp j { π x 1 2 / λ z } ,
I 2 ( x 2 ) = F { υ 0 ( x 1 ) υ 0 * ( x 1 r 1 ) × exp [ ψ ( x 2 ; x 1 ) + ψ * ( x 2 ; x 1 r 1 ) ] d x 1 } .
I 2 ( x 2 ) = F { υ 0 ( x 1 ) υ 0 * ( x 1 r 1 ) × exp [ ψ ( x 2 ; x 1 ) + ψ * ( x 2 ; x 1 r 1 ) ] d x 1 } .
ψ ( x 2 ; x 1 ) + ψ * ( x 2 ; x 1 r 1 ) = [ L ( x 2 ; x 1 ) + L ( x 2 ; x 1 r 1 ) ] + j [ Φ ( x 2 ; x 1 ) Φ ( x 2 ; x 1 r 1 ) ] ,
[ L ( x 2 ; x 1 ) + L ( x 2 ; x 1 r 1 ) ] = 2 L ¯ ,
[ Φ ( x 2 ; x 1 ) Φ ( x 2 ; x 1 r 1 ) ] = 0.
C G ( r 2 ; r 1 ) = [ G ( x 2 ; x 1 ) G ¯ ] [ G ( x 2 r 2 ; x 1 r 1 ) G ¯ ] ,
D G ( r 2 ; r 1 ) = [ G ( x 2 ; x 1 ) G ( x 2 r 2 ; x 1 r 1 ) ] 2 = 2 C G ( 0 ; 0 ) 2 C G ( r 2 ; r 1 ) ,
exp { ψ ( x 2 ; x 1 ) + ψ * ( x 2 ; x 1 r 1 ) } = exp { 1 2 D L ( 0 ; r 1 ) 1 2 D Φ ( 0 ; r 1 ) } = exp { 1 2 D ( 0 ; r 1 ) } ,
exp { 1 2 D ( 0 ; r 1 ) } m ( r 1 ) ,
I 2 ( x 2 ) = F { υ 0 ( x 1 ) υ 0 * ( x 1 r 1 ) m ( r 1 ) d x 1 } = F { m ( r 1 ) [ υ 0 ( r 1 ) υ 0 ( r 1 ) ] } ,
E 2 ( x 2 ) = τ I 2 ( x 2 ) .
t 2 ( x 2 ) = t 0 χ E 2 ( x 2 ) = t 0 χ τ I 2 ( x 2 ) ,
u 3 ( x 3 ) = F { t 2 ( x 2 ) p 2 ( x 2 ) } = T 2 ( ξ 3 ) * P 2 ( ξ 3 ) ,
T 2 ( ξ 3 ) = F { t 0 χ τ I 2 ( x 2 ) } = t 0 δ ( ξ 3 ) χ τ F { I 2 ( x 2 ) } ,
Δ T 2 ( ξ 3 ) = χ τ F ( F { m ( r 1 ) [ υ 0 ( r 1 ) υ 0 ( r 1 ) ] } ) = χ τ λ z [ m ( x 3 z f ) ] [ υ 0 ( x 3 z f ) υ 0 ( x 3 z f ) ] .
Δ u 3 ( x 3 ) = { m ( x 3 z f ) [ υ 0 ( x 3 z f ) υ 0 ( x 3 z f ) ] } * P 2 ( x 2 λ f ) ,
Δ I 3 ( x 3 ) = Δ u 3 ( x 3 ) Δ u 3 * ( x 3 ) = | { m ( x 3 z f ) × [ υ 0 ( x 3 z f ) υ 0 ( x 3 z f ) ] } * P 2 ( x 3 λ f ) | 2 ,
u 1 ( x 1 ) = a r δ [ x 1 ( d / 2 ) ] + a s δ [ x 1 + ( d / 2 ) ] .
υ 0 ( β ) = u 1 ( β ) exp { j ( π β 2 / λ z ) } = { a r δ [ β ( d / 2 ) ] + a s δ [ β + ( d / 2 ) ] } × exp { j ( π d 2 / 4 λ z ) } .
υ 0 ( β ) υ 0 ( β ) = [ a r 2 + a s 2 ] δ ( β ) + a r a s [ δ ( β d ) + δ ( β + d ) ] .
Δ I 3 ( x 3 ) = | { m ( β ) [ δ ( β d ) + δ ( β + d ) ] } * P 2 ( β / λ z ) | 2 = | m ( d ) P 2 [ ( β d ) / λ z ] + m ( d ) P 2 [ ( β + d ) / λ z ] | 2 .
Δ I 3 ( x 3 ) = m 2 ( d ) [ P 2 2 ( x 3 λ f d λ z ) + P 2 2 ( x 3 λ f + d λ z ) ] .
m 2 ( d ) = exp { D ( 0 ; d ) } .
υ ( x 2 ; x 1 ) u 1 ( x 1 ) exp { j ( π x 1 2 / λ z ) + ψ ( x 2 ; x 1 ) } .
w ( x 2 ; x 1 ) υ ( x 2 ; x 1 ) υ ( x 2 ; x 1 ) ,
I 2 ( x 2 ) = F { υ ( x 2 ; x 1 ) υ * ( x 2 ; x 1 r 1 ) d x 1 } = w ( x 2 ; r 1 ) exp { j 2 π ξ 2 r 1 } d r 1 .
t 2 ( x 2 ) = t 0 χ τ I 2 ( x 2 ) ,
u 3 ( x 3 ) = t 2 ( x 2 ) p 2 ( x 2 ) exp { j 2 π ξ 3 x 2 } d x 2 ,
Δ u 3 ( x 3 ) = I 2 ( x 2 ) p 2 ( x 2 ) exp { j 2 π ξ 3 x 2 } d x 2 = w ( x 2 ; r 1 ) p 2 ( x 2 ) × exp { j 2 π [ ( r 1 / λ z ) + ξ 3 ] x 2 } d r 1 d x 2 .
Δ I 3 ( x 3 ) = w ( x 2 ; r 1 ) w * ( x 2 ; r 1 ) p 2 ( x 2 ) p 2 ( x 2 ) × exp { j 2 π [ ( r 1 λ z + ξ 3 ) x 2 ( r 1 λ z + ξ 3 ) x 2 ] } × d r 1 d x 2 d r 1 d x 2 .
u 1 ( x 1 ) = a r δ [ x 1 ( d / 2 ) ] + a s δ [ x 1 + ( d / 2 ) ] ,
υ ( x 2 ; x 1 ) = a r δ ( x 1 d 2 ) exp { j π d 2 4 λ z + ψ ( x 2 ; d 2 ) } + a s δ ( x 1 + d 2 ) exp { j π d 2 4 λ z + ψ ( x 2 ; d 2 ) } .
w ( x 2 ; x 1 ) = a r 2 δ ( x 1 ) exp { ψ ( x 2 ; d 2 ) + ψ * ( x 2 ; d 2 ) } + a s 2 δ ( x 1 ) exp { ψ ( x 2 ; d 2 ) + ψ * ( x 2 ; d 2 ) } + a r a s δ ( x 1 d ) exp { ψ ( x 2 ; d 2 ) + ψ * ( x 2 ; d 2 ) } + a r a s δ ( x 1 + d ) exp { ψ ( x 2 ; d 2 ) + ψ * ( x 2 ; d 2 ) } .
exp { ψ [ x 2 ; ( d / 2 ) ] + ψ * [ x 2 ; ( d / 2 ) ] } θ ( x 2 )
Δ w ( x 2 ; x 1 ) = a r a s [ δ ( x 1 d ) θ ( x 2 ) + δ ( x 1 + d ) θ * ( x 2 ) ] .
Δ w ( x 2 ; x 1 ) Δ w * ( x 2 ; x 1 ) = a r 2 a s 2 [ δ ( x 1 d ) δ ( x 1 d ) θ ( x 2 ) θ * ( x 2 ) + δ ( x 1 + d ) δ ( x 1 + d ) θ * ( x 2 ) θ ( x 2 ) + δ ( x 1 d ) δ ( x 1 + d ) θ ( x 2 ) θ ( x 2 ) + δ ( x 1 + d ) δ ( x 1 d ) θ * ( x 2 ) θ * ( x 2 ) ] .
Δ I 3 ( x 3 ) = [ θ ( x 2 ) θ * ( x 2 ) exp { j 2 π ( d λ z ) ( x 2 x 2 ) } + θ * ( x 2 ) θ ( x 2 ) exp { j 2 π ( d λ z ) ( x 2 x 2 ) } ] p 2 ( x 2 ) p 2 ( x 2 ) exp { j 2 π ξ 3 ( x 2 x 2 ) } d x 2 d x 2 .
Δ I 3 ( x 3 ) = [ θ ( x 2 ) θ * ( x 2 r 2 ) exp { j 2 π α r 2 } = + θ * ( x 2 ) θ ( x 2 r 2 ) exp { j 2 π α r 2 } ] p 2 ( x 2 ) p 2 ( x 2 r 2 ) × exp { j 2 π ξ 3 r 2 } d x 2 d r 2 .
θ ( x 2 ) θ * ( x 2 r 2 ) m ˜ ( r 2 ; d ) ,
Δ I 3 ( x 3 ) = [ m ˜ ( r 2 ; d ) exp { j 2 π α r 2 } + m ˜ * ( r 2 ; d ) × exp { j 2 π α r 2 } ] p 2 ( x 2 ) p 2 ( x 2 r 2 ) × exp { j 2 π ξ 3 r 2 } d x 2 d r 2 .
Δ I 3 ( x 3 ) = | P 2 ( ξ 3 ) | 2 * [ M ˜ ( ξ 3 + α ; d ) + M ˜ * ( ξ 3 + α ; d ) ] .
Δ I 3 ( x 3 ) = P 2 2 ( x 3 λ f ) * [ M ˜ ( x 3 λ f + d λ z ; d ) + M ˜ ( x 3 λ f d λ z ; d ) ] .
m ˜ ( r 2 ; d ) = exp { 2 C L ( 0 ; 0 ) + 2 C L ( r 2 ; 0 ) + 2 C L ( 0 ; d ) + C L ( r 2 ; d ) + C L ( r 2 ; d ) 2 C Φ ( 0 ; 0 ) + 2 C Φ ( r 2 ; 0 ) + 2 C Φ ( 0 ; d ) C Φ ( r 2 ; d ) C Φ ( r 2 ; d ) + j [ 2 C L Φ ( r 2 ; d ) 2 C L Φ ( r 2 ; d ) ] } ,
m ˜ ( r 2 ; 0 ) = exp { 4 C L ( r 2 ; 0 ) } .
m ˜ ( r 2 ; ) = exp { D ( r 2 ; 0 ) } ,
MTF = exp { 1 2 D ( r 2 ; 0 ) }
D ( r 2 ; 0 ) = A | r 2 | 5 / 3 ,
G = G ¯
H = H ¯
μ j k = ( G G ¯ ) j ( H H ¯ ) k .
exp { G + j H } = exp { G ¯ + j H ¯ + 1 2 [ μ 20 + 2 j μ 11 μ 02 ] } .
G = L ( x 2 ; d 2 ) + L ( x 2 ; d 2 ) + L ( x 2 r 2 ; d 2 ) + L ( x 2 r 2 ; d 2 )
H = Φ ( x 2 ; d 2 ) Φ ( x 2 ; d 2 ) Φ ( x 2 r 2 ; d 2 ) + Φ ( x 2 r 2 ; d 2 ) ,
G ¯ = 4 L ¯
H ¯ = 0
μ 20 = 4 C L ( 0 ; 0 ) + 4 C L ( r 2 ; 0 ) + 4 C L ( 0 ; d ) + 2 C L ( r 2 ; d ) + 2 C L ( r 2 ; d )
μ 11 = 2 C L Φ ( r 2 ; d ) 2 C L Φ ( r 2 ; d )
μ 02 = 4 C Φ ( 0 ; 0 ) 4 C Φ ( r 2 ; 0 ) 4 C Φ ( 0 ; d ) + 2 C Φ ( r 2 ; d ) + 2 C Φ ( r 2 ; d ) ,
C L Φ ( r 2 ; r 1 ) = [ L ( x 2 ; x 1 ) L ¯ ] [ Φ ( x 2 r 2 ; x 1 r 1 ) Φ ¯ ] .
exp { G + j H } = exp { 2 C L ( 0 ; 0 ) + 2 C L ( r 2 ; 0 ) + 2 C L ( 0 ; d ) + C L ( r 2 ; d ) + C L ( r 2 ; d ) 2 C Φ ( 0 ; 0 ) + 2 C Φ ( r 2 ; 0 ) + 2 C Φ ( 0 ; d ) C Φ ( r 2 ; d ) C Φ ( r 2 ; d ) + j [ 2 C L Φ ( r 2 ; d ) 2 C L Φ ( r 2 ; d ) ] } .