Abstract

The use of weak phase-retarding diffusers in coherent imaging systems is analyzed. It is shown that ringing caused by diffraction from dust or blemishes can be significantly reduced by a diffuser that scatters light slightly beyond the first interference minimum of the diffraction pattern, provided that the specular transmission of the diffuser is small. Weak phase diffusers, whose phase standard deviations are a small part of a wavelength, accomplish this is an optimal manner. The characteristics of the speckle patterns that consititute their spatially filtered images are found to depend not only on the size of the aperture of the imaging system, but on the Wiener spectrum of the phase and the statistical law that the phase obeys. It is shown that random diffusers with normally distributed phase can easily be constructed that produce image speckle patterns whose Wiener spectra consist primarily of high spatial frequencies, if the size of the aperture of the imaging system is greater than a specified minimum. Speckle patterns are also found to exist near the image of a phase diffuser even when the aperture of the imaging system is large, and their Wiener spectra are functions of their distances from the image of the diffuser.

© 1975 Optical Society of America

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References

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  1. L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
    [CrossRef]
  2. H. J. Gerritsen, W. J. Hannan, and E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
    [CrossRef] [PubMed]
  3. J. Upatnieks and R. W. Lewis, Appl. Opt. 12, 2161 (1973).
    [CrossRef] [PubMed]
  4. J. Upatnieks, Appl. Opt. 6, 1905 (1967).
    [CrossRef] [PubMed]
  5. J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966), p. 79.
  6. E. D. Rainville, Special Functions (Macmillan, New York, 1960), p. 56.
  7. J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.
  8. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 101.
  9. J. C. Dainty and W. T. Welford, Opt. Commun. 3, 289 (1971).
    [CrossRef]
  10. H. J. Caulfield, in Proceedings of the SPIE Seminar on Developments in Holography, Vol. 25 (Redondo Beach, Calif., 1971), p. 111.
    [CrossRef]
  11. R. C. Waag and K. T. Knox, J. Opt. Soc. Am. 62, 877 (1972).
    [CrossRef]
  12. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  13. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [CrossRef]
  14. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [CrossRef]
  15. J. C. Dainty, J. Opt. Soc. Am. 62, 595 (1972).
    [CrossRef]
  16. H. H. Hopkins, Proc. R. Soc. (London) A 217, 408 (1953).
    [CrossRef]
  17. J. C. Dainty, Opt. Acta 18, 327 (1971).
    [CrossRef]
  18. R. Barakat, Opt. Acta 20, 729 (1973).
    [CrossRef]
  19. A. A. Scribot, Opt. Commun. 11, 348 (1974).
    [CrossRef]
  20. C. N. Kurtz, J. Opt. Soc. Am. 62, 982 (1972).
    [CrossRef]
  21. M. J. Beran and G. P. Parrent, Theory of Partial Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), p. 118.
  22. P. S. Considine, J. Opt. Soc. Am. 56, 1001 (1966).
    [CrossRef]
  23. S. Lowenthal and D. Joyeux, J. Opt. Soc. Am. 61, 847 (1971)
    [CrossRef]

1974 (1)

A. A. Scribot, Opt. Commun. 11, 348 (1974).
[CrossRef]

1973 (2)

1972 (3)

1971 (3)

S. Lowenthal and D. Joyeux, J. Opt. Soc. Am. 61, 847 (1971)
[CrossRef]

J. C. Dainty, Opt. Acta 18, 327 (1971).
[CrossRef]

J. C. Dainty and W. T. Welford, Opt. Commun. 3, 289 (1971).
[CrossRef]

1970 (1)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

1968 (1)

1967 (2)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
[CrossRef]

J. Upatnieks, Appl. Opt. 6, 1905 (1967).
[CrossRef] [PubMed]

1966 (1)

1965 (2)

1953 (1)

H. H. Hopkins, Proc. R. Soc. (London) A 217, 408 (1953).
[CrossRef]

Barakat, R.

R. Barakat, Opt. Acta 20, 729 (1973).
[CrossRef]

Battin, R. H.

J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966), p. 79.

Beran, M. J.

M. J. Beran and G. P. Parrent, Theory of Partial Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), p. 118.

Caulfield, H. J.

H. J. Caulfield, in Proceedings of the SPIE Seminar on Developments in Holography, Vol. 25 (Redondo Beach, Calif., 1971), p. 111.
[CrossRef]

Considine, P. S.

Dainty, J. C.

J. C. Dainty, J. Opt. Soc. Am. 62, 595 (1972).
[CrossRef]

J. C. Dainty and W. T. Welford, Opt. Commun. 3, 289 (1971).
[CrossRef]

J. C. Dainty, Opt. Acta 18, 327 (1971).
[CrossRef]

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Enloe, L. H.

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
[CrossRef]

Gerritsen, H. J.

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Hannan, W. J.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. (London) A 217, 408 (1953).
[CrossRef]

Joyeux, D.

Knox, K. T.

Kurtz, C. N.

Laning, J. H.

J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.

Lewis, R. W.

Lowenthal, S.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 101.

Parrent, G. P.

M. J. Beran and G. P. Parrent, Theory of Partial Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), p. 118.

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966), p. 79.

Rainville, E. D.

E. D. Rainville, Special Functions (Macmillan, New York, 1960), p. 56.

Ramberg, E. G.

Scribot, A. A.

A. A. Scribot, Opt. Commun. 11, 348 (1974).
[CrossRef]

Upatnieks, J.

Waag, R. C.

Welford, W. T.

J. C. Dainty and W. T. Welford, Opt. Commun. 3, 289 (1971).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Acta (3)

J. C. Dainty, Opt. Acta 18, 327 (1971).
[CrossRef]

R. Barakat, Opt. Acta 20, 729 (1973).
[CrossRef]

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Opt. Commun. (2)

A. A. Scribot, Opt. Commun. 11, 348 (1974).
[CrossRef]

J. C. Dainty and W. T. Welford, Opt. Commun. 3, 289 (1971).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

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

H. H. Hopkins, Proc. R. Soc. (London) A 217, 408 (1953).
[CrossRef]

Other (6)

M. J. Beran and G. P. Parrent, Theory of Partial Coherence (Prentice–Hall, Englewood Cliffs, N. J., 1964), p. 118.

H. J. Caulfield, in Proceedings of the SPIE Seminar on Developments in Holography, Vol. 25 (Redondo Beach, Calif., 1971), p. 111.
[CrossRef]

J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966), p. 79.

E. D. Rainville, Special Functions (Macmillan, New York, 1960), p. 56.

J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956), p. 83.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 101.

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Figures (9)

FIG. 1
FIG. 1

Imaging system that includes a diffuser. The coherent source S illuminates a diffuser D through the lens L1, and the image of the diffuser is in the plane F. A dust particle is in plane P.

FIG. 2
FIG. 2

Comparison of diffraction patterns produced by a particle. Curve (a) is the irradiance distribution of the pattern without a diffuser and curve (b) is the irradiance distribution when the system contains a diffuser with a scattered-light distribution shown by the dotted lines. The particle is a square 0.2 mm on a side, the distance to the observation plane is 100 mm, and the wavelength of the light is 500 nm.

FIG. 3
FIG. 3

Regions of integration in the ωx1ωy1 domain of Eq. (14). The solid circles in (a) and (c) outline the regions where s1(ω1) and s1(ω1 + ω) are not zero and the solid circles in (b) and (d) outline the region where s2(ω) is not zero. Within the dotted circles, F(ω1) and F(ω1 + ω) are unity.

FIG. 4
FIG. 4

Speckle Wiener spectra IN(ωx, 0) of a diffuser whose phase variance is 1.0 and whose phase Wiener spectrum is as indicated by the dotted lines. Curve (a) was calculated with Ω2 equal to 12.5 and curve (b) with Ω2 equal to 18.5. For both curves, Ω1, is 13.5.

FIG. 5
FIG. 5

Contrast of the speckle pattern as a function of Ω2. The diffuser has a phase variance of 1.0 and the phase Wiener spectrum is as described in Fig. 4. Curve (a) is the contrast without spatial filtering and curve (b) is the contrast after removal of irradiance variations with spatial frequencies greater than 13.5.

FIG. 6
FIG. 6

Fraction of light scattered as a function of the standard deviation of the phase of a weak diffuser. Curve (a) characterizes a random diffuser whose phase is normally distributed, and curve (b) a diffuser whose phase distribution obeys the negative-exponential law.

FIG. 7
FIG. 7

Scattered-light flux as a function of the spatial frequency of the field at the surface of a weak diffuser. Curves (a), (b), (c), and (d) correspond to phase variances of 0.5, 1.0, 2.0, and 4.0, respectively. The phase Wiener spectrum T(ω) is indicated by the dotted line.

FIG. 8
FIG. 8

Speckle-pattern Wiener spectra IN(ωx, 0) produced by diffusers whose phases have the same Wiener spectrum, but which obey different statistical laws. Curves (a) and (b) characterize a negative-exponential diffuser with Ω2 = 13.5 and 16.5, respectively. Curves (c) and (d) characterize normally distributed diffuser with the same Ω2. The phase variance is 0.5 is all cases.

FIG. 9
FIG. 9

Speckle Wiener spectra IN(ωx, 0) for three distances from diffuser to observation plane. Curves (a), (b), and (c) describe Wiener spectra for β = 0.02, 0.05, and 0.1, respectively. The phase Wiener spectrum is shown by the dotted line and the phase variance is 0.5.

Equations (42)

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h ( x , ω ) = α ( ω ) [ 1 - f ( x - λ ω z 2 π ) ] exp ( i π x 2 λ z 0 ) ,
I ( x ) = 1 ( 2 π ) 4 h ( x , ω ) h * ( x , ω ) d ω d ω ,
α * ( ω ) α ( ω ) = ( 2 π ) 2 A ( ω ) δ ( ω - ω ) ,
I ( x ) = 1 ( 2 π ) 2 A ( ω ) { 1 + | f ( x - λ ω z 2 π ) | 2 - 2 Re [ exp ( - i π x 2 λ z 0 ) f ( x - λ ω z 2 π ) ] } d ω .
A ( ω ) = ( 2 π ) 2 δ ( ω )
A ( ω ) = 1 3 ( 2 π ) 2 δ ( ω ) + 2 3 ( 4 π / Ω c 2 ) , if ω Ω c = 0 , if ω > Ω c .
f ( x ) = - i b λ R ( x ) e i k R ( x ) sinc ω b 2 ,
g ( ω ) = 1 ( 2 π ) 2 α * ( ω 1 ) α ( ω 1 + ω ) F * ( ω 1 ) F ( ω 1 + ω ) d ω 1 .
a ( x ) = e i θ ( x ) ,
α ( ω ) = n = 0 ( i ) n n ! s n ( ω ) .
s 0 ( ω ) = ( 2 π ) 2 δ ( ω ) .
g ( ω ) = 1 ( 2 π ) 2 n = 0 k = 0 n i 2 k - n k ! ( n - k ) ! s n - k * ( ω 1 ) s k ( ω 1 + ω ) × F * ( ω 1 ) F ( ω 1 + ω ) d ω 1 .
g 0 ( ω ) = ( 2 π ) 2 δ ( ω ) .
g 2 ( ω ) = 1 ( 2 π ) 2 s 1 * ( ω 1 ) s 1 ( ω 1 + ω ) F ( ω 1 ) F ( ω 1 + ω ) d ω 1 - 1 ( 2 π ) 2 s 2 * ( ω 1 ) s 0 ( ω 1 + ω ) F ( ω 1 ) F ( ω 1 + ω ) d ω 1 .
g 2 ( ω ) = 1 ( 2 π ) 2 s 1 * ( ω 1 ) s 1 ( ω 1 + ω ) F ( ω 1 ) F ( ω 1 + ω ) d ω 1 - s 2 ( ω ) F ( ω ) .
s n ( ω ) = 0 ,             if             ( ω x 2 + ω y 2 ) 1 / 2 > n Ω 1 .
F ( ω ) = cyl ( ω Ω 2 ) = 1 ,             if             ( ω x 2 + ω y 2 ) 1 / 2 Ω 2 = 0 ,             if             ( ω x 2 + ω y 2 ) 1 / 2 > Ω 2 ,
( 2 π ) 2 I ( ω ) δ ( ω - ω ) = m = 0 n = 0 g m * ( ω ) g n ( ω ) ,
g m * ( ω ) g n ( ω ) = 1 ( 2 π ) 4 l = 0 m k = 0 n i [ 2 ( k - l ) + ( m - n ) ] k ! l ! ( n - k ) ! ( m - l ) ! × s m - l ( ω 1 ) s l * ( ω 1 + ω ) s n - k * ( ω 2 ) s k ( ω 2 + ω ) × F ( ω 1 ) F * ( ω 1 + ω ) F * ( ω 2 ) F ( ω 2 + ω ) d ω 1 d ω 2 .
s n ( ω ) = 1 ( 2 π ) 2 s k * ( ω ) s n - k ( ω + ω ) d ω .
I ( ω ) = 2 ( 2 π ) 2 T ( ω 1 + ω ) T ( ω 1 ) [ cyl ( ω 1 + ω Ω 2 ) cyl ( ω 1 Ω 2 ) - cyl ( ω Ω 2 ) ] 2 d ω 1 + ( 2 π ) 2 δ ( ω ) { 1 + { 1 ( 2 π ) 2 T ( ω 1 ) [ cyl ( ω 1 Ω 2 ) - 1 ] d ω 1 } 2 } + .
F ( ω ) = 1 ,             if ω x Ω 2 and ω y Ω 2 = 0 ,             if ω x > Ω 2 and ω y > Ω 2 ,
σ I = [ 1 ( 2 π ) 2 I N ( ω ) d ω ] 1 / 2 .
ϕ ( x ) = e - σ 2 exp [ τ ( x ) ] ,
G ( σ ) = 1 - e - σ 2 .
p ( θ ) = 1 σ exp [ - θ ( x ) σ ] .
s 1 ( ω ) = 1 ( 2 π ) 2 y * ( ω 1 ) y ( ω 1 + ω ) d ω 1 .
I ( ω ) = 2 ( 2 π ) 6 [ Y ( ω 1 ) Y ( ω 1 + ω 2 ) d ω 1 ] [ Y ( ω 3 ) Y ( ω 2 + ω 3 + ω ) d ω 3 ] [ cyl ( ω 2 + ω Ω 2 ) cyl ( ω 2 Ω 2 ) - cyl ( ω Ω 2 ) ] 2 d ω 2 + 4 ( 2 π ) 6 { Y ( ω 1 + ω 2 + ω ) [ cyl ( ω 1 + ω Ω 2 ) cyl ( ω 1 Ω 2 ) - cyl ( ω Ω 2 ) ] d ω 1 } 2 Y ( ω 2 + ω ) Y ( ω 2 ) d ω 2 + 2 ( 2 π ) 6 Y ( ω 1 + ω 2 ) Y ( ω 1 + ω 2 + ω 3 + ω ) Y ( ω 2 + ω 3 ) Y ( ω 2 ) [ cyl ( ω 1 + ω Ω 2 ) cyl ( ω 1 Ω 2 ) - cyl ( ω Ω 2 ) ] × [ cyl ( ω 3 + ω Ω 2 ) cyl ( ω 3 Ω 2 ) - cyl ( ω Ω 2 ) ] d ω 1 d ω 2 d ω 3 + ( 2 π ) 2 δ ( ω ) { 1 + { 1 ( 2 π ) 4 Y ( ω 1 ) Y ( ω 1 + ω 2 ) [ cyl ( ω 1 Ω 2 ) - 1 ] d ω 1 d ω 2 } 2 } + .
T ( ω ) = 1 ( 2 π ) 2 Y ( ω 1 ) Y ( ω 1 + ω ) d ω 1 + 1 ( 2 π ) 2 δ ( ω ) [ Y ( ω 1 ) d ω 1 ] 2 .
ϕ ( x ) = 1 1 + 2 σ 2 - τ ( x ) ,
G ( σ ) = σ 2 1 + σ 2 .
I ( s ) = ρ ( s - s ) I 0 ( s ) d s .
F ( ω ) = exp ( i λ ω 2 z 4 π ) ,
I ( ω ) = ( 2 π ) 2 δ ( ω ) + e - 2 σ 2 { 4 T ( ω ) sin 2 [ λ z ω 2 4 π ] + 8 T ( ω ) sin 2 [ λ z ω 2 4 π ] 1 ( 2 π ) 2 T ( ω 1 ) cos [ 2 λ z ω · ω 1 4 π ] d ω 1 + 8 1 ( 2 π ) 2 T ( ω 1 ) T ( ω 1 + ω ) sin 2 [ λ z ω · ω 1 4 π ] × sin 2 [ λ z ω · ( ω 1 - ω ) 4 π ] d ω 1 + } .
β = λ z Ω 1 2 / 4 π 2 ,
1 ( 2 π ) 2 s 1 ( ω 1 ) s 1 * ( ω 1 + ω ) s 1 * ( ω 2 ) s 1 ( ω 2 + ω ) × cyl ( ω 1 Ω 1 ) cyl ( ω 1 + ω Ω 2 ) cyl ( ω 2 Ω 2 ) cyl ( ω 2 + ω Ω 2 ) d ω 1 d ω 2 .
s 1 ( ω 1 ) s 1 * ( ω 1 + ω ) s 1 * ( ω 2 ) s 1 ( ω 2 + ω ) + s 1 ( ω 1 ) s 1 * ( ω 2 ) s 1 * ( ω 1 + ω ) s 1 ( ω 2 + ω ) + s 1 ( ω 1 ) s 1 ( ω 2 + ω ) s 1 * ( ω 1 + ω ) s 1 * ( ω 2 ) .
s 1 * ( ω ) s 1 ( ω ) = ( 2 π ) 2 T ( ω ) δ ( ω - ω ) .
s 1 ( ω ) = s 1 * ( - ω ) .
( 2 π ) 4 δ 2 ( ω ) [ T ( ω 1 ) cyl ( ω 1 Ω 2 ) d ω 1 ] 2 + 2 ( 2 π ) 2 δ ( 0 ) T ( ω 1 ) T ( ω 1 + ω ) cyl ( ω 1 Ω 2 ) cyl ( ω 1 + ω Ω 2 ) d ω 1 .
ĝ ( x ) = B ( x - x 1 ) B * ( x - x 2 ) exp { [ i θ ( x 1 ) - θ ( x 2 ) ] } d x 1 d x 2 ,
I ( x ) = B * ( x 1 + x ) B ( x 2 + x ) B * ( x 3 ) B ( x 4 ) × exp { i [ θ ( x 1 ) - θ ( x 2 ) + θ ( x 3 ) - θ ( x 4 ) ] } d x 1 d x 2 d x 3 d x 4 .