Abstract

The output speckle statistics are affected by deterministic aberrations present in an imaging system when the speckle is only partially developed. The speckle-power spectral density and intensity contrast are shown to depend on these aberrations. The rough target surface is modeled as a perfectly transmitting (reflecting) surface with uniform phase distribution over (−α, α), where απ. The effects of the type and severity of the primary aberrations on the normalized power spectral density and intensity contrast are examined.

© 1986 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
    [CrossRef]
  2. H. M. Pedersen, “Theory of speckle dependence on surface roughness,”J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [CrossRef]
  3. K. A. Stetson, “The vulnerability of speckle photography to lens aberrations,”J. Opt. Soc. Am. 67, 1587–1590 (1977).
    [CrossRef]
  4. R. Barakat, P. Nisenson, “Influence of the wavefront correlation function and deterministic wavefront aberrations on the speckle image-reconstruction problem in the high light level regime,”J. Opt. Soc. Am. 71, 1930–1402 (1981).
  5. K. A. O’Donnell, “Correlations of time-varying speckle near the focal plane,”J. Opt. Soc. Am. 72, 191–197 (1982).
    [CrossRef]
  6. R. D. Bahuguna, K. K. Gupta, K. Singh, “Speckle patterns of weak diffusers: effect of spherical aberrations,” Appl. Opt. 19, 1874–1878 (1980).
    [CrossRef] [PubMed]
  7. F. Thon, “Phase contrast electron microscopy,” in Electron Microscopy in Materials Science, U. Valdre, ed. (Academic, London, 1971).
    [CrossRef]
  8. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York1978), Chap. I.
  9. F. Thon, B. M. Siegel, “Experiments with optical image reconstruction of high resolution electron micrographs,” Ber. Bunsen Ges. Phys. Chem. 74, 1116–1120 (1970).
  10. B. Stoffregen, “Speckle statistics for general scattering objects: I. general relations for speckle amplitude and intensity,” Optik 52, 305–312 (1979).
  11. B. Stoffregen, “Speckle statistics for general scattering objects: II. mean, covariance and power spectrum of image speckle patterns,” Optik 52, 385–399 (1978/79).
  12. J. Ohtsubo, “Non-Gaussian speckle: a computer simulation,” Appl. Opt. 21, 4167–4175 (1982).
    [CrossRef] [PubMed]
  13. R. Barakat, “Direct derivation of intensity and phase statistics of speckle produced by a weak scatterer from the random sinusoid model,”J. Opt. Soc. Am. 71, 86–90 (1981).
    [CrossRef]
  14. J. Uozumi, T. Asakura, “First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region,” Appl. Opt. 20, 1454–1466 (1981).
    [CrossRef] [PubMed]
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Ch. 4.
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  18. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,”J. Opt. Soc. Am. 55, 247–254 (1965).
    [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 539–542.

1982

1981

1980

1979

B. Stoffregen, “Speckle statistics for general scattering objects: I. general relations for speckle amplitude and intensity,” Optik 52, 305–312 (1979).

1977

1976

1970

F. Thon, B. M. Siegel, “Experiments with optical image reconstruction of high resolution electron micrographs,” Ber. Bunsen Ges. Phys. Chem. 74, 1116–1120 (1970).

1965

Asakura, T.

Bahuguna, R. D.

Barakat, R.

R. Barakat, P. Nisenson, “Influence of the wavefront correlation function and deterministic wavefront aberrations on the speckle image-reconstruction problem in the high light level regime,”J. Opt. Soc. Am. 71, 1930–1402 (1981).

R. Barakat, “Direct derivation of intensity and phase statistics of speckle produced by a weak scatterer from the random sinusoid model,”J. Opt. Soc. Am. 71, 86–90 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Ch. 4.

Goldfischer, L. I.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 539–542.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[CrossRef]

Gupta, K. K.

Nisenson, P.

R. Barakat, P. Nisenson, “Influence of the wavefront correlation function and deterministic wavefront aberrations on the speckle image-reconstruction problem in the high light level regime,”J. Opt. Soc. Am. 71, 1930–1402 (1981).

O’Donnell, K. A.

Ohtsubo, J.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Pedersen, H. M.

Saxton, W. O.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York1978), Chap. I.

Siegel, B. M.

F. Thon, B. M. Siegel, “Experiments with optical image reconstruction of high resolution electron micrographs,” Ber. Bunsen Ges. Phys. Chem. 74, 1116–1120 (1970).

Singh, K.

Stetson, K. A.

Stoffregen, B.

B. Stoffregen, “Speckle statistics for general scattering objects: I. general relations for speckle amplitude and intensity,” Optik 52, 305–312 (1979).

B. Stoffregen, “Speckle statistics for general scattering objects: II. mean, covariance and power spectrum of image speckle patterns,” Optik 52, 385–399 (1978/79).

Thon, F.

F. Thon, B. M. Siegel, “Experiments with optical image reconstruction of high resolution electron micrographs,” Ber. Bunsen Ges. Phys. Chem. 74, 1116–1120 (1970).

F. Thon, “Phase contrast electron microscopy,” in Electron Microscopy in Materials Science, U. Valdre, ed. (Academic, London, 1971).
[CrossRef]

Uozumi, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Ch. 4.

Appl. Opt.

Ber. Bunsen Ges. Phys. Chem.

F. Thon, B. M. Siegel, “Experiments with optical image reconstruction of high resolution electron micrographs,” Ber. Bunsen Ges. Phys. Chem. 74, 1116–1120 (1970).

J. Opt. Soc. Am.

Optik

B. Stoffregen, “Speckle statistics for general scattering objects: I. general relations for speckle amplitude and intensity,” Optik 52, 305–312 (1979).

B. Stoffregen, “Speckle statistics for general scattering objects: II. mean, covariance and power spectrum of image speckle patterns,” Optik 52, 385–399 (1978/79).

Other

F. Thon, “Phase contrast electron microscopy,” in Electron Microscopy in Materials Science, U. Valdre, ed. (Academic, London, 1971).
[CrossRef]

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York1978), Chap. I.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 539–542.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Ch. 4.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Monochromatic coherent optical-imaging geometry.

Fig. 2
Fig. 2

Normalized intensity power spectral density S ¯ I(f) for the five primary aberrations for aberration severity K0 = 2π, surface correlation a = 0.15, and phase parameter α = 0.75π.

Fig. 3
Fig. 3

Effect of varying surface phase parameter α on the normalized intensity power spectral density for a spherical aberration with an aberration severity K0 = 2π and a surface correlation a = 0.15.

Fig. 4
Fig. 4

Effect of varying surface phase parameter α on the intensity contrast for a spherical aberration with an aberration severity K0 = 2π and surface correlation a = 0.15.

Fig. 5
Fig. 5

Effect of varying aberration severity K0 on the normalized intensity power spectral density of a spherical aberration α = 0.75π and surface correlation a = 0.15.

Fig. 6
Fig. 6

Effect of varying aberration severity K0 on the intensity contrast for a spherical aberration α = 0.75π and surface correlation a = 0.15.

Tables (1)

Tables Icon

Table 1 Contrast Values for the Five Primary Aberrations for Severity K0 = 2π, Surface Correlation a = 0.15, and Phase Parameter α s= 0.757r

Equations (38)

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A ( x , y , z ) = 1 N k = 1 N h k ( x , y , z ) exp ( i Θ k ) ,
A ( r ) = Re [ A ( x , y , z ) ] = 1 N k = 1 N Re [ h k exp ( i Θ k ) ] , A ( i ) = Im [ A ( x , y , z ) ] = 1 N k = 1 N Im [ h k exp ( i Θ k ) ] ,
E [ A ( r ) ] = m r = S α 1 N k = 1 N Re [ h k ] , Var [ A ( r ) ] = σ r 2 = 1 2 N 2 [ ( 1 - S α 2 ) k = 1 N h k 2 + ( S 2 α - S α 2 ) k = 1 N Re h k 2 ] , E [ A ( i ) ] = m i = S α 1 N k = 1 N Im [ h k ] , Var [ A ( i ) ] = σ i 2 = 1 2 N 2 [ ( 1 - S α 2 ) k = 1 N h k 2 - ( S 2 α - S α 2 ) k = 1 N Re [ h k 2 ] ] ,
Cov [ A ( r ) , A ( i ) ] = σ r σ i ρ = 1 2 N 2 ( S 2 α - S α 2 ) k = 1 N Im [ h k 2 ] .
E [ I ] = E [ ( A ( r ) ) 2 ] + E [ ( A ( i ) ) 2 ] = 1 N 2 S α 2 | k = 1 N h k | 2 + 1 N 2 ( 1 - S α 2 ) k = 1 N h k 2 .
Var [ I ] = 2 N 4 ( S 2 α S α - S α 4 ) Re [ ( k = 1 N h k 2 ) ( j = 1 N h * j ) 2 ] + 2 N 4 ( S α 2 - S α 4 ) ( k = 1 N h k 2 ) | j = 1 N h j | 2 + 4 N 4 ( 2 S α 4 - S α 2 - S 2 α S α 2 ) × Re [ ( k = 1 N h k 2 h k ) ( j = 1 N h * j ) ] + 1 N 4 ( 4 S 2 α S α 2 + 4 S α 2 - S 2 α - 1 - 6 S α 4 ) k = 1 N h k 4 + 1 N 4 ( S α 2 - S 2 α ) 2 | k = 1 N h k 2 | 2 + 1 N 4 ( 1 - S α 2 ) 2 [ k = 1 N h k 2 ] 2 .
E [ I ] = 1 N 2 | k = 1 N h k | 2 , Var [ I ] = 0 ,
H ( f x , f y ) = P ( f x , f y ) exp [ i k W j ( f x , f y ) ] ,             j = 1 , 2 , 5 ,
P ( f x , f y ) = { 1 ( f x 2 + f y 2 ) 1 / 2 1 0 elsewhere ,
W 1 ( f x , f y ) = ( f x 2 + f y 2 ) 2 ( spherical ) , W 2 ( f x , f y ) = ( f x 2 + f y 2 ) ( curvature of field ) , W 3 ( f x , f y ) = f x 2 ( astigmatism ) , W 4 ( f x , f y ) = ( f x 2 + f y 2 ) f x ( coma ) , W 5 ( f x , f y ) = f x ( distortion ) .
t ^ ( x , y ) = m = - N - 1 2 N - 1 2 n = - N - 1 2 N - 2 2 exp [ i Θ ( m a , n a ) ] × rect ( x - m a a ) rect ( y - n a a ) ,
rect ( x ) 1 x 1 / 2 0 elsewhere .
A ( x , y ) = t ^ ( x , y ) * h ( x , y ) ,
h ( x , y ) = - - H ( f x , f y ) exp [ i 2 π ( x f x + y f y ) ] d f x d f y .
I ( x , y ) = A ( x , y ) 2 = t ^ ( x , y ) * h ( x , y ) 2 ,
R I ( x 1 x 2 ) = E [ I ( x 1 ) I ( x 2 ) ] = - - - - E [ t ^ ( u 1 ) t ^ * ( u 2 ) t ^ ( u 3 ) t ^ * ( u 4 ) ] × h ( x 1 - u 1 ) h * ( x 1 - u 2 ) h ( x 2 - u 3 ) × h * ( x 2 - u 4 ) d u 1 d u 2 d u 3 d u 4
S I ( f ) = E { | - I ( x ) exp [ - i 2 π f · x ] d x | 2 } .
S I ( f ) = δ ( f ) { S α 4 + 2 ( S α 2 - S α 4 ) a 2 H a ( 0 ) + ( 1 - S α 2 ) 2 a 4 [ H a ( 0 ) ] 2 } + 2 ( S 2 α S α 2 - S α 4 ) a 2 ( sinc a f ) 2 Re [ H ( f ) H ( - f ) ] + 2 ( S α 2 - S α 4 ) a 2 ( sinc a f ) 2 H ( f ) 2 + 2 ( 2 S α 4 - S α 2 - S 2 α S α 2 ) a 4 sinc a f × { Re [ H ( - f ) H a ( f ) ] + Re [ H a ( f ) H * ( f ) ] } + ( 4 S 2 α S α 2 + 4 S α 2 - S 2 α - 1 - 6 S α 2 ) a 6 H a ( f ) 2 + ( S α 2 - S 2 α ) 2 a 4 [ H ( f ) H ( - f ) ( sinc a f ) 2 * H * ( f ) H * ( - f ) ( sinc a f ) 2 ] + ( 1 - S α 2 ) 2 a 4 [ H ( f ) 2 ( sinc a f ) 2 * H ( f ) 2 ( sinc a f ) 2 ] ,
H a ( f ) = H ( f ) sinc a f * H * ( - f ) sinc a f ,
sinc ( β f ) sin ( π β f x ) π β f x sin ( π β f y ) π β f y ,
E [ I ( x ) ] = S α 2 + ( 1 - S α 2 ) a 2 [ H a ( 0 ) ] 2 .
N e f f max [ π ( 0.610 ) 2 a 2 , 1 ] max ( 1.17 a 2 , 1 ) .
μ t ( x 2 - x 1 ) = E [ t ^ ( x 1 ) t ^ * ( x 2 ) ] - E [ t ^ ( x 1 ) ] E [ t ^ * ( x 2 ) ] = E [ t ^ ( x 1 ) t ^ * ( x 2 ) ] - S α 2 = Λ ( Δ x a ) ,
Λ ( x ) { [ 1 - x ] [ 1 - y ] x , y 1 0 elsewhere .
S ¯ I ( f ) S I ( f ) - S α 4 δ ( f ) σ I 2
S ¯ I ( f ) { a 4 δ ( f ) [ - H ( u ) 2 d u ] 3 + a 4 - H ( f - u ) 2 H ( u ) 2 d u - a 6 | - H ( f - u ) H * ( - u ) d u | 2 } × { a 4 ( - H ( u ) 2 d u ) 2 - a 6 - H ( u ) 4 d u } - 1 δ ( f ) + - H ( f - u ) 2 H ( u ) 2 d u [ - H ( u ) 2 d u ] 2 .
C σ I E [ I ] [ a 4 ( - H ( u ) 2 d u ) 2 - a 6 - H ( u ) 4 d u ] 1 / 2 a 2 - H ( u ) 2 d u - H ( u ) 2 d u - H ( u ) 2 d u = 1.
S ¯ I ( f ) 1 K [ ( S α 2 - S α 4 ) δ ( f ) - H ( u ) 2 d u + ( S 2 α S α 2 - S α 4 ) Re [ H ( f ) H ( - f ) ] + ( S α 2 - S α 4 ) H ( f ) 2 ] ,
C [ 2 a 2 K ] 1 / 2 S α 2 + ( 1 - S α 2 ) a 2 - H ( u ) 2 d u ,
K = ( S 2 α S α 2 - S α 4 ) Re [ - H ( u ) H ( - u ) d u ] + ( S α 2 - S α 4 ) - H ( u ) 2 d u .
Re { [ H ( f ) ] 2 } = P ( f ) cos [ 2 k W j ( f ) ] ,             j = 1 , 2 , 3.
r = S α 2 a 2 ( 1 - S α 2 ) - H ( u ) 2 d u
C 0 [ 2 ( S α 2 - S α 4 ) a 2 π + ( 1 - S α 2 ) 2 a 4 π 2 ] 1 / 2 S α 2 + ( 1 - S α 2 ) a 2 π .
S I ( f ) = E { - - t ^ ( x 1 ) * h ( x 1 ) 2 t ^ ( x 2 ) * h ( x 2 ) 2 × exp [ - i 2 π f · ( x 1 - x 2 ) ] d x 1 d x 2 } , = E { [ T ^ ( f ) H ( f ) * T ^ * ( - f ) H * ( - f ) ] 2 } ,
T ^ ( f ) = F [ t ^ ( x ) ] = m 1 = - N - 1 2 N - 1 2 exp [ i Θ m 1 ] a 2 × sinc a f exp [ - i 2 π a m 1 · f ] .
S I ( f ) = a 8 m 1 = - N - 1 2 N - 1 2 m 2 = - N - 1 2 N - 1 2 m 3 = - N - 1 2 N - 1 2 m 4 = - N - 1 2 N - 1 2 × E { exp [ i ( Θ m 1 - Θ m 2 + Θ m 3 - Θ 4 ) ] } × - - H ( f - u 1 ) H * ( - u 1 ) H ( - f + u 2 ) H * ( u 2 ) × sinc a ( f - u 1 ) sinc a u 1 sinc a ( f - u 2 ) sinc a u 2 × exp [ - i 2 π a [ m 1 · ( f - u 1 ) + m 2 · u 1 - m 3 · ( f - u 2 ) - m 4 · u 4 ] ] d u 1 d u 2 .
S I ( f ) = S α 4 L 8 H L ( f ) 2 + 2 ( S α 2 - S α 4 ) a 2 L 6 sinc L f sinc a f Re [ H a ( f ) H L * ( f ) ] + ( 1 - S α 2 ) 2 a 4 L 4 ( sinc L f sinc a f ) 2 H a ( f ) 2 + 2 ( S 2 α S α 2 - S α 4 ) a 2 L 6 Re [ G 1 ( f ) ] + 2 ( S α 2 - S α 4 ) a 2 L 6 Re [ G 2 ( f ) ] + 2 ( 2 S α 4 - S α 2 - S 2 α S α 2 ) a 4 L 4 Re [ H a ( f ) H a L 2 ( - f ) ] + 2 ( 2 S α 4 - S α 2 - S 2 α S α 2 ) a 4 L 4 Re [ H a ( f ) H * a L 2 ( f ) ] + ( 4 S 2 α S α 2 + 4 S α 2 - S 2 α - 1 - 6 S α 4 ) a 6 L 2 H a ( f ) 2 + 2 ( S α 2 - S 2 α ) 2 a 4 L 4 G 3 ( f ) + ( 1 - S α 2 ) 2 a 4 L 4 G 4 ( f ) ,
H L ( f ) = H ( f ) sinc L f * H * ( - f ) sinc L f , H a ( f ) = H ( f ) sinc a f * H * ( - f ) sinc a f , H a L 2 ( f ) = H ( f ) sinc a f * H * ( - f ) sinc L f 2 sinc a f , G 1 ( f ) = - - H ( f - u 1 ) H * ( - u 1 ) H ( - f + u 2 ) H * ( u 2 ) × sinc a ( f - u 1 ) sinc L u 1 sinc a ( f - u 2 ) sinc L u 2 × sinc L ( u 1 - u 2 ) sinc a ( u 1 - u 2 ) d u 1 d u 2 , G 2 ( f ) = - - H ( f - u 1 ) H * ( - u 1 ) H ( - f + u 2 ) H * ( u 2 ) × sinc a ( f - u 1 ) sinc L u 1 sinc L ( f - u 2 ) sinc a u 2 × sinc L ( f - u 1 - u 2 ) sinc a ( f - u 1 - u 2 ) d u 1 d u 2 , G 3 ( f ) = - - H ( f - u 1 ) H * ( - u 1 ) × H ( - f + u 2 ) H * ( u 2 ) sinc a ( f - u 1 ) × sinc a u 1 sinc a ( f - u 2 ) sinc a u 2 × [ sinc L ( u 1 - u 2 ) sinc a ( u 1 - u 2 ) ] 2 d u 1 d u 2 , G 4 ( f ) = - - H ( f - u 1 ) H * ( - u 1 ) × H ( - f + u 2 ) H * ( u 2 ) sinc a ( f - u 1 ) × sinc a u 1 sinc a ( f - u 2 ) sinc a u 2 × [ sinc L ( f - u 1 - u 2 ) sinc a ( f - u 1 - u 2 ) ] 2 d u 1 d u 2 .

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