Abstract

This paper is a study of the principles of a new method of rough-surface interferometry (ROSI). We review some properties of the higher-order probability density function (pdf) of correlated speckle fields and discuss some assumptions that are usually made silently. This pdf is used to investigate statistical properties of intensities and phases in a dichromatic speckle field; the results are applied to ROSI. The experiments described confirm the theoretical results.

© 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, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).
  2. C. Wykes, “Decorrelation effects in speckle-pattern interferometry. I: Wavelength change dependent decorrelation with application to contouring and surface roughness measurement,” Opt. Acta 24, 517 (1977).
    [CrossRef]
  3. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 5.
  4. A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181 (1985).
    [CrossRef]
  5. A. F. Fercher, H. Z. Hu, “Two-wavelength heterodyne interferometry,” in Optoelectronics in Engineering; Proceedings of the 6th International Congress Laser 83, W. Waidelich, ed. (Springer-Verlag, Berlin, 1984).
  6. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  7. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (University Microfilms, Ann Arbor, Mich., 1958).
  8. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  9. B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  10. P. F. Steeger, “Probability density function of the intensity in partially polarized speckle fields,” Opt. Lett. 8, 523 (1983).
    [CrossRef]
  11. J. W. Goodman, “Statistical properties of laser speckle patterns,” Stanford Electronics Lab. Rep. TR2303-1, SEL-63-140 (Stanford University, Stanford, Calif., 1963).
  12. S. Donati, G. Martini, “Speckle-pattern intensity and phase: second-order conditional statistics,” J. Opt. Soc. Am. 69, 1690 (1979).
    [CrossRef]
  13. H. Fujii, W. Y. Lit, “Measurement of surface roughness using dichromatic speckle,” Opt. Commun. 22, 231 (1977).
    [CrossRef]
  14. M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166 (1979).
    [CrossRef]
  15. N. George, A. Jain, “Speckle in microscopy,” Opt. Commun. 6, 253 (1972).
    [CrossRef]
  16. N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202 (1973).
    [CrossRef] [PubMed]
  17. N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201 (1974).
    [CrossRef]
  18. E. G. Rawson, J. W. Goodman, R. E. Norton, “Frequency dependence of modal noise in multimode optical fibers,” J. Opt. Soc. Am. 70, 968 (1980).
    [CrossRef]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  20. R. Jones, C. Wykes, “The comparison of complex object geometries using a combination of electronic speckle pattern interferometric difference contouring and holographic illumination elements,” Opt. Acta 25, 449 (1978).
    [CrossRef]
  21. R. Crane, “Interferometric phase measurements,” Appl. Opt. 8, 538 (1969).
  22. M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975), App. III.

1985 (1)

1983 (1)

1980 (1)

1979 (2)

S. Donati, G. Martini, “Speckle-pattern intensity and phase: second-order conditional statistics,” J. Opt. Soc. Am. 69, 1690 (1979).
[CrossRef]

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

1978 (1)

R. Jones, C. Wykes, “The comparison of complex object geometries using a combination of electronic speckle pattern interferometric difference contouring and holographic illumination elements,” Opt. Acta 25, 449 (1978).
[CrossRef]

1977 (2)

C. Wykes, “Decorrelation effects in speckle-pattern interferometry. I: Wavelength change dependent decorrelation with application to contouring and surface roughness measurement,” Opt. Acta 24, 517 (1977).
[CrossRef]

H. Fujii, W. Y. Lit, “Measurement of surface roughness using dichromatic speckle,” Opt. Commun. 22, 231 (1977).
[CrossRef]

1974 (1)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

1973 (1)

1972 (1)

N. George, A. Jain, “Speckle in microscopy,” Opt. Commun. 6, 253 (1972).
[CrossRef]

1969 (1)

R. Crane, “Interferometric phase measurements,” Appl. Opt. 8, 538 (1969).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975), App. III.

Crane, R.

R. Crane, “Interferometric phase measurements,” Appl. Opt. 8, 538 (1969).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (University Microfilms, Ann Arbor, Mich., 1958).

Donati, S.

Fercher, A. F.

A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181 (1985).
[CrossRef]

A. F. Fercher, H. Z. Hu, “Two-wavelength heterodyne interferometry,” in Optoelectronics in Engineering; Proceedings of the 6th International Congress Laser 83, W. Waidelich, ed. (Springer-Verlag, Berlin, 1984).

Fujii, H.

H. Fujii, W. Y. Lit, “Measurement of surface roughness using dichromatic speckle,” Opt. Commun. 22, 231 (1977).
[CrossRef]

George, N.

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202 (1973).
[CrossRef] [PubMed]

N. George, A. Jain, “Speckle in microscopy,” Opt. Commun. 6, 253 (1972).
[CrossRef]

Giglio, M.

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Goodman, J. W.

E. G. Rawson, J. W. Goodman, R. E. Norton, “Frequency dependence of modal noise in multimode optical fibers,” J. Opt. Soc. Am. 70, 968 (1980).
[CrossRef]

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

J. W. Goodman, “Statistical properties of laser speckle patterns,” Stanford Electronics Lab. Rep. TR2303-1, SEL-63-140 (Stanford University, Stanford, Calif., 1963).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

Hu, H. Z.

A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181 (1985).
[CrossRef]

A. F. Fercher, H. Z. Hu, “Two-wavelength heterodyne interferometry,” in Optoelectronics in Engineering; Proceedings of the 6th International Congress Laser 83, W. Waidelich, ed. (Springer-Verlag, Berlin, 1984).

Jain, A.

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

N. George, A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202 (1973).
[CrossRef] [PubMed]

N. George, A. Jain, “Speckle in microscopy,” Opt. Commun. 6, 253 (1972).
[CrossRef]

Jones, R.

R. Jones, C. Wykes, “The comparison of complex object geometries using a combination of electronic speckle pattern interferometric difference contouring and holographic illumination elements,” Opt. Acta 25, 449 (1978).
[CrossRef]

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 5.

Lit, W. Y.

H. Fujii, W. Y. Lit, “Measurement of surface roughness using dichromatic speckle,” Opt. Commun. 22, 231 (1977).
[CrossRef]

Martini, G.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Musazzi, S.

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Norton, R. E.

Papoulis, A.

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

Perini, U.

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Rawson, E. G.

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (University Microfilms, Ann Arbor, Mich., 1958).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

Steeger, P. F.

Vry, U.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975), App. III.

Wykes, C.

R. Jones, C. Wykes, “The comparison of complex object geometries using a combination of electronic speckle pattern interferometric difference contouring and holographic illumination elements,” Opt. Acta 25, 449 (1978).
[CrossRef]

C. Wykes, “Decorrelation effects in speckle-pattern interferometry. I: Wavelength change dependent decorrelation with application to contouring and surface roughness measurement,” Opt. Acta 24, 517 (1977).
[CrossRef]

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 5.

Appl. Opt. (3)

Appl. Phys. (1)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (2)

C. Wykes, “Decorrelation effects in speckle-pattern interferometry. I: Wavelength change dependent decorrelation with application to contouring and surface roughness measurement,” Opt. Acta 24, 517 (1977).
[CrossRef]

R. Jones, C. Wykes, “The comparison of complex object geometries using a combination of electronic speckle pattern interferometric difference contouring and holographic illumination elements,” Opt. Acta 25, 449 (1978).
[CrossRef]

Opt. Commun. (3)

H. Fujii, W. Y. Lit, “Measurement of surface roughness using dichromatic speckle,” Opt. Commun. 22, 231 (1977).
[CrossRef]

M. Giglio, S. Musazzi, U. Perini, “Surface roughness measurements by means of speckle wavelength decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

N. George, A. Jain, “Speckle in microscopy,” Opt. Commun. 6, 253 (1972).
[CrossRef]

Opt. Lett. (1)

Other (10)

J. W. Goodman, “Statistical properties of laser speckle patterns,” Stanford Electronics Lab. Rep. TR2303-1, SEL-63-140 (Stanford University, Stanford, Calif., 1963).

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 5.

A. F. Fercher, H. Z. Hu, “Two-wavelength heterodyne interferometry,” in Optoelectronics in Engineering; Proceedings of the 6th International Congress Laser 83, W. Waidelich, ed. (Springer-Verlag, Berlin, 1984).

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

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (University Microfilms, Ann Arbor, Mich., 1958).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

B. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

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, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975), App. III.

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Figures (17)

Fig. 1
Fig. 1

Principal parts of the rough-surface interferometer.

Fig. 2
Fig. 2

Geometry for the calculation of the correlation coefficient μ.

Fig. 3
Fig. 3

The pdf of the statistical error ϕ of the phase difference of the two speckle fields for various values of |μ|.

Fig. 4
Fig. 4

Domains of integration for the conditional pdf’s of Eq. (42): ///, condition (I); |||, condition (II); \\\, condition (III).

Fig. 5
Fig. 5

Some conditional pdf’s of the statistical error ϕ for |μ| = 0.5. I0 is normalized to the mean intensity.

Fig. 6
Fig. 6

Conditional densities of the statistical error ϕ for condition (I) and three different correlations.

Fig. 7
Fig. 7

The standard deviations σϕ of some conditional pdf’s as a function of the correlation |μ| for the conditions (I)–(III).

Fig. 8
Fig. 8

The probabilities P(I1, I2, > I0) (dashed line), P(I1 > I0) (solid line), and P(I1 > I0 or I2 > I2) (dotted line) as a function of the correlation for various values of the intensity threshold I0.

Fig. 9
Fig. 9

Two-wavelength heterodyne interferometer used for experimental measurements.

Fig. 10
Fig. 10

Comparison of the measured pdf’s of I1 (histogram) with the theoretical negative exponential function (solid line) for the five ground glasses used.

Fig. 11
Fig. 11

The experimental (dots) and theoretical (solid lines) probabilities P(I1, I2 > I0) (lower curve), P(I1 > I0) (intermediate curve), and P(I1 > I0 or I2 > I0) (upper curve) for two cases: (a) ground glass 1; correlation of the theoretical curves |μ| = 0.9; (b) ground glass 3; correlation of the theoretical curves |μ| = 0.8.

Fig. 12
Fig. 12

Unconditional pdf of the statistical phase error ϕ obtained from ground glass 3 (histogram) and theoretical curve of |μ| = 0.85 (solid line).

Fig. 13
Fig. 13

Conditional pdf’s of the statistical phase error ϕ for conditions (I)–(III), I0 = 1, obtained from ground glass 3 (histograms) and theoretical curves of |μ| = 0.85 (solid lines).

Fig. 14
Fig. 14

Unconditional pdf’s (solid lines) and conditional pdf’s (dashed lines) of the statistical phase error ϕ for conditions (I)–(III) and I0 = 1, obtained from ground glass 3.

Fig. 15
Fig. 15

Standard deviations σϕ obtained from ground glass 3 as a function of the intensity threshold I0: condition (I), lower curve; condition (II), intermediate curve; and condition (III), upper curve.

Fig. 16
Fig. 16

Measured standard deviations σϕ (dots) and some theoretical curves (solid lines: the corresponding correlation |μ| is noted at each curve) for condition (I). The numbers on the right-hand margin denote the ground glasses used.

Fig. 17
Fig. 17

Illustration of the applicability of the method of stationary phase.

Tables (2)

Tables Icon

Table 1 Estimated Values of the Correlation |μ| and the Surface Roughness σh Obtained from the Statistical Measurements

Tables Icon

Table 2 Comparison of the Values of the Surface Roughness Obtained by Using Different Methods

Equations (77)

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i m I m ( r ) + I m ( s ) + 2 [ I m ( r ) I m ( s ) ] 1 / 2 cos ( ω H t ϕ m ) ( m = 1 , 2 )
A m ( r ) = [ I m ( r ) ] 1 / 2 exp [ i ( 2 π c λ m + ω H ) t ]
A m ( s ) = [ I m ( s ) ] 1 / 2 exp [ i ( 2 π c λ m t + ϕ m ) ]
z ( ξ , η ) = z 0 ( ξ , η ) + h ( ξ , η ) .
ϕ 2 ϕ 1 = 4 π Λ z 0 ( ξ , η ) + ϕ .
Λ = λ 1 λ 2 | λ 1 λ 2 |
A m = A m ( r ) + i A m ( i ) ( m = 1 , , N )
X 2 m 1 = A m ( r ) , X 2 m = A m ( i ) .
p ( X 1 , , X 2 N ) = 1 ( 2 π ) N ( det R ) 1 / 2 exp ( 1 2 r , s = 1 2 N b r s X r X s ) ,
R = ( R m n ) = ( X m X n ) , B = ( b m n ) = R 1 .
R 2 m , 2 n = R 2 m 1 , 2 n 1 , R 2 m 1 , 2 n = R 2 m , 2 n 1 ( m , n = 1 , , N ) ,
p ( A 1 , , A N ) = 1 π N det ( H ) exp ( m , n = 1 N A m A n * g m n ) = 1 π N det ( H ) exp ( A G A + ) ,
G 1 = H = ( μ m n ) = ( A m * A n ) , A = ( A 1 , , A N ) ,
μ ˜ m n = A m A n = 0.
μ = μ 12 = A 1 * A 2 = | μ | exp ( i ψ ) .
A m ( r ) = I m cos ( ϕ m ) , A m ( i ) = I m sin ( ϕ m ) ( m = 1 , 2 ) ,
ϕ = ϕ 2 ϕ 1 ψ ,
p ( I 1 , I 2 , ϕ ) = exp [ I 1 + I 2 2 ( I 1 I 2 ) 1 / 2 | μ | cos ( ϕ ) 1 | μ | 2 ] 2 π ( 1 | μ | 2 ) .
A m ( x , y ) = exp ( i k m f ) i λ m f d ξ d η A 0 m ( ξ , η ) × exp { i k m 2 f [ ( x ξ ) 2 + ( y η ) 2 ] } ,
L m ( x , y ) = exp [ i k m 2 f ( x 2 + y 2 ) ] .
A m ( x , y ) = L m ( x , y ) A m ( x , y ) = d ξ d η A 0 m ( ξ , η ) × exp [ i k m 2 f ( ξ 2 + η 2 2 ξ x 2 η y ) ] .
A 0 m ( ξ , η ) = δ ( ξ ξ 0 , η η 0 ) .
A m ( x , y ) = exp [ i k m 2 f ( ξ 0 2 + η 0 2 2 ξ 0 x 2 η 0 y ) ] .
k m x = k m ξ 0 f , k m y = k m η 0 f , k m z = ( k m 2 k m x 2 k m y 2 ) 1 / 2 = k m ( 1 ξ 0 2 + η 0 2 f 2 ) 1 / 2 k m ( 1 ξ 0 2 + η 0 2 2 f 2 ) .
A m ( u , υ , d ) = exp { i k m [ ξ 0 2 + η 0 2 2 f + ( 1 ξ 0 2 + η 0 2 2 f 2 ) d 1 f ( ξ 0 u + η 0 υ ) ] } = exp { i k m [ d + 1 2 f 2 ( ξ 0 2 + η 0 2 ) ( f d ) 1 f ( ξ 0 u + η 0 υ ) ] } ,
A m ( u , υ , d ) = exp ( i k m d ) d ξ d η A 0 m ( ξ , η ) × exp { i k m [ 1 2 f 2 ( ξ 2 + η 2 ) ( f d ) 1 f ( ξ u + η υ ) ] } .
A m ( u , υ , d ) = d ξ d η P m ( ξ , η ) exp [ 2 i k m z ( ξ , η ) ] × exp { i k m [ 1 2 f 2 ( ξ 2 + η 2 ) ( f d ) 1 f ( ξ u + η υ ) ] } .
A m ( u , υ , d ) = d ξ d η P m ( ξ , η ) exp [ 2 i k m z ( ξ , η ) ] × exp { i k m f ( ξ u + η υ ) } .
p ( h , ξ , η ) = p ( h ) = 1 2 π σ h exp ( h 2 2 σ h 2 ) ,
h ( ξ 1 , η 1 ) h ( ξ 2 , η 2 ) = σ h 2 δ ( ξ 1 ξ 2 , η 1 η 2 ) ,
exp { i [ a 2 h ( ξ 2 , η 2 ) a 1 h ( ξ 1 , η 1 ) ] } = exp [ ( a 2 a 1 ) 2 2 σ h 2 ] δ ( ξ 1 ξ 2 , η 1 η 2 ) .
z 0 ( ξ , η ) = h 0 + s ξ .
P m ( ξ , η ) = exp ( ξ 2 + η 2 2 a 2 ) ,
A m * A n = A m * ( 0 , 0 , d ) A n ( 0 , 0 , d ) = d ξ 1 d ξ 2 d η 1 d η 2 P m * ( ξ 1 , η 1 ) P n ( ξ 2 , η 2 ) × exp { 2 i [ k n z ( ξ 2 , η 2 ) k m z ( ξ 1 , η 1 ) ] } = exp [ 2 i ( k n k m ) h 0 ] × d ξ 1 d ξ 2 d η 1 d η 2 exp ( ξ 1 2 + ξ 2 2 + η 1 2 + η 2 2 2 a 2 ) × exp [ 2 i s ( k n ξ 1 k m ξ 2 ) ] × exp { 2 i [ k n h ( ξ 2 , η 2 ) k m h ( ξ 1 , η 1 ) ] } = exp ( 2 i K n m h 0 ) × d ξ 1 d ξ 2 d η 1 d η 2 exp [ 2 i s ( k n ξ 2 k m ξ 1 ) ] × exp ( ξ 1 2 + ξ 2 2 + η 1 2 + η 2 2 2 a 2 ) exp ( 2 K n m 2 σ h 2 ) × δ ( ξ 1 ξ 2 , η 1 η 2 ) = π a 2 exp [ 2 i K n m h 0 ] exp [ K n m 2 ( 2 σ h 2 + s 2 a 2 ) ] ,
μ = A 1 * A 2 ( A 1 * A 1 A 2 * A 2 ) 1 / 2 = exp ( 4 π i Λ h 0 ) × exp [ 4 π 2 Λ 2 ( 2 σ h 2 + s 2 a 2 ) ] ,
ψ = 4 π Λ h 0 .
μ ˜ = A 1 A 2 ( A 1 * A 1 A 2 * A 2 ) 1 / 2 ,
μ ˜ = exp [ 2 i ( k m + k n ) h 0 ] exp [ ( k 1 + k 2 ) 2 ( 2 σ h 2 + s 2 a 2 ) ] .
p ( ϕ ) = 1 | μ | 2 2 π ( 1 β 2 ) 3 / 2 [ β sin 1 ( β ) + π β 2 + ( 1 β 2 ) 1 / 2 ] ,
β = | μ | cos ( ϕ ) .
σ ϕ = ( π 2 3 π arcsin | μ | + arcsin 2 | μ | 1 2 n = 1 μ 2 n n 2 ) 1 / 2
p ( ϕ | I 1 , I 2 > I 0 ) = I 0 I 0 d I 1 d I 2 p ( I 1 , I 2 , ϕ ) π π I 0 I 0 d I 1 d I 2 d ϕ p ( I 1 , I 2 , ϕ ) .
p ( ϕ | I 1 > I 0 ) = 0 I 0 d I 1 d I 2 p ( I 1 , I 2 , ϕ ) π π 0 I 0 d I 1 d I 2 d ϕ p ( I 1 , I 2 , ϕ ) .
p ( ϕ | I 1 > I 0 or I 2 > I 0 ) = ( 0 I 0 + I 0 0 I 0 ) d I 1 d I 2 p ( I 1 , I 2 , ϕ ) π π ( 0 I 0 + I 0 0 I 0 ) d I 1 d I 2 d ϕ p ( I 1 , I 2 , ϕ ) .
p ( I 1 , I 2 ) = exp ( I 1 + I 2 1 | μ | 2 ) 1 | μ | 2 I ˜ 0 [ 2 ( I 1 I 2 ) 1 / 2 | μ | 1 | μ | 2 ] .
P ( I 1 > I 0 ) = I 0 d I 1 p ( I 1 ) = I 0 d I 1 exp ( I 1 ) = exp ( I 0 ) .
p ( I 1 , I 2 ) = p ( I 1 ) δ ( I 1 I 2 ) = exp ( I 1 ) δ ( I 1 I 2 )
P ( I 1 , I 2 > I 0 ) = P ( I 1 > I 0 ) = P ( I 1 > I 0 or I 2 > I 0 ) = exp ( I 0 ) .
k a 2 | f d | 2 f 2 0.05 1 ,
| μ | = exp ( 2 K 2 σ h 2 ) .
g m n = g m n ( r ) + i g m n ( i ) ( m , n = 1 , , N )
g m n ( r ) = 1 2 b 2 m , 2 n = 1 2 b 2 m 1 , 2 n 1 , g m n ( i ) = 1 2 b 2 m , 2 n 1 = 1 2 b 2 m 1 , 2 n .
δ m n = l = 1 N g m l H l n = l = 1 N [ g m l ( r ) + i g m l ( i ) ] [ μ l n ( r ) + i μ l n ( i ) ] = l = 1 N { [ g m l ( r ) μ l n ( r ) g m l ( i ) μ l n ( i ) ] + i [ g m l ( i ) μ l n ( r ) + g m l ( r ) μ l n ( i ) ] } .
l = 1 N [ g m l ( r ) μ l n ( r ) g m l ( i ) μ l n ( i ) ] = δ m n ,
l = 1 N [ g m l ( i ) μ l n ( r ) + g m l ( r ) μ l n ( i ) ] = 0.
b ˜ 2 m , 2 n = b ˜ 2 m 1 , 2 n 1 = 2 g m l ( r ) , b ˜ 2 m , 2 n 1 = b ˜ 2 m 1 , 2 n = 2 g m l ( i ) .
1 2 l = 1 N [ b ˜ 2 m , 2 l μ l n ( r ) + b ˜ 2 m , 2 l 1 μ l n ( i ) ] = δ m n = δ 2 m , 2 n
1 2 l = 1 N [ b ˜ 2 m 1 , 2 l 1 μ l n ( r ) b ˜ 2 m 1 , 2 l μ l n ( i ) ] = δ m n = δ 2 m 1 , 2 n 1 .
1 2 l = 1 N [ b ˜ 2 m , 2 l 1 μ l n ( r ) + b ˜ 2 m , 2 l μ l n ( i ) ] = 0 ,
1 2 l = 1 N [ b ˜ 2 m 1 , 2 l μ l n ( r ) + b ˜ 2 m 1 , 2 l 1 μ l n ( i ) ] = 0.
1 2 l = 1 N [ b ˜ r , 2 l μ l n ( r ) + b ˜ r , 2 l 1 μ l n ( i ) ] = δ r , 2 n ( r = 1 , , 2 N ) ,
1 2 l = 1 N [ b ˜ r , 2 l 1 μ l n ( r ) b ˜ r , 2 l μ l n ( i ) ] = δ r , 2 n 1 .
μ m m ( r ) = 2 R 2 m , 2 n = 2 R 2 m 1 , 2 n 1 , μ m n ( i ) = 2 R 2 m 1 , 2 n = 2 R 2 m , 2 n 1 ,
l = 1 N ( b ˜ r , 2 l R 2 l , 2 n + b ˜ r , 2 l 1 R 2 l 1 , 2 n ) = t = 1 2 N b ˜ r t R t , 2 n = δ r , 2 n ,
l = 1 N ( b ˜ r , 2 l 1 R 2 l 1 , 2 n 1 + b ˜ r , 2 l R 2 l , 2 n 1 ) = t = 1 2 N b ˜ r t R t , 2 n 1 = δ r , 2 n 1 .
t = 1 2 N b ˜ r t R t s = δ r s ( r , s = 1 , , 2 N ) ,
1 2 r , s = 1 2 N b r s X r X s = A G A + ,
A G A + = m , n = 1 N A m A n * g m n = m , n = 1 N [ A m ( r ) + i A m ( i ) ] [ A n ( r ) i A n ( i ) ] [ g m n ( r ) + i g m n ( i ) ] = m , n = 1 N { [ A m ( r ) A n ( r ) g m n ( r ) + A m ( r ) A n ( i ) g m n ( i ) A m ( i ) A n ( r ) g m n ( i ) + A m ( i ) A n ( i ) g m n ( r ) ] + i [ A m ( r ) A n ( r ) g m n ( i ) A m ( r ) A n ( i ) g m n ( r ) + A m ( i ) A n ( r ) g m n ( r ) + A m ( i ) A n ( i ) g m n ( i ) ] } .
A G A + = 1 2 m , n = 1 N ( X 2 m 1 X 2 n 1 b 2 m 1 , 2 n 1 + X 2 m 1 X 2 n b 2 m 1 , 2 n + X 2 m X 2 n 1 b 2 m , 2 n 1 + X 2 m X 2 n b 2 m , 2 n ) = 1 2 r , s = 1 2 N X r X s b r s ,
A m ( u , υ , d ) = exp ( i k m d ) i λ m d lens d x d y A m ( x , y ) × exp { i k m 2 d [ ( u x ) 2 + ( υ y 2 ) ] } .
A m ( x , y ) = exp [ i k m 2 f ( ξ 0 2 + η 0 2 2 ξ 0 x 2 η 0 y ) ] ,
A m ( u , υ , d ) = exp ( i k m d ) i λ m d × d x d y exp { i k m 2 f ( ξ 0 2 + η 0 2 2 ξ 0 x 2 η 0 y ) + i k m 2 d [ ( u x ) 2 + ( υ y ) 2 ] } = exp [ i k m ( d + ξ 0 2 + η 0 2 2 f ) ] i λ m d d x d y × exp { i k m 2 [ 2 ξ 0 x 2 η 0 y f + ( u x ) 2 + ( υ y ) 2 d ] } .
0 = x i k m 2 ( 2 ξ 0 x 2 η 0 y f + ( u x ) 2 + ( υ y ) 2 d ) | x = x 0 = i k m 2 ( 2 ξ 0 f + 2 ( x 0 u ) d ) ,
x 0 = d f ξ 0 + u .
y 0 = d f η 0 + υ .
A m ( u , υ , d ) = exp [ i k m ( d + ξ 0 2 + η 0 2 2 f ) ] × exp { i k m 2 [ 2 ξ 0 x 2 η 0 y f + ( u x ) 2 + ( υ y ) 2 d ] } = exp [ i k m ( d + ξ 0 2 + η 0 2 2 f ) ] × exp ( i k m 2 { 2 [ d f ( ξ 0 2 + η 0 2 ) ] + ξ 0 u + η 0 υ f + ( ξ 0 2 + η 0 2 ) d f 2 } ) = exp { i k m [ d + 1 2 f 2 ( ξ 0 2 + η 0 2 ) ( f d ) 1 f ( ξ 0 u + η 0 υ ) ] } .
λ | r ± x 0 | , | r ± y 0 | ,

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