Abstract

Optical aberration effects on Gaussian speckle contrast are theoretically examined in an imaging system exhibiting partial spatial coherence. Analysis includes phase-perturbed random fields from a rough object illuminated by an extended source that generate speckle in the image plane. Results indicate that, unlike coherent illumination, speckle contrast in this partially coherent system depends on odd-functional aberrations, such as coma. In addition, calculations show that speckle contrast reduction as a function of coherence factor exhibits a stronger dependence on aberrations than for an aberration-free case.

© 2009 Optical Society of America

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References

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  1. Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984).
  2. J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).
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    [CrossRef]
  4. R. Barakat and 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. 72, 191-197 (1982).
    [CrossRef]
  5. D. Kang, E. Clarkson, and T. D. Milster, “Effect of optical aberrations on Gaussian laser speckle,” Opt. Express 17, 3084-3100 (2009).
    [CrossRef] [PubMed]
  6. R. D. Bahuguna, K. K. Gupta, and K. Singh, “Study of laser speckles in the presence of spherical aberration,” J. Opt. Soc. Am. 69, 6, 877-882 (1979).
    [CrossRef]
  7. R. D. Bahuguna, K. K. Gupta, and K. Singh, “Speckle patterns of weak diffusers: effect of spherical aberration,” Appl. Opt. 19, 1874-1878 (1980).
    [CrossRef] [PubMed]
  8. A. Kumar and K. Singh, “Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations,” Optik 96, 115-119 (1994).
  9. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204-1210 (1976).
    [CrossRef]
  10. D. G. Voelz, P. S. Idell, and K. A. Bush, “Illumination coherence effects in laser-speckle imaging,” Proc. SPIE 1416, 260-265 (1991).
    [CrossRef]
  11. T. A. Leskova, “Generation of partially coherent light in rough surface scattering and suppression of the speckle it produces,” Proc. SPIE 6672, 667201 (2007).
    [CrossRef]
  12. R. K. Raney, “Quadratic filter theory and partially coherent optical systems,” J. Opt. Soc. Am. 59, 1149-1154 (1969).
    [CrossRef]
  13. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761-772 (1970).
    [CrossRef]
  14. H. Lichte and M. Lehmann, “Electron holography-basis and applications,” Rep. Prog. Phys. 71, 016102 (2008).
    [CrossRef]
  15. D. Kang and T. D. Milster, “Simulation method for non-Gaussian speckle in a partially coherent system,” J. Opt. Soc. Am. A 26, 1954-1960 (2009).
    [CrossRef]
  16. I. R. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. TheoryIT-8, 194-195 (1962).
  17. F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory 39, 1293-1302 (1993).
    [CrossRef]
  18. D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).
  19. J. Goodman, Statistical Optics (Wiley, 2000).
  20. J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
    [CrossRef] [PubMed]
  21. G. Palasantzas and J. Krim, “Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface,” Phys. Rev. B 48, 2873 (1993).
    [CrossRef]
  22. E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
    [CrossRef]
  23. J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
    [CrossRef]
  24. E. Jakeman and W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72-79 (1977).
    [CrossRef]
  25. D. Kang and T. D. Milster, “Effect of fractal rough surface Hurst exponent on speckle in imaging systems,” Opt. Lett. 34, 3247-3249 (2009).
    [CrossRef] [PubMed]
  26. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  27. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer, 2001).
    [CrossRef]
  28. Integral variable ξ′ for correlation functions in the case of Gaussian laser speckle from is changed to ρ′.

2009 (3)

2008 (1)

H. Lichte and M. Lehmann, “Electron holography-basis and applications,” Rep. Prog. Phys. 71, 016102 (2008).
[CrossRef]

2007 (1)

T. A. Leskova, “Generation of partially coherent light in rough surface scattering and suppression of the speckle it produces,” Proc. SPIE 6672, 667201 (2007).
[CrossRef]

1994 (1)

A. Kumar and K. Singh, “Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations,” Optik 96, 115-119 (1994).

1993 (3)

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
[CrossRef] [PubMed]

G. Palasantzas and J. Krim, “Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface,” Phys. Rev. B 48, 2873 (1993).
[CrossRef]

F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory 39, 1293-1302 (1993).
[CrossRef]

1991 (2)

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[CrossRef]

D. G. Voelz, P. S. Idell, and K. A. Bush, “Illumination coherence effects in laser-speckle imaging,” Proc. SPIE 1416, 260-265 (1991).
[CrossRef]

1988 (1)

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

1982 (1)

1980 (1)

1979 (1)

1977 (2)

K. A. Stetson, “The vulnerability of speckle photography to lens aberrations,” J. Opt. Soc. Am. 67, 1587-1590 (1977).
[CrossRef]

E. Jakeman and W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72-79 (1977).
[CrossRef]

1976 (1)

1975 (1)

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[CrossRef]

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761-772 (1970).
[CrossRef]

1969 (1)

Asakura, T.

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[CrossRef]

Bahuguna, R. D.

Barakat, R.

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Bruynseraede, Y.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
[CrossRef] [PubMed]

Bush, K. A.

D. G. Voelz, P. S. Idell, and K. A. Bush, “Illumination coherence effects in laser-speckle imaging,” Proc. SPIE 1416, 260-265 (1991).
[CrossRef]

Church, E. L.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[CrossRef]

Clarkson, E.

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761-772 (1970).
[CrossRef]

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer, 2001).
[CrossRef]

Goodman, D. S.

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 2000).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

Gupta, K. K.

Heyvaert, I.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
[CrossRef] [PubMed]

Idell, P. S.

D. G. Voelz, P. S. Idell, and K. A. Bush, “Illumination coherence effects in laser-speckle imaging,” Proc. SPIE 1416, 260-265 (1991).
[CrossRef]

Jakeman, E.

E. Jakeman and W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72-79 (1977).
[CrossRef]

Kang, D.

Krim, J.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
[CrossRef] [PubMed]

G. Palasantzas and J. Krim, “Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface,” Phys. Rev. B 48, 2873 (1993).
[CrossRef]

Kumar, A.

A. Kumar and K. Singh, “Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations,” Optik 96, 115-119 (1994).

Lehmann, M.

H. Lichte and M. Lehmann, “Electron holography-basis and applications,” Rep. Prog. Phys. 71, 016102 (2008).
[CrossRef]

Leskova, T. A.

T. A. Leskova, “Generation of partially coherent light in rough surface scattering and suppression of the speckle it produces,” Proc. SPIE 6672, 667201 (2007).
[CrossRef]

Lichte, H.

H. Lichte and M. Lehmann, “Electron holography-basis and applications,” Rep. Prog. Phys. 71, 016102 (2008).
[CrossRef]

Massey, J. L.

F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory 39, 1293-1302 (1993).
[CrossRef]

Milster, T. D.

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Neeser, F. D.

F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory 39, 1293-1302 (1993).
[CrossRef]

Nisenson, P.

Ohtsubo, J.

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[CrossRef]

Palasantzas, G.

G. Palasantzas and J. Krim, “Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface,” Phys. Rev. B 48, 2873 (1993).
[CrossRef]

Pedersen, H. M.

Raney, R. K.

Reed, I. R.

I. R. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. TheoryIT-8, 194-195 (1962).

Rosenbluth, A. E.

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

Singh, K.

Stetson, K. A.

Takacs, P. Z.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[CrossRef]

Van Haesendonck, C.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
[CrossRef] [PubMed]

Voelz, D. G.

D. G. Voelz, P. S. Idell, and K. A. Bush, “Illumination coherence effects in laser-speckle imaging,” Proc. SPIE 1416, 260-265 (1991).
[CrossRef]

Welford, W. T.

E. Jakeman and W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72-79 (1977).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Inf. Theory (1)

F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory 39, 1293-1302 (1993).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761-772 (1970).
[CrossRef]

Opt. Commun. (2)

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[CrossRef]

E. Jakeman and W. T. Welford, “Speckle statistics in imaging systems,” Opt. Commun. 21, 72-79 (1977).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

A. Kumar and K. Singh, “Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations,” Optik 96, 115-119 (1994).

Phys. Rev. B (1)

G. Palasantzas and J. Krim, “Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface,” Phys. Rev. B 48, 2873 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 1, 57-60 (1993).
[CrossRef] [PubMed]

Proc. SPIE (4)

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

D. G. Voelz, P. S. Idell, and K. A. Bush, “Illumination coherence effects in laser-speckle imaging,” Proc. SPIE 1416, 260-265 (1991).
[CrossRef]

T. A. Leskova, “Generation of partially coherent light in rough surface scattering and suppression of the speckle it produces,” Proc. SPIE 6672, 667201 (2007).
[CrossRef]

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[CrossRef]

Rep. Prog. Phys. (1)

H. Lichte and M. Lehmann, “Electron holography-basis and applications,” Rep. Prog. Phys. 71, 016102 (2008).
[CrossRef]

Other (7)

I. R. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. TheoryIT-8, 194-195 (1962).

J. Goodman, Statistical Optics (Wiley, 2000).

Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer, 2001).
[CrossRef]

Integral variable ξ′ for correlation functions in the case of Gaussian laser speckle from is changed to ρ′.

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Figures (4)

Fig. 1
Fig. 1

Simple scheme for a Kohler illumination system and a 4f imaging system.

Fig. 2
Fig. 2

Speckle contrasts in 1D Kohler illumination system for defocus from the developed theory. (a) and (b) are for σ h = 0.3 λ and 0.2 λ , respectively. H = 0.5 , L cor = 0.25 μ m , and λ = 193 nm . σ c in legends is coherence factor, which is higher for a more incoherent system.

Fig. 3
Fig. 3

Calculated speckle contrasts with Seidel aberrations of (a) defocus and tilt and (b) spherical and coma in 1D Kohler illumination system for σ h = 0.2 λ from the developed theory. Open points are from simulation, which agree well with calculation from theory.

Fig. 4
Fig. 4

Simple schemes showing physical situations of speckle imaging for (a) on-axis point source and (b) off-axis point source.

Equations (50)

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C s = [ I t 2 I t 2 ] 1 2 I t = σ I t I t ,
σ I t = [ I ( ρ s 1 ) I ( ρ s 2 ) I ( ρ s 1 ) I ( ρ s 2 ) ] d ρ s 1 d ρ s 2 .
σ I t = 2 { g R ( ρ s 1 ) g R ( ρ s 2 ) 2 + g I ( ρ s 1 ) g I ( ρ s 2 ) 2 + g I ( ρ s 1 ) g R ( ρ s 2 ) 2 + g R ( ρ s 1 ) g I ( ρ s 2 ) 2 c I 2 ( ρ s 1 ) c I 2 ( ρ s 2 ) c R 2 ( ρ s 1 ) c R 2 ( ρ s 2 ) c R 2 ( ρ s 1 ) c I 2 ( ρ s 2 ) c I 2 ( ρ s 1 ) c R 2 ( ρ s 2 ) } d ρ s 1 d ρ s 2 .
g ( x i , ρ s ) = 1 | m T | h coh ( x i α ) U ( α m T , ρ s ) f ( α m T ) d α ,
U ( x o , ρ s ) = S ( ρ s ) exp ( i 2 π ρ s x o ) ,
T R ( ρ ) = T ( ρ ) exp [ i 2 π W o ( ρ ) ] cos [ 2 π W e ( ρ ) ]
T I ( ρ ) = T ( ρ ) exp [ i 2 π W o ( ρ ) ] sin [ 2 π W e ( ρ ) ] ,
h coh , R ( x i ) = T R ( ρ ) exp [ i 2 π x i ρ ] d ρ ,
h coh , I ( x i ) = T I ( ρ ) exp [ i 2 π x i ρ ] d ρ .
ψ ( Δ , x i , ρ s ) = h coh ( Δ ) U ( x i Δ m T , ρ s ) ,
Ψ R ( ρ , ρ s , x i ) = S ( m T ρ s ) { 1 2 [ T R ( ρ + ρ s ) exp ( i 2 π x i ρ s ) + T R ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] + 1 2 i [ T I ( ρ + ρ s ) exp ( i 2 π x i ρ s ) T I ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] }
Ψ I ( ρ , ρ s , x i ) = S ( m T ρ s ) { 1 2 [ T I ( ρ + ρ s ) exp ( i 2 π x i ρ s ) + T I ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] 1 2 i [ T R ( ρ + ρ s ) exp ( i 2 π x i ρ s ) T R ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] } ,
c R ( ρ s , x i ) = m R | m T | S ( m T ρ s ) T ( ρ s ) cos { 2 π [ x i ρ s + W e ( ρ s ) W o ( ρ s ) ] } ,
c I ( ρ s , x i ) = m R | m T | S ( m T ρ s ) T ( ρ s ) sin { 2 π [ x i ρ s + W e ( ρ s ) W o ( ρ s ) ] } ,
g R ( ρ s 1 ) g R ( ρ s 2 ) = m R 2 S ( m T ρ s 1 ) S ( m T ρ s 2 ) 2 | m T | 2 × { T R α ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 1 ( ρ , ρ s 1 , ρ s 2 ) ] + T R β ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 2 ( ρ , ρ s 1 , ρ s 2 ) ] } d ρ ,
g I ( ρ s 1 ) g I ( ρ s 2 ) = m R 2 S ( m T ρ s 1 ) S ( m T ρ s 2 ) 2 | m T | 2 × { T R α ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 1 ( ρ , ρ s 1 , ρ s 2 ) ] T R β ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 2 ( ρ , ρ s 1 , ρ s 2 ) ] } d ρ ,
g R ( ρ s 1 ) g I ( ρ s 2 ) = m R 2 S ( m T ρ s 1 ) S ( m T ρ s 2 ) 2 | m T | 2 × { T R α ( ρ , ρ s 1 , ρ s 2 ) sin [ φ 1 ( ρ , ρ s 1 , ρ s 2 ) ] T R β ( ρ , ρ s 1 , ρ s 2 ) sin [ φ 2 ( ρ , ρ s 1 , ρ s 2 ) ] } d ρ ,
g I ( ρ s 1 ) g R ( ρ s 2 ) = m R 2 S ( m T ρ s 1 ) S ( m T ρ s 2 ) 2 | m T | 2 { T R α ( ρ , ρ s 1 , ρ s 2 ) sin [ φ 1 ( ρ , ρ s 1 , ρ s 2 ) ] T R β ( ρ , ρ s 1 , ρ s 2 ) sin [ φ 2 ( ρ , ρ s 1 , ρ s 2 ) ] } d ρ ,
T R α ( ρ , ρ s 1 , ρ s 2 ) = T ( ρ + ρ s 1 ) T ( ρ + ρ s 2 ) F ρ 1 [ R α ( Δ x o m T ) ] ,
T R β ( ρ , ρ s 1 , ρ s 2 ) = T ( ρ + ρ s 1 ) T ( ρ ρ s 2 ) F ρ 1 [ R β ( Δ x o m T ) ] ,
R α ( Δ x o ) = exp { ( 2 π λ ) 2 K u ( Δ x o ) }
R β ( Δ x o ) = exp { ( 2 π λ ) 2 K u ( Δ x o ) } .
φ 1 ( ρ , ρ s 1 , ρ s 2 ) = 2 π [ x i ( ρ s 1 ρ s 2 ) W e ( ρ + ρ s 1 ) + W e ( ρ + ρ s 2 ) + W o ( ρ + ρ s 1 ) W o ( ρ + ρ s 2 ) ] ,
φ 2 ( ρ , ρ s 1 , ρ s 2 ) = 2 π [ x i ( ρ s 1 + ρ s 2 ) W e ( ρ + ρ s 1 ) W e ( ρ ρ s 2 ) + W o ( ρ + ρ s 1 ) W o ( ρ ρ s 2 ) ] .
T cos ( ρ + ρ s 1 ) F ρ 1 [ R α ( Δ x o m T ) ] T cos ( ρ + ρ s 2 ) d ρ + T sin ( ρ + ρ s 1 ) F ρ 1 [ R α ( Δ x o m T ) ] T sin ( ρ + ρ s 2 ) d ρ ,
T cos ( ρ ) = T ( ρ ) cos { 2 π [ W e ( ρ ) W o ( ρ ) ] } ,
T sin ( ρ ) = T ( ρ ) sin { 2 π [ W e ( ρ ) W o ( ρ ) ] } ,
K u ( Δ x o ) = σ h 2 exp [ ( Δ x o L cor ) 2 H ] ,
I ( ρ s 1 ) I ( ρ s 2 ) [ T R α ( ρ , ρ s 1 , ρ s 2 ) d ξ ] 2 + [ T R β ( ρ , ρ s 1 , ρ s 2 ) d ρ ] 2 ,
I ( ρ s 1 ) I ( ρ s 2 ) [ T R α ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 1 ( ρ , ρ s 1 , ρ s 2 ) ] d ρ ] 2 + [ T R β ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 2 ( ρ , ρ s 1 , ρ s 2 ) ] d ρ ] 2 ,
ψ R ( Δ , x i , ρ s ) = S ( m T ρ s ) { cos [ 2 π ρ s ( x i Δ ) ] h coh , R ( Δ ) + sin [ 2 π ρ s ( x i Δ ) ] h coh , I ( Δ ) }
ψ I ( Δ , x i , ρ s ) = S ( m T ρ s ) { cos [ 2 π ρ s ( x i Δ ) ] h coh , I ( Δ ) sin [ 2 π ρ s ( x i Δ ) ] h coh , R ( Δ ) } ,
T R ( ρ ) = h coh , R ( Δ ) exp ( i 2 π Δ ρ ) d Δ
T I ( ρ ) = h coh , I ( Δ ) exp ( i 2 π Δ ρ ) d Δ .
Ψ R ( ρ , ρ s , x i ) = S ( m T ρ s ) { cos [ 2 π ρ s ( x i Δ ) ] h coh , R ( Δ ) + sin [ 2 π ρ s ( x i Δ ) ] h coh , I ( Δ ) } exp ( i 2 π Δ ρ ) d Δ .
Ψ R ( ρ , ρ s , x i ) = S ( m T ρ s ) { 1 2 [ T R ( ρ + ρ s ) exp ( i 2 π x i ρ s ) + T R ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] + 1 2 i [ T I ( ρ + ρ s ) exp ( i 2 π x i ρ s ) T I ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] } .
Ψ I ( ρ , ρ s , x i ) = S ( m T ρ s ) { 1 2 [ T I ( ρ + ρ s ) exp ( i 2 π x i ρ s ) + T I ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] 1 2 i [ T R ( ρ + ρ s ) exp ( i 2 π x i ρ s ) T R ( ρ ρ s ) exp ( i 2 π x i ρ s ) ] } .
c R = m R | m T | T R ( 0 ) and c I = m R | m T | T I ( 0 ) ,
R R R i = m R 2 2 | m T | 2 { [ T R ( ρ ) T R ( ρ ) + T I ( ρ ) T I ( ρ ) ] F ρ 1 [ R α ( Δ x o m T ) ] + [ T R ( ρ ) T R ( ρ ) T I ( ρ ) T I ( ρ ) ] F ρ 1 [ R β ( Δ x o m T ) ] } d ρ ,
R I I i = m R 2 2 | m T | 2 { [ T R ( ρ ) T R ( ρ ) + T I ( ρ ) T I ( ρ ) ] F ρ 1 [ R α ( Δ x o m T ) ] [ T R ( ρ ) T R ( ρ ) T I ( ρ ) T I ( ρ ) ] F ρ 1 [ R β ( Δ x o m T ) ] } d ρ ,
R R I i = m R 2 2 | m T | 2 { [ T R ( ρ ) T I ( ρ ) T I ( ρ ) T R ( ρ ) ] F ρ 1 [ R α ( Δ x o m T ) ] + [ T R ( ρ ) T I ( ρ ) T I ( ρ ) T R ( ρ ) ] F ρ 1 [ R β ( Δ x o m T ) ] } d ρ ,
R I R i = m R 2 2 | m T | 2 { [ T I ( ρ ) T R ( ρ ) T R ( ρ ) T I ( ρ ) ] F ρ 1 [ R α ( Δ x o m T ) ] + [ T I ( ρ ) T R ( ρ ) + T R ( ρ ) T I ( ρ ) ] F ρ 1 [ R β ( Δ x o m T ) ] } d ρ ,
c R ( ρ s , x i ) = g R ( ρ s ) = m R | m T | Ψ R ( 0 , ρ s , x i ) = m R | m T | S ( m T ρ s ) T ( m T ρ s ) cos { 2 π [ x i ρ s + W e ( ρ s ) W o ( ρ s ) ] } .
g R ( ρ s 1 ) g R ( ρ s 2 ) = m R 2 2 | m T | 2 × { [ Ψ R ( ρ , ρ s 1 ) Ψ R ( ρ , ρ s 2 ) + Ψ I ( ρ , ρ s 1 ) Ψ I ( ρ , ρ s 2 ) ] F ρ 1 [ R α ( Δ x o m T ) ] + [ Ψ R ( ρ , ρ s 1 ) Ψ R ( ρ , ρ s 2 ) Ψ I ( ρ , ρ s 1 ) Ψ I ( ρ , ρ s 2 ) ] F ρ 1 [ R β ( Δ x o m T ) ] } d ρ .
Ψ R ( ρ , ρ s 1 ) Ψ R ( ρ , ρ s 2 ) + Ψ I ( ρ , ρ s 1 ) Ψ I ( ρ , ρ s 2 ) = S ( m T ρ s ) 2 ( T ( ρ + ρ s 1 ) T ( ρ + ρ s 2 ) exp { i 2 π [ x i ( ρ s 1 ρ s 2 ) W e ( ρ + ρ s 1 ) + W e ( ρ + ρ s 2 ) + W o ( ρ + ρ s 1 ) W o ( ρ + ρ s 2 ) ] } + T ( ρ ρ s 1 ) T ( ρ ρ s 2 ) exp { i 2 π [ x i ( ρ s 1 ρ s 2 ) W e ( ρ ρ s 1 ) + W e ( ρ ρ s 2 ) W o ( ρ ρ s 1 ) + W o ( ρ ρ s 2 ) ] } ) ,
Ψ R ( ρ , ρ s 1 ) Ψ R ( ρ , ρ s 2 ) Ψ I ( ρ , ρ s 1 ) Ψ I ( ρ , ρ s 2 ) = S ( m T ρ s ) 2 ( T ( ρ + ρ s 1 ) T ( ρ ρ s 2 ) exp { i 2 π [ x i ( ρ s 1 + ρ s 2 ) W e ( ρ + ρ s 1 ) W e ( ρ ρ s 2 ) + W o ( ρ + ρ s 1 ) W o ( ρ ρ s 2 ) ] } + T ( ρ ρ s 1 ) T ( ρ + ρ s 2 ) exp { i 2 π [ x i ( ρ s 1 + ρ s 2 ) W e ( ρ ρ s 1 ) W e ( ρ + ρ s 2 ) W o ( ρ ρ s 1 ) + W o ( ρ + ρ s 2 ) ] } ) ,
S ( m T ρ s ) 2 { A ( ρ , ρ s 1 , ρ s 2 ) B ( ρ , ρ s 1 , ρ s 2 ) + A ( ρ , ρ s 1 , ρ s 2 ) B * ( ρ , ρ s 1 , ρ s 2 ) } ,
A ( ρ , ρ s 1 , ρ s 2 ) = T ( ρ + ρ s 1 ) T ( ρ + ρ s 2 ) ,
B ( ρ , ρ s 1 , ρ s 2 ) = exp { i 2 π [ x i ( ρ s 1 ρ s 2 ) W e ( ρ + ρ s 1 ) + W e ( ρ + ρ s 2 ) + W o ( ρ + ρ s 1 ) W o ( ρ + ρ s 2 ) ] } ,
g R ( ρ s 1 ) g R ( ρ s 2 ) = m R 2 S ( m T ρ s ) 2 | m T | 2 × { T R α ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 1 ( ρ , ρ s 1 , ρ s 2 ) ] + T R β ( ρ , ρ s 1 , ρ s 2 ) cos [ φ 2 ( ρ , ρ s 1 , ρ s 2 ) ] } d ρ ,

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