Abstract

A generalized form of spectral representation theory is developed and used with the ABCD formulation of the Huygens–Fresnel integral for studying optical wave propagation through a random medium in the presence of any complex paraxial optical system that can be characterized by an ABCD ray matrix. Formal expressions are developed for the basic optical field moments and various related second-order statistical quantities in terms of three fundamental moments of the first- and second-order complex phase perturbations. Special propagation environments include line-of-sight propagation, single-pass propagation through arbitrary ABCD optical systems, and double-pass propagation through the same random medium in the presence of an ABCD optical system. For illustrative purposes the method is used in the development of expressions for the mean and the normalized variance of the irradiance associated with the Fourier-transform-plane geometry of a lens and the enhanced backscatter effect (EBS) associated with irradiance and phase fluctuations of a reflected Gaussian-beam wave from a Gaussian mirror. The EBS analysis accounts for both finite size and finite focal length of the mirror.

© 1995 Optical Society of America

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  1. L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, transl. (McGraw-Hill, New York, 1960).
  2. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  3. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  4. R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  5. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
    [CrossRef]
  6. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. II.
  8. R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  9. V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics III, E. Wolf, ed. (Elsevier, New York, 1980).
    [CrossRef]
  10. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
    [CrossRef]
  11. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
    [CrossRef]
  12. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  13. H. T. Yura, S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564–575 (1989).
    [CrossRef]
  14. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
    [CrossRef]
  15. C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
    [CrossRef]
  16. A. E. Seigman, Lasers (University Science, Mill Valley, Calif., 1986).
  17. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex paraxial optical systems,” Appl. Opt. 32, 5918–5929 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  20. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
    [CrossRef]
  21. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [CrossRef]
  22. V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).
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    [CrossRef] [PubMed]
  24. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991).
    [CrossRef] [PubMed]
  25. L. C. Andrews, “Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9, 597–600 (1992).
    [CrossRef]
  26. M. S. Belen’kii, “Effect of residual turbulent scintillation and a remote-sensing technique for simultaneous determination of turbulence and scattering parameters of the atmosphere,” J. Opt. Soc. Am. A 11, 1150–1158 (1994).
    [CrossRef]
  27. V. P. Lukin, “Efficiency of the compensation of phase distortions of optical waves,” Sov. J. Quantum Electron. 7, 522–524 (1977).
    [CrossRef]
  28. G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
    [CrossRef]
  29. J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
    [CrossRef]
  30. J. Smith, “Folded-path weighting function for a high-frequency spherical wave,” J. Opt. Soc. Am. 63, 1095–1097 (1973).
    [CrossRef]

1994 (4)

1993 (3)

1992 (2)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

L. C. Andrews, “Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9, 597–600 (1992).
[CrossRef]

1991 (1)

1989 (1)

1987 (1)

1983 (1)

H. T. Yura, C. C. Sung, S. C. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. A 73, 500–502 (1983).
[CrossRef]

1980 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

1979 (1)

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

1978 (1)

G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

1977 (1)

V. P. Lukin, “Efficiency of the compensation of phase distortions of optical waves,” Sov. J. Quantum Electron. 7, 522–524 (1977).
[CrossRef]

1975 (2)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

1973 (1)

1972 (1)

J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
[CrossRef]

1971 (1)

1970 (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Andrews, L. C.

Banakh, V. A.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).

Belen’kii, M. S.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, transl. (McGraw-Hill, New York, 1960).

Churnside, J. H.

Clifford, S. C.

H. T. Yura, C. C. Sung, S. C. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. A 73, 500–502 (1983).
[CrossRef]

Dashen, R.

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
[CrossRef]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Hamiter, M. A.

J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
[CrossRef]

Hanson, S. G.

Hill, R. J.

H. T. Yura, C. C. Sung, S. C. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. A 73, 500–502 (1983).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. II.

Kravtsov, Yu. A.

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Lukin, V. P.

V. P. Lukin, “Efficiency of the compensation of phase distortions of optical waves,” Sov. J. Quantum Electron. 7, 522–524 (1977).
[CrossRef]

Lutomirski, R. F.

Miller, W. B.

Mironov, V. L.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).

Patrushev, G. Ya.

G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

Pries, T. H.

J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
[CrossRef]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Ricklin, J. C.

Seigman, A. E.

A. E. Seigman, Lasers (University Science, Mill Valley, Calif., 1986).

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Skipka, K. J.

J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
[CrossRef]

Smith, J.

J. Smith, “Folded-path weighting function for a high-frequency spherical wave,” J. Opt. Soc. Am. 63, 1095–1097 (1973).
[CrossRef]

J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
[CrossRef]

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Sung, C. C.

H. T. Yura, C. C. Sung, S. C. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. A 73, 500–502 (1983).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics III, E. Wolf, ed. (Elsevier, New York, 1980).
[CrossRef]

Young, C. Y.

C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
[CrossRef]

Yura, H. T.

Zavorotnyi, V. U.

V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics III, E. Wolf, ed. (Elsevier, New York, 1980).
[CrossRef]

Appl. Opt. (4)

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,”J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

J. Smith, T. H. Pries, K. J. Skipka, M. A. Hamiter, “High-frequency plane-wave filter functions for a folded path,” J. Opt. Soc. Am. A 62, 1183–1187 (1972).
[CrossRef]

L. C. Andrews, “Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9, 597–600 (1992).
[CrossRef]

M. S. Belen’kii, “Effect of residual turbulent scintillation and a remote-sensing technique for simultaneous determination of turbulence and scattering parameters of the atmosphere,” J. Opt. Soc. Am. A 11, 1150–1158 (1994).
[CrossRef]

H. T. Yura, C. C. Sung, S. C. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. A 73, 500–502 (1983).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

H. T. Yura, S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564–575 (1989).
[CrossRef]

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
[CrossRef]

Proc. IEEE (4)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68, 1424–1443 (1980).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. P. Lukin, “Efficiency of the compensation of phase distortions of optical waves,” Sov. J. Quantum Electron. 7, 522–524 (1977).
[CrossRef]

G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

Waves Random Media (1)

C. Y. Young, L. C. Andrews, “Effects of a modified spectral model on the spatial coherence of a laser beam,” Waves Random Media 4, 385–397 (1994).
[CrossRef]

Other (7)

A. E. Seigman, Lasers (University Science, Mill Valley, Calif., 1986).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. II.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics III, E. Wolf, ed. (Elsevier, New York, 1980).
[CrossRef]

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
[CrossRef]

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech, Dedham, Mass., 1987).

L. A. Chernov, Wave Propagation in a Random Medium, R. A. Silverman, transl. (McGraw-Hill, New York, 1960).

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Figures (4)

Fig. 1
Fig. 1

Propagation geometry for a Gaussian beam originating at distance L1 to the left of a thin Gaussian lens of real focal length FG and effective transmission radius WG. The observation plane is located at distance L2 to the right of the lens.

Fig. 2
Fig. 2

Normalized variance of irradiance (scaled by the spherical-wave irradiance variance) as a function of Fresnel ratio ΩG = 2FG/kWG2 for the Fourier-transform-plane propagation geometry. The curves corresponding to r = 0 denote the normalized variance at the center line of the beam, and those corresponding to r = W denote the normalized variance at the diffractive beam edge.

Fig. 3
Fig. 3

Normalized irradiance variance [scaled by the Rytov variance σ12(L)] at the center line of an optical wave reflected from a point target as a function of the Fresnel ratio Ω = 2L/kW02. The solid curve depicts a collimated beam, and the dashed curve depicts a convergent beam.

Fig. 4
Fig. 4

Schematic representation of optical wave propagation through a general cross-link propagation system consisting of a train of optical elements O1, O2, …, ON arbitrarily located along the propagation path.

Equations (112)

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

U 0 ( r , 0 ) = exp ( - 1 2 α k r 2 ) ,             α = 2 k W 0 2 + i 1 F 0 ,
U 0 ( r , L ) = - i k 2 π B exp ( i k L ) - d 2 s U 0 ( s , 0 ) × exp [ i k 2 B ( A s 2 - 2 s · r + D r 2 ) ] = 1 p ( L ) exp ( i k L ) exp ( - 1 2 β ( L ) k r 2 ) ,
p ( L ) = A + i α B ,
β ( L ) = α D - i C A + i α B = 2 k W 2 + i 1 F .
Θ = 1 + L F ,             Λ = 2 L k W 2 ,             Θ ˜ = 1 - Θ ,
U 0 ( r , l ) = ( Θ - i Λ ) exp ( i k L ) exp [ i k 2 L ( Θ ˜ + i Λ ) r 2 ] ,
U ( r , L ) = U 0 ( r , L ) exp [ ψ ( r , L ) ] ,
ψ ( r , L ) = ψ 1 ( r , L ) + ψ 2 ( r , L ) + = χ ( r , L ) + i S ( r , L ) ,
M 1 ( r ) = exp [ ψ ( r , L ) ] ,
M 2 ( r 1 , r 2 ) = exp [ ψ ( r 1 , L ) + ψ * ( r 2 , L ) ] ,
M 4 ( r 1 , r 2 , r 3 , r 4 ) = exp [ ψ ( r 1 , L ) + ψ * ( r 2 , L ) + ψ ( r 3 , L ) + ψ * ( r 4 , L ) ] ,
exp ( ψ ) = exp [ ψ + 1 2 ( ψ - ψ ) 2 ] .
U ( r , L ) = U 0 ( r , L ) M 1 ( r ) = U 0 ( r , L ) exp [ E 1 ( 0 , 0 ) ] ,
E 1 ( 0 , 0 ) = ψ 2 ( r , L ) + 1 2 ψ 1 2 ( r , L ) = - π k 2 0 L - Φ n ( κ , z ) d 2 κ d z .
ψ 1 ( r , L ) = i k 0 L - exp [ i γ κ · r - i κ 2 γ 2 k B ( z ; L ) ] × d ν ( κ , z ) d z ,
n 1 ( r , z ) = - exp ( i κ · r ) d ν ( κ , z ) ,
Γ 2 ( r 1 , r 2 , L ) = U 0 ( r 1 , L ) U 0 * ( r 2 , L ) M 2 ( r 1 , r 2 ) = Γ 2 0 ( r 1 , r 2 , L ) exp [ 2 E 1 ( 0 , 0 ) + E 2 ( r 1 , r 2 ) ] ,
Γ 2 0 ( r 1 , r 2 , L ) = W 0 2 W 2 exp ( - 2 r 12 2 W 2 - ρ 12 2 2 W 2 - i k F ρ 12 · r 12 ) .
Γ 4 ( r 1 , r 2 , r 3 , r 4 , L ) = U 0 ( r 1 , L ) U 0 * ( r 2 , L ) U 0 ( r 3 , L ) U 0 * ( r 4 , L ) × M 4 ( r 1 , r 2 , r 3 , r 4 ) = Γ 2 0 ( r 1 , r 2 , L ) Γ 2 0 ( r 3 , r 4 , L ) exp [ 4 E 1 ( 0 , 0 ) + E 2 ( r 1 , r 2 ) + E 2 ( r 1 , r 4 ) + E 2 ( r 3 , r 2 ) + E 2 ( r 3 , r 4 ) + E 3 ( r 1 , r 3 ) + E 3 * ( r 2 , r 4 ) ] .
I ( r , L ) = Γ 2 0 ( r , r , L ) exp [ 2 E 1 ( 0 , 0 ) + E 2 ( r , r ) ] ,
I 2 ( r , L ) = I ( r , L ) 2 exp { 2 Re [ E 2 ( r , r ) + E 3 ( r , r ) ] } ,
σ I 2 ( r , L ) σ ln I 2 ( r , L ) = 2 Re [ E 2 ( r , r ) + E 3 ( r , r ) ] ,
B χ , S ( r 1 , r 2 , L ) = 1 2 Re [ E 2 ( r 1 , r 2 ) ± E 3 ( r 1 , r 2 ) ] ,
σ S 2 ( r , L ) = 1 2 Re [ E 2 ( r , r ) - E 3 ( r , r ) ] ,
D ( r 1 , r 2 , L ) = Re [ E 2 ( r 1 , r 2 ) + E 2 ( r 2 , r 2 ) - 2 E 2 ( r 1 , r 2 ) ] ,
γ = p ( z ) p ( L ) = 1 + i α z 1 + i α L ,             0 z L ,
γ = 1 - ( Θ ˜ + i Λ ) ξ ,             ξ = 1 - z / L .
E 2 ( r 1 , r 2 , ) = ψ 1 ( r 1 , L ) ψ 1 * ( r 2 , L ) = 2 π k 2 0 L - Φ n ( κ , z ) × exp [ i κ · ( γ r 1 - γ * r 2 ) - i κ 2 2 k ( γ - γ * ) ( L - z ) ] d 2 κ d z ,
E 3 ( r 1 , r 2 , ) = ψ 1 ( r 1 , L ) ψ 1 ( r 2 , L ) = - 2 π k 2 0 L - Φ n ( κ , z ) × exp [ i γ κ · ( r 1 - r 2 ) - i κ 2 γ k ( L - z ) ] d 2 κ d z ,
E 1 ( 0 , 0 ) = - 2 π 2 k 2 L 0 κ Φ n ( κ ) d κ ,
E 2 ( r 1 , r 2 , ) = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ L ξ 2 κ 2 / k ) × J 0 [ κ ( 1 - Θ ˜ ξ ) ρ 12 - 2 i Λ ξ r 12 ] d κ d ξ ,
E 3 ( r 1 , r 2 , ) = - 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ L ξ 2 κ 2 / k ) × exp [ - i κ 2 L k ξ ( 1 - Θ ˜ ξ ) ] × J 0 [ ( 1 - Θ ˜ ξ - i Λ ξ ) κ ρ 12 ] d κ d ξ .
r 12 = 1 2 ( r 1 + r 2 ) ,             ρ 12 = r 1 - r 2 .
α G = 2 k W G 2 + i 1 F G ,
[ A B C D ] = [ 1 L 2 0 1 ] [ 1 0 i α G 1 ] [ 1 L 1 0 1 ] = [ 1 + i α G L 2 L 1 + L 2 ( 1 + i α G L 1 ) i α G 1 + i α G L 1 ] ,
p ( L 1 + L 2 ) = A + i α B = 1 + i α L 1 + i α L 2 + i α G L 2 ( 1 + i α L 1 ) = ( 1 + i α L 1 ) [ 1 + ( Θ ˜ 1 + i Λ 1 ) L 2 / L 1 + i α G L 2 ] .
Θ 1 - i Λ 1 = 1 1 + i α L 1 = 1 + L 1 F 1 - i 2 L 1 k W 1 2 ,
Θ 2 - i Λ 2 = 1 1 + ( Θ ˜ 1 + i Λ 1 ) L 2 / L 1 + i α G L 2 = L 1 L 2 [ L 1 / L 2 - L 1 / F G + Θ ˜ 1 - i ( Λ 1 + Ω G ) ( L 1 / L 2 - L 1 / F G + Θ ˜ 1 ) 2 + ( Λ 1 + Ω G ) 2 ] ,
1 p ( L 1 + L 2 ) = ( Θ 1 - i Λ 1 ) ( Θ 2 - i Λ 2 ) = Θ - i Λ ,
Θ = Θ 1 Θ 2 - Λ 1 Λ 2 ,             Λ = Λ 1 Θ 2 + Θ 1 Λ 2 .
ψ 1 ( r , L ) = i k 0 L 1 + L 2 - exp [ i γ κ · r - i κ 2 γ 2 k B ( z ; L ) ] × d ν ( κ , z ) d z = i k 0 L 1 - exp [ i γ 1 κ · r - i κ 2 γ 1 2 k B ( z ; L ) ] × d ν ( κ , z ) d z + i k 0 L 2 - exp [ i γ 2 κ · r - i κ 2 γ 2 2 k B ( z ; L 2 ) ] d ν ( κ , z ) d z ,
[ A 1 B 1 C 1 D 1 ] = [ 1 z 0 1 ] = [ 1 L 1 ( 1 - ξ ) 0 1 ] ,
γ 1 = p ( z ) p ( L 1 + L 2 ) = ( Θ - i Λ ) ξ + ( Θ 2 - i Λ 2 ) ( 1 - ξ ) .
[ A 2 B 2 C 2 D 2 ] = [ 1 z 0 1 ] [ 1 0 i α G 1 ] [ 1 L 1 0 1 ] = [ 1 + i α G L 2 η L 1 + L 2 η ( 1 + i α G L 1 ) i α G 1 + i α G L 1 ] ,
γ 2 = p ( L 1 + z ) p ( L 1 + L 2 ) = - ( Θ - i Λ ) L 2 η / L 1 + ( Θ 2 - i Λ 2 ) × ( 1 - L 2 η / L 1 + i α G L 2 η ) .
B ( z ; L ) B ( ξ ) = L 2 + L 1 ( 1 + i α G L 2 ) ξ ,             0 ξ 1 ,
B ( z ; L 2 ) = L 2 ( 1 - η ) ,             0 η 1.
E 2 ( r 1 , r 2 ) = 2 π k 2 L 1 0 1 - Φ n ( κ , ξ ) × exp { i κ · ( γ 1 r 1 - γ 1 * r 2 ) - i κ 2 2 k [ γ 1 B ( ξ ) - γ 1 * B * ( ξ ) ] } d 2 κ d ξ + 2 π k 2 L 2 0 1 - Φ n ( κ , ξ ) × exp [ i κ · ( γ 2 r 1 - γ 2 * r 2 ) - i κ 2 L 2 2 k ( γ 2 - γ 2 * ) ( 1 - ξ ) ] d 2 κ d ξ ,
E 3 ( r 1 , r 2 ) = - 2 π k 2 L 1 0 1 - Φ n ( κ , ξ ) × exp [ i γ 1 κ · ( r 1 - r 2 ) - i κ 2 k γ 1 B ( ξ ) ] d 2 κ d ξ - 2 π k 2 L 2 0 1 - Φ n ( κ , ξ ) × exp [ i γ 2 κ · ( r 1 - r 2 ) - i κ 2 L 2 k γ 2 ( 1 - ξ ) ] × d 2 κ d ξ .
γ 1 = ( Θ 2 - i Λ 2 ) ( 1 - ξ ) ,             B ( ξ ) = F G ( 1 - i ξ Ω G ) ,
Θ 2 = 1 1 + Ω G 2 ,             Λ 2 = Ω G 1 + Ω G 2 .
E 2 ( r 1 , r 2 ) = 4 π 2 k 2 F G 0 1 0 κ Φ n ( κ ) × J 0 ( Θ 2 ρ 12 - 2 i Λ 2 r 12 κ ξ ) × exp ( - Λ 2 F G κ 2 ξ 2 k ) d κ d ξ ,
E 3 ( r 1 , r 2 ) = - 4 π 2 k 2 F G 0 1 0 κ Φ n ( κ ) × J 0 [ ( Θ 2 - i Λ 2 ) κ ξ ρ 12 ] exp ( - Λ 2 F G κ 2 ξ 2 k ) × exp [ - i κ 2 F G k ξ ( 1 - Θ ˜ 2 ξ ) ] d κ d ξ .
I ( r , L ) = W G 2 W 2 exp ( - 2 r 2 / W 2 ) × exp { - 4 π 2 k 2 F G 0 1 0 κ Φ n ( κ ) × [ 1 - exp ( - Λ 2 F G κ 2 ξ 2 / k ) × I 0 ( 2 Λ 2 r κ ξ ) ] d κ d ξ } ,
I ( r , L ) W G 2 W 2 [ 1 + 1.33 σ 1 2 ( F G ) Λ 2 5 / 6 ] × exp { - 2 r 2 W 2 [ 1 + 1.11 σ 1 2 ( F G ) Λ 2 5 / 6 ] } ,             r W ,
σ I 2 ( r , L ) = 8 π 2 k 2 F G 0 1 0 κ Φ n ( κ ) exp ( - Λ 2 F G κ 2 ξ 2 k ) × { I 0 ( 2 Λ 2 r κ ξ ) - cos [ κ 2 F G k ξ ( 1 - Θ ˜ 2 ξ ) ] } d κ d ξ .
σ I 2 ( r , L ) 4.42 σ 1 2 ( F G ) ( Ω G 1 + Ω G 2 ) 5 / 6 r 2 W 2 + 3.86 σ 1 2 ( F G ) { ( 1 + Ω G 2 / 9 1 + Ω G 2 ) 5 / 12 × cos [ 5 6 tan - 1 ( 3 + Ω G 2 2 Ω G ) ] - 11 16 ( Ω G 1 + Ω G 2 ) 5 / 6 } ,             r W .
E 2 ( r 1 , r 2 ) = 4 π 2 k 2 F G 0 κ Φ n ( κ ) I 0 ( 2 κ r 12 Ω G ) × exp ( - κ 2 F G k Ω G ) d κ ,
E 3 ( r 1 , r 2 ) = - 4 π 2 k 2 F G 0 1 0 κ Φ n ( κ ) I 0 ( κ ρ 12 Ω G ) × exp ( - κ 2 F G k Ω G ) exp ( - i κ 2 F G ξ k ) d κ d ξ .
I ( r , L ) = 1 Ω G 2 exp ( - 2 r 2 / W 2 ) exp { - 4 π 2 k 2 F G 0 κ Φ n ( κ ) × [ 1 - exp ( - κ 2 F G / k Ω G ) I 0 ( 2 κ r / Ω G ) ] d κ } 1 Ω G 2 [ 1 + 3.54 σ 1 2 ( F G ) Ω G - 5 / 6 ] × exp { - 2 r 2 W 2 [ 1 + 2.95 σ 1 2 ( F G ) Ω G - 5 / 6 ] } ,             r W ,
σ I 2 ( r , L ) = 8 π 2 k 2 F G 0 1 0 κ Φ n ( κ ) exp ( - κ 2 F G k Ω G ) × [ I 0 ( 2 κ r Ω G ) - cos ( κ 2 F G ξ k ) ] d κ d ξ 11.79 σ 1 2 ( F G ) Ω G - 5 / 6 r 2 W 2 + 3.86 σ 1 2 ( F G ) × [ ( 1 + 1 / Ω G 2 ) 11 / 12 sin ( 11 6 tan - 1 Ω G ) - 11 6 Ω G - 5 / 6 ] ,             r W ,
γ 2 = 1 - ( Θ ˜ 2 + i Λ 2 ) ξ ,
Ω 0 = 1 - L F 0 ,             Ω = 2 L k W 0 2 ,
Θ 1 = Ω 0 Ω 0 2 + Ω 2 ,             Λ 1 = Ω Ω 0 2 + Ω 2 .
Θ 2 = 2 - Θ 1 - L / F R ( 2 - Θ 1 - L / F R ) 2 + ( Λ 1 + Ω R ) 2 , Λ 2 = Λ 1 + Ω R ( 2 - Θ 1 - L / F R ) 2 + ( Λ 1 + Ω R ) 2 ,
E 2 ( r 1 , r 2 ) = 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp { i κ · ( γ 1 r 1 - γ 1 * r 2 ) - i κ 2 2 k [ γ 1 B ( ξ ) - γ 1 * B * ( ξ ) ] } d 2 κ d ξ + 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp [ i κ · ( γ 2 r 1 - γ 2 * r 2 ) - i L κ 2 2 k ( γ 2 - γ 2 * ) ( 1 - ξ ) ] d 2 κ d ξ + 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp { i κ · ( γ 1 r 1 - γ 2 * r 2 ) - i κ 2 2 k [ γ 1 B ( ξ ) - γ 2 * L ( 1 - ξ ) ] } d 2 κ d ξ + 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp { i κ · ( γ 2 r 1 - γ 1 * r 2 ) - i κ 2 2 k [ γ 2 L ( 1 - ξ ) - γ 1 * B * ( ξ ) ] } d 2 κ d ξ ,
E 3 ( r 1 , r 2 ) = - 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp [ i γ 1 κ · ( r 1 - r 2 ) - i κ 2 k γ 1 B ( ξ ) ] d 2 κ d ξ - 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp [ i γ 2 κ · ( r 1 - r 2 ) - i L κ 2 k γ 2 ( 1 - ξ ) ] d 2 κ d ξ - 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp { i κ · ( γ 1 r 1 - γ 2 r 2 ) - i κ 2 2 k [ γ 1 B ( ξ ) + γ 2 L ( 1 - ξ ) ] } d 2 κ d ξ - 2 π k 2 L 0 1 - Φ n ( κ , ξ ) exp { i κ · ( γ 2 r 1 - γ 1 r 2 ) - i κ 2 2 k [ γ 2 L ( 1 - ξ ) + γ 1 B ( ξ ) ] } d 2 κ d ξ .
γ 1 = ( Θ - i Λ ) ξ + ( Θ 2 - i Λ 2 ) ( 1 - ξ ) ,
γ 2 = ξ + ( Θ 2 - i Λ 2 ) ( 1 - ξ ) ,
B ( ξ ) = L ( 1 + ξ + i ξ α R L ) ,
σ I 2 ( r , 2 L ) 2 Re [ E 2 ( r , r ) + E 3 ( r , r ) ] ,
σ S 2 ( r , 2 L ) = 1 2 Re [ E 2 ( r , r ) - E 3 ( r , r ) ] ,
σ I 2 ( r , 2 L ) = σ I , b 2 ( 0 , L ) + σ I , sph 2 ( L ) + 2 C 1 ( r , L ) ,             Ω R 1.
σ I , b 2 ( 0 , L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ 1 L κ 2 ξ 2 / k ) × { 1 - cos [ L κ 2 k ξ ( 1 - Θ ˜ 1 ξ ) ] } d κ d ξ ,
C 1 ( r , L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) J 0 ( r κ ξ ) × exp ( - Λ 1 L κ 2 ξ 2 / 2 k ) { cos ( L κ 2 ξ 2 Θ 1 2 k ) - cos [ L κ 2 2 κ ξ ( 2 - 2 ξ + Θ 1 ξ ) ] } d κ d ξ .
σ I , b 2 ( 0 , L ) 3.86 σ 1 2 ( L ) { 0.4 [ ( 1 + 2 Θ 1 ) 2 + 4 Λ 1 2 ] 5 / 12 × cos [ 5 6 tan - 1 ( 1 + 2 Θ 1 2 Λ 1 ) ] - 11 6 Λ 1 5 / 6 } ,
C 1 ( 0 , L ) 1.55 σ 1 2 ( L ) { [ ( 1 + Θ 1 ) 2 + Λ 1 2 ] 5 / 12 × cos [ 5 6 tan - 1 ( 1 + Θ 1 Λ 1 ) ] - 0.96 ( Θ 1 2 + Λ 1 2 ) 5 / 12 cos [ 5 6 tan - 1 ( Θ 1 Λ 1 ) ] } .
σ S 2 ( r , 2 L ) = σ S , b 2 ( 0 , L ) + σ S , sph 2 ( L ) + 2 C 2 ( r , L ) ,             Ω R 1 ,
σ S , b 2 ( 0 , L ) = 2 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ 1 L κ 2 ξ 2 / k ) × { 1 + cos [ L κ 2 k ξ ( 1 - Θ ˜ 1 ξ ) ] } d κ d ξ ,
C 2 ( r , L ) = 2 π 2 k 2 L 0 1 0 κ Φ n ( κ ) J 0 ( r κ ξ ) × exp ( - Λ 1 L κ 2 ξ 2 / 2 k ) { cos ( L κ 2 ξ 2 Θ 1 2 k ) + cos [ L κ 2 2 k ξ ( 2 - 2 ξ + Θ 1 ξ ) ] } d κ d ξ .
Θ 2 = 2 - Θ 1 ( 2 - Θ 1 ) 2 + Λ 1 2 ,             Λ 2 = Λ 1 ( 2 - Θ 1 ) 2 + Λ 1 2 ,
σ I 2 ( r , 2 L ) = 16 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - 2 Λ L κ 2 ξ 2 k ) × { I 0 ( 2 Λ κ ξ r ) - cos [ 2 L κ 2 k ξ ( 1 - Θ ˜ ξ ) ] } d κ d ξ + 16 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp [ - 2 Λ 2 L κ 2 k ( 1 + ξ 2 ) ] × Re { J 0 [ 2 κ r ( Θ ˜ 2 ξ + i Λ 2 ) ] exp ( - i L κ 2 ξ Θ k ) - J 0 [ 2 κ r ξ ( Θ ˜ 2 + i Λ 2 ) ] exp [ - i L κ 2 k ( Θ 2 - Θ ˜ 2 ξ 2 ) ] } × d κ d ξ
σ I 2 ( r , 2 L ) = σ I , b 2 ( r , 2 L ) + 2 C 3 ( r , L ) ,             Ω R 1 ,             L / F R = 0.
σ S 2 ( r , 2 L ) = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - 2 Λ L κ 2 ξ 2 k ) × { I 0 ( 2 Λ κ ξ r ) + cos [ 2 L κ 2 k ξ ( 1 - Θ ˜ ξ ) ] } d κ d ξ + 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp [ - 2 Λ 2 L κ 2 k ( 1 + ξ 2 ) ] × Re { J 0 [ 2 κ r ( Θ ˜ 2 ξ + i Λ 2 ) ] exp ( - i L κ 2 ξ Θ k ) + J 0 [ 2 κ r ξ ( Θ ˜ 2 + i Λ 2 ) ] exp [ - i L κ 2 k ( Θ 2 - Θ ˜ 2 ξ 2 ) ] } d κ d ξ = σ S , b 2 ( r , 2 L ) + 2 C 4 ( r , L ) ,             Ω R 1 , L / F R = 0 ,
Θ 2 = 2 - L / F R ( 2 - L / F R ) 2 + Ω R 2 ,             Λ 2 = Ω R ( 2 - L / F R ) 2 + Ω R 2 .
γ 1 = ( Θ 2 - i Λ 2 ) ( 1 - ξ ) ,             γ 2 = ξ + ( Θ 2 - i Λ 2 ) ( 1 - ξ ) , B ( ξ ) = L ( 1 + ξ - ξ L / F R + i ξ Ω R ) ,
σ I 2 ( r , 2 L ) = 16 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ 2 L κ 2 ξ 2 k ) × { I 0 ( 2 Λ 2 κ ξ r ) - cos [ L κ 2 k ξ ( 1 - Θ ˜ 2 ξ ) ] } d κ d ξ + 16 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - Λ 2 L κ 2 ξ 2 k ) × Re { J 0 [ ( 1 - ξ + 2 i Λ 2 ξ ) κ r ] - J 0 [ ( 1 - ξ ) κ r ] × cos [ L κ 2 k ξ ( 1 - Θ ˜ 2 ξ ) ] } d κ d ξ .
Θ 2 = 1 1 + Ω R 2 ,             Λ 2 = Ω R 1 + Ω R 2 ,
σ I 2 ( 0 , 2 L ) = 32 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( - L κ 2 ξ 2 k Ω R ) × { 1 - cos [ L κ 2 k ξ ( 1 - ξ ) ] } d κ d ξ .
Θ = Θ 2 = 1 - L / F R ( 1 - L / F R ) 2 + Ω R 2 , Λ = Λ 2 = Ω R ( 1 - L / F R ) 2 + Ω R 2 ,
γ 1 = Θ 2 - i Λ 2 ,             γ 2 = ξ + ( Θ 2 - i Λ 2 ) ( 1 - ξ ) , B ( ξ ) = L ( 1 + ξ - ξ L / F R + i ξ Ω R ) .
σ I 2 ( r , 2 L ) = σ I , pl 2 ( L ) + σ I , sph 2 ( L ) + 2 C 1 ( r , L ) ,             Ω R 1 ,
σ I 2 ( r , 2 L ) = 8 π 2 κ 2 L 0 1 0 κ Φ n ( κ ) × { 2 - cos [ L κ 2 k ( Θ 2 + ξ ) ] - cos [ L κ 2 k ξ ( 1 - Θ ˜ 2 ξ ) ] } d κ d ξ + 16 π 2 k 2 L 0 1 0 κ Φ n ( κ ) J 0 ( Θ ˜ 2 κ ξ r ) × { cos [ L κ 2 2 k ξ ( 2 Θ 2 + Θ ˜ 2 ξ ) ] - cos [ L κ 2 2 k ( 2 Θ 2 + Θ ˜ 2 ξ - Θ ˜ 2 ξ 2 ) ] } d κ d ξ .
σ I 2 ( r , 2 L ) = σ I , pl 2 ( 2 L ) + 2 C 3 ( 0 , L ) ,             Ω R 1 ,             L / F R = 0 ,
σ I 2 ( r , 2 L ) = { 1.40 σ 1 2 ( L ) , Ω R 1 , r ( L / k ) 1 / 2 2.06 σ 1 2 ( L ) , Ω R 1 , r = 0 5.23 σ 1 2 ( L ) , Ω R 1 , L / F R = 0 , all r .
σ I 2 ( 0 , 2 L ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) ( exp ( - L κ 2 k Ω R ) × [ 1 - cos ( L κ 2 ξ k ) ] + exp ( - L κ 2 k Ω R ξ 2 ) × { 1 - cos [ L κ 2 k ξ ( 1 - ξ ) ] } + 2 exp [ - L κ 2 2 k Ω R ( 2 - 2 ξ + ξ 2 ) ] × { cos ( L κ 2 ξ 2 2 k ) - cos [ L κ 2 2 k ( 2 ξ - ξ 2 ) ] } ) d κ d ξ .
σ S 2 ( r , 2 L ) = σ S , pl 2 ( 2 L ) + 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) × [ cos ( L κ 2 ξ k ) + cos ( L κ 2 k ) ] d κ d ξ = 8 π 2 k 2 L 0 κ Φ n ( κ ) cos 2 ( L κ 2 2 k ) × [ 1 + k L κ 2 sin ( L κ 2 k ) ] d κ ,
1 p ( L ) = m = 1 N + 1 ( Θ m - i Λ m ) = Θ - i Λ ,
ψ 1 ( r , L ) = i k m = 1 N + 1 0 L m - exp [ i γ m κ · r - i κ 2 γ m 2 k × B ( z m ; L - ζ m - 1 ) ] d ν ( κ , z m ) d z m ,
ζ 0 = 0 ,             ζ m = j = 1 m L j ,             m = 1 , 2 , , N .
E 2 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 * ( r 2 , L ) = 2 π k 2 m = 1 N + 1 0 L m - Φ n ( κ , z m ) × exp { i κ · ( γ m r 1 - γ m * r 2 ) - i κ 2 2 k × [ γ m B ( z m ; L - ζ m - 1 ) - γ m * × B * ( z m ; L - ζ m - 1 ) ] } d 2 κ d z m ,
E 3 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 ( r 2 , L ) = - 2 π k 2 m = 1 N + 1 0 L m - Φ n ( κ , z m ) × exp [ i γ m κ · ( r 1 - r 2 ) - i κ 2 k γ m × B ( z m ; L - ζ m - 1 ) ] d 2 κ d z m .
[ A B C D ] = [ A B C D ] [ 1 0 - 1 f + i 2 k W 0 2 1 ] [ 1 L 0 1 ] = [ A - B L ( 1 + L F 0 - i 2 L k W 0 2 ) A L - B ( L F 0 - i 2 L k W 0 2 ) C - D L ( 1 + L F 0 - i 2 L k W 0 2 ) C L - D ( L F 0 - i 2 L k W 0 2 ) ] ,
ψ 1 ( r , L ) = 0 L - exp ( i γ κ · r ) H ( κ , z ) d ν ( κ , z ) d z ,
h ( r , z ) = 2 k 2 G ( 0 , r ; z , L ) U 0 ( r , z ) U 0 ( 0 , L ) .
G ( 0 , r ; z , L ) = 1 4 π B ( z ; L ) × exp [ i k ( L - z ) + i k D ( z ; L ) r 2 2 B ( z ; L ) ] .
h ( r , z ) = k 2 2 π γ B ( z ; L ) exp [ i k r 2 γ B ( z ; L ) ] ,
H ( κ , z ) = - exp ( - κ · r ) h ( r , z ) d 2 r = i k exp [ - i κ 2 2 k γ B ( z ; L ) ] .
H ( - κ , z ) = H ( κ , z ) ,
d ν ( κ , z ) = d ν * ( - κ , z ) ,
d ν ( κ , z ) d ν * ( κ , z ) = F n ( κ , z - z ) δ ( κ - κ ) d 2 κ d 2 κ ,
- F n ( κ , z ) d z = 2 π Φ n ( κ ) .

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