Abstract

The problem of the propagation of narrow radiation beams in a scattering medium is considered. The previously formulated small-angle approximation solution accounting for the path length spread is further developed. The numerical scheme for practical calculations is implemented, and the simulation results are presented and discussed. Applicability of the new solution to certain problems of optical communications and data transfer techniques is shown.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. W. S. Helliwell, “Finite-difference solution to the radiative-transfer equation for in-water radiance,” J. Opt. Soc. Am. A 2, 1325–1330 (1985).
    [CrossRef]
  7. P. L. Walker, “Beam propagation through slab scattering media in the small angle approximation,” Appl. Opt. 26, 524–528(1987).
    [CrossRef] [PubMed]
  8. J. Tessendorf, “Time dependent radiative transfer and pulse evolution,” J. Opt. Soc. Am. A 6, 280–297 (1989).
    [CrossRef]
  9. F. Hanson and M. Lasher, “Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber,” Appl. Opt. 49, 3224–3230 (2010).
    [CrossRef] [PubMed]
  10. Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
    [CrossRef]
  11. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).
  12. B. V. Kaul and I. V. Samokhvalov, “Double scattering approximation of the atmospheric laser location equation taking polarization effects into account,” Russ. Phys. J. 19, 64–67 (1976).
    [CrossRef]
  13. S. Duntley, “Light in the sea,” J. Opt. Soc. Am. 53, 214–233(1963).
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  14. V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
    [CrossRef] [PubMed]
  15. H. S. Snyder and W. T. Scott, “Multiple scattering of fast charged particles,” Phys. Rev. 76, 220–225 (1949).
    [CrossRef]
  16. L. Tsang and A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
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    [CrossRef] [PubMed]
  18. E. W. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. 37, 2464–2472 (1998).
    [CrossRef]
  19. S. Ito, “Theory of beam light pulse propagation through thick clouds: effects of beamwidth and scatterers behind the light source on pulse broadening,” Appl. Opt. 20, 2706–2715 (1981).
    [CrossRef] [PubMed]
  20. I. Sreenivasiah and A. Ishimaru, “Beam wave two-frequency mutual-coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618(1979).
    [CrossRef] [PubMed]
  21. K. K. Benke and B. H. J. McKellar, “Modulation transfer function of photographic emulsion: the small-angle approximation in radiative transfer theory,” Appl. Opt. 29, 151–156 (1990).
    [CrossRef] [PubMed]
  22. W. G. Tam and A. Zardecki, “Multiple scattering corrections to the Beer-Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
    [CrossRef] [PubMed]
  23. M. A. Box and A. Deepak, “Limiting cases of the small-angle scattering approximation solutions for the propagation of laser beams in anisotropic scattering media,” J. Opt. Soc. Am. 71, 1534–1539 (1981).
    [CrossRef]
  24. J. W. McLean, J. D. Freeman, and R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
    [CrossRef]
  25. R. E. Walker and J. W. McLean, “Lidar equations for turbid media with pulse stretching,” Appl. Opt. 38, 2384–2397 (1999).
    [CrossRef]
  26. V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Opt. 18, 32–37 (2005).
  27. Y. Ilyushin and V. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182, 940–945 (2011).
    [CrossRef]
  28. V. P. Budak and Y. A. Ilyushin, “Separation of the singular part of the light field on the basis of the small angle solutions of the transfer theory,” Atmos. Opt. 24, 93–100 (2011).
  29. M. D. Alexandrov and V. S. Remizovich, “Depth mode characteristics of light propagation in real turbid media and media with two-dimensional scattering,” J. Opt. Soc. Am. A 12, 2726–2735(1995).
    [CrossRef]
  30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  31. E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959).
  32. S. D. Gedzelman, M.  Á. López-Álvarez, J. Hernandez-Andrés, and R. Greenler, “Quantifying the “milky sky” experiment,” Appl. Opt. 47, H128 –H132 (2008).
    [CrossRef] [PubMed]
  33. N. L. Swanson, V. M. Gehman, B. D. Billard, and T. L. Gennaro, “Limits of the small-angle approximation to the radiative transport equation,” J. Opt. Soc. Am. A 18, 385–391 (2001).
    [CrossRef]
  34. N. L. Swanson, B. D. Billard, V. M. Gehman, and T. L. Gennaro, “Application of the small-angle approximation to ocean water types,” Appl. Opt. 40, 3608–3613 (2001).
    [CrossRef]
  35. R. A. Elliott, “Multiple scattering of optical pulses in scale model clouds,” Appl. Opt. 22, 2670–2681 (1983).
    [CrossRef] [PubMed]
  36. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 –R93 (1999).
    [CrossRef]
  37. F. Hanson and S. Radic, “High bandwidth underwater optical communication,” Appl. Opt. 47, 277–283 (2008).
    [CrossRef] [PubMed]

2011

Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
[CrossRef]

Y. Ilyushin and V. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182, 940–945 (2011).
[CrossRef]

V. P. Budak and Y. A. Ilyushin, “Separation of the singular part of the light field on the basis of the small angle solutions of the transfer theory,” Atmos. Opt. 24, 93–100 (2011).

2010

2008

2005

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Opt. 18, 32–37 (2005).

2001

1999

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 –R93 (1999).
[CrossRef]

R. E. Walker and J. W. McLean, “Lidar equations for turbid media with pulse stretching,” Appl. Opt. 38, 2384–2397 (1999).
[CrossRef]

1998

1995

1990

K. K. Benke and B. H. J. McKellar, “Modulation transfer function of photographic emulsion: the small-angle approximation in radiative transfer theory,” Appl. Opt. 29, 151–156 (1990).
[CrossRef] [PubMed]

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
[CrossRef] [PubMed]

1989

1987

1985

1984

1983

1982

1981

1979

1977

1976

B. V. Kaul and I. V. Samokhvalov, “Double scattering approximation of the atmospheric laser location equation taking polarization effects into account,” Russ. Phys. J. 19, 64–67 (1976).
[CrossRef]

1972

1971

1963

1949

H. S. Snyder and W. T. Scott, “Multiple scattering of fast charged particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Alexandrov, M. D.

Arnush, D.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 –R93 (1999).
[CrossRef]

Benke, K. K.

Billard, B. D.

Box, M. A.

Budak, V.

Y. Ilyushin and V. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182, 940–945 (2011).
[CrossRef]

Budak, V. P.

V. P. Budak and Y. A. Ilyushin, “Separation of the singular part of the light field on the basis of the small angle solutions of the transfer theory,” Atmos. Opt. 24, 93–100 (2011).

Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
[CrossRef]

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Opt. 18, 32–37 (2005).

Deepak, A.

Duntley, S.

Elliott, R. A.

Eloranta, E. W.

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
[CrossRef] [PubMed]

Freeman, J. D.

Gedzelman, S. D.

Gehman, V. M.

Gennaro, T. L.

Greenler, R.

Hanson, F.

Harris, J. F. S.

Helliwell, W. S.

Hernandez-Andrés, J.

Ilyushin, Y.

Y. Ilyushin and V. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182, 940–945 (2011).
[CrossRef]

Ilyushin, Y. A.

V. P. Budak and Y. A. Ilyushin, “Separation of the singular part of the light field on the basis of the small angle solutions of the transfer theory,” Atmos. Opt. 24, 93–100 (2011).

Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
[CrossRef]

Ishimaru, A.

Ito, S.

Kaul, B. V.

B. V. Kaul and I. V. Samokhvalov, “Double scattering approximation of the atmospheric laser location equation taking polarization effects into account,” Russ. Phys. J. 19, 64–67 (1976).
[CrossRef]

Kozelskii, A. V.

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Opt. 18, 32–37 (2005).

Lasher, M.

López-Álvarez, M. Á.

McKellar, B. H. J.

McLean, J. W.

Morton, K. W.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

Patterson, M. S.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
[CrossRef] [PubMed]

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
[CrossRef] [PubMed]

Radic, S.

Remizovich, V. S.

Richtmyer, R. D.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

Samokhvalov, I. V.

B. V. Kaul and I. V. Samokhvalov, “Double scattering approximation of the atmospheric laser location equation taking polarization effects into account,” Russ. Phys. J. 19, 64–67 (1976).
[CrossRef]

Scott, W. T.

H. S. Snyder and W. T. Scott, “Multiple scattering of fast charged particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Snyder, H. S.

H. S. Snyder and W. T. Scott, “Multiple scattering of fast charged particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Sreenivasiah, I.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Stotts, L. B.

Swanson, N. L.

Tam, W. G.

Tessendorf, J.

Tsang, L.

Walker, P. L.

Walker, R. E.

Wigner, E.

E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959).

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
[CrossRef] [PubMed]

Zardecki, A.

Appl. Opt.

J. F. S. Harris, “Water and ice cloud discrimination by laser beam scattering,” Appl. Opt. 10, 732–737 (1971).
[CrossRef] [PubMed]

P. L. Walker, “Beam propagation through slab scattering media in the small angle approximation,” Appl. Opt. 26, 524–528(1987).
[CrossRef] [PubMed]

F. Hanson and M. Lasher, “Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber,” Appl. Opt. 49, 3224–3230 (2010).
[CrossRef] [PubMed]

W. G. Tam, “Aerosol backscattering of a laser beam,” Appl. Opt. 22, 2965–2969 (1983).
[CrossRef] [PubMed]

E. W. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. 37, 2464–2472 (1998).
[CrossRef]

S. Ito, “Theory of beam light pulse propagation through thick clouds: effects of beamwidth and scatterers behind the light source on pulse broadening,” Appl. Opt. 20, 2706–2715 (1981).
[CrossRef] [PubMed]

I. Sreenivasiah and A. Ishimaru, “Beam wave two-frequency mutual-coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618(1979).
[CrossRef] [PubMed]

K. K. Benke and B. H. J. McKellar, “Modulation transfer function of photographic emulsion: the small-angle approximation in radiative transfer theory,” Appl. Opt. 29, 151–156 (1990).
[CrossRef] [PubMed]

W. G. Tam and A. Zardecki, “Multiple scattering corrections to the Beer-Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
[CrossRef] [PubMed]

J. W. McLean, J. D. Freeman, and R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
[CrossRef]

R. E. Walker and J. W. McLean, “Lidar equations for turbid media with pulse stretching,” Appl. Opt. 38, 2384–2397 (1999).
[CrossRef]

S. D. Gedzelman, M.  Á. López-Álvarez, J. Hernandez-Andrés, and R. Greenler, “Quantifying the “milky sky” experiment,” Appl. Opt. 47, H128 –H132 (2008).
[CrossRef] [PubMed]

N. L. Swanson, B. D. Billard, V. M. Gehman, and T. L. Gennaro, “Application of the small-angle approximation to ocean water types,” Appl. Opt. 40, 3608–3613 (2001).
[CrossRef]

R. A. Elliott, “Multiple scattering of optical pulses in scale model clouds,” Appl. Opt. 22, 2670–2681 (1983).
[CrossRef] [PubMed]

F. Hanson and S. Radic, “High bandwidth underwater optical communication,” Appl. Opt. 47, 277–283 (2008).
[CrossRef] [PubMed]

Atmos. Opt.

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Opt. 18, 32–37 (2005).

V. P. Budak and Y. A. Ilyushin, “Separation of the singular part of the light field on the basis of the small angle solutions of the transfer theory,” Atmos. Opt. 24, 93–100 (2011).

Comput. Phys. Commun.

Y. Ilyushin and V. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182, 940–945 (2011).
[CrossRef]

Inverse Probl.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 –R93 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Med. Biol.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334(1990).
[CrossRef] [PubMed]

Phys. Rev.

H. S. Snyder and W. T. Scott, “Multiple scattering of fast charged particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Russ. Phys. J.

B. V. Kaul and I. V. Samokhvalov, “Double scattering approximation of the atmospheric laser location equation taking polarization effects into account,” Russ. Phys. J. 19, 64–67 (1976).
[CrossRef]

Other

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959).

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Figures (5)

Fig. 1
Fig. 1

Schematic view of the geometry of the problem.

Fig. 2
Fig. 2

Radiance distribution of the backscattered radiation (point A in Fig. 1). z 1 = 1.0 , z 2 = 5.0 , R = 0.4 , ε = 1 , g = 0.9 , and Λ = 1 .

Fig. 3
Fig. 3

Radiance distributions of the backscattered radiation (point A in Fig. 1) in the beam plane at different distances R from the beam axis (shown by the labels). z 1 = 1.0 , z 2 = 5.0 , ε = 1 , g = 0.9 , and Λ = 1 . Solid curves, total radiance; dashed curves, singly scattered radiance; dashed–dotted curves, doubly scattered radiance.

Fig. 4
Fig. 4

Radiance distributions of the backscattered radiation (point A in Fig. 1) in the beam plane at different distances R from the beam axis (shown by the labels). z 1 = 1.0 , z 2 = 1.5 , ε = 1 , g = 0.9 , and Λ = 1 . Solid curves, total radiance; dashed curves, singly scattered radiance; dashed–dotted curves, doubly scattered radiance.

Fig. 5
Fig. 5

Frequency spectra of the angular radiance distributions of the forward scattered radiation (point B in Fig. 1) in the beam plane at different distances R from the beam axis (shown by the labels). z 1 = 0.0 , z 2 = 10.0 , ε = 1 , g = 0.9 , and Λ = 1 . Solid curves, ω = 0 ; dashed curves, ω = 10 ; dashed–dotted curves, ω = 35 . The apodization kernel [Eq. (18)] M ( θ ) ( s = 5000 ) is shown in the separate graph at the left of the figure.

Equations (22)

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( Ω · ) L = - ε L + Λ ε 4 π L ( r , Ω ) x ( Ω , Ω ) d Ω + f ( r , Ω ) ,
( Ω · ) L 1 = - L 1 + M 1 ( Ω ) Λ 4 π L 1 ( r , Ω ) x ( Ω , Ω ) d Ω + M 1 ( Ω ) f ( r ) ,
( Ω · ) L 2 = - L 2 + M 2 ( Ω ) Λ 4 π L 2 ( r , Ω ) x ( Ω , Ω ) d Ω + M 2 ( Ω ) f ( r ) + M 2 ( Ω ) Λ 4 π L 1 ( r , Ω ) x ( Ω , Ω ) d Ω .
μ z - 1 + ( 1 - μ z ) + ( 1 - μ z ) 2 + + ( 1 - μ z ) n + o ( ( 1 - μ z ) n ) 1 μ z | μ z + 1 ,
μ z + - 1 - ( 1 + μ z ) - ( 1 + μ z ) 2 - - ( 1 + μ z ) n + o ( ( 1 - μ z ) n ) 1 μ z | μ z - 1 .
z L 1 + μ z - μ x x L 1 + μ z - μ y y L 1 = - μ z - L 1 + μ z - M 1 ( Ω ) Λ 4 π L 1 ( r , Ω ) x ( Ω , Ω ) d Ω + μ z - M 1 ( Ω ) f ( r ) ,
z L 2 + μ z + μ x x L 2 + μ z + μ y y L 2 = - μ z + L 2 + μ z + M 2 ( Ω ) Λ 4 π ( L 1 ( r , Ω ) + L 2 ( r , Ω ) ) x ( Ω , Ω ) d Ω + μ z + M 2 ( Ω ) f ( r ) .
f D = ( μ z μ z - - 1 ) ( μ x x L 1 + μ y y L 1 + L 1 ) + ( μ z μ z + - 1 ) ( μ x x L 2 + μ y y L 2 + L 2 ) + ( 1 - μ z μ z - M 1 ( Ω ) - μ z μ z + M 2 ( Ω ) ) Λ 4 π L 1 ( r , Ω ) x ( Ω , Ω ) d Ω + ( 1 - μ z μ z + M 2 ( Ω ) ) Λ 4 π L 2 ( r , Ω ) x ( Ω , Ω ) d Ω + ( 1 - μ z μ z - M 1 ( Ω ) - μ z μ z + M 2 ( Ω ) ) f .
L = n = 0 m = - n n c n m Y n m ( θ , ϕ ) ,
x ( Ω , Ω ) = 4 π n = 0 m = - n n x n Y n m ( θ , ϕ ) Y n m ( θ , ϕ ) .
L 0 = δ ( x ) δ ( y ) δ ( Ω ) exp ( ε d z ) ,
f ( r , Ω ) = Λ 4 π L 0 ( r , Ω ) x ( Ω , Ω ) d Ω = δ ( x ) δ ( y ) Λ 4 π x ( Ω ) exp ( - ε d z ) .
L ( x , y , z , Ω ) = 1 ( 2 π ) 2 - - n = 0 m = - n n c ˜ n m ( k x , k y , z ) Y n m ( θ , ϕ ) exp ( i k x x + i k y y ) d k x d k y ,
z C 1 + μ ^ z - μ ^ x i k x C 1 + μ ^ z - μ ^ y i k y C 1 = - μ ^ z - C 1 + μ ^ z - M ^ 1 Λ x ^ C 1 + μ ^ z - M ^ 1 Λ x ^ f ,
z C 2 + μ ^ z + μ ^ x i k x C 2 + μ ^ z + μ ^ y i k y C 2 = - μ ^ z + C 2 + μ ^ z + M ^ 2 Λ x ^ ( C 2 + C 1 ) + μ ^ z + M ^ 2 Λ x ^ f ,
M 1 ( μ z ) = - 5 32 ( μ z + 1 ) 7 + 35 32 ( μ z + 1 ) 6 - 21 8 ( μ z + 1 ) 5 + 35 16 ( μ z + 1 ) 4 = π Y 0 , 0 ( θ , ϕ ) + 4 3 π 3 Y 1 , 0 ( θ , ϕ ) - 2 7 π 33 Y 3 , 0 ( θ , ϕ ) + 4 39 π 11 Y 5 , 0 ( θ , ϕ ) - 1 429 5 π 3 Y 7 , 0 ( θ , ϕ ) ,
M 2 ( μ z ) = - 5 32 ( 1 - μ z ) 7 + 35 32 ( 1 - μ z ) 6 - 21 8 ( 1 - μ z ) 5 + 35 16 ( 1 - μ z ) 4 = π Y 0 , 0 ( θ , ϕ ) - 4 3 π 3 Y 1 , 0 ( θ , ϕ ) + 2 33 7 π Y 3 , 0 ( θ , ϕ ) - 4 39 π 11 Y 5 , 0 ( θ , ϕ ) + 1 429 5 π 3 Y 7 , 0 ( θ , ϕ ) .
M ( θ ) = ( s + 1 ) ( cos θ + 1 ) s 2 s + 2 π ,
m n = Γ ( s + 1 ) Γ ( s + 2 ) Γ ( s - n + 1 ) Γ ( n + s + 2 ) ( s + 1 ) s + 1 / 2 ( s + 2 ) s + 3 / 2 ( s - n + 1 ) n - s - 1 / 2 ( n + s + 2 ) - n - s - 3 / 2 .
c t L + ( Ω · ) L = - ε L + Λ ε 4 π L ( r , Ω ) x ( Ω , Ω ) d Ω + f ( r , Ω ) ,
L 1 ( x , y , z , Ω , t ) = 1 ( 2 π ) 3 exp ( i k x x + i k y y + i ω ( t - z ) ) m , n c ˜ n m ( 1 ) ( k x , k y , z , ω ) Y n m ( θ , ϕ ) d k x d k y d ω
i ω ( 1 - μ ^ ) C 1 + z C 1 + μ ^ z - μ ^ x i k x C 1 + μ ^ z - μ ^ y i k y C 1 = - μ ^ z - C 1 + μ ^ z - M ^ 1 Λ x ^ C 1 + μ ^ z - M ^ 1 Λ x ^ f .

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