Abstract

The concepts of statistical electromagnetics are used to derive the general radiative transfer equation (RTE) that describes multiple scattering of polarized light by sparse discrete random media consisting of arbitrarily shaped and arbitrarily oriented particles. The derivation starts with the volume integral and Lippmann-Schwinger equations for the electric field scattered by a fixed N-particle system and proceeds to the vector form of the Foldy-Lax equations and their approximate far-field version. I then assume that particle positions are completely random and derive the vector RTE by applying the Twersky approximation to the coherent electric field and the Twersky and ladder approximations to the coherency dyad of the diffuse field in the limit N → ∞. The concluding section discusses the physical meaning of the quantities that enter the general vector RTE and the assumptions made in its derivation.

© 2002 Optical Society of America

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References

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  1. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  2. V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).
  3. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  4. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, London, 1974).
  5. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  6. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  7. J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).
  8. J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres (Deepak, Hampton, Va., 1985).
  9. A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Norwood, Mass., 1994).
  10. A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, Propagation and Polarization of Radiation in Cosmic Media (Gordon Breach, Basel, Switzerland, 1995).
  11. E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer-Verlag, Berlin, 1997).
  12. G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, New York, 1999).
  13. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, San Diego, Calif., 2002).
  14. J. W. Hovenier, C. V. M. van der Mee, H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, Dordrecht, The Netherlands, to be published).
  15. A. G. Borovoy, “Method of iterations in multiple scattering: the transfer equation,” Izv. Vyssh. Uchebn. Zaved. Fiz., No. 6, 50–54 (1966).
  16. Yu. N. Barabanenkov, V. M. Finkel’berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).
  17. A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 10, 707–754 (1970).
    [CrossRef]
  18. L. A. Apresyan, Yu. A. Kravtsov, Radiation Transfer (Gordon Breach, Basel, Switzerland, 1996).
  19. A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
    [CrossRef]
  20. A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997).
  21. L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).
  22. V. P. Tishkovets, “Multiple scattering of light by a layer of discrete random medium: backscattering,” J. Quant. Spectrosc. Radiat. Transfer 72, 123–137 (2002).
    [CrossRef]
  23. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, New York, 2002).
  24. A. P. Prishivalko, V. A. Babenko, V. N. Kuzmin, Scattering and Absorption of Light by Inhomogeneous and Anisotropic Spherical Particles (Nauka i Tekhnika, Minsk, USSR, 1984), in Russian.
  25. L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
    [CrossRef]
  26. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
    [CrossRef]
  27. V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).
  28. D. S. Saxon, “Lectures on the scattering of light,” Science Report No. 9 (Department of Meteorology, University of California, Los Angeles, Los Angeles, Calif., 1955).
  29. G. V. Rozenberg, “Stokes vector-parameter,” Usp. Fiz. Nauk. 56(1), 77–110 (1955).
    [CrossRef]
  30. M. I. Mishchenko, “Multiple scattering of light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
    [CrossRef]

2002

V. P. Tishkovets, “Multiple scattering of light by a layer of discrete random medium: backscattering,” J. Quant. Spectrosc. Radiat. Transfer 72, 123–137 (2002).
[CrossRef]

1996

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

1990

M. I. Mishchenko, “Multiple scattering of light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
[CrossRef]

1983

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

1974

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1970

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 10, 707–754 (1970).
[CrossRef]

1968

Yu. N. Barabanenkov, V. M. Finkel’berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

1966

A. G. Borovoy, “Method of iterations in multiple scattering: the transfer equation,” Izv. Vyssh. Uchebn. Zaved. Fiz., No. 6, 50–54 (1966).

1964

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

1955

G. V. Rozenberg, “Stokes vector-parameter,” Usp. Fiz. Nauk. 56(1), 77–110 (1955).
[CrossRef]

1951

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

1945

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

1905

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Apresyan, L. A.

L. A. Apresyan, Yu. A. Kravtsov, Radiation Transfer (Gordon Breach, Basel, Switzerland, 1996).

Babenko, V. A.

A. P. Prishivalko, V. A. Babenko, V. N. Kuzmin, Scattering and Absorption of Light by Inhomogeneous and Anisotropic Spherical Particles (Nauka i Tekhnika, Minsk, USSR, 1984), in Russian.

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, V. M. Finkel’berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

Borovoy, A. G.

A. G. Borovoy, “Method of iterations in multiple scattering: the transfer equation,” Izv. Vyssh. Uchebn. Zaved. Fiz., No. 6, 50–54 (1966).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Dolginov, A. Z.

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 10, 707–754 (1970).
[CrossRef]

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, Propagation and Polarization of Radiation in Cosmic Media (Gordon Breach, Basel, Switzerland, 1995).

Domke, H.

J. W. Hovenier, C. V. M. van der Mee, H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, Dordrecht, The Netherlands, to be published).

Finkel’berg, V. M.

Yu. N. Barabanenkov, V. M. Finkel’berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Fung, A. K.

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Norwood, Mass., 1994).

Gnedin, Yu. N.

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 10, 707–754 (1970).
[CrossRef]

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, Propagation and Polarization of Radiation in Cosmic Media (Gordon Breach, Basel, Switzerland, 1995).

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

J. W. Hovenier, C. V. M. van der Mee, H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, Dordrecht, The Netherlands, to be published).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997).

Kong, J. A.

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).

Kravtsov, Yu. A.

L. A. Apresyan, Yu. A. Kravtsov, Radiation Transfer (Gordon Breach, Basel, Switzerland, 1996).

Kuzmin, V. N.

A. P. Prishivalko, V. A. Babenko, V. N. Kuzmin, Scattering and Absorption of Light by Inhomogeneous and Anisotropic Spherical Particles (Nauka i Tekhnika, Minsk, USSR, 1984), in Russian.

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, New York, 2002).

Lagendijk, A.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Lax, M.

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

Liou, K. N.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, San Diego, Calif., 2002).

Mishchenko, M. I.

M. I. Mishchenko, “Multiple scattering of light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, New York, 2002).

Prishivalko, A. P.

A. P. Prishivalko, V. A. Babenko, V. N. Kuzmin, Scattering and Absorption of Light by Inhomogeneous and Anisotropic Spherical Particles (Nauka i Tekhnika, Minsk, USSR, 1984), in Russian.

Rozenberg, G. V.

G. V. Rozenberg, “Stokes vector-parameter,” Usp. Fiz. Nauk. 56(1), 77–110 (1955).
[CrossRef]

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” Science Report No. 9 (Department of Meteorology, University of California, Los Angeles, Los Angeles, Calif., 1955).

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Silant’ev, N. A.

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 10, 707–754 (1970).
[CrossRef]

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, Propagation and Polarization of Radiation in Cosmic Media (Gordon Breach, Basel, Switzerland, 1995).

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, London, 1974).

Stamnes, K.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, New York, 1999).

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, New York, 1999).

Tishkovets, V. P.

V. P. Tishkovets, “Multiple scattering of light by a layer of discrete random medium: backscattering,” J. Quant. Spectrosc. Radiat. Transfer 72, 123–137 (2002).
[CrossRef]

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. 16, 527–610 (1974).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, New York, 2002).

Tsang, L.

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).

Twersky, V.

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

van der Mee, C. V. M.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

J. W. Hovenier, C. V. M. van der Mee, H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, Dordrecht, The Netherlands, to be published).

van Tiggelen, B. A.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Yanovitskij, E. G.

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer-Verlag, Berlin, 1997).

Astron. Astrophys.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

Astrophys. J.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Fiz.

A. G. Borovoy, “Method of iterations in multiple scattering: the transfer equation,” Izv. Vyssh. Uchebn. Zaved. Fiz., No. 6, 50–54 (1966).

J. Quant. Spectrosc. Radiat. Transfer

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, “Photon polarization and frequency change in multiple scattering,” J. Quant. Spectrosc. Radiat. Transfer 10, 707–754 (1970).
[CrossRef]

V. P. Tishkovets, “Multiple scattering of light by a layer of discrete random medium: backscattering,” J. Quant. Spectrosc. Radiat. Transfer 72, 123–137 (2002).
[CrossRef]

Phys. Rep.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[CrossRef]

Phys. Rev.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Proc. Symp. Appl. Math.

V. Twersky, “On propagation in random media of discrete scatterers,” Proc. Symp. Appl. Math. 16, 84–116 (1964).

Rev. Mod. Phys.

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

Sov. Phys. JETP

Yu. N. Barabanenkov, V. M. Finkel’berg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

Space. Sci. Rev.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Transp. Theory Stat. Phys.

M. I. Mishchenko, “Multiple scattering of light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
[CrossRef]

Usp. Fiz. Nauk.

G. V. Rozenberg, “Stokes vector-parameter,” Usp. Fiz. Nauk. 56(1), 77–110 (1955).
[CrossRef]

Other

D. S. Saxon, “Lectures on the scattering of light,” Science Report No. 9 (Department of Meteorology, University of California, Los Angeles, Los Angeles, Calif., 1955).

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, New York, 2002).

A. P. Prishivalko, V. A. Babenko, V. N. Kuzmin, Scattering and Absorption of Light by Inhomogeneous and Anisotropic Spherical Particles (Nauka i Tekhnika, Minsk, USSR, 1984), in Russian.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997).

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, New York, 2001).

L. A. Apresyan, Yu. A. Kravtsov, Radiation Transfer (Gordon Breach, Basel, Switzerland, 1996).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon, Oxford, 1952).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, London, 1974).

J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres (Deepak, Hampton, Va., 1985).

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Norwood, Mass., 1994).

A. Z. Dolginov, Yu. N. Gnedin, N. A. Silant’ev, Propagation and Polarization of Radiation in Cosmic Media (Gordon Breach, Basel, Switzerland, 1995).

E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres (Springer-Verlag, Berlin, 1997).

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, New York, 1999).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, San Diego, Calif., 2002).

J. W. Hovenier, C. V. M. van der Mee, H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, Dordrecht, The Netherlands, to be published).

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Figures (20)

Fig. 1
Fig. 1

Local coordinate system used to describe the direction of propagation and the polarization state of a transverse electromagnetic wave at an observation point.

Fig. 2
Fig. 2

Scattering in the far-field zone.

Fig. 3
Fig. 3

Scattering by a fixed group of N finite particles.

Fig. 4
Fig. 4

Scattering by widely separated particles. The local origins O i and O j were chosen arbitrarily inside particles i and j, respectively.

Fig. 5
Fig. 5

(a) Incident field, (b) single scattering, (c) double scattering, (d) triple scattering through a self-avoiding path, and (e) triple scattering through a path that goes through particle i twice.

Fig. 6
Fig. 6

(a) Self-avoiding scattering paths and (b)–(e) paths that involve four scattering events and go through a particle more than once.

Fig. 7
Fig. 7

Diagrammatic representations of (a) Eq. (64) and (b) approximation (66).

Fig. 8
Fig. 8

Geometry showing the quantities used in the derivation of Eq. (86).

Fig. 9
Fig. 9

Twersky representation of the dyadic correlation function.

Fig. 10
Fig. 10

Classification of terms that enter the Twersky expansion of the dyadic correlation function.

Fig. 11
Fig. 11

Calculation of the total contribution of diagrams with no connectors.

Fig. 12
Fig. 12

Diagrams with one or more vertical connectors.

Fig. 13
Fig. 13

Summation of the tails.

Fig. 14
Fig. 14

Derivation of the ladder approximation for the dyadic correlation function.

Fig. 15
Fig. 15

Calculation of the integrals in Eq. (107).

Fig. 16
Fig. 16

Ladder approximation for the dyadic correlation function.

Fig. 17
Fig. 17

Geometry showing the quantities used in Eq. (111).

Fig. 18
Fig. 18

Geometry showing the quantities used in the derivation of the VRTE.

Fig. 19
Fig. 19

Physical meaning of the diffuse specific intensity vector.

Fig. 20
Fig. 20

Physical interpretation of the RTE.

Equations (131)

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

Er=E0 expiknˆ · r, E0 · nˆ=0,
ρ=ρ11ρ12ρ21ρ22=12εμE0θE0θ*E0θE0ϕ*E0ϕE0θ*E0ϕE0ϕ*,
J=ρ11ρ12ρ21ρ22=12εμE0θE0θ*E0θE0ϕ*E0ϕE0θ*E0ϕE0ϕ*.
I=DJ=12εμE0θE0θ*+E0ϕE0ϕ*E0θE0θ*-E0ϕE0ϕ*-E0θE0ϕ*-E0ϕE0θ*iE0ϕE0θ*-E0θE0ϕ*=IQUV,
D=1001100-10-1-100-ii0.
×Er=iωμ1Hr×Hr=-iωε1Er for rVEXT,
×Er=iωμ2rHr×Hr=-iωε2rEr for rVINT,
××Er-k12Er=0, rVEXT,
××Er-k22rEr=0, rVINT,
Er=Eincr+k12VINTd3rGr, r · Erm2r-1, rR3,
Gr, r=I+1k12   expik1|r-r|4π|r-r|
Escar=VINTd3rGr, r · VINTd3rTr, r· Eincr, rR3.
Tr, r=k12m2r-1δr-rI+k12m2r-1VINTd3rGr, r·Tr, r, r, rVINT.
Escar =r expik1rrk124πI-rˆ  rˆ·VINTd3rm2r-1Erexp-ik1rˆ · r.
Escar =r expik1rrE1scarˆ, rˆ · E1scarˆ=0,
E1scanˆsca=Anˆsca, nˆinc · E0inc,
nˆsca · Anˆsca, nˆinc=0.
Anˆsca, nˆinc · nˆinc=0.
Escarnˆsca =r expik1rr Snˆsca, nˆincE0inc,
S11=θˆsca · A · θˆinc, S12=θˆsca · A · ϕˆinc, S21=ϕˆsca · A · θˆinc, S22=ϕˆsca · A · ϕˆinc.
A-nˆinc, -nˆsca=ATnˆsca, nˆinc,
S-nˆinc, -nˆsca=S11nˆsca, nˆinc-S21nˆsca, nˆinc-S12nˆsca, nˆincS22nˆsca, nˆinc,
Jscarnˆsca=1r2ZJnˆsca, nˆincJinc,
Jinc=12ε1μ0E0θincE0θinc*E0θincE0ϕinc*E0ϕincE0θinc*E0ϕincE0ϕinc*, Jscarnˆsca=1r212ε1μ0E1θscanˆscaE1θscanˆsca*E1θscanˆscaE1ϕscanˆsca*E1ϕscanˆscaE1θscanˆsca*E1ϕscanˆscaE1ϕscanˆsca*,
ZJ=|S11|2S11S12*S12S11*|S12|2S11S21*S11S22*S12S21*S12S22*S21S11*S21S12*S22S11*S22S12*|S21|2S21S22*S22S21*|S22|2.
Iscarnˆsca=1r2Znˆsca, nˆincIinc,
Znˆsca, nˆinc=DZJnˆsca, nˆincD-1.
Jrrˆ=12ε1μ0EθrrˆEθrrˆ*EθrrˆEϕrrˆ*EϕrrˆEθrrˆ*EϕrrˆEϕrrˆ*,
JrnˆincΔS=JincΔS-KJnˆincJinc+Or-2,
KJ=i2πk1×S11*-S11S12*-S120S21*S22*-S110-S12-S210S11*-S22S12*0-S21S21*S22*-S22.
IrnˆincΔS=IincΔS-KnˆincIinc+Or-2,
Knˆinc=DKJnˆincD-1.
K-nˆinc=Δ3KnˆincTΔ3,
Er=Eincr+R3d3rUrGr, r· Er, rR3,
Ur=i=1N Uir, rR3,
Uir=0,rVi,k12mi2r-1,rVi,
Er=Eincr+i=1NVid3rGr, r· Vid3rTir, r · Eir, rR3,
Eir=Eincr+ji=1NEijexcr,
Eijexcr=Vjd3rGr, r · Vjd3rTjr, r· Ejr, rVi,
Tir, r=Uirδr-rI+UirVid3rGr, r· Tir, r, r, rVi.
Uir, r=Uirδr-rI,
E=Einc+ĜÛE,
E=Einc+i=1N ĜTˆiEi,
Ei=Einc+ji=1N ĜTˆjEj,
Tˆi=Ûi+ÛiĜTˆi,
Û=i=1N Ûi,
BˆE= d3rBr, r · Er.
ÛĜTˆi=ÛiĜTˆi+ji=1N ÛjĜTˆi=Tˆi-Ûi+ji=1N ÛjĜTˆi.
Einc+ĜÛE=Einc+ĜÛEinc+i=1N ĜTˆiEi =Einc+ĜÛEinc+Ĝ i=1N×Tˆi-Ûi+ji=1N ÛjĜTˆiEi =Einc+i=1N ĜTˆiEi+ĜÛEinc+Ĝ i=1N Ûiji=1N ĜTˆjEj-Ĝ i=1N ÛiEi =Einc+i=1N ĜTˆiEi+ĜÛEinc+Ĝ i=1N Ûiji=1N ĜTˆjEj-Ei =Einc+i=1N ĜTˆiEi+ĜÛEinc-Ĝ i=1N ÛiEinc=E.
EijexcrGrjE1ijrˆjexp-ik1Rˆij · RiEij×expik1Rˆij · r, rVi,
Gr=expik1rr,
Eij=GRijE1ijRˆij, Eij · Rˆij=0, rˆj=rj/rj, Rˆij=Rij/Rij, rj=|Rij+r-Ri| =Rij Rij+Rˆij · r-Ri,
EirE0inc expik1ŝ · r+ji=1Nexp-ik1Rˆij · RiEij×expik1Rˆij · r, rVi,
Eincr=E0inc expik1ŝ · r, E0inc · ŝ=0.
EjrEincRjexpik1ŝ · rj+lj=1NEjl×expik1Rˆjl · rj, rVj.
GRij×AjRˆij, ŝ · EincRj+lj=1NAjRˆij, Rˆjl · Ejl.
Eij=GRijAjRˆij, ŝ · EincRj+lj=1NAjRˆij, Rˆjl · Ejl, i, j=1,, N, ji.
EirEincRiexpik1ŝ · ri+ji=1NEij×expik1Rˆij · ri, rVi
EiRi=EincRi+ji=1NEij.
Er=Eincr+i=1N GriAirˆi, ŝ · EincRi+i=1N Griji=1NAirˆi, Rˆij · Eij,
E=Einc+i=1NBri0 · Eiinc+i=1Nji=1NBrij · Eij,
Eij=Bij0 · Ejinc+lj=1NBijl · Ejl,
Bri0=GriAirˆi, ŝ, Brij=GriAirˆi, Rˆij, Bij0=GRijAjRˆij, ŝ, Bijl=GRijAjRˆij, Rˆjl.
Eij=Bij0 · Ejinc+l=1ljNBijl · Bjl0 · Elinc+l=1ljNm=1mlNBijl · Bjlm · Blm0 · Eminc+,
E=Einc+i=1NBri0 · Eiinc+i=1Nj=1jiNBrij · Bij0 · Ejinc+i=1Nj=1jiNl=1ljNBrij · Bijl · Bjl0 · Elinc +i=1Nj=1jiNl=1ljNm=1mlNBrij · Bijl · Bjlm· Blm0 · Eminc+
i=1Nj=1jiNl=1liljNBrij · Bijl · Bjl0 · Elinc+i=1Nj=1jiNBrij · Biji · Bji0 · Eiinc.
EEinc+i=1NBri0 · Eiinc+i=1Nj=1jiNBrij · Bij0 · Ejinc +i=1Nj=1jiNl=1liljNBrij · Bijl · Bjl0 · Elinc +i=1Nj=1jiNl=1liljNm=1mimjmlNBrij · Bijl · Bjlm · Blm0·Eminc+.
pR1, ξ1;; Ri, ξi;; RN, ξNi=1Nd3Ridξi.
 pR1, ξ1;; Ri, ξi;; RN, ξNi=1Nd3Ridξi=1,
f= fR1, ξ1;; Ri, ξi;; RN, ξN×pR1, ξ1;; Ri, ξi;; RN, ξNi=1Nd3Ridξi.
pR1, ξ1;; Ri, ξi;; RN, ξN=i=1N piRi, ξi.
piRi, ξi=pRiRipξiξi.
piRi, ξipRi, ξi=pRRipξξi,
 pRRd3R=1,  pξξdξ=1.
pRRd3R=probability of finding a particle within volume d3R centered at R=number of particles within d3Rtotal number of particles=n0Rd3RN,
pRR=n0R/N.
n0Rn0=N/V, pRR=1/V.
Er=Ecr+Efr, Ecr=Er, Efr=0.
Ec=Einc+i=1N  Arˆi, ŝ · EiincGripRRid3Ri+i=1Nj=1jiN  Arˆi, Rˆij · ARˆij, ŝ·EjincGriGRijpRRipRRjd3Rid3Rj+,
Ec=Einc+NN  Arˆi, ŝ·EiincGrin0Rid3Ri +NN-1N2  Arˆi, Rˆij·ARˆij, ŝ ·EjincGriGRijn0Rin0Rjd3Rid3Rj +
=NEinc+ Arˆi, ŝ·EiincGrin0Rid3Ri + Arˆi, Rˆij·ARˆij, ŝ ·EjincGriGRijn0Rin0Rjd3Rid3Rj+,
I1=n0Vd3Ri expik1ŝ·Ri ×expik1RiRi A-Rˆi, ŝ·Eincr.
expik1ŝ · Ri=k1Rii2πk1Riδŝ+Rˆiexp-ik1Ri -δŝ-Rˆiexpik1Ri.
I1=i2πn0k14πdRˆi  dRiA- Rˆi, ŝ·Eincrδŝ+Rˆi-δŝ-Rˆiexp2ik1Ri i2πn0k1 srAŝ, ŝ · Eincr,
I2=n02  dRiRi2GRi4πdRˆi  dRjiRji2GRji×4πdRˆjiA-Rˆi, -Rˆji · A-Rˆji, ŝ · Ejinc,
Ejinc=expik1ŝ · RjE0inc=expik1ŝ · Ri×expik1ŝ · RjiEincr =i2πk121Riδŝ+Rˆiexp-ik1Ri-δŝ-Rˆiexpik1Ri×1Rjiδŝ+Rˆjiexp-ik1Rji-δŝ-Rˆjiexpik1RjiEincr.
I2=12i2πn0k1 sr2Aŝ, ŝ · Aŝ, ŝ · Eincr.
Ecr=expi2πn0k1 srAŝ, ŝ · Eincr,
Ecr=expiκŝsr · EincrA =ηŝ, sr · EincrA,
κŝ=k1I+2πn0k1 Aŝ, ŝ
ηŝ, s=expiκŝs
dEcrds=iκŝ·Ecr.
2Ecr+k12εŝ·Ecr=0,
dEcrds=ikŝEcr,
k11ŝ=θˆŝ·κŝ·θˆŝ, k12ŝ=θˆŝ·κŝ·ϕˆŝ, k21ŝ=ϕˆŝ·κŝ·θˆŝ, k22ŝ=ϕˆŝ·κŝ·ϕˆŝ.
kŝ=k1diag1, 1+2πn0k1 Sŝ, ŝ,
Ecs=hŝ, s · Ec0,
hŝ, s=expikŝs
η-ŝ, s=ηŝ, sT,
h-ŝ, s=diag1, -1hŝ, sTdiag1, -1.
Jc=12ε1μ0EcθEcθ*EcθEcϕ*EcϕEcθ*EcϕEcϕ*
dJcrds=-n0KJŝJcr,
dIcrds=-n0KŝIcr,
Icr=Hŝ, srIcrA,
Hŝ, s=exp-n0Kŝs
H-ŝ, s=Δ3Hŝ, sTΔ3.
Icr=exp-n0CextsrIcrA =exp-αextsrIcrA,
GRpqARˆpq, ŝ · Eqinc +N-nN n0GRpq  GRqiARˆpq, Rˆqi ·ARˆqi, ŝ · Eiincd3Ri +N-nN-n-1N2 n02GRpq × GRqiARˆpq, Rˆqi · ARˆqi, Rˆij ·ARˆij, ŝ · Ejincd3Rid3Rj + =N GRpqARˆpq, ŝ · EcRq,
Er=GRrpGRpqApRˆrp, Rˆpq · AqRˆpq, Rˆqu · Eq+i GRrpGRpiGRiqApRˆrp, Rˆpi ·AiRˆpi, Rˆiq · AqRˆiq, Rˆqu · Eq +ij GRrpGRpiGRijGRjqApRˆrp, Rˆpi ·AiRˆpi, Rˆij · AjRˆij, Rˆjq · AqRˆjq, Rˆqu·Eq + ,
Er=GRrpGRpqApRˆrp, Rˆpq · AqRˆpq, Rˆqu · Eq+n0GRrpVd3RiGRpiGRiqApRˆrp, Rˆpi · ARˆpi, Rˆiq · AqRˆiq, Rˆqu · Eq +n02GRrpVd3Rid3RjGRpiGRijGRjq ×ApRˆrp, Rˆpi · ARˆpi, Rˆij · ARˆij, Rˆjq · AqRˆjq, Rˆqu · Eq +.
I1=n0GRrp-+dxi-+dyi  dzi×expik1Rpi+RiqRpiRiqApRˆrp, Rˆpi·ARˆpi, Rˆiq · AqRˆiq, Rˆqu · Eq,
I1=GRrpexpik1RpqRpqi2πn0Rpqk1ApRˆrp, Rˆpq·ARˆpq, Rˆpq·AqRˆpq, Rˆqu · Eq.
Er=GRrpApRˆrp, Rˆpq·ηRˆpq, RpqRpq·AqRˆpq, Rˆqu · Eq,
ηRˆ1p, R1pR1p · ApRˆ1p, Rˆpq·ηRˆpq, RpqRpq·AqRˆpq, Rˆqt·ηRˆqt, RqtRqt·AtRˆqt, ŝ · EcRt ηRˆ2p, R2pR2p·ApRˆ2p, Rˆpq·ηRˆpq, RpqRpq·AqRˆpq, Rˆqt·ηRˆqt, RqtRqt·AtRˆqt, ŝ · EcRt*,
Cr=Ccr+n0  d3R1dξ1ηrˆ1, r1r1 · A1rˆ1, ŝ · CcR1 · A1T*rˆ1, ŝ · ηT*rˆ1, r1r1 +n02  d3R1dξ1  d3R2dξ2ηrˆ1, r1r1 · A1rˆ1, Rˆ12 · ηRˆ12, R12R12 · A2Rˆ12, ŝ · CcR2 · A2T*Rˆ12, ŝ · ηT*Rˆ12, R12R12 · A1T*rˆ1, Rˆ12 · ηT*rˆ1, r1r1+,
Cr=4πdpˆr, -pˆ,
r, -pˆ=δpˆ+ŝCcr+n0  dpdξ1η-pˆ, p · A1-pˆ, ŝ · Ccr+p · A1T*-pˆ, ŝ · ηT*-pˆ, p+n02  dpdξ1 × dR21dRˆ21dξ2η-pˆ, p · A1-pˆ, -Rˆ21 · η-Rˆ21, R21 · A2-Rˆ21, ŝ · Ccr+p+R21 · A2T*-Rˆ21, ŝ · ηT*-Rˆ21, R21 · A1T*-pˆ, -Rˆ21 · ηT*-pˆ, p+.
r, -pˆ=δpˆ+ŝCcr+n0  dpdξdpˆη-pˆ, p · A-pˆ, -pˆ · r+p, -pˆ · AT*-pˆ, -pˆ · ηT*-pˆ, p.
Q, qˆ=δqˆ-ŝCcQ+n00Qdq  dξ 4πdqˆηq, Q-a · Aqˆ, qˆ · q, qˆ · AT*qˆ, qˆ · ηT*qˆ, Q-q.
dQ, qˆ=n00Qdq  dξηqˆ, Q-q· Aqˆ, ŝCcq · AT*qˆ, ŝ · ηT*qˆ, Q-q +n00Qdq  dξ 4πdqˆηqˆ, Q-q · Aqˆ, qˆ · dq, qˆ · AT*qˆ, qˆ · ηT*qˆ, Q-q.
ddQ, qˆdQ=iκqˆ · dQ, qˆ-idQ, qˆ · κT*qˆ+n0  dξ 4πdqˆAqˆ, qˆ · dQ, qˆ · AT*qˆ, qˆ+n0  dξAqˆ, ŝ · CcQ · AT*qˆ, ŝ.
ddr, qˆdq=iκqˆ · dr, qˆ-idr, qˆ · κT*qˆ+n0  dξ 4πdqˆAqˆ, qˆ · dr, qˆ · AT*qˆ, qˆ+n0  dξAqˆ, ŝ · Ccr · AT*qˆ, ŝ,
ρ˜dr, qˆ=12ε1μ0θˆqˆ · dr, qˆ · θˆqˆθˆqˆ · dr, qˆ · ϕˆqˆϕˆqˆ · dr, qˆ · θˆqˆϕˆqˆ · dr, qˆ · ϕˆqˆ.
dρ˜dr, qˆdq=ikqˆρ˜dr, qˆ-iρ˜dr, qˆkT*qˆ+n0  dξ 4πdqˆSqˆ, qˆρ˜dr, qˆST*qˆ, qˆ +n0  dξSqˆ, ŝρcrST*qˆ, ŝ,
ρr=12×ε1μ0θˆŝ · Ccr · θˆŝθˆŝ · Ccr · ϕˆŝϕˆŝ · Ccr · θˆŝϕˆŝ · Ccr · ϕˆŝ.
J˜dr, qˆ=ρ˜d11r, qˆρ˜d12r, qˆρ˜d21r, qˆρ˜d22r, qˆ, Jcr=ρc11rρc12rρc21rρc22r
dJ˜dr, qˆdq=-n0KJqˆJ˜dr, qˆ+n04πdqˆZJqˆ, qˆJ˜dr, qˆ+n0ZJqˆ, ŝJcr,
dĨdr, qˆdq=-n0KqˆĨdr, qˆ+n04πdqˆZqˆ, qˆĨdr, qˆ+n0Zqˆ, ŝIcr,
ΔΩĨdr, qˆn0ΔVd3p1p2Hqˆ, p×Zqˆ, ŝIcr+p+4πdqˆZqˆ, qˆĨdr+p, qˆ,
qˆ · Ir, qˆ= · qˆIr, qˆ=-n0KqˆIr, qˆ+n04πdqˆZqˆ, qˆIr, qˆ,
- · Fr=n04πdqˆCabsqˆIr, q.

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