Abstract

We use the T-matrix approach and the analytical orientation-averaging technique to formulate the problem of light scattering by an ensemble of rotationally symmetric particles in arbitrary orientation. The mathematical formulation yields analytical expressions for the elements of the ensemble-averaged scattering matrix that involve no more than four nested summations. An expansion into generalized spherical functions is used in the particular case where the scatterers are partially aligned along the direction of incidence. A computer code that implements the analytical expressions derived is publicly available on the World Wide Web at http://irctr.et.tudelft.nl/~Skaropoulos/T-matrix.htm.

© 2002 Optical Society of America

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  1. M. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).[errata: 9, 497 (1992)].
    [CrossRef]
  2. M. Mishchenko, L. Travis, D. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
    [CrossRef]
  3. M. Mishchenko, J. Hovenier, L. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).
  4. M. Mishchenko, L. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60, 309–324 (1988).
    [CrossRef]
  5. M. Mishchenko, L. Travis, D. Mackowski, “T-matrix codes for computing electromagnetic scattering by nonspherical and aggregated particles,” http://www.giss.nasa.gov/~crmim/t_matrix.html .
  6. M. Mishchenko, “Extinction and polarization of transmitted light by partially aligned nonspherical grains,” Astrophys. J. 367, 561–574 (1991).
    [CrossRef]
  7. M. Mishchenko, “Coherent propagation of polarized millimeter waves through falling hydrometeors,” J. Electromagn. Waves Appl. 6, 1341–1351 (1992).
  8. L. Paramonov, “T-matrix approach and the angular momentum theory in light-scattering problems by ensembles of arbitrarily shaped particles,” J. Opt. Soc. Am. A 12, 2698–2707 (1995).
    [CrossRef]
  9. W. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” (NASA, Washington, D.C., 1986).
  10. P. Barber, S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  11. J. Vivekanandan, W. Adams, V. Bringi, “Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions,” J. Appl. Meteorol. 30, 1053–1063 (1991).
    [CrossRef]
  12. A. Battaglia, F. Prodi, O. Sturniolo, “Radar and scattering parameters through falling hydrometeors with axisymmetric shapes,” Appl. Opt. 40, 3092–3100 (2001).
    [CrossRef]
  13. H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
    [CrossRef]
  14. J. Hovenier, C. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).
  15. N. Skaropoulos, “T-matrix codes for computing the scattering of electromagnetic waves by partially aligned particles,” http://irctr.et.tudelft.nl/~Skaropoulos/T-matrix.htm .
  16. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  17. C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  18. J. Hovenier, C. van der Mee, “Scattering of polarized light: properties of the elements of the phase matrix,” Astron. Astrophys. 196, 287–295 (1988).
  19. I. Gelfand, R. Minlos, Z. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, UK, 1963).
  20. D. Varshalovich, A. Moskalev, V. Khersonksii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  21. P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  22. P. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  23. L. Tsang, J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  24. P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  25. D. Wielaard, M. Mishchenko, A. Macke, B. Carlson, “Improved T-matrix computations for large, nonabsorbing, and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4317 (1997).
    [CrossRef] [PubMed]

2001

1997

1996

M. Mishchenko, L. Travis, D. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

1995

1992

M. Mishchenko, “Coherent propagation of polarized millimeter waves through falling hydrometeors,” J. Electromagn. Waves Appl. 6, 1341–1351 (1992).

1991

J. Vivekanandan, W. Adams, V. Bringi, “Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions,” J. Appl. Meteorol. 30, 1053–1063 (1991).
[CrossRef]

M. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).[errata: 9, 497 (1992)].
[CrossRef]

M. Mishchenko, “Extinction and polarization of transmitted light by partially aligned nonspherical grains,” Astrophys. J. 367, 561–574 (1991).
[CrossRef]

1988

M. Mishchenko, L. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60, 309–324 (1988).
[CrossRef]

J. Hovenier, C. van der Mee, “Scattering of polarized light: properties of the elements of the phase matrix,” Astron. Astrophys. 196, 287–295 (1988).

1983

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

1974

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
[CrossRef]

1971

P. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1965

P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Adams, W.

J. Vivekanandan, W. Adams, V. Bringi, “Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions,” J. Appl. Meteorol. 30, 1053–1063 (1991).
[CrossRef]

Barber, P.

P. Barber, S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Battaglia, A.

Bohren, C.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bringi, V.

J. Vivekanandan, W. Adams, V. Bringi, “Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions,” J. Appl. Meteorol. 30, 1053–1063 (1991).
[CrossRef]

Carlson, B.

Domke, H.

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
[CrossRef]

Feshbach, H.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Gelfand, I.

I. Gelfand, R. Minlos, Z. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, UK, 1963).

Hill, S.

P. Barber, S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Hovenier, J.

J. Hovenier, C. van der Mee, “Scattering of polarized light: properties of the elements of the phase matrix,” Astron. Astrophys. 196, 287–295 (1988).

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

M. Mishchenko, J. Hovenier, L. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Huffman, D.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Khersonksii, V.

D. Varshalovich, A. Moskalev, V. Khersonksii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Kong, J.

L. Tsang, J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Macke, A.

Mackowski, D.

M. Mishchenko, L. Travis, D. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

Minlos, R.

I. Gelfand, R. Minlos, Z. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, UK, 1963).

Mishchenko, M.

D. Wielaard, M. Mishchenko, A. Macke, B. Carlson, “Improved T-matrix computations for large, nonabsorbing, and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4317 (1997).
[CrossRef] [PubMed]

M. Mishchenko, L. Travis, D. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

M. Mishchenko, “Coherent propagation of polarized millimeter waves through falling hydrometeors,” J. Electromagn. Waves Appl. 6, 1341–1351 (1992).

M. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).[errata: 9, 497 (1992)].
[CrossRef]

M. Mishchenko, “Extinction and polarization of transmitted light by partially aligned nonspherical grains,” Astrophys. J. 367, 561–574 (1991).
[CrossRef]

M. Mishchenko, L. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60, 309–324 (1988).
[CrossRef]

M. Mishchenko, J. Hovenier, L. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Morse, P.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Moskalev, A.

D. Varshalovich, A. Moskalev, V. Khersonksii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Mugnai, A.

W. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” (NASA, Washington, D.C., 1986).

Paramonov, L.

Prodi, F.

Shapiro, Z.

I. Gelfand, R. Minlos, Z. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, UK, 1963).

Shin, R.

L. Tsang, J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Sturniolo, O.

Travis, L.

M. Mishchenko, L. Travis, D. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

M. Mishchenko, L. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60, 309–324 (1988).
[CrossRef]

M. Mishchenko, J. Hovenier, L. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

Tsang, L.

L. Tsang, J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van der Mee, C.

J. Hovenier, C. van der Mee, “Scattering of polarized light: properties of the elements of the phase matrix,” Astron. Astrophys. 196, 287–295 (1988).

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

Varshalovich, D.

D. Varshalovich, A. Moskalev, V. Khersonksii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Vivekanandan, J.

J. Vivekanandan, W. Adams, V. Bringi, “Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions,” J. Appl. Meteorol. 30, 1053–1063 (1991).
[CrossRef]

Waterman, P.

P. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Wielaard, D.

Wiscombe, W.

W. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” (NASA, Washington, D.C., 1986).

Appl. Opt.

Astron. Astrophys.

J. Hovenier, C. van der Mee, “Scattering of polarized light: properties of the elements of the phase matrix,” Astron. Astrophys. 196, 287–295 (1988).

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

Astrophys. J.

M. Mishchenko, “Extinction and polarization of transmitted light by partially aligned nonspherical grains,” Astrophys. J. 367, 561–574 (1991).
[CrossRef]

Astrophys. Space Sci.

H. Domke, “The expansion of scattering matrices for an isotropic medium in generalized spherical functions,” Astrophys. Space Sci. 29, 379–386 (1974).
[CrossRef]

J. Appl. Meteorol.

J. Vivekanandan, W. Adams, V. Bringi, “Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions,” J. Appl. Meteorol. 30, 1053–1063 (1991).
[CrossRef]

J. Electromagn. Waves Appl.

M. Mishchenko, “Coherent propagation of polarized millimeter waves through falling hydrometeors,” J. Electromagn. Waves Appl. 6, 1341–1351 (1992).

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

M. Mishchenko, L. Travis, D. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

M. Mishchenko, L. Travis, “Capabilities and limitations of a current fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60, 309–324 (1988).
[CrossRef]

Phys. Rev. D

P. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Proc. IEEE

P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Other

L. Tsang, J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

M. Mishchenko, L. Travis, D. Mackowski, “T-matrix codes for computing electromagnetic scattering by nonspherical and aggregated particles,” http://www.giss.nasa.gov/~crmim/t_matrix.html .

M. Mishchenko, J. Hovenier, L. Travis, Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).

W. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” (NASA, Washington, D.C., 1986).

P. Barber, S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

I. Gelfand, R. Minlos, Z. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon, Oxford, UK, 1963).

D. Varshalovich, A. Moskalev, V. Khersonksii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

N. Skaropoulos, “T-matrix codes for computing the scattering of electromagnetic waves by partially aligned particles,” http://irctr.et.tudelft.nl/~Skaropoulos/T-matrix.htm .

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (1)

Fig. 1
Fig. 1

Geometric configuration.

Equations (98)

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cos σ1=cos θLsca-cos θLinc cos θSscasin θLinc sin θSsca,
sin σ1=sin θLsca sin(ϕLinc-ϕLsca)sin θSsca.
cos θSsca=cos θLsca cos θLinc+sin θLsca sin θLinc×cos(ϕLinc-ϕLsca).
cos σ2=cos θLinc-cos θLsca cos θSscasin θLsca sin θSsca,
sin σ2=sin θLinc sin(ϕLinc-ϕLsca)sin θSsca.
E+1E-1=PEθEϕ,P=121i1-1.
EθscaEϕsca=exp(ikR-iωt-ikninc·R)R×S(nsca; ninc; α, β, γ)EθincEϕinc,
C(nsca; ninc; α, β, γ)C+1+1C+1-1C-1+1C-1-1=PS(nsca; ninc; α, β, γ)P-1.
ISsca=1R2ZS(nsca; ninc; α, β, γ)ISinc,
IS=[IQUV]T,
I=EθEθ*+EϕEϕ*,Q=EθEθ*-EϕEϕ*,
U=-EθEϕ*-EϕEθ*,V=i(EϕEθ*-EθEϕ*).
ICsca=1R2ZC(nsca; ninc; α, β, γ)ICinc,
IC=[I2I0I-0I-2]T,
I2=E-1E+1*=(Q+iU)/2,
I0=E+1E+1*=(I+V)/2,
I-0=E-1E-1*=(I-V)/2,
I-2=E+1E-1*=(Q-iU)/2.
ZC(nsca; ninc; α, β, γ)=C-1-1C+1+1*C-1+1C+1+1*C-1-1C+1-1*C-1+1C+1-1*C+1-1C+1+1*C+1+1C+1+1*C+1-1C+1-1*C+1+1C+1-1*C-1-1C-1+1*C-1+1C-1+1*C-1-1C-1-1*C-1+1C-1-1*C+1-1C-1+1*C+1+1C-1+1*C+1-1C-1-1*C+1+1C-1-1*,
ZS=A-1ZCA,
A=1201i01001100-101-i0,A-1=01101001-i00i01-10.
ZS,C(nsca; ninc)=n0ZS,C(nsca; ninc)dv,
ZS,C(nsca; ninc)=02πdα0πdβ sin β02πdγ×ZS,C(nsca; ninc; α, β, γ)p(α, β, γ).
02πdα0πdβ sin β02πdγp(α, β, γ)=1.
FS,C(θSsca)=4πCscaZS,C(θSsca, 0; 0, 0),
Csca=2π0πdθSsca sin θSscaZ11S(θSsca, 0; 0, 0),
120πdθSsca sin θSscaF11S(θSsca)=1.
ZS,C(nLsca; nLinc)=Csca4πLS,C(π-σ2)FS,C(θSsca)LS,C(-σ1),
LS(n)=10000cos 2nsin 2n00-sin 2ncos 2n00001,
LC(n)=exp(-i2n)00001000010000exp(i2n).
FklC=4πCscaCpqCpˆqˆ*,
k=pˆ-pifppˆsgn(p)0ifp=pˆ,
l=qˆ-qifqqˆsgn(q)0ifq=qˆ.
FkkC=F-k-kC*,F-kkC=Fk-kC* ifk=±2,
F±0kC=F±0-kC*,Fk±0C=F-k±0C* ifk=±2,
FklC real ifk,l=±0.
FS=a1b100c1a20000a3b200c2a4,
FC=12a2+a3c1+ib2c1-ib2a2-a3b1-ic2a1+a4a1-a4b1+ic2b1+ic2a1-a4a1+a4b1-ic2a2-a3c1-ib2c1+ib2a2+a3,
FklC(θSsca)=s=max(|k|,|l|) gklsPkls(cos θSsca),
gkls=2s+12-11d(cos θSsca)FklC(θSsca)Pkls(cos θSsca).
gkls=g-k-ls.
c1=b1,c2=-b2,
gkls=glks.
p(α, β, γ)=n=0m=-nnm=-nn2n+18π2×pmmnDmmn(α, β, γ),
pmmn=02πdα0πdβ sinβ 02πdγ×Dmmn*(α, β, γ)p(α, β, γ).
p(α, β, γ)=12πp(α, β),
p(α, β)=n=0m=-nn2n+14πpmn exp(-ima)dm0n(β),
pmmn=δm0pmn,pmn=02πdα0πdβ sin β exp(ima)dm0n(β)p(α, β).
p(αSP, βSP)=12πp(βSP),
p(βSP)=n=02n+12pnd00n(βSP),
pmn=δm0pn,pn=0πdβ sin βd00n(β)p(β).
p(βSP)=12,
pn=δn0
0πdβ sin βd00n(β)=2δn0.
Dmmn(αLP, βLP, γLP)=mˆ=-nnDmmˆn(αLS, βLS, γLS)Dmˆmn(αSP, βSP, γSP),
pmn=Lpnd0mn(βLS)exp(-imγLS).
Einc(R)=n=1m=-nn[amnRgMmn(kR)+bmmRgNmn(kR)],
Esca(R)=n=1m=-nn[pmnMmn(kR)+qmnNmn(kR)],
pmn=n=1m=-nn(Tmnmn11amn+Tmnmn12bmn),
qmn=n=1m=-nn(Tmnmn21amn+Tmnmn22bmn).
S(nsca; ninc)=4πkn=1m=-nnn=1m=-nnin-n-1(-1)m+mdndn×exp[i(mϕsca-mϕinc)]×{[Tmnmn11Cmn(θsca)+Tmnmn21iBmn(θsca)]Cmn*(θinc)+[Tmnmn12Cmn(θsca)+Tmnmn22iBmn(θsca)]Bmn*(θinc)/i},
dn=2n+14πn(n+1)1/2
Tmnmnij=δmmTmnnij,
Tmnnij=(-1)i+jT-mnnij.
Tmnmnij(αSP, βSP, γSP)=m1=-MM exp(imαSP)dmm1n(βSP)Tm1nnij×exp(-imαSP)dmm1n(βSP),
md0mn(θ)sin θ=[n(n+1)]1/22[d1mn(θ)+d-1mn(θ)],
d[d0mn(θ)]dθ=[n(n+1)]1/22[d1mn(θ)-d-1mn(θ)],
dpmn(0)=δpm.
Cpq(θSsca, 0; 0, 0; αSP, βSP, 0)=n=1m=-nnn=1tmnnd-pmn(θSsca)×Tmn-qnpq(αSP, βSP, 0),
tmnn=12kin-n-1(-1)m+1[(2n+1)(2n+1)]1/2,
Tpq=T11-qT12-pT21+pqT22.
dmmn(θ)dm1m1n(θ)=n1=|n-n|n+nCnmnm1n1 m+m1Cnmnm1n1 m+m1dm+m1 m+m1n1(θ);
Cn1m1n2m2nm=(-1)n1+m12n+12n2+11/2Cn1m1n-mn2-m2,
Cn1m1n2m2nm=(-1)n1+n2+nCn1-m1 n2-m2n-m;
dmmn(θ)=(-1)m+md-m-mn(θ).
Cpq=n=1m=-nnn1=|m-1|fnn1d-p-qmn(θSsca)×exp[-iαSPq(1-m)]dq(1-m)0n1(βSP)Bmnn1p·q,
Bmnn1p=n=max(|n-n1|, 1)n+n1Cnmn1 1-mn1Annn1p,
Annn1p=in-n(2n+1)1/2m1=-MMCnm1n10nm1Tm1nnp,
Tmnnp=Tmnn11+Tmnn12+pTmnn21+pTmnn22,
fnn1=(2n+1)1/2(2n1+1)/2ik.
CpqCpˆqˆ*=n=1 m=-nn n1=|m-1| nˆ=1 mˆ=-nn nˆ1=|mˆ-1| fnn1fnˆnˆ1*×d-p-qmn(θSsca)d-pˆ-qˆmˆnˆ(θSsca)Bmnn1p·qBmˆnˆnˆ1pˆ·qˆ*×Q-q(1-m)qˆ(1-mˆ)n1nˆ1,
Qmmˆnnˆ=02πdα0πdβ sin β exp[i(m+mˆ)a]d-m0n(β)×dmˆ0nˆ(β)p(α, β).
Qmmˆnnˆ=(-1)mn=|n-nˆ|n+nˆCnmnˆmˆn m+mˆCn0nˆ0n0pm+mˆn.
FklC=n=1 m=-nn nˆ=1 mˆ=-nˆn hnnˆ×d-p-qmn(θSsca)d-pˆ-qˆmˆnˆ(θSsca)Dmmˆnnˆkl,
Dmmˆnnˆkl=n1=|m-1| nˆ1=|mˆ-1| (2n1+1)×(2nˆ1+1)Bmnn1p·qBmˆnˆnˆ1pˆ·qˆ*Q-q(1-m)qˆ(1-mˆ)n1nˆ1,
hnnˆ=πk2Csca[(2n+1)(2nˆ+1)]1/2.
Qmmˆnnˆ=δ-mmˆQmnnˆ,
Qmnnˆ=(-1)mn=|n-nˆ|n+nˆCnmnˆ-mn0Cn0nˆ0n0pn.
Dmmˆnnˆkl=δq-qm qˆ-qˆmˆDmnnˆkl,
Dmnnˆkl=n1=|m-1|nˆ1=|m-1| (2n1+1)×(2nˆ1+1)Bmnn1p·qB1+qqˆ(m-1)nˆ nˆ1pˆ·qˆ*Qm-1 n1 nˆ1.
FklC=n=1 nˆ=1 m=mminmmaxhnnˆd-p-qmn(θSsca)×d-pˆ-l-qmnˆ(θSsca)Dmnnˆkl,
dmmn(θ)=im-mPmmn(cos θ)
FklC(θ)=s=max(|k|,|l|) gklsPkls(cos θ),
gkls=il-kn=1 nˆ=1hsnnˆCnpsknˆpˆ m=mminmmaxCn qm slnˆ qm+lDmnnˆkl.
hsnnˆ=(2s+1)πk2Csca2n+12nˆ+11/2
Cn1m1n2m200=(-1)n1-m1(2n1+1)-1/2δn1n2δm1-m2
Qmnnˆ=(2n+1)-1δnnˆ;
Dmnnˆkl=n1=|m-1|(2n1+1)Bmnn1p·qB1+qqˆ(m-1)nˆ n1pˆ·qˆ*.

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