Abstract

An asymptotic analysis of the radiative transfer equation with polarization is developed that leads to a renormalized scalar equation for the total specific intensity of radiation I, the first Stokes parameter, in three-dimensional geometries. The resulting scalar equation can be used without the complexity of performing vector radiative computations since it merely requires an adjustment of the coefficients of the scattering phase matrix. The equation is accurate to first order in the smallness parameter of the asymptotic analysis. Asymptotically consistent quadrature results are obtained for Q, U, and V, the other three Stokes parameters. Numerical results demonstrate the improved accuracy of the renormalized scalar equation for the intensity compared with the usual unpolarized approximation and illustrate that small effects of polarization can be propagated to large optical depths within a medium.

© 1998 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 2–12 (1990).
    [CrossRef]
  4. G. W. Kattawar, C. N. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface, L. Estep, ed., Proc. SPIE1749, 2–22 (1992).
  5. C. N. Adams, G. W. Kattawar, “Effect of volume-scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 32, 4610–4617 (1993).
    [CrossRef] [PubMed]
  6. G. W. Kattawar, “Irradiance invariance for scattering according to a Rayleigh phase function compared to a Rayleigh phase matrix for a plane-parallel medium,” Appl. Opt. 29, 2365–2367 (1990); “Irradiance invariance for scattering according to a Rayleigh phase function compared to a Rayleigh phase matrix for a plane-parallel medium: erratum,” 30, 4288 (1991).
    [CrossRef] [PubMed]
  7. J. W. Hovenier, “Multiple scattering of polarized light in planetary atmospheres,” Astron. Astrophys. 13, 7–29 (1971).
  8. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  9. C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).
  10. G. C. Pomraning, “A renormalized equation of transfer for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Trans. (to be published).
  11. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
    [CrossRef]
  12. C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
  13. I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
    [CrossRef]
  14. I. M. Gel’fand, R. A. Minlos, Z. Ya. Šapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963).
  15. W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).
  16. R. D. M. Garcia, C. E. Siewert, “The discrete spectrum for radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transf. 38, 295–301 (1987).
    [CrossRef]
  17. N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
    [CrossRef]
  18. R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36, 401–423 (1986).
    [CrossRef]
  19. N. J. McCormick, “Particle-size-distribution retrieval from backscattered polarized radiation measurements: a proposed method,” J. Opt. Soc. Am. A 7, 1811–1816 (1990).
    [CrossRef]
  20. E. W. Larsen, “The amplitude and radius of radiation pencil beams,” Transp. Theory Stat. Phys. 26, 533–554 (1997).
    [CrossRef]
  21. G. C. Pomraning, B. D. Ganapol, “Simplified radiative transfer for combined Rayleigh and isotropic scattering,” Astrophys. J. (to be published).

1997 (1)

E. W. Larsen, “The amplitude and radius of radiation pencil beams,” Transp. Theory Stat. Phys. 26, 533–554 (1997).
[CrossRef]

1993 (1)

1990 (3)

1987 (1)

R. D. M. Garcia, C. E. Siewert, “The discrete spectrum for radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transf. 38, 295–301 (1987).
[CrossRef]

1986 (1)

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36, 401–423 (1986).
[CrossRef]

1984 (1)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

1983 (1)

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

1982 (1)

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).

1981 (1)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

1974 (1)

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

1971 (1)

J. W. Hovenier, “Multiple scattering of polarized light in planetary atmospheres,” Astron. Astrophys. 13, 7–29 (1971).

1959 (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Adams, C. N.

C. N. Adams, G. W. Kattawar, “Effect of volume-scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 32, 4610–4617 (1993).
[CrossRef] [PubMed]

G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 2–12 (1990).
[CrossRef]

G. W. Kattawar, C. N. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface, L. Estep, ed., Proc. SPIE1749, 2–22 (1992).

Chandrasekhar, S.

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

de Rooij, W. A.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Ganapol, B. D.

G. C. Pomraning, B. D. Ganapol, “Simplified radiative transfer for combined Rayleigh and isotropic scattering,” Astrophys. J. (to be published).

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “The discrete spectrum for radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transf. 38, 295–301 (1987).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36, 401–423 (1986).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, R. A. Minlos, Z. Ya. Šapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963).

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.

C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

J. W. Hovenier, “Multiple scattering of polarized light in planetary atmospheres,” Astron. Astrophys. 13, 7–29 (1971).

Kattawar, G. W.

C. N. Adams, G. W. Kattawar, “Effect of volume-scattering function on the errors induced when polarization is neglected in radiance calculations in an atmosphere–ocean system,” Appl. Opt. 32, 4610–4617 (1993).
[CrossRef] [PubMed]

G. W. Kattawar, “Irradiance invariance for scattering according to a Rayleigh phase function compared to a Rayleigh phase matrix for a plane-parallel medium,” Appl. Opt. 29, 2365–2367 (1990); “Irradiance invariance for scattering according to a Rayleigh phase function compared to a Rayleigh phase matrix for a plane-parallel medium: erratum,” 30, 4288 (1991).
[CrossRef] [PubMed]

G. W. Kattawar, C. N. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface, L. Estep, ed., Proc. SPIE1749, 2–22 (1992).

G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 2–12 (1990).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kušcer, I.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Larsen, E. W.

E. W. Larsen, “The amplitude and radius of radiation pencil beams,” Transp. Theory Stat. Phys. 26, 533–554 (1997).
[CrossRef]

McCormick, N. J.

N. J. McCormick, “Particle-size-distribution retrieval from backscattered polarized radiation measurements: a proposed method,” J. Opt. Soc. Am. A 7, 1811–1816 (1990).
[CrossRef]

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

Minlos, R. A.

I. M. Gel’fand, R. A. Minlos, Z. Ya. Šapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963).

Pomraning, G. C.

G. C. Pomraning, B. D. Ganapol, “Simplified radiative transfer for combined Rayleigh and isotropic scattering,” Astrophys. J. (to be published).

G. C. Pomraning, “A renormalized equation of transfer for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Trans. (to be published).

Ribaric, M.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Sanchez, R.

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

Šapiro, Z. Ya.

I. M. Gel’fand, R. A. Minlos, Z. Ya. Šapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963).

Siewert, C. E.

R. D. M. Garcia, C. E. Siewert, “The discrete spectrum for radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transf. 38, 295–301 (1987).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36, 401–423 (1986).
[CrossRef]

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

Travis, L. D.

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

van der Mee, C. V. M.

C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

van der Stap, C. C. A. H.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Appl. Opt. (2)

Astron. Astrophys. (4)

J. W. Hovenier, “Multiple scattering of polarized light in planetary atmospheres,” Astron. Astrophys. 13, 7–29 (1971).

C. V. M. van der Mee, J. W. Hovenier, “Expansion coefficients in polarized light transfer,” Astron. Astrophys. 228, 559–568 (1990).

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Astrophys. J. (1)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transf. (3)

R. D. M. Garcia, C. E. Siewert, “The discrete spectrum for radiative transfer with polarization,” J. Quant. Spectrosc. Radiat. Transf. 38, 295–301 (1987).
[CrossRef]

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36, 401–423 (1986).
[CrossRef]

Opt. Acta (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Space Sci. Rev. (1)

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

Transp. Theory Stat. Phys. (1)

E. W. Larsen, “The amplitude and radius of radiation pencil beams,” Transp. Theory Stat. Phys. 26, 533–554 (1997).
[CrossRef]

Other (7)

G. C. Pomraning, B. D. Ganapol, “Simplified radiative transfer for combined Rayleigh and isotropic scattering,” Astrophys. J. (to be published).

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

G. W. Kattawar, C. N. Adams, “Errors in radiance calculations induced by using scalar rather than Stokes vector theory in a realistic atmosphere–ocean system,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 2–12 (1990).
[CrossRef]

G. W. Kattawar, C. N. Adams, “Errors induced when polarization is neglected in radiance calculations for an atmosphere–ocean system,” in Optics of the Air–Sea Interface, L. Estep, ed., Proc. SPIE1749, 2–22 (1992).

I. M. Gel’fand, R. A. Minlos, Z. Ya. Šapiro, Representations of the Rotation and Lorentz Groups and their Applications (Pergamon, Oxford, 1963).

G. C. Pomraning, “A renormalized equation of transfer for Rayleigh scattering,” J. Quant. Spectrosc. Radiat. Trans. (to be published).

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Tables (1)

Tables Icon

Table 1 Asymptotic Diffuse Attenuation Coefficient κ versus Single Scattering Albedo ω and Depolarization Factor ρ Computed without and with Polarization Effects Included in the Scalar Radiative Transfer Equation and from the Exact Vector Equation

Equations (64)

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Ω·Ψ(r, μ, ϕ)+σΨ(r, μ, ϕ)=S(r, μ, ϕ)+σs4π02πdϕ-11dμ×P(μ, μ, ϕ-ϕ)Ψ(r, μ, ϕ),
S(r, μ, ϕ)=[σaB(r), 0, 0, 0]T,
P(μ, μ, ϕ-ϕ)=C0(μ, μ)/2+m=1L[Cm(μ, μ)cos m(ϕ-ϕ)+Sm(μ, μ)sin m(ϕ-ϕ)],
Cm(μ, μ)=Am(μ, μ)+DAm(μ, μ)D,
Sm(μ, μ)=Am(μ, μ)D-DAm(μ, μ),
Am(μ, μ)=l=mLAlm(μ)BlAlm(μ),
Alm(μ)=ΓlmΠlm(μ),
Γlm=[(l-m)!/(l+m)!]1/2.
Πlm(μ)=Plm(μ)0000Rlm(μ)-Tlm(μ)00-Tlm(μ)Rlm(μ)0000Plm(μ),
Plm(μ)=(1-μ2)m/2dmdμmPl(μ),
Rlm(μ)=-im2Γlm[Pm,2l(μ)+Pm, -2l(μ)],lm,
Tlm(μ)=-im2Γlm[Pm,2l(μ)-Pm,-2l(μ)],lm,
Pm,nl(μ)=Am,nl(1-μ)(m-n)/2(1+μ)-(m+n)/2×dl-ndμl-n[(1-μ)l-m(1+μ)l+m],
Am,nl=(-1)l-min-m2l(l-m)!ΓlmΓln.
Bl=βlγl00γlαl0000ζl-εl00εlδl,l=0 to L,
β0=1 and γl=αl=ζl=εl=0 for l=0 and 1.
-11Alm(μ)Alm(μ)dμ=[2/(2l+1)]U˜lδl,l,
Ψ(r, μ, ϕ)=m=0L[Φ1m(ϕ)Ψ1m(r,μ)+Φ2m(ϕ)Ψ2m(r, μ)],
Φ1m(ϕ)=diag[cos mϕ, cos mϕ, sin mϕ, sin mϕ],
Φ2m(ϕ)=diag[-sin mϕ, -sin mϕ, cos mϕ, cos mϕ].
Ψjm(r, μ)=[π(1+δm,0)]-102πΦjm(ϕ)Ψ(r, μ, ϕ)dϕ,
j=1, 2.
LjmΨ(r, μ, ϕ)+Ψjm(r, μ)=σ-1S(r)δm,0δj,1+ω2l=mLAlm(μ)Bl×-11Alm(μ)Ψjm(r, μ)dμ,
LjmΨ(r, μ, ϕ)=[π(1+δm,0)σ]-1×02πΦjm(ϕ)Ω·Ψ(r, μ, ϕ)dϕ,
j=1, 2.
I(r, μ, ϕ)=B(r)+O(),
Ω·Ψ(r, μ, ϕ)=O(1).
σ, σa, σs=O(-1).
LmΨ(r, μ, ϕ)+ΛΨm(r, μ)=(1-ω)B(r)[1, 0, 0, 0]Tδm,0δj,1+ω2l=mLAlm(μ)Bl-11Alm(μ)ΛΨm(r, μ)dμ,
Ψm(r, μ)n=0nΨm(n)(r, μ),
0Qm(0)(r, μ)Um(0)(r, μ)Vm(0)(r, μ)=ω2l=mLAlm(μ)Bl-11Alm(μ)0Qm(0)(r, μ)Um(0)(r, μ)Vm(0)(r, μ)dμ.
hl-11Alm(μ)0Qm(0)(r, μ)Um(0)(r, μ)Vm(0)(r, μ)dμ=0,
hl=U-(2l+1)-1ωBl,
-11Alm(μ)0Qm(0)(r, μ)Um(0)(r, μ)Vm(0)(r, μ)dμ=0,
LmI0(r, μ, ϕ)000+Im(0)(r, μ)Qm(1)(r, μ)Um(1)(r, μ)Vm(1)(r, μ)=(1-ω)B(r)000δm,0δj,1+ω2l=mLAlm(μ)Bl×-11Alm(μ)Im(0)(r, μ)Qm(1)(r, μ)Um(1)(r, μ)Vm(1)(r, μ)dμ.
Lm-11Alm(μ)I0(r, μ, ϕ)000dμ+hlI˜m(0)Q˜m(1)U˜m(1)V˜m(1)=2(1-ω)B(r)000δl,0δm,0δj,1,
I˜m(0)Q˜m(1)U˜m(1)V˜m(1)=-11Alm(μ)Im(0)(r, μ)Qm(1)(r, μ)Um(1)(r, μ)Vm(1)(r, μ)dμ.
I˜m(0)(r)=Γlm-11Plm(μ)Im(0)(r, μ)dμ
hˆlQ˜m(1)U˜m(1)V˜m(1)+kˆlI˜m(0)=0,
Q˜m(1)U˜m(1)V˜m(1)=-(hˆl)-1kˆlI˜m(0)=ωγl2l+1-ωαl00I˜m(0).
Θˆlm(μ)=ΓlmαlRlm(μ)-ζlTlm(μ)εlTlm(μ)-αlTlm(μ)ζlRlm(μ)-εlRlm(μ)0εlPlm(μ)δlPlm(μ).
θˆlm(μ)=ΓlmγlRlm(μ)-Tlm(μ)0.
Qm(1)(r, μ)Um(1)(r, μ)Vm(1)(r, μ)=ω2l=mL[θˆlm(μ)-Θˆlm(μ)×(hˆl)-1kˆl]I˜m(0).
Qm(r, μ)Um(r, μ)Vm(r, μ)=ω2l=mLΓlm[θˆlm(μ)-Θˆlm(μ)(hˆl)-1kˆl]×-11Plm(μ)Im(r, μ)dμ+O(2).
Qm(r, μ)Um(r, μ)Vm(r, μ)=ω2l=mL(2l+1)(Γlm)2γl2l+1-ωαlRlm(μ)-Tlm(μ)0×-11Plm(μ)Im(r, μ)dμ+O(2).
Φˆ1m(ϕ)=diag[cos mϕ, sin mϕ, sin mϕ],
Φˆ2m(ϕ)=diag[-sin mϕ, cos mϕ, cos mϕ].
Q(r, μ, ϕ)U(r, μ, ϕ)V(r, μ, ϕ)=ω2πm=0Ll=mL(2l+1)(Γlm)2γl(1+δm,0)(2l+1-ωαl)×02π-11Plm(μ)Rlm(μ)cos m(ϕ-ϕ)-Tlm(μ)sin m(ϕ-ϕ)0I(r, μ,ϕ)dμdϕ+O(2)
tˆlm(μ)=Γlmγl[Plm(μ), 0, 0].
LmI0(r, μ, ϕ)+Im(0)(r, μ)=(1-ω)B(r)δm,0δj,1+ω2l=mLΓlmβlPlm(μ)I˜m(0)+tˆlm(μ)Q˜m(1)U˜m(1)V˜m(1).
LmI(r, μ, ϕ)+Im(r, μ)=(1-ω)B(r)δm,0δj,1+ω2l=mLΓlm[ΓlmβlPlm(μ)-tˆlm(μ)(hˆl)-1kˆl]×-11Plm(μ)I(r, μ)dμ+O().
LmI(r, μ, ϕ)+Im(r, μ)=(1-ω)B(r)δm,0δj,1+ω2l=mL(Γlm)2βlpolarPlm(μ)×-11Plm(μ)I(r, μ)dμ+O(),
βlpolar=βl+ωγl22l+1-ωαl.
Ω·I(r, μ, ϕ)+σI(r, μ, ϕ)=σaB(r)+σs4π02π-11Ppolar(μ, μ, ϕ, ϕ)×I(r, μ, ϕ)dμdϕ+O()
Ppolar(μ, μ, ϕ, ϕ)=l=0Lm=0l2(Γlm)21+δm,0βlpolarPlm(μ)Plm(μ)cos m(ϕ-ϕ).
Ppolar(μ, μ, ϕ, ϕ)=Ppolar(Ω·Ω)=l=0LβlpolarPl(Ω·Ω).
β1=0,β2=c/2,γ2=-6c/2,α2=3c,
c=2(1-ρ)/(2+ρ)
β2polar=c255-3ωc,
κ=ωq0(1/κ)+cG(κ)43(1-ω)κ2-1q2(1/κ),
q0(1/κ)=12ln1+κ1-κ,
q2(1/κ)=143κ2-1ln1+κ1-κ-32κ.
1G(κ)=1-3cω4κ21+(1-κ2)κ×[q2(1/κ)-q0(1/κ)].
Error=κapprox-κexactκexact×100,

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