Abstract

Light scattering by ensembles of independently scattering, randomly oriented, axially symmetric particles is considered. The elements of the scattering matrices are expanded in (combinations of) generalized spherical functions; this is advantageous in computations of both single and multiple light scattering. Waterman’s T-matrix approach is used to develop a rigorous analytical method to compute the corresponding expansion coefficients. The main advantage of this method is that the expansion coefficients are expressed directly in some basic quantities that depend on only the shape, morphology, and composition of the scattering axially symmetric particle; these quantities are the elements of the T matrix calculated with respect to the coordinate system with the z axis along the axis of particle symmetry. Thus the expansion coefficients are calculated without computing beforehand the elements of the scattering matrix for a large set of particle orientations and scattering angles, which minimizes the numerical calculations. Like the T-matrix approach itself, the method can be used in computations for homogeneous and composite isotropic particles of sizes not too large compared with a wavelength. Computational aspects of the method are discussed in detail, and some illustrative numerical results are reported for randomly oriented homogeneous dielectric spheroids and Chebyshev particles. Results of timing tests are presented; it is found that the method described is much faster than the commonly used method of numerical angle integrations.

© 1991 Optical Society of America

Full Article  |  PDF Article

Corrections

M. I. Mishchenko, "Light scattering by randomly oriented axially symmetric particles: errata," J. Opt. Soc. Am. A 9, 497-497 (1992)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-9-3-497

References

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    [CrossRef]

1990 (7)

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

H. Domke, E. G. Yanovitskij, “On a new form of the radiative transfer equation with applications to multiple scattering of polarized light,”J. Quant. Spectrosc. Radiat. Transfer 43, 61–73 (1990).
[CrossRef]

M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,”J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990).
[CrossRef]

M. I. Mishchenko, “Expansion of the scattering matrix for radially inhomogeneous spherical particles in generalized spherical functions,” Kinem. Fiz. Nebes. Tel 6(1), 91–93 (1990).

M. I. Mishchenko, “Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990).
[CrossRef]

M. I. Mishchenko, “Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990).

R. Schiffer, “Perturbation approach for light scattering by an ensemble of irregular particles of arbitrary material,” Appl. Opt. 29, 1536–1550 (1990).
[CrossRef] [PubMed]

1989 (4)

R. Schiffer, “Light scattering by perfectly conducting statistically irregular particles,” J. Opt. Soc. Am. A 6, 385–402 (1989).
[CrossRef]

W. Zheng, S. Ström, “The null field approach to electromagnetic scattering from composite objects: the case of concavo-convex constituents,” IEEE Trans. Antennas Propag. 37, 373–383 (1989).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “The FNmethod for radiative transfer models that include polarization effects,”J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[CrossRef]

P. Stammes, J.F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

1988 (4)

E. A. Ustinov, “Method of spherical harmonics: application to the transfer of polarized radiation in a vertically inhomogeneous planetary atmosphere. Mathematical formalism,” Kosm. Issled. 26, 550–562 (1988).

J. B. Schneider, I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36, 1317–1321 (1988).
[CrossRef]

S. Ström, W. Zheng, “The null field approach to electromagnetic scattering from composite objects,” IEEE Trans. Antennas Propag. 36, 376–382 (1988).
[CrossRef]

W. Zheng, “The null field approach to electromagnetic scattering from composite objects: the case of three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400(1988).
[CrossRef]

1987 (1)

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

1986 (2)

H. Domke, E. G. Yanovitskij, “Principles of invariance applied to the computation of internal polarized radiation in multilayered atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 36, 175–186 (1986).
[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. Transfer 36, 401–423 (1986).
[CrossRef]

1985 (1)

1984 (4)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by highlyaspherical targets: EBCM coupled with reinforced orthogonalization,” Appl. Opt. 23, 3502–3504 (1984).
[CrossRef]

W. A. de Rooij, H. Domke, “On the nonuniqueness of solutions for nonlinear integral equations in radiative transfer theory,”J. Quant. Spectrosc. Radiat. Transfer 31, 285–299 (1984).
[CrossRef]

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).

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

1983 (2)

H. Domke, “Biorthogonality and radiative transfer in finite slab atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 30, 119–129 (1983).
[CrossRef]

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).

1982 (2)

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

A. Ishimaru, R. Woo, J. W. Armstrong, D. C. Blackman, “Multiple scattering calculations of rain effects,” Radio Sci. 17, 1425–1433 (1982).
[CrossRef]

1981 (2)

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

P. W. Barber, D.-S. Y. Wang, M. B. Long, “Scattering calculations using a microcomputer,” Appl. Opt. 20, 1121–1123 (1981).
[CrossRef] [PubMed]

1980 (3)

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter waves due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Ann. Telecommun. 35, 380–389 (1980).

1979 (1)

1976 (2)

H. Domke, “Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations,”J. Quant. Spectrosc. Radiat. Transfer 16, 973–982 (1976).
[CrossRef]

O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Akad. Nauk. SSSR Ser. Fiz. Atmos. Okeana 12, 603–611 (1976).

1975 (6)

H. Domke, “Fourier expansion of the phase matrix for Mie scattering,”Z. Meteorol. 25, 357–361 (1975).

H. Domke, “Transfer of polarized light in an isotropic medium. Singular eigensolutions of the transfer equation,”J. Quant. Spectrosc. Radiat. Transfer 15, 669–679 (1975).
[CrossRef]

H. Domke, “Transfer of polarized light in an isotropic medium. Biorthogonality and the solution of transfer problems in semi-infinite media,”J. Quant. Spectrosc. Radiat. Transfer 15, 681–694 (1975).
[CrossRef]

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[PubMed]

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
[CrossRef]

1974 (3)

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatterers,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

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

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

1971 (1)

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

1965 (1)

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

1964 (1)

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. 135, 1193–1201 (1964).
[CrossRef]

1959 (1)

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

Ambarzumian, V. A.

V. A. Ambarzumian, Nauchnye Trudy (Armenian SSR Academy of Sciences, Erevan, 1960), Vol. 1, pp. 181–205.

Armstrong, J. W.

A. Ishimaru, R. Woo, J. W. Armstrong, D. C. Blackman, “Multiple scattering calculations of rain effects,” Radio Sci. 17, 1425–1433 (1982).
[CrossRef]

Asano, S.

Barber, P.

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

P. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. dissertation (University of California, Los Angeles, Los Angeles, Calif., 1973).

Barber, P. W.

Blackman, D. C.

A. Ishimaru, R. Woo, J. W. Armstrong, D. C. Blackman, “Multiple scattering calculations of rain effects,” Radio Sci. 17, 1425–1433 (1982).
[CrossRef]

Bohren, C. F.

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

Bosma, P. B.

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

Bugaenko, O. I.

O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Akad. Nauk. SSSR Ser. Fiz. Atmos. Okeana 12, 603–611 (1976).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

de Haan, J. F.

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).

J. F. de Haan, Free University, Amsterdam, The Netherlands (personal communication).

de Haan, J.F.

P. Stammes, J.F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

de Rooij, W. A.

W. A. de Rooij, H. Domke, “On the nonuniqueness of solutions for nonlinear integral equations in radiative transfer theory,”J. Quant. Spectrosc. Radiat. Transfer 31, 285–299 (1984).
[CrossRef]

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).

W. A. de Rooij, “Reflection and transmission of polarized light by planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1985).

Domke, H.

H. Domke, E. G. Yanovitskij, “On a new form of the radiative transfer equation with applications to multiple scattering of polarized light,”J. Quant. Spectrosc. Radiat. Transfer 43, 61–73 (1990).
[CrossRef]

H. Domke, E. G. Yanovitskij, “Principles of invariance applied to the computation of internal polarized radiation in multilayered atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 36, 175–186 (1986).
[CrossRef]

W. A. de Rooij, H. Domke, “On the nonuniqueness of solutions for nonlinear integral equations in radiative transfer theory,”J. Quant. Spectrosc. Radiat. Transfer 31, 285–299 (1984).
[CrossRef]

H. Domke, “Biorthogonality and radiative transfer in finite slab atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 30, 119–129 (1983).
[CrossRef]

H. Domke, “Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations,”J. Quant. Spectrosc. Radiat. Transfer 16, 973–982 (1976).
[CrossRef]

H. Domke, “Transfer of polarized light in an isotropic medium. Singular eigensolutions of the transfer equation,”J. Quant. Spectrosc. Radiat. Transfer 15, 669–679 (1975).
[CrossRef]

H. Domke, “Transfer of polarized light in an isotropic medium. Biorthogonality and the solution of transfer problems in semi-infinite media,”J. Quant. Spectrosc. Radiat. Transfer 15, 681–694 (1975).
[CrossRef]

H. Domke, “Fourier expansion of the phase matrix for Mie scattering,”Z. Meteorol. 25, 357–361 (1975).

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

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “The FNmethod for radiative transfer models that include polarization effects,”J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[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. Transfer 36, 401–423 (1986).
[CrossRef]

Gelfand, I. M.

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

Geller, P. E.

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).

P. Stammes, J.F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

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).

Huffman, D. R.

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

Ishimaru, A.

A. Ishimaru, R. Woo, J. W. Armstrong, D. C. Blackman, “Multiple scattering calculations of rain effects,” Radio Sci. 17, 1425–1433 (1982).
[CrossRef]

Khersonskij, V. K.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskij, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975).

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

Kušcer, I.

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

Lakhtakia, A.

Long, M. B.

Lopatin, V. N.

L. E. Paramonov, V. N. Lopatin, “Light scattering by non-spherical particles (algorithm, computational procedure, programs),” Preprint No. 67B (Institute for Physics, Krasnojarsk, USSR, 1987).

Minlos, R. A.

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

Mishchenko, M. I.

M. I. Mishchenko, “Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990).
[CrossRef]

M. I. Mishchenko, “Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990).

M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,”J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990).
[CrossRef]

M. I. Mishchenko, “Expansion of the scattering matrix for radially inhomogeneous spherical particles in generalized spherical functions,” Kinem. Fiz. Nebes. Tel 6(1), 91–93 (1990).

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskij, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975).

Mugnai, A.

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Md., 1986).

Oguchi, T.

T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter waves due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Ann. Telecommun. 35, 380–389 (1980).

Paramonov, L. E.

L. E. Paramonov, V. N. Lopatin, “Light scattering by non-spherical particles (algorithm, computational procedure, programs),” Preprint No. 67B (Institute for Physics, Krasnojarsk, USSR, 1987).

Peden, I. C.

J. B. Schneider, I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36, 1317–1321 (1988).
[CrossRef]

Peterson, B.

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatterers,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

Ribaric, M.

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

Sato, M.

Schaefer, R. W.

R. W. Schaefer, “Calculations of the light scattered by randomly oriented ensembles of spheroids of size comparable to the wavelength,” Ph.D. dissertation (State University of New York, Albany, N.Y., 1980).

Schiffer, R.

Schneider, J. B.

J. B. Schneider, I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36, 1317–1321 (1988).
[CrossRef]

Shapiro, Z. Ya.

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

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

Siewert, C. E.

R. D. M. Garcia, C. E. Siewert, “The FNmethod for radiative transfer models that include polarization effects,”J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[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. Transfer 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]

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

Stammes, P.

P. Stammes, J.F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).

Ström, S.

W. Zheng, S. Ström, “The null field approach to electromagnetic scattering from composite objects: the case of concavo-convex constituents,” IEEE Trans. Antennas Propag. 37, 373–383 (1989).
[CrossRef]

S. Ström, W. Zheng, “The null field approach to electromagnetic scattering from composite objects,” IEEE Trans. Antennas Propag. 36, 376–382 (1988).
[CrossRef]

S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
[CrossRef]

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatterers,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

Travis, L. D.

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

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

Tsuei, T. G.

Ustinov, E. A.

E. A. Ustinov, “Method of spherical harmonics: application to the transfer of polarized radiation in a vertically inhomogeneous planetary atmosphere. Mathematical formalism,” Kosm. Issled. 26, 550–562 (1988).

van de Hulst, H. C.

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

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

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).

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).

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).

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering by highlyaspherical targets: EBCM coupled with reinforced orthogonalization,” Appl. Opt. 23, 3502–3504 (1984).
[CrossRef]

V. K. Varadan, “Multiple scattering of acoustic, electromagnetic and elastic waves,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), pp. 103–104.

Varadan, V. V.

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskij, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975).

Wang, D.-S.

Wang, D.-S. Y.

Waterman, P.

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

Waterman, P. C.

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

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, Oxford, 1973), Vol. 7, pp. 97–157.

Wiscombe, W. J.

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Md., 1986).

Woo, R.

A. Ishimaru, R. Woo, J. W. Armstrong, D. C. Blackman, “Multiple scattering calculations of rain effects,” Radio Sci. 17, 1425–1433 (1982).
[CrossRef]

Yamamoto, G.

Yanovitskij, E. G.

H. Domke, E. G. Yanovitskij, “On a new form of the radiative transfer equation with applications to multiple scattering of polarized light,”J. Quant. Spectrosc. Radiat. Transfer 43, 61–73 (1990).
[CrossRef]

H. Domke, E. G. Yanovitskij, “Principles of invariance applied to the computation of internal polarized radiation in multilayered atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 36, 175–186 (1986).
[CrossRef]

Yeh, C.

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. 135, 1193–1201 (1964).
[CrossRef]

Zheng, W.

W. Zheng, S. Ström, “The null field approach to electromagnetic scattering from composite objects: the case of concavo-convex constituents,” IEEE Trans. Antennas Propag. 37, 373–383 (1989).
[CrossRef]

S. Ström, W. Zheng, “The null field approach to electromagnetic scattering from composite objects,” IEEE Trans. Antennas Propag. 36, 376–382 (1988).
[CrossRef]

W. Zheng, “The null field approach to electromagnetic scattering from composite objects: the case of three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400(1988).
[CrossRef]

Am. J. Phys. (1)

S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
[CrossRef]

Ann. Telecommun. (1)

T. Oguchi, “Effect of incoherent scattering on attenuation and cross-polarization of millimeter waves due to rain: preliminary calculations at 34.8 and 82 GHz for spherical raindrops,” Ann. Telecommun. 35, 380–389 (1980).

Appl. Opt. (8)

Astron. Astrophys. (6)

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

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).

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

J. F. de Haan, P. B. Bosma, J. W. Hovenier, “The adding method for multiple scattering calculations of polarized light,” Astron. Astrophys. 183, 371–391 (1987).

P. Stammes, J.F. de Haan, J. W. Hovenier, “The polarized internal radiation field of a planetary atmosphere,” Astron. Astrophys. 225, 239–259 (1989).

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]

Astrophys. Space Sci. (2)

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

M. I. Mishchenko, “Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

J. B. Schneider, I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36, 1317–1321 (1988).
[CrossRef]

S. Ström, W. Zheng, “The null field approach to electromagnetic scattering from composite objects,” IEEE Trans. Antennas Propag. 36, 376–382 (1988).
[CrossRef]

W. Zheng, “The null field approach to electromagnetic scattering from composite objects: the case of three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400(1988).
[CrossRef]

W. Zheng, S. Ström, “The null field approach to electromagnetic scattering from composite objects: the case of concavo-convex constituents,” IEEE Trans. Antennas Propag. 37, 373–383 (1989).
[CrossRef]

Izv. Akad. Nauk. SSSR Ser. Fiz. Atmos. Okeana (1)

O. I. Bugaenko, “Generalized spherical functions in the Mie problem,” Izv. Akad. Nauk. SSSR Ser. Fiz. Atmos. Okeana 12, 603–611 (1976).

J. Atmos. Sci. (1)

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transfer (10)

H. Domke, E. G. Yanovitskij, “On a new form of the radiative transfer equation with applications to multiple scattering of polarized light,”J. Quant. Spectrosc. Radiat. Transfer 43, 61–73 (1990).
[CrossRef]

M. I. Mishchenko, “The fast invariant imbedding method for polarized light: computational aspects and numerical results for Rayleigh scattering,”J. Quant. Spectrosc. Radiat. Transfer 43, 163–171 (1990).
[CrossRef]

H. Domke, “Transfer of polarized light in an isotropic medium. Singular eigensolutions of the transfer equation,”J. Quant. Spectrosc. Radiat. Transfer 15, 669–679 (1975).
[CrossRef]

H. Domke, “Transfer of polarized light in an isotropic medium. Biorthogonality and the solution of transfer problems in semi-infinite media,”J. Quant. Spectrosc. Radiat. Transfer 15, 681–694 (1975).
[CrossRef]

H. Domke, “Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations,”J. Quant. Spectrosc. Radiat. Transfer 16, 973–982 (1976).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “The FNmethod for radiative transfer models that include polarization effects,”J. Quant. Spectrosc. Radiat. Transfer 41, 117–145 (1989).
[CrossRef]

H. Domke, “Biorthogonality and radiative transfer in finite slab atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 30, 119–129 (1983).
[CrossRef]

W. A. de Rooij, H. Domke, “On the nonuniqueness of solutions for nonlinear integral equations in radiative transfer theory,”J. Quant. Spectrosc. Radiat. Transfer 31, 285–299 (1984).
[CrossRef]

H. Domke, E. G. Yanovitskij, “Principles of invariance applied to the computation of internal polarized radiation in multilayered atmospheres,”J. Quant. Spectrosc. Radiat. Transfer 36, 175–186 (1986).
[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. Transfer 36, 401–423 (1986).
[CrossRef]

Kinem. Fiz. Nebes. Tel (2)

M. I. Mishchenko, “Expansion of the scattering matrix for radially inhomogeneous spherical particles in generalized spherical functions,” Kinem. Fiz. Nebes. Tel 6(1), 91–93 (1990).

M. I. Mishchenko, “Calculation of the total optical cross sections for an ensemble of randomly oriented nonspherical particles,” Kinem. Fiz. Nebes. Tel 6(5), 95–96 (1990).

Kosm. Issled. (1)

E. A. Ustinov, “Method of spherical harmonics: application to the transfer of polarized radiation in a vertically inhomogeneous planetary atmosphere. Mathematical formalism,” Kosm. Issled. 26, 550–562 (1988).

Opt. Acta (1)

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

Phys. Rev. (1)

C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. 135, 1193–1201 (1964).
[CrossRef]

Phys. Rev. D (2)

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatterers,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

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

Proc. IEEE (1)

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

Radio Sci. (2)

A. Ishimaru, R. Woo, J. W. Armstrong, D. C. Blackman, “Multiple scattering calculations of rain effects,” Radio Sci. 17, 1425–1433 (1982).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

Space Sci. Rev. (1)

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

Z. Meteorol. (1)

H. Domke, “Fourier expansion of the phase matrix for Mie scattering,”Z. Meteorol. 25, 357–361 (1975).

Other (19)

W. J. Wiscombe, A. Mugnai, “Single scattering from non-spherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Md., 1986).

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

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

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskij, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975).

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

V. A. Ambarzumian, Nauchnye Trudy (Armenian SSR Academy of Sciences, Erevan, 1960), Vol. 1, pp. 181–205.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

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

W. A. de Rooij, “Reflection and transmission of polarized light by planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1985).

V. K. Varadan, “Multiple scattering of acoustic, electromagnetic and elastic waves,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), pp. 103–104.

V. K. Varadan, V. V. Varadan, eds., Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach (Pergamon, New York, 1980).

P. Stammes, “Light scattering properties of aerosols and the radiation inside a planetary atmosphere,” Ph.D. dissertation (Free University, Amsterdam, 1989).

L. E. Paramonov, V. N. Lopatin, “Light scattering by non-spherical particles (algorithm, computational procedure, programs),” Preprint No. 67B (Institute for Physics, Krasnojarsk, USSR, 1987).

R. W. Schaefer, “Calculations of the light scattered by randomly oriented ensembles of spheroids of size comparable to the wavelength,” Ph.D. dissertation (State University of New York, Albany, N.Y., 1980).

J. F. de Haan, “Effects of aerosols on the brightness and polarization of cloudless planetary atmospheres,” Ph.D. dissertation (Free University, Amsterdam, 1987).

J. F. de Haan, Free University, Amsterdam, The Netherlands (personal communication).

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, Oxford, 1973), Vol. 7, pp. 97–157.

P. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. dissertation (University of California, Los Angeles, Los Angeles, Calif., 1973).

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Tables (6)

Tables Icon

Table 1 Expansion Coefficients for Model 1

Tables Icon

Table 2 Expansion Coefficients for Model 2

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Table 3 Elements of the Scattering Matrix for Model 1

Tables Icon

Table 4 Elements of the Scattering Matrix for Model 2

Tables Icon

Table 5 Computed Values of Extinction Efficiency Factor Qext, Scattering Efficiency Factor Qsca, Absorption Efficiency Factor Qabs, Albedo for Single Scattering w, and Asymmetry Parameter of the Phase Function 〈cos θ

Tables Icon

Table 6 Computer Times for Calculating the T(A) Matrix, tT, the Expansion Coefficients, ta and the Elements of the Scattering Matrix for One Value of the Scattering Angle tFa

Equations (140)

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

E i ( r ) = ( E 1 i θ ^ i + E 2 i φ ^ i ) exp ( i k n ^ i r ) ,
E s ( r ) = E 1 s ( r , n ^ s ) θ ^ s + E 2 s ( r , n ^ s ) φ ^ s ,             n ^ s = r / r , E s ( r ) · r = 0 , [ E 1 s E 2 s ] = exp ( i k r ) / r S ( n ^ s ; n ^ j ) [ E 1 i E 2 i ] ,
[ E + 1 E - 1 ] = 1 / 2 [ 1 i 1 - i ] [ E 1 E 2 ] = 1 / 2 [ E 1 + i E 2 E 1 - i E 2 ] .
C = [ C + 1 + 1 C + 1 - 1 C - 1 + 1 C - 1 - 1 ] = 1 2 [ S 11 - i S 12 + i S 21 + S 22 S 11 + i S 12 + i S 21 - S 22 S 11 - i S 12 - i S 21 - S 22 S 11 + i S 12 - i S 21 + S 22 ] ,
I = E 1 E 1 * + E 2 E 2 * ,
Q = E 1 E 1 * - E 2 E 2 * ,
U = - E 1 E 2 * - E 2 E 1 * ,
V = i ( E 2 E 1 * - E 1 E 2 * )
I 2 = E - 1 E + 1 * = ½ ( Q + i U ) ,
I 0 = E + 1 E + 1 * = ½ ( I + V ) ,
I - 0 = E - 1 E - 1 * = ½ ( I - V ) ,
I - 2 = E + 1 E - 1 * = ½ ( Q - i U ) ,
I S = ( I , Q , U , V ) T = [ I Q U V ] ,
I C = ( I 2 , I 0 , I - 0 , I - 2 ) T = [ I 2 I 0 I - 0 I - 2 ] ,
I s S ( n ^ s ) = ( 1 / r 2 ) Z S ( n ^ s ; n ^ i ) I i S ( n ^ i ) ,
I s C ( n ^ s ) = ( 1 / r 2 ) Z C ( n ^ s ; n ^ i ) I i C ( n ^ i ) ,
Z C = Z p q C = [ C - 1 - 1 C + 1 + 1 * C - 1 + 1 C + 1 + 1 * C - 1 - 1 C + 1 - 1 * C - 1 + 1 C + 1 - 1 * C + 1 - 1 C + 1 + 1 * C + 1 + 1 C + 1 + 1 * C + 1 - 1 C + 1 - 1 * C + 1 + 1 C + 1 - 1 * C - 1 - 1 C - 1 + 1 * C - 1 + 1 C - 1 + 1 * C - 1 - 1 C - 1 - 1 * C - 1 + 1 C - 1 - 1 * C + 1 - 1 C - 1 + 1 * C + 1 + 1 C - 1 + 1 * C + 1 - 1 C - 1 - 1 * C + 1 + 1 C - 1 - 1 * ] ,             p , q = 2 , 0 , - 0 , - 2.
Z S = A - 1 Z C A ,
A = 1 2 [ 0 1 i 0 1 0 0 1 1 0 0 - 1 0 1 - i 0 ] ,             A - 1 = [ 0 1 1 0 1 0 0 1 - i 0 0 i 0 1 - 1 0 ] .
Z S , C ( n ^ s ; n ^ i ) = N Z S , C ( n ^ s ; n ^ i ) ,
Z S , C ( n ^ s ; n ^ i ) = 1 N n = 1 N Z n S , C ( n ^ s ; n ^ i ) ,
Z S , C ( n ^ s ; n ^ i ) = 1 8 π 2 0 2 π d α 0 π d β × sin β 0 2 π d γ Z S , C ( n ^ s ; n ^ i ; α β γ ) ,
F S , C ( θ ) = 4 π C sca Z S , C ( θ , 0 ; 0 , 0 ) ,
C sca = 4 π d Ω Z 11 S ( θ , 0 ; 0 , 0 ) .
1 4 π 4 π d Ω F 11 S ( θ ) = 1.
F C = F p q C = 1 2 [ a 2 + a 3 b 1 + i b 2 b 1 - i b 2 a 2 - a 3 b 1 + i b 2 a 1 + a 4 a 1 - a 4 b 1 - i b 2 b 1 - i b 2 a 1 - a 4 a 1 + a 4 b 1 + i b 2 a 2 - a 3 b 1 - i b 2 b 1 + i b 2 a 2 + a 3 ] ,             p , q = 2 , 0 , - 0 , - 2 ,
F p q C ( θ ) = s = max ( p , q ) g p q s P p q s ( cos θ ) ,             p , q = 2 , 0 , - 0 , - 2 ,
g p q s = 2 s + 1 2 - 1 + 1 d ( cos θ ) F p q C ( θ ) P p q s ( cos θ ) .
g p q s = g q p s = g - p - q s ,
g p p s , g p - p s = real ,             g 20 s = [ g 2 - 0 s ] * .
F S = [ a 1 b 1 0 0 b 1 a 2 0 0 0 0 a 3 b 2 0 0 - b 2 a 4 ] .
a 1 ( θ ) = s = 0 a 1 s P 00 s ( cos θ ) ,
a 2 ( θ ) + a 3 ( θ ) = s = 2 ( a 2 s + a 3 s ) P 22 s ( cos θ ) ,
a 2 ( θ ) - a 3 ( θ ) = s = 2 ( a 2 s - a 3 s ) P 2 - 2 s ( cos θ ) ,
a 4 ( θ ) = s = 0 a 4 s P 00 s ( cos θ ) ,
b 1 ( θ ) = s = 2 b 1 s P 02 s ( cos θ ) ,
b 2 ( θ ) = s = 2 b 2 s P 02 s ( cos θ ) ,
a 1 s = g 00 s + g 0 - 0 s ,
a 2 s = g 22 s + g 2 - 2 s ,
a 3 s = g 22 s - g 2 - 2 s ,
a 4 s = g 00 s - g 0 - 0 s ,
b 1 s = 2 Re g 02 s ,
b 2 s = 2 Im g 02 s .
a 1 0 = 1.
E i ( r ) = n = 1 m = - n n [ a m n Rg M m n ( k r ) + b m n Rg N m n ( k r ) ] ,
E s ( r ) = n = 1 m = - n n [ p m n M m n ( k r ) + q m n N m n ( k r ) ] ,             r > r 0 ,
M m n ( k r ) = ( - 1 ) m d n h n ( 1 ) ( k r ) C m n ( θ ) exp ( i m φ ) ,
N m n ( k r ) = ( - 1 ) m d n { n ( n + 1 ) k r h n ( 1 ) ( k r ) P m n ( θ ) + 1 k r [ k r h n ( 1 ) ( k r ) ] B m n ( θ ) } exp ( i m φ ) ,
B m n ( θ ) = θ ^ d d θ d 0 m n ( θ ) + φ ^ i m sin θ d 0 m n ( θ ) ,
C m n ( θ ) = θ ^ i m sin θ d 0 m n ( θ ) - φ ^ d d θ d 0 m n ( θ ) ,
P m n ( θ ) = r ^ d 0 m n ( θ ) ,
d n = [ 2 n + 1 4 π n ( n + 1 ) ] 1 / 2 ,
d m n n ( θ ) = i m - m P m m n ( cos θ ) .
P m n = n = 1 m = - n n ( T m n m n 11 a m n + T m n m n 12 b m n ) ,
q m n = n = 1 m = - n n ( T m n m n 21 a m n + T m n m n 22 b m n ) .
E i ( r ) = E i exp ( i k n ^ i r ) ,
a m n = 4 π ( - 1 ) m i n d n C m n * ( θ i ) E i exp ( - i m φ i ) ,
b m n = 4 π ( - 1 ) m i n - 1 d n B m n * ( θ i ) E i exp ( - i m φ i ) .
h n ( 1 ) ( k r ) ( - i ) n + 1 exp ( i k r ) k r ,             k r n 2 ,
S ( n ^ s ; n ^ i ) = 4 π k n m n m i n - n - 1 ( - 1 ) m + m d n d n exp [ i ( m φ s - m φ i ) ] × { [ T m n m n 11 C m n ( θ s ) + T m n m n 21 i B m n ( θ s ) ] C m n * ( θ i ) + [ T m n m n 12 C m n ( θ s ) + T m n m n 22 i B m n ( θ s ) ] B m n * ( θ i ) / i } .
M m n ( k r , θ 1 , φ 1 ) = m = - n n D m m n ( α β γ ) M m n ( k r , θ 2 , φ 2 ) ,
D m m n ( α β γ ) = exp ( - i m α ) d m m n ( β ) exp ( - i m γ )
T 2 m n m n i j = m 1 = - n n m 2 = - n n D m 2 m - 1 n ( α β γ ) × T 1 m 1 n m 2 n i j D m m 1 n ( α β γ ) ,             i , j = 1 , 2 ,
D m 2 m - 1 n ( α β γ ) = [ D m m 2 n ( α β γ ) ] * = D m 2 m n ( - γ - β - α ) .
m = - n n d m m 1 n ( β ) d m m 2 n ( β ) = δ m 1 m 2 ,
n = 1 m = - n n T 2 m n m n i j = n = 1 m = - n n T 1 m n m n i j ,             i , j = 1 , 2.
C ext = - 2 π k 2 Re n = 1 m = - n n ( T m n m n 11 + T m n m n 22 ) ,
r ( θ , φ ) = r ( θ ) ,
m r ( r , θ , φ ) = m r ( r , θ ) .
S ( n ^ s ; n ^ i ) = S ( θ s , θ i , φ s - φ i ) ,
S ( θ s , θ i , φ s - φ i ) = QS ( θ s , θ i , φ i - φ s ) Q ,
d m m n ( θ ) = ( - 1 ) m + m d - m - m n ( θ ) ,
T m n m n i j ( A ) = δ m m T m n n i j ( A ) ,
T m n n i j ( A ) = ( - 1 ) i + j T - m n n i j ( A ) .
m sin θ d 0 m n ( θ ) θ = 0 = δ m ± 1 ½ [ n ( n + 1 ) ] 1 / 2 ,
d d θ d 0 m n ( θ ) θ = 0 = m δ m ± 1 ½ [ n ( n + 1 ) ] 1 / 2 ,
m sin θ d 0 m n ( θ ) = ½ [ n ( n + 1 ) ] 1 / 2 [ d 1 m n ( θ ) + d - 1 m n ( θ ) ] ,
d d θ d 0 m n ( θ ) = ½ [ n ( n + 1 ) ] 1 / 2 [ d 1 m n ( θ ) - d - 1 m n ( θ ) ] ,
C + 1 + 1 ( θ , 0 ; 0 , 0 ; α β γ ) = n = 1 n = 1 m = - n n t m n n d - 1 m n ( θ ) × [ T m n - 1 n 11 ( α β γ ) - T m n - 1 n 12 ( α β γ ) - T m n - 1 n 21 ( α β γ ) + T m n - 1 n 22 ( α β γ ) ] ,
C + 1 - 1 = n n m t m n n d - 1 m n [ T m n 1 n 11 + T m n 1 n 12 - T m n 1 n 21 - T m n 1 n 22 ] ,
C - 1 + 1 = n n m t m n n d 1 m n [ T m n - 1 n 11 - T m n - 1 n 12 + T m n - 1 n 21 - T m n - 1 n 22 ] ,
C - 1 - 1 = n n m t m n n d 1 m n [ T m n 1 n 11 + T m n 1 n 12 + T m n 1 n 21 + T m n 1 n 22 ] ,
t m n n = 1 2 k i n - n - 1 ( - 1 ) m + 1 [ ( 2 n + 1 ) ( 2 n + 1 ) ] 1 / 2 .
d m m n ( θ ) d m 1 m 1 n ( θ ) = n 1 = n - n n + n C n m n m 1 n 1 m + m 1 C n m n m 1 n 1 m + m 1 × d m + m 1 m + m 1 n 1 ( θ ) ,
C n 1 m 1 n 2 m 2 n m = ( - 1 ) n 1 + m 1 ( 2 n + 1 2 n 2 + 1 ) 1 / 2 C n 1 m 1 n - m n 2 - m 2 ,
C n 1 m 1 n 2 m 2 n m = ( - 1 ) n + n 1 + n 2 C n 1 - m 1 n 2 - m 2 n - m
C + 1 + 1 ( θ , 0 ; 0 , 0 ; α β γ ) = n = 1 m = - n n n 1 = m - 1 f n n 1 d - 1 - m n ( θ ) × d 1 - m 0 n 1 ( β ) exp [ - i α ( 1 - m ) ] B m n n 1 1 ,
C + 1 - 1 = m n n 1 f n n 1 d - 1 m n d m - 10 n 1 exp [ - i α ( m - 1 ) ] B m n n 1 2 ,
C - 1 + 1 = n m n 1 f n n 1 d - 1 m n d m - 0 n 1 exp [ - i α ( 1 - m ) ] B m n n 1 2 ,
C - 1 - 1 = n m n 1 f n n 1 d 1 - m n d 1 - m 0 n 1 exp [ - i α ( m - 1 ) ] B m n n 1 1 ,
B m n n 1 j = n = max ( 1 , n - n 1 ) n + n 1 C n m n 1 1 - m n 1 A n n n 1 j ,             j = 1 , 2 ,
A n n n 1 j = i n - n ( 2 n + 1 ) 1 / 2 m 1 = - M 1 M 1 C n m 1 n 1 0 n m 1 T m 1 n n j ,             M 1 = min ( n , n ) ,
T m n n 1 = T m n n 11 ( A ) + T m n n 12 ( A ) + T m n n 21 ( A ) + T m n n 22 ( A ) ,
T m n n 2 = T m n n 11 ( A ) + T m n n 12 ( A ) - T m n n 21 ( A ) - T m n n 22 ( A ) .
C n 1 m 1 n 2 m 2 n m = ( - 1 ) n 1 + n 2 + m ( 2 n + 1 ) 1 / 2 [ n 1 n 2 n m 1 m 2 - m ] .
0 π d β sin β d m m 1 n ( β ) d m m 1 n ( β ) = δ n n 2 2 n + 1 ,
g 00 s = n = 1 n ^ = max ( 1 , n - s ) n + s h s n n ^ C n 1 s 0 m = - M M C n m s 0 n ^ m D m n n ^ 00 ,
g 0 - 0 s = n n ^ h s n n ^ ( - 1 ) n + n ^ + s C n 1 s 0 n ^ 1 m = - M M C n m s 0 n ^ m D m n n ^ 0 - 0 ,
g 22 s = n n ^ h s n n ^ C n - 1 s 2 n ^ 1 m = m min m max C n - m s 2 n ^ 2 - m D m n n ^ 22 ,
g 2 - 2 s = n n ^ h s n n ^ ( - 1 ) n + n ^ + s C n - 1 s 2 n ^ 1 m = m min m max C n - m s 2 n ^ 2 - m D m n n ^ 2 - 2 ,
g 02 s = - n n ^ h s n n ^ C n 1 s 0 n ^ 1 m = m min m max C n - m s 2 n ^ 2 - m D m n n ^ 02 ,
h s n n ^ = ( 2 s + 1 ) π k 2 C sca ( 2 n + 1 2 n ^ + 1 ) 1 / 2 ,
D m n n ^ 00 = n 1 = m - 1 ( 2 n 1 + 1 ) B m n n 1 1 ( B m n ^ n 1 1 ) * ,
D m n n ^ 0 - 0 = n 1 ( 2 n 1 + 1 ) B m n n 1 2 ( B m n ^ n 1 2 ) * ,
D m n n ^ 22 = n 1 ( 2 n 1 + 1 ) B m n n 1 1 ( B 2 - m n ^ n 1 1 ) * ,
D m n n ^ 2 - 2 = n 1 ( 2 n 1 + 1 ) B m n n 1 2 ( B 2 - m n ^ n 1 2 ) * ,
D m n n ^ 02 = n 1 ( 2 n 1 + 1 ) B m n n 1 2 ( B 2 - m n ^ n 1 1 ) * .
C n 1 m 1 00 n m = δ n n 1 δ m m 1 ,
m 1 m 2 C n 1 m 1 n 2 m 2 n m C n 1 m 1 n 2 m 2 n m = δ n n δ m m ,
n m C n 1 m 1 n 2 m 2 n m C n 1 m 1 n 2 m 2 n m = δ m 1 m 1 δ m 2 m 2 .
C sca = 2 π k 2 n = 1 n = 1 m = 0 min ( n , n ) i , j = 1 , 2 ( 2 - δ m 0 ) T m n n i j ( A ) 2 .
T m n n 11 ( A ) = - δ n n b n ,
T m n n 22 ( A ) = - δ n n a n ,
T m n n 12 ( A ) = T m n n 21 ( A ) = 0 ,
r ( θ , φ ) = a ( sin 2 θ + d 2 cos 2 θ ) - 1 / 2 ,             d = a / b ,
r ( θ , φ ) = r 0 ( 1 + E cos n θ ) .
Q ext = C ext / S ,
Q sca = C sca / S ,
Q abs = Q ext - Q sca ,
w = C sca / C ext ,
cos θ = ½ - 1 + 1 d ( cos θ ) a 1 ( θ ) cos θ = a 1 / 3 ,
F C ( θ ) = 1 2 π C sca 0 2 π d α 0 π d β sin β 0 2 π d γ Z C ( θ , 0 ; 0 , 0 ; α β γ )
P p q s ( x ) = A p q s ( 1 - x ) ( p - q ) / 2 ( 1 + x ) - ( p + q ) / 2 × d s - q d x s - q [ ( 1 - x ) s - p ( 1 + x ) s + p ]             for             s s * = max ( p , q )
P p q s ( x ) = 0             for             s < s * ,
A p q s = ( - 1 ) s - p i q - p 2 s [ ( s + q ) ! ( s - p ) ! ( s + p ) ! ( s - q ) ! ] 1 / 2 .
P 00 s ( x ) = P s ( x ) ,
P 0 q s ( x ) = i q [ ( s - q ) ! ( s + q ) ! ] 1 / 2 P s q ( x ) ,
P s ( x ) = 1 2 s s ! d s d x s ( x 2 - 1 ) s
P s q ( x ) = ( - 1 ) q 2 s s ! ( 1 - x 2 ) q / 2 d s + q d x s + q ( x 2 - 1 ) s .
P p q s ( x ) = P q p s ( x ) = P - p - q s ( x ) = ( - 1 ) p + q [ P p q s ( x ) ] *
- 1 + 1 d x P p q s ( x ) P p q s ( x ) = 2 2 s + 1 ( - 1 ) p + q δ s s .
s [ ( s + 1 ) 2 - p 2 ] 1 / 2 [ ( s + 1 ) 2 - q 2 ] 1 / 2 P p q s + 1 ( x ) = ( 2 s + 1 ) [ s ( s + 1 ) x - p q ] P p q s ( x ) - ( s + 1 ) ( s 2 - p 2 ) 1 / 2 ( s 2 - q 2 ) 1 / 2 P p q s - 1 ( x ) ,
P p q s * - 1 ( x ) = 0 ,
P p q s * ( x ) = ( - i ) p - q 2 s * [ ( 2 s * ) ! ( p - q ) ! ( p + q ) ! ] 1 / 2 × ( 1 - x ) p - q / 2 ( 1 + x ) p + q / 2 .
C n m n 1 m - m n m = 0.
C n m n 1 m - m n m = [ 4 n 2 ( 2 n + 1 ) ( 2 n - 1 ) ( n + m ) ( n - m ) ( n 1 - n + n ) ( n - n 1 + n ) ( n + n 1 - n + 1 ) ( n + n 1 + n + 1 ) ] 1 / 2 × { ( 2 m - m ) n ( n - 1 ) - m n ( n + 1 ) + m n 1 ( n 1 + 1 ) 2 n ( n - 1 ) C n m n 1 m - m n - 1 m - [ ( n - m - 1 ) ( n + m - 1 ) ( n 1 - n + n - 1 ) ( n - n 1 + n - 1 ) ( n + n 1 - n + 2 ) ( n + n 1 + n ) 4 ( n - 1 ) 2 ( 2 n - 3 ) ( 2 n - 1 ) ] 1 / 2 C n m n 1 m - m n - 2 m } .
C n m n 1 m - m n - n 1 m = ( - 1 ) n 1 + m + m [ ( n + m ) ! ( n - m ) ! ( 2 n 1 ) ! ( 2 n - 2 n 1 + 1 ) ! ( 2 n + 1 ) ! ( n 1 + m - m ) ! ( n 1 - m + m ) ! ( n - n 1 + m ) ! ( n - n 1 - m ) ! ] 1 / 2 .
C n m n 1 m - m n - n 1 m = C n 1 m - m n m n 1 - n m ,
C n m n 1 m - m n m = ( - 1 ) n + m [ ( 2 m + 1 ) ! ( n + n 1 - m ) ! ( n + m ) ! ( n 1 + m - m ) ! ( n + n 1 + m + 1 ) ! ( n - n 1 + m ) ! ( n 1 - n + m ) ! ( n - m ) ! ( n 1 - m + m ) ! ] 1 / 2 .
C n m n 1 m - m - m m = ( - 1 ) n + n 1 + m C n - m n 1 m - m - m - m ,

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