Abstract

A method other than the extended-boundary-condition method (EBCM) to compute the T matrix for electromagnetic scattering is presented. The separation-of-variables method (SVM) is used to solve the electromagnetic scattering problem for a spheroidal particle and to derive its T matrix in spheroidal coordinates. A transformation is developed for transforming the T matrix in spheroidal coordinates into the corresponding T matrix in spherical coordinates. The T matrix so obtained can be used for analytical calculation of the optical properties of ensembles of randomly oriented spheroids of arbitrary shape by use of an existing method to average over orientational angles. The optical properties obtained with the SVM and the EBCM are compared for different test cases. For mildly aspherical particles the two methods yield indistinguishable results. Small differences appear for highly aspherical particles. The new approach can be used to compute optical properties for arbitrary values of the aspect ratio. To test the accuracy of the expansion coefficients of the spheroidal functions for arbitrary arguments, a new testing method based on the completeness relation of the spheroidal functions is developed.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. M. I. Mishchenko, “Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4665 (1993).
    [CrossRef] [PubMed]
  4. M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).
  5. K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
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  6. A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
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  9. P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21,825–21,835 (1997).
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  10. P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
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  15. M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
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    [CrossRef] [PubMed]
  33. P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
    [CrossRef]
  34. M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
    [CrossRef] [PubMed]
  35. M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
    [CrossRef] [PubMed]
  36. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  37. D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
    [CrossRef] [PubMed]
  38. M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
    [CrossRef]
  39. M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–952 (1984).
    [CrossRef] [PubMed]
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  43. R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
    [CrossRef]
  44. R. H. Hackman, “Development and application of the spheroidal coordinate based T matrix solution to elastic wave scattering,” Radio Sci. 29, 1035–1049 (1994).
    [CrossRef]
  45. 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]
  46. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
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  48. J. W. Hovenier, “Multiple scattering of polarized light in planetary atmospheres,” Astron. Astrophys. 13, 7–29 (1971).
  49. J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
    [CrossRef]
  50. C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).
  51. V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
    [CrossRef]
  52. N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
    [CrossRef] [PubMed]

1998 (3)

J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

1997 (5)

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14, 2278–2289 (1997).
[CrossRef]

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21,825–21,835 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

1996 (8)

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape,” J. Geophys. Res. 101, 16,973–16,985 (1996).
[CrossRef]

N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transfer 55, 627–636 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

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

1995 (4)

J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

P. N. Francis, “Some aircraft observations of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

1994 (3)

R. H. Hackman, “Development and application of the spheroidal coordinate based T matrix solution to elastic wave scattering,” Radio Sci. 29, 1035–1049 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

1993 (2)

1992 (1)

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

1991 (1)

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 (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. Transfer 36, 401–423 (1986).
[CrossRef]

1984 (4)

M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–952 (1984).
[CrossRef] [PubMed]

R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
[CrossRef]

V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (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 (1)

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

1980 (2)

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

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

1979 (1)

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

1975 (2)

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

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

1971 (1)

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

1970 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
[CrossRef]

1965 (1)

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

1947 (1)

C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).

Arnott, W. P.

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21,825–21,835 (1997).
[CrossRef]

Asano, S.

Baier, R. V.

S. Hanish, R. V. Baier, A. L. van Buren, B. J. King, Tables of Radial Spheroidal Wave Functions (Naval Research Laboratory, Washington, D.C., 1970), Vols. 1–6.

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).

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).

Bouwkamp, C. J.

C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).

Carlson, B. E.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
[CrossRef] [PubMed]

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).

Doose, L. R.

R. A. West, M. G. Tomasko, L. R. Doose, “Optical properties of small mineral dust particles at visible-near IR wavelengths: numerical calculations and laboratory measurements,” in Eighth Conference on Atmospheric Radiation, Nashville, Tennessee, January 23–28, 1994 (American Meteorological Society, Boston, Mass., 1994), pp. 341–343.

Durney, C. H.

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Eide, H. A.

J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

Farafonov, V. G.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (1984).

Fast, P.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Feshbach, H.

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

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Francis, P. N.

P. N. Francis, “Some aircraft observations of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
[CrossRef]

Garcia, R. D. M.

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]

Giarola, A. J.

A. J. Giarola, “Dyadic Green’s functions in the prolate spheroidal coordinate system,” in IEEE Antennas and Propagation Society International Symposium, 1995 (Institute of Electrical and Electronic Engineers, New York, 1995), Vol. 2, pp. 826–829.

Hackman, R. H.

R. H. Hackman, “Development and application of the spheroidal coordinate based T matrix solution to elastic wave scattering,” Radio Sci. 29, 1035–1049 (1994).
[CrossRef]

R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
[CrossRef]

Hanish, S.

S. Hanish, R. V. Baier, A. L. van Buren, B. J. King, Tables of Radial Spheroidal Wave Functions (Naval Research Laboratory, Washington, D.C., 1970), Vols. 1–6.

Hovenier, J. W.

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

Iskander, M. F.

M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–952 (1984).
[CrossRef] [PubMed]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Kahn, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

Khlebtsov, N. G.

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

King, B. J.

S. Hanish, R. V. Baier, A. L. van Buren, B. J. King, Tables of Radial Spheroidal Wave Functions (Naval Research Laboratory, Washington, D.C., 1970), Vols. 1–6.

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]

Lacis, A. A.

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape,” J. Geophys. Res. 101, 16,973–16,985 (1996).
[CrossRef]

Lakhtakia, A.

M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–952 (1984).
[CrossRef] [PubMed]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14, 2278–2289 (1997).
[CrossRef]

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21,825–21,835 (1997).
[CrossRef]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by ice crystals of complex shapes,” in Ninth Conference on Atmospheric Radiation, Long Beach, California, February 2–7, 1997 (American Meteorological Society, Boston, Mass., 1997), pp. 373–377.

Lumme, K.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Macke, A.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape,” J. Geophys. Res. 101, 16,973–16,985 (1996).
[CrossRef]

Mackowski, D. W.

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

Mishchenko, M. I.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape,” J. Geophys. Res. 101, 16,973–16,985 (1996).
[CrossRef]

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

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

M. I. Mishchenko, “Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4665 (1993).
[CrossRef] [PubMed]

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

Morse, P. M.

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

Mueller, J.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Muinonen, K.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Nousiainen, T.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Peltoniemi, J. I.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Raschke, E.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Rossow, W. B.

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape,” J. Geophys. Res. 101, 16,973–16,985 (1996).
[CrossRef]

Sato, M.

Schulz, F. M.

F. M. Schulz, K. Stamnes, F. Weng, “VDISORT: an improved and generalized discrete ordinate radiative transfer model for polarized (vector) radiative transfer computations,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

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, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transfer 36, 401–423 (1986).
[CrossRef]

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

Stamnes, J.

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

Stamnes, K.

F. M. Schulz, K. Stamnes, F. Weng, “VDISORT: an improved and generalized discrete ordinate radiative transfer model for polarized (vector) radiative transfer computations,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1993).

Takano, Y.

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

Tomasko, M. G.

R. A. West, M. G. Tomasko, L. R. Doose, “Optical properties of small mineral dust particles at visible-near IR wavelengths: numerical calculations and laboratory measurements,” in Eighth Conference on Atmospheric Radiation, Nashville, Tennessee, January 23–28, 1994 (American Meteorological Society, Boston, Mass., 1994), pp. 341–343.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

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

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

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]

van Buren, A. L.

S. Hanish, R. V. Baier, A. L. van Buren, B. J. King, Tables of Radial Spheroidal Wave Functions (Naval Research Laboratory, Washington, D.C., 1970), Vols. 1–6.

Varadan, V. K.

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

Varadan, V. V.

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

Voshchinnikov, N. V.

N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transfer 55, 627–636 (1996).
[CrossRef]

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
[CrossRef]

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

Weng, F.

F. M. Schulz, K. Stamnes, F. Weng, “VDISORT: an improved and generalized discrete ordinate radiative transfer model for polarized (vector) radiative transfer computations,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

R. A. West, M. G. Tomasko, L. R. Doose, “Optical properties of small mineral dust particles at visible-near IR wavelengths: numerical calculations and laboratory measurements,” in Eighth Conference on Atmospheric Radiation, Nashville, Tennessee, January 23–28, 1994 (American Meteorological Society, Boston, Mass., 1994), pp. 341–343.

Wielaard, D. J.

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
[CrossRef] [PubMed]

Yamamoto, G.

Yang, P.

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21,825–21,835 (1997).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14, 2278–2289 (1997).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by ice crystals of complex shapes,” in Ninth Conference on Atmospheric Radiation, Long Beach, California, February 2–7, 1997 (American Meteorological Society, Boston, Mass., 1997), pp. 373–377.

Yeh, C.

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

Appl. Opt. (4)

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

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
[CrossRef] [PubMed]

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

Appl. Opt. (6)

Astron. Astrophys. (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).

Astron. Astrophys. (1)

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

Astrophys. Space Sci. (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Atmos. Res. (1)

K. N. Liou, Y. Takano, “Light scattering by nonspherical particles: remote sensing and climatic implications,” Atmos. Res. 31, 271–298 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

J. Geophys. Res. (1)

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21,825–21,835 (1997).
[CrossRef]

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

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14, 2278–2289 (1997).
[CrossRef]

J. Acoust. Soc. Am. (1)

R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

J. Atmos. Sci. (2)

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

P. N. Francis, “Some aircraft observations of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
[CrossRef]

J. Geophys. Res. (3)

M. I. Mishchenko, W. B. Rossow, A. Macke, A. A. Lacis, “Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape,” J. Geophys. Res. 101, 16,973–16,985 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16,831–16,847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16,989–17,013 (1997).

J. Math. Phys. (1)

C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).

J. Opt. Soc. Am. A (2)

J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

H. A. Eide, J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transfer (1)

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

J. Quant. Spectrosc. Radiat. Transfer (1)

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transfer 55, 627–636 (1996).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (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. Transfer 36, 401–423 (1986).
[CrossRef]

Phys. Rev. D (2)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
[CrossRef]

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

Proc. IEEE (1)

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

Pure Appl. Opt. (2)

J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips, and slits,” Pure Appl. Opt. 4, 841–855 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

Radio Sci. (1)

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]

Radio Eng. Electron. Phys. (1)

V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (1984).

Radio Sci. (1)

R. H. Hackman, “Development and application of the spheroidal coordinate based T matrix solution to elastic wave scattering,” Radio Sci. 29, 1035–1049 (1994).
[CrossRef]

Other (9)

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

S. Hanish, R. V. Baier, A. L. van Buren, B. J. King, Tables of Radial Spheroidal Wave Functions (Naval Research Laboratory, Washington, D.C., 1970), Vols. 1–6.

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

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1993).

A. J. Giarola, “Dyadic Green’s functions in the prolate spheroidal coordinate system,” in IEEE Antennas and Propagation Society International Symposium, 1995 (Institute of Electrical and Electronic Engineers, New York, 1995), Vol. 2, pp. 826–829.

P. Yang, K. N. Liou, “Light scattering by ice crystals of complex shapes,” in Ninth Conference on Atmospheric Radiation, Long Beach, California, February 2–7, 1997 (American Meteorological Society, Boston, Mass., 1997), pp. 373–377.

F. M. Schulz, K. Stamnes, F. Weng, “VDISORT: an improved and generalized discrete ordinate radiative transfer model for polarized (vector) radiative transfer computations,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

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

R. A. West, M. G. Tomasko, L. R. Doose, “Optical properties of small mineral dust particles at visible-near IR wavelengths: numerical calculations and laboratory measurements,” in Eighth Conference on Atmospheric Radiation, Nashville, Tennessee, January 23–28, 1994 (American Meteorological Society, Boston, Mass., 1994), pp. 341–343.

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Figures (8)

Fig. 1
Fig. 1

Prolate spheroid with aspect ratio a/ b = 2 [left, cases (1) and (3)], oblate spheroid with aspect ratio a/ b = 2 [middle, cases (2) and (4)], and prolate spheroid with aspect ratio a/ b = 8.33 [right, case (5)].

Fig. 2
Fig. 2

Difference between the elements of the Stokes scattering matrix computed with the EBCM and the SVM for test case (1).

Fig. 3
Fig. 3

As in Fig. 2 but for test case (2).

Fig. 4
Fig. 4

As in Fig. 2 but for test case (3).

Fig. 5
Fig. 5

As in Fig. 2 but for test case (4).

Fig. 6
Fig. 6

As in Fig. 2 but for test case (5).

Fig. 7
Fig. 7

Elements of the Stokes scattering matrix for test case (4) computed with the EBCM (solid curves) and the SVM (dashed curves).

Fig. 8
Fig. 8

As in Fig. 7 but for test case (5).

Tables (1)

Tables Icon

Table 1 Comparison of the T-Matrix and Derived Optical Properties Obtained with the SVM and the EBCM in the Five Test Cases

Equations (149)

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

E i = n = 1 m = - n n a n , m 1 M n , m 1 k 2 r + a n , m 2 N n , m 1 k 2 r ,
E s = n = 1 m = - n n p n , m 1 M n , m 3 k 2 r + p n , m 2 N n , m 3 k 2 r ,
E int = n = 1 m = - n n b n , m 1 M n , m 1 k 1 r + b n , m 2 N n , m 1 k 1 r ,
ψ n , m j = P n m cos   θ z n j kr exp im ϕ ,     j = 1 , ,   4 ,
p n , m k = n = 1 m = - n n k = 1 2   T n , m , n , m k , k a n , m k
p = T · a ,
a = Q 1 · b ,
p = Q 2 · b = Q 2 · Q 1 - 1 · a .
T = Q 2 · Q 1 - 1 .
E i = n = 1 m = - n n α n , m 1 V n , m 1 + α n , m 2 W n , m 1 ,
E s = n = 1 m = - n n κ n , m 1 V n , m 3 + κ n , m 2 W n , m 3 ,
κ n , m k = k = 1 2 n = 1 m = - n n   T n , m , n , m k , k α n , m k .
α n , m k j = δ n , n j δ m , m j δ k , k j ,     k j = 1 ,   2 , m j = 0 ,   1 , ,     n j = m j ,   m j + 1 , ,
T n , m , n j , m j k , k j = κ n , m k j .
E j i = V n j , m j 1 if   k j = 1 W n j , m j 1 if   k j = 2 .
T n 1 , m , n 2 , m k , k = i n 1 - n 2 n 1 n 1 + 1 n 2 n 2 + 1 1 / 2 n = | m | , | m | + 1   n = | m | , | m | + 1 × N m , n N m , n 1 / 2   i n - n f n 1 - | m | m , n c f n 2 - | m | m , n * c T n , m , n , m k , k .
C ext = - 2 π / k 2 Re n = 1 m = - n n T m , n , m , n 1,1 + T m , n , m , n 2,2 ,
C ext = - 2 π / k 2 ReSp T .
Sp T = Sp 1 - F T
C ext = - 2 π / k 2 ReSp 1 - F T .
δ 2 C ext   : =   C ext spherical - C ext spheroidal C ext spherical ,
n = | m | , | m | + 1   f s m , n * c f t m , n c = 0
n = | m | , | m | + 1   f s m , n * c f s m , n c = 1 .
δ 1 C ext   : =   C ext SVM - C ext EBCM C ext SVM ,
F s Θ = F 11 F 12 0 0 F 12 F 22 0 0 0 0 F 33 F 34 0 0 - F 34 F 44 .
x = d 2 1 - η 2 ξ 2 ± 1 1 / 2   cos   ϕ ,
y = d 2 1 - η 2 ξ 2 ± 1 1 / 2   sin   ϕ ,
z = d / 2 η ξ ,
c ξ kr ,
η cos   θ ,
ξ ˆ r ˆ
η ˆ - θ ˆ ,
Φ m ϕ = exp im ϕ ,
Φ e , m ϕ = cos   m ϕ ,
Φ o , m ϕ = sin   m ϕ .
φ n , m j c ;   η ,   ξ ,   ϕ = S n , m c ;   η R n , m j c ;   ξ exp im ϕ
ξ i ξ ,
c - ic .
S n , m c ;   η = r = 0,1   d r m , n c P | m | + r m η ,
d r m , n 0 = const .   δ | m | + r , n ,
S n , m 0 ,   η = P n m η .
- 1 1 d η S n , m * c ;   η S n , m c ;   η = N m , n δ n , n ,
N m , n = r = 0,1   | m | + m + r ! | m | - m + r ! 2 2 | m | + r + 1 × d r m , n * c d r m , n c ,
f r m , n c = 1 N m , n 2 2 | m | + r + 1 | m | + m + r ! | m | - m + r ! 1 / 2 d r m , n c r   and   n - m   have   the   same   parity   and   | m | n 0 otherwise ,
r = 0,1   f r m , n * c f r m , n c = δ n , n ,
Z n , m c ;   η ,   ϕ = 1 2 π N m , n   S n , m c ;   η exp im ϕ .
| j ,   m = Y j , m θ ,   ϕ = 2 j + 1 4 π j - m ! j + m ! 1 / 2 P j m cos   θ exp im ϕ
| Z n , m = r = 0,1 f r m , n c m | + r ,   m .
| ψ = n , m   α n , m | n ,   m .
| ψ = n , m   β n , m | Z n , m
q = 0,1   α q + | m | , m f q m , n * c = β n , m .
n = 0 m = - n n   | Z n , m Z n , m | = 1 ˆ ,
n = | m | , | m | + 1   f s m , n * c f t m , n c = δ s , t .
V n , m j = × a · φ n , m j ,
W n , m j = k - 1 · × V n , m j ,
V σ , n , m j = × a · φ σ , n , m j ,
W σ , n , m j = k - 1 · × V σ , n , m j ,
V n , m j = V e , n , m j + i V o , n , m j ,
V n , - m j = - 1 m n - m ! n + m ! V e , n , m j - i V o , n , m j ,
E = n , m n , m 1 V n , m j + n , m 2 W n , m j = σ = e , o n , m η σ , n , m 1 V σ , n , m j + η σ , n , m 2 W σ , n , m j ,
n , m k = η e , n , m k - i η o , n , m k 2 ,
n , - m k = - 1 m n + m ! n - m ! η e , n , m k + i η o , n , m k 2 ,
n , 0 k = η e , n , 0 k
E i = m = 0 n = m   i n f n , m 2 ζ V e , n , m 1 k 2 r + if n , m 1 ζ W o , n , m 1 k 2 r ,
E i = m = 0 n = m   i n f n , m 1 ζ V o , n , m 1 k 2 r - if n , m 2 ζ W e , n , m 1 k 2 r .
f n , m 1 ζ = 4 m N m , n r = 0,1   d r m , n r + m r + m + 1 P m + r m cos   ζ sin   ζ ,
f n , m 2 ζ = 2 2 - δ 0 , m N m , n r = 0,1   d r m , n r + m r + m + 1 × dP m + r m cos   ζ d ζ .
E s = m = 0 n = m   i n π 1 , n , m 2 V e , n , m 3 k 2 r + i π 1 , n , m 1 W o , n , m 3 k 2 r ,     ξ > ξ 0 ,
E int = m = 0 n = m   i n χ 1 , n , m 2 V e , n , m 1 k 1 r + i χ 1 , n , m 1 W o , n , m 1 k 1 r ,     ξ < ξ 0
E s = m = 0 n = m   i n π 2 , n , m 1 V o , n , m 3 k 2 r - i π 2 , n , m 2 W e , n , m 3 k 2 r ,     ξ > ξ 0 ,
E int = m = 0 n = m   i n χ 2 , n , m 1 V o , n , m 1 k 1 r - i χ 2 , n , m 2 W e , n , m 1 k 1 r ,     ξ < ξ 0 ,
E η i + E η s = E η int     at   ξ = ξ 0 ,
E ϕ i + E ϕ s = E ϕ int     at   ξ = ξ 0 ,
A k m · x k m = b k m ,     k = 1 ,   2 ,
x k m = π k , m , m 1 ,   π k , m + 1 , m 1 , ,   π k , m , m 2 ,   π k , m + 1 , m 2 , , χ k , m , m 1 ,   χ k , m + 1 , m 1 , ,   χ k , m , m 2 ,   χ k , m + 1 , m 2 , T
f n , m 1 = 0 ,     f n , m 2 = i - n 0 δ n , n 0 δ m , m 0 corresponds   to   E i = V e , n 0 , m 0 1 ,
f n , m 1 = i - n 0 δ n , n 0 δ m , m 0 ,     f n , m 2 = 0 corresponds   to   E i = V o , n 0 , m 0 1 ,
f n , m 1 = 0 ,     f n , m 2 = - i - n 0 + 1 δ n , n 0 δ m , m 0 corresponds   to   E i = W e , n 0 , m 0 1 ,
f n , m 1 = i - n 0 + 1 δ n , n 0 δ m , m 0 ,     f n , m 2 = 0 corresponds   to   E i = W o , n 0 , m 0 1 .
i n π 1 , n , m 0 2 = T e , n , m 0 , e , n 0 , m 0 1,1 ,
i n + 1 π 1 , nm 0 1 = T o , n , m 0 , e , n 0 , m 0 2,1 ;
i n π 2 , n , m 0 1 = T o , n , m 0 , o , n 0 , m 0 1,1 ,
- i n + 1 π 2 , n , m 0 2 = T e , n , m 0 , o , n 0 , m 0 2,1 ;
i n π 2 , n , m 0 1 = T o , n , m 0 e , n 0 , m 0 1,2 ,
- i n + 1 π 2 , n , m 0 2 = T e , n , m 0 , e , n 0 , m 0 2,2 ;
i n π 1 , n , m 0 2 = T e , n , m 0 , o , n 0 , m 0 1,2 ,
i n + 1 π 1 , n , m 0 1 = T o , n , m 0 , o , n 0 , m 0 2,2 .
T σ , n , m , σ , n , m k , k = δ m , m T σ , n , m , σ , n , m k , k .
T e , n , m , o , n , m 1,1 = T o , n , m , e , n , m 1,1 = T e , n , m , o , n , m 2,2 = T o , n , m , e , n , m 2,2 = T enmen m 1,2 = T onmon m 1,2 = T enmen m 2,1 = T onmon m 2,1 = 0 n ,   m ,   n .
E i = n = 0 m = - n n β n , m 1 V n , m 1 + β n , m 2 W n , m 1 ,
E s = n = 0 m = - n n ρ n , m 1 V n , m 3 + ρ n , m 2 W n , m 3 ,
ρ n , m k = k = 1 2 n = 0 m = - n n   T n , m , n , m k , k β n , m k .
T n , m , n , m k , k = 1 2 T e , n , m , e , n , m k , k + i T e , n , m , o , n , m k , k - i T o , n , m , e , n , m k , k + T o , n , m , o , n , m k , k ,
T n , - m , n , m k , k = - 1 m 2 T e , n , m , e , n , m k , k + i T e , n , m , o , n , m k , k + i T o , n , m , e , n , m k , k - T o , n , m , o , n , m k , k ,
T n , m , n , - m k , k = - 1 m 2 T e , n , m , e , n , m k , k - i T e , n , m , o , n , m k , k - i T o , n , m , e , n , m k , k - T o , n , m , o , n , m k , k ,
T n , - m , n , - m k , k = - 1 m + m 2 T e , n , m , e , n , m k , k - i T e , n , m , o , n , m k , k + i T o , n , m , e , n , m k , k + T o , n , m , o , n , m k , k
T n , 0 , n , 0 k , k = T e , n , 0 , e , n , 0 k , k .
T n , - m , n , - m k , k = - 1 k + k T nmn m k , k .
T nmn m k , k = δ m , m T nmn m k , k .
V n , m ; η 3 - - i n + 1 imS n , m c ;   θ sin   θ exp ikr kr exp im ϕ ,
V n , m ; ϕ 3 - - i n + 1 dS n , m c ;   θ d θ exp ikr kr exp im ϕ ;
M n , m ; θ 3 - i n + 1 γ n , m imP n m θ sin   θ exp ikr kr exp im ϕ ,
M n , m ; ϕ 3 - - i n + 1 γ n , m dP n m θ d θ exp ikr kr exp im ϕ ,
γ n , m = 2 n + 1 4 π n n + 1 n - m ! n + m ! 1 / 2 .
- V n , m ; η 3 = V n , m ; θ 3 = - i n r = 0,1   i | m | + r γ | m | + r , m   d r m , n c M | m | + r , m ; θ 3 .
E ξ i E η i E ϕ i = n , m α n , m 1 0 V n , m ; η 1 V n , m ; ϕ 1 + α n , m 2 0 W n , m ; η 1 W n , m ; ϕ 1 = E r i - E θ i E ϕ i ,
E r i E θ i E ϕ i = n , m a n , m 1 0 M n , m ; θ 1 M n , m ; ϕ 1 + a n , m 2 0 N n , m ; θ 1 N n , m ; ϕ 1
n = 1 m = - n n a n , m 1 M n , m 1 + a n , m 2 N n , m 1 = n = 0 m = - n n q = | m | , | m | + 1 × i q - n γ q , m   d q - | m | m , n c α n , m 1 M q , m 1 + α n , m 2 N q , m 1 .
a n , m k = s = | m | , | m | + 1   i n - s γ n , m   d n - | m | ms c α s , m k
a n , m k = s = | m |   i n - s 2 π n n + 1 N m , s 1 / 2 f n - | m | m , s α s , m k ,
a k m = a n , m k = a M m , m k ,   a M m + 1 , m k , T ,
α k m = α n , m k = α | m | , m k ,   α | m | + 1 , m k , T
G 0 m = g M m , | m | m g M m , | m | + 1 m   g M m + 1 , | m | m g M m + 1 , | m | + 1 m   ,
g n , s m = i n - s 2 π n n + 1 N m , s 1 / 2 f n - | m | m , s ,
M m = max 1 ,   | m | .
a k m = G 0 m · α k m ,
a = a n , 0 1 ,   a n , 1 1 , ,   a n , 0 2 ,   a n , 1 2 , T , n = M m ,   M m + 1 , ,
α = α n , 0 1 ,   α n , 1 1 , ,   α n , 0 2 ,   α n , 1 2 , T , n = | m | , | m | + 1 , ,
G = diag G 0 0 ,   G 0 1 , ,   G 0 0 ,   G 0 1 , .
a = G · α .
p = G · κ .
h n , s m = i n - s 2 π s s + 1 N m , n 1 / 2   f s - | m | m , n * , n = | m | ,   | m | + 1 , , s = M m ,   M m + 1 ,
H = diag H 0 0 ,   H 0 1 , ,   H 0 0 ,   H 0 1 , .
G · H = 1 ,
s = M m , M m + 1   h n , s m g s , l m = δ n , l
s = M 0 , M 0 + 1   h n , s 0 g s , l 0 = i n - l N 0 , l N 0 , n 1 / 2 s = 1   f s 0 , n * c f s 0 , l c = i n - l N 0 , l N 0 , n 1 / 2 s = 0   f s 0 , n * c × f s 0 , l c - f 0 0 , n * c f 0 0 , l c = δ n , l - F n , l c ;   0 ,
F n , l c ;   m = i n - 1 N 0 , l N 0 , n 1 / 2 f 0 0 , n * c f 0 0 , l c n , l   even   and   m = 0 0 otherwise
H · G = 1 - F .
F · F = F .
G = G · H · G = G · 1 - F
G · F = 0 .
F · H = 0 .
T = T n , m , n , m 1,1 T n , m , n , m 1,2 T n , m , n , m 2,1 T n , m , n , m 2,2 ,
T = T n , m , n , m 1,1 T n , m , n , m 1,2 T n , m , n , m 2,1 T n , m , n , m 2,2 ,
p = T · a ,
κ = T · α .
T = G · T · H .
U ˆ α ,   β ,   γ = exp - i J ˆ z α exp - i J ˆ y β exp - i J ˆ z γ .
U ˆ P r = P r .
U ˆ M n , m j = U ˆ × a · ψ n , m j = × a · U ˆ ψ n , m j ,
J ˆ z | j ,   m = m | j ,   m ,
J ˆ 2 | j ,   m = j j + 1 | j ,   m
U ˆ α ,   β ,   γ | j ,   m = m   D m , m j α ,   β ,   γ | j ,   m ;
D m , m j α ,   β ,   γ = j ,   m | U ˆ α ,   β ,   γ | j ,   m = exp - im α d m , m j β exp - im γ ,
d m , m j β = jm | exp - i J ˆ y β | jm
T n , m , n , m k , k α ,   β ,   γ = m 1 = - n n m 2 = - n n   D m , m 1 n α ,   β ,   γ × T n , m 1 , n , m 2 k , k 0 ,   0 ,   0 D - 1 m 2 , m n α ,   β ,   γ .
U ˆ α ,   β ,   γ | Z n , m = n , m   D m , m n , n α ,   β ,   γ | Z n , m ,
D m , m n , n α ,   β ,   γ = Z n , m | U ˆ α ,   β ,   γ | Z n , m = q = | m | , | m | + 1   f q - | m | m , n * c f q - | m | m , n c D m , m q α ,   β ,   γ .
T n , m , n , m k , k α ,   β ,   γ = n 1 , m 1 n 2 , m 2   D m , m 1 n , n 1 α ,   β ,   γ × T n 1 , m 1 , n 2 , m 2 k , k 0 ,   0 ,   0 D - 1 m 2 , m n 2 , n α ,   β ,   γ ,

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