Abstract

The single-scattering properties of the Platonic shapes, namely, the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron, are investigated by use of the finite-difference time-domain method. These Platonic shapes have different extents of asphericity in terms of the ratios of their volumes (or surface areas) to those of their circumscribed spheres. We present the errors associated with four types of spherical equivalence that are defined on the basis of (a) the particle’s geometric dimension (b) equal surface area (A), (c) equal volume (V), and (d) equal-volume-to-surface-area ratio (V/A). Numerical results show that the derivations of the scattering properties of a nonspherical particle from its spherical counterpart depend on the definition of spherical equivalence. For instance, when the Platonic and spherical particles have the same geometric dimension, the phase function for a dodecahedron is more similar than that for an icosahedron to the spherical result even though an icosahedron has more faces than a dodecahedron. However, when the nonspherical and spherical particles have the same volume, the phase function of the icosahedral particle essentially converges to the phase function of the sphere, whereas the result for the dodecahedron is quite different from its spherical counterpart. Furthermore, the present scattering calculation shows that the approximation of a Platonic solid with a sphere based on V/A leads to larger errors than the spherical equivalence based on either volume or projected area.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2003

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1083–1090 (2003).
[CrossRef]

P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
[CrossRef]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

2002

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

T. Wriedt, “Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 4, 256–268 (2002).
[CrossRef]

2001

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

S. Havemann, A. J. Baran, “Extension of T-matrix to scattering of electromagnetic plane waves by non-axisymmetric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transfer 70, 139–158 (2001).
[CrossRef]

2000

1999

W. B. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
[CrossRef]

Q. Fu, W. Sun, P. Yang, “On modeling of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelengths,” J. Atmos. Sci. 56, 2937–2947 (1999).
[CrossRef]

T. C. Grenfell, S. G. Warren, “Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation,” J. Geophys. Res. 104, 31697–31709 (1999).
[CrossRef]

1998

K. Wyser, P. Yang, “Average crystal size and bulk shortwave single scattering properties in ice clouds,” Atmos. Res. 49, 315–335 (1998).
[CrossRef]

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

G. Videen, W. Sun, Q. Fu, “Light scattering from irregular tetrahedral aggregates,” Opt. Commun. 156, 5–9 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

1997

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21825–21835 (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, 16831–16847 (1997).
[CrossRef]

1996

1995

1994

J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for light calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

P. N. Francis, A. Jones, R. W. Saunders, K. P. Shine, A. Slingo, Z. Sun, “An observational and theoretical study of the radiative properties of cirrus: some results from ICE’89,” Q. J. R. Meteorol. Soc. 120, 809–848 (1994).
[CrossRef]

D. L. Mitchell, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus cloud. II. Dependence of absorption and extinction on ice crystal morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

1991

1988

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
[CrossRef] [PubMed]

J. S. Foot, “Some observations of the optical properties of clouds. II: cirrus,” Q. J. R. Meteorol. Soc. 114, 145–164 (1988).
[CrossRef]

1986

1984

1980

P. J. Barrett, “The shape of rock particles, a critical review,” Sedimentology 27, 291–303 (1980).
[CrossRef]

1966

K. S. Yee, “Numerical solution of initial boundary problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas. Propag. AP-14, 302–307 (1966).

1948

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

1758

L. Euler, “Elementa Doctrinae Solidorum,” Novi Commentarii Academiae Scientiarum Petropolitanae 4, 109–140 (1758).

Ackerman, S.

P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
[CrossRef]

Arnott, W. P.

W. P. Arnott, Y. Y. Dong, J. Hallett, “Extinction efficiency in the infrared (2–18 μm) of laboratory ice clouds: observations of scattering minima in the Christiansen bands of ice,” Appl. Opt. 34, 541–551 (1995).
[CrossRef] [PubMed]

D. L. Mitchell, W. P. Arnott, “A model predicting the evolution of ice particle size spectra and radiative properties of cirrus cloud. II. Dependence of absorption and extinction on ice crystal morphology,” J. Atmos. Sci. 51, 817–832 (1994).
[CrossRef]

Baran, A. J.

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

S. Havemann, A. J. Baran, “Extension of T-matrix to scattering of electromagnetic plane waves by non-axisymmetric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transfer 70, 139–158 (2001).
[CrossRef]

Barber, P. W.

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

Barrett, P. J.

P. J. Barrett, “The shape of rock particles, a critical review,” Sedimentology 27, 291–303 (1980).
[CrossRef]

Baum, B. A.

P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bohren, C. F.

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

Chen, Z.

Cromwell, P. R.

P. R. Cromwell, Polyhedra (Cambridge U. Press, Cambridge, UK, 1997).

Dong, Y. Y.

Draine, B. T.

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for light calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Euler, L.

L. Euler, “Elementa Doctrinae Solidorum,” Novi Commentarii Academiae Scientiarum Petropolitanae 4, 109–140 (1758).

Flatau, P. J.

Foot, J. S.

J. S. Foot, “Some observations of the optical properties of clouds. II: cirrus,” Q. J. R. Meteorol. Soc. 114, 145–164 (1988).
[CrossRef]

Francis, P. N.

P. N. Francis, A. Jones, R. W. Saunders, K. P. Shine, A. Slingo, Z. Sun, “An observational and theoretical study of the radiative properties of cirrus: some results from ICE’89,” Q. J. R. Meteorol. Soc. 120, 809–848 (1994).
[CrossRef]

Fu, Q.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1083–1090 (2003).
[CrossRef]

W. B. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
[CrossRef]

Q. Fu, W. Sun, P. Yang, “On modeling of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelengths,” J. Atmos. Sci. 56, 2937–2947 (1999).
[CrossRef]

G. Videen, W. Sun, Q. Fu, “Light scattering from irregular tetrahedral aggregates,” Opt. Commun. 156, 5–9 (1998).
[CrossRef]

Fuller, K. A.

K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 33, 4716–4731 (1991).
[CrossRef]

Gao, B. C.

Grenfell, T. C.

T. C. Grenfell, S. G. Warren, “Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation,” J. Geophys. Res. 104, 31697–31709 (1999).
[CrossRef]

Grunbaum, B.

B. Grunbaum, Convex Polytopes (Wiley, London, 1967).

Hallett, J.

Havemann, S.

S. Havemann, A. J. Baran, “Extension of T-matrix to scattering of electromagnetic plane waves by non-axisymmetric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transfer 70, 139–158 (2001).
[CrossRef]

Heymsfield, A. J.

P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
[CrossRef]

Hill, S. C.

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

Hovenier, J. W.

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 29–60.

Hu, Y. X.

P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
[CrossRef]

Huang, H.-L.

P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
[CrossRef]

Huffman, D. R.

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

Jones, A.

P. N. Francis, A. Jones, R. W. Saunders, K. P. Shine, A. Slingo, Z. Sun, “An observational and theoretical study of the radiative properties of cirrus: some results from ICE’89,” Q. J. R. Meteorol. Soc. 120, 809–848 (1994).
[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, 16831–16847 (1997).
[CrossRef]

Kahnert, F. M.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

Lacis, A. A.

M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Liou, K. N.

P. Yang, K. N. Liou, M. I. Mishchenko, B. C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[CrossRef]

P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21825–21835 (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]

K. N. Liou, An Introduction to Atmospheric Radiation, 2nd ed. (Academic, San Diego, Calif., 2002).

K. N. Liou, Y. Takano, P. Yang, “Light scattering and radiative transfer in ice crystal clouds: applications to climate research,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 417–1449.

Liu, L.

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

Loeb, N. G.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1083–1090 (2003).
[CrossRef]

Lyusternik, L. A.

L. A. Lyusternik, Convex Figures and Polyhedra (Heath, Boston, Mass., 1966).

Macke, A.

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

Mackowski, D. W.

McFarquhar, G. M.

G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
[CrossRef]

T. Nousiainen, G. M. McFarquhar, “Light scattering by small quasi-spherical ice crystals: preliminary results,” in Proceedings of the 7th Conference on Electromagnetic and Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Universitäts-Buchhandlung, Bremen, Germany, 2003), pp. 271–274.

Mischenko, M. I.

M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Mishchenko, M. I.

L. Liu, M. I. Mishchenko, “Constraints on PSC particle microphysics derived from lidar observations,” J. Quant Spectrosc. Radiat. Transfer 70, 817–831 (2001).
[CrossRef]

P. Yang, K. N. Liou, M. I. Mishchenko, B. C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[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, 16831–16847 (1997).
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Mitchell, D. L.

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W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1083–1090 (2003).
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P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. 102, 21825–21835 (1997).
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P. N. Francis, A. Jones, R. W. Saunders, K. P. Shine, A. Slingo, Z. Sun, “An observational and theoretical study of the radiative properties of cirrus: some results from ICE’89,” Q. J. R. Meteorol. Soc. 120, 809–848 (1994).
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W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1083–1090 (2003).
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P. N. Francis, A. Jones, R. W. Saunders, K. P. Shine, A. Slingo, Z. Sun, “An observational and theoretical study of the radiative properties of cirrus: some results from ICE’89,” Q. J. R. Meteorol. Soc. 120, 809–848 (1994).
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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, 16831–16847 (1997).
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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, 16831–16847 (1997).
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T. Wriedt, “Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 4, 256–268 (2002).
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T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
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K. Wyser, P. Yang, “Average crystal size and bulk shortwave single scattering properties in ice clouds,” Atmos. Res. 49, 315–335 (1998).
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P. Yang, B. A. Baum, A. J. Heymsfield, Y. X. Hu, H.-L. Huang, S.-C. Tsay, S. Ackerman, “Single-scattering properties of droxtals,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1159–1180 (2003).
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G. M. McFarquhar, P. Yang, A. Macke, A. J. Baran, “A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions,” J. Atmos. Sci. 59, 2458–2478 (2002).
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P. Yang, K. N. Liou, M. I. Mishchenko, B. C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[CrossRef]

Q. Fu, W. Sun, P. Yang, “On modeling of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelengths,” J. Atmos. Sci. 56, 2937–2947 (1999).
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K. Wyser, P. Yang, “Average crystal size and bulk shortwave single scattering properties in ice clouds,” Atmos. Res. 49, 315–335 (1998).
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W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988).
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P. Yang, K. N. Liou, M. I. Mishchenko, B. C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
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Q. Fu, W. Sun, P. Yang, “On modeling of scattering and absorption by cirrus nonspherical ice particles at thermal infrared wavelengths,” J. Atmos. Sci. 56, 2937–2947 (1999).
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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, 16831–16847 (1997).
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J. Quant Spectrosc. Radiat. Transfer

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W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1083–1090 (2003).
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T. Wriedt, “Using the T-matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 4, 256–268 (2002).
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H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), pp. 29–60.

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M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

A. Taflove, Advances in Computational Electromagnetics (Artech House, Boston, Mass., 1998).

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