Abstract

The transverse wave condition is not applicable to the refracted electromagnetic wave within the context of geometrical optics when absorption is involved. Either the TM or the TE wave condition can be assumed for the wave to locally satisfy the electromagnetic boundary condition in a ray-tracing calculation. The assumed wave mode affects both the reflection and the refraction coefficients. As a result, nonunique solutions for these coefficients are inevitable. In this study the appropriate solutions for the Fresnel reflection–refraction coefficients are identified in light-scattering calculations based on the ray-tracing technique. In particular, a 3 × 2 refraction or transmission matrix is derived to account for the inhomogeneity of the refracted wave in an absorbing medium. An asymptotic solution that completely includes the effect of medium absorption on Fresnel coefficients is obtained for the scattering properties of a general polyhedral particle. Numerical results are presented for hexagonal plates and columns with both preferred and random orientations.

© 2001 Optical Society of America

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2000

D. L. Mitchell, “Parameterization of the Mie extinction and absorption coefficients for water clouds,” J. Atmos. Sci. 57, 1311–1326 (2000).
[CrossRef]

1999

A. J. Baran, S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transfer 63, 499–519 (1999).
[CrossRef]

Q. Han, W. B. Rossow, J. Chou, K-S. Kuo, R. M. Welch, “The effect of aspect ratio and surface roughness on satellite retrieval of ice-cloud properties,” J. Quant. Spectrosc. Radiat. Transfer 63, 559–583 (1999).
[CrossRef]

M.-D. Chou, K-T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmosphere models,” J. Climate 12, 159–169 (1999).
[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]

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (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, 31,697–31,709 (1999).
[CrossRef]

1998

Q. Fu, P. Yang, W. B. Sun, “An accurate parameterization of the infrared radiative properties of cirrus clouds for climate models,” J. Climate 11, 2223–2237 (1998).
[CrossRef]

P. Yang, K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223–248 (1998).

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effect and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Res. 103, 1799–1805 (1998).
[CrossRef]

1997

1996

1995

1994

B. A. Baum, R. F. Arduini, B. A. Wielicki, P. Minnis, S.-C. Tsay, “Multilevel cloud retrieval using multispectral HIRS and AVHRR data: nighttime oceanic analysis,” J. Geophys. Res. 99, 5499–5514 (1994).
[CrossRef]

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

1993

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, 623–625 (1993).

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. I. Parameterization of radiance fields,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

1992

M. D. King, Y. J. Kaufman, W. P. Menzel, D. Tanre, “Remote sensing of cloud, aerosol, and water vapor properties from the Moderate Resolution Imaging Spectrometer (MODIS),” IEEE Trans. Geosci. Remote Sens. 30, 2–26 (1992).
[CrossRef]

1991

1989

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

K. Muinonen, K. Lumme, J. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

K.-D. Rockwitz, “Scattering properties of horizontally oriented ice crystal columns in cirrus clouds,” Appl. Opt. 28, 4103–4110 (1989).
[CrossRef] [PubMed]

1985

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Clim. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

1984

1980

1973

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1971

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

1908

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Arduini, R. F.

B. A. Baum, R. F. Arduini, B. A. Wielicki, P. Minnis, S.-C. Tsay, “Multilevel cloud retrieval using multispectral HIRS and AVHRR data: nighttime oceanic analysis,” J. Geophys. Res. 99, 5499–5514 (1994).
[CrossRef]

Arking, A.

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Clim. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

Arnott, W. P.

Asano, S.

Baran, A. J.

A. J. Baran, S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transfer 63, 499–519 (1999).
[CrossRef]

Baum, B. A.

B. A. Baum, R. F. Arduini, B. A. Wielicki, P. Minnis, S.-C. Tsay, “Multilevel cloud retrieval using multispectral HIRS and AVHRR data: nighttime oceanic analysis,” J. Geophys. Res. 99, 5499–5514 (1994).
[CrossRef]

Best, F. A.

W. L. Smith, H. E. Revercomb, R. O. Knuteson, F. A. Best, R. Dedecker, H. B. Howell, H. M. Woolf, “Cirrus cloud properties derived from high spectral resolution infrared spectrometry during FIRE II. I. The high resolution interferometer sounder (HIS) systems,” J. Atmos. Sci. 52, 4238–4245 (1995).
[CrossRef]

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).

Carlson, B. E.

Chen, Z.

Childs, J. D.

A. Arking, J. D. Childs, “Retrieval of cloud cover parameters from multispectral satellite images,” J. Clim. Appl. Meteorol. 24, 322–333 (1985).
[CrossRef]

Chou, J.

Q. Han, W. B. Rossow, J. Chou, K-S. Kuo, R. M. Welch, “The effect of aspect ratio and surface roughness on satellite retrieval of ice-cloud properties,” J. Quant. Spectrosc. Radiat. Transfer 63, 559–583 (1999).
[CrossRef]

Chou, M.-D.

M.-D. Chou, K-T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmosphere models,” J. Climate 12, 159–169 (1999).
[CrossRef]

Dedecker, R.

W. L. Smith, H. E. Revercomb, R. O. Knuteson, F. A. Best, R. Dedecker, H. B. Howell, H. M. Woolf, “Cirrus cloud properties derived from high spectral resolution infrared spectrometry during FIRE II. I. The high resolution interferometer sounder (HIS) systems,” J. Atmos. Sci. 52, 4238–4245 (1995).
[CrossRef]

Dong, Y. Y.

Draine, B. T.

Farafonov, V. G.

Flatau, P. J.

Fu, Q.

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]

M.-D. Chou, K-T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmosphere models,” J. Climate 12, 159–169 (1999).
[CrossRef]

Q. Fu, P. Yang, W. B. Sun, “An accurate parameterization of the infrared radiative properties of cirrus clouds for climate models,” J. Climate 11, 2223–2237 (1998).
[CrossRef]

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, 31,697–31,709 (1999).
[CrossRef]

Hallett, J.

Han, Q.

Q. Han, W. B. Rossow, J. Chou, K-S. Kuo, R. M. Welch, “The effect of aspect ratio and surface roughness on satellite retrieval of ice-cloud properties,” J. Quant. Spectrosc. Radiat. Transfer 63, 559–583 (1999).
[CrossRef]

Hansen, J. E.

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Havemann, S.

A. J. Baran, S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transfer 63, 499–519 (1999).
[CrossRef]

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).

Howell, H. B.

W. L. Smith, H. E. Revercomb, R. O. Knuteson, F. A. Best, R. Dedecker, H. B. Howell, H. M. Woolf, “Cirrus cloud properties derived from high spectral resolution infrared spectrometry during FIRE II. I. The high resolution interferometer sounder (HIS) systems,” J. Atmos. Sci. 52, 4238–4245 (1995).
[CrossRef]

Huffman, D. R.

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

Irvine, W. M.

Kaufman, Y. J.

M. D. King, Y. J. Kaufman, W. P. Menzel, D. Tanre, “Remote sensing of cloud, aerosol, and water vapor properties from the Moderate Resolution Imaging Spectrometer (MODIS),” IEEE Trans. Geosci. Remote Sens. 30, 2–26 (1992).
[CrossRef]

King, M. D.

M. D. King, Y. J. Kaufman, W. P. Menzel, D. Tanre, “Remote sensing of cloud, aerosol, and water vapor properties from the Moderate Resolution Imaging Spectrometer (MODIS),” IEEE Trans. Geosci. Remote Sens. 30, 2–26 (1992).
[CrossRef]

Knuteson, R. O.

W. L. Smith, H. E. Revercomb, R. O. Knuteson, F. A. Best, R. Dedecker, H. B. Howell, H. M. Woolf, “Cirrus cloud properties derived from high spectral resolution infrared spectrometry during FIRE II. I. The high resolution interferometer sounder (HIS) systems,” J. Atmos. Sci. 52, 4238–4245 (1995).
[CrossRef]

Kuo, K-S.

Q. Han, W. B. Rossow, J. Chou, K-S. Kuo, R. M. Welch, “The effect of aspect ratio and surface roughness on satellite retrieval of ice-cloud properties,” J. Quant. Spectrosc. Radiat. Transfer 63, 559–583 (1999).
[CrossRef]

Lee, K-T.

M.-D. Chou, K-T. Lee, S.-C. Tsay, Q. Fu, “Parameterization for cloud longwave scattering for use in atmosphere models,” J. Climate 12, 159–169 (1999).
[CrossRef]

Liou, K. N.

P. Yang, K. N. Liou, “Single-scattering properties of complex ice crystals in terrestrial atmosphere,” Contrib. Atmos. Phys. 71, 223–248 (1998).

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: solution by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14, 2278–2289 (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, “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. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. I. Parameterization of radiance fields,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

K. N. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Liu, Y.

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. II. Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

Lock, J. A.

Lumme, K.

Macke, A.

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effect and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Res. 103, 1799–1805 (1998).
[CrossRef]

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. II. Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

A. Macke, M. I. Mishchenko, K. Muinonen, B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20, 1934–1936 (1995).
[CrossRef] [PubMed]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

Menzel, W. P.

M. D. King, Y. J. Kaufman, W. P. Menzel, D. Tanre, “Remote sensing of cloud, aerosol, and water vapor properties from the Moderate Resolution Imaging Spectrometer (MODIS),” IEEE Trans. Geosci. Remote Sens. 30, 2–26 (1992).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Minnis, P.

B. A. Baum, R. F. Arduini, B. A. Wielicki, P. Minnis, S.-C. Tsay, “Multilevel cloud retrieval using multispectral HIRS and AVHRR data: nighttime oceanic analysis,” J. Geophys. Res. 99, 5499–5514 (1994).
[CrossRef]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. I. Parameterization of radiance fields,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effect and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Res. 103, 1799–1805 (1998).
[CrossRef]

A. Macke, M. I. Mishchenko, K. Muinonen, B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20, 1934–1936 (1995).
[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, 623–625 (1993).

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

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).

Mitchell, D. L.

D. L. Mitchell, “Parameterization of the Mie extinction and absorption coefficients for water clouds,” J. Atmos. Sci. 57, 1311–1326 (2000).
[CrossRef]

D. L. Mitchell, A. Macke, Y. Liu, “Modeling cirrus clouds. II. Treatment of radiative properties,” J. Atmos. Sci. 53, 2967–2988 (1996).
[CrossRef]

Muinonen, K.

Peltoniemi, J.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Revercomb, H. E.

W. L. Smith, H. E. Revercomb, R. O. Knuteson, F. A. Best, R. Dedecker, H. B. Howell, H. M. Woolf, “Cirrus cloud properties derived from high spectral resolution infrared spectrometry during FIRE II. I. The high resolution interferometer sounder (HIS) systems,” J. Atmos. Sci. 52, 4238–4245 (1995).
[CrossRef]

Rockwitz, K.-D.

Rossow, W. B.

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[CrossRef]

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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, 31,697–31,709 (1999).
[CrossRef]

B. A. Baum, R. F. Arduini, B. A. Wielicki, P. Minnis, S.-C. Tsay, “Multilevel cloud retrieval using multispectral HIRS and AVHRR data: nighttime oceanic analysis,” J. Geophys. Res. 99, 5499–5514 (1994).
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J. Quant. Spectrosc. Radiat. Transfer

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Figures (14)

Fig. 1
Fig. 1

Comparison of the inherent and apparent refractive indices for ice at 11 and 12 µm.

Fig. 2
Fig. 2

Scattering geometry for a large sphere with strong absorption.

Fig. 3
Fig. 3

Phase function and the degree of linear polarization computed from geometrical optics compared with the Mie solution at λ = 12 µm.

Fig. 4
Fig. 4

The same as Fig. 3, except that here the wavelength is 11 µm.

Fig. 5
Fig. 5

Comparison of the phase functions computed from Mie theory and from the geometrical-optics method for three moderate sizes at wavelengths of 11 and 12 µm.

Fig. 6
Fig. 6

Comparison of the phase function computed from the FDTD technique and from the present asymptotic theory for hexagonal ice crystals.

Fig. 7
Fig. 7

[figure omitted on the printed page]

Fig. 8
Fig. 8

Comparison of the phase function of hexagonal columns with that of equivalent spheres.

Fig. 9
Fig. 9

As for Fig. 8 but for hexagonal plates.

Fig. 10
Fig. 10

Comparison of the single-scattering albedo values computed for hexagonal ice crystals and equivalent spheres.

Fig. 11
Fig. 11

Incident and scattering geometries for an ice crystal with preferred orientation.

Fig. 12
Fig. 12

Phase functions computed for hexagonal plates with preferred orientations.

Fig. 13
Fig. 13

As for Fig. 12 but for columns.

Fig. 14
Fig. 14

Single-scattering albedo for ice crystals that have preferred orientations compared with the results for randomly oriented crystals.

Equations (53)

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Eir=Eio expikeˆi·r,
Err=Ero expikeˆr·r,
Etr=Eto expikNreˆt+iNnnˆ·r,
Nr=2-1/2mr2-mi2+sin2 ζi+mr2-mi2-sin2 ζi2+4mr2mi21/21/2,
Nn=2-1/2-mr2-mi2-sin2 ζi+mr2-mi2-sin2 ζi2+4mr2mi21/21/2,
eˆr=eˆi+2eˆi·nˆnˆ,
eˆt=eˆi/Nr+cos ζt-cos ζi/Nrnˆ,
Etleˆt=Eto exp-kNnnˆ·eˆtlexpikNrl.
Ni=Nnnˆ·eˆt=Nn cos ζt=mrmi/Nr.
αˆi=eˆi×βˆ,  αˆr=eˆr×βˆ,  αˆt=eˆt×βˆ.
Eio=Eio,ααˆi+Eio,ββˆ,
Ero=Ero,ααˆr+Ero,ββˆ.
Er=×Hr/-ikε,
Hr=×Er/ik,
Hio=eˆi×Eio,ααˆi+Eio,ββˆ=Eio,βαˆi-Eio,αβˆ,
Hro=eˆr×Ero,ααˆr+Ero,ββˆ=Ero,βαˆr-Ero,αβˆ.
Eto=Eto,ααˆt+Eto,ββˆ.
Htor=Nreˆt+iNnnˆ×Eto,ααˆt+Eto,ββˆ=Nr+iNiEto,βαˆt-Nr+iNiEto,αβˆ+iNn sin ζtEto,βeˆt.
Eio,β+Ero,β=Eto,β,
Eio,βcos ζi-Ero,βcos ζi=Nr+iNicos ζt+iNn sin2 ζtEto,β,
Eio,α+Ero,α=Nr+iNiEto,α,
Eio,αcos ζi-Ero,αcos ζi=cos ζtEto,α.
RTE,=Ero,β/Eio,β=cos ζi-Nr+iNicos ζt+iNn sin2 ζtcos ζi+Nr+iNicos ζt+iNn sin2 ζt=cos ζi-Nr cos ζt+iNncos ζi+Nr cos ζt+iNn,
RTE,=Ero,α/Eio,α=Nr+iNicos ζi-cos ζtNr+iNicos ζi+cos ζt.
RTM,=Nr+iNicos ζi-ε cos ζtNr+iNicos ζi+ε cos ζt,
RTM,=ε cos ζi-Nr cos ζt+iNnε cos ζi+Nr cos ζt+iNn.
Is=I0|R|2r2dssin θdθdφ=I0|R|2a24r2,
Is=σs4πr2 PI0,
Prθ=12|R|2+|R|2πa2σs=12|R|2+|R|2Qe-Qa,
S2S3S4S1=ka242J1ka sin θka sin θ×cos θ1+cos θ001+cos θ,
Pθ=ka28Qe-Qa2J1ka sin θka sin θ2×1+cos2 θ1+cos θ2+12|R|2+|R|2Qe-Qa.
DLPθ=Idθ-Idθ+Irθ-IrθIdθ+Idθ+Irθ+Irθ=ka22J1ka sin θ/ka sin θ21+cos θ21-cos2 θ+4|R|2-|R|2ka22J1ka sin θ/ka sin θ21+cos θ21+cos2 θ+4|R|2+|R|2,
Et,o=Et,o,ααˆt+Et,o,ββˆ+Et,o,γeˆt,
Eto,αEto,βEto,γ=Tα00TβTγ0Eio,αEio,β,
Tα=2Nr+iNicos ζiε cos ζi+Nr cos ζt+iNn,
Tβ=2 cos ζicos ζi+Nr cos ζt+iNn,
Tγ=i2Nn sin ζt cos ζiε cos ζi+Nr cos ζt+iNn.
Pθs, φs=3k2 La8πσssinkL/2sin θs cos φskL/2sin θs cos φssin3 ka/2sin θs sin φs3 ka/2sin θs sin φs2×1+cos2 θs1+cos θs2+3 πaLσsδφs-π/2+δφs+π/2δθs-2π/3]|RTM,,ζi=π/6|2+|RTE,,ζi=π/6|2,
σs=3aL2-|RTM,,ζi=π/6|2-|RTE,,ζi=π/6|2,
Esr=k2 expikr4πrε-1νEr-rˆrˆ·Er×exp-ikrˆ·rd3r,
σext=Imk|Ei|ε-1νErEird3r,
σabs=k|Ei|2 εiνEr·Eird3r,
S2eˆsS3eˆsS4eˆsS1eˆs=k2ε-14πsurfacecos ζiNr+iNi-eˆs·eˆtαˆs·αˆtαˆs·βˆαˆs·eˆsβˆs·αˆtβˆs·βˆβˆs·eˆsTα00TβTγ0αˆi·αˆ0αˆi·βˆsβˆ·αˆ0βˆ·βˆs×exp-ikeˆi-eˆs·r1-exp-kNilrexpikNr-eˆs·eˆtlrd2r,
σext=2πk2ReS1eˆi+S2eˆi,
σabs=12particle surfacecos ζtNr|Tα|2+|Tβ|2+|Tγ|2×1-exp-2kNilrd2r,
S˜2S˜3S˜4S˜1j=cos ζiNr+iNi-eˆs·eˆtαˆs·αˆtαˆs·βˆαˆs·eˆsβˆs·αˆtβˆs·βˆβˆs·eˆs×Tα00TβTγ0αˆi·αˆoαˆi·βˆsβˆ·αˆoβˆ·βˆs.
r=r0+ξr1+ηr2parallelogramr0+ξ1-ηr1+ηξr2triangle,
Di=face jexpikeˆi-eˆs·rjd2rj=|rj,1×rj,2| 0101expikeˆi-eˆsrj,0+ηrj,1+ξrj,2dηdξ=|rj,1×rj,2|expikeˆi-eˆs·rj,0+rj,1/2+rj,2/2×sinkeˆi-eˆs·rj,1/2keˆi-eˆs·rj,1/2sinkeˆi-eˆs·rj,1/2keˆi-eˆs·rj,1/2.
Di=face jexpikeˆi-eˆs·rjd2rj=|rj,1×rj,2| expikeˆi-eˆs·rj,0ikeˆi-eˆsrj,2-rj,1×expikeˆi-eˆs·rj,2/2sinkeˆi-eˆs·rj,2/2keˆi-eˆs·rj,2/2-expikeˆi-eˆs·rj,1/2sinkeˆi-eˆs·rj,1/2keˆi-eˆs·rj,1/2.
S2eˆsS3eˆsS4eˆsS1eˆs=k2ε-14πj heˆi·nˆj×DjS˜2eˆsS˜3eˆsS˜4eˆsS˜1eˆsj,
heˆi·nˆj=1 eˆi·nˆj>00 eˆi·nˆj0.
ω˜=σext-σabsσext,
re=34VA=323/2aL3/2a+L,

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