Abstract

A semi-empirical high-frequency formula is developed to efficiently and accurately compute the extinction efficiencies of spheroids in the cases of moderate and large size parameters under either fixed or random orientation condition. The formula incorporates the semi-classical scattering concepts formulated by extending the complex angular momentum approximation of the Lorenz-Mie theory to the spheroid case on the basis of the physical rationales associated with changing the particle morphology from a sphere to a spheroid. The asymptotic edge-effect expansion is truncated with an optimal number of terms based on a priori knowledge obtained from comparing the semi-classical Mie extinction efficiencies with the Lorenz-Mie solutions. The present formula is fully tested in comparison with the T-matrix results for spheroids with the aspect ratios from 0.5 to 2.0, and for various refractive indices mr + imi, with mr from 1.0 to 2.0 and mi from 0 to 0.5.

© 2014 Optical Society of America

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2013 (2)

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013).
[CrossRef]

2012 (1)

L. Wang, X. Sun, J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012).
[CrossRef]

2011 (3)

K. N. Liou, Y. Takano, P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011).
[CrossRef]

P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011).
[CrossRef]

K. F. Ren, F. Onofri, C. Rozé, T. Girasole, “Vectorial complex ray model and application to two-dimensional scattering of plane wave by a spheroidal particle,” Opt. Lett. 36(3), 370–372 (2011).
[CrossRef] [PubMed]

2010 (2)

L. Bi, P. Yang, G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010).
[CrossRef] [PubMed]

K. N. Liou, Y. Takano, P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1980–1989 (2010).
[CrossRef]

2009 (1)

2007 (1)

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

2006 (1)

2003 (1)

2001 (1)

M. Z. Li, D. Wilkinson, “Particle size distribution determination from spectral extinction using evolutionary programming,” Chem. Eng. Sci. 56(10), 3045–3052 (2001).
[CrossRef]

2000 (1)

1999 (1)

A. J. Baran, S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transf. 63(2-6), 499–519 (1999).
[CrossRef]

1997 (3)

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(D18), 21825–21835 (1997).
[CrossRef]

G. Crawley, M. Cournil, D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997).
[CrossRef]

A. A. Kokhanovsky, E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. 28(1), 1–21 (1997).
[CrossRef]

1996 (1)

1994 (2)

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero–order rainbow,” Am. J. Phys. 62(12), 1082–1089 (1994).
[CrossRef]

B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33(24), 5796–5804 (1994).
[CrossRef] [PubMed]

1993 (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[CrossRef]

1991 (2)

1989 (2)

G. E. Elicable, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. 129(1), 192–200 (1989).
[CrossRef]

T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28(19), 4096–4102 (1989).
[CrossRef] [PubMed]

1988 (1)

1984 (1)

1980 (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980).
[CrossRef]

1975 (1)

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971).
[CrossRef]

1957 (1)

D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957).
[CrossRef]

1908 (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

Arnott, W. P.

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(D18), 21825–21835 (1997).
[CrossRef]

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. Transf. 63(2-6), 499–519 (1999).
[CrossRef]

Barber, P. W.

Benedetto, D. D.

G. Crawley, M. Cournil, D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997).
[CrossRef]

Bi, L.

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013).
[CrossRef]

P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010).
[CrossRef] [PubMed]

L. Bi, P. Yang, G. W. Kattawar, R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. 48(1), 114–126 (2009).
[CrossRef] [PubMed]

Cai, X.

Chen, T. W.

Chylek, P.

Cournil, M.

G. Crawley, M. Cournil, D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997).
[CrossRef]

Crawley, G.

G. Crawley, M. Cournil, D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997).
[CrossRef]

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

Dubovik, O.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

Elicable, G. E.

G. E. Elicable, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. 129(1), 192–200 (1989).
[CrossRef]

Evans, B. T. N.

Farafonov, V. G.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[CrossRef]

Feng, Q.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

Fournier, G. R.

Garcia-Rubio, L. H.

G. E. Elicable, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. 129(1), 192–200 (1989).
[CrossRef]

Girasole, T.

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. Transf. 63(2-6), 499–519 (1999).
[CrossRef]

Hill, A. C.

Hill, S. C.

Hong, G.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

Hu, Y.

P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011).
[CrossRef]

Hu, Y. Q.

Johnson, B. R.

Jones, D. S.

D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957).
[CrossRef]

Kahn, R.

Kattawar, G. W.

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013).
[CrossRef]

P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010).
[CrossRef] [PubMed]

L. Bi, P. Yang, G. W. Kattawar, R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. 48(1), 114–126 (2009).
[CrossRef] [PubMed]

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

Klett, J. D.

Kokhanovsky, A. A.

A. A. Kokhanovsky, E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. 28(1), 1–21 (1997).
[CrossRef]

Laszlo, I.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

Li, M. Z.

M. Z. Li, D. Wilkinson, “Particle size distribution determination from spectral extinction using evolutionary programming,” Chem. Eng. Sci. 56(10), 3045–3052 (2001).
[CrossRef]

Liou, K. N.

K. N. Liou, Y. Takano, P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011).
[CrossRef]

K. N. Liou, Y. Takano, P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1980–1989 (2010).
[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(D18), 21825–21835 (1997).
[CrossRef]

Lock, J. A.

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35(3), 515–531 (1996).
[CrossRef] [PubMed]

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero–order rainbow,” Am. J. Phys. 62(12), 1082–1089 (1994).
[CrossRef]

McCollum, T. A.

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero–order rainbow,” Am. J. Phys. 62(12), 1082–1089 (1994).
[CrossRef]

Mishchenko, M. I.

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[CrossRef]

P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011).
[CrossRef]

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. 39(6), 1026–1031 (2000).
[CrossRef] [PubMed]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980).
[CrossRef]

Onofri, F.

Ren, K. F.

Rozé, C.

Shen, J.

Sokolik, I. N.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007).
[CrossRef]

Sun, X.

L. Wang, X. Sun, J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012).
[CrossRef]

Takano, Y.

K. N. Liou, Y. Takano, P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011).
[CrossRef]

K. N. Liou, Y. Takano, P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1980–1989 (2010).
[CrossRef]

Voshchinnikov, N. V.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[CrossRef]

Wang, L.

L. Wang, X. Sun, J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971).
[CrossRef]

Wendisch, M.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Prolate spheroid (a) and oblate spheroid (b). The semi-axis of rotation is denoted as c . The other axis is denoted as a . The size parameters for the two axes are denoted as β c and β a .

Fig. 2
Fig. 2

Comparison of the extinction efficiencies of a sphere computed from the Lorenz-Mie theory and Eq. (6). The accuracy of the inclusion of the edge-effect terms is illustrated.

Fig. 3
Fig. 3

Shape factor associated with the divergence of the first order transmitted ray. (a) prolate spheroid. (b) oblate spheroid. Note that, for a prolate spheroid, the shape factor has delta peaks, which correspond to the forward glory.

Fig. 4
Fig. 4

A schematic diagram to illustrate the oblique incidence. The wave front blocked by the geometry is an ellipse. The red curve is a boundary separating the illuminated and shadow side. The location within the ellipse is denoted as ( η , ξ ).

Fig. 5
Fig. 5

Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formula. The incident light is aligned with the symmetric axis. 10 aspect ratios are selected for illustration. The refractive index is assumed to be 1.1 + i10−7.

Fig. 6
Fig. 6

Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formula. The refractive index is 1.1 + i0.5. (a) The extinction efficiency is plotted against the particle orientation. The aspect ratio is 0.8. Three size parameters (20, 50, and 100) are selected. (b) Similar to (a) except that the aspect ratio is 1.6. (c) The extinction efficiency is plotted against the particle aspect ratio. The size parameter is 50. Three orientations (θ = 0°, 45 o, 90 o) are selected. (d) Similar to (c) except that the size parameter is 100.

Fig. 7
Fig. 7

Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formulas. The incident light is aligned with the symmetric axis. The refractive index is 1.3 + i10−7.

Fig. 8
Fig. 8

Comparison between the extinction efficiencies of spheroids as a function of particle orientation computed from the II-TM, the FE approximation, and the present new approximate formula. The refractive index is 1.3 + i10−7. The aspect ratio and the size parameters are indicated in the figure.

Fig. 9
Fig. 9

Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formula. The aspect ratio is 0.8.

Fig. 10
Fig. 10

The ratios of the extinction efficiencies computed from the present formula and the II-TM.

Fig. 11
Fig. 11

A schematic diagram of the trajectory of the first-order refracted ray.

Tables (4)

Tables Icon

Table 1 Computational Wall Time of the II-TM for Spheroids with the End-on Orientation and Random Orientations for Different Aspect Ratios and Size Parameters

Tables Icon

Table 2 Coefficients for Edge-effect Terms in Eq. (45)a

Tables Icon

Table 3 Extinction Efficiencies of Spheroids Computed from the II-TM, the Present Formula, and the FE Formula

Tables Icon

Table 4 Extinction Efficiencies of Randomly Oriented Spheroids for Four Size Parameters Shown in Fig. 8

Equations (63)

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Q e x t = 2 β a 2 Re [ n = 1 ( 2 n + 1 ) n ' = 1 2 n ' + 1 2 n + 1 i n ' n ( T 1 n 1 n ' 11 + T 1 n 1 n ' 12 + T 1 n 1 n ' 21 + T 1 n 1 n ' 22 ) ] .
Q e x t ( θ ) = 4 π k 2 A ( θ ) Re [ S 11 ( 0 ) + S 22 ( 0 ) ] ,
A ( θ ) = π a ( c 2 sin 2 θ + a 2 cos 2 θ ) 1 / 2 .
Q e x t = 2 π k 2 A Re l = 1 ( T l l 11 + T l l 22 ) ,
< A > = { π a 2 2 ( 1 + 1 e 2 e arc tan h(e) ) , e = 1 c 2 a 2 , if a > c π a 2 2 ( 1 + c a e arc sin ( e ) ) , e = 1 a 2 c 2 , if a < c ,
Q ext = Q ext,geo + Q ext,edge ,
Q ext,geo =2 8 β Im{ m 2 ( m1 ) ( m+1 ) 2 e 2i(m1)β × [ 1 i 16β ( 51 6 + 1 2m 9m m1 ) j1 (m1) 2j+1 (m+1) 2j e 4imjβ 2jm+1 +O( β 2 ) ] },
Q ext,edge = c 1 β 2/3 + c 2 β 1 + c 3 β 4/3 + c 4 β 5/3 ,
c 1 =1.9924,
c 2 =2Im[ ( m 2 +1 )/N ],
c 3 =0.7154,
c 4 =0.6641Im[ e iπ/3 ( m 2 +1 )( 2 m 4 6 m 2 +3 ) / N 3 ].
m r , 0 = ( 1 m i , 0 2 ) + 2 1 m i , 0 2 , 0 < m i , 0 < 1.
S( Θ )=ka ( η Θ η sinΘ ) 1/2 ε,
S=ka lim η0 η Θ ε,
ε=[ 1 ( 1m 1+m ) 2 ].
S sphere =ka m 2( m1 ) ε=ka 2 m 2 ( m+1 )( m 2 1 ) .
S spheroid =ka m 2( m1 ) ε | mc/a (m1) c 3 / a 3 | .
Q ext,geo =2 8 β c ( c a )Im{ m 2 (m1) (m+1) 2 e 2i(m1) β c mc/a(m1) c 3 / a 3 × [ 1 i 16 β c ( 51 6 + 1 2m 9m m1 ) j1 (m1) 2j+1 (m+1) 2j e 4imj β c 2jm+1 +O( β c 2 ) ].
Q ext,geo =24Im[ e iρ ρ ( 1+i 9 8ρ ) ].
Q ext,geo =24Im[ e iρ ρ +i e iρ 1 ρ 2 ].
C 1 ext,edge =( 0.9962 k 2/3 ( c 2 a ) 1/3 )ds.
Q 1 ext,edge = 1.9924 β c 2/3 ( c a ) 4/3 .
Q ext,edge = c 1 β c 2/3 ( c a ) 4/3 + c 2 β c ( c a )+ c 3 β c 4/3 ( c a ) 2/3 + c 4 β c 5/3 ( c a ) 1/3 .
Q ext =24Im[ εfF m exp( ρ ) ρ ( 1+ i ρ ( 1+ m 2 +1 16m ) ) ]+ Q ext,edge .
ρ=2k(m1)a, ε=1 ( 1m 1+m ) 2 , f=1, F=1.
ρ=2k(m1)c, ε=1 ( 1m 1+m ) 2 f= [ mc/a(m1) c 3 / a 3 ] 1 , F=c/a.
x = a ¯ η cos ξ ,
y = c ¯ η sin ξ ,
a ¯ = a ,
c ¯ = c 2 sin 2 θ + a 2 cos 2 θ .
ρ = 2 a ( m cos ϕ ) { c a p [ p 2 cos ϕ + s sin ϕ p 2 cos 2 ϕ + q 2 sin 2 ϕ + s sin (2 ϕ ) ] } ,
ϕ = θ i θ t ,
cos ϕ = s 2 + p 2 Δ m ( p 4 + s 2 ) ,
sin ϕ = s ( Δ p 2 ) m ( p 4 + s 2 ) ,
Δ = [ m 2 ( p 4 + s 2 ) s 2 ] 1 / 2 ,
s = ( p 2 q 2 c 2 a 2 ) 1 / 2 ,
q = ( c 2 a 2 cos 2 ( θ ) + sin 2 ( θ ) ) 1 / 2 ,
p = ( cos 2 ( θ ) + c 2 a 2 sin 2 ( θ ) ) 1 / 2 .
ε = 1 1 2 ( r 1 2 + r 2 2 ) 1 1 2 ( r ' 1 2 + r ' 2 2 ) ,
r 1 = m cos θ i cos θ t m cos θ i + cos θ t , r 2 = cos θ i m cos θ t cos θ i + m cos θ t .
f = 2 ( m 1 ) m lim η 0 [ | Θ / η Θ / ξ Φ / η Φ / ξ | sin Θ / η ] 1 / 2 .
F = { c p [ p 2 cos ϕ + s sin ϕ p 2 cos 2 ϕ + q 2 sin 2 ϕ + s sin (2 ϕ ) ] } a c 2 sin 2 θ + a 2 cos 2 θ .
c 2 a ( sin 2 ξ + p 2 cos 2 ξ ) 1 / 2 p 3 .
Q e x t , e d g e = q 1 c 1 β c 2 / 3 ( c a ) 4 / 3 + q 2 c 2 β c ( c a ) + q 3 c 3 β c 4 / 3 ( c a ) 2 / 3 + q 4 c 4 β c 5 / 3 ( c a ) 1 / 3 .
0 π / 2 [ sin 2 ξ + p 2 cos 2 ξ ] n d ξ = { π 2 p 2 n F 2 1 ( n , 1 / 2 ; 1 ; 1 p 2 ) , p > 1 π 2 F 2 1 ( n , 1 / 2 ; 1 ; 1 p 2 ) , p < 1 .
Q ext = 0 π/2 Q ext (θ) A(θ)sinθdθ 0 π/2 A(θ)sinθdθ .
Q ext = 0 1 Q ext (μ) f(μ)dμ,
f(μ)= πac <A> [ 1+( a 2 c 2 1 ) μ 2 ] 1/2 ,
x 0 = c si n δ cos 2 δ + c 2 / a 2 sin 2 δ c δ ( 1 1 2 c 2 a 2 δ 2 ) ,
y 0 = c cos δ cos 2 δ + c 2 / a 2 sin 2 δ c ( 1 1 2 c 2 a 2 δ 2 ) .
θ i = arctan ( c 2 a 2 tan δ ) c 2 a 2 δ .
Δ d = θ i θ t ( 1 1 m ) c 2 a 2 δ .
cos Δ d 1 1 2 ( 1 1 m ) 2 c 4 a 4 δ 2 ,
sin Δ d ( 1 1 m ) c 2 a 2 δ .
L 2 c + Ο ( δ ) .
x 1 = x 0 L sin Δ d c δ [ 1 2 ( 1 1 m ) c 2 a 2 ] + Ο ( δ 2 ) ,
y 1 = y 0 L cos Δ d c + Ο ( δ 2 ) .
θ ¯ i = arctan ( c 2 a 2 x 1 y 1 ) + Δ d c 2 a 2 [ 1 2 ( 1 1 m ) c 2 a 2 ] δ + Δ d .
Θ = θ ¯ t θ ¯ i + Δ d = [ 2 ( m 1 ) c 2 a 2 2 ( m 1 ) 2 m c 4 a 4 ] δ .
η c a δ .
Θ = 2 ( m 1 ) m [ m c a ( m 1 ) c 3 a 3 ] η .
lim η0 η Θ = m 2(m1) 1 [ mc/a(m1) c 3 / a 3 ] .

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