Abstract

A new geometric-optics model has been developed for the calculation of the single-scattering and polarization properties for arbitrarily oriented hexagonal ice crystals. The model uses the ray-tracing technique to solve the near field on the ice crystal surface, which is then transformed to the far field on the basis of the electromagnetic equivalence theorem. From comparisons with the results computed by the finite-difference time domain method, we show that the novel geometric-optics method can be applied to the computation of the extinction cross section and single-scattering albedo for ice crystals with size parameters along the minimum dimension as small as ~6. Overall agreement has also been obtained for the phase function when size parameters along the minimum dimension are larger than ~20. We demonstrate that the present model converges to the conventional ray-tracing method for large size parameters and produces single-scattering results close to those computed by the finite-difference time domain method for size parameters along the minimum dimension smaller than ~20. The present geometric-optics method can therefore bridge the gap between the conventional ray-tracing and the exact numerical methods that are applicable to large and small size parameters, respectively.

© 1996 Optical Society of America

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References

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    [CrossRef]
  2. A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
    [CrossRef]
  3. L. M. Miloshevich, A. J. Heymsfied, P. M. Norris, “Microphysical measurements in cirrus clouds from ice crystals replicator sonders launched during FIRE II,” in Proceedings of the 11th International Conference on Clouds and Precipitation (17–21 August 1992), Vol. 1, pp. 525–528.
  4. A. Taflove, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech, Boston, 1995).
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    [CrossRef]
  6. 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]
  7. E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. C. W. Oseen, “Uber die Wechselwirkung zwischen zwei elektrichen Dipolen und uber die Drenhung der Polarisationsebene in Kristallen und Flussigkeiten,” Ann. Phys. 48, 1–15 (1915).
    [CrossRef]
  22. 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]
  23. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
    [CrossRef] [PubMed]
  24. 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).
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  25. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection,” Appl. Opt. 35, 500–514 (1996); “II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
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1996

1995

1994

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

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

1993

1991

1990

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

1989

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

K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
[CrossRef] [PubMed]

1982

1980

1975

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirriform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

1973

E. M. Purcell, C. P. 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]

1966

1915

C. W. Oseen, “Uber die Wechselwirkung zwischen zwei elektrichen Dipolen und uber die Drenhung der Polarisationsebene in Kristallen und Flussigkeiten,” Ann. Phys. 48, 1–15 (1915).
[CrossRef]

Cai, Q.

Carlson, B. E.

Draine, B. T.

Flatau, P. J.

Greenberg, J. M.

Hage, J. I.

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]

Heymsfied, A. J.

L. M. Miloshevich, A. J. Heymsfied, P. M. Norris, “Microphysical measurements in cirrus clouds from ice crystals replicator sonders launched during FIRE II,” in Proceedings of the 11th International Conference on Clouds and Precipitation (17–21 August 1992), Vol. 1, pp. 525–528.

Heymsfield, A. J.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirriform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

Karczewski, B.

Liou, K. N.

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]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

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

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

Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

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]

Lock, J. A.

Macke, A.

Miller, K. M.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

Miloshevich, L. M.

L. M. Miloshevich, A. J. Heymsfied, P. M. Norris, “Microphysical measurements in cirrus clouds from ice crystals replicator sonders launched during FIRE II,” in Proceedings of the 11th International Conference on Clouds and Precipitation (17–21 August 1992), Vol. 1, pp. 525–528.

Mishchenko, M. I.

Muinonen, K.

Norris, P. M.

L. M. Miloshevich, A. J. Heymsfied, P. M. Norris, “Microphysical measurements in cirrus clouds from ice crystals replicator sonders launched during FIRE II,” in Proceedings of the 11th International Conference on Clouds and Precipitation (17–21 August 1992), Vol. 1, pp. 525–528.

Oseen, C. W.

C. W. Oseen, “Uber die Wechselwirkung zwischen zwei elektrichen Dipolen und uber die Drenhung der Polarisationsebene in Kristallen und Flussigkeiten,” Ann. Phys. 48, 1–15 (1915).
[CrossRef]

Pennypacker, C. P.

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

Purcell, E. M.

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

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

Sphinhirne, J. D.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech, Boston, 1995).

Tai, C. T.

C. T. Tai, Dyadic Green’s Function in Electromagnetic Theory (International Textbook, Scranton, 1971), Chap. 4, pp. 48–49.

Takano, Y.

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

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

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

Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
[CrossRef] [PubMed]

Tanaka, M.

van de Hulst, H. C.

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

Wang, R. T.

Wolf, E.

B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
[CrossRef]

E. Wolf, “A generalized extinction theorem and its role in scattering theory,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1973), pp. 339–359.
[CrossRef]

Yang, P.

Ann. Phys.

C. W. Oseen, “Uber die Wechselwirkung zwischen zwei elektrichen Dipolen und uber die Drenhung der Polarisationsebene in Kristallen und Flussigkeiten,” Ann. Phys. 48, 1–15 (1915).
[CrossRef]

Appl. Opt.

Astrophys. J.

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

Atmos. Res.

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

J. Atmos. Sci.

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]

A. J. Heymsfield, “Cirrus uncinus generating cells and the evolution of cirriform clouds. Part I: Aircraft observations of the growth of the ice phase,” J. Atmos. Sci. 32, 799–808 (1975).
[CrossRef]

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

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mon. Weather Rev.

A. J. Heymsfield, K. M. Miller, J. D. Sphinhirne, “The 27–28 October 1986 FIRE IFO cirrus case study: cloud microstructure,” Mon. Weather Rev. 118, 2313–2328 (1990).
[CrossRef]

Opt. Lett.

Other

L. M. Miloshevich, A. J. Heymsfied, P. M. Norris, “Microphysical measurements in cirrus clouds from ice crystals replicator sonders launched during FIRE II,” in Proceedings of the 11th International Conference on Clouds and Precipitation (17–21 August 1992), Vol. 1, pp. 525–528.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech, Boston, 1995).

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

C. T. Tai, Dyadic Green’s Function in Electromagnetic Theory (International Textbook, Scranton, 1971), Chap. 4, pp. 48–49.

E. Wolf, “A generalized extinction theorem and its role in scattering theory,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1973), pp. 339–359.
[CrossRef]

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

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Figures (10)

Fig. 1
Fig. 1

(a) Conceptual diagram of the geometric-optics–integral-equation method. (b) The polarization configuration of the localized waves for external reflection (p = 1) and two refraction and internal reflections (p ≥ 2). The directions of various unit vectors are defined.

Fig. 2
Fig. 2

Conceptual geometry for the active and passive sources in light scattering by a dielectric particle.

Fig. 3
Fig. 3

(a) Large sphere on which the near field is mapped to the far field by the intensity mapping algorithm. The ice particle is located at the center. A number of unit vectors are also defined. (b) Phase delay of a ray inside an ice crystal.

Fig. 4
Fig. 4

Extinction efficiency computed by GOM1 and GOM2 at λ = 0.55 and 3.7 μm. The refractive indices of ice at these wavelengths are m = 1.311 + i3.11 × 10−9 and m = 1.4005 + i7.1967 × 10−3. Also shown are the single-scattering albedos computed by GOM1 and GOM2 at λ = 3.7 μm. The exact FDTD results for size parameters less than 30 are also presented.

Fig. 5
Fig. 5

Phase functions computed by the intensity and field mapping algorithms for randomly oriented plate crystals for three size parameters at two wavelengths. The vertical scale is applied to ka = 20; for ka = 40 and 60, this scale should be decreased by 102 and 103, respectively.

Fig. 6
Fig. 6

Comparison of the phase function and degree of linear polarization computed by GOM1 and GOM2 for the size parameter of 200 at the 0.55-μm and 3.7-μm wavelengths.

Fig. 7
Fig. 7

(a) Geometry for an outgoing ray in 2-D space. (b) Angular distribution of the far-field intensity produced by the ray for different size parameters and scattering angles.

Fig. 8
Fig. 8

Phase functions and degrees of linear polarization for three size parameters and two wavelengths computed by GOM2.

Fig. 9
Fig. 9

Remaining nonzero elements of the phase matrix associated with the results presented in Fig. 8.

Fig. 10
Fig. 10

Comparison of the phase functions computed by GOM2 and the FDTD method at the 0.55-μm wavelength for solid and hollow columns, plates, and bullet rosettes.

Equations (99)

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J = n ^ s × H ,
M = E × n ^ s ,
j m ( r ) = s M ( r ) G ( r , r ) d 2 r ,
j e ( r ) = s J ( r ) G ( r , r ) d 2 r ,
G ( r , r ) = exp ( i k r - r ) 4 π r - r .
E s ( r ) = - × j m ( r ) + i k × × j e ( r ) .
E s ( r ) k r = exp ( i k r ) - i k r k 2 4 π r ^ × s { n ^ s × E ( r ) - r ^ × [ n ^ s × H ( r ) ] } exp ( - i k r ^ · r ) d 2 r ,
e ^ 2 = N r - 1 { e ^ 1 - ( e ^ 1 · n ^ 1 ) n ^ 1 - [ N r 2 - 1 + ( e ^ 1 · n ^ 1 ) 2 ] 1 / 2 n ^ 1 } ,
e ^ p = e ^ p - 1 - 2 ( e ^ p - 1 · n ^ p - 1 ) n ^ p - 1 ,             p = 3 , 4 , 5 , .
e ^ 1 s = e ^ 1 - 2 ( e ^ 1 · n ^ 1 ) n ^ 1 ,
e ^ p s = N r { e ^ p - ( e ^ p · n ^ p ) n ^ p - [ N r - 2 - 1 + ( e ^ p · n ^ p ) 2 ] 1 / 2 n ^ p } , p = 2 , 3 , 4 , ,
e ^ i = β ^ i × α ^ i .
β ^ p = ( n ^ p × e ^ p ) [ 1 - ( e ^ p · n ^ p ) 2 ] - 1 / 2 ,             p = 1 , 2 , 3 , .
α ^ p s = e ^ p s × β ^ p ,             p = 1 , 2 , 3 , ,
α ^ p = e ^ p × β ^ p ,             p = 1 , 2 , 3 , .
E p s ( r ) = α ^ p s E p , α s ( r ) + β ^ p E p , β s ( r ) .
E p ( r ) = α ^ p E p , α ( r ) + β ^ p E p , β ( r ) .
A = [ A α A β ] = α ^ i A α + β ^ i A β .
E 1 ( r ) = [ E 1 , α ( r ) E 1 , β ( r ) ] = U 1 A exp [ i k ( e ^ 1 · r ) ] ,
E 1 s ( r ) = [ E 1 , α s ( r ) E 1 β s ( r ) ] = U 1 s A exp { i k [ e ^ 1 · r Q 1 + e ^ 1 s · ( r - r Q 1 ) ] } ,
E p ( r ) = [ E p , α ( r ) E p , β ( r ) ] = U p A exp { i k [ e ^ 1 · r Q 1 + N j = 1 P - 2 d j + N e ^ p · ( r - r Q p - 1 ) ] } , p = 2 , 3 , 4 , ,
E p s ( r ) = [ E p , α s ( r ) E p , β s ( r ) ] = U p s A exp { i k [ e ^ 1 · r Q 1 + N j = 1 P - 1 d j + e ^ p s · ( r - r Q p ) ] } , p = 2 , 3 , 4 , ,
U 1 = Λ 1 ,
U 2 = T 1 U 1 ,
U p = R p - 1 Λ p - 1 U p - 1 , p = 3 , 4 , 5 , ,
U 1 s = R 1 U 1 ,
U p s = T p Λ p U p , p = 2 , 3 , 4 , ,
R p = [ R p , α 0 0 R p , β ] , p = 1 , 2 , 3 , ,
T p = [ T p , α 0 0 T p , β ] , p = 1 , 2 , 3 , ,
E ( r ) = { E a ( r ) + E b ( r ) , r illuminated side E b ( r ) , r shadowed side ,
E a ( r ) = E 1 ( r ) + E 1 s ( r ) ,
E b ( r ) = γ p = 2 E p s ( r ) .
H p s ( r ) = e ^ p s × E p s ( r ) ,             for r outside the particle .
Δ σ ˜ 1 s = - Δ σ 0 ( n ^ 1 · e ^ i ) - 1 .
Δ σ ˜ p s = - Δ σ 0 ( n 1 · e ^ 2 ) [ ( n ^ 1 · e ^ i ) ( n ^ p · e ^ p ) ] - 1 ,             for p = 2 , 3 , 4 ,
E b s ( r ) k r = exp ( i k r ) - i k r k 2 4 π r ^ × s { γ p = 2 { n ^ s , p × E p s ( r ) - r ^ × [ n ^ s , p × H p s ( r ) ] } exp ( - i k r ^ · r ) d 2 r ,
E b s ( r ) = α ^ s E b , α s ( r ) + β ^ s E b , β s ( r ) ,
r ^ = β ^ s × α ^ s .
μ ^ p × ν ^ p = n ^ s , p .
E b s ( r ) = [ E b , α s ( r ) E b , β s ( r ) ] = exp ( i k r ) - i k r S b i A ,
S b i ( r ^ ) = k 2 4 π γ p = 2 Δ σ ˜ p s [ J K p J Y p s + K p J Y p s J ] U p s × exp [ i k ( e ^ i · r Q 1 + N j = 1 p - 1 d j - r ^ · r Q p ) ] ,
S b i ( r ^ ) = k 2 4 π n Δ S n { γ p = 2 ( Δ σ ˜ p s / Δ S n ) × [ J K p J Y p s + K p J Y p s J ] U p s × exp [ i k ( e ^ i · r Q 1 + N j = 1 p - 1 d j - r ^ · r Q p ) ] } ,
S b ( r ^ ) = S b i ( r ^ ) Γ i ,
S a ( r ) = j = 1 4 D j S a , j ( r ^ ) ,
S a , j ( r ^ ) = k 2 4 π [ J K 1 J ( Y 1 s U 1 s + Y 1 U 1 ) + K 1 J × ( Y 1 s J U 1 s + Y 1 J U 1 ) ] Γ i ,             γ γ ˜ j ,
D j = # j face exp [ i k ( e ^ i - r ^ ] · r ] d 2 r .
S ( r ^ ) = [ S 2 S 3 S 4 S 1 ] = S a ( r ^ ) + S b ( r ) .
σ ¯ e = ( σ e , / / = + σ e , ) / 2 = 2 π k 2 Re [ S 1 ( e ^ i ) + S 2 ( e ^ i ) ] ,
σ a = k ɛ i E i 2 v E ( r ) · E * ( r ) d 3 r ,
σ ¯ a = 1 2 ( σ a , / / + σ a , ) = 1 2 γ p = 2 Δ σ 0 ( n ^ 1 · e ^ i ) - 1 ( n ^ 1 · e ^ 2 ) N r × exp ( - 2 k N i j = 1 p - 1 d j ) [ 1 - exp ( - 2 k N i d p ) ] × [ U p , 11 2 + U p , 12 2 + U p , 21 2 + U p , 22 2 ] ,
v ( Q · × × P - P · × × Q ) d v = v ( P · 2 Q - Q · 2 P ) d v + s n ^ s · ( Q · P - P · Q ) d s ,
v ( ϕ 2 P - P 2 ϕ ) d v = s ( ϕ P n s - P ϕ n s ) d s ,
P = a · G ,             Q = E ,             ϕ = G ,
G ( r , r ) = ( I + 1 k 2 r r ) G ( r , r ) ,
E ( r ) = 1 k 2 × × [ S + S + S 0 ] [ E ( r ) G ( r , r ) n s - G ( r , r ) E ( r ) n s ] d 2 r ,             r V ,
lim r r [ × G - i k r ^ × G ] = 0.
E s ( r ) = 1 k 2 × × s [ E ( r ) G ( r , r ) n s - G ( r , r ) E ( r ) n s ] d 2 r ,             r V .
E s ( r ) k r = exp ( i k r ) - i k r k 2 4 π r ^ × { r ^ × s [ n ^ s · r ^ E ( r ) + 1 i k E ( r ) n s ] exp ( - i k r ^ · r ) d 2 r } .
r G ( r , r ) = - r G ( r , r ) .
1 i k E p s ( r ) n s = ( e ^ p s · n ^ s ) E p s ( r ) .
S i ( r ^ ) = j = 1 4 D j S ˜ j i ( r ^ ) ,
S ˜ j i ( r ^ ) = - k 2 4 π ( n ^ s , 1 · r ^ + n ^ s , 1 · e ^ 1 ) Π 1 U 1 Γ i ,             γ γ ˜ j ,
σ s i = 1 2 k 2 0 2 π 0 π [ S 1 i 2 + S 2 i 2 + S 3 i 2 + S 4 i 2 ] sin θ d θ d φ ,
δ p = N j = 1 p - 1 d j - ( d ˜ 1 + d ˜ 2 ) ,
E p s = [ E p , η s E p , ξ s ] = η ^ p E p , η s + ξ ^ p E p , ξ s = Y ˜ p s U p s A exp [ i k ( ρ + δ p ) ] ,
A = [ A α A β ] = Γ ˜ p i Γ p s A s ,
A s = [ A α s A β s ] = ( e ^ i × β ^ s ) A α s + β ^ s A β s .
E p s k r = exp ( i k r ) - i k r S p s A s ,
S p s = k 2 4 π q exp ( i k ζ ) ( 1 + r ^ · e ^ p s ) K ˜ p S ˜ p s Γ p s ,
ζ = ρ ( 1 - r ^ · e ^ p s ) + δ p ,
q = ray cross section exp ( - i k r ^ · r ) d 2 r = Δ σ p s 2 J 1 { k ( Δ σ p s / π ) 1 / 2 sin [ cos - 1 ( e ^ p s · r ^ ) ] } k ( Δ σ p s / π ) 1 / 2 sin [ cos - 1 ( e ^ p s · r ^ ) ] ,
S ˜ p s = Y ˜ p s U p s Γ ˜ p i
S p s = - k 2 4 π exp ( i k ζ [ f · S ˜ p s + g · S ˜ p s J ] ,
f = h cos φ t Ξ ,             g = h sin φ t Ξ ,
h = Δ σ p s ( 1 + cos Θ ) J 1 ( χ ) / χ , cos Θ = cos θ cos θ t + sin θ sin θ t cos φ t ,
χ = k ( Δ σ p s / π ) 1 / 2 sin Θ .
P = [ P 11 P 12 0 0 P 12 P 22 0 0 0 0 P 33 - P 43 0 0 P 43 P 44 ] .
p s = ( 1 - f δ ) π p ˜ s + 4 π f δ p δ ,
P c ( θ t ) = ( 1 - f δ ) P ˜ c ( θ t ) + 2 f δ I δ ( 1 - cos θ t ) ,
P ˜ 11 s ( θ ) = 2 0 π { [ F 11 ( θ , θ t ) + G 11 ( θ , θ t ) ] P ˜ 11 c ( θ t ) + [ F 12 ( θ , θ t ) + G 12 ( θ , θ t ) ] P ˜ 12 c ( θ t ) } sin θ t d θ t ,
P 11 δ ( θ ) = F 11 ( θ , 0 ) + G 11 ( θ , 0 ) .
X 11 = c ( x 1 2 + x 2 2 + x 3 2 + x 4 2 ) / 2 ,
X 12 = c ( x 2 2 - x 3 2 + x 4 2 - x 1 2 ) / 2 ,
x 1 2 = 1 2 π 0 2 π x 1 2 ( θ , θ t , φ t ) d φ t .
P 11 s ( θ ) = 2 π 0 π { [ F 11 ( θ , θ t ) + G 11 ( θ , θ t ) ] P 11 c ( θ t ) + [ F 12 ( θ , θ t ) + G 12 ( θ , θ t ) ] P 12 c ( θ t ) ) sin θ t d θ t .
P = σ s i σ s i + σ s s P i + σ s s σ s i + σ s s P s ,
Λ p = { [ β ^ 1 · β ^ i - β ^ i · α ^ i β ^ 1 · α ^ i β ^ 1 · β ^ i ] for p = 1 [ β ^ p · β ^ p - 1 - β ^ p · α ^ p - 1 β ^ p · α ^ p - 1 β ^ p · β ^ p - 1 ] for p 2 ,
K p = [ α ^ s · ν ^ p α ^ s · μ ^ p β ^ s · ν ^ p β ^ s · μ ^ p ] ,             K ˜ p = [ α ^ s · η ^ p α ^ s · ξ ^ p β ^ s · η ^ p β ^ s · ξ ^ p ] .
Y p s = [ ν ^ p · α ^ p s ν ^ p · β ^ p μ ^ p · α ^ p s μ ^ p · β ^ p ] ,             Y ˜ p s = [ η ^ p · α ^ p s η ^ p · β ^ p ξ ^ p · α ^ p s ξ ^ p · β ^ p ] .
Y 1 = [ ν ^ 1 · α ^ 1 ν ^ 1 · β ^ 1 μ ^ 1 · α ^ 1 μ ^ 1 · β ^ 1 ] ,             J = [ 0 1 - 1 0 ] .
Π 1 = [ α ^ s · α ^ 1 α ^ s · β ^ 1 β ^ s · α ^ 1 β ^ s · β ^ 1 ] ,             Π 2 s = [ α ^ s · α ^ 2 s α ^ s · β ^ 2 β ^ s · α ^ 2 s β ^ s · β ^ 2 ]
Γ i = [ β ^ s · β ^ i β ^ s · α ^ i - β ^ s · α ^ i β ^ s · β ^ i ] ,             Γ ˜ p i = [ ξ ^ p · β ^ i ξ ^ p · α ^ i - ξ ^ p · α ^ i ξ ^ p · β ^ i ] .
Γ p s = [ ξ ^ p · β ^ s - ξ ^ p · ( e ^ i × β ^ s ) ξ ^ p · ( e ^ i × β ^ s ) ξ ^ p · β ^ s ] .
Ξ = [ cos θ cos θ t cos φ t + sin θ sin θ t - cos θ sin φ t cos θ t sin φ t cos φ t ] .
S p = 2 = - k 2 4 π exp [ i ( δ 0 - r ^ · r Q 2 ) ] ( n ^ s , 2 · r ^ + n ^ s , 2 · e ^ 2 s ) Π 2 s T 2 T 1 Γ i κ x κ y ,
κ y = - exp ( - i k r y y ) d y = 2 π κ δ ( r y ) ,
κ x = - l l exp [ i k ( e 2 , x s - r x ) x ] d x = 2 l sin [ k l ( sin φ - sin φ t ) ] k l ( sin φ - sin φ t ) ,
E ( φ - φ t ) = ( n ^ s , 2 · r ^ + n ^ s , 2 · e 2 s ) κ x = ( cos φ + cos φ t ) κ x .
I ( φ , φ t ) = E ( φ , φ t ) 2 / c ,

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