Abstract

An analytical solution to the scattering of an off-axis Gaussian beam obliquely incident on a uniaxial anisotropic sphere is obtained in the particle-centered system. Based on the local approximation to the off-axis beam shape coefficients and the coordinate rotation theory, the off-axis obliquely incident Gaussian beam is expanded with the spherical vector wave functions in the primary coordinate of the uniaxial anisotropic sphere. The internal fields of the uniaxial anisotropic sphere are proposed in the integrating form of the spherical vector wave functions by introducing the Fourier transform. By matching the fields on the boundary and solving matrix equations, the expansion coefficients are analytically derived. The influences of the beam waist center positioning and the obliquely incident angles, as well as the permittivity tensors on the far scattered field distributions, are numerically presented. The correctness of the theory is verified by comparing our numerical results in special cases with results from the references and with calculations by other algorithms.

© 2010 Optical Society of America

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References

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  1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
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    [CrossRef] [PubMed]
  4. C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1998), pp. 93–97.
  5. A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  6. J. H. Richmond, “Scattering by a ferrite-coated conducting sphere,” IEEE Trans. Antennas Propag. 35, 73–79 (1987).
    [CrossRef]
  7. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  8. Y. L. Xu and B. Å. S. Gustafson, “Experimental and theoretical results of light scattering by aggregates of spheres,” Appl. Opt. 36, 8026–8030 (1997).
    [CrossRef]
  9. W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664–673 (1993).
    [CrossRef]
  10. X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996–2003 (1997).
    [CrossRef]
  11. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112 (1997).
    [CrossRef]
  12. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
    [CrossRef]
  13. C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag. 55, 3515–3523 (2007).
    [CrossRef]
  14. J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag. 41, 994–999 (1993).
    [CrossRef]
  15. R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
    [CrossRef]
  16. S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A 7, 991–997 (1990).
    [CrossRef]
  17. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  18. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouestbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188–5198 (1997).
    [CrossRef] [PubMed]
  19. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  20. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  21. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  22. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  23. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [CrossRef] [PubMed]
  24. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.
  25. Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients for arbitrary shaped beam in oblique illumination,” Opt. Express 15, 735–746 (2007).
    [CrossRef] [PubMed]
  26. Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A 26, 1778–1788 (2009).
    [CrossRef]

2009 (1)

2007 (2)

Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients for arbitrary shaped beam in oblique illumination,” Opt. Express 15, 735–746 (2007).
[CrossRef] [PubMed]

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag. 55, 3515–3523 (2007).
[CrossRef]

2004 (1)

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

1998 (1)

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1998), pp. 93–97.

1997 (5)

1994 (2)

1993 (2)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664–673 (1993).
[CrossRef]

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag. 41, 994–999 (1993).
[CrossRef]

1991 (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990 (1)

1989 (2)

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1987 (1)

J. H. Richmond, “Scattering by a ferrite-coated conducting sphere,” IEEE Trans. Antennas Propag. 35, 73–79 (1987).
[CrossRef]

1980 (1)

1979 (1)

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

1957 (2)

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.

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

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1998), pp. 93–97.

Capsalis, C. N.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Doicu, A.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.

Geng, Y. L.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Gouesbet, G.

Gouestbet, G.

Graglia, R. D.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Grehan, G.

Guan, B. R.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Guo, L. X.

Gustafson, B. Å. S.

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112 (1997).
[CrossRef]

Han, G. X.

Han, Y. P.

Hudson, S.

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag. 41, 994–999 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1998), pp. 93–97.

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Li, L. W.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Li, Z. J.

Lock, J. A.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Papadakis, S. N.

Peng, Y.

Qiu, C. W.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag. 55, 3515–3523 (2007).
[CrossRef]

Razek, A.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag. 55, 3515–3523 (2007).
[CrossRef]

Ren, K. F.

Ren, W.

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664–673 (1993).
[CrossRef]

Richmond, J. H.

J. H. Richmond, “Scattering by a ferrite-coated conducting sphere,” IEEE Trans. Antennas Propag. 35, 73–79 (1987).
[CrossRef]

Sarkar, D.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112 (1997).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Schneider, J.

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag. 41, 994–999 (1993).
[CrossRef]

Uslenghi, P. L. E.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Uzunoglu, N. K.

van de Hulst, H. C.

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

Wang, Y. P.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wiscombe, W. J.

Wriedt, T.

Wu, X. B.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996–2003 (1997).
[CrossRef]

Wu, Z. S.

Xu, Y. L.

Yasumoto, K.

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996–2003 (1997).
[CrossRef]

Yuan, Q. K.

Zhang, H. Y.

Zich, R. S.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Zouhdi, S.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag. 55, 3515–3523 (2007).
[CrossRef]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (3)

J. H. Richmond, “Scattering by a ferrite-coated conducting sphere,” IEEE Trans. Antennas Propag. 35, 73–79 (1987).
[CrossRef]

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag. 55, 3515–3523 (2007).
[CrossRef]

J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag. 41, 994–999 (1993).
[CrossRef]

J. Appl. Phys. (4)

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996–2003 (1997).
[CrossRef]

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

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

Opt. Express (1)

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. E (3)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664–673 (1993).
[CrossRef]

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102–1112 (1997).
[CrossRef]

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 056609 (2004).
[CrossRef]

Proc. IEEE (1)

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750–760 (1989).
[CrossRef]

Radio Sci. (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other (3)

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

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1998), pp. 93–97.

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Figures (9)

Fig. 1
Fig. 1

Uniaxial anisotropic sphere illuminated by an off-axis obliquely incident Gaussian beam.

Fig. 2
Fig. 2

Coordinate rotations described by the Euler angles.

Fig. 3
Fig. 3

Results reduced to the on-axis case compared with those in [26] [ k 0 a = 4 π , w 0 = 3.0 λ ε t = 4 ε 0 , ε z = 2 ε 0 , μ t = μ z = μ 0 , ( x 0 , y 0 , z 0 ) = ( 0 , 0 , 0 ) λ , α = β = γ = 0 ° ].

Fig. 4
Fig. 4

Results reduced to the case of an isotropic sphere compared with those by the generalized Mie theory [ k 0 a = 2 π , w 0 = 2 λ , ε t = ε z = 3.25 ε 0 , μ t = μ z = μ 0 , ( x 0 , y 0 , z 0 ) = ( 1.5 , 1.0 , 3.0 ) λ , α = β = γ = 0 ° ].

Fig. 5
Fig. 5

Results reduced to the case of an obliquely incident plane wave compared with those by the CST simulation [ k 0 a = 2 π , w 0 = 50.0 λ , ε t = 5.3495 ε 0 , ε z = 4.9284 ε 0 , μ t = μ z = μ 0 , ( x 0 , y 0 , z 0 ) = ( 1.0 , 2.0 , 1.5 ) λ , α = γ = 0 ° ]: (a) β = 30 ° ; (b) β = 60 ° .

Fig. 6
Fig. 6

Effects of the beam waist center positioning along the x-axis on the RCS ( k 0 a = 2 π , w 0 = 2.0 λ , ε t = 5.3495 ε 0 , ε z = 3.9284 ε 0 , μ t = μ z = μ 0 , α = β = γ = 0 ° ): (a) E-plane; (b) H-plane.

Fig. 7
Fig. 7

Effects of the beam waist center positioning along the y-axis on the RCS (all the parameters are the same as those of the case in Fig. 6): (a) E-plane; (b) H-plane.

Fig. 8
Fig. 8

Effects of the obliquely incident angle on the RCS ( k 0 a = 2 π , w 0 = 1.5 λ , ε t = 5.3495 ε 0 , ε z = 4.9284 ε 0 , μ t = μ z = μ 0 , α = γ = 0 ° ): (a) on-center case, ( x 0 , y 0 , z 0 ) = ( 0 , 0 , 0 ) λ ; (b) off-center case, ( x 0 , y 0 , z 0 ) = ( 1.0 , 0.0 , 1.0 ) λ .

Fig. 9
Fig. 9

Calculations made for an off-axis Gaussian beam obliquely incident on a titanium dioxide sphere ( ε t = 5.913 ε 0 , ε z = 7.197 ε 0 ) and a silicon dioxide sphere ( ε t = 2.3 ε 0 , ε z = 2.25 ε 0 ) [ k 0 a = 2 π , w 0 = 3.0 λ , μ t = μ z = μ 0 , ( x 0 , y 0 , z 0 ) = ( 1.0 , 1.0 , 0.5 ) λ , α = 8 ° , β = 6 ° , γ = 5 ° ]: (a) E-plane; (b) H-plane.

Equations (47)

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E i n c ( r ) = n = 1 m = n n C n m [ i g n , TE m M m n ( 1 ) ( r , θ , ϕ , k 0 ) + g n , TM m N m n ( 1 ) ( r , θ , ϕ , k 0 ) ] ,
H i n c ( r ) = k 0 ω μ 0 n = 1 m = n n C n m [ i g n , TM m M m n ( 1 ) ( r , θ , ϕ , k 0 ) g n , TE m N m n ( 1 ) ( r , θ , ϕ , k 0 ) ] ,
C n m = { i n 1 2 n + 1 n ( n + 1 ) , m 0 ( 1 ) | m | ( n + | m | ) ! ( n | m | ) ! i n 1 2 n + 1 n ( n + 1 ) , m < 0 , }
g n , TM m = ( 1 ) m 1 K n m Ψ ¯   exp ( i k 0 z 0 ) 1 2 { exp [ i ( m 1 ) φ 0 ] J m 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) + exp [ i ( m + 1 ) φ 0 ] J m + 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) } ,
i g n , TE m = ( 1 ) m 1 K n m Ψ ¯   exp ( i k 0 z 0 ) 1 2 { exp [ i ( m 1 ) φ 0 ] J m 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) exp [ i ( m + 1 ) φ 0 ] J m + 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) } ,
Ψ ¯ = i Q ¯   exp ( i Q ¯ ρ 0 2 / w 0 2 ) exp ( i Q ¯ ( n + 0.5 ) 2 / k 0 2 w 0 2 ) ,    
Q ¯ = 1 / ( i 2 z 0 / l ) ,     l = k 0 w 0 2 ,
ρ 0 = ( x 0 2 + y 0 2 ) 1 / 2 ,     ρ n = ( n + 0.5 ) / k 0 ,     φ 0 = arctan ( x 0 / y 0 ) ,
K n m = { ( i ) m i ( n + 0.5 ) m 1 , m 0 n ( n + 1 ) n + 0.5 , m = 0 , }
( x 0 , y 0 , z 0 ) = A γ A β A α ( x 0 , y 0 , z 0 ) ,
A α = [ cos   α sin   α 0 sin   α cos   α 0 0 0 1 ] ,     A β = [ cos   β 0 sin   β 0 1 0 sin   β 0 cos   β ] ,    
A γ = [ cos   γ sin   γ 0 sin   γ cos   γ 0 0 0 1 ] .
P n m ( cos   θ ) e i m ϕ = s = n n ρ ( m , s , n ) P n s ( cos   θ ) e i s ϕ ,
ρ ( m , s , n ) = ( 1 ) s + m [ ( n + m ) ! ( n s ) ! ( n m ) ! ( n + s ) ! ] 1 / 2 e i m α u s m ( n ) ( β ) e i s γ ,
u s m ( n ) ( β ) = [ ( n + s ) ! ( n s ) ! ( n + m ) ! ( n m ) ! ] 1 / 2 σ ( n + m n s σ ) ( n m σ ) ( 1 ) n s σ ( cos β 2 ) 2 σ + s + m ( sin β 2 ) 2 n 2 σ s m .
( M , N ) m n ( l ) ( r , θ , ϕ , k 0 ) = s = n n ρ ( m , s , n ) ( M , N ) s n ( l ) ( r , θ , ϕ , k 0 ) .
E i n c = n = 1 m = n n s = n n ρ ( m , s , n ) C n m [ i g n , TE m M s n ( 1 ) ( r , θ , ϕ , k 0 ) + g n , TM m N s n ( 1 ) ( r , θ , ϕ , k 0 ) ] ,
H i n c = k 0 ω μ 0 n = 1 m = n n s = n n ρ ( m , s , n ) C n m [ i g n , TM m M s n ( 1 ) ( r , θ , ϕ , k 0 ) g n , TE m N s n ( 1 ) ( r , θ , ϕ , k 0 ) ] .
E i n c = n = 1 m = n n s = n n ρ ( s , m , n ) C n s [ a m n i x M m n ( 1 ) ( k 0 , r , θ , ϕ ) + b m n i x N m n ( 1 ) ( k 0 , r , θ , ϕ ) ] ,
H i n c = k 0 i ω μ 0 n = 1 m = n n s = n n ρ ( s , m , n ) C n s [ a m n i x N m n ( 1 ) ( k 0 , r , θ , ϕ ) + b m n i x M m n ( 1 ) ( k 0 , r , θ , ϕ ) ] ,
( a m n i x , b m n i x ) = s = n n ρ ( s , m , n ) C n s ( i g n , TE s , g n , TM s ) ,
ρ ( s , m , n ) = ( 1 ) m + s ( n m ) ! ( n + s ) ! e i m α e i s γ σ = a b ( 1 ) n m σ ( cos β 2 ) 2 σ + s + m ( sin β 2 ) 2 n 2 σ s m ( n s σ ) ! ( m + s + σ ) ! ( n m σ ) ! σ ! ,
a = max ( s m , 0 ) ,     b = min ( n s , n m ) .
× ( μ ¯ ¯ 1 × E ) ω 2 ε ¯ ¯ E = 0 ,
ε ¯ ¯ = [ ε t 0 0 0 ε t 0 0 0 ε z ] ,     μ ¯ ¯ = [ μ t 0 0 0 μ t 0 0 0 μ z ] .
E i n t ( r ) = q = 1 2 n = 1 m = n n n = 1 2 π G m n q 0 π [ A m n q e M m n ( 1 ) ( r , k q ) + B m n q e N m n ( 1 ) ( r , k q ) + C m n q e L m n ( 1 ) ( r , k q ) ] p n m ( cos   θ k ) k q 2   sin   θ k d θ k ,
H i n t ( r ) = q = 1 2 n = 1 m = n n n = 1 2 π G m n q 0 π [ A m n q h M m n ( 1 ) ( r , k q ) + B m n q h N m n ( 1 ) ( r , k q ) + C m n q h L m n ( 1 ) ( r , k q ) ] p n m ( cos   θ k ) k q 2   sin   θ k d θ k .
E s = n = 1 m = n n [ A m n s M m n ( 3 ) ( r , k 0 ) + B m n s N m n ( 3 ) ( r , k 0 ) ] ,
H s = k 0 i ω μ 0 n = 1 m = n n [ A m n s N m n ( 3 ) ( r , k 0 ) + B m n s M m n ( 3 ) ( r , k 0 ) ] .
E i n t t = E i n c t + E s t ,     H i n t t = H i n c t + H s t .
a m n i x j n ( k 0 r ) + A m n s h n ( 1 ) ( k 0 r ) = q = 1 2 n = 1 2 π G m n 1 0 π A m n q e j n ( k q r ) p n m ( cos   θ k ) k q 2   sin   θ k d θ k     ( r = a ) ,
b m n i x 1 k 0 r d ( r j n ( k 0 r ) ) d r + B m n s 1 k 0 r d ( r h n ( 1 ) ( k 0 r ) ) d r = q = 1 2 n = 1 2 π G m n q 0 π [ B m n q e 1 k q r d ( r j n ( k q r ) ) d r + C m n q e j n ( k q r ) r ] p n m ( cos   θ k ) k q 2   sin   θ k d θ k     ( r = a ) ,
k 0 i ω μ 0 a m n i x 1 k 0 r d ( r j n ( k 0 r ) ) d r + k 0 i ω μ 0 A m n s 1 k 0 r d ( r h n ( 1 ) ( k 0 r ) ) d r = q = 1 2 n = 1 2 π G m n q 0 π [ B m n q h 1 k q r d ( r j n ( k q r ) ) d r + C m n q h j n ( k q r ) r ] p n m ( cos   θ k ) k q 2   sin   θ k d θ k     ( r = a ) ,
k 0 i ω μ 0 b m n i x j n ( k 0 r ) + k 0 i ω μ 0 B m n s h n ( 1 ) ( k 0 r ) = q = 1 2 n = 1 2 π G m n q 0 π A m n q h j n ( k q r ) p n m ( cos   θ k ) k q 2   sin   θ k d θ k     ( r = a ) .
q = 1 2 n = 0 2 π G m n q 0 π U m n q P n m ( cos   θ k ) k q 2   sin   θ k d θ k = a m n i x i ( k 0 a ) 2 ,
q = 1 2 n = 0 2 π G m n q 0 π V m n q P n m ( cos   θ k ) k q 2   sin   θ k d θ k = b m n i x i ( k 0 a ) 2 ,
U m n q = { A m n q e 1 k 0 r d d r [ r h n ( 1 ) ( k 0 r ) ] j n ( k q r ) i ω μ 0 k 0 [ B m n q h 1 k q r d d r [ r j n ( k q r ) ] + C m n q h j n ( k q r ) r ] h n ( 1 ) ( k 0 r ) } r = a ,
V m n q = { i ω μ 0 k 0 A m n q h 1 k 0 r d d r [ r h n ( 1 ) ( k 0 r ) ] j n ( k q r ) [ B m n q e 1 k q r d d r [ r j n ( k q r ) ] + C m n q e j n ( k q r ) r ] h n ( 1 ) ( k 0 r ) } r = a .
A m n s = 1 h n ( 1 ) ( k 0 a ) { q = 1 2 n = 0 2 π G m n q 0 π A m n q e j n ( k q a ) P n m ( cos   θ k ) k q 2   sin   θ k d θ k a m n i x j n ( k 0 a ) } ,
B m n s = 1 h n ( 1 ) ( k 0 a ) { i ω μ 0 k 0 q = 1 2 n = 0 2 π G m n q 0 π A m n q h j n ( k q a ) P n m ( cos   θ k ) k q 2   sin   θ k d θ k b m n i x j n ( k 0 a ) } .
σ = lim r 4 π r 2 | E s | 2 / | E i n c | 2 = 4 π k 0 2 { | n = 1 m = n n ( i ) n e i m ϕ [ m A m n s π n m + B m n s τ n m ] | 2 + | n = 1 m = n n ( i ) n + 1 e i m ϕ [ A m n s τ n m + m B m n s π n m ] | 2 } .
π n m ( θ ) = p n m ( cos   θ ) / sin   θ ,     τ n m ( θ ) = d p n m ( cos   θ ) / d θ .
ψ ( x ) = x j n ( x ) ,     ξ ( x ) = x h n ( 1 ) ( x ) .
U m n q = 1 ρ 0 ρ q { A m n q e ξ n ( ρ 0 ) ψ n ( ρ q ) i ω μ 0 k 0 [ B m n q h ψ n ( ρ q ) + C m n q h ξ n ( ρ q ) a ] ξ n ( ρ 0 ) } ,
V m n q = 1 ρ 0 ρ q { i ω μ 0 k 0 A m n q h ξ n ( ρ 0 ) ψ n ( ρ q ) [ B m n q e ψ n ( ρ q ) + C m n q e ξ n ( ρ q ) a ] ξ n ( ρ 0 ) } ,
A m n s = { q = 1 2 n = 0 2 π k 0 G m n q 0 π A m n q e ψ ( ρ q ) ξ ( ρ 0 ) P n m ( cos   θ k ) k q   sin   θ k d θ k a m n i x ψ ( ρ 0 ) ξ ( ρ 0 ) } ,
B m n s = { i ω μ 0 k 0 q = 1 2 n = 0 2 π k 0 G m n q 0 π A m n q h ψ ( ρ q ) ξ ( ρ 0 ) P n m ( cos   θ k ) k q   sin   θ k d θ k b m n i x ψ ( ρ 0 ) ξ ( ρ 0 ) } ,

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