Abstract

Scattering of an on-axis Gaussian beam by a uniaxial anisotropic sphere is studied. The incident on-axis Gaussian beam is expanded in terms of spherical vector wave functions, and the beam shape coefficients are obtained by applying the local approximation. The internal fields of a uniaxial anisotropic sphere are proposed in the integrating form of the spherical vector wave functions by introducing the Fourier transform. Utilizing the continuous tangential boundary conditions, both the scattered and the internal field coefficients are derived analytically. Numerical calculations are presented. The effects of the beam width, beam waist center positioning, and anisotropy on scattering properties are analyzed. The internal and near-surface field distributions are also discussed, and the two eigenmodes are characterized. The continuity on the surface of a uniaxial anisotropic sphere is well confirmed.

© 2009 Optical Society of America

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References

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  1. G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377-455 (1908).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
    [CrossRef]
  5. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E 56, 1102-1112 (1997).
    [CrossRef]
  6. 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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    [CrossRef]
  8. 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]
  9. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
    [CrossRef]
  21. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 1-8 (2004).
    [CrossRef]
  22. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
    [CrossRef]
  23. Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
    [CrossRef]
  24. Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572-576 (2009).
    [CrossRef]
  25. 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]

2009

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572-576 (2009).
[CrossRef]

2007

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]

2006

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[CrossRef]

2005

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

2004

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[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, 1-8 (2004).
[CrossRef]

2003

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

1997

1994

1993

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

1992

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc., Part H: Microwaves, Antennas Propag. 139, 314-318 (1992).
[CrossRef]

1991

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

1990

1989

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]

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[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]

1988

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

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

1980

1979

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

1908

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377-455 (1908).
[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.

Chen, H. T.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc., Part H: Microwaves, Antennas Propag. 139, 314-318 (1992).
[CrossRef]

Davis, L. W.

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

Doicu, A.

Dou, L. X.

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[CrossRef]

Geng, Y. L.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572-576 (2009).
[CrossRef]

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[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, 1-8 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Gouesbet, 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, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[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, 1-8 (2004).
[CrossRef]

Guo, L. X.

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]

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.

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

Li, L. W.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[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, 1-8 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Lock, J. A.

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377-455 (1908).
[CrossRef]

Papadakis, S. N.

Qiu, C. W.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572-576 (2009).
[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]

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.

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]

Sebak, A. R.

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[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.

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

Varadran, V. K.

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

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.

Wong, K. L.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc., Part H: Microwaves, Antennas Propag. 139, 314-318 (1992).
[CrossRef]

Wriedt, T.

Wu, X. B.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[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, 1-8 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Wu, Z. S.

Yuan, N.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572-576 (2009).
[CrossRef]

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.

G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377-455 (1908).
[CrossRef]

Appl. Opt.

IEE Proc., Part H: Microwaves, Antennas Propag.

K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc., Part H: Microwaves, Antennas Propag. 139, 314-318 (1992).
[CrossRef]

IEEE Antennas Wireless Propag. Lett.

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572-576 (2009).
[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]

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic Scattering by an imhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

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

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]

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E 70, 1-8 (2004).
[CrossRef]

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

Proc. IEEE

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.

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

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104 (2003).
[CrossRef]

Other

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 (10)

Fig. 1
Fig. 1

Uniaxial anisotropic sphere illuminated by a Gaussian beam.

Fig. 2
Fig. 2

Effects of the beam width on RCS ( k 0 a = 2 π , ε t = 5.3495 ε 0 , ε z = 4.9284 ε 0 , μ t = μ z = μ 0 , z 0 = 0 ).

Fig. 3
Fig. 3

Results in the special case of plane wave incidence compared with those by the FDTD method ( k 0 a = 2 π , ε t = 1.7 ε 0 , ε z = 1.2 ε 0 , μ t = μ z = μ 0 , z 0 = 0 ).

Fig. 4
Fig. 4

Scattering intensities of a lossless uniaxial anisotropic sphere for different beam waist center positioning (symbols for i s in the E plane, lines for i s in the H plane) ( k 0 a = 2 π , ε t = 1.8 ε 0 , ε z = 1.34 ε 0 , μ t = 2.4 μ 0 , μ z = 1.5 μ 0 , w 0 = 2.0 λ ).

Fig. 5
Fig. 5

Effects of the anisotropy ratio on the scattering intensity ( k 0 a = 2 π , ε t = 2 ε 0 , μ t = μ z = μ 0 , w 0 = 3.0 λ , z 0 = 0 ).

Fig. 6
Fig. 6

RCSs versus scattering angle for large-sized anisotropic spheres ( k 0 a = 4 π , μ t = μ z = μ 0 , w 0 = 3.0 λ , z 0 = 0 ). (a) Lossless, ε t = 4 ε 0 , ε z = 2 ε 0 ; (b) absorbing, ε t = ( 4 + 0.5 i ) ε 0 , ε z = ( 2 + 0.5 i ) ε 0 .

Fig. 7
Fig. 7

Internal and near-field distribution along the radii for the radial and tangential components ( k 0 a = 2 π , ε t = 5.3495 ε 0 , ε z = 4.9284 ε 0 , μ t = μ z = μ 0 , w 0 = 2.0 λ , z 0 = 0 ) (a) E r r ( θ = 90 ° , ϕ = 0 ), (b) E ϕ r ( θ = 90 ° , ϕ = 90 ° ).

Fig. 8
Fig. 8

The two eigenmodes of the internal field components in the H plane. (All the parameters are the same as those in Fig. 7.) (a) ϕ component, mode 1; (b) ϕ component, mode 2; (c) θ component, mode 1; (d) θ component, mode 2; (e) r component, mode 1; (f) r component, mode 2.

Fig. 9
Fig. 9

The twoeigen modes of the internal field components in the E plane at r = 0.5 a for a titanium dioxide sphere ( k 0 a = 2 π , ε t = 5.913 ε 0 , ε z = 7.197 ε 0 , μ t = μ z = μ 0 , w 0 = 2.0 λ , z 0 = 0 ). (a) E x θ , (b) E y θ , (c) E z θ .

Fig. 10
Fig. 10

Internal field distribution in the H plane for a titanium dioxide sphere. (All the parameters are the same as those in Fig. 9.)

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

E inc ( x , y , z 0 ) = E x inc ( x , y , z 0 ) e ̂ x = E 0 exp [ ( x 2 + y 2 ) w 0 2 ] e ̂ x ,
E inc ( r ) = n 1 m = n n C n m [ i g n , TE m M m n ( 1 ) ( r , k 0 ) + g n , TM m N m n ( 1 ) ( r , k 0 ) ] ,
H inc ( r ) = k 0 ω μ 0 n 1 m = n n C n m [ i g n , TM m M m n ( 1 ) ( r , k 0 ) g n , TE m N m n ( 1 ) ( r , 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 } ,
M m n ( l ) ( r , k ) = z n ( l ) ( k r ) [ i m P n m ( cos θ ) sin θ e i m ϕ e ̂ θ d P n m d θ e i m ϕ e ̂ ϕ ] ,
N m n ( l ) ( r , k ) = n ( n + 1 ) z n ( l ) ( k r ) k r P n m ( cos θ ) e i m ϕ e ̂ r + 1 k r d ( r z n ( l ) ( k r ) ) d r [ d P n m ( cos θ ) d θ e ̂ θ + i m P n m ( cos θ ) sin θ e ̂ ϕ ] e i m ϕ ,
g n , TM m = ( 1 ) m 1 K n m Ψ ¯ 0 0 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 ¯ 0 0 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 ¯ 0 0 = 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 } ,
g n , TM 1 = g n , TM 1 = 1 2 g n , g n , TE 1 = g n , TE 1 = i 2 g n ,
g n = i Q ¯ exp { i Q ¯ [ ( n + 0.5 ) λ ( 2 π w 0 ) ] 2 } exp ( i k 0 z 0 ) .
E inc = n = 1 m = n n [ δ m , 1 + δ m , 1 ] [ a m n i x M m n ( 1 ) ( r , k 0 ) + b m n i x N m n ( 1 ) ( r , k 0 ) ] g n ,
H inc = k 0 i ω μ 0 n = 1 m = n n [ δ m , 1 + δ m , 1 ] [ a m n i x N m n ( 1 ) ( r , k 0 ) + b m n i x M m n ( 1 ) ( r , k 0 ) ] g n ,
δ s , l = { 1 , s = l 0 , s l } , a m n i x = { i n 1 2 n + 1 2 n ( n + 1 ) , m = 1 i n 1 2 n + 1 2 , m = 1 } ,
b m n i x = { i n 1 2 n + 1 2 n ( n + 1 ) , m = 1 i n 1 2 n + 1 2 , m = 1 } .
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 ) ] .
× ( 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 int ( 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 int ( 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 int t = E inc t + E s t , H int t = H inc t + H s t .
[ δ m , 1 + δ m , 1 ] a m n i x g n 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 ) ,
[ δ m , 1 + δ m , 1 ] b m n i x g n 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 [ δ m , 1 + δ m , 1 ] a m n i x g n 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 [ δ m , 1 + δ m , 1 ] b m n i x g n 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 = [ δ m , 1 + δ m , 1 ] a m n i x g n 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 = [ δ m , 1 + δ m , 1 ] b m n i x g n 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 [ δ m , 1 + δ m , 1 ] a m n i x g n 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 [ δ m , 1 + δ m , 1 ] b m n i x g n j n ( k 0 a ) }
E r s = n = 1 n ( n + 1 ) h n ( 1 ) ( k 0 r ) k 0 r sin θ π n ( θ ) ( B 1 n s e i ϕ B 1 n s n ( n + 1 ) e i ϕ ) ,
E θ s = n = 1 [ i h n ( 1 ) ( k 0 r ) π n ( θ ) ( A 1 n s e i ϕ + A 1 n s n ( n + 1 ) e i ϕ ) + 1 k r d d r ( r h n ( 1 ) ( k 0 r ) ) τ n ( θ ) ( B 1 n s e i ϕ B 1 n s n ( n + 1 ) e i ϕ ) ]
E ϕ s = n = 1 [ h n ( 1 ) ( k 0 r ) τ n ( θ ) ( A 1 n s e i ϕ A 1 n s n ( n + 1 ) e i ϕ ) + 1 k r d d r ( r h n ( 1 ) ( k 0 r ) ) i π n ( θ ) ( B 1 n s e i ϕ + B 1 n s n ( n + 1 ) e i ϕ ) ]
π n ( θ ) = p n 1 ( cos θ ) sin θ , τ n ( θ ) = d p n 1 ( cos θ ) d θ .
h n ( 1 ) ( k r ) = 1 k r ( i ) n + 1 e i k r , 1 k r d d r ( r h n ( 1 ) ( k r ) ) = 1 k r ( i ) n e i k r .
E s = E θ s = e i k r k 0 r n = 1 ( i ) n [ π n ( θ ) ( A 1 n s e i ϕ + A 1 n s n ( n + 1 ) e i ϕ ) + τ n ( θ ) ( B 1 n s e i ϕ B 1 n s n ( n + 1 ) e i ϕ ) ] ,
E s = E ϕ s = e i k r k 0 r n = 1 ( i ) n + 1 [ τ n ( θ ) ( A 1 n s e i ϕ A 1 n s n ( n + 1 ) e i ϕ ) + π n ( θ ) ( B 1 n s e i ϕ + B 1 n s n ( n + 1 ) e i ϕ ) ] .
i s = i s + i s = n = 1 ( i ) n [ π n ( θ ) ( A 1 n s e i ϕ + A 1 n s n ( n + 1 ) e i ϕ ) + τ n ( θ ) ( B 1 n s e i ϕ B 1 n s n ( n + 1 ) e i ϕ ) ] 2 + n = 1 ( i ) n + 1 [ τ n ( θ ) ( A 1 n s e i ϕ A 1 n s n ( n + 1 ) e i ϕ ) + π n ( θ ) ( B 1 n s e i ϕ + B 1 n s n ( n + 1 ) e i ϕ ) ] 2 ,
i s = E s 2 , i s = E s 2 .
σ = lim r 4 π r 2 E s 2 E inc 2 = 4 π k 0 2 [ n = 1 ( i ) n { π n ( θ ) ( A 1 n s e i ϕ + A 1 n s n ( n + 1 ) e i ϕ ) + τ n ( θ ) ( B 1 n s e i ϕ B 1 n s n ( n + 1 ) e i ϕ ) } 2 + n = 1 ( i ) n + 1 { τ n ( θ ) ( A 1 n s e i ϕ A 1 n s n ( n + 1 ) e i ϕ ) + π n ( θ ) ( B 1 n s e i ϕ + B 1 n s n ( n + 1 ) e i ϕ ) } 2 ] .

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