Abstract

Within the generalized Lorenz-Mie theory (GLMT) framework, an analytical solution to the scattering by a uniaxial anisotropic cylinder, for oblique incidence of an on-axis Gaussian beam, is constructed by expanding the incident Gaussian beam, scattered fields as well as internal fields in terms of appropriate cylindrical vector wave functions (CVWFs). The unknown expansion coefficients are determined by virtue of the boundary conditions. For a localized beam model, numerical results are provided for the normalized internal and near-surface field intensity distributions, and the scattering characteristics are discussed concisely.

©2013 Optical Society of America

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  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(5), 750–760 (1989).
    [Crossref]
  2. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989).
    [Crossref]
  3. 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(6), 991–997 (1990).
    [Crossref]
  4. C. M. Rappaport and B. J. McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991).
    [Crossref]
  5. W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993).
    [Crossref] [PubMed]
  6. X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys. 82(5), 1996–2003 (1997).
    [Crossref]
  7. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
    [Crossref] [PubMed]
  8. 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. Antenn. Propag. 57(2), 572–576 (2009).
    [Crossref]
  9. Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011).
    [Crossref]
  10. Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011).
    [Crossref] [PubMed]
  11. Z. J. Li, Z. S. Wu, Y. Shi, L. Bai, and H.-Y. Li, “Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres,” J. Opt. Soc. Am. A 29(1), 22–31 (2012).
    [Crossref]
  12. 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(8), 1778–1787 (2009).
    [Crossref] [PubMed]
  13. Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010).
    [Crossref] [PubMed]
  14. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Optics (Paris) 26(5), 225–239 (1995).
    [Crossref]
  15. K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14(11), 3014–3025 (1997).
    [Crossref]
  16. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14(3), 640–652 (1997).
    [Crossref]
  17. H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007).
    [Crossref]
  18. H. Y. Zhang and Y. P. Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B 25(2), 131–135 (2008).
    [Crossref]
  19. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979).
    [Crossref]
  20. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999).
    [Crossref]
  21. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011).
    [Crossref]
  22. G. Gouesbet, G. Gréhan, and K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15(2), 511–523 (1998).
    [Crossref]

2012 (1)

2011 (3)

Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011).
[Crossref]

Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011).
[Crossref] [PubMed]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011).
[Crossref]

2010 (1)

2009 (2)

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(8), 1778–1787 (2009).
[Crossref] [PubMed]

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. Antenn. Propag. 57(2), 572–576 (2009).
[Crossref]

2008 (1)

2007 (1)

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 Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

1999 (1)

1998 (1)

1997 (3)

1995 (1)

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Optics (Paris) 26(5), 225–239 (1995).
[Crossref]

1993 (1)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993).
[Crossref] [PubMed]

1991 (1)

C. M. Rappaport and B. J. McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991).
[Crossref]

1990 (1)

1989 (2)

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77(5), 750–760 (1989).
[Crossref]

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989).
[Crossref]

1979 (1)

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979).
[Crossref]

Bai, L.

Capsalis, C. N.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979).
[Crossref]

Geng, Y. L.

Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011).
[Crossref]

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. Antenn. Propag. 57(2), 572–576 (2009).
[Crossref]

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

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(5), 750–760 (1989).
[Crossref]

Gréhan, 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 Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Han, G. X.

Han, Y. P.

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989).
[Crossref]

Li, H. Y.

Li, H.-Y.

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 Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Li, Z. J.

Lock, J. A.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011).
[Crossref]

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14(3), 640–652 (1997).
[Crossref]

McCartin, B. J.

C. M. Rappaport and B. J. McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991).
[Crossref]

Papadakis, S. N.

Peng, Y.

Qiu, C. W.

Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011).
[Crossref]

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. Antenn. Propag. 57(2), 572–576 (2009).
[Crossref]

Rappaport, C. M.

C. M. Rappaport and B. J. McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991).
[Crossref]

Ren, K. F.

Ren, W.

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993).
[Crossref] [PubMed]

Shi, Y.

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(5), 750–760 (1989).
[Crossref]

Uzunoglu, N. K.

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989).
[Crossref]

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989).
[Crossref]

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 Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys. 82(5), 1996–2003 (1997).
[Crossref]

Wu, Z. S.

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(5), 1996–2003 (1997).
[Crossref]

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. Antenn. Propag. 57(2), 572–576 (2009).
[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(5), 750–760 (1989).
[Crossref]

IEEE Trans. Antenn. Propag. (4)

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antenn. Propag. 37(6), 800–802 (1989).
[Crossref]

C. M. Rappaport and B. J. McCartin, “FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes,” IEEE Trans. Antenn. Propag. 39(3), 345–349 (1991).
[Crossref]

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. Antenn. Propag. 57(2), 572–576 (2009).
[Crossref]

Y. L. Geng and C. W. Qiu, “Extended Mie theory for a gyrotropic-coated conducting sphere: An analytical approach,” IEEE Trans. Antenn. Propag. 59(11), 4364–4368 (2011).
[Crossref]

J. Appl. Phys. (1)

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: An analytical solution,” J. Appl. Phys. 82(5), 1996–2003 (1997).
[Crossref]

J. Opt. Soc. Am. A (9)

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999).
[Crossref]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14(11), 3014–3025 (1997).
[Crossref]

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14(3), 640–652 (1997).
[Crossref]

Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011).
[Crossref] [PubMed]

Z. J. Li, Z. S. Wu, Y. Shi, L. Bai, and H.-Y. Li, “Multiple scattering of electromagnetic waves by an aggregate of uniaxial anisotropic spheres,” J. Opt. Soc. Am. A 29(1), 22–31 (2012).
[Crossref]

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(8), 1778–1787 (2009).
[Crossref] [PubMed]

Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010).
[Crossref] [PubMed]

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(6), 991–997 (1990).
[Crossref]

G. Gouesbet, G. Gréhan, and K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15(2), 511–523 (1998).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Optics (Paris) (1)

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Optics (Paris) 26(5), 225–239 (1995).
[Crossref]

JQSRT (1)

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” JQSRT 112(1), 1–27 (2011).
[Crossref]

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(1), 664–673 (1993).
[Crossref] [PubMed]

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(5), 750–760 (1989).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Uniaxial anisotropic cylinder illuminated by an on-axis incident Gaussian beam.

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Fig. 2
Fig. 2

| E w / E 0 | 2 and | ( E i + E s ) / E 0 | 2 for a uniaxial anisotropic cylinder illuminated by a TM polarized Gaussian beam.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3

Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.

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Equations (44)

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

E i = E 0 ∑ m=−∞ ∞ ∫ 0 π [ I m,TE (ζ) m mλ (1) (h)+ I m,TM (ζ) n mλ (1) (h) ] e ihz dζ ,
H i =−i E 0 k 0 ω μ 0 ∑ m=−∞ ∞ ∫ 0 π [ I m,TE (ζ) n mλ (1) (h)+ I m,TM (ζ) m mλ (1) (h) ] e ihz dζ ,
I m,TE = (−i) m+1 k 0 ∑ n=| m | ∞ (n−m)! (n+m)! 2n+1 2n(n+1) × g n [ m 2 P n m (cosβ) sinβ P n m (cosζ) sinζ + d P n m (cosβ) dβ d P n m (cosζ) dζ ],
I m,TM = (−i) m+1 k 0 m ∑ n=| m | ∞ (n−m)! (n+m)! 2n+1 2n(n+1) × g n [ P n m (cosβ) sinβ d P n m (cosζ) dζ + d P n m (cosβ) dβ P n m (cosζ) sinζ ],
g n = 1 1+2is z 0 / w 0 exp(ik z 0 )exp[ − s 2 (n+1/2) 2 1+2is z 0 / w 0 ],
E s = E 0 ∑ m=−∞ ∞ ∫ 0 π [ α m (ζ) m mλ (3) + β m (ζ) n mλ (3) ] e ihz dζ ,
H s =−i E 0 k 0 ω μ 0 ∑ m=−∞ ∞ ∫ 0 π [ α m (ζ) n mλ (3) + β m (ζ) m mλ (3) ] e ihz dζ ,
E w = E 0 ∑ q=1 2 ∑ m=−∞ ∞ ∫ 0 π F mq (ζ)[ α q e (ζ) m m λ q (1) + β q e (ζ) n m λ q (1) + γ q e (ζ) l m λ q (1) ] e i h q z dζ ,
H w =−i E 0 ∑ q=1 2 ∑ m=−∞ ∞ ∫ 0 π k q ω μ 0 F mq (ζ)[ β q e (ζ)(ζ) m m λ q (1) + α q e (ζ) n m λ q (1) ] e i h q z dζ ,
h 1 = h 2 =h= k 0 cosζ,
a 1 2 = ω 2 ε t μ 0 ,   a 2 2 = ω 2 ε z μ 0 ,
k 1 = a 1 , k 2 = 1 a 1 a 1 2 a 2 2 +( a 1 2 − a 2 2 ) k 0 2 cos 2 ζ ,
λ 1 = a 1 2 − k 0 2 cos 2 ζ ,   λ 2 = k 2 sin θ k = a 2 a 1 a 1 2 − k 0 2 cos 2 ζ ,
α 1 e (ζ)=1,   β 1 e (ζ)= γ 1 e (ζ)= α 2 e (ζ)=0,
β 2 e (ζ)=−i a 1 2 a 2 ( a 1 2 − k 0 2 cos 2 ζ)[ a 1 2 a 2 2 +( a 1 2 − a 2 2 ) k 0 2 cos 2 ζ] ,
γ 2 e (ζ)=− a 1 2 − a 2 2 a 1 2 a 1 a 2 k 0 cosζ a 1 2 − k 0 2 cos 2 ζ a 1 2 a 2 2 +( a 1 2 − a 2 2 ) k 0 2 cos 2 ζ ,
E ϕ i + E ϕ s = E ϕ w , E z i + E z s = E z w H ϕ i + H ϕ s = H ϕ w , H z i + H z s = H z w } at r= r 0 ,
ξ d dξ J m (ξ) I m,TE + hm k 0 J m (ξ) I m,TM +ξ d dξ H m (1) (ξ) α m (ζ)+ hm k 0 H m (1) (ξ) β m (ζ) = F m1 (ζ) ξ 1 d d ξ 1 J m ( ξ 1 )+ F m2 (ζ) β 2 e (ζ) hm k 2 J m ( ξ 2 )− F m2 (ζ) γ 2 e (ζ)im J m ( ξ 2 ),
ξ 2 [ J m (ξ) I m,TM + H m (1) (ξ) β m (ζ)]= F m2 (ζ) k 0 k 2 ξ 2 2 J m ( ξ 2 )[ β 2 e (ζ)+ γ 2 e (ζ) ih k 2 λ 2 2 ],
hm k 0 J m (ξ) I m,TE +ξ d dξ J m (ξ) I m,TM + hm k 0 H m (1) (ξ) α m (ζ)+ξ d dξ H m (1) (ξ) β m (ζ) = hm k 0 F m1 (ζ) J m ( ξ 1 )+ k 2 k 0 F m2 (ζ) β 2 e (ζ) ξ 2 d d ξ 2 J m ( ξ 2 ),
ξ 2 [ J m (ξ) I m,TE + H m (1) (ξ) α m (ζ)]= F m1 (ζ) ξ 1 2 J m ( ξ 1 ),
| E w / E 0 | 2 = ( | E r w | 2 + | E ϕ w | 2 + | E z w | 2 ) / | E 0 | 2 ,
| ( E i + E s ) / E 0 | 2 = ( | E r i + E r s | 2 + | E ϕ i + E ϕ s | 2 + | E z i + E z s | 2 ) / | E 0 | 2 ,
E r w = E 0 ∑ m=−∞ ∞ e imϕ ∫ 0 π { F m1 (ζ)i m r J m ( λ 1 r) + F m2 (ζ) ∂ ∂r J m ( λ 2 r)[ β 2 e (ζ)i h k 2 + γ 2 e (ζ) ] } e ihz dζ
E φ w = E 0 ∑ m=−∞ ∞ e imϕ ∫ 0 π { − F m1 (ζ) ∂ ∂r J m ( λ 1 r) + F m2 (ζ) m r J m ( λ 2 r)[ − β 2 e (ζ) h k 2 +i γ 2 e (ζ) ] } e ihz dζ
E z w = E 0 ∑ m=−∞ ∞ e imϕ ∫ 0 π F m2 (ζ) J m ( λ 2 r)[ β 2 e (ζ) λ 2 2 k 2 +ih γ 2 e (ζ) ] e ihz dζ
E w = E 0 ∑ q=1 2 ∫ k q 2 sin θ k d θ k ∫ 0 2π F q e ( θ k , ϕ k ) f q ( θ k , ϕ k ) e i k q ⋅r d ϕ k ,
k 1 = a 1 ,     k 2 = a 1 a 2 1 a 1 2 sin 2 θ k + a 2 2 cos 2 θ k ,
F q e ( θ k , ϕ k ) e i k q ⋅r =( F qx e x ^ + F qy e y ^ + F qz e z ^ ) e i k q ⋅r ,
F qx e ={ −sin ϕ k , q=1 − a 2 2 a 1 2 cos θ k sin θ k cos ϕ k , q=2
F qy e ={ cos ϕ k , q=1 − a 2 2 a 1 2 cos θ k sin θ k sin ϕ k , q=2
F qz e ={ 0 , q=1 1 , q=2
f q ( θ k , ϕ k )= ∑ n=−∞ ∞ G ′ nq ( θ k ) e in ϕ k ,
E w = E 0 ∑ q=1 2 ∑ n=−∞ ∞ ∫ G ′ nq ( θ k ) k q 2 sin θ k d θ k ∫ 0 2π F q e ( θ k , ϕ k ) e i k q ⋅r e in ϕ k d ϕ k ,
x ^ e ik⋅r = ∑ m=−∞ ∞ ( a m x m mλ (1) + b m x n mλ (1) + c m x l mλ (1) ) e ihz ,
y ^ e ik⋅r = ∑ m=−∞ ∞ ( a m y m mλ (1) + b m y n mλ (1) + c m y l mλ (1) ) e ihz ,
z ^ e ik⋅r = ∑ m=−∞ ∞ ( a m z m mλ (1) + b m z n mλ (1) + c m z l mλ (1) ) e ihz ,
[ a m x b m x c m x ]= i m−1 e −im ϕ k k [ 1 sin θ k sin ϕ k i cos θ k sin θ k cos ϕ k sin θ k cos ϕ k ],
[ a m y b m y c m y ]= i m−1 e −im ϕ k k [ − 1 sin θ k cos ϕ k −i cos θ k sin θ k sin ϕ k sin θ k sin ϕ k ],
[ a m z b m z c m z ]= i m−1 e −im ϕ k k [ 0 i cos θ k ].
E w = E 0 ∑ q=1 2 ∑ m=−∞ ∞ ∫ G mq ( θ k )[ A q e ( θ k ) m m λ q (1) + B q e ( θ k ) n m λ q (1) + C q e ( θ k ) l m λ q (1) ] e i h q z d θ k ,
A 1 e ( θ k )=1,   B 1 e ( θ k )= C 1 e ( θ k )= A 2 e ( θ k )=0,
B 2 e ( θ k )=−i a 1 2 sin 2 θ k + a 2 2 cos 2 θ k a 1 2 sin θ k ,
C 2 e ( θ k )=− a 1 2 − a 2 2 a 1 2 sin θ k cos θ k ,

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