Abstract

Scattering of a Hermite–Gaussian beam field by a chiral sphere is analyzed. A Hermite–Gaussian beam field is expressed as a superposition of multipole fields at complex-source points. Electromagnetic fields are expanded in terms of the spherical vector wave functions. The unknown expansion coefficients for the scattered field and the internal field are determined by the boundary conditions. As numerical examples, the scattered near fields of the beam incidence are calculated, and the effects of the chirality and the radius of the chiral sphere on the fields are examined. The results for a Gaussian beam incidence are also compared with those of a plane-wave incidence.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newslett., October1988, pp. 6–12.
  2. A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).
  3. I. V. Lindel, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech, Norwood, Mass., 1994).
  4. K. Matsumoto, K. Rokushima, J. Yamakita, “Analysis of diffracted waves from isotropic chiral gratings,” Trans. Inst. Electron. Inf. Commun. Eng. Jpn. J79-C-I, 165–172 (1996).
  5. K. Kunishi, M. Koshiba, Y. Tsuji, “Finite element analysis of polarization characteristics of chiral grating,” Trans. Inst. Electron. Info. Commun. Eng. Jpn. J82-C-I, 318–325 (1999).
  6. S. He, Y. Hu, “Electromagnetic scattering from a stratified bi-isotropic (nonreciprocal chiral) slab: numerical computations,” IEEE Trans. Antennas Propag. 41, 1057–1061 (1993).
    [Crossref]
  7. S. He, Y. Hu, S. Ström, “Electromagnetic reflection and transmission for a stratified bianisotropic slab,” IEEE Trans. Antennas Propag. 42, 856–858 (1994).
    [Crossref]
  8. M. Tanaka, A. Kusunoki, “Scattering characteristics of stratified chiral slab,” IEICE Trans. Electr. E76-C, 1443–1448 (1993).
  9. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [Crossref]
  10. C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
    [Crossref]
  11. X. B. Wu, K. Yasumoto, “Cylindrical vector-wave function representations of fields in a biaxial Ω-medium,” J. Electromagn. Waves Appl. 11, 1407–1423 (1997).
    [Crossref]
  12. M. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 299–301 (1990).
    [Crossref] [PubMed]
  13. S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
    [Crossref]
  14. I. E. Psarobas, N. Stefanou, A. Modinos, “Photonic crystals of chiral spheres,” J. Opt. Soc. Am. A 16, 343–347 (1999).
    [Crossref]
  15. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [Crossref]
  16. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [Crossref]
  17. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [Crossref]
  18. M. Yokota, T. Takenaka, O. Fukumitsu, “Scattering of a Hermite–Gaussian beam mode by parallel dielectric circular cylinders,” J. Opt. Soc. Am. A 3, 580–586 (1986).
    [Crossref]
  19. J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
    [Crossref]
  20. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  21. K. Umashankar, A. Taflove, S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. AP-34, 758–766 (1986).
    [Crossref]
  22. A. Yariv, Optical Electronics, 3rd ed. (Holt-Saunders International, Tokyo, 1985).
  23. A. H. Sihvola, I. V. Lindel, “Analysis on chiral mixtures,” J. Electromagn. Waves Appl. 6, 553–572 (1992).
    [Crossref]

1999 (3)

K. Kunishi, M. Koshiba, Y. Tsuji, “Finite element analysis of polarization characteristics of chiral grating,” Trans. Inst. Electron. Info. Commun. Eng. Jpn. J82-C-I, 318–325 (1999).

S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
[Crossref]

I. E. Psarobas, N. Stefanou, A. Modinos, “Photonic crystals of chiral spheres,” J. Opt. Soc. Am. A 16, 343–347 (1999).
[Crossref]

1997 (1)

X. B. Wu, K. Yasumoto, “Cylindrical vector-wave function representations of fields in a biaxial Ω-medium,” J. Electromagn. Waves Appl. 11, 1407–1423 (1997).
[Crossref]

1996 (1)

K. Matsumoto, K. Rokushima, J. Yamakita, “Analysis of diffracted waves from isotropic chiral gratings,” Trans. Inst. Electron. Inf. Commun. Eng. Jpn. J79-C-I, 165–172 (1996).

1994 (1)

S. He, Y. Hu, S. Ström, “Electromagnetic reflection and transmission for a stratified bianisotropic slab,” IEEE Trans. Antennas Propag. 42, 856–858 (1994).
[Crossref]

1993 (2)

M. Tanaka, A. Kusunoki, “Scattering characteristics of stratified chiral slab,” IEICE Trans. Electr. E76-C, 1443–1448 (1993).

S. He, Y. Hu, “Electromagnetic scattering from a stratified bi-isotropic (nonreciprocal chiral) slab: numerical computations,” IEEE Trans. Antennas Propag. 41, 1057–1061 (1993).
[Crossref]

1992 (1)

A. H. Sihvola, I. V. Lindel, “Analysis on chiral mixtures,” J. Electromagn. Waves Appl. 6, 553–572 (1992).
[Crossref]

1990 (1)

1988 (1)

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newslett., October1988, pp. 6–12.

1986 (2)

M. Yokota, T. Takenaka, O. Fukumitsu, “Scattering of a Hermite–Gaussian beam mode by parallel dielectric circular cylinders,” J. Opt. Soc. Am. A 3, 580–586 (1986).
[Crossref]

K. Umashankar, A. Taflove, S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. AP-34, 758–766 (1986).
[Crossref]

1983 (1)

1978 (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[Crossref]

1977 (1)

1974 (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[Crossref]

1973 (1)

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

Bohren, C. F.

C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[Crossref]

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[Crossref]

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

Engheta, N.

M. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 299–301 (1990).
[Crossref] [PubMed]

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newslett., October1988, pp. 6–12.

Felsen, L. B.

Fukumitsu, O.

He, S.

S. He, Y. Hu, S. Ström, “Electromagnetic reflection and transmission for a stratified bianisotropic slab,” IEEE Trans. Antennas Propag. 42, 856–858 (1994).
[Crossref]

S. He, Y. Hu, “Electromagnetic scattering from a stratified bi-isotropic (nonreciprocal chiral) slab: numerical computations,” IEEE Trans. Antennas Propag. 41, 1057–1061 (1993).
[Crossref]

Hu, Y.

S. He, Y. Hu, S. Ström, “Electromagnetic reflection and transmission for a stratified bianisotropic slab,” IEEE Trans. Antennas Propag. 42, 856–858 (1994).
[Crossref]

S. He, Y. Hu, “Electromagnetic scattering from a stratified bi-isotropic (nonreciprocal chiral) slab: numerical computations,” IEEE Trans. Antennas Propag. 41, 1057–1061 (1993).
[Crossref]

Jaggard, D. L.

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newslett., October1988, pp. 6–12.

Kim, J. S.

Koshiba, M.

K. Kunishi, M. Koshiba, Y. Tsuji, “Finite element analysis of polarization characteristics of chiral grating,” Trans. Inst. Electron. Info. Commun. Eng. Jpn. J82-C-I, 318–325 (1999).

Kowarz, M.

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Kunishi, K.

K. Kunishi, M. Koshiba, Y. Tsuji, “Finite element analysis of polarization characteristics of chiral grating,” Trans. Inst. Electron. Info. Commun. Eng. Jpn. J82-C-I, 318–325 (1999).

Kusunoki, A.

M. Tanaka, A. Kusunoki, “Scattering characteristics of stratified chiral slab,” IEICE Trans. Electr. E76-C, 1443–1448 (1993).

Laghari, M. S.

S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
[Crossref]

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

Lee, S. S.

Lindel, I. V.

A. H. Sihvola, I. V. Lindel, “Analysis on chiral mixtures,” J. Electromagn. Waves Appl. 6, 553–572 (1992).
[Crossref]

I. V. Lindel, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech, Norwood, Mass., 1994).

Mahmoud, F.

S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
[Crossref]

Matsumoto, K.

K. Matsumoto, K. Rokushima, J. Yamakita, “Analysis of diffracted waves from isotropic chiral gratings,” Trans. Inst. Electron. Inf. Commun. Eng. Jpn. J79-C-I, 165–172 (1996).

Modinos, A.

Psarobas, I. E.

Rao, S. M.

K. Umashankar, A. Taflove, S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. AP-34, 758–766 (1986).
[Crossref]

Rokushima, K.

K. Matsumoto, K. Rokushima, J. Yamakita, “Analysis of diffracted waves from isotropic chiral gratings,” Trans. Inst. Electron. Inf. Commun. Eng. Jpn. J79-C-I, 165–172 (1996).

Samir, S.

S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
[Crossref]

Shin, S. Y.

Siegman, A. E.

Sihvola, A. H.

A. H. Sihvola, I. V. Lindel, “Analysis on chiral mixtures,” J. Electromagn. Waves Appl. 6, 553–572 (1992).
[Crossref]

I. V. Lindel, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech, Norwood, Mass., 1994).

Stefanou, N.

Ström, S.

S. He, Y. Hu, S. Ström, “Electromagnetic reflection and transmission for a stratified bianisotropic slab,” IEEE Trans. Antennas Propag. 42, 856–858 (1994).
[Crossref]

Taflove, A.

K. Umashankar, A. Taflove, S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. AP-34, 758–766 (1986).
[Crossref]

Takenaka, T.

Tanaka, M.

M. Tanaka, A. Kusunoki, “Scattering characteristics of stratified chiral slab,” IEICE Trans. Electr. E76-C, 1443–1448 (1993).

Tariq, S.

S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
[Crossref]

Tretyakov, S. A.

I. V. Lindel, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech, Norwood, Mass., 1994).

Tsuji, Y.

K. Kunishi, M. Koshiba, Y. Tsuji, “Finite element analysis of polarization characteristics of chiral grating,” Trans. Inst. Electron. Info. Commun. Eng. Jpn. J82-C-I, 318–325 (1999).

Umashankar, K.

K. Umashankar, A. Taflove, S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. AP-34, 758–766 (1986).
[Crossref]

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

Viitanen, A. J.

I. V. Lindel, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech, Norwood, Mass., 1994).

Wu, X. B.

X. B. Wu, K. Yasumoto, “Cylindrical vector-wave function representations of fields in a biaxial Ω-medium,” J. Electromagn. Waves Appl. 11, 1407–1423 (1997).
[Crossref]

Yamakita, J.

K. Matsumoto, K. Rokushima, J. Yamakita, “Analysis of diffracted waves from isotropic chiral gratings,” Trans. Inst. Electron. Inf. Commun. Eng. Jpn. J79-C-I, 165–172 (1996).

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Holt-Saunders International, Tokyo, 1985).

Yasumoto, K.

X. B. Wu, K. Yasumoto, “Cylindrical vector-wave function representations of fields in a biaxial Ω-medium,” J. Electromagn. Waves Appl. 11, 1407–1423 (1997).
[Crossref]

Yokota, M.

Chem. Phys. Lett. (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[Crossref]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

IEEE AP-S Newslett. (1)

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newslett., October1988, pp. 6–12.

IEEE Trans. Antennas Propag. (3)

S. He, Y. Hu, “Electromagnetic scattering from a stratified bi-isotropic (nonreciprocal chiral) slab: numerical computations,” IEEE Trans. Antennas Propag. 41, 1057–1061 (1993).
[Crossref]

S. He, Y. Hu, S. Ström, “Electromagnetic reflection and transmission for a stratified bianisotropic slab,” IEEE Trans. Antennas Propag. 42, 856–858 (1994).
[Crossref]

K. Umashankar, A. Taflove, S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. AP-34, 758–766 (1986).
[Crossref]

IEICE Trans. Electr. (1)

M. Tanaka, A. Kusunoki, “Scattering characteristics of stratified chiral slab,” IEICE Trans. Electr. E76-C, 1443–1448 (1993).

J. Colloid Interface Sci. (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[Crossref]

J. Electromagn. Waves Appl. (2)

X. B. Wu, K. Yasumoto, “Cylindrical vector-wave function representations of fields in a biaxial Ω-medium,” J. Electromagn. Waves Appl. 11, 1407–1423 (1997).
[Crossref]

A. H. Sihvola, I. V. Lindel, “Analysis on chiral mixtures,” J. Electromagn. Waves Appl. 6, 553–572 (1992).
[Crossref]

J. Opt. Soc. Am. (3)

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

Opt. Lett. (1)

Radio Sci. (1)

S. Tariq, S. Samir, F. Mahmoud, M. S. Laghari, “Microwave Gaussian beam splitting with a variable split angle by using a chiral lens,” Radio Sci. 34, 9–18 (1999).
[Crossref]

Trans. Inst. Electron. Inf. Commun. Eng. Jpn. (1)

K. Matsumoto, K. Rokushima, J. Yamakita, “Analysis of diffracted waves from isotropic chiral gratings,” Trans. Inst. Electron. Inf. Commun. Eng. Jpn. J79-C-I, 165–172 (1996).

Trans. Inst. Electron. Info. Commun. Eng. Jpn. (1)

K. Kunishi, M. Koshiba, Y. Tsuji, “Finite element analysis of polarization characteristics of chiral grating,” Trans. Inst. Electron. Info. Commun. Eng. Jpn. J82-C-I, 318–325 (1999).

Other (4)

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

I. V. Lindel, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech, Norwood, Mass., 1994).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

A. Yariv, Optical Electronics, 3rd ed. (Holt-Saunders International, Tokyo, 1985).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometry of the beam scattering by a chiral sphere.

Fig. 2
Fig. 2

Equivalent surface current for a dielectric sphere with ka=1 and r=4: (a) Electric currents Jθ (solid curve) and Jϕ (dashed curve) and (b) magnetic currents Mθ (solid curve) and Mϕ (dashed curve).

Fig. 3
Fig. 3

Normalized scattered near field |Ex| of a Gaussian beam at the focal plane, z=f, for κ=0 (solid curve), κ=0.1 (dashed curve), and κ=0.3 (dotted curve). The nonreciprocity parameter is χ=0, and the radius of the sphere is (a) a=λ and (b) a=1.3λ.

Fig. 4
Fig. 4

Normalized scattered near field |Ex| of a plane wave at the focal plane, z=f, for κ=0 (solid curve), κ=0.1 (dashed curve), and κ=0.3 (dotted curve). The nonreciprocity parameter is χ=0, and the radius of the sphere is (a) a=λ and (b) a=1.3λ.

Fig. 5
Fig. 5

Normalized scattered near field |Ex| of a Gaussian beam at the focal plane, z=f, for κ=0 (solid curve), κ=0.1 (dashed curve), and κ=0.3 (dotted curve). The other parameters are z0=2λ, χ=0, and a=1.3λ.

Fig. 6
Fig. 6

Normalized scattered near field |Ex| for the ψ1,0 incidence at the focal plane, z=f, for κ=0 (solid curve), κ=0.1 (dashed curve), and κ=0.3 (dotted curve). The other parameters are z0=f, χ=0, and a=1.0λ.

Fig. 7
Fig. 7

Normalized scattered near field |Ex| of a Gaussian beam at the focal plane, z=f, for χ=0 (solid curve), χ=0.1 (dashed curve), χ=0.2 (dotted curve), χ=0.3 (dotted–dashed curve), and χ=0.4 (dotted–dotted–dashed curve). The other parameters are z0=f, κ=0.2, and a=1.0λ.

Fig. 8
Fig. 8

Normalized scattered near field |Ey| of a Gaussian beam at the focal plane, z=f, for κ=0 (solid curve), κ=0.1 (dashed curve), and κ=0.3 (dotted curve). The other parameters are χ=0 and a=1.0λ.

Tables (1)

Tables Icon

Table 1 Focal Length and the Spot Size of the Incident Beam at the Origin for Different Values of the Radius of the Chiral Sphere

Equations (53)

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

E=iωμ00 ××A,
H=1μ0 ×A,
A=xˆexp(ik0R)ik0R,
Aμ,ν=xˆμ+νxμyνexp(ik0R)ik0R.
xˆψμ,νk0b exp(-k0b)2μ!ν!π1/2exp(-ik0z0)×p=0[μ/2]q=0[ν/2]2-p+q(-w0)μ+ν-2(p+q)w0p!q!(μ-2p)!(ν-2q)!×Aμ-2p,ν-2q,
ψμ,ν=1π2μ+ν+1μ!ν!1/2σw0×Hμ2|σ| xw0Hν2|σ| yw0×exp-σxw02+yw02-i(μ+ν)arg σ+ik0z,
σ=[1+i(z+z0)/b]-1,
Hn(ν)=(-1)nexp(ν2) dndνnexp(-ν2).
D=0rE+ξH,
B=ζE+μ0μrH,
ξ=(χ+iκ)(ε0μ0)1/2,
ζ=(χ-iκ)(ε0μ0)1/2.
χ2+κ2<εrμr.
Ml,m(s)(k, r)=×[rZl(s)(kr)Ylm(θ, ϕ)],
Nl,m(s)(k, r)=1ik ×Ml,m(s)(k, r),s=1,2,
Ylm(θ, ϕ)=2l+14π(l-|m|)!(l+|m|)!1/2Pl|m|(cos θ)exp(imϕ),
Einc=iωl=1+m=-ll12 [α(η)(μ, ν : l, m)-iβ(η)(μ, ν : l, m)]Pl,m(s)(k0, r)+12 [α(η)(μ, ν : l, m)+iβ(η)(η, ν : l, m)]Ql,m(s)(k0, r),
Hinc=iωε0μ01/2i=1+m=-ll-i2 [α(η)(μ, ν : l, m)-iβ(η)(μ, ν : l, m)]Pl,m(s)(k0, r)+i2 [α(n)(μ, ν : l, m)+iβ(η)(μ, ν : l, m)]Ql,m(s)(k0, r),
Pl,m(s)(k, r)=Ml,m(s)(k, r)+iNl,m(s)(k, r),
Ql,m(s)(k, r)=Ml,m(s)(k, r)-iNl,m(s)(k, r),s=1, 2.
η=1,s=2,forr>|r0|,
η=2,s=1,forr<|r0|.
k±=ω(ε0μ0)1/2[(εrμr-χ2)1/2±κ].
Echi=l=1+m=-ll[al,mPl,m(1)(k+, r)+bl,mQl,m(1)(k-, r)],
Esct=l=l+m=-ll[cl,mPl,m(2)(k0, r)+dl,mQl,m(2)(k0, r)].
rˆ×Einc=l=1+m=-ll[αl,mincGl,m(θ, ϕ)+βl,mincRl,m(θ, ϕ)],
rˆ×Hinc=l=1+m=-ll[γl,mincGl,m(θ, ϕ)+δl,mincRl,m(θ, ϕ)],
Gl,m(θ, ϕ)=1[l(l+1)]1/2 S1Ylm(θ, ϕ),
Rl,m(θ, ϕ)=rˆ×Gl,m(θ, ϕ),
rˆ×E=l=1+m=-ll[αl,mGl,m(θ, ϕ)+βl,mRl,m(θ, ϕ)].
f=(r)1/2[(r)1/2-1]2 a.
J(θ, ϕ)=rˆ×H,
M(θ, ϕ)=E×rˆ.
wout=f λπw0.
α(η)(μ, ν : l, m)=-i2l(l+1) [Hμ,ν(η)(l, m-1)+(l-m)(l+m+1)×Hμ,ν(η)(l, m+1)],
β(η)(μ, ν : l, m)=12l(l+1)l+12l-1×[Hμ,ν(η)(l-1, m-1)-(l-m-1)(l-m)×Hμ,ν(η)(l-1, m+1)]-l2l+3[Hμ,ν(η)(l+1, m-1)-(l+m+1)(l+m+2)×Hμ,ν(η)(l+1, m+1)],
H0,0(η)(l, m)=4π2l+1(l-|m|)!(l+|m|)!1/2×Zl(η)(kr0)Plm(cos θ0)×exp(-imϕ0),
r0=[x02+y02+(-z0+ib)2]1/2,
cos θ0=-z0+ibr0,
ϕ0=tan-1y0x0.
Ml,m(s)(k,r)=-[l(l+1)]1/2Zl(s)(kr)Rl,m(θ, ϕ),
Nl,m(s)(k, r)=[l(l+1)]1/2ikr [Zl(s)(kr)+krZl(s)(kr)]Gl,m(θ, ϕ)-l(l+1)ikr×Zl(s)(kr)Ylm(θ, ϕ)rˆ,s=1, 2.
al,m=1[l(l+1)]1/2fl(k-a)αl,m+jl(k-a)βl,mjl(k+a)fl(k-a)+jl(k-a)fl(k+a),
bl,m=1[l(l+1)]1/2fl(k+a)αl,m-jl(k+a)βl,mjl(k+a)fl(k-a)+jl(k-a)fl(k+a),
cl,m=1[l(l+1)]1/2gl(k0a)(αl,m-αl,minc)+hl(1)(k0a)(βl,m-βl,minc)2hl(1)(k0a)gl(k0a),
dl,m=1[l(l+1)]1/2gl(k0a)(αl,m-αl,minc)-hl(1)(k0a)(βl,m-βl,minc)2hl(1)(k0a)gl(k0a),
αl,m=δl,minc-k0iωμ0gl(k0a)hl(1)(k0a) αl,minc(rμr-χ2)1/2iμrε0μ01/2ql-k0iωμ0hl(1)(k0a)gl(k0a)+χ-isl(rμr-χ2)1/2μrε0μ01/2γl,minc-k0iωμ0hl(1)(k0a)gl(k0a) βl,minc/(rμr-χ2)1/2iμrε0μ01/2pl-k0iωμ0gl(k0a)hl(1)(k0a)×(rμr-χ2)1/2iμε0μ01/2ql-k0iωμ0hl(1)(k0a)gl(k0a)-0μr2μ0 [χ2+(rμr-χ2)sl2],
βl,m=(rμr-χ2)1/2iμrε0μ01/2pl-k0iωμ0gl(k0a)hl(1)(k0a)γl,minc-k0iωμ0hl(1)(k0a)gl(k0a) βl,minc+χ+isl(rμr-χ2)1/2μrε0μ01/2δl,minc-k0iωμ0gl(k0a)hl(1)(k0a) αl,minc/(μ-χ2)1/2iμrε0μ01/2pl-k0iωμ0gl(k0a)hl(1)(k0a)×(μ-χ2)1/2iμrε0μ01/2ql-k0iωμ0hl(1)(k0a)gl(k0a)-ε0μr2μ0 [χ2+(rμr-χ2)sl2],
pl=2 fl(k+a)fl(k-a)jl(k+a)fl(k-a)+jl(k-a)fl(k+a),
ql=2jl(k+a)jl(k-a)jl(k+a)fl(k-a)+jl(k-a)fl(k+a),
sl=jl(k+a)fl(k-a)-jl(k-a)fl(k+a)jl(k+a)fl(k-a)+jl(k-a)fl(k+a),
fl(x)=jl(x)x+jl(x),
gl(x)=hl(1)(x)x+hl(1)(x),

Metrics