Abstract

The differential theory of diffraction of light by an arbitrary object described in spherical coordinates is developed. Expanding the fields on the basis of vector spherical harmonics, we reduce the Maxwell equations to an infinite first-order differential set. In view of the truncation required for numerical integration, correct factorization rules are derived to express the components of D in terms of the components of E, a process that extends the fast Fourier factorization to the basis of vector spherical harmonics. Numerical overflows and instabilities are avoided through the use of the S-matrix propagation algorithm for carrying out the numerical integration. The method can analyze any shape and/or material, dielectric or conducting. It is particularly simple when applied to rotationally symmetric objects.

© 2005 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  2. M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  3. L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).
  4. L. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Librairie Lehmann et Stage, Oeuvres scientifique de L. Lorenz, revues et annotées par H. Valentiner, 1898(transl. of Ref. [3]).
  5. G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
  6. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  7. L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
    [CrossRef]
  8. L. Tsang, J. A. Kong, K.-H. Ding, C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
  9. L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 2001).
  10. In the community of grating theoreticians, as well as in the one of waveguides, this matrix is called the S matrix or scattering matrix. It is essentially the S12 block of the S(M−1) matrix appearing later on in Eq. (158); see also Eqs. (163, 165).
  11. W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).
  12. P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
    [CrossRef]
  13. M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction grating problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
    [CrossRef]
  14. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  15. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  16. E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  17. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  18. E. Popov, M. Nevière, N. Bonod, “Factorization of products of discontinuous functions applied to Fourier–Bessel basis,” J. Opt. Soc. Am. A 21, 46–51 (2004).
    [CrossRef]
  19. N. Bonod, E. Popov, M. Neviere, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
    [CrossRef]
  20. M. Nevière, E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, 2003).
  21. C. Cohen-Tannoudji, Photons & Atomes—Introduction à l’électrodynamique quantique (InterEdition/ Editions du CNRS, 1987).
  22. J. D. Jackson, Classical Electrodynamics (Wiley, 1965).
  23. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).
  24. B. Stout, J.-C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).
  25. B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
    [CrossRef]
  26. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  27. D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]
  28. J. A. Gaunt, “On the triplets of Helium,” Philos. Trans. R. Soc. London, Ser. A 228, 151–196 (1929).
    [CrossRef]
  29. Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
    [CrossRef]

2005 (1)

2004 (1)

2002 (1)

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

2001 (2)

B. Stout, J.-C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

2000 (1)

1998 (1)

1996 (4)

1994 (1)

1979 (1)

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

1929 (1)

J. A. Gaunt, “On the triplets of Helium,” Philos. Trans. R. Soc. London, Ser. A 228, 151–196 (1929).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

1890 (1)

L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding, C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).

Auger, J. C.

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

Auger, J.-C.

B. Stout, J.-C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

Bagieu, M.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Bonod, N.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, Photons & Atomes—Introduction à l’électrodynamique quantique (InterEdition/ Editions du CNRS, 1987).

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

Gaunt, J. A.

J. A. Gaunt, “On the triplets of Helium,” Philos. Trans. R. Soc. London, Ser. A 228, 151–196 (1929).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1965).

Kong, J. A.

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 2001).

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Lacis, A. A.

M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Lafait, J.

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

Li, L.

Lorenz, L.

L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

L. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Librairie Lehmann et Stage, Oeuvres scientifique de L. Lorenz, revues et annotées par H. Valentiner, 1898(transl. of Ref. [3]).

Mackowski, D. W.

Maystre, D.

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Mischenko, M. I.

M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Mishchenko, M. I.

Neviere, M.

Nevière, M.

Popov, E.

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Stout, B.

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

Travis, L. D.

M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Tsang, L.

L. Tsang, J. A. Kong, K.-H. Ding, C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 2001).

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

Xu, Y. L.

Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

J. Mod. Opt. (2)

B. Stout, J.-C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

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

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
[CrossRef]

M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction grating problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

E. Popov, M. Nevière, N. Bonod, “Factorization of products of discontinuous functions applied to Fourier–Bessel basis,” J. Opt. Soc. Am. A 21, 46–51 (2004).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
[CrossRef]

Math. Comput. (1)

Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (1)

J. A. Gaunt, “On the triplets of Helium,” Philos. Trans. R. Soc. London, Ser. A 228, 151–196 (1929).
[CrossRef]

Vidensk. Selk. Skr. (1)

L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Other (13)

L. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Librairie Lehmann et Stage, Oeuvres scientifique de L. Lorenz, revues et annotées par H. Valentiner, 1898(transl. of Ref. [3]).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

M. I. Mischenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

L. Tsang, J. A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

L. Tsang, J. A. Kong, K.-H. Ding, C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 2001).

In the community of grating theoreticians, as well as in the one of waveguides, this matrix is called the S matrix or scattering matrix. It is essentially the S12 block of the S(M−1) matrix appearing later on in Eq. (158); see also Eqs. (163, 165).

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).

M. Nevière, E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, 2003).

C. Cohen-Tannoudji, Photons & Atomes—Introduction à l’électrodynamique quantique (InterEdition/ Editions du CNRS, 1987).

J. D. Jackson, Classical Electrodynamics (Wiley, 1965).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

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Figures (2)

Fig. 1
Fig. 1

Description of the diffracting object and notations.

Fig. 2
Fig. 2

Example of the partition of the modulated region in which M = 6 , and an illustration of the notation for the coefficients appearing in Eq. (131) used inside the various homogeneous regions.

Equations (245)

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f ( r , θ , ϕ ) = 0
r = g ( θ , ϕ ) .
ε ( r , θ , ϕ ) = n = 0 m = n n ε n m ( r ) Y n m ( θ , ϕ ) ,
ε n m ( r ) = Y n m ε 0 4 π Y n m * ( θ , ϕ ) ε ( r , θ , ϕ ) d Ω .
g f g * ( x ) f ( x ) d x .
ε n m ( r ) = 0 2 π 0 π ε ( r , θ , ϕ ) Y n m * ( θ , ϕ ) sin θ d θ d ϕ .
Y n m ( θ , ϕ ) r ̂ Y n m ( θ , ϕ ) ,
X n m ( θ , ϕ ) Z n m ( θ , ϕ ) × r ̂ ,
Z n m ( θ , ϕ ) r Y n m ( θ , ϕ ) n ( n + 1 )
if n 0 , Z 0 0 ( θ , ϕ ) 0 .
Z n m ( θ , ϕ ) = r ̂ × X n m ( θ , ϕ ) .
W n m ( i ) W ν μ ( j ) 0 4 π W n m ( i ) * W ν μ ( j ) d Ω = δ i j δ n ν δ m μ ,
Y n m ( θ , ϕ ) = [ 2 n + 1 4 π ( n m ) ! ( n + m ) ! ] 1 2 P n m ( cos θ ) exp ( i m ϕ )
Y n m ( θ , ϕ ) = P ¯ n m ( cos θ ) exp ( i m ϕ ) .
u ¯ n m ( cos θ ) = 1 n ( n + 1 ) m sin θ P ¯ n m ( cos θ ) ,
s ¯ n m ( cos θ ) = 1 n ( n + 1 ) d d θ P ¯ n m ( cos θ ) ,
X n m ( θ , ϕ ) = i u ¯ n m ( cos θ ) exp ( i m ϕ ) θ ̂ s ¯ n m ( cos θ ) exp ( i m ϕ ) ϕ ̂ ,
Z n m ( θ , ϕ ) = s ¯ n m ( cos θ ) exp ( i m ϕ ) θ ̂ + i u ¯ n m ( cos θ ) exp ( i m ϕ ) ϕ ̂ .
curl [ h ( r ) Y n m ] = [ n ( n + 1 ) ] 1 2 h ( r ) r X n m ,
curl [ h ( r ) X n m ] = h ( r ) r [ n ( n + 1 ) ] 1 2 Y n m + [ h ( r ) r + h ( r ) ] Z n m ,
curl [ h ( r ) Z n m ] = [ h ( r ) r + h ( r ) ] X n m .
U ( r , θ , ϕ ) = n = 0 m = n n [ A Y n m ( r ) Y n m ( θ , ϕ ) + A X n m ( r ) X n m ( θ , ϕ ) + A Z n m ( r ) Z n m ( θ , ϕ ) ] .
U ( r , θ , ϕ ) = p = 1 N [ A Y p ( r ) Y p ( θ , ϕ ) + A X p ( r ) X p ( θ , ϕ ) + A Z p ( r ) Z p ( θ , ϕ ) ] ,
p = n ( n + 1 ) + m + 1 .
n = Int p 1 , m = p 1 n ( n + 1 ) ,
[ E l ] { [ E Y , p E X , p E Z , p ] } 3 × ( n max + 1 ) 2 .
N ̂ ( θ , ϕ ) = grad f grad f f = 0 ,
N r = f r grad f f = 0 , N θ = 1 r f θ grad f f = 0 ,
N ϕ = 1 r sin θ f ϕ grad f f = 0 .
N Y n m = Y n m N ̂ = Y n m r ̂ r ̂ N r = 0 4 π Y n m * ( θ , ϕ ) N r ( θ , ϕ ) d Ω = 0 2 π 0 π N r P ¯ n m ( cos θ ) exp ( i m ϕ ) sin θ d θ d ϕ ,
N X n m = 0 2 π 0 π [ i N θ u ¯ n m ( cos θ ) N ϕ s ¯ n m ( cos θ ) ] exp ( i m ϕ ) sin θ d θ d ϕ ,
N Z n m = 0 2 π 0 π [ N θ s ¯ n m ( cos θ ) + i N ϕ u ¯ n m ( cos θ ) ] exp ( i m ϕ ) sin θ d θ d ϕ .
N Y n 0 = 2 π 0 π N r ( θ ) P ¯ n 0 ( cos θ ) sin θ d θ ,
N X n 0 = 0 ,
N Z n 0 = 2 π 0 π N θ ( θ ) s ¯ n 0 ( cos θ ) sin θ d θ ,
ε n , 0 = 2 π 0 π ε ( r , θ ) P ¯ n 0 ( cos θ ) sin θ d θ .
p = 1 N { curl [ E Y p ( r ) Y p ] + curl [ E X p ( r ) X p ] + curl [ E Z p ( r ) Z p ] } = i ω μ μ 0 p = 1 N ( H Y p Y p + H X p X p + H Z p Z p ) .
n ( n + 1 ) E Y p ( r ) r X p + n ( n + 1 ) E X p ( r ) r Y p + [ E X p ( r ) r + d E X p ( r ) d r ] Z p [ E Z p ( r ) r + d E Z p ( r ) d r ] X p = i ω μ μ 0 ( H Y p Y p + H X p X p + H Z p Z p ) .
a p E X p r = i ω μ μ 0 H Y p ,
a p E Y p r E Z p r d E Z p d r = i ω μ μ 0 H X p ,
E X p r + d E X p d r = i ω μ μ 0 H Z p .
a p H X p r = i ω D Y p ,
a p H Y p r H Z p r d H Z p d r = i ω D X p ,
H X p r + d H X p d r = i ω D Z p .
[ D ] = ϵ 0 Q ε [ E ] .
Q ε = ( Q ε Y Y Q ε Y X Q ε Y Z Q ε X Y Q ε X X Q ε X Z Q ε Z Y Q ε Z X Q ε Z Z ) .
1 ϵ 0 [ D Y ] = Q ε Y Y [ E Y ] + Q ε Y X [ E X ] + Q ε Y Z [ E Z ] ,
[ E Y ] = ( Q ε Y Y ) 1 ( 1 ϵ 0 [ D Y ] Q ε Y X [ E X ] Q ε Y Z [ E Z ] ) .
1 ϵ 0 [ D X ] = Q ε X Y [ E Y ] + Q ε X X [ E X ] + Q ε X Z [ E Z ] ,
1 ϵ 0 [ D Z ] = Q ε Z Y [ E Y ] + Q ε Z X [ E X ] + Q ε Z Z [ E Z ] .
E X p r + d E X p d r = i ω μ μ 0 H Z p
a p r ( ( Q ε Y Y ) 1 [ i a ω ϵ 0 r [ H X ] Q ε Y X [ E X ] Q ε Y Z [ E Z ] ] ) p E Z p r d E Z p d r = i ω μ μ 0 H X p ,
H X p r + d H X p d r = i ω ϵ 0 ( Q ε Z X [ E X ] ) p i ω ϵ 0 ( Q ε Z Z [ E Z ] ) p i ω ϵ 0 ( Q ε Z Y Q ε Y Y 1 ( i a ω ϵ 0 r [ H X ] Q ε Y X [ E X ] Q ε Y Z [ E Z ] ) ) p ,
i a p 2 ω μ μ 0 E X p r 2 + H Z p r + d H Z p d r = i ω ϵ 0 ( Q ε X X [ E X ] ) p + i ω ϵ 0 ( Q ε X Z [ E Z ] ) p + i ω ϵ 0 ( Q ε X Y Q ε Y Y 1 ( i a ω ϵ 0 r [ H X ] Q ε Y X [ E X ] Q ε Y Z [ E Z ] ) ) p .
[ F ] = [ [ E X ] [ E Z ] [ H ̃ X ] [ H ̃ Z ] ] ,
H ̃ Z 0 H = μ 0 ϵ 0 H = 1 c ϵ 0 H .
d [ F ] d r = M ( r ) [ F ] ,
M 11 = 1 r , M 12 = M 13 = 0 , M 14 = i μ ω c 1 ,
M 21 = a r Q ε Y Y 1 Q ε Y X , M 22 = 1 r a r Q ε Y Y 1 Q ε Y Z ,
M 23 = i ω c ( ( c r ω ) 2 a Q ε Y Y 1 a μ 1 ) , M 24 = 0 ,
M 31 = i ω c ( Q ε Z Y Q ε Y Y 1 Q ε Y X Q ε Z X ) ,
M 32 = i ω c ( Q ε Z Y Q ε Y Y 1 Q ε Y Z Q ε Z Z ) ,
M 33 = 1 r ( Q ε Z Y Q ε Y Y 1 a 1 ) , M 34 = 0 ,
M 41 = i ω c ( Q ε X X Q ε X Y Q ε Y Y 1 Q ε Y X 1 μ ( a c ω r ) 2 ) ,
M 42 = i ω c ( Q ε X Z Q ε X Y Q ε Y Y 1 Q ε Y Z ) ,
M 43 = Q ε X Y Q ε Y Y 1 a r , M 44 = 1 r ,
h m = φ m h = φ m g f = φ m g p f p φ p .
h m = p φ m g f p φ p = p φ m g φ p f p .
g m p φ m g φ p ,
h m = p g m p f p .
f m = φ m f = φ m 1 g h = φ m 1 g p h p φ p = p φ m 1 g φ p h p ,
g inv , m p φ m 1 g φ p ,
f m = p g inv , m p h p .
h m = p [ ( g inv ) 1 ] m p f p ,
E T = n , m ( E T Y n m Y n m + E T X n m X n m + E T Z n m Z n m ) ,
D T = n , m ( D T Y n m Y n m + D T X n m X n m + D T Z n m Z n m ) .
D T = ϵ 0 ε E T ,
[ D T ] = ϵ 0 { ε ( T ) } [ E T ] ,
{ ε ( T ) } = ( { ε Y Y } { ε Y X } { ε Y Z } { ε X Y } { ε X X } { ε X Z } { ε Z Y } { ε Z X } { ε Z Z } )
{ ε ( T ) } = ( { ε Y Y } 0 0 0 { ε X X } { ε X Z } 0 { ε Z X } { ε Z Z } ) .
n , m ( D T Y n m Y n m + D T X n m X n m + D T Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) ν , μ ( E T Y ν μ Y ν μ + E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
Y n m * n , m ( D T Y n m Y n m + D T X n m X n m + D T Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) Y n m * ν , μ ( E T Y ν μ Y ν μ + E T X ν μ X ν μ + E T Z ν μ Z ν μ ) ,
Y n m * X n m = Y n m * Z n m = Y n m * X ν μ = Y n m * Z ν μ = 0 ,
n , m D T Y n m Y n m * Y n m = ϵ 0 ε ( r , θ , ϕ ) ν , μ E T Y ν μ Y n m * Y ν μ .
n , m D T Y n m 0 4 π d Ω Y n m * Y n m = ϵ 0 ν , μ E T Y ν μ 0 4 π d Ω ε ( r , θ , ϕ ) Y n m * Y ν μ ,
D T Y n m = ϵ 0 ν , μ E T Y ν μ 0 4 π d Ω ε ( r , θ , ϕ ) Y n m * Y ν μ .
ε Y Y n m , ν μ 0 4 π d Ω ε ( r , θ , ϕ ) Y n m * Y ν μ Y n m ε Y ν μ ,
D T Y n m = ϵ 0 ν , μ ε Y Y n m , ν μ E T Y ν μ .
D T Y p = ϵ 0 q ε Y Y p , q E T Y q ,
X n m * n , m ( D T Y n m Y n m + D T X n m X n m + D T Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) X n m * ν , μ ( E T Y ν μ Y ν μ + E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
n , m ( D T X n m X n m * X n m + D T Z n m X n m * Z n m ) = ϵ 0 ε ( r , θ , ϕ ) X n m * ν , μ ( E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
D T X n m = ϵ 0 ν , μ 0 4 π d Ω ε ( r , θ , ϕ ) X n m * ( E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
ε X X n m , ν μ 0 4 π d Ω ε ( r , θ , ϕ ) X n m * X ν μ X n m ε X ν μ ,
ε X Z n m , ν μ 0 4 π d Ω ε ( r , θ , ϕ ) X n m * Z ν μ X n m ε Z ν μ ,
1 ϵ 0 D T X p = q ε X X p , q E T X q + q ε X Z p , q E T Z q ,
Z n m * n , m ( D T Y n m Y n m + D T X n m X n m + D T Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) Z n m * ν , μ ( E T Y ν μ Y ν μ + E T X ν μ X ν μ + E T Z ν μ Z ν μ ) ,
n , m ( D T X n m Z n m * X n m + D T Z n m Z n m * Z n m ) = ϵ 0 ε ( r , θ , ϕ ) Z n m * ν , μ ( E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
D T Z n m = ϵ 0 ν , μ 0 4 π d Ω ε ( r , θ , ϕ ) Z n m * ( E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
ε Z X n m , ν μ 0 4 π d Ω ε ( r , θ , ϕ ) Z n m * X ν μ Z n m ε X ν μ ,
ε Z Z n m , ν μ 0 4 π d Ω ε ( r , θ , ϕ ) Z n m * Z ν μ Z n m ε Z ν μ .
1 ϵ 0 D T Z P = q ε Z X p , q E T X q + q ε Z Z p , q Z T X q ,
{ ε X X } = { ε Z Z } , { ε X Z } = { ε Z X } .
{ ε ( T ) } = ( { ε Y Y } 0 0 0 { ε X X } { ε X Z } 0 { ε X Z } { ε X X } ) .
D N = ϵ 0 ε E N ,
[ D N ] = ϵ 0 { ε ( N ) } [ E N ] ,
{ ε ( N ) } = ( { ε Y Y ( N ) } 0 0 0 { ε X X ( N ) } { ε X Z ( N ) } 0 { ε X Z ( N ) } { ε X X ( N ) } ) .
n , m ( E N Y n m Y n m + E N X n m X n m + E N Z n m Z n m ) = 1 ϵ 0 ε ( r , θ , ϕ ) ν , μ ( D N Y ν μ Y ν μ + D N X ν μ X ν μ + D N Z ν μ Z ν μ ) .
( 1 ε ) Y Y n m ; ν μ 0 4 π d Ω 1 ε ( r , θ , ϕ ) Y n m * Y ν μ Y n m 1 ε Y ν μ ,
ϵ 0 E N Y n m = ν μ ( 1 ε ) Y Y n m , ν μ D N Y ν μ
ϵ 0 E N Y p = q ( 1 ε ) Y Y p , q D N Y q .
D N Y p = ϵ 0 q [ ( 1 ε ) Y Y 1 ] p , q E N Y q ,
{ ε Y Y ( N ) } = ( ( 1 ε ) Y Y ) 1 .
ϵ 0 E N X n m = ν , μ ( 1 ε ) X X n m , ν μ D N X ν μ + ν , μ ( 1 ε ) X Z n m , ν μ D N Z ν μ ,
( 1 ε ) X X n m , ν μ X n m 1 ε X ν μ ,
( 1 ε ) X Z n m , ν μ X n m 1 ε Z ν μ .
ϵ 0 [ E N X ] = ( 1 ε ) X X [ D N X ] + ( 1 ε ) X Z [ D N Z ] ,
ϵ 0 [ E N Z ] = ( 1 ε ) X Z [ D N X ] + ( 1 ε ) X X [ D N Z ] .
[ [ D N X ] [ D N Z ] ] = ϵ 0 ( ( 1 ε ) X X ( 1 ε ) X Z ( 1 ε ) X Z ( 1 ε ) X X ) 1 [ [ E N X ] [ E N Z ] ] ,
{ ε ( N ) } = ( { ε Y Y ( N ) } 0 0 0 { ε X X ( N ) } { ε X Z ( N ) } 0 { ε X Z ( N ) } { ε X X ( N ) } ) = ( ( ( 1 ε ) Y Y ) 1 0 0 0 ( ( 1 ε ) X X ( 1 ε ) X Z ( 1 ε ) X Z ( 1 ε ) X X ) 1 0 ) .
r [ R 1 , R 2 ] , N ̂ ( r , θ , ϕ ) = grad f grad f f = 0 .
E N = N ̂ ( N ̂ E ) ,
E T = E E N = E N ̂ ( N ̂ E ) ;
D = ϵ 0 ε E = ϵ 0 ε ( E T + E N ) = ϵ 0 ε ( E N ̂ ( N ̂ E ) ) + ϵ 0 ε N ̂ ( N ̂ E ) .
1 ϵ 0 [ D ] = { ε ( T ) } [ E T ] + { ε ( N ) } [ E N ] = ( { ε ( T ) } ( 1 { N ̂ N ̂ } ) + { ε ( N ) } { N ̂ N ̂ } ) [ E ] .
Q ε = { ε ( T ) } + ( { ε ( N ) } { ε ( T ) } ) { N ̂ N ̂ } ,
Q ε i j = { ε i j ( T ) } + k ( { ε i k ( N ) } { ε i k ( T ) } ) { N ̂ k N ̂ j } .
Δ = ( Δ Y Y 0 0 0 Δ X X Δ X Z 0 Δ X Z Δ X X ) .
Q ε Y Y = Δ Y Y { N Y N Y } + { ε Y Y } , Q ε Y X = Δ Y Y { N Y N X } ,
Q ε Y Z = Δ Y Y { N Y N Z } ,
Q ε X Y = Δ X X { N X N Y } + Δ X Z { N Z N Y } ,
Q ε X X = Δ X X { N X N X } + Δ X Z { N Z N X } + { ε X X } ,
Q ε X Z = Δ X X { N X N Z } + Δ X Z { N Z N Z } + { ε X Z } ,
Q ε Z Y = Δ X X { N Z N Y } Δ X Z { N X N Y } ,
Q ε Z X = Δ X X { N Z N X } Δ X Z { N X N X } { ε X Z } ,
Q ε Z Z = Δ X X { N Z N Z } Δ X Z { N X N Z } + { ε X X } .
curl ( curl E ) ( ω c ) 2 ε j μ j E = 0 .
Δ E + k j 2 E = 0 ,
k j 2 = ( ω c ) 2 ε j μ j .
1 r 2 r ( r 2 ψ r ) + 1 r 2 sin θ θ ( sin θ ψ θ ) + 1 r 2 sin 2 θ 2 ψ ϕ 2 + k j 2 ψ = 0 .
ψ ( r , θ , ϕ ) = n , m R ( r ) Y n m ( θ , ϕ ) ,
d d r ( r 2 d R d r ) + [ k j 2 r 2 n ( n + 1 ) ] R ( r ) = 0 .
d 2 R ̌ d 2 ρ + 1 ρ d R ̌ d ρ + [ 1 ( n + 1 2 ) 2 ρ 2 ] R ̌ ( ρ ) = 0 .
R ( r ) = j n ( k j r ) π 2 k j r J n + 1 2 ( k j r ) ,
R ( r ) = y n ( k j r ) π 2 k j r Y n + 1 2 ( k j r ) ,
h n + ( ρ ) = j n ( ρ ) + i y n ( ρ ) ,
h n ( ρ ) = j n ( ρ ) i y n ( ρ ) .
ψ n m ( r , θ , φ ) = z n ( k j r ) Y n m ( θ , ϕ ) .
M n m curl ( r ψ n m ) n ( n + 1 ) .
M n m ( ρ , θ , ϕ ) = z n ( ρ ) X n m ( θ , ϕ ) ,
N n m ( ρ , θ , ϕ ) = 1 ρ { n ( n + 1 ) z n ( ρ ) Y n m ( θ , ϕ ) + [ ρ z n ( ρ ) ] Z n m ( θ , ϕ ) } ,
f ( x 0 ) d d x f ( x ) x = x 0 .
E ( r ) = n , m { A h , n m ( j ) j n ( k j r ) X n m + A e , n m ( j ) k j r [ n ( n + 1 ) j n ( k j r ) Y n m + ( k j r j n ( k j r ) ) Z n m ] } + n , m { B h , n m ( j ) h n + ( k j r ) X n m + B e , n m ( j ) k j r [ n ( n + 1 ) h n + ( k j r ) Y n m + ( k j r h n + ( k j r ) ) Z n m ] } .
B e , n m ( 1 ) = 0 = B h , n m ( 1 ) n , m .
E = n , m { A h , n m ( 1 ) j n ( k 1 r ) X n m + A e , n m ( 1 ) k 1 r [ n ( n + 1 ) j n ( k 1 r ) Y n m + ψ n ( k 1 r ) Z n m ] } .
A h , n m i = 4 π i n X n m * ( θ i , ϕ i ) e ̂ i ,
A e , n m i = 4 π i n 1 Z n m * ( θ i , ϕ i ) e ̂ i ,
E = n , m { A h , n m i j n ( k M r ) X n m + A e , n m i k M r × [ n ( n + 1 ) j n ( k M r ) Y n m + ψ n ( k M r ) Z n m ] } + n , m { B h , n m ( M ) h n + ( k M r ) X n m + B e , n m ( M ) k M r × [ n ( n + 1 ) h n + ( k M r ) Y n m + ξ n ( k M r ) Z n m ] } .
[ V ( j ) ] = [ A e , p ( j ) ψ n ( k M r j ) ( k M r j ) A h , p ( j ) j n ( k M r j ) B e , p ( j ) ξ n ( k M r j ) ( k M r j ) B h , p ( j ) h n + ( k M r j ) ] ,
A e , p ( 1 ) = A e , p ( 1 ) ; A h , p ( 1 ) = A h , p ( 1 ) ; B e , p ( 1 ) = B e , p ( 1 ) ; B h , p ( 1 ) = B h , p ( 1 ) ,
A e , p ( M 1 ) = A e , p ( M ) ; A h , p ( M 1 ) = A h , p ( M ) ; B e , p ( M 1 ) = B e , p ( M ) ;
B h , p ( M 1 ) = B h , p ( M ) .
[ V ( j ) ] = T ( j ) [ V ( j 1 ) ] ,
[ B e , p ( j ) ξ n ( k M r j ) ( k M r j ) B h , p ( j ) h n + ( k M r j ) A e , p ( 1 ) ψ n ( k 1 r 1 ) ( k 1 r 1 ) A h , p ( 1 ) j n ( k 1 r 1 ) ] = ( S 11 ( j ) S 21 ( j ) ( S 12 ( j ) S 22 ( j ) ) [ B e , q ( 1 ) ξ n ( k 1 r 1 ) ( k 1 r 1 ) B h , q ( 1 ) h n + ( k 1 r 1 ) A e , q ( j ) ψ n ( k M r j ) ( k M r j ) A h , q ( j ) j n ( k M r j ) ] ,
S 12 ( j ) = ( T 21 ( j ) + T 22 ( j ) S 12 ( j 1 ) ) Z ( j 1 ) ,
S 22 ( j ) = S 22 ( j 1 ) Z ( j 1 ) ,
Z ( j 1 ) = ( T 11 ( j ) + T 12 ( j ) S 12 ( j 1 ) ) 1 .
[ E X ] p = A h , p ( j ) j n ( k M r j ) + B h , p ( j ) h n + ( k M r j ) ,
[ E Z ] p = 1 k M r j { A e , p ( j ) ψ n ( k M r j ) + B e , p ( j ) ξ n ( k M r j ) } .
[ H ̃ X ] p = 1 i ω μ M ϵ 0 μ 0 { n ( n + 1 ) k M r j 2 [ A e , p ( j ) j n ( k M r j ) + B e , p ( j ) h n + ( k M r j ) ] 1 k M r j 2 [ A e , p ( j ) ψ n ( k M r j ) + B e , p ( j ) ξ n ( k M r j ) ] d d r [ A e , p ( j ) ψ n ( k M r j ) k M r j + B e , p ( j ) ξ n ( k M r j ) k M r j ] } = 1 i ω μ M ϵ 0 μ 0 1 k M r j 2 { n ( n + 1 ) [ A e , p ( j ) j n ( k M r j ) + B e , p ( j ) h n + ( k M r j ) ] [ A e , p ( j ) ( k M r j ) 2 j n ( k M r j ) + B e , p ( j ) ( k M r j ) 2 ( h n + ) ( k M r j ) ] } ,
ρ 2 d d ρ ( 1 ρ ( ρ z n ( ρ ) ) ) + ( ρ z n ( ρ ) ) = d d ρ ( ρ 2 z n ( ρ ) ) .
n ( n + 1 ) z n ( ρ ) d d ρ ( ρ 2 z n ( ρ ) ) = ρ 2 z n ( ρ ) .
[ H ̃ X ] p = i ε M μ M [ A e , p ( j ) j n ( k M r j ) + B e , p ( j ) h n + ( k M r j ) ] .
[ H ̃ Z ] p = i ε M μ M 1 k M r j { A h , p ( j ) ψ n ( k M r j ) + B h , p ( j ) ξ n ( k M r j ) } .
[ F ( j ) ] [ [ E X ( j ) ] [ E Z ( j ) ] [ H ̃ X ( j ) ] [ H ̃ Z ( j ) ] ] Ψ ( j ) [ A e , p ( j ) ψ n ( k M r j ) ( k M r j ) A h , p ( j ) j n ( k M r j ) B e , p ( j ) ξ n ( k M r j ) ( k M r j ) B h , p ( j ) h n + ( k M r j ) ] = Ψ ( j ) [ V ( j ) ] ,
Ψ ( j ) = ( 0 1 0 1 1 0 1 0 i ε M μ M p ( j ) 0 i ε M μ M q ( j ) 0 0 i ε M μ M p ( j ) 0 i ε M μ M q ( j ) ) ,
p p , q ( j ) δ p , q ψ n ( z ) ψ n ( z ) z = k M r j , q p , q ( j ) δ p , q ξ n ( z ) ξ n ( z ) z = k M r j .
[ V ̂ ( j 1 ) ] = 1 .
[ V ̂ int ( j ) ] = [ Ψ ( j ) ( r j ) ] 1 [ F ̂ int ( r j ) ] ,
[ V ̂ int ( j ) ] = [ Ψ ( j ) ( r j ) ] 1 [ F ̂ int ( r j ) ] [ V ̂ ( j 1 ) ] .
T ( j ) = [ Ψ ( j ) ( r j ) ] 1 [ F ̂ int ( r j ) ] .
[ [ B e ( M 1 ) ] [ B h ( M 1 ) ] [ A e ( 1 ) ] [ A h ( 1 ) ] ] = S ( M 1 ) [ [ B e ( 1 ) ] [ B h ( 1 ) ] [ A e ( M 1 ) ] [ A h ( M 1 ) ] ] ,
[ [ B e ( M ) ] [ B h ( M ) ] [ A e ( 1 ) ] [ A h ( 1 ) ] ] = ( S 11 S 21 ( S 12 S 22 ) ( M 1 ) [ [ B e ( 1 ) ] [ B h ( 1 ) ] [ A e ( M ) ] [ A h ( M ) ] ] .
[ [ B e ( M ) ] [ B h ( M ) ] ] = S 12 ( M 1 ) [ [ A e ( M ) ] [ A h ( M ) ] ] ,
A e , p ( M ) = A e , p i , A h , p ( M ) = A h , p i ,
[ [ A e ( 1 ) ] [ A h ( 1 ) ] ] = S 22 ( M 1 ) [ [ A e ( M ) ] [ A h ( M ) ] ] .
[ B ( M ) ] = S 12 ( M 1 ) [ A ( M ) ] ,
[ B c ( M ) ] [ , B e q ( M ) , * , , B h q ( M ) , * , ] .
[ B c ( M ) ] t [ A i ] ,
σ s = 1 k M 2 [ B c ] [ B c ] ,
σ e = Re { 1 k M 2 [ B c ] [ A i ] } ,
σ a = σ e σ s .
lim r E s ( r ) = E exp ( i k r ) i k r ( E s , θ θ ̂ + E s , ϕ ϕ ̂ ) ,
( E s , θ E s , ϕ ) = E exp ( i k r ) i k r ( F θ θ F θ ϕ F ϕ θ F ϕ ϕ ) ( e θ e ϕ ) ,
F ¯ ¯ 4 π [ X * ( r ̂ ) , Z * ( r ̂ ) ] t [ X ( k ̂ i ) Z ( k ̂ i ) ] ,
X n m ( r ̂ ) i n X n m * ( r ̂ ) , Z n m ( r ̂ ) i n 1 Z n m * ( r ̂ ) .
F θ θ θ ̂ F ¯ ¯ θ ̂ i , F θ ϕ θ ̂ F ¯ ¯ ϕ ̂ i , F ϕ θ ϕ ̂ F ¯ ¯ θ ̂ i ,
F ϕ ϕ ϕ ̂ F ¯ ¯ ϕ ̂ i .
d σ d Ω ( θ , ϕ ; θ i , ϕ i ) = lim r r 2 E s ( r ) 2 E 2 = E s , θ 2 + E s , ϕ 2 k 2 = F θ θ e θ + F θ ϕ e ϕ 2 + F ϕ θ e θ + F ϕ ϕ e ϕ 2 k 2 .
P ¯ n 0 ( cos θ ) = ( 2 n + 1 4 π ) 1 2 P n 0 ( cos θ ) .
( n + 1 ) P n 0 ( c ) = d d c P n + 1 0 ( c ) c d d c P n 0 ( c ) .
d d c ( c P n 0 ( c ) ) = c d d c P n 0 ( c ) + P n 0 ( c ) ,
c d d c P n 0 ( c ) = d d c ( c P n 0 ( c ) ) P n 0 ( c ) ,
P n 0 ( c ) = 1 n d d c P n + 1 0 ( c ) 1 n d d c ( c P n 0 ( c ) ) .
c 2 c 1 P n 0 ( c ) d c = 1 n c 2 c 1 d d c P n + 1 0 ( c ) d c 1 n c 2 c 1 d d c ( c P n 0 ( c ) ) d c ,
c 2 c 1 P n 0 ( c ) d c = 1 n [ P n + 1 0 ( c 1 ) P n + 1 0 ( c 2 ) ] + 1 n [ c 2 P n 0 ( c 2 ) c 1 P n 0 ( c 1 ) ] .
z 2 a 2 + y 2 b 2 = 1 .
r ( θ ) = a b a 2 + ( b 2 a 2 ) cos 2 θ .
θ ( r ) = arccos ( a r r 2 b 2 a 2 b 2 )
ε ( r , θ ) = ε 1 if θ [ 0 , θ 1 ( r ) ] [ θ 2 ( r ) , π ] ,
ε ( r , θ ) = ε M if θ [ θ 1 ( r ) , θ 2 ( r ) ] .
n , m ( D Y n m Y n m + D X n m X n m + D Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) ν , μ ( E Y ν μ Y ν μ + E X ν μ X ν μ + E Z ν μ Z ν μ ) .
Y n m * n , m ( D Y n m Y n m + D X n m X n m + D Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) Y n m * ν , μ ( E Y ν μ Y ν μ + E X ν μ X ν μ + E Z ν μ Z ν μ ) .
n , m D Y n m Y n m * Y n m = ϵ 0 ε ( r , θ , ϕ ) ν , μ E Y ν μ Y n m * Y ν μ .
X n m * n , m ( D Y n m Y n m + D X n m X n m + D Z n m Z n m ) = ϵ 0 ε ( r , θ , ϕ ) X n m * ν , μ ( E Y ν μ Y ν μ + E X ν μ X ν μ + E Z ν μ Z ν μ ) .
1 ϵ 0 n , m ( D X n m X n m * X n m + D Z n m X n m * Z n m ) = ε ( r , θ , ϕ ) X n m * ν , μ ( E X ν μ X ν μ + E Z ν μ Z ν μ ) .
D X n m = ϵ 0 ν , μ 0 4 π d Ω ε ( r , θ , ϕ ) X n m * ( E X ν μ X ν μ + E Z ν μ Z ν μ ) .
X 1 = 1 2 ( x ̂ + i y ̂ ) , X 0 = z ̂ , X 1 = 1 2 ( x ̂ i y ̂ ) ,
Y n , l m = μ = 1 1 ( l , m μ ; 1 , μ n , m ) Y l , m μ X μ ,
X n m = Y n , n m i ,
Z n m = ( n + 1 2 n + 1 ) 1 2 Y n , n 1 m + ( n 2 n + 1 ) 1 2 Y n , n + 1 m ,
Y n m = ( n 2 n + 1 ) 1 2 Y n , n 1 m ( n + 1 2 n + 1 ) 1 2 Y n , n + 1 m .
X ν μ X n m * = Y ν , ν μ ( Y n , n m ) * .
X ν μ X n m * = [ 1 n ( n + 1 ) ν ( ν + 1 ) ] 1 2 { 1 2 [ ( n + m + 1 ) ( n m ) ( ν + μ + 1 ) ( ν μ ) ] 1 2 Y ν , μ + 1 Y n , m + 1 * + m μ Y ν μ Y n m * + 1 2 [ ( n + m ) ( n m + 1 ) ( ν + μ ) ( ν μ + 1 ) ] 1 2 Y ν , μ 1 Y n , m 1 * } .
( n + 1 2 n + 1 ) 1 2 X ν μ Y n , n + 1 m , * = ( n 2 n + 1 ) 1 2 X ν μ Y n , n 1 m , * ,
X ν μ Y n , n + 1 m , * = ( n n + 1 ) 1 2 X ν μ Y n , n 1 m , * .
X ν μ Z n m * = ( n + 1 2 n + 1 ) 1 2 X ν μ Y n , n 1 m , * + ( n 2 n + 1 ) 1 2 X ν μ Y n , n + 1 m , * = ( 2 n + 1 n + 1 ) 1 2 X ν μ Y n , n 1 m , * = i ( 2 n + 1 n + 1 ) 1 2 Y ν , ν μ Y n , n 1 m , * .
X ν μ Z n m * = i ( 1 n ( n + 1 ) ν ( ν + 1 ) 2 n + 1 2 n 1 ) 1 2 { Y ν , μ 1 Y n 1 , m 1 * 2 [ ( n + m ) ( n + m 1 ) ( ν + μ ) ( ν μ + 1 ) ] 1 2 + μ [ ( n 2 m 2 ) ] 1 2 Y ν , μ Y n 1 , m * + Y ν , μ + 1 Y n 1 , m + 1 * 2 [ ( n m ) ( n m 1 ) ( ν μ ) ( ν + μ + 1 ) ] 1 2 } .
a ¯ ( { ν , μ } , { ν , μ } , { n , m } ) 0 2 π 0 π Y ν μ ( θ , ϕ ) Y ν μ ( θ , ϕ ) Y n m ( θ , ϕ ) sin θ d θ d ϕ .
n m ( D T Y n m Y n m + D T X n m X n m + D T Z n m Z n m ) = ϵ 0 ν , μ ν , μ ε ν μ Y ν μ ( E T Y ν μ Y ν μ + E T X ν μ X ν μ + E T Z ν μ Z ν μ ) .
1 ϵ 0 D T Y n m = ν , μ ν , μ ε ν μ E T Y ν μ Y ν μ ( θ , ϕ ) Y ν μ ( θ , ϕ ) Y n m * ( θ , ϕ ) sin θ d θ d ϕ = ν , μ ν , μ ε ν μ E T Y ν μ ( 1 ) m Y ν μ ( θ , ϕ ) Y ν μ ( θ , ϕ ) Y n , m ( θ , ϕ ) sin θ d θ d ϕ = ν , μ ν , μ ε ν μ E T Y ν μ ( 1 ) m Y ν μ ( θ , ϕ ) Y ν μ ( θ , ϕ ) Y n , m ( θ , ϕ ) sin θ d θ d ϕ = ν , μ ν , μ ε ν μ E T Y ν μ ( 1 ) m a ¯ ( { ν , μ } , { ν , μ } , { n , m } ) .
D T Y n m = ϵ 0 ν = n ν n + ν ν = 0 N μ = ν ν ( 1 ) m a ¯ ( { ν , m μ } , { ν , μ } , { n , m } ) ε ν , m μ E T Y ν μ .
ε Y Y n m , ν μ = ( 1 ) m ν = n ν n + ν a ¯ ( { ν , m μ } , { ν , μ } , { n , m } ) ε ν , m μ ( r ) ,
D T X n m = ϵ 0 ν , μ ν , μ ε ν μ E T X ν μ Y ν μ