Abstract

An exact analytic solution is presented, by using the method of separation of variables, to the problem of electromagnetic wave scattering by an optically active (chiral) spheroid. Fields outside as well as inside the spheroid are expanded in terms of vector spheroidal eigenfunctions, and a set of simultaneous linear equations is obtained by imposing boundary conditions on the surface of the spheroid. Solution of these equations results in the unknown coefficients in the series expansions of the associated fields. The behavior of the scattered fields is illustrated by plots of scattering cross sections for both prolate and oblate spheroids of different sizes and materials in the resonance region.

© 1993 Optical Society of America

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References

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  1. D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
    [CrossRef]
  2. N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newsletter 30, 6–12 (1988).
  3. A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Vol. 335 of Springer-Verlag Lecture Notes in Physics (Springer-Verlag, Berlin, 1988).
  4. D. L. Jaggard, X. Sun, N. Engheta, “Canonical sources and duality in chiral media,”IEEE Trans. Antennas Propag. 36, 1007–1013 (1988).
    [CrossRef]
  5. A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
    [CrossRef]
  6. S. Bassiri, “Electromagnetic waves in chiral media,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos, D. L. Jaggard, eds. (Springer-Verlag, New York, 1990).
    [CrossRef]
  7. N. Engheta, ed., Special issue on wave interaction with chiral and complex media, J. Electromag. Waves Applic.5/6, 537–793 (1992).
  8. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  9. C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
    [CrossRef]
  10. C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,”J. Chem. Phys. 62, 1566–1571 (1975).
    [CrossRef]
  11. M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,”IEEE Trans. Antennas Propag. 39, 91–96 (1991).
    [CrossRef]
  12. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 8.
  13. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chap. 9.
  14. B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting spheroids,”IEEE Trans. Antennas Propag. AP-31, 294–304 (1983).
    [CrossRef]
  15. M. F. R. Cooray, I. R. Ciric, “Electromagnetic wave scattering by a system of two spheroids of arbitrary orientation,”IEEE Trans. Antennas Propag. 37, 608–618 (1989).
    [CrossRef]
  16. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  17. J. Dalmas, “Diffusion d’une onde electromagnétique par un ellipsoïde de révolution allongé, de conduction infinie, en incidence non axiale,” Opt. Acta 28, 933–948 (1981).
    [CrossRef]
  18. J. Dalmas, “Indicatrices de diffusion d’un ellipsoïde de révolution allongé, de conduction infinie, en incidence oblique,” Opt. Acta 28, 1277–1287 (1981).
    [CrossRef]
  19. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Scattering and absorption characteristics of lossy dielectric, chiral, nonspherical objects,” Appl. Opt. 24, 4146–4154 (1985).
    [CrossRef] [PubMed]
  20. M. F. R. Cooray, I. R. Ciric, “Scattering by systems of spheroids in arbitrary configurations,” Comput. Phys. Commun. 68, 279–305 (1991).
    [CrossRef]

1991 (2)

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,”IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

M. F. R. Cooray, I. R. Ciric, “Scattering by systems of spheroids in arbitrary configurations,” Comput. Phys. Commun. 68, 279–305 (1991).
[CrossRef]

1989 (1)

M. F. R. Cooray, I. R. Ciric, “Electromagnetic wave scattering by a system of two spheroids of arbitrary orientation,”IEEE Trans. Antennas Propag. 37, 608–618 (1989).
[CrossRef]

1988 (3)

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newsletter 30, 6–12 (1988).

D. L. Jaggard, X. Sun, N. Engheta, “Canonical sources and duality in chiral media,”IEEE Trans. Antennas Propag. 36, 1007–1013 (1988).
[CrossRef]

A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
[CrossRef]

1985 (1)

1983 (1)

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting spheroids,”IEEE Trans. Antennas Propag. AP-31, 294–304 (1983).
[CrossRef]

1981 (2)

J. Dalmas, “Diffusion d’une onde electromagnétique par un ellipsoïde de révolution allongé, de conduction infinie, en incidence non axiale,” Opt. Acta 28, 933–948 (1981).
[CrossRef]

J. Dalmas, “Indicatrices de diffusion d’un ellipsoïde de révolution allongé, de conduction infinie, en incidence oblique,” Opt. Acta 28, 1277–1287 (1981).
[CrossRef]

1979 (1)

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

1978 (1)

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

1975 (2)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,”J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[PubMed]

1974 (1)

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

Asano, S.

Bassiri, S.

S. Bassiri, “Electromagnetic waves in chiral media,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos, D. L. Jaggard, eds. (Springer-Verlag, New York, 1990).
[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, “Scattering of electromagnetic waves by an optically active spherical shell,”J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

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

Ciric, I. R.

M. F. R. Cooray, I. R. Ciric, “Scattering by systems of spheroids in arbitrary configurations,” Comput. Phys. Commun. 68, 279–305 (1991).
[CrossRef]

M. F. R. Cooray, I. R. Ciric, “Electromagnetic wave scattering by a system of two spheroids of arbitrary orientation,”IEEE Trans. Antennas Propag. 37, 608–618 (1989).
[CrossRef]

Cooray, M. F. R.

M. F. R. Cooray, I. R. Ciric, “Scattering by systems of spheroids in arbitrary configurations,” Comput. Phys. Commun. 68, 279–305 (1991).
[CrossRef]

M. F. R. Cooray, I. R. Ciric, “Electromagnetic wave scattering by a system of two spheroids of arbitrary orientation,”IEEE Trans. Antennas Propag. 37, 608–618 (1989).
[CrossRef]

Dalmas, J.

J. Dalmas, “Diffusion d’une onde electromagnétique par un ellipsoïde de révolution allongé, de conduction infinie, en incidence non axiale,” Opt. Acta 28, 933–948 (1981).
[CrossRef]

J. Dalmas, “Indicatrices de diffusion d’un ellipsoïde de révolution allongé, de conduction infinie, en incidence oblique,” Opt. Acta 28, 1277–1287 (1981).
[CrossRef]

Engheta, N.

D. L. Jaggard, X. Sun, N. Engheta, “Canonical sources and duality in chiral media,”IEEE Trans. Antennas Propag. 36, 1007–1013 (1988).
[CrossRef]

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newsletter 30, 6–12 (1988).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chap. 9.

Jaggard, D. L.

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newsletter 30, 6–12 (1988).

D. L. Jaggard, X. Sun, N. Engheta, “Canonical sources and duality in chiral media,”IEEE Trans. Antennas Propag. 36, 1007–1013 (1988).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Kluskens, M. S.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,”IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Lakhtakia, A.

MacPhie, R. H.

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting spheroids,”IEEE Trans. Antennas Propag. AP-31, 294–304 (1983).
[CrossRef]

Mickelson, A. R.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Newman, E. H.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,”IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Papas, C. H.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 8.

Sinha, B. P.

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting spheroids,”IEEE Trans. Antennas Propag. AP-31, 294–304 (1983).
[CrossRef]

Sun, X.

D. L. Jaggard, X. Sun, N. Engheta, “Canonical sources and duality in chiral media,”IEEE Trans. Antennas Propag. 36, 1007–1013 (1988).
[CrossRef]

Varadan, V. K.

Varadan, V. V.

Yamamoto, G.

Appl. Opt. (2)

Appl. Phys. (1)

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Chem. Phys. Lett. (1)

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

Comput. Phys. Commun. (1)

M. F. R. Cooray, I. R. Ciric, “Scattering by systems of spheroids in arbitrary configurations,” Comput. Phys. Commun. 68, 279–305 (1991).
[CrossRef]

IEEE AP-S Newsletter (1)

N. Engheta, D. L. Jaggard, “Electromagnetic chirality and its applications,” IEEE AP-S Newsletter 30, 6–12 (1988).

IEEE Trans. Antennas Propag. (4)

D. L. Jaggard, X. Sun, N. Engheta, “Canonical sources and duality in chiral media,”IEEE Trans. Antennas Propag. 36, 1007–1013 (1988).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,”IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting spheroids,”IEEE Trans. Antennas Propag. AP-31, 294–304 (1983).
[CrossRef]

M. F. R. Cooray, I. R. Ciric, “Electromagnetic wave scattering by a system of two spheroids of arbitrary orientation,”IEEE Trans. Antennas Propag. 37, 608–618 (1989).
[CrossRef]

J. Chem. Phys. (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,”J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

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. Opt. Soc. Am. A (1)

Opt. Acta (2)

J. Dalmas, “Diffusion d’une onde electromagnétique par un ellipsoïde de révolution allongé, de conduction infinie, en incidence non axiale,” Opt. Acta 28, 933–948 (1981).
[CrossRef]

J. Dalmas, “Indicatrices de diffusion d’un ellipsoïde de révolution allongé, de conduction infinie, en incidence oblique,” Opt. Acta 28, 1277–1287 (1981).
[CrossRef]

Other (5)

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 8.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chap. 9.

S. Bassiri, “Electromagnetic waves in chiral media,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos, D. L. Jaggard, eds. (Springer-Verlag, New York, 1990).
[CrossRef]

N. Engheta, ed., Special issue on wave interaction with chiral and complex media, J. Electromag. Waves Applic.5/6, 537–793 (1992).

A. Lakhtakia, V. K. Varadan, V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Vol. 335 of Springer-Verlag Lecture Notes in Physics (Springer-Verlag, Berlin, 1988).

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Figures (4)

Fig. 1
Fig. 1

Prolate spheroidal geometry.

Fig. 2
Fig. 2

Normalized bistatic cross section for a chiral prolate spheroid of axial ratio 2, with ka = 3, r = 1.33, and different chiral admittances.

Fig. 3
Fig. 3

Normalized bistatic cross section for a chiral oblate spheroid with r = 1.33, of axial ratio 0.5, and of the same major-axis length as the spheroid in Fig. 2.

Fig. 4
Fig. 4

Normalized backscattering cross section for a lossy chiral prolate spheroid of axial ratio 2, relative permittivity r = 2.13 − j0.055, and different sizes.

Equations (74)

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D = E - j ξ c B ,
H = ( 1 / μ ) B - j ξ c E ,
D = c E - j μ ξ c H ,
c = + μ ξ c 2 ,
B = μ H + j μ ξ c E .
× E = ω μ ξ c E - j ω μ H ,
× H = j ω c E + ω μ ξ c H ,
· E = 0 ,
· H = 0.
2 ( E H ) + [ K ] 2 ( E H ) = 0 ,
[ K ] = [ ω μ ξ c - j ω μ j ω c ω μ ξ c ] .
( E H ) = [ 1 1 j / η c - j / η c ] ( E R E L ) ,
2 ( E R E L ) + ( k R 2 E R k L 2 E L ) = 0 ,
k R = ω μ c + ω μ ξ c ,
k L = ω μ c - ω μ ξ c ,
E i = y ^ exp [ - j k ( x sin θ i + z cos θ j ) ] = m = 0 n = m [ c m n ( h , θ i ) M e , m n r ( 1 ) ( h , r ) + d m n ( h , θ i ) N o , m n r ( 1 ) ( h : r ) ] ,
c m n ( h , θ i ) = - 2 ( 2 - δ 0 m ) j n N m n ( h ) r = 0 , 1 d r m n ( h ) ( r + m ) ( r + m + 1 ) × d d θ i P m + r m ( cos θ i ) ,
d m n ( h , θ i ) = 4 m j n - 1 N m n ( h ) r = 0 , 1 d r m n ( h ) ( r + m ) ( r + m + 1 ) × P m + r m ( cos θ i ) sin θ i .
c 1 n ( h , 0 ) = - 2 j n N 1 n ( h ) r = 0 , 1 d r 1 n ( h ) ,
d 1 n ( h , 0 ) = 2 j n - 1 N 1 n ( h ) r = 0 , 1 d r 1 n ( h ) .
R - m n ( i ) ( h , ξ ) = R m n ( i ) ( h , ξ ) , S - m n ( h , η ) = ( - 1 ) m ( n - m ) ! ( n + m ) ! S m n ( h , η ) , N - m n ( h ) = [ ( n - m ) ! ( n + m ) ! ] 2 N m n ( h ) ,
E i = m = - n = m [ f m n ( h , θ i ) M m n r ( 1 ) ( h ; r ) + g m n ( h , θ i ) N m n r ( 1 ) ( h ; r ) ] ,
M m n r ( i ) = M e , m n r ( i ) + j M o , m n r ( i ) ,             N m n r ( i ) = N e , m n r , ( i ) + j N o , m n r ( i ) ,
f m n ( h i , θ i ) = [ 1 / ( 2 - δ o m ) ] c m n ( h , θ i ) ,
g m n ( h , θ i ) = - ( j / 2 ) d m n ( h , θ i ) .
f 1 n ( h , 0 ) = g 1 n ( h , 0 ) = - j n N 1 n ( h ) r = 0 , 1 d r 1 n ( h ) .
E i = [ ( - cos θ i ) x ^ + ( sin θ i ) z ^ ] × exp { - j k [ x ( sin θ i ) + z ( cos θ i ) ] } = - j m = - n = m [ g m n ( h , θ i ) M m n r ( 1 ) ( h ; r ) + f m n ( h , θ i ) N m n r ( 1 ) ( h ; r ) ] .
E i = m = - n = m [ p m n ( h , θ i ) M m n r ( 1 ) ( h ; r ) + q m n ( h , θ i ) N m n r ( 1 ) ( h ; r ) ] ,
p m n ( h , θ i ) = f m n ( h , θ i ) cos γ k - j g m n ( h , θ i ) sin γ k ,
q m n ( h , θ i ) = g m n ( h , θ i ) cos γ k - j f m n ( h , θ i ) sin γ k ,
E i = M ¯ i ( 1 ) T I ¯ p + N ¯ i ( i ) T I ¯ q ,
M ¯ i ( 1 ) T = [ M ¯ 0 ( 1 ) T M ¯ 1 ( 1 ) T M ¯ - 1 ( 1 ) T M ¯ 2 ( 1 ) T M ¯ - 2 ( 1 ) T ] ,
M ¯ w ( 1 ) T = [ M w , w r ( 1 ) ( h ; r ) M w , w + 1 r ( 1 ) ( h ; r ) M w , w + 2 r ( 1 ) ( h ; r ) ] ,
I ¯ p T = [ I ¯ 0 T I ¯ 1 T I ¯ - 1 T I ¯ 2 T I ¯ - 2 T ] ,
I ¯ w T = [ P w , w ( h , θ i ) p w , w + 1 ( h , θ i ) p w , w + 2 ( h , θ i ) ] .
E s = m = - n = m [ α m n M m n r ( 4 ) ( h ; r ) + β m n N m n r ( 4 ) ( h ; r ) ] ,
E s = M ¯ s ( 4 ) T α ¯ + N ¯ s ( 4 ) T β ¯ .
E c = E R + E L ,
E R = m = - n = m γ m n [ M m n r ( 1 ) ( h R ; r ) + N m n r ( 1 ) ( h R ; r ) ] ,
E L = m = - n = m δ m n [ M m n r ( 1 ) ( h L ; r ) - N m n r ( 1 ) ( h L ; r ) ] ,
E c = M ¯ c ( 1 ) T γ ¯ + N ¯ c ( 1 ) T δ ¯ ,
M ¯ c ( 1 ) T = [ M ¯ 0 ( 1 ) T M ¯ 1 ( 1 ) T M ¯ - 1 ( 1 ) T M ¯ 2 ( 1 ) T M ¯ - 2 ( 1 ) T ]
N ¯ c ( 1 ) T = [ N ¯ 0 ( 1 ) T N ¯ - 1 ( 1 ) T N ¯ - 1 ( 1 ) T N ¯ 2 ( 1 ) T N ¯ - 2 ( 1 ) T ] ,
M ¯ w ( 1 ) T = [ M w , w r ( 1 ) ( h R ; r ) + N w , w r ( 1 ) ( h R ; r ) M w , w + 1 r ( 1 ) ( h R ; r ) + N w , w + 1 r ( 1 ) ( h R ; r ) ]
N ¯ w ( 1 ) T = [ M w , w r ( 1 ) ( h L ; r ) - N w , w r ( 1 ) ( h L ; r ) M w , w + 1 r ( 1 ) ( h L ; r ) - N w , w + 1 r ( 1 ) ( h L ; r ) ] .
H = j k - 1 ( 0 μ 0 ) 1 / 2 × E
H i = j ( 0 / μ 0 ) 1 / 2 [ N ¯ i ( 1 ) T I ¯ p + M ¯ i ( 1 ) T I ¯ q ] ,
H s = j ( 0 / μ 0 ) 1 / 2 [ N ¯ s ( 4 ) T α ¯ + M ¯ s ( 4 ) T β ¯ ] ,
H c = j ( c / μ ) 1 / 2 [ - N ¯ c ( 1 ) T δ ¯ + M ¯ c ( 1 ) T γ ¯ ] .
( E s + E i ) × ξ ^ ξ = ξ 0 = E c × ξ ^ ξ = ξ 0 ,
( H s + H i ) × ξ ^ ξ = ξ 0 = H c × ξ ^ ξ = ξ 0 ,
n = m { [ U m , κ , n ( 1 ) ( h R ) + V m , κ , n ( 1 ) ( h R ) ] γ m n + [ U m , κ , n ( 1 ) ( h L ) - V m , κ , n ( 1 ) ( h L ) ] δ m n - U m , κ , n ( 4 ) ( h ) α m n - V m , κ , n ( 4 ) ( h ) β m n } = n = m [ U m , κ , n ( 1 ) ( h ) p m n + V m , κ , n ( 1 ) ( h ) q m n ] ,
n = m { [ X m , κ , n ( 1 ) ( h R ) + Y m , κ , n ( 1 ) ( h R ) ] γ m n + [ X m , κ , n ( 1 ) ( h L ) - Y m , κ , n ( 1 ) ( h L ) ] δ m n - X m , κ , n ( 4 ) ( h ) α m n - Y m , κ , n ( 4 ) ( h ) β m n } = n = m [ X m , κ , n ( 1 ) ( h ) p m n + Y m , κ , n ( 1 ) ( h ) q m n ] ,
n = m { χ [ U m , κ , n ( 1 ) ( h R ) + V m , κ , n ( 1 ) ( h R ) ] γ m n - χ [ U m , κ , n ( 1 ) ( h L ) - V m , κ , n ( 1 ) ( h L ) ] δ m n - V m , κ , n ( 4 ) ( h ) α m n - U m , κ , n ( 4 ) ( h ) β m n } = n = m [ V m , κ , n ( 1 ) ( h ) p m n + U m , κ , n ( 1 ) ( h ) q m n ] ,
n = m { χ [ X m , κ , n ( 1 ) ( h R ) + Y m , κ , n ( 1 ) ( h R ) ] γ m n - χ [ X m , κ , n ( 1 ) ( h L ) - Y m , κ , n ( 1 ) ( h L ) ] δ m n - Y m , κ , n ( 4 ) ( h ) α m n - X m , κ , n ( 4 ) ( h ) β m n } = n = m [ Y m , κ , n ( 1 ) ( h ) p m n + X m , κ , n ( 1 ) ( h ) q m n ] ,
U m , κ , n ( i ) ( c ) = 1 2 π - 1 1 0 2 π l η M m n η r ( i ) ( c ; ξ 0 , η , ϕ ) × exp ( ± j m ϕ ) S m - 1 , m - 1 + κ ( h , η ) d η d ϕ ,
V m , κ , n ( i ) ( c ) = 1 2 π - 1 1 0 2 π l η N m n η r ( i ) ( c ; ξ 0 , η , ϕ ) × exp ( ± j m ϕ ) S m - 1 , m - 1 + κ ( h , η ) d η d ϕ ,
X m , κ , n ( i ) ( c ) = 1 2 π - 1 1 0 2 π l ϕ M m n ϕ r ( i ) ( c ; ξ 0 , η , ϕ ) × exp ( ± j m ϕ ) S m - 1 , m - 1 + κ ( h , η ) d η d ϕ ,
Y m , κ , n ( i ) ( c ) = 1 2 π - 1 1 0 2 π l ϕ N m n ϕ r ( i ) ( c ; ξ 0 , η , ϕ ) × exp ( ± j m ϕ ) S m - 1 , m - 1 + κ ( h , η ) d η d ϕ .
V 0 , κ , n ( i ) ( c ) = 1 2 π - 1 1 0 2 π l η N 0 n η r ( i ) ( c ; ξ 0 , η , ϕ ) S 1 , 1 + κ ( h , η ) d η d ϕ ,
X 0 , κ , n ( i ) ( c ) = 1 2 π - 1 1 0 2 π l ϕ M 0 n ϕ r ( i ) ( c ; ξ 0 , η , ϕ ) S 1 , 1 + κ ( h , η ) d η d ϕ .
E s = { [ exp ( - j k r ) ] / k r } [ F θ ( θ , ϕ ) θ ^ + F ϕ ( θ , ϕ ) ϕ ^ ] ,
F θ ( θ , ϕ ) = m = 1 n = m { - j n m S m n ( h , cos θ ) sin θ [ ( α m n - α - m n ) × cos ( m ϕ ) + j ( α m n + α - m n ) sin ( m ϕ ) ] + j n d d θ S m n ( h , cos θ ) [ ( β m n + β - m n ) cos ( m ϕ ) + j ( β m n - β - m n ) sin ( m ϕ ) ] } + n = 0 j n β 0 n d d θ S 0 n ( h , cos θ ) ,
F ϕ ( θ , ϕ ) = m = 1 n = m { j n + 1 m S m n ( h , cos θ ) sin θ [ ( β m n - β - m n ) × cos ( m ϕ ) + j ( β m n + β - m n ) sin ( m ϕ ) ] - j n + 1 d d θ S m n ( h , cos θ ) [ ( α m n + α - m n ) cos ( m ϕ ) + j ( α m n - α - m n ) sin ( m ϕ ) ] } - n = 0 j n + 1 α 0 n d d θ S 0 n ( h , cos θ ) ,
π σ ( θ , ϕ ) / λ 2 = F θ ( θ , ϕ ) 2 + F ϕ ( θ , ϕ ) 2 .
π σ ( θ i ) / λ 2 = F θ ( θ i , 0 ) 2 + F ϕ ( θ i , 0 ) 2 .
Ψ e m n o ( i ) ( h ; ξ , η , ϕ ) = R m n ( i ) ( h , ξ ) S m n ( h , η ) { cos ( m ϕ ) sin ( m ϕ ) ,
S m n ( h , η ) = r = 0 , 1 d r m n ( h ) P m + r m ( η ) ,
R m n ( 1 ) ( h , ξ ) = ( ξ 2 - 1 ξ 2 ) m / 2 r = 0 , 1 a r m n ( h ) j m + r ( h , ξ ) ,
R m n ( 2 ) ( h , ξ ) = ( ξ 2 - 1 ξ 2 ) m / 2 r = 0 , 1 a r m n ( h ) n m + r ( h , ξ ) ,
R m n ( 3 ) ( h , ξ ) = R m n ( 1 ) ( h , ξ ) + j R m n ( 2 ) ( h , ξ ) ,
R m n ( 4 ) ( h , ξ ) = R m n ( 1 ) ( h , ξ ) - j R m n ( 2 ) ( h , ξ ) .
M e m n o r ( i ) ( h ; ξ , η , ϕ ) = Ψ e m n o ( i ) ( h ; ξ , η , ϕ ) × r ,
N e m n o r ( i ) ( h ; ξ , η , ϕ ) = k - 1 [ × M e m n o r ( i ) ( h ; ξ , η , ϕ ) ] ,

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