Abstract

An efficient procedure is developed for calculating light scattering from a spherical particle on a conducting plane. The problem is converted into an equivalent two-particle scattering problem by the method of images. The scattering solution for this two-particle problem is then obtained by the multipole expansion method. Mirror symmetry is exploited to simplify the calculation. Formulas are developed for the differential-scattering cross section and for the extinction and the absorption cross sections. Several example problems are calculated, and the exact solutions are compared with the results from two approximation procedures.

© 1992 Optical Society of America

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Corrections

B. R. Johnson, "Light scattering from a spherical particle on a conducting plane: I. Normal incidence: errata," J. Opt. Soc. Am. A 10, 766-766 (1993)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-10-4-766

References

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  1. P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
    [Crossref]
  2. R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
    [Crossref]
  3. K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
    [Crossref] [PubMed]
  4. D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
    [Crossref]
  5. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  6. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [Crossref]
  7. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
    [Crossref]
  8. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).
  9. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. I. TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [Crossref]
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  12. Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.
  13. V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
    [Crossref]
  14. M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
    [Crossref]
  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  16. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  17. K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
    [Crossref]
  18. B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27, 4861–4873 (1988).
    [Crossref] [PubMed]
  19. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [Crossref]
  20. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [Crossref] [PubMed]
  21. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962); Phys. Rev. 134, AB1(E) (1964).
    [Crossref]
  22. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [Crossref] [PubMed]
  23. B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: Part I. Normal incidence,” (Aerospace Corporation, El Segundo, Calif., April15, 1991).
  24. M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (MIT Press, Cambridge, Mass., 1959).

1991 (2)

1990 (1)

1989 (1)

1988 (3)

1987 (1)

1986 (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
[Crossref]

1985 (1)

1976 (1)

R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
[Crossref]

1975 (1)

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[Crossref]

1971 (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

1967 (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

1962 (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962); Phys. Rev. 134, AB1(E) (1964).
[Crossref]

1957 (1)

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[Crossref]

1951 (2)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
[Crossref]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

Barakat, R.

Barber, P. W.

Bhandari, R.

Biot, M. A.

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[Crossref]

Bivens, R.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (MIT Press, Cambridge, Mass., 1959).

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Bohren, C. F.

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

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

Fuller, K. A.

Gordon, R. G.

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[Crossref]

Hirleman, E. D.

D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
[Crossref]

Huffman, D. R.

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

Johnson, B. R.

B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27, 4861–4873 (1988).
[Crossref] [PubMed]

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: Part I. Normal incidence,” (Aerospace Corporation, El Segundo, Calif., April15, 1991).

Kattawar, G. W.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lilienfeld, P.

P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
[Crossref]

Lindell, I. V.

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Metropolis, N.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (MIT Press, Cambridge, Mass., 1959).

Muinonen, K. O.

Nahm, K. B.

Rao, T. C.

Rayleigh,

Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

Rotenberg, M. R.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (MIT Press, Cambridge, Mass., 1959).

Schulten, K.

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[Crossref]

Sihvola, A. H.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Taubenblatt, M. A.

Twersky, V.

V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
[Crossref]

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Weber, D. C.

D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
[Crossref]

Wolfe, W. L.

Wooten, J. K.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (MIT Press, Cambridge, Mass., 1959).

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962); Phys. Rev. 134, AB1(E) (1964).
[Crossref]

Young, R. P.

R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
[Crossref]

Aerosol Sci. Technol. (1)

P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

J. Appl. Phys. (3)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
[Crossref]

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[Crossref]

J. Math. Phys. (1)

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962); Phys. Rev. 134, AB1(E) (1964).
[Crossref]

Physica (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 209–242 (1986).

Radio Sci. (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Other (5)

Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: Part I. Normal incidence,” (Aerospace Corporation, El Segundo, Calif., April15, 1991).

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (MIT Press, Cambridge, Mass., 1959).

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Figures (10)

Fig. 1
Fig. 1

Illustration of the method of images: (a) physical system, (b) equivalent method-of-images system.

Fig. 2
Fig. 2

Geometry of the problem of a particle on a mirror.

Fig. 3
Fig. 3

Geometry of the two-particle problem that is equivalent, by the method of images, to the problem shown in Fig. 2.

Fig. 4
Fig. 4

Geometry of the light rays for the Mie approximation.

Fig. 5
Fig. 5

Geometry of the light rays for the SSA.

Fig. 6
Fig. 6

Differential cross section for light scattering from a particle on a mirror. R = 0.2, d = 0.2, n = 1.46. Comparison of the exact solution with the Mie approximation and the SSA.

Fig. 7
Fig. 7

Differential cross section for light scattering from a particle on a mirror. R = 1.0, d = 1.0, n = 1.46. Comparison of the exact solution with the Mie approximation and the SSA.

Fig. 8
Fig. 8

Differential cross section for light scattering from a particle on a mirror. R = 2.5, d = 2.5, n = 1.46. Comparison of the exact solution with the Mie approximation and the SSA.

Fig. 9
Fig. 9

Differential cross section for light scattering from a particle on a mirror. R = 1.0, d = 1.0, n = 1.3. Comparison of the exact solution with the Mie approximation and the SSA.

Fig. 10
Fig. 10

Differential cross section for light scattering from a particle above a mirror. R = 1.0, d = 5.0, n = 1.3. Comparison of the exact solution with the Mie approximation and the SSA.

Equations (59)

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× H + i ω ( ɛ + i σ ω ) E = 0 , × E - i ω μ H = 0 , · ( ɛ E ) = 0 , · ( μ H ) = 0 .
n ^ × E = 0 , n ^ · H = 0 ,
E ( x , y , z ) = E x ( x , y , z ) e ^ x + E y ( x , y , z ) e ^ y + E z ( x , y , z ) e ^ z , H ( x , y , z ) = H x ( x , y , z ) e ^ x + H y ( x , y , z ) e ^ y + H z ( x , y , z ) e ^ z ,
E ˜ ( x , y , z ) = - E x ( x , y , - z ) e ^ x - E y ( x , y , - z ) e ^ y + E z ( x , y , - z ) e ^ z , H ˜ ( x , y , z ) = H x ( x , y , - z ) e ^ x + H y ( x , y , - z ) e ^ y - H z ( x , y , - z ) e ^ z ,
E total = E + E ˜ , H total = H + H ˜ .
E i = ( e ^ x + i e ^ y ) [ exp ( i k z ) - exp ( - i k z ) ] .
E = E i + E s ( 1 ) + E s ( 2 ) .
M n , m ( j ) = z n ( j ) ( k r ) exp ( i m ϕ ) X n , m ( θ ) , N n , m ( j ) = exp ( i m ϕ ) k r { r [ r z n ( j ) ( k r ) ] Y n , m ( θ ) + z n ( j ) ( k r ) Z n m ( θ ) } ,
X n , m ( θ ) = i π n , m ( θ ) e ^ θ - τ n , m ( θ ) e ^ ϕ , Y n , m ( θ ) = τ n , m ( θ ) e ^ θ + i π n , m ( θ ) e ^ ϕ , Z n , m ( θ ) = n ( n + 1 ) P n m ( cos θ ) e ^ r ,
π n , m ( θ ) = m sin ( θ ) P n m ( cos θ ) , τ n , m ( θ ) = θ P n m ( cos θ ) .
( e ^ x + i e ^ y ) exp ( ± i k z ) = n = 1 ( ± i ) n + 1 2 n + 1 n ( n + 1 ) × [ M n ( 1 ) ( r , θ , ϕ ) ± M n ( 1 ) ( r , θ , ϕ ) ] .
E i ( r 1 ) = n = 1 [ p n M n ( 1 ) ( r 1 ) + q n N n ( 1 ) ( r 1 ) ] .
p n = - i n + 1 2 n + 1 n ( n + 1 ) [ exp ( - i k d ) - ( - 1 ) n exp ( i k d ) ] , q n = - i n + 1 2 n + 1 n ( n + 1 ) [ exp ( - i k d ) + ( - 1 ) n exp ( i k d ) ] ,
E s ( j ) ( r j ) = n = 1 [ a n ( j ) M n ( 3 ) ( r j ) + b n ( j ) N n ( 3 ) ( r j ) ] ,             j = 1 , 2.
a n ( 1 ) = - ( - 1 ) n a n ( 2 ) , b n ( 1 ) = ( - 1 ) n b n ( 2 ) .
M n ( 3 ) ( r 2 ) = n = 1 [ A n , n M n ( 1 ) ( r 1 ) + B n , n N n ( 1 ) ( r 1 ) ] , N n ( 3 ) ( r 2 ) = n = 1 [ A n , n N n ( 1 ) ( r 1 ) + B n , n M n ( 1 ) ( r 1 ) ] .
E = n { [ p n + n ( A n , n a n ( 2 ) + B n , n b n ( 2 ) ) ] M n ( 1 ) ( r 1 ) + a n ( 1 ) M n ( 3 ) ( r 1 ) + [ q n + n ( B n , n a n ( 2 ) + A n , n b n ( 2 ) ) ] × N n ( 1 ) ( r 1 ) + b n ( 1 ) N n ( 3 ) ( r 1 ) } .
E = n [ α n M n ( 1 ) + a n ( 1 ) M n ( 3 ) + β n N n ( 1 ) + b n ( 1 ) N n ( 3 ) ] .
u n = a n ( 1 ) / α n , v n = b n ( 1 ) / β n .
a n ( 1 ) = u n { p n + n [ A n , n a n ( 2 ) + B n , n b n ( 2 ) ] } , b n 1 = v n { q n + n [ B n , n a n ( 2 ) + A n , n b n ( 2 ) ] } .
[ A T + g u - 1 B T B T A T - g v - 1 ] [ a ( 2 ) b ( 2 ) ] = - [ p q ] ,
E scat = E s ( 1 ) + E s ( 2 ) .
M n ( 3 ) = ( - i ) n exp ( i k r ) i k r X n ( θ ) exp ( i ϕ ) , N n ( 3 ) = i ( - i ) n exp ( i k r ) i k r Y n ( θ ) exp ( i ϕ ) .
E scat = exp ( i k r 1 ) i k r 1 exp ( i ϕ 1 ) n = 1 N ( - i ) n [ a n ( 1 ) X n ( θ 1 ) + i b n ( 1 ) Y n ( θ 1 ) ] + exp ( i k r 2 ) i k r 2 exp ( i ϕ 2 ) n = 1 N ( - i ) n [ a n ( 2 ) X n ( θ 2 ) + i b n ( 2 ) Y n ( θ 2 ) ] .
r 1 = r + d cos ( θ ) , r 2 = r - d cos ( θ ) , θ 1 = θ 2 = θ , ϕ 1 = ϕ 2 = ϕ .
E scat = exp ( i k r ) i k r exp ( i ϕ ) [ S θ ( θ ) e ^ θ + i S ϕ ( θ ) e ^ ϕ ] ,
S θ ( θ ) = exp [ i k d cos ( θ ) ] S θ ( 1 ) ( θ ) + exp [ - i k d cos ( θ ) ] S θ ( 2 ) ( θ ) , S ϕ ( θ ) = exp [ i k d cos ( θ ) ] S ϕ ( 1 ) ( θ ) + exp [ - i k d cos ( θ ) ] S ϕ ( 2 ) ( θ ) ,
S θ ( j ) ( θ ) = - n = 1 N ( - i ) n + 1 [ a n ( j ) π n ( θ ) + b n ( j ) τ n ( θ ) ] , S ϕ ( j ) ( θ ) = - n = 1 N ( - i ) n + 1 [ a n ( j ) τ n ( θ ) + b n ( j ) π n ( θ ) ] ,
σ ( θ ) = 1 2 k 2 [ S θ ( θ ) 2 + S ϕ ( θ ) 2 ] .
σ ( θ ) = 1 k 2 [ S θ ( θ ) 2 cos 2 ( ϕ ) + S ϕ ( θ ) 2 sin 2 ( ϕ ) ] .
C scat = π k 2 0 π / 2 [ S θ ( θ ) 2 + S ϕ ( θ ) 2 ] sin ( θ ) d θ .
C scat = C ext - C abs ,
C ext = - 4 π k 2 Re S ( 0 ) ,
C ext = 2 π k 2 Im n = 1 N ( - 1 ) n n ( n + 1 ) × { exp ( i k d ) [ a n ( 1 ) + b n ( 1 ) ] + exp ( - i k d ) [ a n ( 2 ) + b n ( 2 ) ] } .
P = F { - 2 π k 2 n = 1 N ( 2 n + 1 ) [ ( u n 2 + Re u n ) + ( v n 2 + Re v n ) ] } ,
P n ( M ) = F [ - 2 π k 2 ( 2 n + 1 ) ( u n 2 + Re u n ) ] , P n ( N ) = F [ - 2 π k 2 ( 2 n + 1 ) ( v n 2 + Re v n ) ] .
f n = - [ i n + 1 2 n + 1 n ( n + 1 ) ] .
P ^ n ( M ) = [ n ( n + 1 ) 2 n + 1 ] 2 P n ( M ) , P ^ n ( N ) = [ n ( n + 1 ) 2 n + 1 ] 2 P n ( N ) .
E = n [ α n M n ( 1 ) + β n N n ( 1 ) ] .
α n = a n ( 1 ) / u n , β n = b n ( 1 ) / v n .
C abs = - 2 π k 2 n = 1 N [ n ( n + 1 ) ] 2 2 n + 1 × [ | a n ( 1 ) u n | 2 ( u n 2 + Re u n ) + | b n ( 1 ) v n | 2 ( v n 2 + Re v n ) ] .
( U - 1 - C ) F = Q ,
C = [ A T B T B T A T ] ,
U = [ - g u 0 0 g v ] ,
F = [ a ( 2 ) b ( 2 ) ] ,
Q = [ p q ] .
F = ( U - 1 - C ) - 1 Q .
( U - 1 - C ) - 1 = U + U C U + U C U C U + .
a n ( 1 ) = u n p n , b n ( 1 ) = v n q n .
r 2 = r 1 , θ 2 = π - θ 1 , ϕ 2 = ϕ 1 .
[ E s ( j ) ] ϕ j = - n = 1 N exp ( i ϕ j ) ( z n ( 3 ) ( k r j ) [ a n ( j ) τ n ( θ j ) ] - i { 1 k r j r j [ r j z n ( 3 ) ( k r j ) ] } [ b n ( j ) π n ( θ j ) ] ) ,
[ E s ( 1 ) ] ϕ 1 + [ E s ( 2 ) ] ϕ 2 = 0.
a n ( 1 ) τ n ( θ 1 ) + a n ( 2 ) τ n ( θ 2 ) = 0 , b n ( 1 ) π n ( θ 1 ) + b n ( 2 ) π n ( θ 2 ) = 0.
π n ( π - θ ) = - ( - 1 ) n π n ( θ ) , τ n ( π - θ ) = ( - 1 ) n τ n ( θ ) ,
A n , n = - i n - n 2 n + 1 2 n ( n + 1 ) ν i - ν [ n ( n + 1 ) + n ( n + 1 ) + ν ( ν + 1 ) ] a ( n , n ; ν ) j ν ( k δ ) ,
B n , n = i n - n 2 n + 1 2 n ( n + 1 ) ν i - ν ( 2 i k δ ) a ( n , n ; ν ) j ν ( k δ ) ,
a ( n , n ; ν ) = ( 2 ν + 1 ) [ n ( n + 1 ) n ( n + 1 ) ] 1 / 2 [ n n ν 0 0 0 ] × [ n n ν 1 - 1 0 ] .
u n = - Ψ n ( m x ) Ψ n ( x ) - m Ψ n ( x ) Ψ n ( m x ) Ψ n ( m x ) ξ n ( x ) - m ξ n ( x ) Ψ n ( m x ) ,
v n = - Ψ n ( x ) Ψ n ( m x ) - m Ψ n ( m x ) Ψ n ( x ) ξ n ( x ) Ψ n ( m x ) - m Ψ n ( m x ) ξ n ( x ) ,

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