Abstract

We investigate the far-field scattering of an eccentric sphere (a host sphere with an eccentric spherical inclusion) arbitrarily located in a shaped beam. The extended beam-shape coefficients of the incident beam of arbitrary direction and location in spherical coordinates are presented and evaluated by using the rotational addition theorems for spherical vector wave functions and the approach of the localized approximation. Based on the generalized Lorenz–Mie theory, a general infinite set of scattering equations of an eccentric sphere arbitrarily illuminated is given. The angular distribution of the scattered intensity is calculated and discussed with comparisons of scattered intensities with respect to different incidence angles, center–center separation distances, and refractive indices of the host and the inclusion.

© 2008 Optical Society of America

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  1. D. S. Wang and P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190-1197 (1979).
    [CrossRef] [PubMed]
  2. D. S. Wang, “Light scattering by nonspherical multilayered particles,” Ph.D. dissertation (University of Utah, 1979).
  3. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13-23 (1954).
  7. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15-24 (1961).
  8. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33-44 (1962).
  9. J. G. Fikioris and N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359-1366 (1979).
    [CrossRef]
  10. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327-1335 (1992).
    [CrossRef]
  11. K. A. Fuller, “Scattering and absorbtion by inhomogeneous spheres and sphere aggregates,” Proc. SPIE 1862, 249-257 (1993).
    [CrossRef]
  12. D. Ngo, “Light scattering from a sphere with a nonconcentirc spherical inclusion,” Ph.D. dissertation (New Mexico State University, 1994).
  13. D. Ngo, “Light scattering from nonconcentric spheres,” Tech. Rep. (New Mexico State University, 1996).
  14. G. Videen, D. Ngo, P. Chylek, and R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922-928 (1995).
    [CrossRef]
  15. G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821-837 (2000).
    [CrossRef]
  16. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [CrossRef]
  17. F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113-7124 (1995).
    [CrossRef] [PubMed]
  18. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithms for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  19. Y. Han, H. Zhang, and G. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express 15, 735-746 (2007).
    [CrossRef] [PubMed]
  20. H. Zhang, Y. Han, and G. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24, 1383-1391 (2007).
    [CrossRef]
  21. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  22. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave function,” Appl. Opt. 36, 2971-2978 (1997).
    [CrossRef] [PubMed]
  23. G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874-4883 (1988).
    [CrossRef] [PubMed]
  24. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998-1003 (1990).
    [CrossRef]
  25. K. F. Ren, G. Gréhan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: fast algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144-150 (1992).
    [CrossRef]
  26. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693-706 (1993).
    [CrossRef]
  27. C. Wu and Z. Wu, “Comparison of beam coefficients gnm when waves vary in time as e−iωt and eiωt respectively,” J. Gezhouba Institute of Hydro-Electric Engineering , China, 18, 41-47 (1996).
  28. R. Mittra, Computer Techniques for Electromagnetics, (Pergamon, 1973).
  29. R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

2007 (2)

2000 (1)

G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821-837 (2000).
[CrossRef]

1997 (2)

1996 (1)

C. Wu and Z. Wu, “Comparison of beam coefficients gnm when waves vary in time as e−iωt and eiωt respectively,” J. Gezhouba Institute of Hydro-Electric Engineering , China, 18, 41-47 (1996).

1995 (2)

1994 (1)

1993 (2)

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693-706 (1993).
[CrossRef]

K. A. Fuller, “Scattering and absorbtion by inhomogeneous spheres and sphere aggregates,” Proc. SPIE 1862, 249-257 (1993).
[CrossRef]

1992 (2)

K. F. Ren, G. Gréhan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: fast algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144-150 (1992).
[CrossRef]

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327-1335 (1992).
[CrossRef]

1991 (1)

1990 (1)

1988 (3)

1979 (2)

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33-44 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15-24 (1961).

1954 (1)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13-23 (1954).

Barber, P. W.

Borghese, F.

Chowdhury, D. Q.

Chylek, P.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33-44 (1962).

Denti, P.

Doicu, A.

Fikioris, J. G.

Friedman, B.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13-23 (1954).

Fuller, K. A.

K. A. Fuller, “Scattering and absorbtion by inhomogeneous spheres and sphere aggregates,” Proc. SPIE 1862, 249-257 (1993).
[CrossRef]

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Han, G.

Han, Y.

Harrington, R.

R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

Hill, S. C.

Jamison, J. M.

Kong, J. A.

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

Lin, C. Y.

Lock, J. A.

Maheu, B.

Mittra, R.

R. Mittra, Computer Techniques for Electromagnetics, (Pergamon, 1973).

Ngo, D.

G. Videen, D. Ngo, P. Chylek, and R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922-928 (1995).
[CrossRef]

D. Ngo, “Light scattering from a sphere with a nonconcentirc spherical inclusion,” Ph.D. dissertation (New Mexico State University, 1994).

D. Ngo, “Light scattering from nonconcentric spheres,” Tech. Rep. (New Mexico State University, 1996).

Onofri, F.

Pinnick, R. G.

Ren, K. F.

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithms for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188-5198 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: fast algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144-150 (1992).
[CrossRef]

Russek, J.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13-23 (1954).

Saija, R.

Shin, R. T.

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

Sindoni, O. I.

Srivstava, V.

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15-24 (1961).

Tsang, L.

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

Uzunoglu, N. K.

Videen, G.

Wang, D. S.

D. S. Wang and P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190-1197 (1979).
[CrossRef] [PubMed]

D. S. Wang, “Light scattering by nonspherical multilayered particles,” Ph.D. dissertation (University of Utah, 1979).

Wang, R. T.

Wriedt, T.

Wu, C.

C. Wu and Z. Wu, “Comparison of beam coefficients gnm when waves vary in time as e−iωt and eiωt respectively,” J. Gezhouba Institute of Hydro-Electric Engineering , China, 18, 41-47 (1996).

Wu, Z.

C. Wu and Z. Wu, “Comparison of beam coefficients gnm when waves vary in time as e−iωt and eiωt respectively,” J. Gezhouba Institute of Hydro-Electric Engineering , China, 18, 41-47 (1996).

Wu, Z. S.

Zhang, H.

Appl. Opt. (7)

J. Gezhouba Institute of Hydro-Electric Engineering (1)

C. Wu and Z. Wu, “Comparison of beam coefficients gnm when waves vary in time as e−iωt and eiωt respectively,” J. Gezhouba Institute of Hydro-Electric Engineering , China, 18, 41-47 (1996).

J. Mod. Opt. (1)

G. Gouesbet and G. Gréhan, “Generalized Lorenz-Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821-837 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Part. Part. Syst. Charact. (1)

K. F. Ren, G. Gréhan, and G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: fast algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144-150 (1992).
[CrossRef]

Proc. SPIE (1)

K. A. Fuller, “Scattering and absorbtion by inhomogeneous spheres and sphere aggregates,” Proc. SPIE 1862, 249-257 (1993).
[CrossRef]

Q. Appl. Math. (3)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13-23 (1954).

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15-24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33-44 (1962).

Other (6)

D. Ngo, “Light scattering from a sphere with a nonconcentirc spherical inclusion,” Ph.D. dissertation (New Mexico State University, 1994).

D. Ngo, “Light scattering from nonconcentric spheres,” Tech. Rep. (New Mexico State University, 1996).

D. S. Wang, “Light scattering by nonspherical multilayered particles,” Ph.D. dissertation (University of Utah, 1979).

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

R. Mittra, Computer Techniques for Electromagnetics, (Pergamon, 1973).

R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the rotation of the coordinates.

Fig. 2
Fig. 2

Geometry of the problem.

Fig. 3
Fig. 3

Comparison of the scattered intensity obtained by our code for d = 0.0 (concentric sphere) with that of the code of Wu et al. [18] for a two-layered sphere.

Fig. 4
Fig. 4

Distribution of the scattered intensity of a concentric sphere for different locations of the beam-waist center.

Fig. 5
Fig. 5

Backscattering total intensity S 11 as a function of the incident angle for an air inclusion inside a water host sphere.

Fig. 6
Fig. 6

Distribution of the scattered intensity of an eccentric sphere for different locations of the beam-waist center.

Fig. 7
Fig. 7

Distribution of the scattered intensity of eccentric sphere for different center–center distances.

Tables (1)

Tables Icon

Table 1 Values of EBSCs

Equations (55)

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m m n ( i ) ( k r , θ , ϕ ) = m = n n m m n ( i ) ( k r 1 , θ 1 , ϕ 1 ) D m m ( n ) ( α β γ ) ,
n m n ( i ) ( k r , θ , ϕ ) = m = n n n m n ( i ) ( k r 1 , θ 1 , ϕ 1 ) D m m ( n ) ( α β γ ) ,
D m m ( n ) ( α β γ ) = e i m γ d m m ( n ) ( β ) e i m α ,
d m m ( n ) ( β ) = ( σ ) [ ( n + m ) ! ( n m ) ! ( n + m ) ! ( n m ) ! ] 1 2 σ ! ( m + m + σ ) ! ( n m σ ) ! ( n m σ ) ! ( 1 ) n m σ ( cos β 2 ) 2 σ + m + m ( sin β 2 ) 2 n 2 σ m m ,
E i = E 0 n = 1 m = n n C n m [ i g n , TE m m m n ( 1 ) ( k r , θ , ϕ ) + g n , TM m n m n ( 1 ) ( k r , θ , ϕ ) ]
C n m = { i n 1 2 n + 1 n ( n + 1 ) , m 0 ( 1 ) m ( n + m ) ! ( n m ) ! i n 1 2 n + 1 n ( n + 1 ) , m < 0 } ,
E i = E 0 n = 1 m = n n [ i G n , TE m ( α , β , γ ) m m n ( 1 ) ( k r 1 , θ 1 , ϕ 1 ) + G n , TM m ( α , β , γ ) n m n ( 1 ) ( k r 1 , θ 1 , ϕ 1 ) ] .
G n , TM m ( α , β , γ ) = m = n n C n m D m m ( n ) ( α β γ ) g n , TM m ,
G n , TE m ( α , β , γ ) = m = n n C n m D m m ( n ) ( α β γ ) g n , TE m .
( g n , TM + m g n , TE + m ) = Z n m ( g n , TM + m , old g n , TE + m , old ) ,
( g n , TM + m , old g n , TE + m , old ) = exp ( i k z 0 ) i Q + exp [ i Q + ( ρ n w 0 ) 2 ] exp ( i Q + x 0 2 + y 0 2 w 0 2 ) 1 2 ( j + = m j p Ψ j p ± j = m j p Ψ j p ) ( 1 i ) ,
Q + = 1 i 2 z 0 l , ρ n = n + 1 2 2 π λ , j p = j = 0 p = 0 j ,
j + j = j 2 p ± 1 ,
Ψ j p = ( i Q + r sin θ w 0 2 ) j ( x 0 i y 0 ) j p ( x 0 + i y 0 ) p ( j p ) ! p ! ,
Z n m = { i n ( n + 1 ) n + 1 2 if m = 0 ( i n + 1 2 ) m 1 if m 0 } .
( g n , TM m g n , TE m ) = Z n m ( g n , TM + m , old g n , TE + m , old ) * .
G n , TM m ( α , β , γ ) = m = n n C n m D m m ( n ) ( α β γ ) Z n m ( g n , TM + m , old ) * ,
G n , TE m ( α , β , γ ) = m = n n C n m D m m ( n ) ( α β γ ) Z n m ( g n , TE + m , old ) * .
G n , TM m ( 0 ° , 0 ° , 0 ° ) C n m = g n , TM m ,
G n , TE m ( 0 ° , 0 ° , 0 ° ) C n m = g n , TE m .
x 2 = x 1 , y 2 = y 1 , z 2 = z 1 d ,
E i = E 0 n = 1 m = n n [ a n m i c m m n ( 1 ) ( k r 1 , θ 1 , ϕ 1 ) + b n m i c n m n ( 1 ) ( k r 1 , θ 1 , ϕ 1 ) ] ,
a n m i c = i G n , TE m ( α , β , γ ) ,
b n m i c = G n , TM m ( α , β , γ ) ,
E int , 1 = E 0 n = 0 m = n n [ c m n m m n r ( 3 ) ( k 1 r 2 , θ 2 , ϕ 2 ) + d m n n m n r ( 3 ) ( k 1 r 2 , θ 2 , ϕ 2 ) + e m n m m n r ( 4 ) ( k 1 r 2 , θ 2 , ϕ 2 ) + f m n n m n r ( 4 ) ( k 1 r 2 , θ 2 , ϕ 2 ) ] ,
E int , 2 = E 0 n = 0 m = n n [ g m n m m n r ( 1 ) ( k 2 r 2 , θ 2 , ϕ 2 ) + h m n n m n r ( 1 ) ( k 2 r 2 , θ 2 , ϕ 2 ) ] ,
E sca = E 0 n = 0 m = n n [ a m n sc n m n r ( 3 ) ( k r 1 , θ 1 , ϕ 1 ) + b m n sc m m n r ( 3 ) ( k r 1 , θ 1 , ϕ 1 ) ] ,
m m n ( i ) ( k 1 r 2 , θ 2 , ϕ 2 ) n m n ( i ) ( k 1 r 2 , θ 2 , ϕ 2 ) = v [ A m v m n 2 , 1 ( i ) m m v ( i ) ( k 1 r 1 , θ 1 , ϕ 1 ) n m v ( i ) ( k 1 r 1 , θ 1 , ϕ 1 ) + B m v m n 2 , 1 ( i ) n m v ( i ) ( k 1 r 1 , θ 1 , ϕ 1 ) m m v ( i ) ( k 1 r 1 , θ 1 , ϕ 1 ) ] ,
A m v m n ( 3 ) = A m v m n ( 4 ) = A m v m , n ( 3 ) ,
B m v m n ( 3 ) = B m v m n ( 4 ) = B m v m , n ( 3 ) .
E int , 1 = E 0 n = 0 m = n n [ c m n m m n ( 3 ) ( k 1 r 1 , θ 1 , ϕ 1 ) + d m n n m n ( 3 ) ( k 1 r 1 , θ 1 , ϕ 1 ) + e m n m m n ( 4 ) ( k 1 r 1 , θ 1 , ϕ 1 ) + f m n n m n ( 4 ) ( k 1 r 1 , θ 1 , ϕ 1 ) ] ,
c m n d m n = v = 0 [ A m n m v ; 2 , 1 ( 3 ) c m v d m v + B m n m v ; 2 , 1 ( 3 ) d m v c m v ] ,
e m n f m n = v = 0 [ A m n m v ; 2 , 1 ( 4 ) e m v f m v + B m n m v ; 2 , 1 ( 4 ) f m v e m v ] .
w m n w m n , w m v w m v , A m n m v ; 2 , 1 ( 3 , 4 ) A m n m v ; 1 , 2 ( 3 , 4 ) ,
B m n m v ; 2 , 1 ( 3 , 4 ) B m n m v ; 1 , 2 ( 3 , 4 ) ,
c m n d m n = v = 0 [ A m n m v ; 1 , 2 ( 3 ) c m v d m v + B m n m v ; 1 , 2 ( 3 ) d m v c m v ] ,
e m n f m n = v = 0 [ A m n m v ; 1 , 2 ( 4 ) e m v f m v + B m n m v ; 1 , 2 ( 4 ) f m v e m v ] .
E int , 1 θ = E int , 2 θ , H int , 1 θ = H int , 2 θ ,
E int , 1 ϕ = E int , 2 ϕ , H int , 1 ϕ = H int , 2 ϕ ,
E i θ + E s θ = E int , 1 θ , H i θ + H s θ = H int , 1 θ ,
E i ϕ + E s ϕ = E int , 1 ϕ , H i ϕ + H s ϕ = H int , 1 ϕ ,
C n v n { [ A m n m v ; 2 , 1 ( 3 ) A m v m n ; 1 , 2 ( 4 ) U m v + B m n m v ; 2 , 1 ( 3 ) B m v m n ; 1 , 2 ( 4 ) V m v ] e m n + [ A m n m v ; 2 , 1 ( 3 ) B m v m n ; 1 , 2 ( 4 ) U m v + B m n m v ; 2 , 1 ( 3 ) A m v m n ; 1 , 2 ( 4 ) V m v ] f m n } + E n e m n = G n ,
D n v n { [ A m n m v ; 2 , 1 ( 3 ) A m v m n ; 1 , 2 ( 4 ) V m v + B m n m v ; 2 , 1 ( 3 ) B m v m n ; 1 , 2 ( 4 ) U m v ] f m n + [ A m n m v ; 2 , 1 ( 3 ) B m v m n ; 1 , 2 ( 4 ) V m v + B m n m v ; 2 , 1 ( 3 ) A m v m n ; 1 , 2 ( 4 ) U m v ] e m n } + F n f m n = H n ,
C n = k ξ n ( 1 ) ( k 1 a ) ξ n ( 1 ) ( k a ) ξ n ( 1 ) ( k 1 a ) k 1 ξ n ( 1 ) ( k a ) ,
D n = k 1 ξ n ( 1 ) ( k a ) ξ n ( 1 ) ( k 1 a ) k ξ n ( 1 ) ( k 1 a ) ξ n ( 1 ) ( k a ) ,
E n = k ξ n ( 2 ) ( k 1 a ) ξ n ( 1 ) ( k a ) k 1 ξ n ( 1 ) ( k a ) ξ n ( 2 ) ( k 1 a ) ,
F n = k 1 ξ n ( 1 ) ( k a ) ξ n ( 2 ) ( k 1 a ) k ξ n ( 1 ) ( k a ) ξ n ( 2 ) ( k 1 a ) ,
G n = a n m i c k 1 ψ n ( k a ) ξ n ( 1 ) ( k a ) a n m i c ψ n ( k a ) k 1 ξ n ( 1 ) ( k a ) ,
H n = b n m i c k 1 ξ n ( 1 ) ( k a ) ψ n ( k a ) b n m i c k 1 ψ n ( k a ) ξ n ( 1 ) ( k a ) ,
U m n = c m n e m n = k 1 ξ n ( 2 ) ( k 1 b ) ψ n ( k 2 b ) k 2 ξ n ( 2 ) ( k 1 b ) ψ n ( k 2 b ) k 2 ξ n ( 1 ) ( k 1 b ) ψ n ( k 2 b ) k 1 ξ n ( 1 ) ( k 1 b ) ψ n ( k 2 b ) ,
V m n = d m n f m n = k 2 ξ n ( 2 ) ( k 1 b ) ψ n ( k 2 b ) k 1 ξ n ( 2 ) ( k 1 b ) ψ n ( k 2 b ) k 1 ξ n ( 1 ) ( k 1 b ) ψ n ( k 2 b ) k 2 ξ n ( 1 ) ( k 1 b ) ψ n ( k 2 b ) ,
C n c m n + E n e m n = G n ,
D n d m n + F n f m n = H n .
a m n sc = c m n k ξ n ( 1 ) ( k 1 a ) + e m n k ξ n ( 2 ) ( k 1 a ) a n m i c k 1 ψ n ( k a ) k 1 ξ n ( 1 ) ( k a ) ,
b m n sc = d m n ξ n ( 1 ) ( k 1 a ) + f m n ξ n ( 2 ) ( k 1 a ) b n m i c ψ n ( k a ) ξ n ( 1 ) ( k a ) .

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