Abstract

A rigorous integral equation analysis of the coupling between a fiber waveguide and an adjacent spherical particle is developed. The solution is obtained by applying the entire-domain Galerkin technique, based on Mie-type spherical wave expansion of the field in the sphere and the use of dyadic Green’s function of the fiber waveguide. The conversion between cylindrical and spherical wave functions is done through classic analytical formulas. The analysis is applied to numerically investigate transmission through silica wires of subwavelength diameter in the presence of particles of comparable size. The results show the possibility of sensing microparticles through the reduction of transmitted power, which is maximum for certain critical values of the fiber diameter.

© 2006 Optical Society of America

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  1. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
    [CrossRef] [PubMed]
  2. M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
    [CrossRef] [PubMed]
  3. L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
    [CrossRef] [PubMed]
  4. D. Hondros and P. Debye, "Electromagnetische wellen an dielektrishen drahten," Ann. Phys. (Leipzig) 32, 465-476 (1910).
  5. J. Lou, L. Tong, and Z. Ye, "Modeling of silica nanowires for optical sensing," Opt. Express 13, 2135-2140 (2005).
    [CrossRef] [PubMed]
  6. L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2004).
    [CrossRef] [PubMed]
  7. M. D. Marazuela and M. C. Moreno-Bondi, "Fiber-optic biosensors--an overview," Anal. Bioanal. Chem. 372, 664-682 (2002).
    [CrossRef] [PubMed]
  8. N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
    [CrossRef]
  9. N. K. Uzunoglu, "Scattering from inhomogeneities inside a fiber waveguide," J. Opt. Soc. Am. 71, 259-273 (1981).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1983).
  13. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).
  14. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  16. R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
    [CrossRef]
  17. T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
    [CrossRef]
  18. S. Kawata and T. Tani, "Optically driven Mie particles in an evanescent field along a channeled waveguide," Opt. Lett. 21, 1768-1770 (1996).
    [CrossRef] [PubMed]
  19. L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
    [CrossRef]

2005 (2)

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

J. Lou, L. Tong, and Z. Ye, "Modeling of silica nanowires for optical sensing," Opt. Express 13, 2135-2140 (2005).
[CrossRef] [PubMed]

2004 (2)

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2004).
[CrossRef] [PubMed]

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

2003 (1)

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

2002 (2)

L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
[CrossRef]

M. D. Marazuela and M. C. Moreno-Bondi, "Fiber-optic biosensors--an overview," Anal. Bioanal. Chem. 372, 664-682 (2002).
[CrossRef] [PubMed]

1999 (3)

1996 (1)

1981 (1)

1980 (1)

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

1976 (1)

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

1910 (1)

D. Hondros and P. Debye, "Electromagnetische wellen an dielektrishen drahten," Ann. Phys. (Leipzig) 32, 465-476 (1910).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Ashcom, J. B.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Chen, X. W.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

Chew, W. C.

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

Cui, T. J.

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

Debye, P.

D. Hondros and P. Debye, "Electromagnetische wellen an dielektrishen drahten," Ann. Phys. (Leipzig) 32, 465-476 (1910).

Gattass, R. R.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Goldberger, J.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Gorodetsky, M. L.

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1983).

Haus, H. A.

He, S. L.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Hondros, D.

D. Hondros and P. Debye, "Electromagnetische wellen an dielektrishen drahten," Ann. Phys. (Leipzig) 32, 465-476 (1910).

Ilchenko, V. S.

Johnson, J. C.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Kawata, S.

Kumagai, N.

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

Laine, J. P.

Law, M.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Little, B. E.

Liu, L.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

Lou, J.

Lou, J. Y.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Lu, B. J.

L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
[CrossRef]

Lun, E.

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

Marazuela, M. D.

M. D. Marazuela and M. C. Moreno-Bondi, "Fiber-optic biosensors--an overview," Anal. Bioanal. Chem. 372, 664-682 (2002).
[CrossRef] [PubMed]

Maxwell, I.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2004).
[CrossRef] [PubMed]

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Moreno-Bondi, M. C.

M. D. Marazuela and M. C. Moreno-Bondi, "Fiber-optic biosensors--an overview," Anal. Bioanal. Chem. 372, 664-682 (2002).
[CrossRef] [PubMed]

Morita, N.

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

Ng, L. N.

L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
[CrossRef]

Pogorzelski, R. J.

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

Saykally, R. J.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Shen, M. Y.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Sirbuly, D. J.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Tai, C. T.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

Tani, T.

Tong, L.

Tong, L. M.

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Uzunoglu, N. K.

Wilkinson, J. S.

L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
[CrossRef]

Yang, P.

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Ye, Z.

Zervas, M. N.

L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
[CrossRef]

Anal. Bioanal. Chem. (1)

M. D. Marazuela and M. C. Moreno-Bondi, "Fiber-optic biosensors--an overview," Anal. Bioanal. Chem. 372, 664-682 (2002).
[CrossRef] [PubMed]

Ann. Phys. (Leipzig) (1)

D. Hondros and P. Debye, "Electromagnetische wellen an dielektrishen drahten," Ann. Phys. (Leipzig) 32, 465-476 (1910).

IEEE Trans. Geosci. Remote Sens. (1)

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

Nano Lett. (1)

L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, "Assembly of silica nanowires on silica aerogels for microphotonic devices," Nano Lett. 5, 259-262 (2005).
[CrossRef] [PubMed]

Nature (1)

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. N. Ng, B. J. Lu, M. N. Zervas, and J. S. Wilkinson, "Propulsion of gold nanoparticles on optical waveguides," Opt. Commun. 208, 117-124 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Radio Sci. (1)

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

Science (1)

M. Law, D. J. Sirbuly, J. C. Johnson, J. Goldberger, R. J. Saykally, and P. Yang, "Nanoribbon waveguides for subwavelength photonics integration," Science 305, 1269-1273 (2004).
[CrossRef] [PubMed]

Other (4)

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1983).

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (6)

Fig. 1
Fig. 1

Geometry of a fiber waveguide coupled to a sphere.

Fig. 2
Fig. 2

Power transmission versus wire diameter for silica spheres of different sizes at λ = 1.5 μ m .

Fig. 3
Fig. 3

Power transmission versus wire diameter for spheres with diameter 1 μ m and different refractive indices at λ = 1.5 μ m .

Fig. 4
Fig. 4

Power transmission through a 0.608 μ m wide wire for x polarization versus sphere radius and index at λ = 1.5 μ m .

Fig. 5
Fig. 5

Power transmission versus sphere refractive index for a sphere with diameter 1.6 μ m and a 0.8 μ m wide wire at λ = 1.5 μ m . Two cases ( g = 0 , 0.1 μ m ) of fiber–sphere separation g = d α s α f are considered.

Fig. 6
Fig. 6

Power transmission versus wire diameter for the system of Fig. 5 under excitation of WG modes TE5 ( n s = 2.37 ) and TM5 ( n s = 2.68 ) . The fiber–sphere separation is zero.

Equations (49)

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

E ( r ) = E inc ( r ) + ( k s 2 k c 2 ) V s G ̱ f ( r , r ) E ( r ) d V ,
G ̱ f ( r f , r f ) = { G ̱ c ( r f , r f ) + G ̱ c c ( r f , r f ) , ρ f > a f G ̱ f c ( r f , r f ) , ρ f < a f } ,
G ̱ c c f c ( r f , r f ) = j 8 π + d k m = + ( 1 ) m a c f 2 W m , k ( 2 1 ) ( r f , k c f ) A c c f c ( m , k ) W m , k ( 2 ) ( r f , k c ) T ,
E ( r s ) = n = 1 + m = n n [ a n m m n , m ( 1 ) ( r s , k s ) + b n m n n , m ( 1 ) ( r s , k s ) ] n , m w n , m ( 1 ) ( r s , k s ) x n m T , x n m = [ a n m , b n m ] ,
G ̱ c ( r , r ) = r ̂ s r ̂ s k c 2 δ ( r s r s ) j k c 4 π n , m ( 1 ) m 2 n + 1 n ( n + 1 ) { w n , m ( 2 ) ( r s , k c ) w n , m ( 1 ) ( r s , k c ) T , r s > r s w n , m ( 1 ) ( r s , k c ) w n , m ( 2 ) ( r s , k c ) T , r s < r s } ,
( k f 2 k c 2 ) V s G ̱ c ( r s , r s ) E ( r s ) d V s = E ( r s ) + j k c α s n , m w n , m ( 1 ) ( r s , k c ) R n ( 2 ) x n m T ,
W m , k ( 2 ) ( r f , k c ) = λ = + H λ m ( 2 ) ( a c d ) W λ , k ( 1 ) ( r s , k c ) ,
W m , k ( 1 ) ( r s , k c ) = k c n = m + j n m ( n m ) ! ( n + m ) ! w n , m ( 1 ) ( r s , k c ) Ω ( n , m , k ) ,
( k f 2 k c 2 ) V s G ̱ c c ( r s , r s ) E ( r s ) d V s = j k c α s n , m w n , m ( 1 ) ( r s , k c ) ( n , m C ( n , m , n , m ) R n ( 1 ) x n m T ) ,
E inc ( r f ) = { W m 0 , β ( 1 ) ( r f , k f ) [ 1 δ 0 ] T , ρ f < α f A 0 W m 0 , β ( 2 ) ( r f , k c ) [ 1 δ 0 k c k f ] T , ρ f > α f } ,
E inc ( r s ) = n , m w n , m ( 1 ) ( r s , k c ) I ( n , m ) ,
I ( n , m ) = k c A 0 j n m ( n m ) ! ( n + m ) ! H m m 0 ( 2 ) ( a c d ) Ω ( n , m , β ) [ 1 δ 0 k c k f ] T .
R n ( 2 ) x n m T + n , m C ( n , m , n , m ) R n ( 1 ) x n m T = j I ( n , m ) ( k c α s ) 1 ,
n 1 , m n .
C E E = C H H = 0 , if n + m + n + m = odd ,
C E H = C H E = 0 , if n + m + n + m = even or m = m = 0 ,
{ C ( n , m , n , m ) C ( n , m , n , m ) } = ( 1 ) m + m ( n + m ) ! ( n m ) ! ( n m ) ! ( n + m ) ! { X C ( n , m , n , m ) X ( 1 ) n + n ( 2 n + 1 ) n ( n + 1 ) n ( n + 1 ) ( 2 n + 1 ) C ( n , m , n , m ) T } , X = [ 1 0 0 1 ] .
E ( r f ) = E inc ( r f ) + j k c α s 2 + d k m = + ( 1 ) m a f 2 W m , k ( 1 ) ( r f , k f ) A f c ( m , k ) [ n , m j 2 n m n ( n + 1 ) 2 n + 1 H m m ( 2 ) ( a c d ) Ω ( n , m , k ) R n ( 1 ) x n m T ] .
E ( r f ) E inc ( r f ) + { p = 1 M T p W m p , β p ( 1 ) ( r f , k f ) [ 1 δ p ( β p ) ] T , z f + p = 1 M R p W m p , β p ( 1 ) ( r f , k f ) [ 1 δ p ( + β p ) ] T , z f } ,
T ( e e ) = T m p ( m 0 ) ( 1 ) m p T m p ( m 0 ) ,
T ( o o ) = T m p ( m 0 ) + ( 1 ) m p T m p ( m 0 ) ,
M m , k ( i ) ( r , k 0 ) = × [ z ̂ C m ( i ) ( a 0 ρ ) exp ( j m φ + j k z ) ] ,
N m , k ( i ) ( r , k 0 ) = k 0 1 × M m , k ( i ) ( r , k 0 ) ,
W m , k ( i ) ( r , k 0 ) [ M m , k ( i ) ( r , k 0 ) N m , k ( i ) ( r , k 0 ) ] .
α 11 M m , k ( i ) ( r , k 0 ) M m , k ( j ) ( r , k 0 ) + α 12 M m , k ( i ) ( r , k 0 ) N m , k ( j ) ( r , k 0 ) + α 21 N m , k ( i ) ( r , k 0 ) M m , k ( j ) ( r , k 0 ) + α 22 N m , k ( i ) ( r , k 0 ) N m , k ( j ) ( r , k 0 )
W m , k ( i ) ( r , k 0 ) [ α 11 α 12 α 21 α 22 ] W m , k ( i ) ( r , k 0 ) T ,
m n , m ( i ) ( r , k 0 ) = × [ r c n ( i ) ( k 0 r ) P n m ( cos θ ) exp ( j m φ ) ] ,
n n , m ( i ) ( r , k 0 ) = k 0 1 × m n , m ( i ) ( r , k 0 ) ,
w n , m ( i ) ( r , k 0 ) [ m n , m ( i ) ( r , k 0 ) n n , m ( i ) ( r , k 0 ) ] .
A c c ( m , k ) = [ a c c b c c b c c c c c ] , A f c ( m , k ) = [ a f c b f c k f b f c k c k f c f c k c ] ,
{ a c c b c c c c c } = J m ( a c α f ) H m ( 2 ) ( a c α f ) { 1 + ( S 1 S 4 Δ ) , S 2 S 4 Δ , 1 + ( S 3 S 4 Δ ) , a f c b f c c f c } = J m ( a c α f ) J m ( a f α f ) { S 1 S 4 Δ S 2 S 4 Δ S 3 S 4 Δ } ,
S 1 = Λ m ( 1 ) ( a f α f ) n f 2 n c 2 Λ m ( 2 ) ( a c α f ) ,
S 2 = m k k c ( 1 n f 2 n c 2 ) ( a c a f α f ) 2 ,
S 3 = Λ m ( 1 ) ( a f α f ) Λ m ( 2 ) ( a c α f ) ,
S 4 = 2 j [ π ( a c α f ) 2 J m ( a c α f ) H m ( 2 ) ( a c α f ) ] ,
Λ m ( i ) ( x ) = C m ( x ) [ x C m ( x ) ] 1 , i = 1 , 2 , Δ m , k = S 1 S 3 S 2 2 ,
a c = ( k c 2 k 2 ) 1 2 , a f = ( k f 2 k 2 ) 1 2 .
R n ( i ) = [ E n ( i ) 0 0 H n ( i ) ] ,
E n ( i ) = x s c n ( i ) ( x c ) j n ( x s ) x c j n ( x s ) c n ( i ) ( x c ) ,
H n ( i ) = x c c n ( i ) ( x c ) j n d ( x s ) x s x s j n ( x s ) c n d ( i ) ( x c ) x c ,
x c = k c α s , x s = k s α s ,
Ω ( n , m , k ) = [ j q n m ( k ) 2 n + 1 n ( n + 1 ) p n m ( k ) 2 n + 1 n ( n + 1 ) p n m ( k ) j q n m ( k ) ] ,
p n m ( k ) = j m P n m ( k k c ) ,
q n m ( k ) = P n 1 m ( k k c ) ( n + m ) n P n + 1 m ( k k c ) ( n m + 1 ) ( n + 1 ) ,
C ( n , m , n , m ) [ C EE C EH C HE C HH ] = k c 2 j n + m n m ( n m ) ! ( n + m ) ! × n ( n + 1 ) 2 n + 1 × + d k Ω ( n , m , k ) Y ( m , m , k ) Ω ( n , m , k ) a c 2 ,
Y ( m , m , k ) = p = + H p m ( 2 ) ( a c d ) H p m ( 2 ) ( a c d ) A c c ( p , k ) .
δ 0 = k f ( a f a c α f ) 2 ( k f 2 k c 2 ) m 0 β [ Λ m 0 ( 2 ) ( a c α f ) Λ m 0 ( 1 ) ( a f α f ) ] ,
A 0 = a f 2 J m 0 ( a f α f ) a c 2 H m 0 ( 2 ) ( a c α f ) , a c = ( k c 2 β 2 ) 1 2 , a f = ( k f 2 β 2 ) 1 2 .
( T p R p ) = π k c α s ( 1 ) m p a f 2 J m p ( a c α f ) J m p ( a f α f ) S 1 S 4 ( Δ k ) k = β p [ 1 δ p ( β p ) k c k f ] n , m j 2 n m n ( n + 1 ) 2 n + 1 H m p m ( 2 ) ( a c d ) Ω ( n , m , β p ) R n ( 1 ) x n m T ,

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