Abstract

The interaction of surface guided modes with scatterers inside a fiber dielectric waveguide is investigated analytically. A volume integral equation technique, based on Green’s function theory for fiber boundaries, is used to formulate the problem. For the case of spherical scatterers, an analytical solution is developed by using an expansion in spherical vector wave functions, when b(k1k2) < 1, where (k1k2) is the difference between the wave numbers of the fiber and the spherical scatterer whose radius is b. Expressions are obtained for the reflection, transmission, and radiated-field quantities up to the order of [b(k1k2)]5. Numerical results are computed and presented for several cases. Coupling between even–odd HE11 modes is treated also.

© 1981 Optical Society of America

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References

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  1. D. Hondros and P. Debye, “Electromagnetische Wellen an dielektrishen Drahten,” Ann. Phys. Leipzig 32, 465–476 (1910).
    [Crossref]
  2. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
    [Crossref]
  3. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
    [Crossref]
  4. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  5. C. Yeh and et al., “Single-mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
    [Crossref] [PubMed]
  6. J. G. Fikioris and N. K. Uzunoglu, “Scattering of plane and guided waves from composite dielectric bodies,” in Proceedings of the Ninth European Microwave Conference (Microwave Exhibition and Publishers Ltd., Kent, England, 1979), pp. 516–520.
  7. A. Safaai-Jazi and G. L. Yip, “Scattering from an off-axis inhomogeneity in step index optical fibers: radiation loss,” J. Opt. Soc. Am. 70, 40–52 (1980).
    [Crossref]
  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. MTT-28, 137–141 (1980).
    [Crossref]
  9. P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part II.
  10. D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964).
  11. A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), Chap. VI.
  12. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.
  13. C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971).
  14. W. M. Elsasser, “Attenuation in a dielectric circular rod,” J. Appl. Phys. 20, 1188–1192 (1949).
    [Crossref]
  15. R. J. Pogorzelski and E. Lun, “On the expansion of cylindrical waves in terms of spherical vector waves,” Radio Sci. 11, 753–761 (1976).
    [Crossref]
  16. R. Mittra, ed., Computer Techniques for Electromagnetics (Pergamon, New York, 1973).
  17. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  18. E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 13.
  19. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

1980 (2)

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. MTT-28, 137–141 (1980).
[Crossref]

A. Safaai-Jazi and G. L. Yip, “Scattering from an off-axis inhomogeneity in step index optical fibers: radiation loss,” J. Opt. Soc. Am. 70, 40–52 (1980).
[Crossref]

1979 (1)

1976 (1)

R. J. Pogorzelski and E. Lun, “On the expansion of cylindrical waves in terms of spherical vector waves,” Radio Sci. 11, 753–761 (1976).
[Crossref]

1969 (2)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
[Crossref]

1949 (1)

W. M. Elsasser, “Attenuation in a dielectric circular rod,” J. Appl. Phys. 20, 1188–1192 (1949).
[Crossref]

1910 (1)

D. Hondros and P. Debye, “Electromagnetische Wellen an dielektrishen Drahten,” Ann. Phys. Leipzig 32, 465–476 (1910).
[Crossref]

Balmain, K. G.

E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 13.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

Debye, P.

D. Hondros and P. Debye, “Electromagnetische Wellen an dielektrishen Drahten,” Ann. Phys. Leipzig 32, 465–476 (1910).
[Crossref]

Elsasser, W. M.

W. M. Elsasser, “Attenuation in a dielectric circular rod,” J. Appl. Phys. 20, 1188–1192 (1949).
[Crossref]

Feschbach, H.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part II.

Fikioris, J. G.

J. G. Fikioris and N. K. Uzunoglu, “Scattering of plane and guided waves from composite dielectric bodies,” in Proceedings of the Ninth European Microwave Conference (Microwave Exhibition and Publishers Ltd., Kent, England, 1979), pp. 516–520.

Goell, J. E.

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
[Crossref]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

Hondros, D.

D. Hondros and P. Debye, “Electromagnetische Wellen an dielektrishen Drahten,” Ann. Phys. Leipzig 32, 465–476 (1910).
[Crossref]

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964).

Jordan, E. C.

E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 13.

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. MTT-28, 137–141 (1980).
[Crossref]

Lun, E.

R. J. Pogorzelski and E. Lun, “On the expansion of cylindrical waves in terms of spherical vector waves,” Radio Sci. 11, 753–761 (1976).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

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. MTT-28, 137–141 (1980).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part II.

Pogorzelski, R. J.

R. J. Pogorzelski and E. Lun, “On the expansion of cylindrical waves in terms of spherical vector waves,” Radio Sci. 11, 753–761 (1976).
[Crossref]

Safaai-Jazi, A.

Sommerfeld, A. R.

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), Chap. VI.

Stratton, J. A.

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

Tai, C.-T.

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971).

Uzunoglu, N. K.

J. G. Fikioris and N. K. Uzunoglu, “Scattering of plane and guided waves from composite dielectric bodies,” in Proceedings of the Ninth European Microwave Conference (Microwave Exhibition and Publishers Ltd., Kent, England, 1979), pp. 516–520.

Yeh, C.

Yip, G. L.

Ann. Phys. Leipzig (1)

D. Hondros and P. Debye, “Electromagnetische Wellen an dielektrishen Drahten,” Ann. Phys. Leipzig 32, 465–476 (1910).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
[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. MTT-28, 137–141 (1980).
[Crossref]

J. Appl. Phys. (1)

W. M. Elsasser, “Attenuation in a dielectric circular rod,” J. Appl. Phys. 20, 1188–1192 (1949).
[Crossref]

J. Opt. Soc. Am. (1)

Radio Sci. (1)

R. J. Pogorzelski and E. Lun, “On the expansion of cylindrical waves in terms of spherical vector waves,” Radio Sci. 11, 753–761 (1976).
[Crossref]

Other (11)

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

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), Chap. 13.

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

P. M. Morse and H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part II.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964).

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), Chap. VI.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 11.

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971).

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

J. G. Fikioris and N. K. Uzunoglu, “Scattering of plane and guided waves from composite dielectric bodies,” in Proceedings of the Ninth European Microwave Conference (Microwave Exhibition and Publishers Ltd., Kent, England, 1979), pp. 516–520.

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Figures (9)

Fig. 1
Fig. 1

(a) Inhomogeneity inside a fiber waveguide. (b) Spherical inhomogeneity inside a fiber.

Fig. 2
Fig. 2

Complex k plane.

Fig. 3
Fig. 3

Modified contour C′ to compute the Kij(n,m|n′,m′) integrals.

Fig. 4
Fig. 4

Complex β plane.

Fig. 5
Fig. 5

Forward [T+(+)] and backward [R+(+)] scattering coefficients versus k2 for k1 = 1.3, k0 = 1.0, d = 0, a = 1.5, and k0b = 1.0.

Fig. 6
Fig. 6

Forward [T+(+)] and backward [R+(+)] scattering coefficients versus k2 for k1 = 1.6, k0 = 1.0, d = 0, a = 1.6, and k0b = 0.5.

Fig. 7
Fig. 7

Forward [T+(+)] and backward [R+(+)] scattering coefficients versus k2 for k1 = 1.6, k0 = 1.0, d = 0, a = 1.6, and k0b = 1.

Fig. 8
Fig. 8

(a) Forward scattering coefficient T+(+) versus k2 for k1 = 1.6, a = 1.6, d = 0, k0 = 1.0, and k0b = 1.2. (b) Backward scattering coefficient R+(+) versus k2 of Fig. 8(a) case.

Fig. 9
Fig. 9

Coupling geometry for even and odd HE11 modes.

Tables (4)

Tables Icon

Table 1 Inner-Field-Expansion Coefficients amn(q) and bmn(q) for a Fiber, Where k1/k0 = 1.6, k0a = 1.6, k0d = 0, k0 = 1, and E = E0+ Incident Wave

Tables Icon

Table 2 Radiation Loss versus k2/k0 for k1/k0 = 1.6, k0 = 1, k0a = 1.2, and Spherical Scatterer k1b = 1.6.

Tables Icon

Table 3 Scattering Coefficients for a Fiber k0a = 1.5, r = k1/k0 = 1.3, d = 0.5, ν = 1.0205015, and Spherical Scatterer k0b = 0.50 for an E = E0+ Incident Wave

Tables Icon

Table 4 Contributions from the Dipole (n = 1) and Quadrapole (n = 2) Terms for k1/k0 = 1.6, k2/k0 = 1.9, k0 = 1, k0d = 0, and k0b = 1.

Equations (122)

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× × - { k 1 2 k 0 2 E 0 ( r ) = 0             for { r ( V 1 + V 2 ) r V 0 ,
ρ ˆ × E 0 ( r ) × E 0 ( r ) } = continuous for ρ = α ,
( × × - k i 2 ) E ( r ) = 0             for i = 0 , 1 , 2,
ρ ˆ × E ( r ) , × E ( r ) = continuous for ρ = α , n ˆ × E ( r ) × E ( r ) = continuous on V 1 , V 2 interface ,
( × × - k 1 2 k 0 2 ) G ( r r ) = J δ ( r - r )             for { r ( V 1 + V 2 ) r V 0 ,
ρ ˆ × G ( r r ) × G ( r r ) } = continuous for ρ = α .
V { × × [ E ( r ) - E 0 ( r ) ] · G ( r r ) - [ E ( r ) - E 0 ( r ) ] · × × G ( r r ) d r = I s ,
I s = s d s { n ˆ s × [ E ( r s ) - E 0 ( r s ) ] · s × G ( r s , r ) - s × [ E ( r s ) - E 0 ( r s ) ] · n ˆ s × G ( r s , r ) } ,
lim s + I s = - E ( r ) + E 0 ( r ) + V 2 [ k 2 2 ( r ) - k 1 2 ( r ) ] G ( r r ) · E ( r ) d r .
E ( r ) = E 0 ( r ) + V 2 [ k 2 2 ( r ) - k 1 2 ( r ) ] G ( r r ) · E ( r ) d r ,
G ( r r ) = { G 0 ( r r ) + G 1 ( r r ) for ρ < α G 2 ( r r ) for ρ > α ,
G 0 ( r r ) = i 8 π - + d k m = - + ( - 1 ) m a 1 2 { M m , k ( 3 ) ( r , k 1 ) M - m , - k ( 1 ) ( r , k 1 ) + N m , k ( 3 ) ( r , k 1 ) N - m , - k ( 1 ) ( r , k 1 ) for ρ > ρ M m , k ( 1 ) ( r , k 1 ) M - m , - k ( 3 ) ( r , k 1 ) + N m , k ( 1 ) ( r , k 1 ) N - m , - k ( 3 ) ( r , k 1 ) for ρ < ρ ,
M m , k ( i ) ( r , k 1 ) = [ ρ ˆ i m ρ C m ( i ) ( a 1 ρ ) - φ ˆ C m ( i ) ( a 1 ρ ) ρ ] e i m φ e i k z ,
N m , k ( i ) ( r , k 1 ) = 1 k 1 [ i k C m ( i ) ( a 1 ρ ) ρ ρ ˆ - m k ρ C m ( i ) ( a 1 ρ ) φ ˆ + a 1 2 C m ( i ) ( a 1 ρ ) z ˆ ] e i m φ e i k z
G 1 ( r r ) = i 8 π - + d k m = - + ( - 1 ) m a 1 2 [ M m , k ( 1 ) ( r , k 1 ) A m , k + N m , k ( 1 ) ( r , k 1 ) B m , k ] ,
G 2 ( r r ) = i 8 π - + d k m = - + ( - 1 ) m a 0 2 [ M m , k ( 3 ) ( r , k 0 ) C m , k + N m , k ( 3 ) ( r , k 0 ) D m , k ] ,
ρ ˆ × [ G 0 ( r r ) + G 1 ( r r ) ] = ρ ˆ × G 2 ( r r ) ,
ρ ˆ × × [ G 0 ( r r ) + G 1 ( r r ) ] = ρ ˆ × × G 2 ( r r ) .
G 1 ( r r ) = i 8 π - + d k m = - + ( - 1 ) m a 1 2 × [ a ( m , k ) M m , k ( 1 ) ( r , k 1 ) M - m , - k ( 1 ) ( r , k 1 ) + b ( m , k ) M m , k ( 1 ) ( r , k 1 ) N - m , - k ( 1 ) ( r , k 1 ) + b ( m , k ) N m , k ( 1 ) ( r , k 1 ) M - m , - k ( 1 ) ( r , k 1 ) + c ( m , k ) N m , k ( 1 ) ( r , k 1 ) N - m , - k ( 1 ) ( r , k 1 ) ] ,
a ( m , k ) = H m ( 1 ) ( a 1 α ) J m ( a 1 α ) 1 Δ m × { r Ψ m ( a 0 α ) Λ m ( a 1 α ) + Ψ m ( a 0 α ) Ψ m ( a 1 α ) - [ Ψ m ( a 0 α ) ] 2 - r Ψ m ( a 1 α ) Λ m ( a 1 α ) + ( r - 1 ) 2 k 0 2 k 2 m 2 ( a 1 a 0 α ) 4 } ,
b ( m , k ) = - 2 i π r ( r - 1 ) m k k 0 ( a 1 a 0 α ) 2 1 [ a 1 J m ( a 1 α ) ] 2 1 Δ m ,
c ( m , k ) = H m ( 1 ) ( a 1 α ) J m ( a 1 α ) 1 Δ m { Λ m ( a 1 α ) Ψ m ( a 0 α ) - [ Ψ m ( a 0 α ) ] 2 + Ψ m ( a 0 α ) Ψ m ( a 1 α ) r - r Ψ m ( a 1 α ) Λ m ( a 1 α ) + ( r - 1 ) 2 k 0 2 k 2 m 2 ( a 1 a 0 α ) 4 } ,
Ψ m ( X ) = H m ( 1 ) ( X ) X H m ( X ) ,             Λ m ( X ) = J m ( X ) X J m ( X ) ,
Δ m = r [ Λ m ( a 1 α ) ] 2 + [ Ψ m ( a 0 α ) ] 2 - ( r + 1 ) Ψ m ( a 0 α ) Λ m ( a 1 α ) - ( r - 1 ) 2 k 2 k 0 2 m 2 ( a 1 a 0 α ) 2 ,
a 1 = ( k 1 2 - k 2 ) 1 / 2 = [ exp ( i π ) ( k - k 1 ) ( k + k 1 ) ] 1 / 2 = exp ( i π / 2 ) exp [ i ( ω 1 + ω 1 ) / 2 ] k 2 - k 1 2 ,
a 0 = ( k 0 2 - k 2 ) 1 / 2 = exp ( i π / 2 ) k 2 - k 0 2 exp [ i ( ω 0 + ω 0 ) / 2 ] ,
E 0 ± ( r ) = d 1 ± M ± 1 , ν ( 1 ) ( r , k 1 ) + d 2 ± N ± 1 , ν ( 1 ) ( r , k 1 ) for ρ < α , = d 3 ± M 1 , ν ( 3 ) ( r , k 0 ) + d 4 ± N ± 1 , ν ( 3 ) ( r , k 0 ) for ρ > α ,
E 0 ± ( r ) = M ± 1 , ν ( 1 ) ( r , k 1 ) ± δ ( ν ) N ± 1 , ν ( 1 ) ( r , k 1 )             for ρ < α ,
δ ( ν ) = k 1 a ν a 0 J 1 ( a 1 α ) H 1 ( 1 ) ( a 0 α ) - H 1 ( 1 ) ( a 0 α ) J 1 ( a 1 α ) a 1 J 1 ( a 1 α ) H 1 ( 1 ) ( a 0 α ) ( k 1 2 - k 0 2 ) a 1 a 0 .
E ( s ) = n = 1 + m = - n + n [ a m n m m n ( 1 ) ( s , k 2 ) + b m n n m n ( 1 ) ( s , k 2 ) ] ,
m m n ( i ) ( r , k 2 ) = z n ( i ) ( k 2 r ) [ n ( n + 1 ) ] 1 / 2 C m n ( r ˆ ) ,
n m n ( i ) ( r , k 2 ) = n ( n + 1 ) z n ( i ) ( k 2 r ) k 2 r exp ( i m φ ) P n m ( cos ϑ ) r ˆ + z n d ( i ) ( k 2 r ) k 2 r [ n ( n + 1 ) ] 1 / 2 B m n ( r ˆ ) = 1 k 2 × m m n ( i ) ( r , k 2 ) ,
C m n ( r ˆ ) = [ n ( n + 1 ) ] - 1 / 2 × [ i m sin ϑ P n m ( cos ϑ ) ϑ ˆ - P n m ( cos ϑ ) ϑ ϑ ˆ ] e i m φ ,
B m n ( r ˆ ) = [ n ( n + 1 ) ] - 1 / 2 [ i m sin ϑ P n m ( cos ϑ ) φ ˆ + P n m ( cos ϑ ) ϑ ϑ ˜ ] e i m φ ,
z n ( i ) ( x ) = { j n ( x ) i = 1 Bessel functions , y n ( x ) i = 2 Neuman functions , h n ( 1 ) ( x ) = j n ( x ) = i y n ( x ) i = 3 Hankel functions , z n ( i ) d ( x ) = d [ x z n ( i ) ( x ) ] / d x .
n m [ a m n m m n ( 1 ) ( s , k 2 ) + b m n n m n ( 1 ) ( s , k 2 ) ] = E 0 ( s - x ˆ d ) + ( k 2 2 - k 1 2 ) V 2 d s [ G 0 ( s s ) + G 1 ( s - x ˆ d s - x ˆ d ) ] × n m [ a m n m m n ( 1 ) ( s , k 2 ) + b m n n m n ( 1 ) ( s , k 2 ) ]
n m = n = 1 m = - n n .
M m , k ( 1 ) ( r , k 1 ) N m , k ( 1 ) ( r , k 1 ) } = m = - + J ± n - m ( a 1 d ) ( - 1 ) n - m × { M ± m , k ( 1 ) ( s , k 1 ) N ± m , k ( 1 ) ( s , k 1 ) .
a m n j n ( k 2 s ) [ n ( n + 1 ) ] 1 / 2 = m ( - 1 ) 1 - m J ± 1 - m ( σ d ) × u d ŝ C - m n ( ŝ ) · [ M m , ν ( 1 ) ( s , k 1 ± δ ( ν ) N m , ν ( 1 ) ( s , k 1 ) ] + ( k 2 2 - k 1 2 ) n m u d ŝ v 2 d s C - m n ( ŝ ) · G 0 ( s s ) · [ a m n m m n ( 1 ) ( s , k 2 ) + b n m n ( s , k 2 ) ] + ( k 2 2 - k 1 2 ) n m u d ŝ v 2 d s C - m n ( ŝ ) · G 1 ( s - x ˆ d s - x ˆ d ) · ( a m n m m n ( 1 ) ( s , k 2 ) + b m n n m n ( 1 ) ( s , k 2 ) ,
m = m = - + ,             u d ŝ = ϑ s = 0 π d ϑ s sin ϑ s φ s = 0 2 π d φ s ,             and σ = ( k 1 2 - ν 2 ) 1 / 2 .
G 0 ( s s ) i k 1 4 π n , m ( - 1 ) m 2 n + 1 n ( n + 1 ) × { m m n ( 3 ) ( s , k 1 ) m - m n ( 1 ) ( s , k 1 ) + n m n ( 3 ) ( s , k 1 ) n - m n ( 1 ) ( s , k 1 ) ; s > s , m m n ( 1 ) ( s , k 1 ) m - m n ( 3 ) ( s , k 1 ) + n m n ( 1 ) ( s , k 1 ) n - m n ( 3 ) ( s , k 1 ) ; s < s ,
0 b s 2 d s j n ( k 2 s ) { j n ( k 1 s ) h n ( k 1 s ) ( s > s ) j n ( k 1 s ) h n ( k 1 s ) ( s < s ) } = i j n ( k 2 s ) k 1 ( k 1 2 - k 2 2 ) + j n ( k 1 s ) b k 1 2 - k 2 2 G n ( x 1 , x 2 ) ,
f n ( α x ) g n ( β x ) x 2 d x = x 2 α 2 - β 2 × [ α f n + 1 ( α x ) g n ( β x ) - β f n ( α x ) g n + 1 ( β x ) ] ,
M m , k ( 1 ) ( r , k 1 ) N m , k ( 1 ) ( r , k 1 ) = i - m ( ν > m ν 0 ( ν - m ) ! ( ν + m ) ! × P ν m ( X k ) i ν i m ν ( ν + 1 ) [ n m , ν ( 1 ) ( r , k 1 ) m m , ν ( 1 ) ( r , k 1 ) ] + ν m + { ( ν - m ) ! ( ν + m ) ! P ν m ( X k ) i ν ν - m + 1 ν + 1 [ m m , ν + 1 ( 1 ) ( r , k 1 ) n m , ν + 1 ( 1 ) ( r , k 1 ) ] + ( ν - m + 1 ) ! ( ν + m + 1 ) ! P ν + 1 m ( X k ) i ν + 1 × ν + m + 1 ν + 1 [ m m , ν ( 1 ) ( r , k 1 ) n m , ν ( 1 ) ( r , k 1 ) ] } ) ,
u d ŝ C - m n ( ŝ ) · [ M m , k ( 1 ) ( s , k 1 ) N m , k ( 1 ) ( s , k 1 ) ] = k 1 i m j n ( k 1 s ) 4 π { [ n ( n + 1 ) ] 1 / 2 2 n + 1 i n - 1 q n m ( k ) 1 [ n ( n + 1 ) ] 1 / 2 i n p n m ( k ) } δ m , m ,
u d ŝ s = 0 b s 2 d s [ M m , - k ( 1 ) ( s , k 1 ) N - m , - k ( 1 ) ( s , k 1 ) ] · m m n ( 1 ) ( s , k 2 ) = i m 4 π x 1 k 1 2 - k 2 2 E n ( x 1 , x 2 ) δ m , m [ i n - 1 r n m ( k ) 2 n + 1 i n s n m ( k ) ] ,
u d ŝ s = 0 b s 2 d s [ M m , - k ( 1 ) ( s , k 1 ) N - m , - k ( 1 ) ( s , k 1 ) ] · n m n ( 1 ) ( s , k 2 ) = i m 4 π x 1 k 1 2 - k 2 2 F n ( x 1 , x 2 ) δ m , m [ i n s n m ( k ) i n - 1 r n m ( k ) 2 n + 1 ] ,
q n m ( k ) = ( n + m ) P n - 1 m ( X k ) / n - ( n - m + 1 ) P n + 1 m ( X k ) / ( n + 1 ) , p n m ( k ) = i m P n m ( X k ) , r n m ( k ) = P n - 1 m ( - X k ) ( n + m ) ( n + 1 ) - p n + 1 m ( - X k ) ( n + 1 - m ) n , s n m ( k ) = - i m P n m ( - X k ) ,
E n ( x 1 , x 2 ) = x 1 j n + 1 ( x 1 ) j n ( x 2 ) - x 2 j n ( x 1 ) j n + 1 ( x 2 ) ,
F n ( x 1 , x 2 ) = x 1 j n ( x 1 ) j n ( x 1 ) j n d ( x 2 ) / x 2 - x 2 j n ( x 2 ) j n d ( x 1 ) / x 1 .
a m n G n ( x 1 , x 2 ) + n m [ K 11 ( n , m n , m ) E n ( x 1 , x 2 ) a m n + K 12 ( n , m n , m ) F n ( x 1 , x 2 ) b m n ] = L 1 ± ( n , m ) for n = 1 , 2 , , ( m < n ) ,
L 1 ± ( n , m ) = - i n - m b ( 2 n + 1 ) [ n ( n + 1 ) ] 1 / 2 k 1 J m 1 ( σ d ) ( n - m ) ! ( n + m ) ! × { [ n ( n + 1 ) ] 1 / 2 2 n + 1 q n m ( ν ) ± i δ ( ν ) p n m ( ν ) [ n ( n + 1 ] 1 / 2 } ,
K 11 ( n , m n , m ) = ζ m n m n × - + d k m 0 ( - 1 ) m a 1 2 J m 0 - m ( a 1 d ) J m 0 - m ( a 1 d ) × ( a ( m 0 , k ) [ n ( n + 1 ) ] 1 / 2 2 n + 1 q n m ( k ) r n m ( k ) 2 n + 1 - c ( m 0 , k ) p n m ( k ) s n m ( k ) [ n ( n + 1 ) ] 1 / 2 + b ( m 0 , k ) { [ n ( n + 1 ) 1 / 2 2 n + 1 q n m ( k ) i s n m ( k ) + i p n m ( k ) [ n ( n + 1 ) ] 1 / 2 ( 2 n + 1 ) r n m ( k ) } ) ,
K 12 ( n , m n , m ) = ζ m n m n × - + d k m 0 ( - 1 ) m a 1 2 J m 0 - m ( a 1 d ) J m 0 - m ( a 1 d ) × ( a ( m 0 k ) [ n ( n + 1 ) ] 1 / 2 ( 2 n + 1 ) q n m ( k ) i s n m ( k ) + c ( m 0 , k ) × i p n m ( k ) [ n ( n + 1 ) ] 1 / 2 r n m ( k ) 2 n + 1 + b ( m 0 , k ) ×     { [ n ( n + 1 ) ] 1 / 2 ( 2 n + 1 ) q n m ( k ) r n m ( k ) 2 n + 1 - p n m ( k ) s n m ( k ) [ n ( n + 1 ) ] 1 / 2 } ) ,
ζ m n m n = - b 2 i - m + m + n + n ( n - m ) ! n + m ! 2 n + 1 [ n ( n + 1 ) ] 1 / 2 .
u d ŝ B - m n ( ŝ ) [ M m , k ( 1 ) ( s , k 1 ) N m , k ( 1 ) ( s , k 1 ) ] = k 1 i m j n d ( k 1 s ) k 1 s × 4 π δ m 1 m { i n p n m ( k ) [ n ( n + 1 ) ] 1 / 2 [ n ( m + 1 ) ] 1 / 2 2 n + 1 i n - 1 q n m ( k ) } ,
s = 0 b s 2 d s { n ( n + 1 ) j n ( k 2 s ) k 2 s [ h n d ( k 1 s ) k 1 s j n ( k 1 s ) k 1 s ( s > s ) h n d ( k 1 s ) k 1 s j n ( k 1 s ) k 1 s ( s < s ) ] + j n d ( k 2 s ) k 2 s [ h n d ( k 1 s ) k 1 s j n d ( k 1 s ) k 1 s ( s > s ) h n d ( k 1 s ) k 1 s j n d ( k 1 s ) k 1 s ( s < s ) ] } = i k 1 1 k 1 2 - k 2 2 j n d ( k 2 s ) k 2 s + j n d ( k 1 s ) k 1 s b k 1 2 - k 2 2 D n ( x 1 , x 2 ) ,
D n ( x 1 , x 2 ) = x 1 h n ( x 1 ) j n d ( x 2 ) / x 2 - x 2 j n ( x 2 ) h n d ( x 1 ) / x 1 .
b m n D n ( x 1 , x 2 ) + n m [ K 21 ( n , m n m ) E n ( x 1 , x 2 ) a m n + K 22 ( n , m n , m ) F n ( x 1 , x 2 ) b m n ] = L 2 ± ( n , m ) for n = 1 , 2 , , ( m < n ) ,
L 2 ± ( n , m ) = - i n - m b k 1 2 n + 1 [ n ( n + 1 ) ] 1 / 2 J m 1 ( σ d ) ( n - m ) ! ( n + m ) ! × { i p n m ( ν ) [ n ( n + 1 ) ] 1 / 2 ± δ ( ν ) [ n ( n + 1 ) ] 1 / 2 q n m ( ν ) 2 n + 1 } ,
K 21 ( n , m n , m ) = ζ m n m n - + d k m 0 ( - 1 ) m a 1 2 × J m 0 - m ( a 1 d ) J m 0 - m ( a 1 d ) ( a ( m 0 k ) i p n m ( k ) [ n ( n + 1 ) ] 1 / 2 × r n m ( k ) + c ( m 0 , k ) [ n ( n + 1 ) ] 1 / 2 2 n + 1 × q n m ( k ) i s n m ( k ) - b ( m 0 , k ) × { p n m ( k ) s n m ( k ) [ n ( n + 1 ) ] 1 / 2 - [ n ( n + 1 ) ] 1 / 2 2 n + 1 × q n m ( k ) r n m ( k ) ( 2 n + 1 ) } ) ,
K 22 ( n , m n , m ) = ζ m n m n - + d k m 0 ( - 1 ) m a 1 2 J m 0 - m ( a 1 d ) × J m 0 - m ( a 1 d ) ( - a ( m 0 , k ) p n m ( k ) s n m ( k ) [ n ( n + 1 ) ] 1 / 2 + c ( m 0 , k ) [ n ( n + 1 ) ] 1 / 2 ( 2 n + 1 ) q n m ( k ) r n m ( k ) 2 n + 1 - b ( m 0 , k ) { - i p n m ( k ) [ n ( n + 1 ) ] 1 / 2 r n m ( k ) 2 n + 1 - [ n ( n + 1 ) ] 1 / 2 2 n + 1 q n m ( k ) i s n m ( k ) } ) .
E n ( x 1 , x 2 ) = ( x 2 - x 1 ) E n ( 1 ) + ( x 2 - x 1 ) 2 E n ( 2 ) + ,
F n ( x 1 , x 2 ) = ( x 2 - x 1 ) F n ( 1 ) + ( x 2 - x 1 ) 2 F n ( 2 ) + ,
G n ( x 1 , x 2 ) = G n ( 1 ) + ( x 2 - x 1 ) G n ( 2 ) + ( x 2 - x 1 ) 2 G n ( 3 ) + ,
D n ( x 1 , x 2 ) = D n ( 1 ) + ( x 2 - x 1 ) D n ( 2 ) + ( x 2 - x 1 ) 2 D n ( 3 ) + ,
a m n = j = 1 + a m n ( J ) ( x 2 - x 1 ) J - 1 , b m n = J = 1 + b m n ( J ) ( x 2 - x 1 ) J - 1 .
a m n ( 1 ) = L 1 ± ( n , m ) G n ( 1 ) ,             b m n ( 1 ) = L 2 ± ( n , m ) D n ( 1 ) ,
a m n ( q + 1 ) = - 1 G n ( 1 ) s = 2 q + 1 G n ( s ) a m n ( q + 2 - s ) - 1 G n ( 1 ) n m { K 11 ( n , m n , m ) [ q = 1 q E n ( s ) a m n ( q + 1 - s ) ] + K 12 ( n , m n , m ) [ s = 1 q E n ( s ) b m n ( q + 1 - s ) ] }
b m n ( q + 1 ) = - 1 D n ( 1 ) s = 2 q + 1 D n ( s ) b m n ( q + 2 - s ) - 1 D n ( 1 ) n { K 21 ( n , m n , m ) [ s = 1 q E n ( s ) a m n ( q + 1 - s ) ] + K 22 ( n , m n , m ) [ s = 1 q F n ( s ) b m n ( q + 1 - s ) ] } ,
C ( - , + ) F ( k ) d k = lim 0 + [ ( - - ν - d k + - ν + ν - d k + ν + + d k ) F ( k ) + ϕ - = π 0 i d ϕ - F ( - ν + e i ϕ - ) e i ϕ - + ϕ + = π 2 π i d ϕ + F ( ν + e i ϕ + ) e i ϕ + ] = - + F ( k ) d k + π i { Res [ F ( ν ) ] - Res [ F ( - ν ) ] } ,
a i = ( k i 2 - k 2 ) 1 / 2             when - < k < - k i and k i < k < + , a i = i ( k 2 - k i 2 ) 1 / 2             when - k i < k < k i ,
E ( r ) = E 0 ± ( r ) + ( k 2 2 - k 1 2 ) × V 2 d s G G 0 ( r s - d x ˆ ) · E ( s ) - k 1 b 2 - + d k n m m 0 ( - 1 ) m 0 a 1 2 J m 0 - m ( a 1 d ) i m + n × [ M m 0 , k ( 1 ) ( r , k 1 ) Q n m 0 m ( k ) × N m 0 , k ( 1 ) ( r , k 1 ) T n m 0 m ( k ) ] ,
Q n m 0 m ( k ) = E n ( x 1 , x 2 ) a m n × [ a ( m 0 , k ) r n m ( k ) 2 n + 1 - i b ( m 0 , k ) p m m ( k ) ] + F n ( x 1 , x 2 ) b m n [ - i a ( m 0 , k ) + b ( m 0 , k ) r n m ( k ) 2 n + 1 ] ,
T n m 0 , m ( k ) = E n ( x 1 , x 2 ) a m n × [ b ( m 0 , k ) r n m ( k ) 2 n + 1 - i c ( m 0 , k ) p n m ( k ) ] + F n ( x 1 , x 0 ) b m n × [ - i b ( m 0 , k ) p n m ( k ) + c ( m 0 , k ) r n m ( k ) 2 n + 1 ] .
G 0 ( r r ) exp ( i k 1 r ) 4 π r ( J - z ˆ z ˆ ) exp ( - i r ˆ k 1 · r ) ,
F ± = ( k 2 2 - k 1 2 ) 4 π V 2 d s ( J - z ˆ z ˆ ) E ( s ) × exp ( i k 1 s cos θ s )
a 1 = ( k 1 2 - k 2 ) 1 / 2 = ± k 1 ( sin ξ cosh η + i cos ξ sinh η ) .
d d β ( - i k 1 z cos β ) = i k 1 z sin β = 0 ,
Im [ i f ( β ) - i f ( π ) ] = 0 ( SDC + ) , Im [ i f ( β ) - i f ( 0 ) ] = 0 ( SDC - ) ,
- cos ( ξ ) cosh ( η ) = ± 1 ( SDC ± ) .
c 0 d β = 2 π i { Residue contribution for k = ν pole of HE 11 mode + SDC + d β + B + d β ,
Res [ b ( 1 , ν ) ] / Res [ a ( 1 , ν ) ] = Res [ c ( 1 , ν ) ] / Res [ b ( 1 , ν ) ] = δ ( ν ) ,
E ( r ) E 0 ± ( r ) + T + ( ± ) [ M 1 , ν ( 1 ) ( r , k 1 ) + δ ( ν ) N 1 , ν ( 1 ) ( r , k 1 ) ] + T - ( ± ) [ M - 1 , ν ( 1 ) ( r , k 1 ) - δ ( ν ) N - 1 , ν ( 1 ) ( r , k 1 ) ] ,
T ± ( s ) = π i x 1 n m J ± 1 - m ( σ d ) i m + n σ 2 Res [ Q n ± 1 m ( ν ) ] ,
E ( r ) = E 0 ± ( r ) + R + ( s ) [ M 1 , - ν ( 1 ) ( r , k 1 ) - δ ( ν ) N 1 , - ν ( 1 ) ( r , k 1 ) ] + R - ( s ) [ M 1 , ν ( r , k 1 ) + δ ( ν ) N 1 , - ν ( r , k 1 ) ] ,
R + ( s ) = π i x 1 n m J ± 1 - m ( σ d ) i m + n σ 2 Res [ Q n ± 1 m ( - ν ) ]
T ( e e ) T ( o o ) } = 1 2 [ T + ( + ) ± T + ( - ) ± T - ( + ) + T - ( - ) ] ,
T ( e o ) T ( o e ) } = i 2 [ T + ( + ) - T + ( - ) - T - ( + ) T - ( - ) ] ,
T ( e e ) T ( o o ) } = T + ( + ) ± T - ( + ) ,             R ( e e ) R ( o o ) } = R + ( + ) ± R - ( + ) , T ( e o ) = T ( o e ) = 0 ,             R ( e o ) = R ( o e ) = 0.
T ( e e ) = T ( o o ) = T + ( + ) ;             R ( o o ) = R ( e e ) = R + ( + ) .
1 + T + ( + ) 2 + R + ( + ) 2 < 1 ,
( cos ϕ sin ϕ ) = ( - cos γ sin γ - sin γ -