Abstract

The coupling between two single-mode optical fibers that are abruptly terminated and are facing each other is investigated analytically by means of a coupled system of integral equations. The electromagnetic fields are described by using mixed-spectrum eigenwave representations inside the waveguides, and Fourier integrals are utilized to describe the field between the two guides. Weak guidance conditions on the fiber guides are assumed to facilitate the formulation of the corresponding boundary-value problem. Numerical results are presented for several coupling geometries, and misalignment and reflection-loss phenomena are also investigated for cases when the two fiber terminal planes are parallel to each other.

© 1988 Optical Society of America

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References

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  1. C M. Miller, “Mechanical optical fiber splices,” IEEE J. Lightwave Technol. LT-4, 1228–1231 (1986).
    [CrossRef]
  2. K. Nawata and N. Suzuki, Optical Devices and Fibers (Ohm and North-Holland, Amsterdam, 1982), pp. 152–171.
  3. I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
    [CrossRef]
  4. W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Joint-loss in single-mode fibres,” Electron. Lett. 14, 491–493 (1978).
    [CrossRef]
  5. J. S. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignment of coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
    [CrossRef]
  6. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  7. S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
    [CrossRef]
  8. A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).
  9. A. D. Yaghijian and E. T. Kornhauser, “A modal analysis of the dielectric rod antenna excited by the HE11 mode,” IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
    [CrossRef]
  10. N. K. Uzunoglu, C. N. Capsalis, and J. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
    [CrossRef]
  11. C. N. Capsalis and N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech.1043–1051 (1987).
    [CrossRef]
  12. J. Senior, Optical Fiber Communications: Principle and Practise (Prentice-Hall, London, 1985).
  13. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 13.
  14. Ref. 13, Chap. 32.
  15. Ref. 7; see footnote on p. 449.

1987 (2)

N. K. Uzunoglu, C. N. Capsalis, and J. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[CrossRef]

C. N. Capsalis and N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech.1043–1051 (1987).
[CrossRef]

1986 (2)

C M. Miller, “Mechanical optical fiber splices,” IEEE J. Lightwave Technol. LT-4, 1228–1231 (1986).
[CrossRef]

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

1985 (1)

A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).

1979 (1)

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

1978 (1)

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Joint-loss in single-mode fibres,” Electron. Lett. 14, 491–493 (1978).
[CrossRef]

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

1973 (1)

J. S. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignment of coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

1972 (1)

A. D. Yaghijian and E. T. Kornhauser, “A modal analysis of the dielectric rod antenna excited by the HE11 mode,” IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[CrossRef]

Arioka, R.

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

Capsalis, C. N.

C. N. Capsalis and N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech.1043–1051 (1987).
[CrossRef]

N. K. Uzunoglu, C. N. Capsalis, and J. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[CrossRef]

Cook, J. S.

J. S. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignment of coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

Gambling, W. A.

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Joint-loss in single-mode fibres,” Electron. Lett. 14, 491–493 (1978).
[CrossRef]

Grow, R. J.

J. S. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignment of coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

Ishida, M.

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

Kornhauser, E. T.

A. D. Yaghijian and E. T. Kornhauser, “A modal analysis of the dielectric rod antenna excited by the HE11 mode,” IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 13.

Makimoto, T.

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Mammel, W. L.

J. S. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignment of coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

Manenkov, A. B.

A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Matsumura, H.

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Joint-loss in single-mode fibres,” Electron. Lett. 14, 491–493 (1978).
[CrossRef]

Miller, C M.

C M. Miller, “Mechanical optical fiber splices,” IEEE J. Lightwave Technol. LT-4, 1228–1231 (1986).
[CrossRef]

Nagasawa, S.

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

Nawata, K.

K. Nawata and N. Suzuki, Optical Devices and Fibers (Ohm and North-Holland, Amsterdam, 1982), pp. 152–171.

Nemoto, S.

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Ragdale, C. M.

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Joint-loss in single-mode fibres,” Electron. Lett. 14, 491–493 (1978).
[CrossRef]

Sankawa, I.

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

Satake, T.

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

Senior, J.

J. Senior, Optical Fiber Communications: Principle and Practise (Prentice-Hall, London, 1985).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 13.

Suzuki, N.

K. Nawata and N. Suzuki, Optical Devices and Fibers (Ohm and North-Holland, Amsterdam, 1982), pp. 152–171.

Tigelis, J. G.

Uzunoglu, N. K.

N. K. Uzunoglu, C. N. Capsalis, and J. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[CrossRef]

C. N. Capsalis and N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech.1043–1051 (1987).
[CrossRef]

Yaghijian, A. D.

A. D. Yaghijian and E. T. Kornhauser, “A modal analysis of the dielectric rod antenna excited by the HE11 mode,” IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[CrossRef]

Bell Syst. Tech. J. (2)

J. S. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignment of coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Electron. Lett. (1)

W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Joint-loss in single-mode fibres,” Electron. Lett. 14, 491–493 (1978).
[CrossRef]

IEEE J. Lightwave Technol. (2)

I. Sankawa, S. Nagasawa, T. Satake, M. Ishida, and R. Arioka, “Low-loss small-size optical-fiber connector with a precision plastic-molded ferrule,” IEEE J. Lightwave Technol. LT-4, 1237–1242 (1986).
[CrossRef]

C M. Miller, “Mechanical optical fiber splices,” IEEE J. Lightwave Technol. LT-4, 1228–1231 (1986).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. D. Yaghijian and E. T. Kornhauser, “A modal analysis of the dielectric rod antenna excited by the HE11 mode,” IEEE Trans. Antennas Propag. AP-20, 122–128 (1972).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. N. Capsalis and N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech.1043–1051 (1987).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).

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

Opt. Quantum Electron. (1)

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Other (5)

K. Nawata and N. Suzuki, Optical Devices and Fibers (Ohm and North-Holland, Amsterdam, 1982), pp. 152–171.

J. Senior, Optical Fiber Communications: Principle and Practise (Prentice-Hall, London, 1985).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 13.

Ref. 13, Chap. 32.

Ref. 7; see footnote on p. 449.

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Figures (4)

Fig. 1
Fig. 1

Coupling geometry between two single-mode fibers.

Fig. 2
Fig. 2

Variation of the |R0| and |Z0| coefficients with interguide w distance when w ≃ 50 μm and for α = b = 3 μm, n2a = n2b = 1.46, Δ = 5 × 10−3.

Fig. 3
Fig. 3

Same as in Fig. 2 except for w ≃ 80 μm.

Fig. 4
Fig. 4

Dependence of the coupling coefficient |Z0| on the h distance between the two guide axes.

Tables (1)

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Table 1 Sample Convergence Patternsa

Equations (51)

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( 2 + k 0 2 n i 2 ) Ψ ( ρ , φ , z ) = 0 ( i = 1 , 2 ) ,
Ψ A ( ρ 1 , φ 1 , z ) = Ψ 0 a ( ρ 1 , φ 1 ) [ exp ( j β 0 a z ) + R 0 exp ( j β 0 a z ) ] + 0 + q d q m = + R m ( q ) Ψ m a ( ρ 1 , φ 1 / q ) exp ( j β a z ) ,
β a = ( k 0 2 n 1 a 2 q 2 ) 1 / 2 ( 0 < q < + )
Ψ B ( ρ 2 , φ 2 , z ) = Z 0 Ψ 0 b ( ρ 2 , φ 2 ) exp ( j β 0 b z ) + 0 + q d q m = + Z m ( q ) Ψ m b ( ρ 2 , φ 2 / q ) exp ( j β b z ) ,
ρ 1 cos φ 1 = ρ 2 cos φ 2 + h , ρ 1 sin φ 1 = ρ 2 sin φ 2 .
Ψ c = 0 + λ d λ m = + J m ( λ ρ ) e j m φ [ C m ( λ ) exp ( j γ z ) + C m ( λ ) exp ( j γ z ) ] ,
γ = ( k 0 2 n 0 2 λ 2 ) 1 / 2 , Re ( γ ) > 0 , Im ( γ ) < 0 .
R 0 = 1 + 1 2 π 0 2 π d φ 1 0 + ρ 1 d ρ 1 E 1 ( ρ 1 , φ 1 ) Ψ 0 a * ( ρ 1 , φ 1 ) ,
R m ( q ) = 1 2 π 0 2 π d φ 1 0 + ρ 1 d ρ 1 E 1 ( ρ 1 , φ 1 ) Ψ m a * ( ρ 1 , φ 1 / q ) ,
Z 0 = 1 2 π 0 2 π d φ 2 0 + ρ 2 d ρ 2 E 2 ( ρ 2 , φ 2 ) Ψ 0 b * ( ρ 2 , φ 2 ) ,
Z m ( q ) = 1 2 π 0 2 π d φ 2 0 + ρ 2 d ρ 2 E 2 ( ρ 2 , φ 2 ) Ψ m b * ( ρ 2 , φ 2 / q ) ,
0 + ρ d ρ J m ( λ ρ ) J m ( λ ρ ) = 1 λ δ ( λ λ ) ,
1 2 π 0 2 π d φ exp [ j ( m m ) φ ] = δ m m ,
C m ( λ ) + C m ( λ ) = 1 2 π 0 2 π d φ 1 0 + ρ 1 d ρ 1 J m ( λ ρ 1 ) × exp ( j m φ 1 ) E 1 ( ρ 1 , φ 1 ) ,
C m ( λ ) exp ( j γ w ) + C m ( λ ) exp ( j γ w ) = 1 2 π 0 2 π d φ 2 0 + ρ 2 d ρ 2 J m ( λ ρ 2 ) exp ( j m φ 2 ) E 2 ( ρ 2 , φ 2 ) ,
Ψ A z = Ψ C z at z = 0
Ψ C z = Ψ B z at z = w .
2 β 0 a Ψ 0 a ( ρ 1 , φ 1 ) = 0 2 π d φ 1 0 + ρ 1 d ρ 1 E 1 ( ρ 1 , φ 1 ) × [ Ξ 1 a ( ρ 1 , φ 1 / ρ 1 , φ 1 ) + Ξ 2 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) ] + 0 2 π d φ 1 0 + ρ 1 d ρ 1 E 2 ( ρ 2 , φ 2 ) × Ξ 3 ( ρ 1 , φ 1 / ρ 1 , φ 1 )
0 2 π d φ 2 0 + ρ 2 d ρ 2 E 2 ( ρ 2 , φ 2 ) × [ Ξ 1 b ( ρ 2 , φ 2 / ρ 2 , φ 2 ) + Ξ 2 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) ] + 0 2 π d φ 1 0 + ρ 1 d ρ 1 E 1 ( ρ 1 , φ 1 ) Ξ 3 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) = 0 ,
Ξ 1 s ( ρ , φ / ρ , φ ) 1 2 π [ β 0 s Ψ 0 s ( ρ , φ ) Ψ 0 s * ( ρ , φ ) + 0 + q d q × M = + β s Ψ m s ( ρ , φ / q ) Ψ m s * ( ρ , φ / q ) ] ( s = a , b ) ,
Ξ 2 ( ρ , φ / ρ , φ ) = 1 2 π 0 + λ d λ m = + J m ( λ ρ ) exp ( j m φ ) J m ( λ ρ ) × exp ( j m φ ) γ exp ( 2 j γ w ) + 1 exp ( 2 j γ w ) 1 ,
Ξ 3 ( ρ , φ / ρ , φ ) = 1 2 π 0 + λ d λ m = + J m ( λ ρ ) exp ( j m φ ) J m ( λ ρ ) × exp ( j m φ ) γ 2 exp ( j γ w ) 1 exp ( j 2 γ w ) .
E j ( ρ j , φ j ) = E j ( 0 ) ( ρ j , φ j ) + 0 2 π d φ j 0 + ρ j d ρ j × i = 1 2 [ K i j ( ρ 1 , φ 1 / ρ 1 , φ 1 ) E i ( ρ j , φ j ) ] for j = 1 , 2 ,
E 1 ( 0 ) ( ρ 1 , φ 1 ) = 2 β 0 a k 0 A 1 ( w ) Ψ 0 a ( ρ 1 , φ 1 ) 2 A 1 ( w ) A 3 ( w ) E 2 ( 0 ) ( ρ 2 , φ 2 ) ,
E 2 ( 0 ) ( ρ 2 , φ 2 ) = 2 A 2 ( w ) A 3 ( w ) E 1 ( 0 ) ( ρ 1 , φ 1 ) ,
K 11 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) = A 1 ( w ) 2 π k 0 { ( β 0 a k 0 n 2 a ) Ψ 0 a ( ρ 1 , φ 1 ) Ψ 0 a * ( ρ 1 , φ 1 ) + 0 + q d q m = + ( β a k 0 n 2 a ) Ψ m a ( ρ 1 , φ 1 / q ) Ψ m a * ( ρ 1 , φ 1 / q ) + 0 + λ d λ m = + J m ( λ ρ 1 ) J m ( λ ρ 1 ) exp [ j m ( φ 1 φ 1 ) ] × [ c 1 ( λ , w ) c 1 ( 0 , w ) ] } ,
K 12 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) = A 1 ( w ) π k 0 0 + λ d λ m = + J m ( λ ρ 1 ) J m ( λ ρ 1 ) × exp [ j m ( φ 1 φ 1 ) ] [ c 2 ( λ , w ) c 2 ( 0 , w ) ] ,
K 21 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) = A 2 ( w ) A 1 ( w ) K 12 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) ,
K 22 ( ρ 1 , φ 1 / ρ 1 , φ 1 ) = A 2 ( w ) 2 π k 0 { 0 + λ d λ M = + J m ( λ ρ 2 ) J m ( λ ρ 2 ) × exp [ j m ( φ 2 φ 2 ) ] [ c 1 ( λ , w ) c 1 ( 0 , w ) ] + ( β 0 b k 0 n 2 b ) Ψ 0 b ( ρ 2 , φ 2 ) Ψ 0 b * ( ρ 2 , φ 2 ) + 0 + q d q m = + ( β b k 0 n 2 b ) × Ψ m b ( ρ 2 , φ 2 / q ) Ψ m b * ( ρ 2 , φ 2 / q ) } ,
A 1 ( w ) = [ n 2 a + n 0 exp ( 2 j k 0 n 0 w ) + 1 exp ( 2 j k 0 n 0 w ) 1 ] 1 ,
A 2 ( w ) = [ n 2 b + n 0 exp ( 2 j k 0 n 0 w ) + 1 exp ( 2 j k 0 n 0 w ) 1 ] 1 ,
A 3 ( w ) = n 0 exp ( j k 0 n 0 w ) + 1 1 exp ( 2 j k 0 n 0 w ) ,
C 1 ( λ , w ) = γ [ exp ( 2 j γ w ) + 1 ] exp ( 2 j γ w ) 1 ,
C 2 ( λ , w ) = γ exp ( j γ w ) exp ( 2 j γ w ) 1 .
E 1 ( ρ 1 , φ 1 ) = E 1 ( 0 ) ( ρ 1 , φ 1 ) , E 2 ( ρ 2 , φ 2 ) = E 2 ( 0 ) ( ρ 2 , φ 2 ) .
R 0 = 1 + 2 β 0 a k 0 A 1 ( w ) 1 4 A 1 ( w ) A 2 ( w ) A 3 2 ( w ) 2 β 0 a k 0 2 [ A 1 ( w ) 1 4 A 1 ( w ) A 2 ( w ) A 3 2 ( w ) ] 2 × ( ( β 0 a k 0 n 2 a ) + ( a C 0 a ) 2 0 + λ d λ S a 2 ( λ ) { [ C 1 ( λ , w ) C 1 ( 0 , w ) ] + 4 A 2 ( w ) A 3 ( w ) [ c 2 ( λ , w ) c 2 ( 0 , w ) ] } ) ,
Z 0 = 4 β 0 a k 0 A 1 ( w ) A 2 ( w ) A 3 ( w ) 1 4 A 1 ( w ) A 2 ( w ) A 3 2 ( w ) 1 2 π 0 2 π d φ 2 0 + ρ 2 d ρ 2 Ψ 0 a ( ρ 1 , φ 1 ) Ψ 0 b * ( ρ 2 , φ 2 ) + 4 β 0 a a C 0 a b C 0 b k 0 2 A 1 ( w ) A 2 ( w ) 1 4 A 1 ( w ) A 2 ( w ) A 3 2 ( w ) × 0 + λ d λ J 0 ( λ h ) S a ( λ ) S b ( λ ) [ c 2 ( λ , w ) c 2 ( 0 , w ) ] + 4 β 0 a a C 0 a b C 0 b k 0 2 A 1 ( w ) A 2 2 ( w ) A 3 ( w ) [ 1 4 A 1 ( w ) A 2 ( w ) A 3 2 ( w ) ] 2 × λ = 0 + λ d λ J 0 ( λ h ) S a ( λ ) S b ( λ ) { [ c 1 ( λ , w ) c 1 ( 0 , w ) ] + 4 A 1 ( w ) A 3 ( w ) [ c 2 ( λ , w ) c 2 ( 0 , w ) ] } + 4 β 0 a ( β 0 b k 0 n 2 b ) k 0 2 A 1 ( w ) A 2 2 ( w ) A 3 ( w ) [ 1 4 A 1 ( w ) A 2 ( w ) A 3 2 ( w ) ] 2 1 2 π 0 2 π d φ 2 0 + ρ 2 d ρ 2 Ψ 0 a ( ρ 1 , φ 1 ) Ψ 0 b * ( ρ 2 , φ 2 ) ,
S s ( λ ) = γ s K 1 ( γ s s ) J 0 ( λ s ) λ K 0 ( γ s s ) J 1 ( λ s ) ( α 0 s 2 λ 2 ) ( γ s 2 λ 2 ) k 0 2 ( n 2 s 2 n 1 s 2 ) , s = α , b .
α = b = 3 μ m , n 2 a = n b 2 b = 1.46 , n 1 a = n 1 b = 1.46 ( 1 Δ ) ,
Ψ 0 ( ρ , φ ) = C 0 { K 0 ( γ α ) J 0 ( α 0 α ) J 0 ( α 0 ρ ) for ρ < α k 0 ( γ ρ ) for ρ < α ,
α 0 2 = k 0 2 n 2 2 β 0 2 , γ 2 = β 02 k 0 2 n 1 2 ,
( α 0 α ) J 1 ( α 0 α ) J 0 ( α 0 α ) = ( γ α ) K 1 ( γ α ) K 0 ( γ α ) .
Ψ m ( ρ , φ / q ) = A m ( q ) exp ( j m φ ) { 1 C m ( q ) J m [ σ ( q ) p ] for ρ < α J m ( q ρ ) + D m ( q ) Y m ( q ρ ) for ρ > α ,
m = 0 , ± 1 , ± 2 , σ 2 + β 2 = k 0 2 n 2 2 , q 2 + β 2 = k 0 2 n 1 2 , Re ( β ) > 0 , Im ( β ) < 0 ,
D m ( q ) = q J m ( σ α ) J m + 1 ( q α ) + σ J m + 1 ( σ α ) J m ( q α ) q J m ( σ α ) Y m + 1 ( q α ) σ J m + 1 ( σ α ) Y m ( q α ) ,
1 C m ( q ) = q J m ( q α ) Y m + 1 ( q α ) q J m + 1 ( q α ) Y m ( q α ) q J m ( σ α ) Y m + 1 ( q α ) σ J m + 1 ( σ α ) Y m ( q α ) .
I 1 = 1 2 π 0 2 π d φ 0 + ρ d ρ Ψ 0 ( ρ , φ ) Ψ 0 * ( ρ , φ ) = 1 ,
I 2 = 1 2 π 0 2 π d φ 0 + ρ d ρ Ψ m ( ρ , φ / q ) Ψ m * ( ρ , φ / q ) = δ m m δ ( q q ) q ,
I 3 = 1 2 π 0 2 π d φ 0 + ρ d ρ Ψ 0 ( ρ , φ ) Ψ m * ( ρ , φ / q ) = 0 .
C 0 = J 0 ( α 0 α ) α [ 2 K 1 2 ( γ α ) J 0 2 ( α 0 α ) + K 0 2 ( γ α ) J 1 2 ( α 0 α ) ] 1 / 2 ,
A m ( q ) = 1 [ 1 + D m 2 ( q ) ] 1 / 2 .

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