Abstract

New forms of boundary integral equations are presented for the exact treatment of wave propagation in two-dimensional complicated dielectric waveguide circuits. Since the new boundary integral equations can be solved numerically by the conventional boundary-element method, they are suitable for the basic theory of computer-aided-design software for dielectric waveguide circuits. The new boundary integral equations are derived for a two-dimensional dielectric waveguide bend. They can be obtained by considering specific conditions at points far away from the bend. The numerical solution of the corner bend by the boundary-element method establishes the validity of the new boundary integral equations. The numerical results are evaluated on the bases of the law of energy conservation and the reciprocity condition. Since the theory is based on the exact theory, the solution is exact if sufficiently large computer memory and computational time are used.

© 1989 Optical Society of America

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References

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  1. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  2. J. van Roey, J. van der Donk, P. E. Lagasse, “Beam propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  3. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
    [CrossRef]
  4. K. Ogusu, “Transmission characteristics of single-mode asymmetric dielectric waveguide Y-junctions,” Opt. Commun. 53, 169–172 (1985).
    [CrossRef]
  5. T. Shiina, K. Shiraishi, S. Kawakami, “Waveguide-bend configuration with low-loss characteristics,” Opt. Lett. 11, 736–738 (1986).
    [CrossRef] [PubMed]
  6. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
  7. H. Yajima, “Dielectric thin film optical branching waveguide,” Appl. Phys. Lett. 2, 647–649 (1973).
    [CrossRef]
  8. W. K. Burns, A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
    [CrossRef]
  9. I. Anderson, “Transmission performance of Y junctions in planar dielectric waveguide,” IEEE J. Microwaves Opt. Acoust. MOA-2, 7–12 (1978).
    [CrossRef]
  10. D. Marcuse, “Length optimization of an S-shaped transition between offset optical waveguides,” Appl. Opt. 17, 763–768 (1978).
    [CrossRef] [PubMed]
  11. H. F. Taylor, “Power loss at directional change in dielectric waveguide,” Appl. Opt. 13, 624–647 (1974).
    [CrossRef]
  12. H. F. Taylor, “Losses at corner bends in dielectric waveguides,” Appl. Opt. 16, 711–715 (1977).
    [CrossRef] [PubMed]
  13. S. Kawakami, K. Baba, “Field distribution near an abrupt bend in single-mode waveguide: a simple model,” Appl. Opt. 24, 3643–3647 (1985).
    [CrossRef] [PubMed]
  14. K. Ogusu, “Transmission characteristics of optical waveguide corners,” Opt. Commun. 55, 149–153 (1985).
    [CrossRef]
  15. E. A. J. Marcatili, “Bends in optical dielectric waveguides,” Bell. Syst. Tech. J. 48, 2103–2132 (1969).
  16. L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MIT-22, 718–727 (1974).
    [CrossRef]
  17. D. Marcuse, “Curvature loss formula for optical fiber,”J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  18. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
    [CrossRef]
  19. W. A. Gambling, H. Matsumura, C. M. Ragdle, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
    [CrossRef]
  20. M. Miyagi, S. Matsuo, S. Nishida, “New formalism of curvature losses of leaky modes in doubly clad slab waveguides,” J. Opt. Soc. Am. A 4, 678–682 (1987).
    [CrossRef]
  21. M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
    [CrossRef]
  22. N. Morita, “An integral equation method for electromagnetic scattering of guided modes by boundary deformations of dielectric slab waveguides,” Radio Sci. 18, 39–47 (1983).
    [CrossRef]
  23. M. Koshiba, M. Suzuki, “Boundary-element analysis of dielectric slab waveguide discontinuities,” Appl. Opt. 25, 828–829 (1986).
    [CrossRef] [PubMed]
  24. E. Nishimura, N. Morita, N. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrary shaped junctions between multilayered dielectric planer waveguides,” IEEE J. Lightwave Technol. LT-3, 887–894 (1985).
    [CrossRef]
  25. M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inform. Commun. Eng. Jpn. J71-C, 1021–1026 (1988).
  26. N. Morita, “Surface integral representations for electromagnetic scattering from dielectric cylinders,”IEEE Trans. Antennas Propag. AP-26, 261–266 (1978).
    [CrossRef]
  27. T. K. Wu, L. L. Tsuai, “Numerical analysis of electromagnetic fields in biological tissues,” Proc. IEEE 62, 1167–1168 (1974).
    [CrossRef]
  28. K. Tanaka, M. Kojima, “Volume integral equations for analysis of dielectric branching waveguides,” IEEE Trans. Microwave Theory Tech. MTT-36, 1239–1245 (1988).
    [CrossRef]
  29. K. Tanaka, M. Kojima, “New boundary integral equation for CAD of waveguide circuit,” Electron. Lett. 24, 807–808 (1988).
    [CrossRef]

1988 (3)

M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inform. Commun. Eng. Jpn. J71-C, 1021–1026 (1988).

K. Tanaka, M. Kojima, “Volume integral equations for analysis of dielectric branching waveguides,” IEEE Trans. Microwave Theory Tech. MTT-36, 1239–1245 (1988).
[CrossRef]

K. Tanaka, M. Kojima, “New boundary integral equation for CAD of waveguide circuit,” Electron. Lett. 24, 807–808 (1988).
[CrossRef]

1987 (1)

1986 (2)

1985 (4)

K. Ogusu, “Transmission characteristics of single-mode asymmetric dielectric waveguide Y-junctions,” Opt. Commun. 53, 169–172 (1985).
[CrossRef]

S. Kawakami, K. Baba, “Field distribution near an abrupt bend in single-mode waveguide: a simple model,” Appl. Opt. 24, 3643–3647 (1985).
[CrossRef] [PubMed]

K. Ogusu, “Transmission characteristics of optical waveguide corners,” Opt. Commun. 55, 149–153 (1985).
[CrossRef]

E. Nishimura, N. Morita, N. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrary shaped junctions between multilayered dielectric planer waveguides,” IEEE J. Lightwave Technol. LT-3, 887–894 (1985).
[CrossRef]

1983 (1)

N. Morita, “An integral equation method for electromagnetic scattering of guided modes by boundary deformations of dielectric slab waveguides,” Radio Sci. 18, 39–47 (1983).
[CrossRef]

1982 (2)

M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
[CrossRef]

R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
[CrossRef]

1981 (1)

1979 (1)

W. A. Gambling, H. Matsumura, C. M. Ragdle, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

1978 (4)

N. Morita, “Surface integral representations for electromagnetic scattering from dielectric cylinders,”IEEE Trans. Antennas Propag. AP-26, 261–266 (1978).
[CrossRef]

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

I. Anderson, “Transmission performance of Y junctions in planar dielectric waveguide,” IEEE J. Microwaves Opt. Acoust. MOA-2, 7–12 (1978).
[CrossRef]

D. Marcuse, “Length optimization of an S-shaped transition between offset optical waveguides,” Appl. Opt. 17, 763–768 (1978).
[CrossRef] [PubMed]

1977 (1)

1976 (2)

1975 (1)

W. K. Burns, A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[CrossRef]

1974 (3)

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MIT-22, 718–727 (1974).
[CrossRef]

H. F. Taylor, “Power loss at directional change in dielectric waveguide,” Appl. Opt. 13, 624–647 (1974).
[CrossRef]

T. K. Wu, L. L. Tsuai, “Numerical analysis of electromagnetic fields in biological tissues,” Proc. IEEE 62, 1167–1168 (1974).
[CrossRef]

1973 (1)

H. Yajima, “Dielectric thin film optical branching waveguide,” Appl. Phys. Lett. 2, 647–649 (1973).
[CrossRef]

1970 (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric waveguides,” Bell. Syst. Tech. J. 48, 2103–2132 (1969).

Anderson, I.

I. Anderson, “Transmission performance of Y junctions in planar dielectric waveguide,” IEEE J. Microwaves Opt. Acoust. MOA-2, 7–12 (1978).
[CrossRef]

Baba, K.

Baets, R.

Burns, W. K.

W. K. Burns, A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Gambling, W. A.

W. A. Gambling, H. Matsumura, C. M. Ragdle, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Kawakami, S.

Kojima, M.

K. Tanaka, M. Kojima, “Volume integral equations for analysis of dielectric branching waveguides,” IEEE Trans. Microwave Theory Tech. MTT-36, 1239–1245 (1988).
[CrossRef]

K. Tanaka, M. Kojima, “New boundary integral equation for CAD of waveguide circuit,” Electron. Lett. 24, 807–808 (1988).
[CrossRef]

Koshiba, M.

M. Koshiba, M. Suzuki, “Boundary-element analysis of dielectric slab waveguide discontinuities,” Appl. Opt. 25, 828–829 (1986).
[CrossRef] [PubMed]

M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
[CrossRef]

Kumagai, N.

E. Nishimura, N. Morita, N. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrary shaped junctions between multilayered dielectric planer waveguides,” IEEE J. Lightwave Technol. LT-3, 887–894 (1985).
[CrossRef]

Lagasse, P. E.

Lewin, L.

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MIT-22, 718–727 (1974).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric waveguides,” Bell. Syst. Tech. J. 48, 2103–2132 (1969).

Marcuse, D.

Matsuhara, M.

M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inform. Commun. Eng. Jpn. J71-C, 1021–1026 (1988).

Matsumura, H.

W. A. Gambling, H. Matsumura, C. M. Ragdle, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Matsuo, S.

Miki, T.

M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
[CrossRef]

Milton, A. F.

W. K. Burns, A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[CrossRef]

Miyagi, M.

Morita, N.

E. Nishimura, N. Morita, N. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrary shaped junctions between multilayered dielectric planer waveguides,” IEEE J. Lightwave Technol. LT-3, 887–894 (1985).
[CrossRef]

N. Morita, “An integral equation method for electromagnetic scattering of guided modes by boundary deformations of dielectric slab waveguides,” Radio Sci. 18, 39–47 (1983).
[CrossRef]

N. Morita, “Surface integral representations for electromagnetic scattering from dielectric cylinders,”IEEE Trans. Antennas Propag. AP-26, 261–266 (1978).
[CrossRef]

Nishida, S.

Nishimura, E.

E. Nishimura, N. Morita, N. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrary shaped junctions between multilayered dielectric planer waveguides,” IEEE J. Lightwave Technol. LT-3, 887–894 (1985).
[CrossRef]

Ogusu, K.

K. Ogusu, “Transmission characteristics of single-mode asymmetric dielectric waveguide Y-junctions,” Opt. Commun. 53, 169–172 (1985).
[CrossRef]

K. Ogusu, “Transmission characteristics of optical waveguide corners,” Opt. Commun. 55, 149–153 (1985).
[CrossRef]

Ooishi, K.

M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
[CrossRef]

Ragdle, C. M.

W. A. Gambling, H. Matsumura, C. M. Ragdle, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Shiina, T.

Shirae, H.

M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inform. Commun. Eng. Jpn. J71-C, 1021–1026 (1988).

Shiraishi, K.

Suzuki, M.

M. Koshiba, M. Suzuki, “Boundary-element analysis of dielectric slab waveguide discontinuities,” Appl. Opt. 25, 828–829 (1986).
[CrossRef] [PubMed]

M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
[CrossRef]

Tanaka, K.

K. Tanaka, M. Kojima, “Volume integral equations for analysis of dielectric branching waveguides,” IEEE Trans. Microwave Theory Tech. MTT-36, 1239–1245 (1988).
[CrossRef]

K. Tanaka, M. Kojima, “New boundary integral equation for CAD of waveguide circuit,” Electron. Lett. 24, 807–808 (1988).
[CrossRef]

Taylor, H. F.

H. F. Taylor, “Losses at corner bends in dielectric waveguides,” Appl. Opt. 16, 711–715 (1977).
[CrossRef] [PubMed]

H. F. Taylor, “Power loss at directional change in dielectric waveguide,” Appl. Opt. 13, 624–647 (1974).
[CrossRef]

Toyota, I.

M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inform. Commun. Eng. Jpn. J71-C, 1021–1026 (1988).

Tsuai, L. L.

T. K. Wu, L. L. Tsuai, “Numerical analysis of electromagnetic fields in biological tissues,” Proc. IEEE 62, 1167–1168 (1974).
[CrossRef]

van der Donk, J.

van Roey, J.

Wu, T. K.

T. K. Wu, L. L. Tsuai, “Numerical analysis of electromagnetic fields in biological tissues,” Proc. IEEE 62, 1167–1168 (1974).
[CrossRef]

Yajima, H.

H. Yajima, “Dielectric thin film optical branching waveguide,” Appl. Phys. Lett. 2, 647–649 (1973).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

H. Yajima, “Dielectric thin film optical branching waveguide,” Appl. Phys. Lett. 2, 647–649 (1973).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

Bell. Syst. Tech. J. (1)

E. A. J. Marcatili, “Bends in optical dielectric waveguides,” Bell. Syst. Tech. J. 48, 2103–2132 (1969).

Electron. Lett. (2)

M. Koshiba, K. Ooishi, T. Miki, M. Suzuki, “Finite-element analysis of the discontinuities in a dielectric slab waveguide bounded by parallel plates,” Electron. Lett. 18, 33–34 (1982).
[CrossRef]

K. Tanaka, M. Kojima, “New boundary integral equation for CAD of waveguide circuit,” Electron. Lett. 24, 807–808 (1988).
[CrossRef]

IEEE J. Lightwave Technol. (1)

E. Nishimura, N. Morita, N. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrary shaped junctions between multilayered dielectric planer waveguides,” IEEE J. Lightwave Technol. LT-3, 887–894 (1985).
[CrossRef]

IEEE J. Microwaves Opt. Acoust. (1)

I. Anderson, “Transmission performance of Y junctions in planar dielectric waveguide,” IEEE J. Microwaves Opt. Acoust. MOA-2, 7–12 (1978).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. K. Burns, A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11, 32–39 (1975).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Morita, “Surface integral representations for electromagnetic scattering from dielectric cylinders,”IEEE Trans. Antennas Propag. AP-26, 261–266 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

K. Tanaka, M. Kojima, “Volume integral equations for analysis of dielectric branching waveguides,” IEEE Trans. Microwave Theory Tech. MTT-36, 1239–1245 (1988).
[CrossRef]

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MIT-22, 718–727 (1974).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (2)

K. Ogusu, “Transmission characteristics of single-mode asymmetric dielectric waveguide Y-junctions,” Opt. Commun. 53, 169–172 (1985).
[CrossRef]

K. Ogusu, “Transmission characteristics of optical waveguide corners,” Opt. Commun. 55, 149–153 (1985).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

W. A. Gambling, H. Matsumura, C. M. Ragdle, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Proc. IEEE (1)

T. K. Wu, L. L. Tsuai, “Numerical analysis of electromagnetic fields in biological tissues,” Proc. IEEE 62, 1167–1168 (1974).
[CrossRef]

Radio Sci. (1)

N. Morita, “An integral equation method for electromagnetic scattering of guided modes by boundary deformations of dielectric slab waveguides,” Radio Sci. 18, 39–47 (1983).
[CrossRef]

Trans. Inst. Electron. Inform. Commun. Eng. Jpn. (1)

M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inform. Commun. Eng. Jpn. J71-C, 1021–1026 (1988).

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Figures (9)

Fig. 1
Fig. 1

Dielectric waveguide bends and four coordinate systems: (x, y) (X1, Y1), (X2, Y2), and (r, θ).

Fig. 2
Fig. 2

Boundaries C = C1 + C2 + C3 + C4 + C5 + C6 between (inner) region II and (outer) region I, and virtual boundaries C11, C12, C21, and C22 between two waveguides and the connecting section.

Fig. 3
Fig. 3

Dielectric waveguide corner bend used in the numerical calculations (2k0a1 = 2k0a2 cos ϕ1, 2k0a2 = 1.0, ϕ2 = π, n1 = 1.0, n2 = 1.5).

Fig. 4
Fig. 4

Numerical examples of the disturbed field EC(x) along the boundary C1 of waveguide 1 for the case in which ϕ1 = 10° and incidence is from port 2. The solid curve shows the real part, and the dashed curve shows the imaginary part. The abscissa k0l is the normalized distance from the junction point O′ in Fig. 3.

Fig. 5
Fig. 5

Numerical examples of the disturbed field EC(x) along the boundary C3 of waveguide 2 for the case in which ϕ1 = 10° and incidence is from port 2. The solid curve shows the real part, and the dashed curve shows the imaginary part. The abscissa k0l is the normalized distance from the junction point O′ in Fig. 3.

Fig. 6
Fig. 6

Numerical examples of the derivative of the disturbed field ∂EC(x)/∂n along the boundaries C1 of waveguide 1 for the case in which ϕ1 = 10° and incidence is from port 2. The solid curve shows the real part, and the dashed curve shows the imaginary part. The abscissa k0l is the normalized distance from the junction point O′ in Fig. 3.

Fig. 7
Fig. 7

Numerical examples of the derivative of the disturbed field ∂EC(x)/∂n along the boundaries of waveguide 2 for the case in C3 which t = 10° and incidence is from port 2. The solid curve shows the real part, and the dashed curve shows the imaginary part. The abscissa k0l is the normalized distance from the junction point O′ in Fig. 3.

Fig. 8
Fig. 8

Scattering patterns for (solid curve) ϕ1 = 5°, (dotted curve) ϕ1 = 10°, and (dashed curve) ϕ1 = 15° for the case in which incidence is from port 1.

Fig. 9
Fig. 9

Scattering patterns for (solid curve) ϕ1 = 5°, (dotted curve) ϕ1 = 10°, and (dashed curve) ϕ1 = 15° for the case in which incidence is from port 2.

Equations (57)

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

E z ( x ) = E ( x , y ) = E ( X 1 , Y 1 ) = E ( X 2 , Y 2 ) = E ( r , θ )
E i ( x ) = E - ( 2 ) ( x ) ,
E r ( x ) = T 22 E + ( 2 ) ( x ) ,
E ± 2 ( x ) = { cos ( κ 2 Y 2 ) exp ( j β 2 X 2 ) Y 2 a 2 cos ( κ 2 a 2 ) exp [ δ 2 ( a 2 - Y 2 ) ] exp ( j β 2 X 2 ) Y 2 > a 2 cos ( κ 2 a 2 ) exp [ δ 2 ( a 2 + Y 2 ) ] exp ( j β 2 X 2 ) Y 2 < - a 2 .
E t ( x ) = T 21 E + ( 1 ) ( x ) ,
E + ( 1 ) ( x ) = { cos ( κ 1 Y 1 ) exp ( - j β 1 X 1 ) Y 1 a 1 cos ( κ 1 a 1 ) exp [ δ 1 ( a 1 - Y 1 ) ] exp ( - j β 1 X 1 ) Y 1 > a 1 cos ( κ 1 a 1 ) exp [ δ 1 ( a 1 + Y 1 ) ] exp ( - j β 1 X 1 ) Y 1 < - a 1 .
κ j sin ( κ j a j ) - δ j cos ( κ j a j ) = 0             ( j = 1 , 2 ) ,
κ j = ( n 2 2 k 0 2 - β j 2 ) 1 / 2 ,             δ j = ( β j 2 - n 1 2 k 0 2 ) 1 / 2             ( j = 1 , 2 )
( E ( x ) x in region II ( 1 / 2 ) E ( x ) x on C 0 x in region I ) = C [ G 2 ( x x ) E ( x ) / n - E ( x ) G 2 ( x x ) / n ] d l .
G 2 ( x x ) = ( - j / 4 ) H 0 ( 2 ) ( n 2 k 0 x - x ) ,
E ( x ) = E C ( x ) + T 21 E + ( 1 ) ( x )             on C 1 and C 2 ,
E ( x ) = E C ( x ) + T 22 E + ( 2 ) ( x ) + E - ( 2 ) ( x )             on C 3 and C 4 ,
E ( x ) = E C ( x )             on C 5 and C 6
E C ( x ) , E C ( x ) / n 0             ( r = , θ = ϕ 1 ) ,
E C ( x ) , E C ( x ) / n 0             ( r = , θ = ϕ 2 ) .
( T 21 E + ( 1 ) ( x ) x in region II ( 1 / 2 ) T 21 E + ( 1 ) ( x ) x on C 0 x in region I ) = C [ G 2 ( x x ) E C / n - E C G 2 ( x x ) / n ] d l + T 21 C 1 + C 2 [ G 2 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 2 ( x x ) / n ] d l + T 22 C 3 + C 4 [ G 2 ( x x ) E + ( 2 ) / n - E + ( 2 ) G 2 ( x x ) / n ] d l + C 3 + C 4 [ G 2 ( x x ) E - ( 2 ) / n - E - ( 2 ) G 2 ( x x ) / n ] d l ,
( E + ( 1 ) ( x ) x to the right of C 12 ( 1 / 2 ) E + ( 1 ) ( x ) x on C to the right of C 12 0 x to the left of C 12 ) - C 1 + C 2 [ G 2 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 2 ( x x ) / n ] d l = C 12 [ G 2 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 2 ( x x ) / n ] d l ,
( E ± ( 2 ) ( x ) x to the left of C 22 ( 1 / 2 ) E ± ( 2 ) ( x ) x on C to the left of C 22 0 x to the rigt of C 22 ) - C 3 + C 4 [ G 2 ( x x ) E ± ( 2 ) / n - E ± ( 2 ) G 2 ( x x ) / n ] d l = C 22 [ G 2 ( x x ) E ± ( 2 ) / n - E ± ( 2 ) G 2 ( x x ) / n ] d l ,
T 21 C 12 [ G 2 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 2 ( x x ) / n ] d l + T 22 C 22 [ G 2 ( x x ) E + ( 2 ) / n - E + ( 2 ) G 2 ( x x ) / n ] d l = C [ G 2 ( x x ) / E C / n - E C G 2 ( x x ) / n ] d l - C 22 [ G 2 ( x x ) E - ( 2 ) / n - E - ( 2 ) G 2 ( x x ) / n ] d l
G 2 ( x x ) = ( - j / 4 ) H 0 ( 2 ) ( n 2 k 0 x - x ) = A ( r ) g 2 ( θ x ) + O [ ( n 2 k 0 r ) - 3 / 2 ]
A ( r ) = ( - j / 4 ) [ 2 j / ( n 2 π k 0 r ) ] 1 / 2 exp ( - j n 2 k 0 r ) ,
g 2 ( θ x ) = exp ( j n 2 k 0 x cos θ + j n 2 k 0 y sin θ ) .
T 21 N + ( 1 ) ( ϕ 1 ) + T 22 N + ( 2 ) ( ϕ 1 ) = C [ g 2 ( ϕ 1 x ) E C / n - E C g 2 ( ϕ 1 x ) / n ] d l - N - ( 2 ) ( ϕ 1 ) ,
N ± ( j ) ( θ ) = C j 2 [ g 2 ( θ x ) E ± ( j ) / n - E ± ( j ) g 2 ( θ x ) / n ] d l             ( j = 1 , 2 ) ,
T 21 N + ( 1 ) ( ϕ 2 ) + T 22 N + ( 2 ) ( ϕ 2 ) = C [ g 2 ( ϕ 2 x ) E C / n - E C g 2 ( ϕ 2 x ) / n ] d l - N - ( 2 ) ( ϕ 2 ) .
T 21 = C [ W ( x ) E C / n - E C W ( x ) / n ] d l / Δ N + [ N + ( 2 ) ( ϕ 1 ) N - ( 2 ) ( ϕ 2 ) - N + ( 2 ) ( ϕ 2 ) N - ( 2 ) ( ϕ 1 ) ] / Δ N ,
T 22 = C [ V ( x ) E C / n - E C V ( x ) / n ] d l / Δ N + [ N + ( 1 ) ( ϕ 2 ) N - ( 2 ) ( ϕ 1 ) - N + ( 1 ) ( ϕ 1 ) N - ( 2 ) ( ϕ 2 ) ] / Δ N ,
W ( x ) = N + ( 2 ) ( ϕ 2 ) g 2 ( ϕ 1 x ) - N + ( 2 ) ( ϕ 1 ) g 2 ( ϕ 2 x ) ,
V ( x ) = N + ( 1 ) ( ϕ 1 ) g 2 ( ϕ 2 x ) - N + ( 1 ) ( ϕ 2 ) g 2 ( ϕ 1 x ) ,
Δ N = N + ( 1 ) ( ϕ 1 ) N + ( 2 ) ( ϕ 2 ) - N + ( 2 ) ( ϕ 1 ) N + ( 1 ) ( ϕ 2 ) .
( E ( x ) x in region I ( 1 / 2 ) E ( x ) x on C 0 x in region II ) = - C [ G 1 ( x x ) E ( x ) / n - E ( x ) G 1 ( x x ) / n ] d l ,
G 1 ( x x ) = ( - j / 4 ) H 0 ( 2 ) ( n 1 k 0 x - x ) .
( E C ( x ) x in region I ( 1 / 2 ) E C ( x ) x on C 0 x in region II ) = - C [ G 1 ( x x ) E C / E C / n - E C G 1 ( x x ) / n ] d l - T 21 C 11 [ G 1 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 1 ( x x ) / n ] d l - T 22 C 21 [ G 1 ( x x ) E + ( 2 ) / n - E + ( 2 ) G 1 ( x x ) / n ] d l - C 21 [ G 1 ( x x ) E - ( 2 ) / n - E - ( 2 ) G 1 ( x x ) / n ] d l .
( E + ( 1 ) ( x ) x to the right of C 11 ( 1 / 2 ) E + ( 1 ) ( x ) x on C to the right of C 11 0 x to the left of C 11 ) + C 1 + C 2 [ G 1 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 1 ( x x ) / n ] d l = C 11 [ G 1 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 1 ( x x ) / n ] d l ,
( E ± ( 2 ) ( x ) x to the left of C 21 ( 1 / 2 ) E ± ( 2 ) ( x ) x on C to the left of C 21 0 x to the right of C 21 ) + C 3 + C 4 [ G 1 ( x x ) E ± ( 2 ) / n - E ± ( 2 ) G 1 ( x x ) / n ] d l = C 21 [ G 1 ( x x ) E ± ( 2 ) / n - E ± ( 2 ) G 1 ( x x ) / n ] d l ,
( E C ( x ) x in region II ( 1 / 2 ) E C ( x ) x on C 0 x in region I ) = C [ G 2 ( x x ) E C / n - E C G 2 ( x x ) / n ] d l - T 21 C 12 [ G 2 ( x x ) E + ( 1 ) / n - E + ( 1 ) G 2 ( x x ) / n ] d l - T 22 C 22 [ G 2 ( x x ) E + ( 2 ) / n - E + ( 2 ) G 2 ( x x ) / n ] d l - C 22 [ G 2 ( x x ) E - ( 2 ) / n - E - ( 2 ) G 2 ( x x ) / n ] d l .
( E C ( x ) x in region II ( 1 / 2 ) E C ( x ) x on C 0 x in region I ) = C [ P 2 ( x x ) E C / E C / n - E C P 2 ( x x ) / n ] d l - S 2 ( x ) ,
P 2 ( x x ) = G 2 ( x x ) - [ U 2 + ( 1 ) ( x ) W ( x ) + U 2 + ( 2 ) ( x ) V ( x ) ] / Δ N ,
S 2 ( x ) = U 2 - ( 2 ) ( x ) + { U 2 + ( 1 ) ( x ) [ N + ( 2 ) ( ϕ 1 ) N - ( 2 ) ( ϕ 2 ) - N + ( 2 ) ( ϕ 2 ) N - ( 2 ) ( ϕ 1 ) ] + U 2 + ( 2 ) ( x ) [ N + ( 1 ) ( ϕ 2 ) N - ( 2 ) ( ϕ 1 ) - N + ( 1 ) ( ϕ 1 ) N - ( 2 ) ( ϕ 2 ) ] } / Δ N ,
U i ± ( j ) ( x ) = C j i [ G i ( x x ) E ± ( j ) / n - E ± ( j ) G i ( x x ) / n ] d l             ( i , j = 1 , 2 )
( E C ( x ) x in region I ( 1 / 2 ) E C ( x ) x on C 0 x in region II ) = C [ P 1 ( x x ) E C / n - E C P 1 ( x x ) / n ] d l - S 1 ( x ) ,
P 1 ( x x ) = - G 1 ( x x ) - [ U 1 + ( 1 ) ( x ) W ( x ) + U 1 + ( 2 ) ( x ) V ( x ) ] / Δ N
S 1 ( x ) = U 1 - ( 2 ) ( x ) + { U 1 + ( 1 ) ( x ) [ N + ( 2 ) ( ϕ 1 ) N - ( 2 ) ( ϕ 2 ) - N + ( 2 ) ( ϕ 2 ) N - ( 2 ) ( ϕ 1 ) ] + U 1 + ( 2 ) ( x ) [ N + ( 1 ) ( ϕ 2 ) N - ( 2 ) ( ϕ 1 ) - N + ( 1 ) ( ϕ 1 ) N - ( 2 ) ( ϕ 2 ) ] } / Δ N .
E S ( r , θ ) = ( - j / 4 ) [ j / ( n 1 π k 0 r ) ] 1 / 2 exp ( - j n 1 k 0 r ) B ( θ ) ,
B ( θ ) = - C [ g 1 ( θ x ) E C / n - E C g 1 ( θ x ) / n ] d l - T 21 M + ( 1 ) ( θ ) - T 22 M + ( 2 ) ( θ ) - M - ( 2 ) ( θ )
g 1 ( θ x ) = exp ( j n 1 k 0 x cos θ + j n 1 k 0 y sin θ ) ,
M ± ( j ) ( θ ) = C j 1 [ g 1 ( θ x ) E ± ( j ) / n - E ± ( j ) g 1 ( θ x ) / n ] d l             ( j = 1 , 2 ) .
Γ i 1 + Γ i 2 + Γ i S = Γ TOTAL = 1             ( i = 1 , 2 ) ,
Γ i j = ( Q j / Q i ) T i j 2 ,
Γ i S = 0 2 π B ( θ ) 2 d θ / Q i ,
Q i = β i [ ( a i / 2 ) - 1 / ( 4 κ i ) sin ( 2 κ i a i ) + 1 / ( 2 δ i ) cos 2 ( κ i a i ) ]             ( i = 1 , 2 ) .
Γ i j = Γ i j             ( i , j = 1 , 2 ) .
E C ( x ) x II C = E C ( x ) x I C ,
E C ( x ) / n x II C = E C ( x ) / n x I C .
Γ 11 = 0.0000 ,     Γ 12 = 0.9853 ,     Γ 1 S = 0.0142 ,     Γ TOTAL = 0.9995 , Γ 21 = 0.9853 ,     Γ 22 = 0.0000 ,     Γ 2 S = 0.0142 ,     Γ TOTAL = 0.9995 ;
Γ 11 = 0.0000 ,     Γ 12 = 0.9425 ,     Γ 1 S = 0.0553 ,     Γ TOTAL = 0.9978 , Γ 21 = 0.9432 ,     Γ 22 = 0.0000 ,     Γ 2 S = 0.0553 ,     Γ TOTAL = 0.9984 ;
Γ 11 = 0.0000 ,     Γ 12 = 0.8754 ,     Γ 1 S = 0.1193 ,     Γ TOTAL = 0.9947 , Γ 21 = 0.8780 ,     Γ 22 = 0.0000 ,     Γ 2 S = 0.1193 ,     Γ TOTAL = 0.9973 ;

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