Abstract

We show examples of accurate computer-aided design of two-dimensional dielectric waveguide bends of finite length by using the boundary-element method (BEM) based on guided-mode extracted integral equations (GMEIE’s) that have been proposed previously. In contrast to integral equations used in previous papers, the unknown function in the GMEIE’s that we discuss is the total field, which is the same as that in the conventional boundary integral equation method. Therefore waveguide discontinuity problems in dielectric open waveguides can be perfectly regarded as problems of scattering by isolated dielectric objects. Various properties in numerical calculations of GMEIE’s are examined. The numerical results obtained satisfy the energy conservation law within an accuracy of 1.0%. It is shown that it is possible to design an optimum bend shape of a dielectric optical waveguide bend that gives small radiation losses by using the BEM based on GMEIE’s.

© 1996 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. P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
    [Crossref]
  3. D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
    [Crossref]
  4. K. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1990).
    [Crossref]
  5. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
    [Crossref] [PubMed]
  6. N. Morita, “An integral equation method for electromagnetic scattering of guided modes by boundary deformation of dielectric slab waveguides,” Radio Sci. 18, 39–47 (1983).
    [Crossref]
  7. M. Koshiba, M. Suzuki, “Boundary-element analysis of dielectric slab waveguide discontinuities,” Appl. Opt. 25, 828–829 (1986).
    [Crossref] [PubMed]
  8. T. G. Livernois, D. N. Nyquist, “Integral-equation formulation for scattering by dielectric discontinuities along open-boundary dielectric waveguides,” J. Opt. Soc. Am. A 4, 1289–1295 (1987).
    [Crossref]
  9. M. Matsuhara, I. Toyota, H. Shirae, “Analysis of a bending dielectric slab waveguide by integral equation formulation,” Trans. Inst. Electron. Inf. Eng. Jpn. J71-C, 1021–1026 (1988).
  10. K. Hirayama, M. Koshiba, “Analysis of discontinuities in an asymmetric dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 40, 686–691 (1992).
    [Crossref]
  11. N. Morita, “A rigorous analytical solution to abrupt dielectric waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. 39, 1272–1278 (1992).
    [Crossref]
  12. K. Tanaka, M. Kojima, “New boundary integral equations for CAD of waveguide circuits,” Electron. Lett. 24, 807–808 (1989).
    [Crossref]
  13. K. Tanaka, M. Kojima, “New boundary integral equations for computer-aided design of dielectric waveguide circuits,” J. Opt. Soc. Am. A 6, 667–674 (1989).
    [Crossref]
  14. K. Tanaka, M. Nakahara, “New boundary integral equations for CAD of waveguide circuits: guided-mode extracted integral equations,” IEEE Trans. Microwave Theory Tech. 40, 1647–1654 (1992).
    [Crossref]
  15. K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
    [Crossref]
  16. T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1990), Chap. 6.
    [Crossref]
  17. R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1991), Chap. 5.
    [Crossref]
  18. H. F. Taykor, “Losses at corner bends in dielectric waveguides,” Appl. Opt. 16, 711–716 (1977).
    [Crossref]
  19. E. G. Newman, “Reducing radiation loss of tilts in dielectric optical waveguides,” Electron. Lett. 17, 3990–3998 (1978).
  20. K. Ogusu, “Transmission characteristics of optical waveguide corners,” Opt. Commun. 55, 149–153 (1985).
    [Crossref]
  21. T. Shiina, K. Shiraishi, S. Kawakami, “Waveguide-bend configuration with low-loss characteristics,” Opt. Lett. 11, 736–738 (1986).
    [Crossref] [PubMed]
  22. M. Sanagi, M. Nakajima, “An optical waveguide bend with enhanced curvature and optimization of its structure,” Fiber Integrated Opt. 17, 329–345 (1991).
  23. S. Safavi-Naeini, Y. L. Chow, “A novel design and analysis of low loss abrupt bends of dielectric slab waveguides,” J. Lightwave Technol. 10, 570–580 (1992).
    [Crossref]
  24. K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bends in dielectric waveguide,” J. Lightwave Technol. 10, 563–568 (1992).
    [Crossref]
  25. R. M. James, “On the use of Fourier series/FFT’s as global basis functions in the solution of boundary integral equations for EM scattering,”IEEE Trans. Antennas Propag. 42, 1309–1316 (1994).
    [Crossref]
  26. R. Coifman, V. Rokhlin, S. Wandzura, “The fast multi-pole method for the wave equation: a pedestrian prescription,”IEEE Antennas Propag. Mag. 35, (June), (1993).
    [Crossref]
  27. G. F. Herrmann, S. M. Strain, “Sampling method using prefiltered band-limited Green’s functions for the solution of electromagnetic integral equations,”IEEE Trans. Antennas Propag. 41, 20–24 (1993).
    [Crossref]

1994 (1)

R. M. James, “On the use of Fourier series/FFT’s as global basis functions in the solution of boundary integral equations for EM scattering,”IEEE Trans. Antennas Propag. 42, 1309–1316 (1994).
[Crossref]

1993 (3)

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multi-pole method for the wave equation: a pedestrian prescription,”IEEE Antennas Propag. Mag. 35, (June), (1993).
[Crossref]

G. F. Herrmann, S. M. Strain, “Sampling method using prefiltered band-limited Green’s functions for the solution of electromagnetic integral equations,”IEEE Trans. Antennas Propag. 41, 20–24 (1993).
[Crossref]

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

1992 (5)

K. Tanaka, M. Nakahara, “New boundary integral equations for CAD of waveguide circuits: guided-mode extracted integral equations,” IEEE Trans. Microwave Theory Tech. 40, 1647–1654 (1992).
[Crossref]

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an asymmetric dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 40, 686–691 (1992).
[Crossref]

N. Morita, “A rigorous analytical solution to abrupt dielectric waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. 39, 1272–1278 (1992).
[Crossref]

S. Safavi-Naeini, Y. L. Chow, “A novel design and analysis of low loss abrupt bends of dielectric slab waveguides,” J. Lightwave Technol. 10, 570–580 (1992).
[Crossref]

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bends in dielectric waveguide,” J. Lightwave Technol. 10, 563–568 (1992).
[Crossref]

1991 (2)

M. Sanagi, M. Nakajima, “An optical waveguide bend with enhanced curvature and optimization of its structure,” Fiber Integrated Opt. 17, 329–345 (1991).

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
[Crossref] [PubMed]

1990 (1)

K. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1990).
[Crossref]

1989 (3)

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

K. Tanaka, M. Kojima, “New boundary integral equations for CAD of waveguide circuits,” Electron. Lett. 24, 807–808 (1989).
[Crossref]

K. Tanaka, M. Kojima, “New boundary integral equations for computer-aided design of dielectric waveguide circuits,” J. Opt. Soc. Am. A 6, 667–674 (1989).
[Crossref]

1988 (2)

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

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[Crossref]

1987 (1)

1986 (2)

1985 (1)

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

1983 (1)

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

1978 (2)

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

E. G. Newman, “Reducing radiation loss of tilts in dielectric optical waveguides,” Electron. Lett. 17, 3990–3998 (1978).

1977 (1)

Chow, Y. L.

S. Safavi-Naeini, Y. L. Chow, “A novel design and analysis of low loss abrupt bends of dielectric slab waveguides,” J. Lightwave Technol. 10, 570–580 (1992).
[Crossref]

Coifman, R.

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multi-pole method for the wave equation: a pedestrian prescription,”IEEE Antennas Propag. Mag. 35, (June), (1993).
[Crossref]

Davies, J. B.

K. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1990).
[Crossref]

Feit, M. D.

Fleck, J. A.

Hadley, G. R.

Hermansson, B.

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

Herrmann, G. F.

G. F. Herrmann, S. M. Strain, “Sampling method using prefiltered band-limited Green’s functions for the solution of electromagnetic integral equations,”IEEE Trans. Antennas Propag. 41, 20–24 (1993).
[Crossref]

Hirayama, K.

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an asymmetric dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 40, 686–691 (1992).
[Crossref]

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bends in dielectric waveguide,” J. Lightwave Technol. 10, 563–568 (1992).
[Crossref]

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1991), Chap. 5.
[Crossref]

James, R. M.

R. M. James, “On the use of Fourier series/FFT’s as global basis functions in the solution of boundary integral equations for EM scattering,”IEEE Trans. Antennas Propag. 42, 1309–1316 (1994).
[Crossref]

Kaczmarski, P.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[Crossref]

Kawakami, S.

Koch, K. B.

K. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1990).
[Crossref]

Kojima, M.

K. Tanaka, M. Kojima, “New boundary integral equations for CAD of waveguide circuits,” Electron. Lett. 24, 807–808 (1989).
[Crossref]

K. Tanaka, M. Kojima, “New boundary integral equations for computer-aided design of dielectric waveguide circuits,” J. Opt. Soc. Am. A 6, 667–674 (1989).
[Crossref]

Koshiba, M.

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bends in dielectric waveguide,” J. Lightwave Technol. 10, 563–568 (1992).
[Crossref]

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an asymmetric dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 40, 686–691 (1992).
[Crossref]

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

Lagasse, P. E.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[Crossref]

Livernois, T. G.

Matsuhara, M.

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

Morita, N.

N. Morita, “A rigorous analytical solution to abrupt dielectric waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. 39, 1272–1278 (1992).
[Crossref]

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

Nakahara, M.

K. Tanaka, M. Nakahara, “New boundary integral equations for CAD of waveguide circuits: guided-mode extracted integral equations,” IEEE Trans. Microwave Theory Tech. 40, 1647–1654 (1992).
[Crossref]

Nakajima, M.

M. Sanagi, M. Nakajima, “An optical waveguide bend with enhanced curvature and optimization of its structure,” Fiber Integrated Opt. 17, 329–345 (1991).

Newman, E. G.

E. G. Newman, “Reducing radiation loss of tilts in dielectric optical waveguides,” Electron. Lett. 17, 3990–3998 (1978).

Nyquist, D. N.

Ogusu, K.

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

Ootera, H.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

Rokhlin, V.

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multi-pole method for the wave equation: a pedestrian prescription,”IEEE Antennas Propag. Mag. 35, (June), (1993).
[Crossref]

Safavi-Naeini, S.

S. Safavi-Naeini, Y. L. Chow, “A novel design and analysis of low loss abrupt bends of dielectric slab waveguides,” J. Lightwave Technol. 10, 570–580 (1992).
[Crossref]

Sanagi, M.

M. Sanagi, M. Nakajima, “An optical waveguide bend with enhanced curvature and optimization of its structure,” Fiber Integrated Opt. 17, 329–345 (1991).

Shiina, T.

Shirae, H.

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

Shiraishi, K.

Strain, S. M.

G. F. Herrmann, S. M. Strain, “Sampling method using prefiltered band-limited Green’s functions for the solution of electromagnetic integral equations,”IEEE Trans. Antennas Propag. 41, 20–24 (1993).
[Crossref]

Suzuki, M.

Tamir, T.

T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1990), Chap. 6.
[Crossref]

Tanaka, K.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

K. Tanaka, M. Nakahara, “New boundary integral equations for CAD of waveguide circuits: guided-mode extracted integral equations,” IEEE Trans. Microwave Theory Tech. 40, 1647–1654 (1992).
[Crossref]

K. Tanaka, M. Kojima, “New boundary integral equations for CAD of waveguide circuits,” Electron. Lett. 24, 807–808 (1989).
[Crossref]

K. Tanaka, M. Kojima, “New boundary integral equations for computer-aided design of dielectric waveguide circuits,” J. Opt. Soc. Am. A 6, 667–674 (1989).
[Crossref]

Tanaka, M.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

Tashima, H.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

Taykor, H. F.

Toyota, I.

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

Wandzura, S.

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multi-pole method for the wave equation: a pedestrian prescription,”IEEE Antennas Propag. Mag. 35, (June), (1993).
[Crossref]

Wickramasinghe, D.

K. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1990).
[Crossref]

Yevick, D.

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

Yoshino, Y.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

Appl. Opt. (3)

Electron. Lett. (4)

E. G. Newman, “Reducing radiation loss of tilts in dielectric optical waveguides,” Electron. Lett. 17, 3990–3998 (1978).

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[Crossref]

K. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element finite difference propagation algorithm for integrated optical device,” Electron. Lett. 25, 514–516 (1990).
[Crossref]

K. Tanaka, M. Kojima, “New boundary integral equations for CAD of waveguide circuits,” Electron. Lett. 24, 807–808 (1989).
[Crossref]

Fiber Integrated Opt. (1)

M. Sanagi, M. Nakajima, “An optical waveguide bend with enhanced curvature and optimization of its structure,” Fiber Integrated Opt. 17, 329–345 (1991).

IEEE Antennas Propag. Mag. (1)

R. Coifman, V. Rokhlin, S. Wandzura, “The fast multi-pole method for the wave equation: a pedestrian prescription,”IEEE Antennas Propag. Mag. 35, (June), (1993).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[Crossref]

IEEE Trans. Antennas Propag. (2)

G. F. Herrmann, S. M. Strain, “Sampling method using prefiltered band-limited Green’s functions for the solution of electromagnetic integral equations,”IEEE Trans. Antennas Propag. 41, 20–24 (1993).
[Crossref]

R. M. James, “On the use of Fourier series/FFT’s as global basis functions in the solution of boundary integral equations for EM scattering,”IEEE Trans. Antennas Propag. 42, 1309–1316 (1994).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an asymmetric dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 40, 686–691 (1992).
[Crossref]

N. Morita, “A rigorous analytical solution to abrupt dielectric waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. 39, 1272–1278 (1992).
[Crossref]

K. Tanaka, M. Nakahara, “New boundary integral equations for CAD of waveguide circuits: guided-mode extracted integral equations,” IEEE Trans. Microwave Theory Tech. 40, 1647–1654 (1992).
[Crossref]

J. Lightwave Technol. (2)

S. Safavi-Naeini, Y. L. Chow, “A novel design and analysis of low loss abrupt bends of dielectric slab waveguides,” J. Lightwave Technol. 10, 570–580 (1992).
[Crossref]

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bends in dielectric waveguide,” J. Lightwave Technol. 10, 563–568 (1992).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

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

Opt. Lett. (2)

Radio Sci. (2)

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

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[Crossref]

Trans. Inst. Electron. Inf. Eng. Jpn. (1)

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

Other (2)

T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1990), Chap. 6.
[Crossref]

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1991), Chap. 5.
[Crossref]

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Figures (14)

Fig. 1
Fig. 1

Geometry of the problem. The problem of transmission through a waveguide bend or discontinuity (solid curves + dashed lines) is treated like the problem of scattering by an isolated dielectric object, whose shape is given by the solid curves.

Fig. 2
Fig. 2

Example of the practical circular bend shape used in the numerical calculations. Numerical calculations by GMEIE’s are similar to those of problems of scattering by the dielectric object truncated by two virtual boundaries C11 + C12 + C13 and C21 + C22 + C23, whose shapes are given by the solid curves. The parameters are k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°.

Fig. 3
Fig. 3

Dependence of results on the discretization region. The abscissa is the normalized distance between the discretization end and the center of the circular bend section. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°.

Fig. 4
Fig. 4

Distribution of the real part of the total field E(x) along the upper boundary of C. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°.

Fig. 5
Fig. 5

Distribution of the real part of the total field ∂E(x)/∂n along the upper boundary of C. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°.

Fig. 6
Fig. 6

Distribution of absolute values of the field EC(x) ∂ E(x) − TE+(1)(x) and its derivative ∂E(x)/∂n along the upper boundary C of the circular bend. The abscissa is the normalized distance from a boundary between the bend section and uniform waveguide 1. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°.

Fig. 7
Fig. 7

Dependence of the results of the circular bend on the average radius of curvature k0R. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, θ1 = 10°, and θ2 = 170°.

Fig. 8
Fig. 8

Rectangular region in which the bend shape is assumed to be changed arbitrarily in order to produce an optimum bend shape that gives small radiation loss. The parameters are k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°, k0a = 101, and k0b = 18.

Fig. 9
Fig. 9

Two waveguides are connected by a straight waveguide whose width is the same as that of waveguides 1 and 2. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, θ1 = 10°, and θ2 = 170°.

Fig. 10
Fig. 10

The lower part of the abrupt corner is cut in order to use total reflection by the lower plane. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, θ1 = 10°, and θ2 = 170°.

Fig. 11
Fig. 11

The inner side of the abrupt corner is inflated in order to use the reduction of the wave-front velocity. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, θ1 = 10°, and θ2 = 170°.

Fig. 12
Fig. 12

Circular bend that connects two waveguides smoothly under given conditions. The inner side of the circular bend is inflated in order to use the reduction of the wave-front velocity. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, k0R = 283, θ1 = 10°, and θ2 = 170°.

Fig. 13
Fig. 13

The lower part of the bend is cut in the optimum bend shape obtained in Fig. 11 in order to use the total reflection. The parameters are n1 = n3 = 1.515, n2 = 1.61, k0a1 = k0a2 = 4.468, θ1 = 10°, and θ2 = 170°.

Fig. 14
Fig. 14

Concrete numerical values of results and practical bend shapes that are obtained in Figs. 12 and 13 by the CAD, with results of the abrupt corner bend shown as dashed lines.

Equations (12)

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

E ( x ) x exists in the waveguides E ( x ) / 2 x exists on the boundaries 0 x exists outside the waveguides } = C + C 1 + C 2 [ G 2 ( x x ) E / n - E G 2 ( x x ) / n ] d l ,
G 2 ( x x ) = - ( j / 4 ) H 0 ( 2 ) ( n 2 k 0 x - x )
E ( x ) = C [ G 2 ( x x ) E / n - E ( x ) G 2 / n ] d l + T C 1 [ G 2 ( x x ) E + ( 1 ) / n - E + ( 1 ) ( x ) G 2 / n ] d l + R C 2 [ G 2 ( x x ) E + ( 2 ) / n - E + ( 2 ) ( x ) G 2 / n ] d l + C 2 [ G 2 ( x x ) E - ( 2 ) / n - E - ( 2 ) ( x ) G 2 / n ] d l .
E ( x ) = C [ G 2 ( x x ) E / n - E ( x ) G 2 / n ] d l - T U 2 + ( 1 ) ( x ) - R U 2 + ( 2 ) ( x ) - U 2 - ( 2 ) ( x ) ,
U i ± ( j ) ( x ) = C j i [ G i ( x x ) E ± ( j ) / n - E ± ( j ) ( x ) G i / n ] d l ,             i , j = 1 , 2.
E ( r , θ ) = - ( j / 4 ) ( 2 j / π k 0 n 2 r ) 1 / 2 exp ( - k 0 n 2 r ) B ( θ ) ,
B ( θ ) = C [ g 2 ( θ x ) E / n - E ( x ) g 2 ( θ x ) / n ] d l - T u 2 + ( 1 ) ( θ ) - R u 2 + ( 2 ) ( θ ) - u 2 - ( 2 ) ( θ ) ,
u i ± ( j ) ( x ) = C j i [ g i ( θ x ) E ± ( j ) ( x ) / n - E ± ( j ) ( x ) g i ( θ x ) / n ] d l ,             i = 2 , j = 1 , 2 ,
g 2 ( θ x ) = exp ( j n 2 k 0 x cos θ + j n 2 k 0 y sin θ ) .
B ( θ 1 ) = 0 ,             B ( θ 2 ) = 0.
E ( x ) / 2 = C [ P j ( x x ) E ( x ) / n - E ( x ) P j ( x x ) / n ] d l - S j ( x ) ,             j = 1 , 2 ,
E C ( x ) = E ( x ) - T E + ( 1 ) ( x ) .

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