Abstract

The abrupt transition coupling between two five-layered symmetric slab waveguides is analyzed by means of an integral equation appropriate for the boundary-value problem. A complete set of modes is employed to describe the fields inside both waveguides, and the integral equation is solved by a Neumann-series iterative procedure giving highly accurate results, especially in the case of small difference (1%) between the refractive indices of the core and cladding regions. The reflection and coupling coefficients of the guided waves as well as the total power of the radiation modes are computed. Numerical results are presented for several coupling geometries, and misalignment losses are also calculated. Finally, a comparison with relevant previously published papers is made.

© 1998 Optical Society of America

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  1. G. A. Hockham, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
    [CrossRef]
  2. K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
    [CrossRef]
  3. P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
    [CrossRef]
  4. T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
    [CrossRef]
  5. K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
    [CrossRef]
  6. A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Invest. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).
  7. C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
    [CrossRef]
  8. C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
    [CrossRef]
  9. D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electrooptic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
    [CrossRef]
  10. M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
    [CrossRef]
  11. Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
    [CrossRef]
  12. M. Szustakowski, M. Marciniak, “Light power division in a monomode Ti:LiNbO3 waveguide Y-junction power-combiner–power-divider sequence,” Opt. Commun. 81, 20–22 (1991).
    [CrossRef]
  13. L. A. Molter-Orr, H. A. Haus, F. J. Leonberger, “20 GHz optical waveguide sampler,” IEEE J. Quantum Electron. QE-19, 1877–1883 (1983).
    [CrossRef]
  14. M. Izutsu, H. Haga, T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulators,” J. Lightwave Technol. LT-1, 285–289 (1983).
    [CrossRef]
  15. H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
    [CrossRef]
  16. Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation-loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
    [CrossRef]
  17. K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
    [CrossRef]
  18. E. Nishimura, N. Morita, K. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrarily shaped junctions between multilayered dielectric planar waveguides,” J. Lightwave Technol. LT-3, 887–893 (1985).
    [CrossRef]
  19. I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layered symmetric slab waveguide,” J. Opt. Soc. Am. A 106, 1260–1267 (1997).
    [CrossRef]

1997

I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layered symmetric slab waveguide,” J. Opt. Soc. Am. A 106, 1260–1267 (1997).
[CrossRef]

1992

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electrooptic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

1991

M. Szustakowski, M. Marciniak, “Light power division in a monomode Ti:LiNbO3 waveguide Y-junction power-combiner–power-divider sequence,” Opt. Commun. 81, 20–22 (1991).
[CrossRef]

1990

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

1989

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation-loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

1988

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
[CrossRef]

C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
[CrossRef]

1987

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

1986

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

1985

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

E. Nishimura, N. Morita, K. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrarily shaped junctions between multilayered dielectric planar waveguides,” J. Lightwave Technol. LT-3, 887–893 (1985).
[CrossRef]

1984

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

1983

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

L. A. Molter-Orr, H. A. Haus, F. J. Leonberger, “20 GHz optical waveguide sampler,” IEEE J. Quantum Electron. QE-19, 1877–1883 (1983).
[CrossRef]

M. Izutsu, H. Haga, T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulators,” J. Lightwave Technol. LT-1, 285–289 (1983).
[CrossRef]

1979

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

1978

H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[CrossRef]

1972

G. A. Hockham, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[CrossRef]

Aoki, K.

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

Benech, P.

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electrooptic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

Burrows, E. C.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Capsalis, C. N.

C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
[CrossRef]

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

Citerne, J.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

Fukumitsu, O.

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

Gelin, P.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

Haga, H.

M. Izutsu, H. Haga, T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulators,” J. Lightwave Technol. LT-1, 285–289 (1983).
[CrossRef]

Hardy, A.

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation-loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

Haus, H. A.

L. A. Molter-Orr, H. A. Haus, F. J. Leonberger, “20 GHz optical waveguide sampler,” IEEE J. Quantum Electron. QE-19, 1877–1883 (1983).
[CrossRef]

Hirai, H.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
[CrossRef]

Hockham, G. A.

G. A. Hockham, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[CrossRef]

Imada, Y.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
[CrossRef]

Izutsu, M.

M. Izutsu, H. Haga, T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulators,” J. Lightwave Technol. LT-1, 285–289 (1983).
[CrossRef]

Kawano, K.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

Khalil, D.

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electrooptic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

Koren, U.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Kumagai, K.

E. Nishimura, N. Morita, K. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrarily shaped junctions between multilayered dielectric planar waveguides,” J. Lightwave Technol. LT-3, 887–893 (1985).
[CrossRef]

Leonberger, F. J.

L. A. Molter-Orr, H. A. Haus, F. J. Leonberger, “20 GHz optical waveguide sampler,” IEEE J. Quantum Electron. QE-19, 1877–1883 (1983).
[CrossRef]

Liou, Kang-Yih

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Manenkov, A. B.

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

Marciniak, M.

M. Szustakowski, M. Marciniak, “Light power division in a monomode Ti:LiNbO3 waveguide Y-junction power-combiner–power-divider sequence,” Opt. Commun. 81, 20–22 (1991).
[CrossRef]

Marom, E.

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation-loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

Martyak, M. J. R.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Mitomi, O.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

Miyazawa, H.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

Molter-Orr, L. A.

L. A. Molter-Orr, H. A. Haus, F. J. Leonberger, “20 GHz optical waveguide sampler,” IEEE J. Quantum Electron. QE-19, 1877–1883 (1983).
[CrossRef]

Morita, N.

E. Nishimura, N. Morita, K. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrarily shaped junctions between multilayered dielectric planar waveguides,” J. Lightwave Technol. LT-3, 887–893 (1985).
[CrossRef]

Munowitz, M.

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

Nishimura, E.

E. Nishimura, N. Morita, K. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrarily shaped junctions between multilayered dielectric planar waveguides,” J. Lightwave Technol. LT-3, 887–893 (1985).
[CrossRef]

Oron, M.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Papakonstantinou, I. A.

I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layered symmetric slab waveguide,” J. Opt. Soc. Am. A 106, 1260–1267 (1997).
[CrossRef]

Petenzi, M.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

Raybon, G.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Sueta, T.

M. Izutsu, H. Haga, T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulators,” J. Lightwave Technol. LT-1, 285–289 (1983).
[CrossRef]

Szustakowski, M.

M. Szustakowski, M. Marciniak, “Light power division in a monomode Ti:LiNbO3 waveguide Y-junction power-combiner–power-divider sequence,” Opt. Commun. 81, 20–22 (1991).
[CrossRef]

Takenaka, T.

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

Tedjini, S.

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electrooptic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

Theodoropoulos, T. G.

I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layered symmetric slab waveguide,” J. Opt. Soc. Am. A 106, 1260–1267 (1997).
[CrossRef]

Tigelis, I. G.

I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layered symmetric slab waveguide,” J. Opt. Soc. Am. A 106, 1260–1267 (1997).
[CrossRef]

C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
[CrossRef]

Tsutsumi, K.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
[CrossRef]

Uchida, K.

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

Uzunoglu, N. K.

C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
[CrossRef]

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

Vezzetti, D. J.

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

Weissman, Z.

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation-loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

Yajima, H.

H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[CrossRef]

Young, M.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

Yuba, Y.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
[CrossRef]

Electron. Lett.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

G. A. Hockham, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[CrossRef]

IEEE J. Quantum Electron.

Kang-Yih Liou, U. Koren, E. C. Burrows, M. Young, M. J. R. Martyak, M. Oron, G. Raybon, “Y-junction power divider in InGaAsP-InP photonic integrated circuits,” IEEE J. Quantum Electron. 26, 1376–1383 (1990).
[CrossRef]

L. A. Molter-Orr, H. A. Haus, F. J. Leonberger, “20 GHz optical waveguide sampler,” IEEE J. Quantum Electron. QE-19, 1877–1883 (1983).
[CrossRef]

H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[CrossRef]

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation-loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electrooptic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

Invest. Vyssh. Uchebn. Zaved. Radiofiz.

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

J. Lightwave Technol.

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. 6, 590–600 (1988).
[CrossRef]

E. Nishimura, N. Morita, K. Kumagai, “An integral equation approach to electromagnetic scattering from arbitrarily shaped junctions between multilayered dielectric planar waveguides,” J. Lightwave Technol. LT-3, 887–893 (1985).
[CrossRef]

M. Izutsu, H. Haga, T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulators,” J. Lightwave Technol. LT-1, 285–289 (1983).
[CrossRef]

J. Opt. Soc. Am. A

I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layered symmetric slab waveguide,” J. Opt. Soc. Am. A 106, 1260–1267 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

M. Szustakowski, M. Marciniak, “Light power division in a monomode Ti:LiNbO3 waveguide Y-junction power-combiner–power-divider sequence,” Opt. Commun. 81, 20–22 (1991).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of the coupling between two five-layered symmetric slab waveguides.

Fig. 2
Fig. 2

Variation of the radiated power Prad with the parameter 2k0D1 for a geometry with d1=d2=0 µm, n21=n22=1.01, n31=n32=1, and D2=0.5D1.

Fig. 3
Fig. 3

Variation with 2D2 of the reflected guided power |R0|2 (solid line), the transmitted guided power |T0|2 (dashed curve), and the radiated power Prad (asterisks) for a geometry with d1 =d2=0 µm, n212=13, n312=12.7, n222=2.35, n322=2.25, and D1=0.3 µm. In this figure λ0=1 µm.

Fig. 4
Fig. 4

Variation with the ratio D1/D2 of the magnitude of the reflection coefficient |R0| (asterisks) and the coupling coefficient |T0| (solid curve) for a geometry with d1=d2=0 µm, n21 =n22=2.36, D2=0.14 µm, and (a) Δ321=Δ322=0.055 and (b) Δ321=Δ322=0.255.

Fig. 5
Fig. 5

Variation with d1 of (a) the reflected guided power |R0|2, (b) the transmitted guided power |T0|2, and (c) the radiated power Prad for the three values Δ321=Δ322=0.5% (asterisks), 1% (dashed curves), and 5% (solid curves) for a geometry with n21=n22=3.6, Δ131=Δ132=0%, d2=1 µm, D2=1.4 µm, and D1-d1=0.4 µm.

Fig. 6
Fig. 6

Variation with d1 of (a) the transmitted guided power |T0|2 and (b) the radiated power Prad for the three values Δ131 =Δ132=0% (asterisks), 5% (dashed curves), and 30% (solid curves) for a geometry with n21=n22=3.6, d2=1 µm, D2 =1.4 µm, D1-d1=0.4 µm, and Δ321=Δ322=1%.

Fig. 7
Fig. 7

Variation with d1 of (a) the transmitted guided power |T0|2 and (b) the radiated power Prad for the three values Δ131 =Δ132=0% (asterisks), 5% (dashed curves), and 30% (solid curves) for a geometry with n21=n22=3.6, d2=1 µm, D2 =1.4 µm, D1-d1=0.4 µm, and Δ321=Δ322=5%.

Fig. 8
Fig. 8

Variation with the ratio n21/n22 of the reflected guided power |R0|2 (solid curve), the transmitted guided power |T0|2 (dashed curve), and the radiated power Prad (asterisks) for a geometry with n22=3.6, d1=d2=1 µm, D1=D2=1.4 µm, Δ321=Δ322=1%, and Δ131=Δ132=1%.

Tables (1)

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Table 1 Convergence Patterns for Several Coupling Geometries

Equations (43)

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E1(x, z)=U01(x)[exp(-jβ01z)+R0 exp(+jβ01z)]+0+R(q)Ψ1(x, q)exp[+jγ(q)z]dq,
γ(q)=(k02n312-q2)1/2(0<q<+)
E2(x, z)=T0U02(x)exp(-jβ02z)+0+T(q)Ψ2(x, q)exp[-jδ(q)z]dq,
R0=-1+-+E(x)U01(x)dx,
R(q)=-+E(x)Ψ1(x, q)dx,
T0=-+E(x)U02(x)dx,
T(q)=-+E(x)Ψ2(x, q)dx.
2β01U01(x)=-+E(x)β01U01(x)U01(x)+0+γ(q)Ψ1(x, q)Ψ1(x, q)dqdx+-+E(x)β02U02(x)U02(x)+0+δ(q)Ψ2(x, q)Ψ2(x, q)dqdx.
-k0n3iU0i(x)U0i(x)+0+Ψi(x, q)Ψi(x, q)dq
+k0n3iδ(x-x)=0(i=1, 2).
E(x)=E(0)(x)-1k0(n31+n32)-+E(x)[K1(x, x)+K2(x, x)]dx,
E(0)(x)=2β01k0(n31+n32)U01(x),
Ki(x, x)=(β0i-k0n3i)U0i(x)U0i(x)+0+[ξ(q)-k0n3i]×Ψi(x, q)Ψi(x, q)dq(i=1, 2),
ξ(q)=γ(q)fori=1δ(q)fori=2.
E(1)(x)=E(0)(x)-2β01k02(n31+n32)2×(β01-k0n31)U01(x)+(β02-k0n32)×UU12U02(x)+0+dq[δ(q)-k0n32]×UΨ12(q)Ψ2(x, q),
UUij=-+U0i(x)U0j(x)dx(ij,i, j=1, 2),
UΨij(q)=-+U0i(x)Ψj(x, q)dx
(ij,i, j=1, 2).
R0(0)=-1+2β01k0(n31+n32),R(0)(q)=0,
T0(0)=2β01k0(n31+n32)UU12,
T(0)(q)=2β01k0(n31+n32)UΨ12(q).
R0(1)=R0(0)-2β01(β01-k0n31)k02(n31+n32)2-2β01(β02-k0n32)k02(n31+n32)2UU122-2β01k02(n31+n32)20+dq[δ(q)-k0n32]×UΨ122(q),
R(1)(q)=R(0)(q)-2β01k02(n31+n32)2(β02-k0n32)×UU12UΨ21(q)-2β01k02(n31+n32)2×0+dq[δ(q)-k0n32]×UΨ12(q)ΨΨ12(q, q),
T0(1)=T0(0)-2β01(β01+β02-k0n31-k0n32)k02(n31+n32)2×UU122,
T(1)(q)=T(0)(q)-2β01(β01-k0n31)k02(n31+n32)2UΨ12(q)-2β01[δ(q)-k0n32]k02(n31+n32)2UΨ12(q),
ΨΨij(q, ξ)=-+Ψi(x, q)Ψj(x, ξ)dx,
(ij,i, j=1, 2).
Prad=Prefl,rad+Ptran,rad=0+|R(q)|2 dq+0+|T(q)|2 dq,
U0(x)=β02ωμ0×A exp[-h3(x-D)],D<x<+B cos(h2x)+Γ sin(h2x),d<x<DΔ cosh(h1x),-d<x<dB cos(h2x)-Γ sin(h2x),-D<x<-dA exp[+h3(x+D)],-<x<-D,
h1 tanh(h1d)=-h2{tan[h2(D-d)+tan-1(h2/h3)]}-1,
B=Acos(h2D)+sin(h2D)h3h2,
Γ=Asin(h2D)-cos(h2D)h3h2,
Δ=A1cosh(h1d)cos[h2(D-d)]+h3h2sin[h2(D-d)],
A=β0ωμ0 1cosh2(h1d)cos[h2(D-d)]+h3h2×sin[h2(D-d)]2sinh(2h1d)4h1+0.5d+12(D-d)1+h32h22+h32h2+sin[2h2(D-d)]4h2-h32h2cos[2h2(D-d)]-h324h23sin[2h2(D-d)]+12h3-1/2.
Ψ(x, q)=1π×cos[q(x-D)+b],D<x<+B(q)cos[σ(x-d)+a],d<x<DC(q)cos(ρx),-d<x<dB(q)cos[σ(x+d)-a],-D<x<-dcos[q(x+D)-b],-<x<-D,
tan a=ρσtan(ρd)
tan b=σqtan[σ(D-d)+a],
B(q)=cos bcos[σ(D-d)+a],
C(q)=B(q)(cos a)/cos(ρd).
-+U02(x)dx=1,
-+Ψ(x, q)Ψ(x, q)dx=δ(q-q),
-+U0(x)Ψ(x, q)dx=0,
U0(x)U0(x)+0+Ψ(x, q)Ψ(x, q)dq=δ(x-x).

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