Abstract

The scattering properties of an abruptly ended buried slab waveguide for both TE and TM modes are examined by an improved iteration technique that is based on the integral equation method with “accelerating” parameters. The waveguide is considered a symmetrical slab, for which the weakly guiding conditions are invalid, and it is embedded in a different dielectric material. The tangential electric field distribution on the terminal plane, the reflection coefficient of the first TE and TM guided modes, and the far-field radiation pattern are computed. Numerical results are presented for several ended waveguides, while special attention is given to the far-field radiation pattern rotation and the terminal field distributions.

© 2004 Optical Society of America

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  1. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
    [CrossRef]
  2. R. M. Knox, P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic Press, Brooklyn, N.Y., 1970), pp. 497–516.
  3. T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
    [CrossRef]
  4. K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
    [CrossRef]
  5. A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
    [CrossRef]
  6. H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
    [CrossRef]
  7. 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]
  8. M. Munowitz, D. J. Vezzetti, “Numerical modeling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. LT-10, 1570–1573 (1992).
    [CrossRef]
  9. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
    [CrossRef] [PubMed]
  10. 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. LT-6, 590–600 (1988).
    [CrossRef]
  11. M. S. Stern, “Semi-vectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 333–338 (1988).
    [CrossRef]
  12. S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
    [CrossRef]
  13. W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
    [CrossRef]
  14. C. J. Smartt, T. M. Benson, P. C. Kendall, “Free-space radiation mode method for the analysis of propagation in optical waveguide devices,” IEE Proc. Optoelectron. 140, 56–61 (1993).
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  15. F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Research Studies Press, Hertfordshire, UK, 1996).
  16. A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half-space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 43–51 (1999).
    [CrossRef]
  17. D. N. Chien, M. Tanaka, K. Tanaka, “Numerical simulation of an arbitrarily ended asymmetrical slab waveguide by guided-mode extracted integral equations,” J. Opt. Soc. Am. A 19, 1649–1657 (2002).
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  18. I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  19. I. G. Tigelis, A. B. Manenkov, “Analysis of mode scattering from an abruptly ended dielectric slab waveguide by an accelerated iteration technique,” J. Opt. Soc. Am. A 17, 2249–2259 (2000).
    [CrossRef]
  20. A. B. Manenkov, G. P. Latsas, I. G. Tigelis, “Scattering of the transverse magnetic modes from an abruptly endedstrongly asymmetrical slab waveguide by an accelerated integral equation technique,” J. Opt. Soc. Am. A 18, 3110–3119 (2001).
    [CrossRef]
  21. H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. QE-30, 2827–2835 (1994).
  22. Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y-junctions and directional couplers,” IEEE J. Quantum Electron. QE-25, 1200–1208 (1989).
    [CrossRef]
  23. S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low-loss applications,” J. Lightwave Technol. LT-11, 567–575 (1993).
    [CrossRef]
  24. D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. MTT-40, 2235–2242 (1992).
    [CrossRef]
  25. I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
    [CrossRef]
  26. I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
    [CrossRef]
  27. A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Spectral method applied to design spot size converters,” Electron. Lett. 33, 2121–2123 (1997).
    [CrossRef]
  28. A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Facet reflectivity in the presence of a diffracting corner,” Electron. Lett. 31, 327–335 (1999).
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    [CrossRef]
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    [CrossRef]
  34. L. Lewin, Theory of Waveguides (Newness-Butterworths, London, 1975), Chap. 9.
  35. A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. II: solution by variational method,” Radiophys. Quantum Electron. 25, 1050–1055 (1982).
    [CrossRef]
  36. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J. Optoelectron. 139, 101–104 (1992).
    [CrossRef]
  37. Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Pade approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
    [CrossRef]
  38. M. Abramowitz, I. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972), p. 887.

2002 (1)

2001 (1)

2000 (1)

1999 (3)

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half-space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 43–51 (1999).
[CrossRef]

I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
[CrossRef]

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Facet reflectivity in the presence of a diffracting corner,” Electron. Lett. 31, 327–335 (1999).

1997 (2)

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Spectral method applied to design spot size converters,” Electron. Lett. 33, 2121–2123 (1997).
[CrossRef]

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Pade approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

1994 (3)

I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
[CrossRef]

I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
[CrossRef]

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. QE-30, 2827–2835 (1994).

1993 (2)

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low-loss applications,” J. Lightwave Technol. LT-11, 567–575 (1993).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Free-space radiation mode method for the analysis of propagation in optical waveguide devices,” IEE Proc. Optoelectron. 140, 56–61 (1993).
[CrossRef]

1992 (4)

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

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

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

A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J. Optoelectron. 139, 101–104 (1992).
[CrossRef]

1991 (1)

A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides (fibres),” Opt. Quantum Electron. 23(5), 621–632 (1991).
[CrossRef]

1989 (2)

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

S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
[CrossRef]

1988 (2)

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. LT-6, 590–600 (1988).
[CrossRef]

M. S. Stern, “Semi-vectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 333–338 (1988).
[CrossRef]

1984 (1)

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]

1982 (2)

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

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. II: solution by variational method,” Radiophys. Quantum Electron. 25, 1050–1055 (1982).
[CrossRef]

1979 (2)

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

1978 (2)

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

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

1970 (1)

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

1957 (1)

C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. 3, 100–109 (1957).
[CrossRef]

Angulo, C. M.

C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. 3, 100–109 (1957).
[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]

Baets, R.

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 electro-optic modulators,” IEEE Trans. Microwave Theory Tech. MTT-40, 2235–2242 (1992).
[CrossRef]

Benson, T. M.

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Facet reflectivity in the presence of a diffracting corner,” Electron. Lett. 31, 327–335 (1999).

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half-space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 43–51 (1999).
[CrossRef]

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Spectral method applied to design spot size converters,” Electron. Lett. 33, 2121–2123 (1997).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Free-space radiation mode method for the analysis of propagation in optical waveguide devices,” IEE Proc. Optoelectron. 140, 56–61 (1993).
[CrossRef]

Burke, S. V.

S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
[CrossRef]

Chang, H. C.

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Pade approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

Chaudhuri, S. K.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low-loss applications,” J. Lightwave Technol. LT-11, 567–575 (1993).
[CrossRef]

Chaundhuri, S. K.

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Chien, D. N.

Chiou, Y. P.

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Pade approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

Chow, Y. L.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low-loss applications,” J. Lightwave Technol. LT-11, 567–575 (1993).
[CrossRef]

Fernandez, F.

F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Research Studies Press, Hertfordshire, UK, 1996).

Goss, A.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low-loss applications,” J. Lightwave Technol. LT-11, 567–575 (1993).
[CrossRef]

Hamid, M.

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

Hardy, A.

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

Harlow, M. J.

I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
[CrossRef]

I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
[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. LT-6, 590–600 (1988).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett. 4, 148–151 (1992).
[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. LT-6, 590–600 (1988).
[CrossRef]

Inagaki, S.

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Ittipiboon, A.

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

Jones, D. S.

D. S. Jones, Theory of Electromagnetism (Macmillan, New York, 1964), Chap. 8.

Kendall, P. C.

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Facet reflectivity in the presence of a diffracting corner,” Electron. Lett. 31, 327–335 (1999).

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half-space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 43–51 (1999).
[CrossRef]

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Spectral method applied to design spot size converters,” Electron. Lett. 33, 2121–2123 (1997).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Free-space radiation mode method for the analysis of propagation in optical waveguide devices,” IEE Proc. Optoelectron. 140, 56–61 (1993).
[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 electro-optic modulators,” IEEE Trans. Microwave Theory Tech. MTT-40, 2235–2242 (1992).
[CrossRef]

Knox, R. M.

R. M. Knox, P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic Press, Brooklyn, N.Y., 1970), pp. 497–516.

Kumagai, N.

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Lagasse, P. E.

Latsas, G. P.

Lealman, I. F.

I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
[CrossRef]

I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
[CrossRef]

Lewin, L.

L. Lewin, Theory of Waveguides (Newness-Butterworths, London, 1975), Chap. 9.

Lin, H.-B.

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. QE-30, 2827–2835 (1994).

Lu, Y.

F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Research Studies Press, Hertfordshire, UK, 1996).

Manenkov, A. B.

A. B. Manenkov, G. P. Latsas, I. G. Tigelis, “Scattering of the transverse magnetic modes from an abruptly endedstrongly asymmetrical slab waveguide by an accelerated integral equation technique,” J. Opt. Soc. Am. A 18, 3110–3119 (2001).
[CrossRef]

I. G. Tigelis, A. B. Manenkov, “Analysis of mode scattering from an abruptly ended dielectric slab waveguide by an accelerated iteration technique,” J. Opt. Soc. Am. A 17, 2249–2259 (2000).
[CrossRef]

I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
[CrossRef]

A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J. Optoelectron. 139, 101–104 (1992).
[CrossRef]

A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides (fibres),” Opt. Quantum Electron. 23(5), 621–632 (1991).
[CrossRef]

A. B. Manenkov, “Propagation of a surface wave along a dielectric waveguide with an abrupt change of parameters. II: solution by variational method,” Radiophys. Quantum Electron. 25, 1050–1055 (1982).
[CrossRef]

Marcuse, D.

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

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

Marom, E.

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

Morishita, K.

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Munowitz, M.

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

Perrin, S. D.

I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
[CrossRef]

I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
[CrossRef]

Rivers, L. J.

I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
[CrossRef]

I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
[CrossRef]

Robertson, M. J.

I. F. Lealman, L. J. Rivers, M. J. Harlow, S. D. Perrin, M. J. Robertson, “1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 857–859 (1994).
[CrossRef]

Rozzi, T. E.

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

Safavi-Naeini, S.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low-loss applications,” J. Lightwave Technol. LT-11, 567–575 (1993).
[CrossRef]

Seltzer, C. P.

I. F. Lealman, C. P. Seltzer, L. J. Rivers, M. J. Harlow, S. D. Perrin, “Low-threshold current 1.56-μm InGaAsP/InP tapered active layer multiquantum-well laser with improved coupling to cleaved singlemode fiber,” Electron. Lett. 30, 973–975 (1994).
[CrossRef]

Sewell, P.

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half-space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 43–51 (1999).
[CrossRef]

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Facet reflectivity in the presence of a diffracting corner,” Electron. Lett. 31, 327–335 (1999).

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A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half-space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 43–51 (1999).
[CrossRef]

A. Vucovic, P. Sewell, T. M. Benson, P. C. Kendall, “Facet reflectivity in the presence of a diffracting corner,” Electron. Lett. 31, 327–335 (1999).

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[CrossRef]

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H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. QE-30, 2827–2835 (1994).

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H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. QE-30, 2827–2835 (1994).

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[CrossRef]

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[CrossRef]

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. QE-30, 2827–2835 (1994).

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

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[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. LT-6, 590–600 (1988).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of an abruptly ended buried slab waveguide.

Fig. 2
Fig. 2

Power reflectivity of the first TE and TM guided modes versus H for a slab waveguide with Δ12=10% (n3=n1=3.24), n4=n0=1, and D=0.5 μm.

Fig. 3
Fig. 3

Power reflectivity of the first TE guided mode versus H for different values of D for a slab waveguide with Δ12=10% (n3=n1=3.24), and n4=n0=1.

Fig. 4
Fig. 4

Power reflectivity of the first TE guided mode versus D for different values of H for a slab waveguide with Δ12=10% (n3=n1=3.24), and n4=n0=1.

Fig. 5
Fig. 5

Power reflectivity of the first TM guided mode versus D for different values of H for a slab waveguide with Δ12=10% (n3=n1=3.24), and n4=n0=1.

Fig. 6
Fig. 6

The tangential electric-field distribution of the first TM guided mode at the terminal plane z=0 for different values of D for a slab waveguide with H=0.25 μm, Δ12=10% (n3=n1=3.24), and n4=n0=1.

Fig. 7
Fig. 7

Power reflectivity of the first TM guided mode versus H for different values of D and n4 for a slab waveguide with Δ12=10% (n3=n1=3.24), and n0=1.

Fig. 8
Fig. 8

Power reflectivity of the dominant TE and TM modes versus H for a slab waveguide with constant and with parabolic refractive-index profile.

Equations (44)

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U0(x)=A4exp(-h4x), 0<x<+A3cosh[h3(x+D)]+B3sinh[h3(x+D)],-D<x<0A2cos[h2(x+H+D)]+B2sin[h2(x+H+D)],-(H+D)<x<-DA1exp[h1(x+H+D)],-<x<-(H+D),
Ψ(x, ρ)=A4(ρ)cos qx+A4(ρ)Fr(ρ)sin qx, 0<x<+A3(ρ)cos ρ(x+D)+B3(ρ)sin ρ(x+D),-D<x<0A2(ρ)cos σ(x+H+D)+B2(ρ)sin σ(x+H+D),-(H+D)<x<-DA1(ρ)cos ρ(x+H+D)+B1(ρ)sin ρ(x+H+D),-<x<-(H+D),
U0(x)U0(x)+m=12ρm+dρΨm(x, ρ)Ψm(x, ρ)
=δ(x-x).
Φ1(x, z)=U0(x)exp(-jβ0z)+R0U0(x)exp(jβ0z)+m=12ρm+dρRm(ρ)Ψm(x, ρ)exp[+jβ(ρ)z],
Φ2(x, z)=l=120+dsTl(s)ϕl(x, s)exp[-jγ(s)z],
-+dxϕi(x, s)ϕj(x, s)=δijδ(s-s),i, j=1, 2,
l=120+dsϕl(x, s)ϕl(x, s)=δ(x-x).
E(x)=E0(x)+-+dxE(x)K(x, x),
E0(x)=2β0(β¯+γ¯) U0(x),
K(x, x)
=1(β¯+γ¯)(β¯-β0)U0(x)U0(x)+m=12ρm+dρ[β¯-β(ρ)]Ψm(x, ρ)Ψm(x, ρ)+l=120+ds[γ¯-γ(s)]ϕl(x, s)ϕl(x, s),
β¯=β0,γ¯=0+dsγ(s)l=12Uϕl(s)2,
E1(x)=E0(x)+2β0(β¯+γ¯)20+ds[γ¯-γ(s)]×l=12Uϕl(s)ϕl(x, s),
E2(x)=E1(x)+2β0(β¯+γ¯)30+ds[γ¯-γ(s)]×l=12Uϕl(s)ρm+dρ[β¯-β(ρ)]×m=12Ψm(x, ρ)Ψmϕl(ρ, s)+2β0(β¯+γ¯)30+ds[γ¯-γ(s)]2×l=12Uϕl(s)ϕl(x, s).
R0(0)=-1+2β0(β¯+γ¯),
Tl(0)(s)=2β0(β¯+γ¯) Uϕl(s).
R0(1)=R0(0),
Tl(1)(s)=Tl(0)(s)+2β0(β¯+γ¯)2 [γ¯-γ(s)]×Uϕl(s).
R0(2)=R0(1)+2β0(β¯+γ¯)30+ds[γ¯-γ(s)]2×l=12Uϕl(s)2,
Tl(2)(s)=Tl(1)(s)+2β0(β¯+γ¯)3 [γ¯-γ(s)]2Uϕl(s)+2β0(β¯+γ¯)30+ds[γ¯-γ(s)]×l=12Uϕl(s)ρm+dρ[β¯-β(ρ)]×m=12Ψmϕl(ρ, s)Ψmϕl(ρ, s).
n(x)=n4,0<x<+n3=n1,-D<x<0n2,-(H+D)<x<-Dn1,-<x<-(H+D).
-+dx Ψm(x, ρ)Ψk(x, ρ)n2(x)=δmkδ(ρ-ρ),m, k=1, 2.
U0(x)U0(x)+m=12ρm+dρΨm(x, ρ)Ψm(x, ρ)
=n2(x)δ(x-x).
-+dxϕi(x, s)ϕj(x, s)=n02δijδ(s-s),i, j=1, 2,
0+dsl=12ϕl(x, s)ϕl(x, s)=n02δ(x-x).
E0(x)=2U0(x)p(x),
E1(x)=E0(x)+2p(x)m=12ρm+dρ1β¯-1β(ρ)Ψm(x, ρ)×U0Ψm(ρ)p(x)+0+ds1γ¯-1γ(s)l=12ϕl(x, s)U0ϕl(s)p(x),
E2(x)=E1(x)+2p(x)m=12ρm+dρ1β¯-1β(ρ)×U0Ψm(ρ)p(x)×m=12ρm+dρ1β¯-1β(ρ)Ψm(x, ρ)×ΨmΨm(ρ, ρ)p(x)+m=12ρm+dρ1β¯-1β(ρ)U0Ψm(ρ)p(x)×0+ds1γ¯-1γ(s)l=12ϕl(x, s)Ψmϕl(ρ, s)p(x)+0+ds1γ¯-1γ(s)×l=12U0ϕl(s)p(x)m=12ρm+dρ1β¯-1β(ρ)Ψm(x, ρ)×Ψmϕl(ρ, s)p(x)+0+ds1γ¯-1γ(s)l=12U0ϕl(s)p(x)×0+ds1γ¯-1γ(s)l=12ϕl(x, s)×ϕlϕl(s, s)p(x),
p(x)=n2(x)/β¯+n02/γ¯.
β¯=β0,γ¯=0+dsl=12Uϕl(s)q(x)20+ds 1γ(s)l=12Uϕl(s)q(x)2,
q(x)=n2(x)/β0+n02/(k0n0)
R0(0)=1-2β0U0U0p(x),
Tl(0)(s)=2γ(s)U0ϕl(s)p(x).
R0(1)=R0(0)-2β0m=12ρm+dρ1β¯-1β(ρ)×U0Ψm(ρ)p(x)2+0+ds1γ¯-1γ(s)l=12U0ϕl(s)p(x)2,
Tl(1)(s)=Tl(0)(s)+2γ(s)m=12ρm+dρ1β¯-1β(ρ)×U0Ψm(ρ)p(x)Ψmϕl(ρ, s)p(x)+0+ds1γ¯-1γ(s)l=12U0ϕl(s)p(x)×ϕlϕl(s, s)p(x).
R0(2)=R0(1)-2β0m=12ρm+dρ1β¯-1β(ρ)U0Ψm(ρ)p(x)×m=12ρm+dρ1β¯-1β(ρ)U0Ψm(ρ)p(x)×ΨmΨm(ρ, ρ)p(x)+2m=12ρm+dρ1β¯-1β(ρ)×U0Ψm(ρ)p(x)0+ds1γ¯-1γ(s)l=12U0ϕl(s)p(x)×Ψmϕl(ρ, s)p(x)+0+ds1γ¯-1γ(s)×l=12U0ϕl(s)p(x)0+ds1γ¯-1γ(s)×l=12U0ϕl(s)p(x)ϕlϕl(s, s)p(x),
Tl(2)(s)=Tl(1)(s)+2γ(s)m=12ρm+dρ1β¯-1β(ρ)×U0Ψm(ρ)p(x)×m=12ρm+dρ1β¯-1β(ρ)ΨmΨm(ρ, ρ)p(x)×Ψmϕl(ρ, s)p(x)+m=12ρm+dρ1β¯-1β(ρ)×U0Ψm(ρ)p(x)0+ds1γ¯-1γ(s)×l=12Ψmϕl(ρ, s)p(x)ϕlϕl(s, s)p(x)+0+ds1γ¯-1γ(s)l=12U0ϕl(s)p(x)×m=12ρm+dρ1β¯-1β(ρ)Ψmϕl(ρ, s)p(x)×Ψmϕl(ρ, s)p(x)+0+ds1γ¯-1γ(s)×l=12U0ϕl(s)p(x)0+ds1γ¯-1γ(s)×l=12ϕlϕl(s, s)p(x)ϕlϕl(s, s)p(x).
Ey(r, θ)=2k0n0r1/2exp(-jk0n0r+jπ/4)k0n0cos θl=12Tl(s=k0n0sin θ),
Eθ(r, θ, ϕ)=cos θ sin ϕEy(r, θ),
Eϕ(r, θ, ϕ)=cos ϕEy(r, θ),
1-R0(υ)1+R0(υ)=1β0l=120+dsγ(s)U0(x), ϕl(x, s)2.
n(x)=nS[1-8Δ(x/H)2]1/2,

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