Abstract

We deal with the scattering phenomenon from an abruptly terminated asymmetrical slab waveguide for the case of transverse magnetic (TM) modes. The analysis uses both the integral equation method and the variational technique. The reflection coefficient of the dominant TM guided mode and the far-field radiation pattern are computed, and the discontinuity of the electric field distribution on the core–clad interface is exhibited. Numerical results are presented for several cases of abruptly ended waveguides, including the three-layer slab guide and the structure with variable profile of the refractive index.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
    [CrossRef]
  2. G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
    [CrossRef]
  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. C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
    [CrossRef]
  9. T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
    [CrossRef]
  10. C. Vassallo, “Reflectivity of multi-dielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
    [CrossRef]
  11. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).
  12. Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Pade approximants,” IEEE Photonics Technol. Lett. 9, 964–966 (1997).
    [CrossRef]
  13. 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]
  14. A. B. Manenkov, “Step discontinuities in dielectric waveguides (fibres),” Opt. Quantum Electron. 22, 65–76 (1990).
    [CrossRef]
  15. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J 139, 101–104 (1992).
  16. T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
    [CrossRef]
  17. A. B. Manenkov, “Comparison of approximate methods of computing diffraction of waves at diameter discontinuity in a dielectric waveguide,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 28, 743–752 (1985).
  18. 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]
  19. 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]
  20. I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  21. D. Marcuse, Theory of Dielectric Optical Waveguide, 2nd ed. (Academic, London, 1991), Chap. 1.
  22. L. Lewin, Theory of Waveguides (Newness-Butterworth, London, 1975).
  23. A. B. Manenkov, “Accuracy of approximations for fibre discontinuity analysis,” Opt. Quantum Electron. 23, 81–90 (1991).
    [CrossRef]
  24. A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides (fibres),” Opt. Quantum Electron. 23, 621–632 (1991).
    [CrossRef]
  25. A. B. Manenkov, “Irregular magneto-optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 906–910 (1981).
    [CrossRef]
  26. A. D. Vasil’ev, A. B. Manenkov, “Diffraction of the surface wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
    [CrossRef]
  27. T. Y. Na, Computational Methods in Engineering Boundary Value Problems (Academic, New York, 1979).
  28. F. K. Reinhart, I. Hayashi, M. B. Panish, “Mode reflectivity and waveguide properties of double-heterostructure injection lasers,” J. Appl. Phys. 42, 4466–4479 (1971).
    [CrossRef]
  29. J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
    [CrossRef]
  30. J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
    [CrossRef]
  31. C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 61–62 (1988).
    [CrossRef]
  32. A. G. Failla, G. P. Bava, I. Montrosset, “Structural design criteria for polarization insensitive semiconductor optical amplifiers,” J. Lightwave Technol. 8, 302–308 (1990).
    [CrossRef]
  33. Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–429 (1991).
    [CrossRef]
  34. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
    [CrossRef]

1999 (1)

1997 (1)

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

1993 (1)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

1992 (1)

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

1991 (3)

A. B. Manenkov, “Accuracy of approximations for fibre discontinuity analysis,” Opt. Quantum Electron. 23, 81–90 (1991).
[CrossRef]

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

Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–429 (1991).
[CrossRef]

1990 (2)

A. G. Failla, G. P. Bava, I. Montrosset, “Structural design criteria for polarization insensitive semiconductor optical amplifiers,” J. Lightwave Technol. 8, 302–308 (1990).
[CrossRef]

A. B. Manenkov, “Step discontinuities in dielectric waveguides (fibres),” Opt. Quantum Electron. 22, 65–76 (1990).
[CrossRef]

1988 (3)

1987 (2)

A. D. Vasil’ev, A. B. Manenkov, “Diffraction of the surface wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
[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]

1985 (1)

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

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 (1)

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]

1981 (2)

A. B. Manenkov, “Irregular magneto-optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 906–910 (1981).
[CrossRef]

J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
[CrossRef]

1980 (1)

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[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]

1976 (1)

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

1974 (1)

J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
[CrossRef]

1972 (2)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
[CrossRef]

T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

1971 (1)

F. K. Reinhart, I. Hayashi, M. B. Panish, “Mode reflectivity and waveguide properties of double-heterostructure injection lasers,” J. Appl. Phys. 42, 4466–4479 (1971).
[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. AP-5, 100–109 (1957).
[CrossRef]

Adams, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Angulo, C. M.

C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. AP-5, 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]

Bava, G. P.

A. G. Failla, G. P. Bava, I. Montrosset, “Structural design criteria for polarization insensitive semiconductor optical amplifiers,” J. Lightwave Technol. 8, 302–308 (1990).
[CrossRef]

Boyd, T. J. M.

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Brooke, G. H.

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

Butler, J. K.

J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
[CrossRef]

Buus, J.

J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
[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]

Chang, H. C.

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

Chew, W. C.

Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–429 (1991).
[CrossRef]

Chiou, Y. P.

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

Failla, A. G.

A. G. Failla, G. P. Bava, I. Montrosset, “Structural design criteria for polarization insensitive semiconductor optical amplifiers,” J. Lightwave Technol. 8, 302–308 (1990).
[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]

Hayashi, I.

F. K. Reinhart, I. Hayashi, M. B. Panish, “Mode reflectivity and waveguide properties of double-heterostructure injection lasers,” J. Appl. Phys. 42, 4466–4479 (1971).
[CrossRef]

Ikegami, T.

T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[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]

Kendall, P. C.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Kharadly, M. M. Z.

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

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]

Lewin, L.

L. Lewin, Theory of Waveguides (Newness-Butterworth, London, 1975).

Liu, Q.

Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–429 (1991).
[CrossRef]

Manenkov, A. B.

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 139, 101–104 (1992).

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

A. B. Manenkov, “Accuracy of approximations for fibre discontinuity analysis,” Opt. Quantum Electron. 23, 81–90 (1991).
[CrossRef]

A. B. Manenkov, “Step discontinuities in dielectric waveguides (fibres),” Opt. Quantum Electron. 22, 65–76 (1990).
[CrossRef]

A. D. Vasil’ev, A. B. Manenkov, “Diffraction of the surface wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
[CrossRef]

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

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]

A. B. Manenkov, “Irregular magneto-optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 906–910 (1981).
[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 Waveguide, 2nd ed. (Academic, London, 1991), Chap. 1.

Meixner, J.

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
[CrossRef]

Montrosset, I.

A. G. Failla, G. P. Bava, I. Montrosset, “Structural design criteria for polarization insensitive semiconductor optical amplifiers,” J. Lightwave Technol. 8, 302–308 (1990).
[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]

Moshkun, I.

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Na, T. Y.

T. Y. Na, Computational Methods in Engineering Boundary Value Problems (Academic, New York, 1979).

Panish, M. B.

F. K. Reinhart, I. Hayashi, M. B. Panish, “Mode reflectivity and waveguide properties of double-heterostructure injection lasers,” J. Appl. Phys. 42, 4466–4479 (1971).
[CrossRef]

Reinhart, F. K.

F. K. Reinhart, I. Hayashi, M. B. Panish, “Mode reflectivity and waveguide properties of double-heterostructure injection lasers,” J. Appl. Phys. 42, 4466–4479 (1971).
[CrossRef]

Roberts, D. A.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Robertson, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Robson, P. N.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

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]

Stephenson, I. M.

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Tigelis, I. G.

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]

Vasil’ev, A. D.

A. D. Vasil’ev, A. B. Manenkov, “Diffraction of the surface wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
[CrossRef]

Vassallo, C.

C. Vassallo, “Reflectivity of multi-dielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

Yajima, H.

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

Zoroofchi, J.

J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
[CrossRef]

Bell Syst. Tech. J. (1)

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

Electron. Lett. (2)

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 61–62 (1988).
[CrossRef]

IEE Proc. J (2)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

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

IEEE J. Quantum Electron. (4)

T. Ikegami, “Reflectivity of mode at facet and oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

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

J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
[CrossRef]

J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

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

IEEE Trans. Antennas Propag. (1)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (6)

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]

A. B. Manenkov, “Irregular magneto-optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 906–910 (1981).
[CrossRef]

Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–429 (1991).
[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]

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]

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]

IRE Trans. Antennas Propag. (1)

C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
[CrossRef]

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

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

J. Appl. Phys. (1)

F. K. Reinhart, I. Hayashi, M. B. Panish, “Mode reflectivity and waveguide properties of double-heterostructure injection lasers,” J. Appl. Phys. 42, 4466–4479 (1971).
[CrossRef]

J. Lightwave Technol. (1)

A. G. Failla, G. P. Bava, I. Montrosset, “Structural design criteria for polarization insensitive semiconductor optical amplifiers,” J. Lightwave Technol. 8, 302–308 (1990).
[CrossRef]

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

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

Opt. Quantum Electron. (4)

A. B. Manenkov, “Accuracy of approximations for fibre discontinuity analysis,” Opt. Quantum Electron. 23, 81–90 (1991).
[CrossRef]

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

A. B. Manenkov, “Step discontinuities in dielectric waveguides (fibres),” Opt. Quantum Electron. 22, 65–76 (1990).
[CrossRef]

T. J. M. Boyd, I. Moshkun, I. M. Stephenson, “Radiation losses due to discontinuities in asymmetric three-layer optical waveguides,” Opt. Quantum Electron. 12, 143–158 (1980).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

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

Radiophys. Quantum Electron. (2)

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]

A. D. Vasil’ev, A. B. Manenkov, “Diffraction of the surface wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
[CrossRef]

Other (3)

T. Y. Na, Computational Methods in Engineering Boundary Value Problems (Academic, New York, 1979).

D. Marcuse, Theory of Dielectric Optical Waveguide, 2nd ed. (Academic, London, 1991), Chap. 1.

L. Lewin, Theory of Waveguides (Newness-Butterworth, London, 1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometry of an abruptly ended asymmetrical slab waveguide.

Fig. 2
Fig. 2

(a) Comparison of the results obtained by using the two methods presented in this paper with others appearing in previously published works for an abruptly terminated symmetrical slab waveguide with λ0=0.86 µm, n2=3.6, n0=1, and n1=n3=3.492 (Δ12=Δ32=3%, Δ13=0%). (b) Similar to (a) but applies to the problem with the parameters λ0=0.86 µm, n2=3.6, n0=1, and n1=n3=3.24 (Δ12=Δ32=10%, Δ13=0%).

Fig. 3
Fig. 3

(a) Difference between the first- and second-order solutions of the integral equation method for an abruptly terminated symmetrical slab waveguide with λ0=0.9 µm, n2=3.6, n0=1, and Δ12=Δ32=0.1%, 0.5%, 1%, and 5%. (b) Similar to (a) but for the difference between the second- and third-order solutions.

Fig. 4
Fig. 4

(a) Variation of the reflectivity |R0|2 of the dominant TM guided mode with the core width D for an abruptly terminated asymmetrical slab waveguide with λ0=0.9 µm, n2=3.61, n0=1, Δ12=1%, and Δ32=10%. (b) Similar to (a) but applies to the problem with the parameters λ0=0.9 µm, n2=3.61, n0=1, Δ12=5%, and Δ32=10%.

Fig. 5
Fig. 5

(a) Variation of the magnitude of the transverse electric field distribution |E(x)| (solid curve) and the transverse electric field distribution of the guided mode, |(1+R0) U0(x)|/n2(x) (dashed curve), at the plane z=0 with the normalized transverse distance x/D for a slab waveguide with λ0=0.9 µm, n2=3.61, D=0.5 µm, n0=1, and Δ12=Δ32=1%. Also given is the variation of |E(x)| (dotted curve) and |(1+R0)U0(x)|/n2(x) (circles) for the same geometry but with Δ12=1%, Δ32=10%. (b) Variation of the magnitude of the transverse electric field distribution |E(x)| (solid curve) and the transverse electric field distribution of the guided mode, |(1+R0)U0(x)|/n2(x) (dashed curve), at the plane z=0 with the normalized transverse distance x/D for a slab waveguide with λ0=0.9 µm, n2=3.61, D=0.25 µm, n0=1, and Δ12=Δ32=5%. Also given is the variation of |E(x)| (dotted curve) and |(1+R0)U0(x)|/n2(x) (circles) for the same geometry but with Δ12=5%,Δ32=10%.

Fig. 6
Fig. 6

(a) Normalized radiation pattern for an abruptly terminated slab waveguide with λ0=0.9µm, n2=3.61, D=0.5 µm, n0=1, and Δ12=Δ32=1% (solid curve) and Δ12=1%, Δ32=10% (dashed curve). (b) Similar to (a) but applies to the problem with the parameters λ0=0.9µm, n2=3.61, D=0.25 µm, n0=1, and Δ12=Δ32=5% (solid curve) and Δ12=5%, Δ32=10% (dashed curve).

Fig. 7
Fig. 7

(a) Variation of the refractive index n(x), which is assumed to change linearly from n1 to n2 on the interval (-D/2,xl). (b) Variation of the reflectivity |R0|2 of the dominant TM guided mode with the core width D for an abruptly terminated symmetrical slab with λ0=0.9 µm, n2=3.61, Δ12=Δ32=10%, and n0=1 and with linearly varying refractive index from n1 to n2 on the interval (-D/2, xl) as shown in (a). (c) Variation of the reflectivity |R0|2 of the dominant TM guided mode with the core width D for an abruptly terminated asymmetrical slab with λ0=0.9 µm, n2=3.61, Δ12=5%, Δ32=10%, and n0=1 and with linearly varying refractive index from n1 to n2 on the interval (-D/2, xl) as shown in (a).

Tables (1)

Tables Icon

Table 1 Convergence of the Reflection Coefficient for an Abruptly Ended Waveguidea

Equations (63)

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

Hyg(x, z)=ω0β0U0(x)exp(-jβ0z),
U0(x)=Aexp[-h3(x-D/2)],xD/2cos[h2(x-D/2)]-(h3/h2)(n2/n3)2 sin[h2(x-D/2)],|x|D/2,[cos(h2D)+(h3/h2)(n2/n3)2 sin(h2D)]exp[h1(x+D/2)],x-D/2
h22n24-h1h3n12n32tan(h2D)=h2n22h1n12+h3n32
h2D=cos-1h2/n22V21+cos-1h2/n22V23,
V21=h22n24+h12n141/2,V23=h22n24+h32n341/2.
-+U02(x)n2(x)dx=1,
n(x)=n3,x>D/2n2,|x|<D/2.n1,x<-D/2
-+Ψm(x, ρ)Ψk(x, ρ)n2(x)dx=δmkδ(ρ-ρ),
m, k=1, 2,
-+U0(x)Ψm(x, ρ)n2(x)dx=0,m=1, 2,
U0(x)U0(x)+m=12ρm+Ψm(x, ρ)Ψm(x, ρ)dρ
=n2(x)δ(x-x),
HyI(x, z)=ω0β0U0(x)[exp(-jβ0z)-R0 exp(+jβ0z)]-m=12ρm+ω0β(ρ)Rm(ρ)Ψm(x, ρ)×exp[+jβ(ρ)z]dρ,
β2(ρ)=k02n12-ρ2(ρm<ρ<+).
HyII(x, z)=l=120+ω0γ(s)Tl(s)φl(x, s)exp[-jγ(s)z]ds,
-+φm(x, s)φk(x, s)n02dx=δmkδ(s-s),m, k=1, 2,
0+φ1(x, s)φ1(x, s)+φ2(x, s)φ2(x, s)n02ds=δ(x-x).
ExI(x, z)=1n2(x)U0(x)[exp(-jβ0z)+R0 exp(+jβ0z)]+1n2(x)m=12ρm+Rm(ρ)Ψm(x, ρ)×exp[+jβ(ρ)z]dρ,
ExII(x, z)=1n02l=120+Tl(s)φl(x, s)exp[-jγ(s)z]ds.
E(x)=E0(x)+-+E(x)K(x, x)dx,
E0(x)=2Y1par(x)U0(x),
K(x, x)=-1par(x)(Y1-Y10)U0(x)U0(x)+m=12ρm+[Y(ρ)-Y10]Ψm(x, ρ)Ψm(x, ρ)dρ+l=120+[Y0(s)-Y00]φl(x, s)φl(x, s)ds,
Y1=ω0β0,Y(ρ)=ω0β(ρ),
Y0(s)=ω0γ(s),Y10=ω0k0n1,Y00=ω0k0n0,
par(x)=Y10n2(x)+Y00n02.
EN(x)=E0(x)+i=1NCi(x),N=1, 2, 3,
Ci(x)=-+dx1-+dx2  -+dxi K(x, x1)K(x1, x2)  K(xi-1, xi)E0(xi).
E1(x)=E0(x)-2Y1par(x)(Y1-Y10)U0(x)UUpar+m=12ρm+[Y(ρ)-Y10]Ψm(x, ρ)UΨmpardρ+l=120+[Y0(s)-Y00]φl(x, s)Uφlpards,
UUpar=-+U0(x)U0(x)par(x)dx,
UΨmpar=-+U0(x)Ψm(x, ρ)par(x)dx,m=1, 2,
Uφlpar=-+U0(x)φl(x, s)par(x)dx,l=1, 2,
R0=-1+-+E(x)U0(x)dx,
Rm(ρ)=-+E(x)Ψm(x, ρ)dx,m=1, 2,
Tl(s)=-+E(x)φl(x, s)dx,l=1, 2.
R00=-1+2Y1UUpar,
R01=R00-2Y1(Y1-Y10)UUpar2+m=12ρm+[Y(ρ)-Y10]UΨmpar2dρ+l=120+[Y0(s)-Y00]Uφlpar2ds,
Prad=m=12ρm+|Rm(ρ)|2 dρ,
ΦII(r, θ)=2k0n0r exp(-jk0n0r+jπ/4)(k0n0 cos θ)×l=12Tl(s=k0n0 sin θ)(r+),
E=[exp(-jβ0z)+R0 exp(+jβ0z)]E0(1)+m=12ρmRm(ρ)Emρ(1)exp[jβ(ρ)z]dρ,
E=m=120Tm(s)Ems(2)exp[-jγ(s)z]ds,
n2(x)ddx1n2(x)dU0dx+[k02n2(x)-β02]U0=0,
E0(1), Hmρ(1)=0,
Emρ(1), Hmρ(1)=δmmδ(ρ-ρ),
E0(1), H0(1)=1,
Ems(2), Hms(2)=δmmδ(s-s),
E, H=z=0(E×H)·ez dxdy.
Ξˆe[E]=2H0(1),
Ξˆe[E]=E, H0(1)H0(1)+m=12E, Hmρ(1)Hmρ(1)dρ+m=12E, Hms(2)Hms(2)ds
R0=-1+E, H0(1),
1-R01+R0=E, Ξˆer[E]E, H0(1)2,
Ξˆer[E]=Ξˆe[E]-E, H0(1)H0(1).
1-R01+R0=β0n02πQ02(1)0+[Qc2(s)+Qs2(s)]×ds(k02n02-s2)1/2,
Qc(s)=-+U0(x)cos(sx)n2(x)dx,
Qs(s)=-+U0(x)sin(sx)n2(x)dx,
Q02(1)=-+U02(x)n2(x)dx,
R0=(n1-n0)/(n1+n0).
Ξˆm[H]=2E0(1),
1+R01-R0=1πβ0n02Q02(1)×0+[Q˜c2(s)+Q˜s2(s)]k02n02-s2 ds,
Q˜c(s)=-+U0(x)cos(sx)dx,
Q˜s(s)=-+U0(x)sin(sx)dx.
Ex(x, 0)=F1(x)+F2(x)Δ12re-a/ln(kre),
re=|x2-D2/4|,a2Δ12/[π(n02+n22)].
|Ex(x, 0)-Ex(D/2, 0)|Δ12re2 ln(kre).

Metrics