Abstract

The problem of the dominant TE guided-mode scattering from an abruptly terminated asymmetrical slab waveguide is examined by both the integral equation method and the variational technique. The reflection coefficient of the guided mode, the far-field radiation pattern, and the field distribution on the terminal plane are computed. Numerical results are presented for several cases of abruptly ended waveguides, including the three-layer guide and the structure with variable profile of the refractive index.

© 1999 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. G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
    [CrossRef]
  3. H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (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. T. E. Rozzi, G. H. in’t Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
    [CrossRef]
  7. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
    [CrossRef] [PubMed]
  8. M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
    [CrossRef]
  9. 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]
  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. 6, 590–600 (1988).
    [CrossRef]
  11. 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]
  12. A. B. Manenkov, “Step discontinuities in dielectric waveguides (fibres),” Opt. Quantum Electron. 22, 65–76 (1990).
    [CrossRef]
  13. 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]
  14. 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: Optoelectron. 140, 49–55 (1993).
  15. Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Pade approximants,” IEEE Photonics Technol. Lett. 9, 964–966 (1997).
    [CrossRef]
  16. C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
    [CrossRef]
  17. 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]
  18. P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
    [CrossRef]
  19. C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
    [CrossRef]
  20. 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]
  21. N. K. Uzunoglu, C. N. Capsalis, I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
    [CrossRef]
  22. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc.-J: Optoelectron. 139, 101–104 (1992).
  23. 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]
  24. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991), Chap. 1.
  25. L. Lewin, Theory of Waveguides (Newness-Butterworth, London, 1975).
  26. I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layer symmetric slab waveguide,” J. Opt. Soc. Am. A 14, 1260–1267 (1997).
    [CrossRef]
  27. A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides (fibres),” Opt. Quantum Electron. 23, 621–632 (1991).
    [CrossRef]
  28. T. Y. Na, Computational Methods in Engineering Boundary Value Problems (Academic, New York, 1979).
  29. 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]
  30. 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]
  31. J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
    [CrossRef]
  32. J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
    [CrossRef]

1997

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

I. G. Tigelis, T. G. Theodoropoulos, I. A. Papakonstantinou, “Radiation properties of an abruptly terminated five-layer symmetric slab waveguide,” J. Opt. Soc. Am. A 14, 1260–1267 (1997).
[CrossRef]

1993

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

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: Optoelectron. 140, 49–55 (1993).

1992

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]

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

1991

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

1990

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

1987

N. K. Uzunoglu, C. N. Capsalis, I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[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]

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

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

1982

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]

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

1981

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

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[CrossRef]

1980

T. E. Rozzi, G. H. in’t Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[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]

1979

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

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

1976

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

1974

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

1972

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

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

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

1957

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: Optoelectron. 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]

Baets, R.

Benson, T. M.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[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.

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]

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]

Citerne, J.

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[CrossRef]

Gelin, P.

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[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]

Haus, H. A.

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[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]

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]

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]

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]

in’t Veld, G. H.

T. E. Rozzi, G. H. in’t Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[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.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

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: Optoelectron. 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]

Kuznetsov, M.

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

Lagasse, P. E.

Lewin, L.

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

Manenkov, A. B.

A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc.-J: Optoelectron. 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, “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, “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, London, 1991), Chap. 1.

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]

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]

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]

Papakonstantinou, I. A.

Petenzi, M.

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[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: Optoelectron. 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: Optoelectron. 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: Optoelectron. 140, 49–55 (1993).

Rozzi, T. E.

T. E. Rozzi, G. H. in’t Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[CrossRef]

Smartt, C. J.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[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]

Theodoropoulos, T. G.

Tigelis, I.

Tigelis, I. G.

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.

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.

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]

Yajima, H.

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

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]

Appl. Opt.

Bell Syst. Tech. J.

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

Electron. Lett.

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

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

IEE Proc.-J: Optoelectron.

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: Optoelectron. 140, 49–55 (1993).

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

IEEE J. Quantum Electron.

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]

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]

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

IEEE Photonics Technol. Lett.

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. Microwave Theory Tech.

T. E. Rozzi, G. H. in’t Veld, “Variational treatment of the diffraction at the facet of d.h. lasers and of dielectric millimeter wave antennas,” IEEE Trans. Microwave Theory Tech. MTT-28, 61–73 (1980).
[CrossRef]

P. Gelin, M. Petenzi, J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-29, 107–114 (1981).
[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]

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.

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

J. Appl. Phys.

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.

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]

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Figures (9)

Fig. 1
Fig. 1

Geometry of an abruptly terminated asymmetrical slab waveguide.

Fig. 2
Fig. 2

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, n1=n3=3.24 (Δ12=Δ32=10% and Δ13=0%), and n0=1.

Fig. 3
Fig. 3

Convergence 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%.

Fig. 4
Fig. 4

Variation of the square magnitude of the reflection coefficient, |R0|2, of the dominant guided TE 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%.

Fig. 5
Fig. 5

Variation of the square magnitude of the reflection coefficient, |R0|2, of the dominant guided TE mode with the core width D for an abruptly terminated asymmetrical slab waveguide with λ0=0.9 µm, n2=3.61, n0=1, Δ12=10%, and Δ32=30%.

Fig. 6
Fig. 6

(a) Variation of the electric-field magnitude |E(x)| (solid curve) and the total electric field of the guided mode, |1+R0|U0(x) (dashed curve), at the terminal plane z=0 with the normalized transverse distance x/D for an abruptly terminated 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) (solid circles) for the same geometry but with Δ12=1% and Δ32=10%. (b) Variation of the electric-field magnitude |E(x)| (solid curve) and the total electric field of the guided mode, |1+R0|U0(x) (dashed curve), at the terminal plane z=0 with the normalized transverse distance x/D for an abruptly terminated slab waveguide with λ0=0.9 µm, n2=3.61, D=0.25 µm, n0=1, and Δ12=Δ32=10%. Also given is the variation of |E(x)| (dotted curve) and |1+R0|U0(x) (solid circles) for the same geometry but with Δ12=10% and Δ32=30%.

Fig. 7
Fig. 7

(a) Normalized radiation pattern for an abruptly terminated slab waveguide with λ0=0.9 µm, n2=3.61, D=0.5 µm, n0=1, Δ12=Δ32=1% (solid curve), and Δ12=1%, Δ32=10% (dashed curve). (b) Similar to (5) but applies to the problem with the parameters λ0=0.9 µm, n2=3.61, D=0.25 µm, n0=1, Δ12=Δ32=10% (solid curve), and Δ12=10%, Δ32=30% (dashed curve).

Fig. 8
Fig. 8

(a) Variation of the square magnitude of the reflection coefficient, |R0|2, of the dominant guided TE mode with the core width D for an abruptly terminated slab waveguide with λ0=0.9 µm, n2=3.61, n0=1, Δ12=Δ32=1% (solid curve), and Δ12=1%, Δ32=10% (dashed curve). These curves have been plotted by using the integral equation method [Eq. (24c)]. (b) Variation of the square magnitude of the reflection coefficient, |R0|2, of the dominant guided TE mode with the core width D for an abruptly terminated slab waveguide with λ0=0.9 µm, n2=3.61, n0=1, Δ12=Δ32=10% (solid curve), and Δ12=10%, Δ32=30% (dashed curve). These curves have been plotted by using the integral equation method [Eq. (24c)].

Fig. 9
Fig. 9

Variation of the square magnitude of the reflection coefficient, |R0|2, of the dominant guided TE 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%.

Equations (48)

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U0(x)=Aexp[-h3(x-D/2)],xD/2cos[h2(x-D/2)]-(h3/h2)sin[h2(x-D/2)],|x|D/2[cos(h2D)+(h3/h2)sin(h2D)]exp[h1(x+D/2)],x-D/2,
(h22-h1h3)tan(h2D)=h2(h1+h3)
h2D=cos-1(h2/V21)+cos-1(h2/V23),
V212=k02(n22-n12),V232=k02(n22-n32).
A=h22(h22+h32)(D+h1-1+h3-1)1/2
-+U02(x)dx=1.
-+Ψm(x, ρ)Ψk(x, ρ)dx=δmkδ(ρ-ρ), m, k=1, 2,
-+U0(x)Ψm(x, ρ)dx=0,m=1, 2,
U0(x)U0(x)+m=12ρm+Ψm(x, ρ)Ψm(x, ρ)dρ=δ(x-x),
ΦI(x, z)=U0(x)[exp(-jβ0 z)+R0 exp(+jβ0 z)]+m=12ρm+Rm(ρ)Ψm(x, ρ)exp[+jβ(ρ) z]d ρ,
β2(ρ)=k02n12-ρ2.
ΦII(x, z)=l=120+Tl(s)ϕl(x, s)exp[-jγ(s)z]ds,
-+ϕk(x, s)ϕl(x, s)d x=δklδ(s-s), k, l=1, 2,
l=120+ϕl(x, s)ϕl(x, s)d s=δ(x-x).
E(x)=E0(x)+-+E(x)K(x, x)dx,
E0(x)=2β0U0(x)k0(n0+n1),
K(x, x)=(β0-k0n1)U0(x)U0(x)+m=12ρm+[β (ρ)-k0n1]×Ψm(x, ρ)Ψm(x, ρ)d ρ+l=120+[γ(s)-k0n0]×ϕl(x, s)ϕl(x, s)d s.
EN(x)=E0(x)+i=1NCi(x),N=1, 2, 3,,
Ci(x)=-+d x1-+d x2-+d xiK(x, x1)×K(x1, x2)K(xi-1, xi)E0(xi).
E1(x)=E0(x)-2β0k02(n0+n1)2(β0-k0 n1)U0(x)+l=120+[γ(s)-k0 n0]Uϕl(s)ϕl(x, s)d s,
E2(x)=E1(x)+2β0(β0-k0 n1)k03(n0+n1)3(β0-k0 n1)U0(x)+l=120+[γ(s)-k0 n0]Uϕl(s)ϕl(x, s)d s+2β0k03(n0+n1)3(β0-k0 n1)U0(x)×l=120+[γ (s)-k0 n0]|Uϕl(s)|2 d s+m=12l=120+[γ(s)-k0 n0]Uϕl(s)d s×ρm+[β(ρ)-k0 n1]Ψmϕl(ρ, s)Ψm(x, ρ)d ρ+l=120+[γ (s)-k0 n0]2×Uϕl(s)ϕl(x, s)d s,
Uϕl(s)=-+U0(x)ϕl(x, s)d x,l=1, 2,
Ψmϕl(ρ, s)=-+Ψm(x, ρ)ϕl(x, s)d x,m, l=1, 2
R0=-1+-+E(x)U0(x)d x,
Rm(ρ)=-+E(x)Ψm(x, ρ)d x,m=1, 2,
Tl(s)=-+E(x)ϕl(x, s)d x,l=1, 2.
R00=-1+2β0k0(n0+n1),
R01=R00-2β0k02(n0+n1)2(β0-k0 n1)+l=120+[γ (s)-k0 n0][Uϕl(s)]2 d s,
R02=R01+2β0(β0-k0 n1)k03(n0+n1)3(β0-k0 n1)+2l=120+[γ (s)-k0 n0][Uϕl(s)]2 d s+2β0k03(n0+n1)3l=120+[γ (s)-k0 n0]2×[Uϕl(s)]2 d s;
Prad=m=12ρm+|Rm(ρ)|2 d ρ
ΦII(r, θ)=2k0 n0r1/2×exp[(-jk0 n0r+jπ/4)](k0 n0 cos θ)×l=12[Tl(s=k0 n0 sin θ)](r+),
Ξˆ(E)=2H0(1),
Ξˆ(E)=E, H0(1)H0(1)+mE, Hmρ(1)Hmρ(1)d ρ+mE, Hms(2)Hms(2)d s,
E, Hmρ=z=0(E×Hmρ)·ez d xd y,
Emρ(1), Hmρ(1)=δmmδ(ρ-ρ), Ems(2), Hms(2)=δmmδ(s-s),
E0(1), H0(1)=1,
E0(1), Hmρ(1)=0,
R0=E, H0(1)-1,
1-R01+R0=E,ΞˆR(E)E, H0(1)2,
ΞˆR(E)=Ξˆ(E)-E, H0(1)H0(1).
1-R01+R0=1πβ0(1)0+(Qc2+Qs2)(k02n02-s2)1/2 d s,
Qc=-+U0(x)cos(sx)d x,
Qs=-+U0(x)sin(sx)d x.
R0=(n1-n0)/(n1+n0).
R0=(n2-n0)/(n2+n0).
n(x)=n3,x>D/2n2,0<x<+D/2[n2+n1+(n2-n1)cos(2πx/D)]/2,-D/2<x<0n1,x<-D/2.
d2U0d x2+[k02n2(x)-β02]U0=0.
R0=(β0-k0 n0)/(β0+k0 n0)

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