Abstract

A new modification of the integral equation method using an iteration technique with “accelerating” parameters is presented to solve the problem of guided-mode scattering from an abruptly ended asymmetrical slab waveguide. The optimal choice of the parameters is shown to be closely connected with the variational principle. The electric-field distribution at the terminal plane, the reflection coefficient of the guided mode, and the far-field radiation pattern are computed. Numerical results are presented for several cases of abruptly ended waveguides, including the systems with constant and variable profiles of the refractive indices. The phenomenon of the radiation pattern rotation is examined in detail.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. M. Angulo, “Diffraction of surface waves by a semi-infinite dielectric slab,” IRE Trans. Antennas Propag. AP-5, 100–109 (1957).
    [CrossRef]
  2. 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]
  3. 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]
  4. 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]
  5. C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. 3, 408–415 (1985).
    [CrossRef]
  6. C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
    [CrossRef]
  7. 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).
  8. 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]
  9. G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
    [CrossRef]
  10. Y. P. Chiou, H. C. Chang, “Analysis of optical wave-guide discontinuities using Padé approximants,” IEEE Photonics Technol. Lett. 9, 964–966 (1997).
    [CrossRef]
  11. I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  12. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 4.
  13. M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).
  14. L. Lewin, Theory of Waveguides (Newness-Butterworths, London, 1975), Chap. 9.
  15. 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]
  16. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J 139, 101–104 (1992).
  17. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991), Chap. 1.
  18. F. G. Tricomi, Integral Equations (Interscience, New York, 1957).
  19. G. Latsas, A. B. Manenkov, I. G. Tigelis, E. Sarri, “Reflectivity properties of an abruptly ended asymmetrical slab waveguide for the case of transverse magnetic modes,” J. Opt. Soc. Am. A 17, 162–172 (2000).
    [CrossRef]
  20. A. D. Vasil’ev, A. B. Manenkov, “Diffraction of the sur-face wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
    [CrossRef]
  21. A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides (fibres),” Opt. Quantum Electron. 23, 621–632 (1991).
    [CrossRef]
  22. A. B. Manenkov, “Radiation modes of a fibre. Part I: construction and properties,” IEE Proc. J. 141, 287–295 (1994).
  23. J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
    [CrossRef]
  24. M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).
  25. R. E. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Chap. 8.
  26. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended dielectric waveguide,” IEE Proc. J. 139, 194–200 (1992).
  27. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  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]

2000 (1)

1999 (1)

1997 (1)

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

1996 (2)

M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).

G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
[CrossRef]

1994 (1)

A. B. Manenkov, “Radiation modes of a fibre. Part I: construction and properties,” IEE Proc. J. 141, 287–295 (1994).

1993 (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).

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]

1992 (2)

A. B. Manenkov, “Reflection of the surface mode from an abruptly ended dielectric waveguide,” IEE Proc. J. 139, 194–200 (1992).

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

1991 (1)

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

1988 (1)

1987 (1)

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

1985 (1)

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. 3, 408–415 (1985).
[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)

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 (1)

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]

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 (1)

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]

1962 (1)

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).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 4.

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]

Benson, T. M.

M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).

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]

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, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. 3, 408–415 (1985).
[CrossRef]

Chang, H. C.

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

Chiou, Y. P.

Y. P. Chiou, H. C. Chang, “Analysis of optical wave-guide discontinuities using Padé 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]

Fikioris, J. G.

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. 3, 408–415 (1985).
[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]

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).

Harrington, R. E.

R. E. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Chap. 8.

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]

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]

Keller, J. B.

Kendall, P. C.

M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).

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).

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]

Kweon, G.

G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
[CrossRef]

Latsas, G.

Lewin, L.

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

Manenkov, A. B.

G. Latsas, A. B. Manenkov, I. G. Tigelis, E. Sarri, “Reflectivity properties of an abruptly ended asymmetrical slab waveguide for the case of transverse magnetic modes,” J. Opt. Soc. Am. A 17, 162–172 (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, “Radiation modes of a fibre. Part I: construction and properties,” IEE Proc. J. 141, 287–295 (1994).

A. B. Manenkov, “Reflection of the surface mode from an abruptly ended dielectric waveguide,” IEE Proc. J. 139, 194–200 (1992).

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. D. Vasil’ev, A. B. Manenkov, “Diffraction of the sur-face 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, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991), Chap. 1.

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]

Park, I.

G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
[CrossRef]

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]

Reed, M.

M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).

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, 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]

Sarri, E.

Sewell, P.

M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).

Shim, J.

G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
[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]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).

Tigelis, I. G.

Tricomi, F. G.

F. G. Tricomi, Integral Equations (Interscience, New York, 1957).

Uzunoglu, N. K.

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. 3, 408–415 (1985).
[CrossRef]

Vasil’ev, A. D.

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

Vassallo, C.

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]

Electron. Lett. (1)

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 (1)

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

IEE Proc. J. (4)

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, “Radiation modes of a fibre. Part I: construction and properties,” IEE Proc. J. 141, 287–295 (1994).

M. Reed, T. M. Benson, P. C. Kendall, P. Sewell, “Antireflection-coated angled facet design,” IEE Proc. J. 143, 214–220 (1996).

A. B. Manenkov, “Reflection of the surface mode from an abruptly ended dielectric waveguide,” IEE Proc. J. 139, 194–200 (1992).

IEEE J. Quantum Electron. (2)

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]

IEEE Photonics Technol. Lett. (1)

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

IEEE Trans. Microwave Theory Tech. (2)

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]

IEEE. J. Quantum Electron. (1)

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

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. (2)

G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
[CrossRef]

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. 3, 408–415 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Quantum Electron. (1)

A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides (fibres),” Opt. Quantum Electron. 23, 621–632 (1991).
[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 sur-face wave at the end of the dielectric tube,” Radiophys. Quantum Electron. 30, 320–326 (1987).
[CrossRef]

Other (6)

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

F. G. Tricomi, Integral Equations (Interscience, New York, 1957).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 4.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).

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

R. E. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Chap. 8.

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 (12)

Fig. 1
Fig. 1

Geometry of an abruptly ended asymmetrical slab waveguide.

Fig. 2
Fig. 2

Comparison of the power reflectivity |R0|2 obtained by employing the integral equation method with accelerating parameters (IEMAP) and the variational electric-field formulation for an abruptly ended symmetrical slab waveguide with λ0=0.86 µm, n2=3.6, n0=1, and Δ12=Δ32=3% (Δ31=0%). The drawn curves are plotted by using the zero- and first-order solutions of the IEMAP [Eqs. (17) and (19)] (solid curve), the second-order solution of the IEMAP [Eq. (22)] (dashed curve), the third-order solution of the IEMAP (squares), the electric-field variational formulation [Eq. (35)] (circles), and the effective refractive-index equation (ERIE) [Eq. (30)] (stars).

Fig. 3
Fig. 3

Same as Fig. 2, but for Δ12=Δ32=10% (Δ31=0%).

Fig. 4
Fig. 4

Same as Fig. 2, but for Δ12=3% and Δ32=10% (Δ31=7.22%).

Fig. 5
Fig. 5

Same as Fig. 2, but for Δ12=3% and Δ32=30% (Δ31=27.84%).

Fig. 6
Fig. 6

Same as Fig. 2, but for Δ12=10% and Δ32=30% (Δ31=22.22%).

Fig. 7
Fig. 7

Same as Fig. 2, but for Δ12=10% and Δ32=72.22% (Δ31=69.14%).

Fig. 8
Fig. 8

Variation of the electric-field magnitude |E(x)| (solid curve) and the total electric field of the guided mode, |(1+R0)U0(x)| (squares), at the terminal plane z=0 with the normalized transverse distance x/D for an abruptly ended slab waveguide with λ0=0.86 µm, n2=3.6, D=0.4 µm, n0=1, and Δ12=Δ32=3% (Δ31=0%). Also given are the variations of |E(x)| (dashed curve) and |(1+R0)U0(x)| (circles) for the same geometry, but with Δ12=3% and Δ32=10% (Δ31=7.22%).

Fig. 9
Fig. 9

Variation of the electric-field magnitude |E(x)| (solid curve) and the total electric field of the guided mode, |(1+R0)U0(x)| (squares), at the terminal plane z=0 with the normalized transverse distance x/D for an abruptly ended slab waveguide with λ0=0.86 µm, n2=3.6, D=0.25 µm, n0=1, and Δ12=Δ32=10% (Δ31=0%). Also given are the variations of |E(x)| (dashed curve) and |(1+R0)U0(x)| (circles) for the same geometry, but with Δ12=10% and Δ32=30% (Δ31=22.22%).

Fig. 10
Fig. 10

Normalized radiation pattern for an abruptly ended slab waveguide with λ0=0.86 µm, n2=3.6, D=0.2 µm, n0=1, and Δ12=Δ32=10% (Δ31=0%, solid curve) and Δ12=10%, Δ32=72.22% (Δ31=30.86%, dashed curve).

Fig. 11
Fig. 11

Variation of the power reflectivity |R0|2 as a function of the width parameter de for the problem in question with the profile described by Eq. (37) and Δ12=Δ32=3% (Δ31=0%). The drawn curves are plotted by using the electric-field variational formulation [Eq. (35)] (solid curve) and the ERIE formula [Eq. (30)] (dashed curve). Also given are the corresponding curves for an asymmetrical waveguide with Δ12=3% and Δ32=10% (Δ31=7.22%) (dotted and dotted–dashed curves).

Fig. 12
Fig. 12

Variation of the power reflectivity |R0|2 as a function of the width parameter de for the problem in question with the profile described by Eq. (37) and Δ12=Δ32=10% (Δ31=0%). The drawn curves are plotted by using the electric-field variational formulation [Eq. (35)] (solid curve) and the ERIE formula [Eq. (30)] (dashed curve). Also given are the corresponding curves for the an asymmetrical waveguide with Δ12=10% and Δ32=30% (Δ31=22.22%) (dotted and dotted–dashed curves).

Tables (1)

Tables Icon

Table 1 Comparison between the Various Methods Used to Calculate the Reflectivity from an Abruptly Ended Symmetrical Slab Waveguide with λ0=0.86 µm, n2=3.6, D=0.25 µm, n1=n3=3.24 (Δ12=Δ32=10%, Δ31=0%), and n0=1

Equations (41)

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

ΦI(x, z)=U0(x)[exp(-jβ0z)+R0 exp(+jβ0z)]+m=12 ρm+Rm(ρ)Ψm(x, ρ)×exp[+jβ(ρ)z]dρ(z<0),
ΦII(x, z)=l=120+Tl(s)ϕl(x, s)×exp[-jγ(s)z]ds(z>0),
-+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,
-+ϕk(x, s)ϕl(x, s)dx=δklδ(s-s),
k, l=1, 2,
U0(x)U0(x)+m=12 ρm+Ψm(x, ρ)
×Ψm(x, ρ)dρ=δ(x-x),
l=120+ϕl(x, s)ϕl(x, s)ds=δ(x-x),
2β0U0(x)=-+E(x)Ξ(x, x)dx,
Ξ(x, x)=β0U0(x)U0(x)+m=12 ρm+β(ρ)Ψm(x, ρ)Ψm(x, ρ)dρ+l=120+γ(s)ϕl(x, s)ϕl(x, s)ds.
E(x)=E0(x)+-+E(x)K(x, x)dx,
E0(x)=2β0U0(x)β¯+γ¯,
K(x, x)=-1β¯+γ¯(β0-β¯)U0(x)U0(x)+m=12 ρm+[β(ρ)-β¯]Ψm(x, ρ)Ψm(x, ρ)dρ+l=12 0+[γ(s)-γ¯]ϕl(x, s)ϕl(x, s)ds.
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),
E(0)(x)=E0(x)=2β0β¯+γ¯U0(x)
R0(0)=2β0-β¯-γ¯β¯+γ¯.
E(1)(x)=E0(x)+-+K(x, x)E0(x)dx=2β0(2β¯+γ¯-β0)(β¯+γ¯)2U0(x)-2β0(β¯+γ¯)2l=120+[γ(s)-γ¯]×ϕl(x, s), U0(x)ϕl(x, s)ds,
R0(1)=-1+2β0(2β¯+γ¯-β0)(β¯+γ¯)2+2β0(β¯+γ¯)2×l=120+[γ(s)-γ¯]ϕl(x, s), U0(x)2 ds,
ϕl(x, s),U0(x)=-+ϕl(x, s)U0(x)dx,l=1, 2,
E(2)(x)=E(1)(x)+2β0(β0-β¯)(β¯+γ¯)3(β0-β¯)U0(x)+l=120+[γ(s)-γ¯]ϕl(x, s), U0(x)ϕl(x, s)ds+2β0(β¯+γ¯)3(β0-β¯)U0(x)×l=120+[γ(s)-γ¯]×ϕl(x, s), U0(x)2 ds+m=12 l=120+[γ(s)-γ¯]×ϕl(x, s), U0(x)ds0+[β(ρ)-β¯]×ϕl(x, s), Ψm(x, ρ)Ψm(x, ρ)dρ+l=120+[γ(s)-γ¯]2×ϕl(x, s),U0(x)ϕl(x, s)ds,
R0(2)=R0(1)+2β0(β0-β¯)(β¯+γ¯)3(β0-β¯)+2l=120+[γ(s)-γ¯]×ϕl(x, s), U0(x)2 ds+2β0(β¯+γ¯)3l=120+[γ(s)-γ¯]2×ϕl(x, s), U0(x)2 ds.
K(f), E=-+K(f)(x, x)E(x)dx=-1β¯+γ¯l=12-+[γ(s)-γ¯]Tl(s)ϕl(x, s)ds.
E(x)=(1+R0)U0(x)+Erad(x),
Erad(x)=F0(x)+Be|n2-n1|(x2-D2/4)2 ln[k02|x2-D2/4|],
β¯=β0,
γ¯=0+γ(s)l=12ϕl(x, s), U0(x)2 ds0+l=12ϕl(x, s), U0(x)2 ds=0+γ(s)l=12ϕl(x, s), U0(x)2 ds
0+l=12ϕl(x, s), U0(x)2 ds=1.
R0(0)=R0(1)=β0-γ¯β0+γ¯.
β¯=k0n1,γ¯=k0n0,
Reff=β0-k0n0β0+k0n0.
-+Ξev(x, x)E(x)dx=β0(1-R0)U0(x),
Ξev(x, x)=2m=12 ρm+β(ρ)Ψm(x, ρ)Ψm(x, ρ)dρ+2l=120+γ(s)ϕl(x, s)ϕl(x, s)ds.
1-R01+R0=E(x), Ξev(x, x), E(x)β0E(x), U0(x)2.
1+R0=E(x), U0(x),
1-R0(ev)1+R0(ev)=1β00+γ(s)ϕl(x, s), U0(x)2 ds.
1-R0(mv)1+R0(mv)=β00+ϕl(x, s), U0(x)2γ(s)ds.
n(x)=n3forx>0n1(1+Δ12)exp(x/de)forx<0,

Metrics