Abstract

We study the problem of the scattering of the first TM guided mode from an abruptly ended strongly asymmetrical slab waveguide by an improved iteration technique, which is based on the integral equation method with “accelerating” parameters. We demonstrate that the values of these parameters are related to the variational principle, and we save approximately 1–2 iterations compared with the case in which these parameters are not employed. The tangential electric-field distribution on the terminal plane, the reflection coefficient of the first TM guided mode, and the far-field radiation pattern are computed. Furthermore, a simple technique based on the Aitken extrapolation procedure is employed for faster computation of the higher-order solutions of the reflection coefficient. Numerical results are presented for several cases of abruptly ended waveguides, including systems with variational profile, while special attention is given to the far-field radiation pattern rotation and its explanation.

© 2001 Optical Society of America

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References

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  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]
  2. H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
    [CrossRef]
  3. 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]
  4. 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]
  5. 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]
  6. A. B. Manenkov, “Step discontinuities in dielectric waveguides (fibres),” Opt. Quantum Electron. 22, 65–76 (1990).
    [CrossRef]
  7. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J. 139, 101–104 (1992).
  8. 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).
  9. 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]
  10. G. Kweon, I. Park, J. Shim, “A computational method of determining reflectance at abrupt waveguide interfaces,” J. Lightwave Technol. 14, 2436–2443 (1996).
    [CrossRef]
  11. Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Padé approximants,” IEEE Photonics Technol. Lett. 9, 964–966 (1997).
    [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. I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  15. G. Latsas, A. B. Manenkov, I. G. Tigelis, E. Sarri, “Reflectivity properties from an abruptly ended asymmetrical slab waveguide for the case of transverse magnetic modes,” J. Opt. Soc. Am. A 17, 162–172 (2000).
    [CrossRef]
  16. I. G. Tigelis, A. B. Manenkov, “Analysis of the mode scattering from an abruptly ended dielectric slab waveguide by an accelerated iteration technique,” J. Opt. Soc. Am. A 17, 2249–2259 (2000).
    [CrossRef]
  17. A. A. Samarskii, A. V. Goolin, Numerical Methods (Nauka, Moscow, 1989) (in Russian).
  18. A. B. Manenkov, “Accuracy of approximation for fibre discontinuity analysis,” Opt. Quantum Electron. 23, 81–90 (1991).
    [CrossRef]
  19. I. G. Tigelis, N. K. Uzunoglu, C. N. Capsalis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Electromagn. Waves Appl. 5, 447–457 (1991).
  20. M. Abramovitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Appl. Math. Ser. 55 (National Bureau of Standards) (U.S. Government Printing Office, Washington, 1964), Chap. 3.
  21. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991), Chap. 1.
  22. F. G. Tricomi, Integral Equations (Interscience, New York, 1957).
  23. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
    [CrossRef]
  24. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  25. 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]
  26. J. K. Butler, J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. QE-10, 809–815 (1974).
    [CrossRef]
  27. J. Buus, “Analytic approximation for the reflectivity of DH lasers,” IEEE J. Quantum Electron. QE-17, 2256–2257 (1981).
    [CrossRef]
  28. C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of a heuristic analysis,” Electron. Lett. 24, 61–62 (1988).
    [CrossRef]
  29. 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]
  30. Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. MTT-39, 422–429 (1991).
    [CrossRef]

2000 (2)

1999 (1)

1997 (1)

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

1996 (1)

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

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 (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 approximation for fibre discontinuity analysis,” Opt. Quantum Electron. 23, 81–90 (1991).
[CrossRef]

I. G. Tigelis, N. K. Uzunoglu, C. N. Capsalis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Electromagn. Waves Appl. 5, 447–457 (1991).

Q. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. MTT-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 (2)

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

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]

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 (1)

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

1979 (1)

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]

1978 (1)

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

1962 (1)

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.

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]

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]

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.

I. G. Tigelis, N. K. Uzunoglu, C. N. Capsalis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Electromagn. Waves Appl. 5, 447–457 (1991).

Chang, H. C.

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using Padé 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. MTT-39, 422–429 (1991).
[CrossRef]

Chiou, Y. P.

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

Ghatak, A. K.

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

Goolin, A. V.

A. A. Samarskii, A. V. Goolin, Numerical Methods (Nauka, Moscow, 1989) (in Russian).

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]

Keller, J. B.

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. 140, 49–55 (1993).

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]

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.

Liu, Q.

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

Manenkov, A. B.

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

G. Latsas, A. B. Manenkov, I. G. Tigelis, E. Sarri, “Reflectivity properties from 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, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J. 139, 101–104 (1992).

A. B. Manenkov, “Accuracy of approximation 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. 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.

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]

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]

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).

Samarskii, A. A.

A. A. Samarskii, A. V. Goolin, Numerical Methods (Nauka, Moscow, 1989) (in Russian).

Sarri, E.

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.

I. G. Tigelis, N. K. Uzunoglu, C. N. Capsalis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Electromagn. Waves Appl. 5, 447–457 (1991).

Vassallo, C.

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]

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]

Electron. Lett. (2)

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]

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

IEE Proc. J. (2)

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

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).

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 Padé 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. (2)

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

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. Electromagn. Waves Appl. (1)

I. G. Tigelis, N. K. Uzunoglu, C. N. Capsalis, “Scattering from an abruptly terminated single-mode fiber waveguide,” J. Electromagn. Waves Appl. 5, 447–457 (1991).

J. Lightwave Technol. (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]

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Quantum Electron. (2)

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

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

Radiophys. Quantum Electron. (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]

Other (6)

A. A. Samarskii, A. V. Goolin, Numerical Methods (Nauka, Moscow, 1989) (in Russian).

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).

M. Abramovitz, I. A. Stegun, eds., Handbook of Mathematical Functions, Appl. Math. Ser. 55 (National Bureau of Standards) (U.S. Government Printing Office, Washington, 1964), Chap. 3.

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

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

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Figures (9)

Fig. 1
Fig. 1

Geometry of an abruptly ended asymmetrical slab waveguide.

Fig. 2
Fig. 2

Comparison of the power reflectivity |R0|2 of the first TM guided mode for an abruptly ended symmetrical slab waveguide with Δ12=Δ32=10%. The curves are plotted by using the successive-order solutions of the IEMAP, i.e., zero-order [Eq. (10)] (dotted curve), first-order [Eq. (13)] (circles), second-order (squares), and third-order (solid curve), and the electric-field variational formulation [Eq. (38) in Ref. 15] (stars).

Fig. 3
Fig. 3

Comparison of power reflectivity |R0|2 of the first TM guided mode for an abruptly ended asymmetrical slab waveguide with Δ12=10% and Δ32=72.22%. Definitions of symbols and curves are as in Fig. 2.

Fig. 4
Fig. 4

Variation of |R0|2 with D for an abruptly ended slab waveguide with Δ32=10% and Δ12=1% (solid curve), Δ12=5% (dashed curve) and Δ12=10% (dotted curve).

Fig. 5
Fig. 5

Variation of |R0|2 with D for an abruptly ended slab waveguide with Δ12=10% and Δ32=10% (solid curve), Δ32=30% (dashed curve) and Δ32=72.22% (dotted curve).

Fig. 6
Fig. 6

Variation of |R0|2 with D for an abruptly ended asymmetrical slab waveguide with Δ12=10% and Δ32=30% for the first TE and first TM guided modes. The curves are plotted by using the second-order solution of the IEMAP for the first TE guided mode [Eq. (21) in Ref. 16] (solid curve), the electric-field variational formulation for the first TE guided mode [Eq. (35) in Ref. 16] (squares), the third-order solution of the IEMAP for the first TM guided mode (dashed curve), and the electric-field variational formulation for the first TM guided mode [Eq. (38) in Ref. 15] (circles).

Fig. 7
Fig. 7

Variation of the magnitudes of the total tangential electric-field magnitude |E(x)| (solid curve) and the tangential electric-field distribution of the guided mode |(1+R0)U0(x)/n2(x)| (dotted curve) on z=0 with the transverse normalized distance x/D for an abruptly ended symmetrical slab waveguide with D=0.2 µm and Δ12=Δ32=10%. Also given are the variations of |E(x)| (solid curve) and |(1+R0)U0(x)/n2(x)| (dotted curve) for the same waveguide geometry, but with Δ12=10% and Δ32=72.22% (asymmetrical slab).

Fig. 8
Fig. 8

Normalized radiation pattern for an abruptly ended slab waveguide with D=0.2 µm, and Δ12=Δ32=5% (solid curve), Δ12=5% and Δ32=10% (squares), Δ12=10% and Δ32=30% (circles), and Δ12=10% and Δ32=72.22% (dashed curve).

Fig. 9
Fig. 9

Variation of |R0|2 with D for an abruptly ended asymmetrical slab waveguide with a linear refractive-index profile n(x) [see Eq. (20)] and n2=3.6, Δ12=10% and Δ32=72.22%. The drawn curves are plotted by using the electric-field variational formulations for the first TE [Eq. (35) in Ref. 16] (solid curve) and the first TM guided mode [Eq. (38) in Ref. 15] (dashed curve).

Tables (2)

Tables Icon

Table 1 Convergence of the IEMAP for an Abruptly Ended Symmetrical Slab Waveguide with λ0=0.86 µm, n2=3.6, D=0.2 µm, Δ12=Δ32=10% and n0=1

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Table 2 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=0.2 µm, Δ12=Δ32=10% and n0=1

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E(x)=E0(x)+-+E(x)K(x, x)dx,
E0(x)=2Y1U0(x)par(x),
K(x, x)=1par(x)(Y¯1-Y1)U0(x)U0(x)+m=12×ρm+[Y¯1-Y(ρ)]Ψm(x, ρ)Ψm(x, ρ)dρ+l=120+[Y¯0-Y0(s)]ϕl(x, s)ϕl(x, s)ds,
par(x)=Y¯1n2(x)+Y¯0n02,
Y1=ω0/β0,Y(ρ)=ω0/β(ρ),
Y0(s)=ω0/γ(s),
Y¯1=Y10=ω0/k0n1 and
Y¯0=Y00=ω0/k0n0.
E(N)(x)=E0(x)+i=lNCi(x),N=1, 2, 3,,
Ci(x)=-+dx1-+dx2×-+dxiK(x, x1)K(x1, x2)K(xi-1, xi)E0(xi),
E(0)(x)=E0(x)=2Y1U0(x)par(x),
R0(0)=-1+2Y1-+U02(x)par(x)dx=-1+2Y1U0(x),U0(x)/par(x).
U(x),V(x)=-+U(x)V(x)dx.
E(1)(x)=E0(x)+2Y1par(x)(Y¯1-Y1)U0(x)U0(x),×U0(x)/par(x)+m=12ρm+dρ[Y¯1-Y(ρ)]×Ψm(x, ρ)U0,Ψm(x, ρ)/par(x)+l=120+ds[Y¯0-Y0(s)]ϕl(x, s)×U0(x), ϕl(x, s)/par(x),
R0(1)=R0(0)+2Y1(Y¯1-Y1)U0(x),U0(x)/par(x)2+m=12ρm+dρ[Y¯1-Y(ρ)]×U0(x),Ψm(x, ρ)/par(x)2+l=120+ds[Y¯0-Y0(s)]×U0(x),ϕl(x, s)/par(x)2.
Y¯1=Y1
Y¯0l=120+dsU0(x),ϕl(x, s)/par(x)=l=120+dsY(s)U0(x), ϕl(x, s)/par(x).
Y¯0l=120+dsU0(x), ϕl(x, s)/par¯(x)=l=120+dsY(s)U0(x), ϕl(x, s)/par¯(x),
par¯(x)=Y1n2(x)+Y00n02.
R0(N)R0()+CRqN,
R˜0(N)R0(N)-[R0(N)-R0(N-1)]2/[R0(N)-2R0(N-1)+R0(N-2)]
n(x)=n1,forx<-D/2n2+(n2-n1)(2x+D)/(2D)for|x|<D/2n3,forx<D/2.

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