Abstract

We examine the influence of the refractive index core profile on the modal scattering of abruptly terminated slab waveguides. The analysis is based on the integral equation method with accelerating parameters, while for the field description in the waveguide core, an appropriate Lanczos–Fourier expansion is employed. The electric-field distribution on the terminal plane, the reflection and transformation coefficient of the TE guided modes, and the far-field radiation pattern are computed. Numerical results are presented for slab waveguides with step, linear, and parabolic refractive index profiles of the core. Finally, several approximate analytical solutions are derived to study the problem in question and to explain the results obtained.

© 2007 Optical Society of America

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  7. J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
    [CrossRef]
  8. F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
    [CrossRef] [PubMed]
  9. R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, "Surface polariton reflection and radiation at end faces," Appl. Phys. Lett. 42, 764-766 (1983).
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    [CrossRef]
  12. K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
    [CrossRef]
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  14. F. Fernandez and Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Research Studies, 1996).
  15. A. V. Brovko, A. B. Manenkov, and A. G. Rozhnev, "FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities," Opt. Quantum Electron. 35, 395-406 (2003).
    [CrossRef]
  16. C. J. Smartt, T. M. Benson, and P. C. Kendall, "Free-space radiation mode method for the analysis of propagation in optical waveguide devices," IEE Proc.: Optoelectron. 140, 56-61 (1993).
    [CrossRef]
  17. A. Vucovic, P. Sewell, T. M. Benson, and P. C. Kendall, "Novel half-space radiation mode method for buried waveguide analysis," Opt. Quantum Electron. 31, 43-51 (1999).
    [CrossRef]
  18. D. N. Chien, M. Tanaka, and K. Tanaka, "Numerical simulation of an arbitrarily ended asymmetrical slab waveguide by guided-mode extracted integral equations," J. Opt. Soc. Am. A 19, 1649-1657 (2002).
    [CrossRef]
  19. I. G. Tigelis and A. B. Manenkov, "Analysis of mode scattering from an abruptly ended dielectric slab waveguide by an accelerated iteration technique," J. Opt. Soc. Am. A 17, 2249-2259 (2000).
    [CrossRef]
  20. A. B. Manenkov, G. P. Latsas, and I. G. Tigelis, "Scattering of the transverse magnetic modes from an abruptly ended strongly asymmetrical slab waveguide by an accelerated integral equation technique," J. Opt. Soc. Am. A 18, 3110-3119 (2001).
    [CrossRef]
  21. 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]
  22. K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
    [CrossRef]
  23. P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmodes expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
    [CrossRef]
  24. A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
    [CrossRef]
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    [CrossRef]
  28. A. B. Manenkov, "Eigenmodes expansion in lossy open waveguides," Opt. Quantum Electron. 23, 621-632 (1991).
    [CrossRef]
  29. C. Lanczos, Applied Analysis (Prentice Hall, 1956).
  30. P. A. Koukoutsaki, I. G. Tigelis, and A. B. Manenkov, "Guided-mode analysis by the Lanczos-Fourier expansion," J. Opt. Soc. Am. A 19, 2293-2300 (2002).
    [CrossRef]
  31. N. F. Dasyras, I. G. Tigelis, A. Tsigopoulos, and A. B. Manenkov, "Calculation of radiation modes using the Lanczos-Fourier expansion," J. Opt. Soc. Am. A 21, 1740-1749 (2004).
    [CrossRef]
  32. I. G. Tigelis and A. B. Manenkov, "Scattering from an abruptly terminated asymmetrical slab waveguide," J. Opt. Soc. Am. A 16, 523-532 (1999).
    [CrossRef]
  33. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).
  34. L. Lewin, Theory of Waveguides (Newness-Butterworths, 1975).
  35. D. Marcuse, "Gaussian approximation of the fundamental modes of graded-index fibers," J. Opt. Soc. Am. 68, 103-109 (1978).
    [CrossRef]

2004 (1)

2003 (1)

A. V. Brovko, A. B. Manenkov, and A. G. Rozhnev, "FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities," Opt. Quantum Electron. 35, 395-406 (2003).
[CrossRef]

2002 (2)

2001 (3)

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmodes expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
[CrossRef]

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

2000 (1)

1999 (3)

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

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

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

1998 (1)

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

1993 (3)

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

E. V. Alieva, V. A. Yakovlev, V. I. Silin, and A. Volkov, "Surface electromagnetic waves excitation from infrared to visible," Opt. Commun. 96, 218-220 (1993).
[CrossRef]

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

1991 (2)

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

A. B. Manenkov, "Eigenmodes expansion in lossy open waveguides," Opt. Quantum Electron. 23, 621-632 (1991).
[CrossRef]

1988 (2)

A. F. Gonvharov, G. N. Zhizhin, S. A. Kiselev, L. A. Kuzik, and V. A. Yakovlev, "Determination of the dielectric constant of Yba2Cu3O7−δ single crystals in the 10 μm spectral range by SEW phase spectroscopy," Phys. Lett. A 133, 163-166 (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]

1984 (1)

J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
[CrossRef]

1983 (1)

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, "Surface polariton reflection and radiation at end faces," Appl. Phys. Lett. 42, 764-766 (1983).
[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]

1978 (1)

1970 (1)

A. B. Manenkov, "Excitation of open homogeneous waveguides," Radiophys. Quantum Electron. 13, 578-586 (1970).
[CrossRef]

Adams, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

Afsar, M. N.

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

Alieva, E. V.

E. V. Alieva, V. A. Yakovlev, V. I. Silin, and A. Volkov, "Surface electromagnetic waves excitation from infrared to visible," Opt. Commun. 96, 218-220 (1993).
[CrossRef]

Avetissian, Yu. H.

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

Baets, R.

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmodes expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Banajanian, A.

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

Benson, T. M.

A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
[CrossRef]

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

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

Bienstman, P.

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmodes expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

Brovko, A. V.

A. V. Brovko, A. B. Manenkov, and A. G. Rozhnev, "FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities," Opt. Quantum Electron. 35, 395-406 (2003).
[CrossRef]

Chester, A. N.

A. N. Chester, S. Mastellucci, and A. M. Verga Scheggi, Optical Fiber Sensors (Martinus Nijhoff, 1967).

Chien, D. N.

Ciocci, F.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

Dasyras, N. F.

Dattoli, G.

J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
[CrossRef]

Doria, A.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

Fernandez, F.

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

Gallerano, G. P.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

Giabbai, I.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

Gonvharov, A. F.

A. F. Gonvharov, G. N. Zhizhin, S. A. Kiselev, L. A. Kuzik, and V. A. Yakovlev, "Determination of the dielectric constant of Yba2Cu3O7−δ single crystals in the 10 μm spectral range by SEW phase spectroscopy," Phys. Lett. A 133, 163-166 (1988).
[CrossRef]

Hayashi, K.

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer, 1984).

Ishimura, A.

A. Ishimura, Wave Propagation and Scattering in Random Media (Academic, 1978).

Johnson, B.

J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
[CrossRef]

Kaji, R.

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

Kendall, P. C.

A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
[CrossRef]

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

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

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

Kimmitt, M. F.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

Kiselev, S. A.

A. F. Gonvharov, G. N. Zhizhin, S. A. Kiselev, L. A. Kuzik, and V. A. Yakovlev, "Determination of the dielectric constant of Yba2Cu3O7−δ single crystals in the 10 μm spectral range by SEW phase spectroscopy," Phys. Lett. A 133, 163-166 (1988).
[CrossRef]

Kocharyn, K. N.

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

Koshiba, M.

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

Koukoutsaki, P. A.

Kuzik, L. A.

A. F. Gonvharov, G. N. Zhizhin, S. A. Kiselev, L. A. Kuzik, and V. A. Yakovlev, "Determination of the dielectric constant of Yba2Cu3O7−δ single crystals in the 10 μm spectral range by SEW phase spectroscopy," Phys. Lett. A 133, 163-166 (1988).
[CrossRef]

Lanczos, C.

C. Lanczos, Applied Analysis (Prentice Hall, 1956).

Latsas, G. P.

Lewin, L.

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

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Lu, Y.

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

Manenkov, A. B.

N. F. Dasyras, I. G. Tigelis, A. Tsigopoulos, and A. B. Manenkov, "Calculation of radiation modes using the Lanczos-Fourier expansion," J. Opt. Soc. Am. A 21, 1740-1749 (2004).
[CrossRef]

A. V. Brovko, A. B. Manenkov, and A. G. Rozhnev, "FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities," Opt. Quantum Electron. 35, 395-406 (2003).
[CrossRef]

P. A. Koukoutsaki, I. G. Tigelis, and A. B. Manenkov, "Guided-mode analysis by the Lanczos-Fourier expansion," J. Opt. Soc. Am. A 19, 2293-2300 (2002).
[CrossRef]

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

A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
[CrossRef]

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

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

A. B. Manenkov, "Eigenmodes expansion in lossy open waveguides," Opt. Quantum Electron. 23, 621-632 (1991).
[CrossRef]

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

A. B. Manenkov, "Excitation of open homogeneous waveguides," Radiophys. Quantum Electron. 13, 578-586 (1970).
[CrossRef]

Maradudin, A. A.

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, "Surface polariton reflection and radiation at end faces," Appl. Phys. Lett. 42, 764-766 (1983).
[CrossRef]

Marcuse, D.

Mastellucci, S.

A. N. Chester, S. Mastellucci, and A. M. Verga Scheggi, Optical Fiber Sensors (Martinus Nijhoff, 1967).

Messina, G.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

Renieri, A.

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
[CrossRef]

Roberts, D. A.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

Robertson, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

Robson, P. N.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

Rozhnev, A. G.

A. V. Brovko, A. B. Manenkov, and A. G. Rozhnev, "FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities," Opt. Quantum Electron. 35, 395-406 (2003).
[CrossRef]

Sarkissian, R. L.

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

Sewell, P.

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

Sewell, P. D.

A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
[CrossRef]

Silin, V. I.

E. V. Alieva, V. A. Yakovlev, V. I. Silin, and A. Volkov, "Surface electromagnetic waves excitation from infrared to visible," Opt. Commun. 96, 218-220 (1993).
[CrossRef]

Smartt, C. J.

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

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Stegeman, G. I.

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, "Surface polariton reflection and radiation at end faces," Appl. Phys. Lett. 42, 764-766 (1983).
[CrossRef]

Tanaka, K.

Tanaka, M.

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K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

Tsigopoulos, A.

Tsuji, Y.

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

Vassallo, C.

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[CrossRef]

Vucovic, A.

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

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A. Vukovic, "Fourier transformation analysis of optoelectronics components and circuits," Ph.D. thesis (University of Nottingham, 2000), (http://www.nottingham.ac.uk/ggiemr/publications/avlowbarthesis.htm).

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[CrossRef]

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F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
[CrossRef]

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E. V. Alieva, V. A. Yakovlev, V. I. Silin, and A. Volkov, "Surface electromagnetic waves excitation from infrared to visible," Opt. Commun. 96, 218-220 (1993).
[CrossRef]

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[CrossRef]

Yoneta, S.

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

Zhizhin, G. N.

A. F. Gonvharov, G. N. Zhizhin, S. A. Kiselev, L. A. Kuzik, and V. A. Yakovlev, "Determination of the dielectric constant of Yba2Cu3O7−δ single crystals in the 10 μm spectral range by SEW phase spectroscopy," Phys. Lett. A 133, 163-166 (1988).
[CrossRef]

Appl. Phys. Lett. (1)

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, "Surface polariton reflection and radiation at end faces," Appl. Phys. Lett. 42, 764-766 (1983).
[CrossRef]

IEE Proc.: Optoelectron. (2)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M. J. Robertson, "Semiconductor laser facet reflectivities using free-space radiation modes," IEE Proc.: Optoelectron. 140, 49-55 (1993).
[CrossRef]

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

IEEE Trans. Microwave Theory Tech. (1)

K. N. Kocharyn, M. N. Afsar, Yu. H. Avetissian, R. L. Sarkissian, A. Banajanian, and I. I. Tkachov, "A novel quasi-optical resonator for the surface near-millimeter waves," IEEE Trans. Microwave Theory Tech. 47, 27-33 (1999).
[CrossRef]

J. Lightwave Technol. (1)

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis," J. Lightwave Technol. LT-16, 2040-2045 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

E. V. Alieva, V. A. Yakovlev, V. I. Silin, and A. Volkov, "Surface electromagnetic waves excitation from infrared to visible," Opt. Commun. 96, 218-220 (1993).
[CrossRef]

Opt. Quantum Electron. (5)

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[CrossRef]

P. Bienstman and R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmodes expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

A. B. Manenkov, T. M. Benson, P. D. Sewell, and P. C. Kendall, "Applicability of scalar theory for analysis of mode reflection from an optical dielectric waveguide end," Opt. Quantum Electron. 33, 1195-1204 (2001).
[CrossRef]

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

A. V. Brovko, A. B. Manenkov, and A. G. Rozhnev, "FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities," Opt. Quantum Electron. 35, 395-406 (2003).
[CrossRef]

Phys. Lett. A (1)

A. F. Gonvharov, G. N. Zhizhin, S. A. Kiselev, L. A. Kuzik, and V. A. Yakovlev, "Determination of the dielectric constant of Yba2Cu3O7−δ single crystals in the 10 μm spectral range by SEW phase spectroscopy," Phys. Lett. A 133, 163-166 (1988).
[CrossRef]

Phys. Rev. Lett. (2)

J. E. Walsh, B. Johnson, G. Dattoli, and A. Renieri, "Undulator and Cherenkov free-electron lasers: a preliminary comparison," Phys. Rev. Lett. 53, 779-782 (1984).
[CrossRef]

F. Ciocci, A. Doria, G. P. Gallerano, I. Giabbai, M. F. Kimmitt, G. Messina, A. Renieri, and J. E. Walsh, "Observation of coherent millimeter and submillimeter emission from a microtron-driven Cherenkov free-electron laser," Phys. Rev. Lett. 66, 699-702 (1991).
[CrossRef] [PubMed]

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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, "Excitation of open homogeneous waveguides," Radiophys. Quantum Electron. 13, 578-586 (1970).
[CrossRef]

Other (11)

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

C. Lanczos, Applied Analysis (Prentice Hall, 1956).

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

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T.Tamir, ed., Integrated Optics, 2nd ed. (Springer,1979).

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer, 1984).

A. Vukovic, "Fourier transformation analysis of optoelectronics components and circuits," Ph.D. thesis (University of Nottingham, 2000), (http://www.nottingham.ac.uk/ggiemr/publications/avlowbarthesis.htm).

A. N. Chester, S. Mastellucci, and A. M. Verga Scheggi, Optical Fiber Sensors (Martinus Nijhoff, 1967).

A. Ishimura, Wave Propagation and Scattering in Random Media (Academic, 1978).

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Figures (9)

Fig. 1
Fig. 1

Geometry of an abruptly terminated slab waveguide with an arbitrary refractive index profile of the core.

Fig. 2
Fig. 2

Variation of R n ( 2 ) 2 ( n = 0 , 1 , 2 ) with the core thickness 2 d for an abruptly ended symmetric slab waveguide with n 2 m = 3.6 , n 1 = n 3 = 3.24 , and n 0 = 1 for two different profiles: constant and parabolic.

Fig. 3
Fig. 3

Variation of R n ( 2 ) 2 ( n = 0 , 1 , 2 , 3 , 4 , 5 ) with 2 d for an abruptly ended slab waveguide with n 2 m = 3.6 , n 1 = n 3 = 3.24 , n 0 = 1 , and a linear refractive index profile.

Fig. 4
Fig. 4

Variation of R n ( 2 ) 2 ( n = 0 , 1 , 2 ) with 2 d for the slab waveguide of Fig. 2, but with n 0 = 3.42 .

Fig. 5
Fig. 5

Electric-field distributions at z = 0 for an abruptly ended slab waveguide with n 2 m = 3.6 , n 1 = n 3 = 3.24 , d = 0.125 μ m , and n 0 = 1 for three different profiles: constant, linear, and parabolic.

Fig. 6
Fig. 6

Far-field radiation patterns for an abruptly ended slab waveguide with n 2 m = 3.6 , n 1 = n 3 = 3.24 , d = 1.5 μ m , and n 0 = 1 for three different profiles: constant, linear, and parabolic.

Fig. 7
Fig. 7

Variation of R n ( 2 ) 2 ( n = 0 , 1 , 2 ) with 2 d for an abruptly ended asymmetric slab waveguide with n 2 m = 3.6 , n 1 = 3.24 , n 3 = 2.52 , and n 0 = 1 for two different profiles: constant and linear.

Fig. 8
Fig. 8

Electric-field distributions at z = 0 for an abruptly ended asymmetric slab waveguide with n 2 m = 3.6 , n 1 = 3.24 , n 3 = 2.52 , d = 0.125 μ m , and n 0 = 1 for three different profiles: constant, linear, and parabolic.

Fig. 9
Fig. 9

Approximate solutions of the dominant TE 0 guided-mode power reflectivity for an abruptly ended symmetric slab waveguide with n 2 m = 3.6 , Δ 12 = 10 % ( n 1 = n 3 3.22 ) , and a parabolic refractive index profile.

Tables (1)

Tables Icon

Table 1 Convergence of the Dominant TE 0 Guided-Mode Power Reflectivity R 0 ( 2 ) 2 with the Number of Nodes N

Equations (27)

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Φ I ( x , z ) = U 0 ( x ) exp ( j β 0 z ) + n = 0 K R n U n ( x ) exp ( + j β n z ) + m = 1 2 ρ m + d ρ R m ( ρ ) Ψ m ( x , ρ ) exp [ + j β ( ρ ) z ] , ( for z < 0 ) ,
Φ II ( x , z ) = l = 1 2 0 + d s T l ( s ) ϕ l ( x , s ) exp [ j γ ( s ) z ] , ( for z > 0 ) ,
+ d x U n ( x ) U n ( x ) = δ n n ,
+ d x Ψ m ( x , ρ ) Ψ m ( x , ρ ) = δ m m δ ( ρ ρ ) ,
+ d x U n ( x ) Ψ m ( x , ρ ) = 0 ,
n = 0 K U n ( x ) U n ( x ) + 0 + d ρ m = 1 2 Ψ m ( x , ρ ) Ψ m ( x , ρ ) = δ ( x x ) ,
+ d x ϕ l ( x , s ) ϕ q ( x , s ) = δ l q δ ( s s ) ,
0 + d s l = 1 2 ϕ l ( x , s ) ϕ l ( x , s ) = δ ( x x ) ,
Ψ m ( x , ρ ) = { C 1 { exp [ j ρ ( x + d ) ] + s m exp [ j ρ ( x + d ) ] } , x < d A 0 + B 0 ( x d ) + n = 1 N B n sin [ n π ( x + d ) 2 d ] , x < d C 3 { exp [ j q ( x d ) ] + s m exp [ j q ( x d ) ] } , x > d } ,
β ¯ = β 0 , γ ¯ = 0 + d s γ ( s ) l = 1 2 U 0 ( x ) , ϕ l ( x , s ) 2
f ( x ) , g ( x ) = + d x f ( x ) g ( x ) .
E 0 ( x ) = 2 β 0 β ¯ + γ ¯ U 0 ( x ) ,
E 1 ( x ) = E 0 ( x ) + 2 β 0 ( β ¯ + γ ¯ ) 2 0 + d s [ γ ¯ γ ( s ) ] l = 1 2 ϕ l ( x , s ) U 0 ( x ) , ϕ l ( x , s ) ,
E 2 ( x ) = E 1 ( x ) + 2 β 0 ( β ¯ + γ ¯ ) 3 n = 0 K ( β ¯ β n ) U n ( x ) 0 + d s [ γ ¯ γ ( s ) ] l = 1 2 U 0 ( x ) , ϕ l ( x , s ) U n ( x ) , ϕ l ( x , s ) + 2 β 0 ( β ¯ + γ ¯ ) 3 0 + d s [ γ ¯ γ ( s ) ] l = 1 2 U 0 ( x ) , ϕ l ( x , s ) m = 1 2 ρ m + d ρ [ β β ( ρ ) ] Ψ m ( x , ρ ) Ψ m ( x , ρ ) , ϕ l ( x , s ) + 2 β 0 ( β ¯ + γ ¯ ) 3 0 + d s [ γ ¯ γ ( s ) ] 2 l = 1 2 ϕ l ( x , s ) U 0 ( x ) , ϕ l ( x , s ) ,
R n ( 0 ) = δ n 0 + 2 β β ¯ + γ ¯ δ n 0 ,
R n ( 1 ) = R n ( 0 ) + 2 β 0 ( β ¯ + γ ¯ ) 2 0 + d s [ γ ¯ γ ( s ) ] l = 1 2 U 0 ( x ) , ϕ l ( x , s ) U n ( x ) , ϕ l ( x , s ) ,
R n ( 2 ) = R n ( 1 ) + 2 β 0 ( β ¯ + γ ¯ ) 3 ( β ¯ β n ) 0 + d s [ γ ¯ γ ( s ) ] l = 1 2 U 0 ( x ) , ϕ l ( x , s ) U n ( x ) , ϕ l ( x , s ) + 2 β 0 ( β ¯ + γ ¯ ) 3 0 + d s [ γ ¯ γ ( s ) ] 2 l = 1 2 U 0 ( x ) , ϕ l ( x , s ) U n ( x ) , ϕ l ( x , s ) .
n ( x ) = n 2 m 1 2 Δ 12 ( x d ) 2 , x < d ,
n ( x ) = n 2 m 1 [ 1 ( n 1 n 2 ) 2 ] [ ( d x ) 2 d ] , x < d .
1 R 0 ( v ) 1 + R 0 ( v ) = 1 β 0 0 + d s γ ( s ) l = 1 2 U 0 ( x ) , ϕ l ( x , s ) 2 ,
min + U 0 ( x ) V G ( x , w ) 2 d x ;
1 R 0 ( v ) 1 + R 0 ( v ) = k 0 w 2 π n 2 m 2 2 w 2 0 + γ τ exp [ ( k 0 w ) 2 τ 2 2 ] d τ ,
R 0 ( v ) R 0 = ( n 2 m n 0 ) ( n 2 m + n 0 ) , k 0 w .
1 R 0 ( v ) 1 + R 0 ( v ) = ( n 0 n 2 m ) 1 0.5 ( k 0 w n 0 ) 2 1 1 ( k 0 w n 2 m ) 2 .
E exp [ j k 0 n 0 r ( k 0 w sin θ ) 2 4 ] k 0 r ,
R 0 n 1 n 0 n 1 + n 0 + h 1 2 k 0 2 ( n 1 + n 0 ) ( n 1 n 0 + n 0 n 1 ) .
( 2 Δ 12 ) 1 2 k 0 d n 2 m = ( k 0 w ) 2 .

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