Abstract

For construction of the TE radiation modes of planar waveguides several methods are employed that are based on collocation techniques. The field representation in the core is based on the Lanczos–Fourier sinusoidal series. The numerical codes are very simple and give accurate results. The validity of these methods is checked for constant refractive-index profiles, while numerical results are also given for parabolic profiles. Furthermore, the validity of the orthogonality condition between the guided and the radiation modes is checked. These methods are demonstrated to be effective and can also be employed to study the TM case and waveguides of lossy media, as well as anisotropic and chiral structures.

© 2004 Optical Society of America

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References

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  1. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
    [CrossRef]
  2. R. M. Knox, P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, Brooklyn, N.Y., 1970), pp. 497–516.
  3. A. B. Manenkov, “Reflection of the surface mode from an abruptly ended W-fibre,” IEE Proc. J. Optoelectron. 139, 101–104 (1992).
    [CrossRef]
  4. T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
    [CrossRef]
  5. H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
    [CrossRef]
  6. M. Munowitz, D. J. Vezzetti, “Numerical modeling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. LT-10, 1570–1573 (1992).
    [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. S. Stern, “Semi-vectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 333–338 (1988).
    [CrossRef]
  9. S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
    [CrossRef]
  10. W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4, 148–151 (1992).
    [CrossRef]
  11. C. J. Smartt, T. M. Benson, P. C. Kendall, “Free-space radiation mode method for the analysis of propagation in optical waveguide devices,” IEE Proc. J. Optoelectron. 140, 56–61 (1993).
    [CrossRef]
  12. F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Research Studies Press, Hertfordshire, UK, 1996).
  13. A. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).
  14. D. N. Chien, M. Tanaka, 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]
  15. A. B. Manenkov, “Reflection of the surface mode from an abrupt ended dielectric tube waveguide,” IEE Proc.-J: Optoelectron. 139, 194–200 (1992).
  16. A. V. Brovko, A. B. Manenkov, A. G. Rozhnev, “FDTD-analysis of the wave diffraction from dielectric waveguide discontinuities,” Opt. Quantum Electron. 35, 395–406 (2003).
    [CrossRef]
  17. I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  18. I. G. Tigelis, 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]
  19. A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
    [CrossRef]
  20. V. V. Shevchenko, “On the completeness of spectral expansion of the electromagnetic field in the set of dielectric circular rod waveguide eigenwaves,” Radio Sci. 17, 229–231 (1982).
    [CrossRef]
  21. I. G. Tigelis, C. N. Capsalis, N. K. Uzunoglu, “Computation of the dielectric rod waveguide radiation modes,” Int. J. Infrared Millim. Waves 8, 1053–1068 (1987).
    [CrossRef]
  22. N. Morita, “Radiation modes of a circular dielectric waveguide,” J. Electromagn. Waves Appl. 2, 445–457 (1988).
    [CrossRef]
  23. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991), pp. 19–30.
  24. J. J. Burke, “Simple formulation of radiation modes in planar multilayer waveguides,” J. Opt. Soc. Am. A 11, 2481–2484 (1994).
    [CrossRef]
  25. T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1810–1824 (1995).
    [CrossRef]
  26. A. B. Manenkov, “Excitation of open homogeneous waveguides,” Radiophys. Quantum Electron. 13, 578–586 (1970).
    [CrossRef]
  27. A. B. Manenkov, “Orthogonality relation for the eigenmodes of lossy anisotropic waveguides (fibres),” IEE Proc. J. Optoelectron. 140, 206–212 (1993).
    [CrossRef]
  28. D. P. Nyquist, D. R. Johnson, S. V. Hsu, “Orthogonality and amplitude spectrum of radiation modes along open-boundary waveguides,” J. Opt. Soc. Am. 71, 49–54 (1981).
    [CrossRef]
  29. R. A. Sammut, “Orthogonality and normalization of radiation modes in dielectric waveguides,” J. Opt. Soc. Am. 72, 1335–1337 (1982).
    [CrossRef]
  30. C. Vassallo, “Orthogonality and normalization of radiation modes in dielectric waveguides: an alternate approach,” J. Opt. Soc. Am. 73, 680–683 (1983).
    [CrossRef]
  31. C. Lanczos, Applied Analysis (Prentice Hall, Englewood Cliffs, N. J., 1956).
  32. P. A. Koukoutsaki, I. G. Tigelis, A. B. Manenkov, “Guided-mode analysis by the Lanczos-Fourier expansion,” J. Opt. Soc. Am. A 19, 2293–2300 (2002).
    [CrossRef]
  33. A. B. Manenkov, “Eigenmodes expansion in lossy open waveguides,” Opt. Quantum Electron. 23, 621–632 (1991).
    [CrossRef]
  34. G. E. Forsythe, W. R. Wasow, Finite Difference Methods for Partial Differential Equations (Wiley, New York, 1960).
  35. J. L. Altman, Microwave Circuits (Nostrand, Princeton, N.J., 1964).

2003 (1)

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

2002 (2)

2000 (1)

1999 (2)

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. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).

1995 (1)

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1810–1824 (1995).
[CrossRef]

1994 (1)

1993 (2)

A. B. Manenkov, “Orthogonality relation for the eigenmodes of lossy anisotropic waveguides (fibres),” IEE Proc. J. Optoelectron. 140, 206–212 (1993).
[CrossRef]

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

1992 (4)

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

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

M. Munowitz, D. J. Vezzetti, “Numerical modeling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. LT-10, 1570–1573 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4, 148–151 (1992).
[CrossRef]

1991 (1)

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

1989 (1)

S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
[CrossRef]

1988 (2)

M. S. Stern, “Semi-vectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 333–338 (1988).
[CrossRef]

N. Morita, “Radiation modes of a circular dielectric waveguide,” J. Electromagn. Waves Appl. 2, 445–457 (1988).
[CrossRef]

1987 (1)

I. G. Tigelis, C. N. Capsalis, N. K. Uzunoglu, “Computation of the dielectric rod waveguide radiation modes,” Int. J. Infrared Millim. Waves 8, 1053–1068 (1987).
[CrossRef]

1983 (1)

1982 (3)

1981 (1)

1978 (2)

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

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

1971 (1)

A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
[CrossRef]

1970 (2)

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

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

Altman, J. L.

J. L. Altman, Microwave Circuits (Nostrand, Princeton, N.J., 1964).

Baets, R.

Benson, T. M.

A. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).

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

Brovko, A. V.

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

Burke, J. J.

Burke, S. V.

S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
[CrossRef]

Capsalis, C. N.

I. G. Tigelis, C. N. Capsalis, N. K. Uzunoglu, “Computation of the dielectric rod waveguide radiation modes,” Int. J. Infrared Millim. Waves 8, 1053–1068 (1987).
[CrossRef]

Chaundhuri, S. K.

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Chien, D. N.

Fernandez, F.

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

Forsythe, G. E.

G. E. Forsythe, W. R. Wasow, Finite Difference Methods for Partial Differential Equations (Wiley, New York, 1960).

Hsu, S. V.

Huang, W. P.

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Johnson, D. R.

Kendall, P. C.

A. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).

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

Knox, R. M.

R. M. Knox, P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, Brooklyn, N.Y., 1970), pp. 497–516.

Koukoutsaki, P. A.

Lagasse, P. E.

Lanczos, C.

C. Lanczos, Applied Analysis (Prentice Hall, Englewood Cliffs, N. J., 1956).

Lu, Y.

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

Manenkov, A. B.

A. V. Brovko, A. B. Manenkov, 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, A. B. Manenkov, “Guided-mode analysis by the Lanczos-Fourier expansion,” J. Opt. Soc. Am. A 19, 2293–2300 (2002).
[CrossRef]

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

A. B. Manenkov, “Orthogonality relation for the eigenmodes of lossy anisotropic waveguides (fibres),” IEE Proc. J. Optoelectron. 140, 206–212 (1993).
[CrossRef]

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

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

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

A. B. Manenkov, “Excitation of open homogeneous waveguides,” Radiophys. Quantum Electron. 13, 578–586 (1970).
[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), pp. 19–30.

Morita, N.

N. Morita, “Radiation modes of a circular dielectric waveguide,” J. Electromagn. Waves Appl. 2, 445–457 (1988).
[CrossRef]

Munowitz, M.

M. Munowitz, D. J. Vezzetti, “Numerical modeling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. LT-10, 1570–1573 (1992).
[CrossRef]

Nyquist, D. P.

Rozhnev, A. G.

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

Rozzi, T. E.

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

Sammut, R. A.

Sewell, P.

A. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).

Shevchenko, V. V.

V. V. Shevchenko, “On the completeness of spectral expansion of the electromagnetic field in the set of dielectric circular rod waveguide eigenwaves,” Radio Sci. 17, 229–231 (1982).
[CrossRef]

Smartt, C. J.

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

Snyder, A. W.

A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
[CrossRef]

Stern, M. S.

M. S. Stern, “Semi-vectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 333–338 (1988).
[CrossRef]

Tanaka, K.

Tanaka, M.

Theodoropoulos, T. G.

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1810–1824 (1995).
[CrossRef]

Tigelis, I. G.

Toulios, P. P.

R. M. Knox, P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, Brooklyn, N.Y., 1970), pp. 497–516.

Uzunoglu, N. K.

I. G. Tigelis, C. N. Capsalis, N. K. Uzunoglu, “Computation of the dielectric rod waveguide radiation modes,” Int. J. Infrared Millim. Waves 8, 1053–1068 (1987).
[CrossRef]

Vassallo, C.

Vezzetti, D. J.

M. Munowitz, D. J. Vezzetti, “Numerical modeling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. LT-10, 1570–1573 (1992).
[CrossRef]

Vukovic, A.

A. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).

Wasow, W. R.

G. E. Forsythe, W. R. Wasow, Finite Difference Methods for Partial Differential Equations (Wiley, New York, 1960).

Xu, C. L.

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4, 148–151 (1992).
[CrossRef]

Yajima, H.

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

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

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

Electron. Lett. (1)

S. V. Burke, “Spectral index method applied to coupled rib waveguides,” Electron. Lett. 25, 605–606 (1989).
[CrossRef]

IEE Proc. J. Optoelectron. (4)

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

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

M. S. Stern, “Semi-vectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 333–338 (1988).
[CrossRef]

A. B. Manenkov, “Orthogonality relation for the eigenmodes of lossy anisotropic waveguides (fibres),” IEE Proc. J. Optoelectron. 140, 206–212 (1993).
[CrossRef]

IEE Proc.-J: Optoelectron. (1)

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

IEEE J. Quantum Electron. (1)

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

IEEE Photonics Technol. Lett. (1)

W. P. Huang, C. L. Xu, S. K. Chaundhuri, “A finite difference vector beam propagation method for three dimensional waveguide structures,” IEEE Photonics Technol. Lett. 4, 148–151 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
[CrossRef]

Int. J. Infrared Millim. Waves (2)

I. G. Tigelis, C. N. Capsalis, N. K. Uzunoglu, “Computation of the dielectric rod waveguide radiation modes,” Int. J. Infrared Millim. Waves 8, 1053–1068 (1987).
[CrossRef]

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1810–1824 (1995).
[CrossRef]

J. Electromagn. Waves Appl. (1)

N. Morita, “Radiation modes of a circular dielectric waveguide,” J. Electromagn. Waves Appl. 2, 445–457 (1988).
[CrossRef]

J. Lightwave Technol. (1)

M. Munowitz, D. J. Vezzetti, “Numerical modeling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. LT-10, 1570–1573 (1992).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Quantum Electron. (3)

A. Vukovic, P. Sewell, T. M. Benson, P. C. Kendall, “Novel half space radiation mode method for buried waveguide analysis,” Opt. Quantum Electron. 31, 41–51 (1999).

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

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

Radio Sci. (1)

V. V. Shevchenko, “On the completeness of spectral expansion of the electromagnetic field in the set of dielectric circular rod waveguide eigenwaves,” Radio Sci. 17, 229–231 (1982).
[CrossRef]

Radiophys. Quantum Electron. (1)

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

Other (6)

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991), pp. 19–30.

G. E. Forsythe, W. R. Wasow, Finite Difference Methods for Partial Differential Equations (Wiley, New York, 1960).

J. L. Altman, Microwave Circuits (Nostrand, Princeton, N.J., 1964).

C. Lanczos, Applied Analysis (Prentice Hall, Englewood Cliffs, N. J., 1956).

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

R. M. Knox, P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proceedings of the MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, Brooklyn, N.Y., 1970), pp. 497–516.

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Figures (10)

Fig. 1
Fig. 1

Geometry of a dielectric slab waveguide with inhomogeneous core; the arrows describe incident, reflected, and scattered waves in the substrate and cladding.

Fig. 2
Fig. 2

Shifted phase ψ˜(p)=ψ(p)+2pd versus p/(k0n1) for a symmetrical slab waveguide with n2=3.6 and n1=n3=3.24.

Fig. 3
Fig. 3

Variation of Re[Ey(x)] of the even radiation mode with the normalized transverse distance x/d for a slab waveguide with the parameters of Fig. 2 and p=0.1k0n1.

Fig. 4
Fig. 4

Eigenfunction Ψ(x, p) (single eigenvalue ψ=4.026349 for p=0.5k0n1) for an asymmetrical slab waveguide with n(x)=n2=3.6, n1=3.24, and n3=2.52.

Fig. 5
Fig. 5

Eigenfunction Ψ(x, p) (first eigenvalue ψ=0.262228 for p=2k0n1) for an asymmetrical slab waveguide with n(x)=n2=3.6, n1=3.24, and n3=2.52.

Fig. 6
Fig. 6

Variation of Q(p) with N for an asymmetrical slab waveguide with n(x)=n2=3.6, n1=3.24, n3=2.52, for the (a) first and (b) second eigenvalue for p=2k0n1.

Fig. 7
Fig. 7

Eigenfunction Ψ(x, p) (single eigenvalue ψ=4.026349 for p=0.5k0n1) for an asymmetrical slab waveguide with n(x)=n2=3.6, n1=3.24, and n3=2.52.

Fig. 8
Fig. 8

Eigenfunction Ψ(x, p) (first eigenvalue ψ=0.262228 for p=2k0n1) for an asymmetrical slab waveguide with n(x)=n2=3.6, n1=3.24, and n3=2.52.

Fig. 9
Fig. 9

Eigenfunction Ψ(x, p) (even mode, first eigenvalue ψ=4.466911 for p=2k0n1) for a symmetrical slab waveguide with n2=3.6 and n1=n3=3.24.

Fig. 10
Fig. 10

Eigenfunction Ψ(x, p) (odd mode, second eigenvalue ψ=1.965969 for p=2k0n1) for a symmetrical slab waveguide with n2=3.6 and n1=n3=3.24.

Tables (3)

Tables Icon

Table 1 Convergence of the First Approach with the Number of Nodes N for p=0.5k0n1

Tables Icon

Table 2 Convergence of the First Approach with N for p=2k0n1

Tables Icon

Table 3 Convergence of the FDM with N for p= 0.5k0n1 for a Parabolic Profile

Equations (44)

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

d2Ψ(x, p)dx2+[k02n(x)-β2(p)]Ψ(x, p)=0.
Ψ(x, p)=A0+B0xd+n=1NBnsinnπ2d (x+d),
xj=-d+j 2dN+1,j=1,2,,N.
A0Rj+B0xjdRj+n=1NBnTj,n=0,j=1,2,,N,
Rj=k02n2(xj)-k02n12+p2,
Tj,n=sinnπ2d (xj+d) Rj-nπ2d2.
Ψ(x, p)
=u1exp[-jp(x+d)]+υ1exp[+jp(x+d)],x<-du3exp[+jq(x-d)]+υ3exp[-jq(x-d)],x>d.
SU=V,
S(C1, C3)T=s(p)(C1, C3)T,
s1,2=12{S11+S33±[(S11-S33)2+4S13S31]1/2}.
Ψ(x, p)
=C1{exp[-jp(x+d)]+s(p)exp[+jp(x+d)]},x<-dC3{exp[+jp(x-d)]+s(p)exp[-jp(x-d)]},x>d
Ψ(x, p)
=exp[-jp(x+d)]+S11exp[jp(x+d)],x<-dA0+B0xd+n=1NBnsinnπ(x+d)2d,|x|<dS31exp[-jq(x-d)],x>d.
1+S11=A0-B0,
-jp(1-S11)=B0d+n=1NBnnπ2d,
S31=A0+B0,
-jqS31=B0d+n=1NBnnπ2d (-1)n.
Ψ(x, p)
=S13exp[jp(x+d)],x<-dA0+B0xd+n=1NBnsinnπ(x+d)2d,|x|<dexp[jq(x-d)]+S33exp[-jq(x-d)],x>d,
Ψ(x, p)
=C1{exp[-jp(x+d)]+s(p)exp[jp(x+d)]},x<-dA0+B0xd+n=1NBnsinnπ(x+d)2d,|x|<dC3{exp[jq(x-d)]+s(p)exp[-jq(x-d)]},x>d.
C3/C1=[s(p)-S11]/S13.
C1(1+s)=A0-B0,C3(1+s)=A0+B0.
Ψ(x, p)
=u1exp[-jp(x+d)]+υ1exp[+jp(x+d)],x<-dυ3exp[-t(x-d)],x>d,
Ψ(x, p)
=exp[-jp(x+d)]+S11exp[jp(x+d)],x<-dA0+B0xd+n=1NBnsinnπ(x+d)2d,|x|<dS31exp[-t(x-d)],x>d.
Ψ(x, p)
=C1{exp[-jp(x+d)]+s0(p)exp[jp(x+d)]},x<-dA0+B0xd+n=1NBnsinnπ(x+d)2d,|x|<dC3exp[-t(x-d)],x>d. 
Det{M[s(p)]}=0,
Ψj+1+(Rj-2)Ψj+Ψj-1=0.
Ψ(x, p)
=C2cos(σ0x),|x|<dC1{s2(p)exp[-jp(|x|-d)]+exp[jp(|x|-d)]},|x|>d
s2(p)=p cos σ0d-jσ0sin σ0dp cos σ0d+jσ0sin σ0d,
C2=[1+s2(p)]C1cos σ0d,
C1=1[4πs2(p)]1/2.
Ψm(x, p), Ψm(x, p)=-+Ψm(x, p)Ψm(x, p)dx=2πsm(p)(C12+qC32/p)×δmmδ(p-p),
limxbsin[(p-p)xb](p-p)=πδ(p-p),
δ(q-q)=|dp/dq|δ(p-p).
Ψm(x, p), Ψm(x, p)=δmmδ(p-p).
Ψm(x, p), Ψm*(x, p)=δmmδ(p-p).
Q(p)=1Ki=1K|ΨN(xi, p)-ΨNmax(xi, p)|2,

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