Abstract

Nonlinear optical channel waveguides are modeled. It is demonstrated that a saturable nonlinear permittivity model is essential mathematically when strong nonlinear effects are simulated, so that the nonlinear wave equation possesses realistic solutions. Several precautionary factors in the numerical simulation of self-focusing behavior are addressed. For example, care must be exercised when the threshold power is calculated for some nonlinear structure exhibiting an abrupt all-optical switching phenomenon.

© 1993 Optical Society of America

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  1. C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593–599 (1985).
  2. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
    [CrossRef]
  3. G. I. Stegeman, R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  4. X. H. Wang, “Finite element methods for nonlinear optical waveguides,” Ph.D. dissertation (Monash University, Clayton, Victoria, Australia, 1992).
  5. K. Hayata, M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
    [CrossRef]
  6. X. H. Wang, L. N. Binh, G. K. Cambrell, “Vectorial finite-element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1988), pp. 129–132.
  7. X. H. Wang, G. K. Cambrell, L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of the IREECON International 1989 (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), pp. 551–554.
  8. N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
    [CrossRef]
  9. R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
    [CrossRef]
  10. N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
    [CrossRef]
  11. N. N. Akhmediev, R. F. Naviev, Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
    [CrossRef]
  12. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
    [CrossRef] [PubMed]
  13. B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
    [CrossRef]
  14. X. H. Wang, G. K. Cambrell, L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed. (Computational Mechanics Publications, Boston, Mass., 1990), pp. 151–162.
  15. All the powers in the power-dispersion relations in Ref. 5 should be scaled down by a factor of 2 [K. Hayata, Department of Electrical Engineering, Hokkaido University, Sapporo, Hokkaido 060, Japan (personal communication)]. The corrected values are used here to facilitate the comparison.
  16. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).
  17. U. Langbein, F. Lederer, T. Peschel, H.-E. Ponath, “Nonlinear guided waves in saturable nonlinear media,” Opt. Lett. 10, 571–573 (1985).
    [CrossRef] [PubMed]
  18. G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
    [CrossRef]
  19. S. J. Al–Bader, H. A. Jamid, “Nonlinear waves in saturable self-focusing thin films bounded by linear media,” IEEE J. Quantum Electron. 24, 2052–2058 (1988).
    [CrossRef]
  20. R. Cuykendall, K. H. Strobl, “Effects of soft saturation on nonlinear interface switching,” Phys. Rev. A 41, 352–358 (1990).
    [CrossRef] [PubMed]
  21. X. H. Wang, L. N. Binh, G. K. Cambrell, “Numerical analysis of a nonlinear optical channel waveguide,” in Proceedings of the 14th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), p. 225–228.
  22. S. J. Al–Bader, H. A. Jamid, “Guided waves in nonlinear saturable self-focusing thin films,” IEEE J. Quantum Electron. QE-23, 1947–1955 (1987).
    [CrossRef]
  23. W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
    [CrossRef]
  24. S. Y. Auyang, P. A. Wolff, “Free-carrier-induced third-order optical nonlinearities in semiconductors,” J. Opt. Soc. Am. B 6, 595–605 (1989).
    [CrossRef]
  25. J. L. Coutaz, M. Kull, “Saturation of the nonlinear index of refraction in semiconductor-doped glass,” J. Opt. Soc. Am. B 8, 95–98 (1991).
    [CrossRef]
  26. X. H. Wang, G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I Power dispersion relations,” in Proceedings of the 16th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1991), pp. 314–317.
  27. X. H. Wang, G. K. Cambrell, “Full vectorial simulation of bistability phenomena in nonlinear optical channel waveguides,” J. Opt. Soc. Am. B 10, 1090–1095 (1993).
    [CrossRef]

1993

1991

J. L. Coutaz, M. Kull, “Saturation of the nonlinear index of refraction in semiconductor-doped glass,” J. Opt. Soc. Am. B 8, 95–98 (1991).
[CrossRef]

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

1990

R. Cuykendall, K. H. Strobl, “Effects of soft saturation on nonlinear interface switching,” Phys. Rev. A 41, 352–358 (1990).
[CrossRef] [PubMed]

1989

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

S. Y. Auyang, P. A. Wolff, “Free-carrier-induced third-order optical nonlinearities in semiconductors,” J. Opt. Soc. Am. B 6, 595–605 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Naviev, Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[CrossRef]

G. I. Stegeman, R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

1988

K. Hayata, M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

S. J. Al–Bader, H. A. Jamid, “Nonlinear waves in saturable self-focusing thin films bounded by linear media,” IEEE J. Quantum Electron. 24, 2052–2058 (1988).
[CrossRef]

1987

S. J. Al–Bader, H. A. Jamid, “Guided waves in nonlinear saturable self-focusing thin films,” IEEE J. Quantum Electron. QE-23, 1947–1955 (1987).
[CrossRef]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[CrossRef] [PubMed]

1986

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

1985

U. Langbein, F. Lederer, T. Peschel, H.-E. Ponath, “Nonlinear guided waves in saturable nonlinear media,” Opt. Lett. 10, 571–573 (1985).
[CrossRef] [PubMed]

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593–599 (1985).

1984

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Naviev, Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Al–Bader, S. J.

S. J. Al–Bader, H. A. Jamid, “Nonlinear waves in saturable self-focusing thin films bounded by linear media,” IEEE J. Quantum Electron. 24, 2052–2058 (1988).
[CrossRef]

S. J. Al–Bader, H. A. Jamid, “Guided waves in nonlinear saturable self-focusing thin films,” IEEE J. Quantum Electron. QE-23, 1947–1955 (1987).
[CrossRef]

Auyang, S. Y.

Banyai, W. C.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

Binh, L. N.

X. H. Wang, L. N. Binh, G. K. Cambrell, “Numerical analysis of a nonlinear optical channel waveguide,” in Proceedings of the 14th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), p. 225–228.

X. H. Wang, G. K. Cambrell, L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed. (Computational Mechanics Publications, Boston, Mass., 1990), pp. 151–162.

X. H. Wang, G. K. Cambrell, L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of the IREECON International 1989 (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), pp. 551–554.

X. H. Wang, L. N. Binh, G. K. Cambrell, “Vectorial finite-element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1988), pp. 129–132.

Boardman, A. D.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[CrossRef] [PubMed]

Cambrell, G. K.

X. H. Wang, G. K. Cambrell, “Full vectorial simulation of bistability phenomena in nonlinear optical channel waveguides,” J. Opt. Soc. Am. B 10, 1090–1095 (1993).
[CrossRef]

X. H. Wang, G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I Power dispersion relations,” in Proceedings of the 16th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1991), pp. 314–317.

X. H. Wang, L. N. Binh, G. K. Cambrell, “Vectorial finite-element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1988), pp. 129–132.

X. H. Wang, G. K. Cambrell, L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of the IREECON International 1989 (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), pp. 551–554.

X. H. Wang, G. K. Cambrell, L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed. (Computational Mechanics Publications, Boston, Mass., 1990), pp. 151–162.

X. H. Wang, L. N. Binh, G. K. Cambrell, “Numerical analysis of a nonlinear optical channel waveguide,” in Proceedings of the 14th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), p. 225–228.

Coutaz, J. L.

Cullen, T. G.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

Cuykendall, R.

R. Cuykendall, K. H. Strobl, “Effects of soft saturation on nonlinear interface switching,” Phys. Rev. A 41, 352–358 (1990).
[CrossRef] [PubMed]

Davies, J. B.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Ettinger, R. D.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

Fernandez, F. A.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

Finlayson, N.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Hayata, K.

K. Hayata, M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
[CrossRef]

All the powers in the power-dispersion relations in Ref. 5 should be scaled down by a factor of 2 [K. Hayata, Department of Electrical Engineering, Hokkaido University, Sapporo, Hokkaido 060, Japan (personal communication)]. The corrected values are used here to facilitate the comparison.

Ironside, C. N.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

Jamid, H. A.

S. J. Al–Bader, H. A. Jamid, “Nonlinear waves in saturable self-focusing thin films bounded by linear media,” IEEE J. Quantum Electron. 24, 2052–2058 (1988).
[CrossRef]

S. J. Al–Bader, H. A. Jamid, “Guided waves in nonlinear saturable self-focusing thin films,” IEEE J. Quantum Electron. QE-23, 1947–1955 (1987).
[CrossRef]

Koshiba, M.

Kull, M.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Langbein, U.

Lederer, F.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Mai, Xu

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593–599 (1985).

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Moloney, J. V.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Nabiev, R. F.

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Naviev, R. F.

N. N. Akhmediev, R. F. Naviev, Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[CrossRef]

O’Neill, M.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

Peschel, T.

Ponath, H.-E.

Popov, Yu. M.

N. N. Akhmediev, R. F. Naviev, Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[CrossRef]

N. N. Akhmediev, R. F. Nabiev, Yu. M. Popov, “Stripe nonlinear surface waves,” Solid State Commun. 66, 981–985 (1988).
[CrossRef]

Rahman, B. M. A.

R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
[CrossRef]

B. M. A. Rahman, J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Seaton, C. T.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593–599 (1985).

Shen, T. P.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Stegeman, G. I.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
[CrossRef]

G. I. Stegeman, R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593–599 (1985).

Stolen, R. H.

Strobl, K. H.

R. Cuykendall, K. H. Strobl, “Effects of soft saturation on nonlinear interface switching,” Phys. Rev. A 41, 352–358 (1990).
[CrossRef] [PubMed]

Twardowski, T.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[CrossRef] [PubMed]

Wallis, R. F.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Wang, X. H.

X. H. Wang, G. K. Cambrell, “Full vectorial simulation of bistability phenomena in nonlinear optical channel waveguides,” J. Opt. Soc. Am. B 10, 1090–1095 (1993).
[CrossRef]

X. H. Wang, G. K. Cambrell, “All-optical switching and bistability phenomena in nonlinear optical waveguides: Part I Power dispersion relations,” in Proceedings of the 16th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1991), pp. 314–317.

X. H. Wang, L. N. Binh, G. K. Cambrell, “Numerical analysis of a nonlinear optical channel waveguide,” in Proceedings of the 14th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), p. 225–228.

X. H. Wang, G. K. Cambrell, L. N. Binh, “A package for nonlinear optical waveguides based on E-vector finite elements,” in Advances in Electrical Engineering Software, P. P. Silvester, ed. (Computational Mechanics Publications, Boston, Mass., 1990), pp. 151–162.

X. H. Wang, “Finite element methods for nonlinear optical waveguides,” Ph.D. dissertation (Monash University, Clayton, Victoria, Australia, 1992).

X. H. Wang, G. K. Cambrell, L. N. Binh, “Scalar and vector formulations of nonlinear optical waveguides: a comparison,” in Proceedings of the IREECON International 1989 (Institution of Radio and Electronics Engineers, Sydney, Australia, 1989), pp. 551–554.

X. H. Wang, L. N. Binh, G. K. Cambrell, “Vectorial finite-element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1988), pp. 129–132.

Winful, H. G.

C. T. Seaton, Xu Mai, G. I. Stegeman, H. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593–599 (1985).

Wolff, P. A.

Wright, E. M.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Appl. Phys. Lett.

W. C. Banyai, N. Finlayson, C. T. Seaton, G. I. Stegeman, M. O’Neill, T. G. Cullen, C. N. Ironside, “Saturation of the nonlinear refractive-index change in a semiconductor-doped glass channel waveguide,” Appl. Phys. Lett. 54, 481–483 (1989).
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S. J. Al–Bader, H. A. Jamid, “Nonlinear waves in saturable self-focusing thin films bounded by linear media,” IEEE J. Quantum Electron. 24, 2052–2058 (1988).
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R. D. Ettinger, F. A. Fernandez, B. M. A. Rahman, J. B. Davies, “Vector finite element solution of saturable nonlinear strip-loaded optical waveguides,” IEEE Photon. Technol. Lett. 3, 147–149 (1991).
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G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
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X. H. Wang, L. N. Binh, G. K. Cambrell, “Vectorial finite-element methods for nonlinear optical waveguides,” in Proceedings of the 13th Australian Conference on Optical Fibre Technology (Institution of Radio and Electronics Engineers, Sydney, Australia, 1988), pp. 129–132.

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All the powers in the power-dispersion relations in Ref. 5 should be scaled down by a factor of 2 [K. Hayata, Department of Electrical Engineering, Hokkaido University, Sapporo, Hokkaido 060, Japan (personal communication)]. The corrected values are used here to facilitate the comparison.

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Figures (15)

Fig. 1
Fig. 1

Classical structure of a rectangular channel waveguide with electric wall boundaries.

Fig. 2
Fig. 2

Mesh network of the rectangular channel waveguide structure (116 second-order elements, 265 nodes).

Fig. 3
Fig. 3

Magnetic field distributions of the fundamental mode ( H 11 x or E 11 y) for the rectangular channel waveguide in the linear case.

Fig. 4
Fig. 4

Power dispersion curve of the fundamental nonlinear mode. The effective index is β/k0, and the optical power is in units of microwatts.

Fig. 5
Fig. 5

Magnetic field distributions of the fundamental nonlinear mode at P = 80 μW, with the mesh shown in Fig. 2.

Fig. 6
Fig. 6

Refined mesh network of the rectangular channel waveguide structure (176 second-order elements, 389 nodes).

Fig. 7
Fig. 7

Power-dispersion curve of the fundamental nonlinear mode corresponding to both the original and the refined meshes. The effective index is β/k0, and the optical power is in units of microwatts.

Fig. 8
Fig. 8

Convergence behavior of the effective index during the nonlinear iteration procedure for P = 80 μW, with both the original and the refined meshes.

Fig. 9
Fig. 9

Magnetic field distributions of the fundamental nonlinear mode at P = 80 μW, with the refined mesh after 23 nonlinear iterations.

Fig. 10
Fig. 10

Comparison of the magnetic field magnitude distributions of the fundamental nonlinear mode ||H||2 at P = 80 μW resulting from (a) the original mesh and (b) the refined mesh after 23 nonlinear iterations.

Fig. 11
Fig. 11

Convergence behavior of the effective index with respect to mesh refinement for P = 60 μW.

Fig. 12
Fig. 12

Convergence behavior of the effective index during the nonlinear iteration procedure for P = 80 μW with three different mesh structures, where the three mesh structures of second-order triangular elements are 166 elements with 367 nodes, 216 elements with 475 nodes, and 270 elements with 585 nodes.

Fig. 13
Fig. 13

Power-dispersion curve of the fundamental nonlinear mode corresponding to both the original and the refined meshes for the model with saturation. The optical power is in units of microwatts.

Fig. 14
Fig. 14

Magnetic field distributions of the fundamental nonlinear mode for the saturating model at P = 80 μW, with the original mesh.

Fig. 15
Fig. 15

Magnetic field distributions of the fundamental nonlinear mode for the saturating model at P = 80 μW, with the refined mesh.

Equations (13)

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ˆ r = ˆ r l = ˆ r n .
ˆ r n = [ 1 n 0 0 0 2 n 0 0 0 3 n ] .
i n = a ( | E i | 2 + b j = i ( j i ) 3 | E j | 2 ) , i = 1 , 2 , 3 ,
a = c 0 0 r l n 2 ,
b = { 1 electrostriction and heating 1 3 electronic distortion 1 2 molecular orientation .
× ( ˆ r 1 · × H ) p μ ˆ r · · ( μ ˆ r · H ) = k 0 2 μ ˆ r · H ,
P = Z 0 2 k 0 ( { Ω [ H * × ( ˆ r 1 · × H ) ] · a z d x d y } ) ,
r l = { 1.55 2 core 1.55 2 cladding .
λ i + 1 λ i λ i + 1 1.0 × 10 6 ,
r n = a | E | 2 1 + α a | E | 2
r n = 1 α [ 1 exp ( α a | E | 2 ) ] ,
i n = a f i ( E ) 1 + α a f i ( E ) , i = 1 , 2 , 3
i n = a ( E 2 ) 2 1 + α a ( E 2 ) 2 , i = 1 , 2 , 3 ,

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