Abstract

TM modes in a nonlinear isotropic Kerr film embedded between two linear media are analyzed. It has been proved that an exact analytical solution that corresponds to a certain class of modes does exist in such a structure. This solution describes balanced modes: the guided electromagnetic fields have two electric components of equal amplitude. An analogous analytical but approximate solution has been derived for the fields close to the balanced modes. The closed-form mode equation and the power flow relation that correspond to such quasi-balanced modes have been obtained. The results are compared with the exact numerical solution and the transverse uniaxial approximation.

© 1995 Optical Society of America

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  1. C. T. Seaton, Xu Mai, G. I. Stegeman, and H. G. Winful, "Nonlinear guided waves applications," Opt. Eng. 24, 593–599 (1985).
  2. A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. QE-22, 319–324 (1986).
    [CrossRef]
  3. W. Chen and A. A. Maradudin, "s-Polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film," J. Opt. Soc. Am. B 5, 529–537 (1988).
    [CrossRef]
  4. S. Okafuji and Y. Nakai, "Nonlinear waves in asymmetric dielectric hollow waveguides with self-focusing Kerr-like nonlinear film," Inst. Electr. Eng. Proc. J 138, 204–210 (1991).
  5. J. Jasiński and K. Gniadek, "A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution," Opt. Quantum Electron. 26, 865–876 (1994).
    [CrossRef]
  6. F. Lederer, U. Langbein, and H.-E. Ponath, "Nonlinear waves guided by a dielectric slab: II. TM-polarization," Appl. Phys. B 31, 187–190 (1983).
    [CrossRef]
  7. C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, "Comparison of solutions for TM-polarized nonlinear guided waves," Opt. Lett. 10, 149–150 (1985).
    [CrossRef] [PubMed]
  8. G. I. Stegeman, C. T. Seaton, and J. Ariyasu, "Nonlinear electromagnetic waves guided by a single interface," J. Appl. Phys. 58, 2543–2549 (1985).
    [CrossRef]
  9. U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).
  10. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, "Exact theory of nonlinear p-Polarized optical waves," Phys. Rev. A 35, 1159–1164 (1987).
    [CrossRef] [PubMed]
  11. M. Yokota, "Guided transverse-magnetic waves supported by a weakly nonlinear slab waveguide," J. Opt. Soc. Am. B 10, 1096–1101 (1993).
    [CrossRef]
  12. K. M. Leung, "p-Polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions," Phys. Rev. B 32, 5093–5101 (1985).
    [CrossRef]
  13. Q. Chen and Zi Hua Wang, "Exact dispersion relations for TM waves guided by thin dielectric films bounded by nonlinear media," Opt. Lett. 18, 260–262 (1993).
    [CrossRef] [PubMed]
  14. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
    [CrossRef] [PubMed]
  15. R. I. Joseph and D. N. Christodoulides, "Exact field decomposition for TM waves in nonlinear media," Opt. Lett. 12, 826–828 (1987).
    [CrossRef] [PubMed]
  16. G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T.-P. Shen, A. A. Maradudin, and R. F. Wallis, "Nonlinear slab-guided waves in non-Kerr-like media," IEEE J. Quantum Electron. QE-22, 977–983 (1986).
    [CrossRef]
  17. A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
    [CrossRef]
  18. A. Groza and V. L. Strizhevskii, "p-Polarized nonlinear surface polaritons near the surface of a saturable defocusing medium," Opt. Commun. 91, 111–114 (1992).
    [CrossRef]
  19. K. Ogusu, "Nonlinear TE waves guided by graded-index planar waveguides," Opt. Commun. 63, 380–384 (1987).
    [CrossRef]
  20. S. J. Al-Bader, "TM waves in nonlinear saturable thin films: a multilayered approach," J. Lightwave Technol. 7, 717–725 (1989).
    [CrossRef]
  21. K. Ogusu, "TM waves guided by nonlinear planar waveguides," IEEE Trans. Microwave Theory Tech. 37, 941–946 (1989).
    [CrossRef]
  22. J. Jasiński, "Specific analytical TM solution in nonlinearly isotropic Kerr-like media," Opt. Appl. 24, 287–292 (1994).
  23. A. D. Boardman and T. Twardowski, "Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides," Phys. Rev. A 39, 2481–2492 (1989).
    [CrossRef] [PubMed]
  24. H. Kogelnik, "Theory of dielectric waveguides," in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), Chap. 2, pp. 34–39.

1994 (2)

J. Jasiński and K. Gniadek, "A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution," Opt. Quantum Electron. 26, 865–876 (1994).
[CrossRef]

J. Jasiński, "Specific analytical TM solution in nonlinearly isotropic Kerr-like media," Opt. Appl. 24, 287–292 (1994).

1993 (2)

1992 (1)

A. Groza and V. L. Strizhevskii, "p-Polarized nonlinear surface polaritons near the surface of a saturable defocusing medium," Opt. Commun. 91, 111–114 (1992).
[CrossRef]

1991 (1)

S. Okafuji and Y. Nakai, "Nonlinear waves in asymmetric dielectric hollow waveguides with self-focusing Kerr-like nonlinear film," Inst. Electr. Eng. Proc. J 138, 204–210 (1991).

1989 (3)

S. J. Al-Bader, "TM waves in nonlinear saturable thin films: a multilayered approach," J. Lightwave Technol. 7, 717–725 (1989).
[CrossRef]

K. Ogusu, "TM waves guided by nonlinear planar waveguides," IEEE Trans. Microwave Theory Tech. 37, 941–946 (1989).
[CrossRef]

A. D. Boardman and T. Twardowski, "Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides," Phys. Rev. A 39, 2481–2492 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (5)

K. Ogusu, "Nonlinear TE waves guided by graded-index planar waveguides," Opt. Commun. 63, 380–384 (1987).
[CrossRef]

U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).

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

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

R. I. Joseph and D. N. Christodoulides, "Exact field decomposition for TM waves in nonlinear media," Opt. Lett. 12, 826–828 (1987).
[CrossRef] [PubMed]

1986 (2)

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

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

1985 (4)

K. M. Leung, "p-Polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions," Phys. Rev. B 32, 5093–5101 (1985).
[CrossRef]

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

G. I. Stegeman, C. T. Seaton, and J. Ariyasu, "Nonlinear electromagnetic waves guided by a single interface," J. Appl. Phys. 58, 2543–2549 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, "Comparison of solutions for TM-polarized nonlinear guided waves," Opt. Lett. 10, 149–150 (1985).
[CrossRef] [PubMed]

1983 (1)

F. Lederer, U. Langbein, and H.-E. Ponath, "Nonlinear waves guided by a dielectric slab: II. TM-polarization," Appl. Phys. B 31, 187–190 (1983).
[CrossRef]

Al-Bader, S. J.

S. J. Al-Bader, "TM waves in nonlinear saturable thin films: a multilayered approach," J. Lightwave Technol. 7, 717–725 (1989).
[CrossRef]

Ariyasu, J.

G. I. Stegeman, C. T. Seaton, and J. Ariyasu, "Nonlinear electromagnetic waves guided by a single interface," J. Appl. Phys. 58, 2543–2549 (1985).
[CrossRef]

Boardman, A.

A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
[CrossRef]

Boardman, A. D.

A. D. Boardman and T. Twardowski, "Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides," Phys. Rev. A 39, 2481–2492 (1989).
[CrossRef] [PubMed]

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

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

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

Chen, Q.

Chen, W.

Christodoulides, D. N.

Egan, P.

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
[CrossRef]

Gniadek, K.

J. Jasiński and K. Gniadek, "A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution," Opt. Quantum Electron. 26, 865–876 (1994).
[CrossRef]

Groza, A.

A. Groza and V. L. Strizhevskii, "p-Polarized nonlinear surface polaritons near the surface of a saturable defocusing medium," Opt. Commun. 91, 111–114 (1992).
[CrossRef]

Jasinski, J.

J. Jasiński and K. Gniadek, "A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution," Opt. Quantum Electron. 26, 865–876 (1994).
[CrossRef]

J. Jasiński, "Specific analytical TM solution in nonlinearly isotropic Kerr-like media," Opt. Appl. 24, 287–292 (1994).

Joseph, R. I.

Kogelnik, H.

H. Kogelnik, "Theory of dielectric waveguides," in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), Chap. 2, pp. 34–39.

Langbein, U.

U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).

F. Lederer, U. Langbein, and H.-E. Ponath, "Nonlinear waves guided by a dielectric slab: II. TM-polarization," Appl. Phys. B 31, 187–190 (1983).
[CrossRef]

A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
[CrossRef]

Lederer, F.

U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).

F. Lederer, U. Langbein, and H.-E. Ponath, "Nonlinear waves guided by a dielectric slab: II. TM-polarization," Appl. Phys. B 31, 187–190 (1983).
[CrossRef]

A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
[CrossRef]

Leung, K. M.

K. M. Leung, "p-Polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions," Phys. Rev. B 32, 5093–5101 (1985).
[CrossRef]

Mai, Xu

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

Maradudin, A. A.

W. Chen and A. A. Maradudin, "s-Polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film," J. Opt. Soc. Am. B 5, 529–537 (1988).
[CrossRef]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and 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, and R. F. Wallis, "Nonlinear slab-guided waves in non-Kerr-like media," IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Mazilu, D.

U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).

Mihalache, D.

U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
[CrossRef]

Moloney, J. V.

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

Nakai, Y.

S. Okafuji and Y. Nakai, "Nonlinear waves in asymmetric dielectric hollow waveguides with self-focusing Kerr-like nonlinear film," Inst. Electr. Eng. Proc. J 138, 204–210 (1991).

Ogusu, K.

K. Ogusu, "TM waves guided by nonlinear planar waveguides," IEEE Trans. Microwave Theory Tech. 37, 941–946 (1989).
[CrossRef]

K. Ogusu, "Nonlinear TE waves guided by graded-index planar waveguides," Opt. Commun. 63, 380–384 (1987).
[CrossRef]

Okafuji, S.

S. Okafuji and Y. Nakai, "Nonlinear waves in asymmetric dielectric hollow waveguides with self-focusing Kerr-like nonlinear film," Inst. Electr. Eng. Proc. J 138, 204–210 (1991).

Ponath, H.-E.

F. Lederer, U. Langbein, and H.-E. Ponath, "Nonlinear waves guided by a dielectric slab: II. TM-polarization," Appl. Phys. B 31, 187–190 (1983).
[CrossRef]

Seaton, C. T.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T.-P. Shen, A. A. Maradudin, and 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, and H. G. Winful, "Nonlinear guided waves applications," Opt. Eng. 24, 593–599 (1985).

G. I. Stegeman, C. T. Seaton, and J. Ariyasu, "Nonlinear electromagnetic waves guided by a single interface," J. Appl. Phys. 58, 2543–2549 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, "Comparison of solutions for TM-polarized nonlinear guided waves," Opt. Lett. 10, 149–150 (1985).
[CrossRef] [PubMed]

Shen, T.-P.

G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T.-P. Shen, A. A. Maradudin, and 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.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and 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, and 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, and H. G. Winful, "Nonlinear guided waves applications," Opt. Eng. 24, 593–599 (1985).

G. I. Stegeman, C. T. Seaton, and J. Ariyasu, "Nonlinear electromagnetic waves guided by a single interface," J. Appl. Phys. 58, 2543–2549 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, "Comparison of solutions for TM-polarized nonlinear guided waves," Opt. Lett. 10, 149–150 (1985).
[CrossRef] [PubMed]

Strizhevskii, V. L.

A. Groza and V. L. Strizhevskii, "p-Polarized nonlinear surface polaritons near the surface of a saturable defocusing medium," Opt. Commun. 91, 111–114 (1992).
[CrossRef]

Svenson, B.

Twardowski, T.

A. D. Boardman and T. Twardowski, "Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides," Phys. Rev. A 39, 2481–2492 (1989).
[CrossRef] [PubMed]

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

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

Valera, J. D.

Wallis, R. F.

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

Wang, Zi Hua

Winful, H. G.

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

Wright, E. M.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface," Opt. Lett. 12, 187–189 (1987).
[CrossRef] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and 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, and R. F. Wallis, "Nonlinear slab-guided waves in non-Kerr-like media," IEEE J. Quantum Electron. QE-22, 977–983 (1986).
[CrossRef]

Yokota, M.

Zanoni, R.

Appl. Phys. B (1)

F. Lederer, U. Langbein, and H.-E. Ponath, "Nonlinear waves guided by a dielectric slab: II. TM-polarization," Appl. Phys. B 31, 187–190 (1983).
[CrossRef]

IEEE J. Quantum Electron. (2)

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

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Ogusu, "TM waves guided by nonlinear planar waveguides," IEEE Trans. Microwave Theory Tech. 37, 941–946 (1989).
[CrossRef]

Inst. Electr. Eng. Proc. J (1)

S. Okafuji and Y. Nakai, "Nonlinear waves in asymmetric dielectric hollow waveguides with self-focusing Kerr-like nonlinear film," Inst. Electr. Eng. Proc. J 138, 204–210 (1991).

J. Appl. Phys. (1)

G. I. Stegeman, C. T. Seaton, and J. Ariyasu, "Nonlinear electromagnetic waves guided by a single interface," J. Appl. Phys. 58, 2543–2549 (1985).
[CrossRef]

J. Lightwave Technol. (1)

S. J. Al-Bader, "TM waves in nonlinear saturable thin films: a multilayered approach," J. Lightwave Technol. 7, 717–725 (1989).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Appl. (1)

J. Jasiński, "Specific analytical TM solution in nonlinearly isotropic Kerr-like media," Opt. Appl. 24, 287–292 (1994).

Opt. Commun. (2)

A. Groza and V. L. Strizhevskii, "p-Polarized nonlinear surface polaritons near the surface of a saturable defocusing medium," Opt. Commun. 91, 111–114 (1992).
[CrossRef]

K. Ogusu, "Nonlinear TE waves guided by graded-index planar waveguides," Opt. Commun. 63, 380–384 (1987).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (4)

Opt. Quantum Electron. (1)

J. Jasiński and K. Gniadek, "A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution," Opt. Quantum Electron. 26, 865–876 (1994).
[CrossRef]

Phys. Rev. A (2)

A. D. Boardman and T. Twardowski, "Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides," Phys. Rev. A 39, 2481–2492 (1989).
[CrossRef] [PubMed]

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

Phys. Rev. B (1)

K. M. Leung, "p-Polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions," Phys. Rev. B 32, 5093–5101 (1985).
[CrossRef]

Physica (1)

U. Langbein, F. Lederer, D. Mihalache, and D. Mazilu, "Nonlinear TM-polarized waves in non-Kerr media," Physica 145C, 377–385 (1987).

Other (2)

A. Boardman, P. Egan, U. Langbein, F. Lederer, and D. Mihalache, "Third-order nonlinear electromagnetic TE and TM waves," in Modern Problems in Condensed Matter Physics, V. M. Agranovich, A. A. Maradudin, H. E. Ponath, and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), pp. 73–287.
[CrossRef]

H. Kogelnik, "Theory of dielectric waveguides," in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), Chap. 2, pp. 34–39.

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Figures (6)

Fig. 1
Fig. 1

Exact solution of TM mode equation (9) for the 1st-order mode with β2 = 0.54L in the guided-modes region as a function of the waveguide thickness. m and i are the extrema and bs and bs are the boundary values of permittivity. The material parameters are L = 2.4025, c = 0.4L, and s = 0.52L, which give bal = 1.08L, hlin = 0.3479 μm, and hbal = 0.3209 μm. (T) and (L) are regions of the transverse and the longitudinal modes, respectively.

Fig. 2
Fig. 2

Power flow of the 0th-order mode in a hollow-waveguide configuration against the propagation constant, Pf, Ps, and Pc are the components of the total power P supported by the film, the substrate, and the cover. The material parameters are α = 6.4 μm2/V2, L = 2.4025, c = 1.017L, and s = 1.02L, and the waveguide thickness is h = 0.951 μm.

Fig. 3
Fig. 3

Solutions of the balanced (solid curves) and the linear (dashed curves) TM mode equations in a waveguide with α = 6.4 μm2/V2, L = 2.4025, c = 0.4L, and s = −10L. (T) and (L) are regions of the transverse and the longitudinal modes, respectively.

Fig. 4
Fig. 4

Permittivity profile of the 1st-order mode with β2 = 0.49L in a waveguide of thickness h = 0.258 μm. The material parameters are the same as in Fig. 3. The solid curve is the exact solution, the dotted curve is the quasi-balanced approximation, and the dashed curve is the transverse uniaxial approximation.

Fig. 5
Fig. 5

Magnetic field profile of the exact, the quasi-balanced, and the transverse uniaxial approximated solutions. The configuration is the same as in Fig. 4.

Fig. 6
Fig. 6

Comparison of the total power flow against the propagation constant for the exact, the quasi-balanced, and the transverse uniaxial approximated solutions. The 1st-order mode is guided in the configuration as in Fig. 4. β lin 2 = 0.4841 L , β upp 2 = 0.5682 L.

Equations (28)

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= L + α ( E x 2 + E z 2 ) ,
β H y = E x , d H y d x = E z , - d E z d x + β E x = H y .
H y 2 = 2 β 2 - 2 - L 2 - C 2 α , ( d H y d x ) 2 = 2 ( - L ) α - β 2 H y 2 ,
( d d x ) 2 = 2 ( 2 β 2 - ) 2 ( 2 - L 2 - C ) [ ( - L ) ( 3 β 2 - 2 2 - β 2 L ) + C β 2 ] [ β 2 ( 2 - L 2 - C ) + ( 2 β 2 - ) ] 2 .
x ( ) - x m = ± m d x d d ,
( m - L ) ( 3 β 2 m - 2 m 2 - β 2 L ) + C β 2 = 0 , i = L 2 + C .
H y = H c exp [ - κ c ( x - h ) ] ,
( b - L ) [ b + L - 2 b c 2 ( 2 β 2 - b ) β 2 ( c 2 + b 2 ) - c b 2 ] = ( m - L ) ( 2 m 2 β 2 - 3 m + L ) .
h = ( b c m + b s m + 2 n i m ) d x d d .
P = P c + P s + ( b c m + b s m + 2 n i m ) d P d d ,
d P d = β 2 μ 0 ω α β 2 ( 2 - L 2 - C ) + 2 ( 2 β 2 - ) 2 ( 2 β 2 - ) 2 × [ 2 - L 2 - C ( - L ) ( 3 β 2 - 2 2 - β 2 L ) + C β 2 ] 1 / 2
P c , s = 1 2 μ 0 ω α β 2 β 2 - c , s c , s b c , s 2 ( b - L ) β 2 ( c , s 2 + b c , s 2 ) - c , s b c , s 2 ,
bal = 2 β 2 .
h = 1 2 β [ 0 1 d u ( cot 2 ϕ c + u ) u + 0 1 d u ( cot 2 ϕ s + u ) u + 4 n 0 1 d u ( 1 + u ) u ] ,
ϕ c , s = arctan 2 β κ c , s c , s ,
h bal = 1 β ( ϕ c + ϕ s + n π ) .
E x = E 0 cos ( β x + ϕ ) , H y = H 0 cos ( β x + ϕ ) , E z = - E 0 sin ( β x + ϕ ) .
E 0 2 = 2 β 2 - L α , H 0 2 = 4 β 2 ( 2 β 2 - L ) α .
P = β ( 2 β 2 - L ) 2 μ 0 ω α [ h + 2 c ( 2 β 2 - c ) κ c + 2 s ( 2 β 2 - s ) κ s ] .
i = 2 β 2 + δ , m = 2 β 2 + δ β 2 k f 2 , C = 4 β 4 - L 2 + 4 β 2 δ ,
= 2 β 2 + 4 β 4 δ 4 β 4 + α H y 2 , ( d H y d x ) 2 = β 2 [ 4 β 2 ( 2 β 2 - L + δ ) - α H y 2 ] ( 4 β 4 + 4 β 2 δ + α H y 2 ) α ( 4 β 4 + α H y 2 ) .
H m 2 = 4 β 2 ( 2 β 2 - L + δ ) α .
± β d x d H y = - 1 H m 2 - H y 2 + 2 β 2 δ ( 4 β 4 + α H y 2 ) H m 2 - H y 2 ,
± ( x - x 0 ) = 1 β arccos H y H m - δ 2 β 2 k f arctan ( β H m 2 - H y 2 k f H y ) .
H y = H m cos [ β x + δ 2 β κ f arctan ( β κ f tan β x ) ] .
h = 1 β ( ϕ c + ϕ s + n π ) - δ 2 β 2 k f ( Ψ c + Ψ s + n π ) ,
Ψ c , s = arctan 2 β 2 κ c , s c , s k f .
P = β 2 μ 0 ω α [ ( 2 β 2 - L - δ ) h + α H c H m 2 β 2 κ c + α H s H m 2 β 2 κ c + δ ( 6 β 2 - L ) 2 β 2 k f ( Ψ c + Ψ s + n π ) ] .

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