Boundary Eigenvalue Problem for Maxwell Equations in a Nonlinear Dielectric Layer

Affiliation(s)

Department Mathematics and Supercomputer Modeling, Penza State University, Penza, Russia.

Department Mathematics and Supercomputer Modeling, Penza State University, Penza, Russia.

ABSTRACT

The propagation of TM-polarized electromagnetic waves in a dielectric layer filled with lossless, nonmagnetic, and isotropic medium is considered. The permittivity in the layer defines by Kerr law. We look for eigenvalues of the problem and reduce the issue to the analysis of the corresponding dispersion equation. The equivalence of the boundary eigenvalue problem and the dispersion equation is proved. We show that the solution of the problem exists and from dispersion equation it can be numerically obtained. Using this solution the components of electromagnetic field in the layer can be numerically obtained as well. Transition to the limit in the case of a linear medium in the layer is proved. Some numerical results are presented also.

The propagation of TM-polarized electromagnetic waves in a dielectric layer filled with lossless, nonmagnetic, and isotropic medium is considered. The permittivity in the layer defines by Kerr law. We look for eigenvalues of the problem and reduce the issue to the analysis of the corresponding dispersion equation. The equivalence of the boundary eigenvalue problem and the dispersion equation is proved. We show that the solution of the problem exists and from dispersion equation it can be numerically obtained. Using this solution the components of electromagnetic field in the layer can be numerically obtained as well. Transition to the limit in the case of a linear medium in the layer is proved. Some numerical results are presented also.

KEYWORDS

Maxwell Equations, Nonlinear Boundary Eigenvalue Problem, Dispersion Equation, Nonlinear Slab (Film)

Maxwell Equations, Nonlinear Boundary Eigenvalue Problem, Dispersion Equation, Nonlinear Slab (Film)

Cite this paper

nullY. Smirnov and D. Valovik, "Boundary Eigenvalue Problem for Maxwell Equations in a Nonlinear Dielectric Layer,"*Applied Mathematics*, Vol. 1 No. 1, 2010, pp. 29-36. doi: 10.4236/am.2010.11005.

nullY. Smirnov and D. Valovik, "Boundary Eigenvalue Problem for Maxwell Equations in a Nonlinear Dielectric Layer,"

References

[1] P. N. Eleonskii, L. G. Oganes’yants and V. P. Silin, “Cy-lindrical Nonlinear Waveguides,” Soviet Physics-Journal of Experimental and Theoretical Physics, Vol. 35, No. 1, 1972, pp. 44-47.

[2] R. I. Joseph and D. N. Christodoulides, “Exact Field De-composition for TM Waves in Nonlinear Media,” Optics Letters, Vol. 12, No. 10, 1987, pp. 826-828.

[3] K. M. Leung, “Р-polarized Nonlinear Surface Polaritons in Materials with Intensity-Dependent Dielectric Functions,” Physical Review B, Vol. 32, No. 8, 1985, pp. 5093- 5101.

[4] K. M. Leung and R. L. Lin, “Scattering of Transverse- Magnetic Waves with a Nonlinear Film: Formal Field Solutions in Quadratures,” Physical Review B, Vol. 44, No. 10, 1991, pp. 5007-5012.

[5] H. W. Schürmann, V. S. Serov and Y. V. Shestopalov, “TE-Polarized Waves Guided by a Lossless Nonlinear Three-Layer Structure,” Physical Review E, Vol. 58, No. 1, 1998, pp. 1040-1050.

[6] H. W. Schürmann, V. S. Serov and Y. V. Shestopalov, “Solutions to the Helmholtz Equation for TE-Guided Waves in a Three-Layer Structure with Kerr-Type Non-linearity,” Journal of physics A, Mathematical and Gen-eral, Vol. 35, No. 50, 2002, pp. 10789-10801.

[7] H. W. Schürmann, Y. G. Smirnov and Y. V. Shestopalov, “Propagation of TE-Waves in Cylindrical Nonlinear Di-electric Waveguides,” Physical Review E, Vol. 71, No. 1, 2005, pp. 016614(1-10).

[8] Y. G. Smirnov, H. W. Schürmann and Y. V. Schestopalov, “Integral Equation Approach for the Propagation of TE-Waves in a Nonlinear Dielectric Cylinrical Wave-guide,” Journal of Nonlinear Mathematical Physics, Vol. 11, No. 2, 2004, pp. 256-268.

[9] G. A. Korn and T. M. Korn, “Mathematical Handbook for Scientists and Engineers,” McGraw Hill Book Company, 1968.

[10] H. F. Baker, “Abelian Functions. Abel’s Theorem and the Allied Theory of Theta Functions,” Cambridge University Press, Cambridge, 1897.

[11] A. I. Markushevich, “Introduction to the Classical Theory of Abelian Functions,” American Mathematical Society, Providence, 2006.

[12] I. T. Gokhberg and M. G. Krein, “Introduction in the Theory of Linear Nonselfadjoint Operators in Hilbert Space,” American Mathematical Society, Providence, 1969.

[13] A. Snyder and J. Love, “Optical Waveguide Theory,” Chapmen and Hall, London, 1983.

[14] D. V. Valovik and Y. G. Smirnov, “Calculation of the Propagation Constants and Fields of Polarized Electro-magnetic TM Waves in a Nonlinear Anisotropic Layer,” Journal of Communications Technology and Electronics, Vol. 54, No. 4, 2009, pp. 391-398.

[15] D. V. Valovik and Y. G. Smirnov, “Calculation of the Propagation Constants of TM Electromagnetic Waves in a Nonlinear Layer,” Journal of Communications Technology and Electronics, Vol. 53, No. 8, 2008, pp. 883-889.

[1] P. N. Eleonskii, L. G. Oganes’yants and V. P. Silin, “Cy-lindrical Nonlinear Waveguides,” Soviet Physics-Journal of Experimental and Theoretical Physics, Vol. 35, No. 1, 1972, pp. 44-47.

[2] R. I. Joseph and D. N. Christodoulides, “Exact Field De-composition for TM Waves in Nonlinear Media,” Optics Letters, Vol. 12, No. 10, 1987, pp. 826-828.

[3] K. M. Leung, “Р-polarized Nonlinear Surface Polaritons in Materials with Intensity-Dependent Dielectric Functions,” Physical Review B, Vol. 32, No. 8, 1985, pp. 5093- 5101.

[4] K. M. Leung and R. L. Lin, “Scattering of Transverse- Magnetic Waves with a Nonlinear Film: Formal Field Solutions in Quadratures,” Physical Review B, Vol. 44, No. 10, 1991, pp. 5007-5012.

[5] H. W. Schürmann, V. S. Serov and Y. V. Shestopalov, “TE-Polarized Waves Guided by a Lossless Nonlinear Three-Layer Structure,” Physical Review E, Vol. 58, No. 1, 1998, pp. 1040-1050.

[6] H. W. Schürmann, V. S. Serov and Y. V. Shestopalov, “Solutions to the Helmholtz Equation for TE-Guided Waves in a Three-Layer Structure with Kerr-Type Non-linearity,” Journal of physics A, Mathematical and Gen-eral, Vol. 35, No. 50, 2002, pp. 10789-10801.

[7] H. W. Schürmann, Y. G. Smirnov and Y. V. Shestopalov, “Propagation of TE-Waves in Cylindrical Nonlinear Di-electric Waveguides,” Physical Review E, Vol. 71, No. 1, 2005, pp. 016614(1-10).

[8] Y. G. Smirnov, H. W. Schürmann and Y. V. Schestopalov, “Integral Equation Approach for the Propagation of TE-Waves in a Nonlinear Dielectric Cylinrical Wave-guide,” Journal of Nonlinear Mathematical Physics, Vol. 11, No. 2, 2004, pp. 256-268.

[9] G. A. Korn and T. M. Korn, “Mathematical Handbook for Scientists and Engineers,” McGraw Hill Book Company, 1968.

[10] H. F. Baker, “Abelian Functions. Abel’s Theorem and the Allied Theory of Theta Functions,” Cambridge University Press, Cambridge, 1897.

[11] A. I. Markushevich, “Introduction to the Classical Theory of Abelian Functions,” American Mathematical Society, Providence, 2006.

[12] I. T. Gokhberg and M. G. Krein, “Introduction in the Theory of Linear Nonselfadjoint Operators in Hilbert Space,” American Mathematical Society, Providence, 1969.

[13] A. Snyder and J. Love, “Optical Waveguide Theory,” Chapmen and Hall, London, 1983.

[14] D. V. Valovik and Y. G. Smirnov, “Calculation of the Propagation Constants and Fields of Polarized Electro-magnetic TM Waves in a Nonlinear Anisotropic Layer,” Journal of Communications Technology and Electronics, Vol. 54, No. 4, 2009, pp. 391-398.

[15] D. V. Valovik and Y. G. Smirnov, “Calculation of the Propagation Constants of TM Electromagnetic Waves in a Nonlinear Layer,” Journal of Communications Technology and Electronics, Vol. 53, No. 8, 2008, pp. 883-889.