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Maxwell’s Equations in a Plane Waveguide with Nonhomogeneous Nonlinear Permittivity: Analytical and Numerical Approaches

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Abstract

The paper focuses on the problem on monochromatic electromagnetic waves propagation in a shielded planar dielectric waveguide. The waveguide is filled with nonhomogeneous nonlinear medium. The nonlinearity is expressed by nonnegative unbounded monotonically increasing function with power growth. Such nonlinearity is a generalization of the well-known Kerr nonlinear law. The existence of propagation constants and eigenwaves is proved. Besides, it is proved that the studied problem has nonlinearized solutions as well as linearized ones.

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References

  • Adams, M.J.: An Introduction to Optical Waveguides. John Wiley & Sons, Chichester - New York - Brisbane - Toronto (1981)

    Google Scholar 

  • Akhmediev, N.N., Ankevich, A.: Solitons, Nonlinear Pulses and Beams. Chapman and Hall, London (1997)

    Google Scholar 

  • Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14(4), 349–381 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  • Boardman, A.D., Egan, P., Lederer, F., Langbein, U., Mihalache, D.: Third-order nonlinear electromagnetic TE and TM guided waves. In: Ponath, H.-E., Stegeman, G.I. (eds.) Reprinted from Nonlinear Surface Electromagnetic Phenomena. Elsevier Sci. Publ, North-Holland, Amsterdam London New York Tokyo (1991)

    Google Scholar 

  • Cazenave, T.: Semilinear Schrödinger Equations, Volume 10 of Courant Lecture Notes in Mathematics. American Mathematical Society, Providence (2003)

    Google Scholar 

  • Chen, Q., Wang, Z.H.: Exact dispersion relations for tm waves guided by thin dielectrics films bounded by nonlinear media. Opt. Lett. 18(4), 260–262 (1993)

    Article  Google Scholar 

  • Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience Publishers Inc., New York (1953)

    MATH  Google Scholar 

  • Huang, J.H., Chang, R., Leung, P.T., Tsai, D.P.: Nonlinear dispersion relation for surface plasmon at a metal-Kerr medium interface. Opt. Commun. 282, 1412–1415 (2009)

    Article  Google Scholar 

  • Kielich, S.: Molekularna Optyka Nieliniowa (Molecular Nonlinear Optics). PWN, Warszawa (in Polish) (1977)

    Google Scholar 

  • Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford - London - New York - Paris (1964)

    MATH  Google Scholar 

  • Kurseeva, V.Y., Smirnov, G.Y.: On the existence of infinitely many eigenvalues in a nonlinear Sturm-Liouville problem arising in the theory of waveguides. Differ. Equ. 53(11), 1419–1427 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Course of Theoretical Physics. Electrodynamics of Continuous Media, vol. 8. Butterworth-Heinemann, Oxford (1993)

    Google Scholar 

  • Marcuse, D.: Light Transmission Optics. Van Nostrand Reinhold Company, New York (1972)

    Google Scholar 

  • Marcuse, D.: Theory of Dielectric Optical Waveguides, 2nd edn. Academic Press, Cambridge (1991)

    Google Scholar 

  • Mihalache, D., Stegeman, G.I., Seaton, C.T., Wright, E.M., Zanoni, R., Boardman, A.D., Twardowki, T.: Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface. Opt. Lett. 12(3), 187–189 (1987)

    Article  Google Scholar 

  • Moskaleva, M.A., Kurseeva, V.Y., Valovik, D.V.: Asymptotical analysis of a nonlinear Sturm–Liouville problem: linearisable and non-linearisable solutions. Asymptot. Anal. 119(1–2), 39–59 (2020)

    MathSciNet  MATH  Google Scholar 

  • Naimark, M.A.: Linear Differential Operators, Part I: Elementary Theory of Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space. Frederick Ungar Publishing Co., New York (1967)

    MATH  Google Scholar 

  • Osmolovskii, V.G.: Nonlinear Sturm–Liouville Problem. Saint Petersburg University Press, Saint Petersburg (2003)

    Google Scholar 

  • Petrovsky, I.G.: Lectures on the Theory of Ordinary Differential Equations. Moscow State University, Moscow (1984). ((in Russian))

    Google Scholar 

  • Schürmann, H.W., Smirnov, Yu.G., Shestopalov, Yu.V.: Propagation of TE-waves in cylindrical nonlinear dielectric waveguides. Phys. Rev. E 71(1), 016614 (2005)

    Article  Google Scholar 

  • Shen, Y.R.: The Principles of Nonlinear Optics. John Wiley and Sons, New York-Chicester-Brisbane-Toronto-Singapore (1984)

    Google Scholar 

  • Smol’kin, E.Y., Valovik, D.V.: Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity. Adv. Math. Phys. (2015). https://doi.org/10.1155/2015/614976

    Article  MathSciNet  MATH  Google Scholar 

  • Stretton, J.A.: Electromagnetic Theory. McGraw Hill, New york (1941)

    Google Scholar 

  • Tikhov, S., Valovik, D.: Theory of nonlinear transverse-magnetic guided waves in a plane waveguide filled with nonhomogeneous nonlinear medium. Waves Random Complex Media (2022). https://doi.org/10.1080/17455030.2022.2102694

    Article  Google Scholar 

  • Tikhov, Stanislav V., Valovik, Dmitry V.: Nonlinearizable solutions in an eigenvalue problem for maxwell’s equations with nonhomogeneous nonlinear permittivity in a layer. Stud. Appl. Math. 149(3), 565–587 (2022)

    Article  MathSciNet  Google Scholar 

  • Unger, H.-G.: Planar Optical Waveguides and Fibres. Clarendon Press, Oxford (1977)

    Google Scholar 

  • Vainberg, M.M.: Variational Methods for the Study of Nonlinear Operators. Holden-Day Series in Mathematical Physics, 1st edn. Holden-Day, San Francisco (1964)

    Google Scholar 

  • Vainstein, L.A.: Electromagnetic Waves, p. 440. Radio i svyaz, Moscow (1988)

    Google Scholar 

  • Valovik, D.V.: On a nonlinear eigenvalue problem related to the theory of propagation of electromagnetic waves. Differ. Equ. 54(2), 168–179 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Valovik, D.V.: On spectral properties of the Sturm–Liouville operator with power nonlinearity. Monatshefte für Math. 188(2), 369–385 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation [grant number 18-71-10015].

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  1. Penza State University, Krasnaya Street 40, Penza, Russian Federation

    S. V. Tikhov & D. V. Valovik

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  1. S. V. Tikhov
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  2. D. V. Valovik
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SV is responsible for numerical study (calculations, software implementations, visualization), writing and editing of the manuscript; DV is responsible for analytical investigation and supervision of the whole research.

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Correspondence to D. V. Valovik.

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Tikhov, S.V., Valovik, D.V. Maxwell’s Equations in a Plane Waveguide with Nonhomogeneous Nonlinear Permittivity: Analytical and Numerical Approaches. J Nonlinear Sci 33, 105 (2023). https://doi.org/10.1007/s00332-023-09962-6

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  • DOI: https://doi.org/10.1007/s00332-023-09962-6

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