ON THE NON-INERT PERTURBATION METHOD FOR PROVING THE EXISTENCE OF NON-LINEARIZABLE SOLUTIONS IN A NONLINEAR EIGENVALUE PROBLEM ARISING IN THE THEORY OF WAVEGUIDES

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Abstract

The problem of propagation of electromagnetic waves in a plane dielectric waveguide is studied. The waveguide is filled with a nonlinear inhomogeneous medium; the nonlinearity is characterized by an arbitrary monotone positive continuously differentiable function with a stepwise increase on infinity. The inhomogeneity of the medium is characterized by small (non-monotone) perturbations of the linear part of the dielectric permeability, as well as the coefficient at the nonlinear term. From a mathematical point of view, this problem is equivalent to the eigenvalue problem for a system of Maxwell equations with mixed boundary conditions. To study the problem, a perturbation method is proposed, in which a simpler nonlinear problem is used as the main problem. The existence of both linearizable and non-linearizable solutions is proved.

About the authors

D. V Valovik

Penza State University

Email: dvalovik@mail.ru
Penza

A. A Dyundyaeva

S. V Tikhov

Penza State University

Email: tik.stanislav2015@yandex.ru
Penza

References

  1. Eleonskii V.M., Silin V.P. Theory of self-trapping of an electromagnetic field in a nonlinear medium // Sov. Phys. JETP. 1970. V. 31. № 5. P. 918–923.
  2. Boardman A.D., Egan P., Lederer F., et al. Third-order nonlinear electromagnetic te and tm guided waves. Elsevier sci. Publ. North-Holland, Amsterdam London New York Tokyo, 1991. Reprinted from Nonlinear Surface Electromagnetic Phenomena, Eds. H.-E. Ponath and G. I. Stegeman.
  3. Ахмедиев Н.Н., Анкевич А. Солитоны. М.: Физматлит, 2003.
  4. Кившарь Ю.С., Агравал Г.П. Оптические солитоны. От волоконных световодов к фотонным кристаллам. М.: Физматлит, 2005.
  5. Christian J.M., McDonald G.S., Potton R.J., Chamorro-Posada P. Helmholtz solitons in power-law optical materials // Phys. Rev. A. 2007. V. 76. № 3. P. 033834.
  6. Boardman A.D., Egan P. Novel nonlinear surface and guided te waves in asymmetric lhm waveguides // J. Optic. A: Pure and Appl. Optic. 2009. V. 11. № 11. P. 114032(10).
  7. Валовик Д.В. Распространение электромагнитных волн в открытом плоском диэлектрическом волноводе, заполненном нелинейной средой i: ТЕ-волны // Ж. вычисл. матем. и матем. физ. 2019. Т. 59. № 5. С. 838–858.
  8. Валовик Д.В. Распространение электромагнитных волн в открытом плоском диэлектрическом волноводе, заполненном нелинейной средой i: ТМ-волны // Ж. вычисл. матем. и матем. физ. 2020. Т. 60 № 3. С. 429–450.
  9. Tikhov S.V., Valovik D.V. Nonlinearizable solutions in an eigenvalue problem for maxwell’s equations with nonhomogeneous nonlinear permittivity in a layer // Studies in Appl. Math. 2022. V. 149. № 3. P. 565–587.
  10. Valovik D.V. Maxwell’s equations with nonlinear inhomogeneous constitutive relation: Guided waves in a film filled with inhomogeneous kerr medium // SIAM J. Appl. Math. 2023. V. 83. № 2. P. 553–575.
  11. Tikhov S.V., Valovik D.V. Maxwell’s equations in a plane waveguide with nonhomogeneous nonlinear permittivity: Analytical and numerical approaches // J. Nonlinear Sci. 2023. V. 33. № 105.
  12. Содха М.С., Гхатак А.К. Неоднородные оптические волноводы. М.: Связь, 1980.
  13. Гончаренко А.М., Карпенко В.А. Основы теории оптических волноводов. Минск: Наука и техника, 1983.
  14. Адамс М. Введение в теорию оптических волноводов. М.: Мир, 1984.
  15. Dyundyaeva A., Tikhov S., Valovik D. Transverse electric guided wave propagation in a plane waveguide with kerr nonlinearity and perturbed inhomogeneity in the permittivity function // Photonics. 2023. Vl. 10. № 4.
  16. Eleonskii P.N., Oganes’yants L.G., Silin V.P. Cylindrical nonlinear waveguides // Sov. Phys. JETP. 1972. V. 35. № 1. P. 44–47.
  17. Unger H.G. Planar optical waveguides and fibres. Oxford: Clarendon Press, 1977.
  18. Marcuse D. Theory of dielectric optical waveguides. 2nd ed. Acad. Press, 1991.
  19. Смирнов Ю.Г. Задачи сопряжения на собственные значения, описывающие распространение ТЕ- и ТМволн в двухслойных неоднородных анизотропных цилиндрических и плоских волноводах // Ж. вычисл. матем. и матем. физ. 2015. Т. 55. № 3. С. 460–468.
  20. Курант Р., Гильберт Д. Методы математической физики. Т. 1. М.-Л.: ГИТТЛ, 1951.
  21. Schurmann H.W., Smirnov Yu.G., Shestopalov Yu.V. Propagation of te-waves in cylindrical nonlinear dielectric waveguides // Phys. Rev. E. 2005. V. 71. № 1. P. 016614(10).
  22. Smolkin E.Yu., Valovik D.V. Coupled electromagnetic wave propagation in a cylindrical dielectric waveguide filled with kerr medium: nonlinear effects // J. Modern Optic. 2017. V. 64. № 4. P. 396–406.
  23. Mihalache D., Stegeman G.I., Seaton C.T., et al. Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface // Optic. Lett. 1987. V. 12. № 3. P. 187–189.
  24. Chen Q., Wang Z.H. Exact dispersion relations for tm waves guided by thin dielectrics films bounded by nonlinear media // Optic. Lett. 1993. V. 18. № 4. P. 260–262.
  25. Huang J.H., Chang R., Leung P.T., et al. Nonlinear dispersion relation for surface plasmon at a metal-kerr medium interface // Optic. Communicat. 2009. V. 282. P. 1412–1415.
  26. Понтрягин Л.С. Обыкновенные дифференциальные уравнения. М.: ГИФМЛ, 1961.

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