Abstract
We consider the problem of leaky waves in an inhomogeneous waveguide structure covered with a layer of graphene, which is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. A variational statement of the problem is used to determine the solution. The variational problem is reduced to the study of an operator function. The properties of the operator function necessary for the analysis of its spectral properties are investigated. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
Geim, A.K. and Novoselov, K.S., The rise of graphene, Nature Mater., 2007, vol. 6, pp. 183–191.
Hanson, G.W., Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene, J. Appl. Phys., 2008, vol. 103, p. 064302.
Falkovsky, L.A., Optical properties of graphene, J. Phys.: Conf. Ser., 2008, vol. 129, p. 012004.
Mikhailov, S.A., Quantum theory of the third-order nonlinear electrodynamic effects of graphene, Phys. Rev. B, 2016, vol. 93, p. 085403.
Smirnov, Yu.G., Matematicheskie metody issledovaniya zadach elektrodinamiki (Mathematical Methods for Studying Electrodynamic Problems), Penza: Penz. Gos. Univ., 2009.
Shestopalov, Y., Smirnov, Y., and Smolkin, E., Optical Waveguide Theory. Mathematical Models, Spectral Theory and Numerical Analysis, vol. 237 of Springer Ser. Opt. Sci., Singapore: Springer, 2022.
Hajian, H., Rukhlenko, I.D., Leung, P.T., Caglayan, H., and Ozbay, E., Guided plasmon modes of a graphene-coated Kerr slab, Plasmonics, 2016, vol. 11, pp. 735–741.
Smirnov, Y. and Tikhov, S., The nonlinear eigenvalue problem of electromagnetic wave propagation in a dielectric layer covered with graphene, Photonics, 2023, vol. 10, p. 523.
Smirnov, Yu.G., Tikhov, S.V., and Gusarova, E.V., On the propagation of electromagnetic waves in a dielectric layer coated with graphene, Izv. VUZov. Povolzhsk. Reg. Fiz.-Mat. Nauki, 2022, no. 3, pp. 11–18.
Nikiforov, A.F. and Uvarov, V.B., Spetsial’nye funktsii matematicheskoi fiziki (Special Functions of Mathematical Physics), Moscow: Nauka, 1978.
Adams, R., Sobolev Spaces, New York: Acad. Press, 1975.
Kato, T., Perturbation Theory for Linear Operators, Berlin–Heidelberg–New York: Springer, 1965. Translated under the title: Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.
Smirnov, Yu.G. and Smolkin, E.Yu., Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide, Differ. Equations, 2017, vol. 53, no. 10, pp. 1262–1273.
Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Ser., 1966. Translated under the title: Spravochnik po spetsial’nym funktsiyam, Moscow, 1979.
Gohberg, I.C. and Sigal, E.I., An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR-Sb., 1971, vol. 13, no. 4, pp. 603–625.
Funding
This work was financially supported by the Russian Science Foundation, project 20-11-20087, https://rscf.ru/en/project/20-11-20087/.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Smirnov, Y.G., Smolkin, E.Y. On the Existence of an Infinite Spectrum of Damped Leaky TE-Polarized Waves in an Open Inhomogeneous Cylindrical Metal–Dielectric Waveguide Coated with a Graphene Layer. Diff Equat 59, 1193–1198 (2023). https://doi.org/10.1134/S0012266123090057
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266123090057