Studying the surface wave spectrum of an open inhomogeneous rectangular dielectric waveguide. (English. Russian original) Zbl 1478.78046

Differ. Equ. 57, No. 9, 1150-1164 (2021); translation from Differ. Uravn. 57, No. 9, 1177-1190 (2021).
Summary: We consider the problem of surface waves in a regular open inhomogeneous waveguide structure of rectangular cross-section. This problem 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 find the solution. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of an operator function on the complex plane are proved. The characteristic numbers of the problem correspond to the waveguide propagation constants.


78A50 Antennas, waveguides in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
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[1] Il’inskii, A. S.; Shestopalov, Yu. V., Primenenie metodov spektral’noi teorii v zadachakh rasprostraneniya voln (Application of Methods of Spectral Theory in Problems of Wave Propagation) (1989), Moscow: Izd. Mos. Univ., Moscow
[2] Smirnov, Yu. G., The method of operator pencils in boundary value problems of transmission for a system of elliptic equations, Differ. Uravn., 27, 1, 140-147 (1991)
[3] Smirnov, Yu. G., Application of the method of operator pencils to the problem of eigenwaves of a partially filled waveguide, Dokl. Akad. Nauk SSSR, 312, 3, 597-599 (1990)
[4] Delitsin, A. L., An approach to the completeness of normal waves in a waveguide with magnetodielectric filling, Differ. Equations, 36, 5, 695-700 (2000) · Zbl 1033.78512
[5] Smirnov, Yu. G., Matematicheskie metody issledovaniya zadach elektrodinamiki (Mathematical Methods for Studying Problems of Electrodynamics) (2009), Penza: Perz. Gos. Univ., Penza
[6] Lozhechko, V. V.; Shestopalov, Yu. V., Problems of the excitation of open cylindrical resonators with an irregular boundary, Comput. Math. Math. Phys., 35, 1, 53-61 (1995) · Zbl 0844.65093
[7] Dautov, R. Z.; Karchevskii, E. M., Metod integral’nykh uravnenii i tochnye nelokal’nye granichnye usloviya v teorii (The Method of Integral Equations and Exact Nonlocal Boundary Conditions in Theory) (2009), Kazan: Kazan. Gos. Univ., Kazan
[8] Smirnov, Yu. G.; Smolkin, E. Yu., Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide, Differ. Equations, 53, 10, 1262-1273 (2017) · Zbl 1384.78008
[9] Smirnov, Yu. G.; Smol’kin, E. Yu., Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section, Dokl. Math., 97, 86-89 (2018) · Zbl 1397.78043
[10] Smirnov, Yu. G.; Smol’kin, E. Yu., Operator function method in the problem of normal waves in an inhomogeneous waveguide, Differ. Equations, 54, 9, 1168-1179 (2018) · Zbl 1414.78010
[11] Smirnov, Y.; Smolkin, E., Mathematical theory of normal waves in an open metal-dielectric regular waveguide of arbitrary cross section, Math. Model. Anal., 25, 3, 391-408 (2020)
[12] Adams, M. J., An Introduction to Optical Waveguides (1981), Chichester-New York-Brisbane-Toronto: John Wiley &Sons, Chichester-New York-Brisbane-Toronto
[13] Snyder, A.; Love, J., Optical Waveguide Theory (1983), London-New York: Chapman and Hall, London-New York
[14] Vainshtein, L. A., Elektromagnitnye volny (Electromagnetic Waves) (1988), Moscow: Radio Svyaz’, Moscow
[15] Marcuse, D., Light Transmission Optics (1972), New York: Van Nostrand Reinhold, New York
[16] Vladimirov, V. S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics) (1981), Moscow: Nauka, Moscow
[17] Kolmogorov, A. N.; Fomin, S. V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis) (2004), Moscow: Fizmatlit, Moscow
[18] Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Natl. Bureau Stand., 1964. Translated under the title: Spravochnik po spetsial’nym funktsiyam, Moscow: Nauka, 1979. · Zbl 0171.38503
[19] Kato, T., Perturbation Theory for Linear Operators (1966), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0148.12601
[20] Shestopalov, Yu. V.; Smirnov, Yu. G.; Chernokozhin, E. V., Logarithmic Integral Equations in Electromagnetics (2000), Amsterdam: VSP, Amsterdam
[21] Adams, R. A., Sobolev Spaces (1975), New York: Academic Press, New York · Zbl 0314.46030
[22] Il’inskii, A. S.; Smirnov, Yu. G., Difraktsiya elektromagnitnykh voln na provodyashchikh tonkikh ekranakh: psevdodifferentsial’nye operatory v zadachakh difraktsii (Diffraction of Electromagnetic Waves by Thin Conducting Screens: Pseudodifferential Operators in Diffraction Problems) (1996), Moscow: Radiotekhnika, Moscow
[23] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), Berlin: Der Deutscher Verlag der Wissenschaften, Berlin · Zbl 0387.46033
[24] Gokhberg, I. Ts.; Krein, M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil’bertovom prostranstve (Introduction to the Theory of Linear Nonself-Adjoint Operators in a Hilbert Space) (1965), Moscow: Nauka, Moscow
[25] Hirsch, M. W., Differential Topology (1976), New York: Springer, New York · Zbl 0356.57001
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