Mathematical theory of normal waves in an open metal-dielectric regular waveguide of arbitrary cross section
Abstract
The problem of normal waves in an open metal-dielectric regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operatorfunction on the complex plane is found.
Keyword : non-linear eigenvalue problem, Maxwell's equations, operator-function, Sobolev spaces, discrete spectrum
How to Cite
Smolkin, E., & Smirnov, Y. (2020). Mathematical theory of normal waves in an open metal-dielectric regular waveguide of arbitrary cross section. Mathematical Modelling and Analysis, 25(3), 391-408. https://doi.org/10.3846/mma.2020.10682
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P.E. Krasnushkin and E.I. Moiseev. On the excitation of oscillations in layered radiowaveguide. Doklady AN SSSR, 264:1123–1127, 1982.
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Y. Shestopalov and Y. Smirnov. Eigenwaves in waveguides with dielectric inclusions: Completeness. Applicable Analysis, 93(9):1824–1845, 2014. https://doi.org/10.1080/00036811.2013.850494
Y. Shestopalov and Y. Smirnov. Eigenwaves in waveguides with dielectric inclusions: Spectrum. Applicable Analysis, 93(2):408–427, 2014. https://doi.org/10.1080/00036811.2013.778980
Yu.V. Shestopalov, Yu.G. Smirnov and E.V. Chernokozhin. Logarithmic Integral Equations in Electromagnetics. Holland: De Gruyter, 2000. https://doi.org/10.1515/9783110942057
Yu.G. Smirnov. Application of the operator pencil method in the eigenvalue problem for partially. Doklady AN SSSR, 312:597–599, 1990.
Yu.G. Smirnov. The method of operator pencils in the boundary transmission problems for elliptic system of equations. Differ. Equ., 27:140–147, 1991.
Yu.G. Smirnov. Mathematical Methods for Electromagnetic Problems. Penza: PSU Press, 2009.
Yu.G. Smirnov and E. Smolkin. Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide. Differential Equations, 53:1262–1273, 2017. https://doi.org/10.1134/S0012266117100032
Yu.G. Smirnov and E. Smolkin. Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section. Doklady Mathematics, 97(1):86–89, 2017. https://doi.org/10.1134/S1064562418010271
Yu.G. Smirnov and E. Smolkin. Eigenwaves in a lossy metaldielectric waveguide. Applicable Analysis, 99(1):1–12, 2018. https://doi.org/10.1080/00036811.2018.1478084
Yu.G. Smirnov and E. Smolkin. Operator function method in the problem of normal waves in an inhomogeneous waveguide. Differential Equations, 54(9):1168– 1179, 2018. https://doi.org/10.1134/S0012266118090057
Yu.G. Smirnov, E. Smolkin and M.O. Snegur. Analysis of the spectrum of azimuthally symmetric waves of an open inhomogeneous anisotropic waveguide with longitudinal magnetization. Computational Mathematics and Mathematical Physics, 58(11):1887–1901, 2018. https://doi.org/10.1134/S096554251811012X
E. Smolkin. Numerical method for electromagnetic wave propagation problem in a cylindrical inhomogeneous metal dielectric waveguiding structures. Mathematical Modelling and Analysis, 22(3):271–282, 2017. https://doi.org/10.3846/13926292.2017.1306809
A.W. Snyder and J. Love. Optical waveguide theory. Springer, 1983.
H. Triebel. Theory of Function Spaces. Birkhauser Verlag, 1983. https://doi.org/10.1007/978-3-0346-0416-1
V.S. Vladimirov. Equations of Mathematical Physics. Nauka, 1985.
G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1995.
A.S. Zilbergleit and Yu.I. Kopilevich. Spectral theory of guided waves. London: Inst. of Phys. Publ., 1966.
R.A. Adams. Sobolev Spaces. Academic Press, 1975.
M. Costabel. Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal., 19(3):613–626, 1988. https://doi.org/10.1137/0519043
A.L. Delitsyn. An approach to the completeness of normal waves in a waveguide with magnitodielectric filling. Differential Equations, 36:695–700, 2000. https://doi.org/10.1007/BF02754228
I.C Gohberg and M.G. Krein. Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Nauka, 1965.
L.L. Helms. Introduction to potential theory. R. E. Krieger Pub. Co, 1975.
A.S. Il’inskii and Yu.G. Smirnov. Diffraction of Electromagnetic Waves on Conductive Thin Screens: Pseudodifferential Operators in Diffraction Problems. Radiotehnika, 1996. (in Russian)
T. Kato. Perturbation Theory for Linear Operators. New York: Springer-Verlag, 1980.
M.V. Keldysh. On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators. Doklady AN SSSR, 77:4–11, 1951.
A.N Kolmogorov and S.V. Fomin. Elements of the Theory of Functions and Functional Analysis. Dover Publications, 1999.
P.E. Krasnushkin and E.I. Moiseev. On the excitation of oscillations in layered radiowaveguide. Doklady AN SSSR, 264:1123–1127, 1982.
L. Levin. Theory of waveguides. London: Newnes-Butterworths, 1975.
V.V. Lozhechko and Yu.V. Shestopalov. Problems of the excitation of open cylindrical resonators with an irregular boundary. Computational Mathematics and Mathematical Physics, 35(1):53–61, 1995.
Y. Shestopalov and Y. Smirnov. Eigenwaves in waveguides with dielectric inclusions: Completeness. Applicable Analysis, 93(9):1824–1845, 2014. https://doi.org/10.1080/00036811.2013.850494
Y. Shestopalov and Y. Smirnov. Eigenwaves in waveguides with dielectric inclusions: Spectrum. Applicable Analysis, 93(2):408–427, 2014. https://doi.org/10.1080/00036811.2013.778980
Yu.V. Shestopalov, Yu.G. Smirnov and E.V. Chernokozhin. Logarithmic Integral Equations in Electromagnetics. Holland: De Gruyter, 2000. https://doi.org/10.1515/9783110942057
Yu.G. Smirnov. Application of the operator pencil method in the eigenvalue problem for partially. Doklady AN SSSR, 312:597–599, 1990.
Yu.G. Smirnov. The method of operator pencils in the boundary transmission problems for elliptic system of equations. Differ. Equ., 27:140–147, 1991.
Yu.G. Smirnov. Mathematical Methods for Electromagnetic Problems. Penza: PSU Press, 2009.
Yu.G. Smirnov and E. Smolkin. Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide. Differential Equations, 53:1262–1273, 2017. https://doi.org/10.1134/S0012266117100032
Yu.G. Smirnov and E. Smolkin. Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section. Doklady Mathematics, 97(1):86–89, 2017. https://doi.org/10.1134/S1064562418010271
Yu.G. Smirnov and E. Smolkin. Eigenwaves in a lossy metaldielectric waveguide. Applicable Analysis, 99(1):1–12, 2018. https://doi.org/10.1080/00036811.2018.1478084
Yu.G. Smirnov and E. Smolkin. Operator function method in the problem of normal waves in an inhomogeneous waveguide. Differential Equations, 54(9):1168– 1179, 2018. https://doi.org/10.1134/S0012266118090057
Yu.G. Smirnov, E. Smolkin and M.O. Snegur. Analysis of the spectrum of azimuthally symmetric waves of an open inhomogeneous anisotropic waveguide with longitudinal magnetization. Computational Mathematics and Mathematical Physics, 58(11):1887–1901, 2018. https://doi.org/10.1134/S096554251811012X
E. Smolkin. Numerical method for electromagnetic wave propagation problem in a cylindrical inhomogeneous metal dielectric waveguiding structures. Mathematical Modelling and Analysis, 22(3):271–282, 2017. https://doi.org/10.3846/13926292.2017.1306809
A.W. Snyder and J. Love. Optical waveguide theory. Springer, 1983.
H. Triebel. Theory of Function Spaces. Birkhauser Verlag, 1983. https://doi.org/10.1007/978-3-0346-0416-1
V.S. Vladimirov. Equations of Mathematical Physics. Nauka, 1985.
G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1995.
A.S. Zilbergleit and Yu.I. Kopilevich. Spectral theory of guided waves. London: Inst. of Phys. Publ., 1966.