Operator Pencil Approach in an Electromagnetic Problem of Symmetric Wave Propagation in a Plane Shielded Waveguide
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Operator Pencil Approach in an Electromagnetic Problem of Symmetric Wave Propagation in a Plane Shielded Waveguide V. Yu. Martynova1* , M. A. Moskaleva1** , D. V. Raschetova1*** , and D. V. Valovik1**** (Submitted by E. K. Lipachev) 1
Penza State University, Penza, 440026 Russia
Received March 7, 2020; revised March 17, 2020; accepted March 20, 2020
Abstract—The paper focuses on the problem of electromagnetic wave propagation in plane shielded dielectric waveguide. The waveguide is characterized by an isotropic inhomogeneous permittivity and constant permeability. We study symmetric guided waves that characterised by a pair of (coupled) propagation constants. Using an operator pencil approach, we prove a discreteness property for the searched-for propagation constants. DOI: 10.1134/S1995080220070276 Keywords and phrases: Maxwell’s equations, wave propagation, coupled propagation constants, inhomogeneous waveguide, operator pencil, multiparameter eigenvalue problem.
1. STATEMENT OF THE PROBLEM AND INTRODUCTORY REMARKS This paper continues the investigation of symmetric guided electromagnetic waves (SGWs) that were introduced in [1]. Let Σ be a plane waveguide, where Σ := {(x, y, z) : 0 x h, (y, z) ∈ R2 }. The waveguide Σ is assumed to be closed having perfectly conducting screens σ0 := {(x, y, z) : x = 0, (y, z) ∈ R2 },
σh := {(x, y, z) : x = h, (y, z) ∈ R2 }.
at the boundaries and located in the Cartesian coordinates Oxyz, see Fig. 1. The layer Σ is filled with a nonmagnetic isotropic inhomogeneous medium characterized by the permittivity ε ≡ ε0 ε(x), where ε(x) ∈ C 1 [0, h] and ε(x) δ > 0 for all x ∈ [0, h] with arbitrary fixed δ > 0. We assume that inside the layer μ = μ0 > 0 is the permeability of free space. As is well known monochromatic transverse-electric (TE) and transverse-magnetic (TM) guided waves that the waveguide Σ supports can be searched for in the forms E = (0, ey , 0) eiγz ,
H = (hx , 0, hz ) eiγz ,
x Vh h 6
H
z
V0 0 Fig. 1. Geometry of the problem. *
E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] **
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(1)
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E = (ex , 0, ez ) eiγz ,
H = (0, hy , 0) eiγz ,
(2)
respectively, where γ is a real unknown (spectral) parameter, ei , hi (i ∈ {x, y, z}) depend only on x, and the time factor e−iωt is omitted [2–5]. Let us consider a monochromatic wave (E, H)e−iωt propagating in the waveguide Σ and such that tangential components of the electric field of the wave vanish at the walls σ0 and σh , where ω is the circular frequency and E, H have the form E = (ex , ey , ez ) ei(γy y+γz z) ,
H = (hx , hy , hz ) ei(γy y+γz z) ,
(3)
with ex ≡ ex (x), ey ≡ ey (x), ez ≡ ez (x), hx ≡ hx (x), hy ≡ hy (x), hz ≡ hz (x), and γy , γz are unknown (spectral) parameters, ( ) is the transposition operation. Such kind of waves are called SGWs [1]. Obviously, SGWs generalises TE and TM guided waves given by (1) and (2), respectively. In paper [1] one can also find some background, additional comment
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