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Title:  Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation 
Authors:  Nickelson, Liudmila Pomarnacki, Raimondas Sledevič, Tomyslav Plonis, Darius 
Keywords:  Maxwell’s equations Helmholtz equation boundary conditions constitutive relations dispersion characteristic Singular Integral Equation (SIE) method Cauchy type singular integral waveguide arbitrary shape of crosssection dielectric semiconductor metal propagation constant phase constant hybrid modes numerical solution comparison with experiments 
Issue Date:  2021 
Publisher:  MDPI 
Citation:  Nickelson, L.; Pomarnacki, R.; Sledevič, T.; Plonis, D. Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation. Mathematics 2021, 9, 140. 
Series/Report no.:  9;2 
Abstract:  This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all crosssection elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the abovementioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a wellconditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper. 
Description:  This article belongs to the Section Mathematical Physics 
URI:  http://dspace.vgtu.lt/handle/1/4173 
ISSN:  22277390 
Appears in Collections:  Moksliniai straipsniai / Research articles

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