Sturm-Liouville Matrix Equation for the Study of Electromagnetic-Waves Propagation in Layered Anisotropic Media

Abstract

Abstract-We obtain a Sturm-Lioville matrix equation of motion (SLME) for the study of electromagnetic wave propagation in layered anisotropic structures. Conducting media were taken into account so that ohmic loss is considered. This equation can be treated using a 4 × 4 associated transfer matrix (T) in layered anisotropic structures, where the tensors: permittivity, permeability and the electric conductivity have a piecewise dependence on the coordinate perpendicular to the layered structure. We use the SLME eigenfunctions and eigenvalues to analyze qualitatively the numerical instability (Ωd problem) which potentially affects practical applications of the transfer matrix method. By means of the SLME coefficients we show analytically that T determinant value can be used to keep a check on the numerical accuracy of calculations. We derive equations to analyze wave propagation in linear layered isotropic structures. The SLME approach is applied on two typical layered structures to verify theoretical predictions and experimental results.

Hybrid matrix method for stable numerical analysis of the propagation of Dirac electrons in gapless bilayer graphene superlattices

“…Having obtained the eigenfunctions, the coefficient transfer matrix method can be formulated applying the proper boundary conditions at the interfaces of the structure, see for example references [40,41,42]. In our case the continuity conditions for the wavefunction (11) and its derivative, along the x axis, are well supported.…”

“…The name associated with this numerical instability derives from the elastic waves studies where this instability is present at high frequencies Ω and/or big thicknesses (d) of the structure layers. This is because the matrix elements contain a mixture of exponentially growing and decaying terms that lead to loss of precision during computations [38,39,40,41,42]. In GBG this mixture is given by the presence of evanescent-divergent states in the eigenfunctions linear superposition representing the Dirac spinors.…”

“…Other approaches employ transfer matrices with dimensions independent of the number of layers. Among them we can find the scattering matrix method, the impedance matrix method, the compound matrix, and the hybrid compliance-stiffness matrix or simply the hybrid matrix [38,39,40,41,42]. From all these possibilities, the hybrid matrix method is an excellent option, since it is numerically stable, well-conditioned, and accurate irrespective of the thickness (large, small or even zero) of the system [38,39,37].…”