Formulation of the Riccati equation for the impedance operator in cylindrical coordinates for inhomogeneous anisotropic waveguides with the example of rectilinear anisotropy

Abstract

The analysis of cylindrical waveguides with arbitrary anisotropy and inhomogeneities is more difficult than for isotropic or cylindrically anisotropic. The reason for this difficulty is because the independent consideration of different Fourier components of the wavefield is no longer possible. Taking into account the coupling between different Fourier components of the wavefield implies that the impedance tensor should be treated as the operator. Treating the impedance this way is the key difference between the isotropic and the cylindrical anisotropy case in which the independent consideration of the partial impedance matrices for each circumferential number n is possible. The natural approach to computing the impedance operator is to use the Riccati equation, which is presented by the authors in cylindrical coordinates. The matrix representations of the impedance operator and the Riccati equation over the basis of the time-harmonic cylindrical waves result in the infinite set of ordinary differential equations. The nondiagonal partial impedance matrices, which describe the coupling of different Fourier components, become nonzero. For the problems, which possess a symmetry, the formulation can be simplified by projecting the Fourier basis on that of the irreducible representations of the symmetry group. The numerical solution of the Riccati equation and its application to finding the dispersion curves of the eigenmodes of the waveguide have peculiarities due to the fact that the matrix representation is infinite. The feasibility of the proposed approach is demonstrated by computing the dispersion curves for the first monopole and dipole normal modes of a cylindrical waveguide in a homogeneous media possessing tilted transverse isotropy type of anisotropy.