This paper studies the generalized equations of electromagnetic fields and their solutions in long waveguides with several regions made of materials with Hermitian dielectric tensors and elliptical cross-sections. The special conditions under which the quadratic homogeneous equations of the hybrid modes can be reduced to second-order equations are presented.
IntroductionIt is well known that the dispersion relation of electromagnetic waves in a waveguide essentially depends on its geometrical dimensions, and the dielectric permittivity and permeability coefficients of its materials [1-3]. There are waveguides with different cross-sections, made of different materials and different configurations. Many years ago these instruments with elliptical cross-sections were used as accelerators of charged particles [4], were used for microwave heating [5], and were also used as transmission lines [6]. Recently an elliptical waveguide made of metamaterials in which the phase velocity and energy flow of an electromagnetic wave can be antiparallel has also been investigated [7]. In [4][5][6][7], it was assumed that the waveguides were made of solid materials with isotropic scaler dielectric permittivity and permeability. The propagation characteristics of ferrite-filled elliptical waveguides have been investigated as an anisotropic problem only in [8,9], where the anisotropic properties refer to the permeability tensor. Recently, ideal stability of an elliptical plasma column [10] and propagation of charge-space and slow waves in elliptical waveguides [11,12] have been investigated. Furthermore, the theory of the equilibrium and stability of a high-intensity beam and the attenuation characteristics of elliptic waveguides have been analysed in [13] and [14], respectively. In the other words, plasma waveguides with elliptical cross-section are a new application of elliptic waveguides. Generally, there are two methods for investigating wave propagation in a material. In the first method, the perturbed values of the macroscopic quantities of the system are matched with the governing fundamental electromagnetic equations of the system [15]. In the second method, the perturbed values of the microscopic