Mathematical modeling of bi-isotropic waveguide using the finite elements method

The artificial materials, or metamaterials, that are strongly interacting with electromagnetic field, are currently actively developed and created. They include materials with negative permittivity and permeability, photonic crystals, bi-isotropic media etc. Bi-isotropic medium is the most general case of linear isotropic medium. Its distinctive feature is the existence of the electromagnetic coupling, so the electric or magnetic field acting on such medium produces both polarization and magnetization, unlike the ordinary dielectrics and magnetics Antireflecting coverings, waveguides, antennas, depolarizers etc. can be fabricated on the basis of metamaterials. Thereby the development of effective algorithms for calculating electromagnetic field in such media is a challenging problem. These algorithms can then be used for multiple solving the direct problem, when solving the inverse problem of synthesis of a material with preset features. One of the widely used methods is the finite elements method that permits to consider rather difficult domains. However in the case of electromagnetic problems in the full vector statement the finite elements method can give spurious solutions. There exist two basic ways to struggle against them: a posteriori, when the real solution is separated from spurious ones after calculating, and a priori, when such statements of a problem are used, that nonphysical solutions do not appear. In the second case the method of mixed finite elements is often used. This method effectively suppresses spurious solutions, but it possesses less accuracy than the Lagrangian finite elements method. In this work the numerical algorithm for calculating propagation constants for the spectral electromagnetic problem in the waveguide with perfectly conducting walls and bi-isotropic filling is proposed. This algorithm is based on specific generalized statement of the vector problem, and it permits to use the Lagrangian finite elements without nonphysical solu- ions.