Mathematical Modeling of Electromagnetic Fields in Hypermaterials

Abstract

The development of technologies in the field of artificial materials with specially specified properties is a priority in the study of the properties of materials and their synthesis. In the presented work, electromagnetic fields arising in various types of hypermaterials are investigated: one-dimensional, two-dimensional, and three-dimensional crystal structures characterized by both positive and negative values of dielectric constant, which solves the Helmholtz equation by the finite element method. At the same time, a vector approach was used based on basis functions of the first order of the second type, and Dirichlet and Neumann conditions were set at the boundaries of the computational domain. The peculiarity of the applied mathematical model is the presence of a large kernel at the rotor operator, which creates poorly conditioned and sign-indeterminate systems of linear algebraic equations. In the course of research, it was found that negative magnetic permeability leads to the emergence of a left-handed medium, in which the vectors of the electric field, the magnetic field and the wave vector form a left triple. At the same time, the negative permittivity, according to the calculations, does not affect the formation of the left-hand medium. As a result of the study, it was concluded that hypermaterials with negative values of both dielectric and magnetic constant are of the greatest interest. Such materials make it possible to create left-handed environments that are promising for use in optical devices, transistors in optical computers.