Numerical problem research of electromagnetic oscillations of open nonuniform spherical resonators

Background. Dielectric spherical resonators, due to the wide possibilities of their application, are increasingly becoming a subject for scientific research. Studies of the spectral characteristics of an open two-layer spherical resonator have shown that when measuring the dielectric properties of a liquid with losses occupying a small volume, it becomes necessary to introduce a middle layer. Therefore, the purpose of this work was to analytically and numerically investigate a dielectric resonator made in the form of a radial three-layer ball. Materials and methods. First, a review of the theory of natural waves of a dielectric sphere is given. Special attention is paid to modes with large radial and azimuthal indices. The system of Maxwell's equations is solved in the case of a space with a dielectric ball, which is reduced to solving a scalar equation for the so-called Debye potentials. Results. The initial model problem is reduced to solving a scalar equation for Debye potentials. The obtained characteristic equation is investigated for the case when the parameters of the structure of the inner and outer ball coincide. The solutions of this equation allow us to study the dependence of the complex permittivity of the substance under study on the parameters of the resonator with its known spectral characteristics. Conclusions. A numerical method based on finding the root equation using the ranging method is proposed. An algorithm was developed in the Maple mathematical package and applied to study the structure of a layer filled with gasoline, air, ethyl alcohol, transformer and fused quartz. The dependences of the solution of the system on frequencies, as well as the values of the wave number / frequency on the radius are illustrated