COUPLING OSCILLATIONS OF LATTICES OF DIFFERENT DIELECTRIC RESONATORS
DOI:
https://doi.org/10.20535/2411-2976.12025.75-86Keywords:
dielectric resonator, eigen oscillations, lattice, tridiagonal matrix, circulantAbstract
Background. The development of many elements of modern communication systems is increasingly based on the use of various types of dielectric resonators (DR). The theory of coupled oscillations of resonators is the basis for further calculations and optimisation of the scattering matrices of electromagnetic waves on various devices. When calculating devices built on a large number of resonators, direct numerical methods are often not effective. They usually require the use of powerful computers, therefore, the calculation of elements on a large number of DR is impossible without building analytical models of complex structures based on electrodynamic modelling.
Objective. The study aims to find analytical expressions for the frequencies and distributions of electromagnetic fields of natural oscillations of lattices, consisting of a large number of various types of dielectric resonators for use in various devices of optical communication systems. To solve this problem, a linear system of equations, which relates the complex amplitudes and frequencies of the resonators, obtained earlier from the perturbation theory, was used.
Methods. To find analytical expressions, methods of matrix theory are used. In this case, both known methods of calculating the determinants of tri-diagonal and circulant matrices are used, as well as their modifications related to the calculations of more complex matrices, which, after transformations, are reduced to much simpler formulas. The final result is the receipt of new general analytical formulas for describing coupled oscillations of lattices consisting of a large number of dielectric resonators of various types.
Results. Coupled oscillations of one-dimensional linear lattices of two types of dielectric resonators are considered. New analytical expressions for complex frequencies and amplitudes of resonators, as well as Q-factor expressions without restrictions on their number, are obtained. A new model of natural oscillations of two-dimensional lattices, consisting of dielectric resonators of two different types, is constructed. General analytical solutions are found for the frequencies and amplitudes of coupled oscillations for two types of two-dimensional lattices with different arrangements of resonators. Analytical solutions are found for the amplitudes and frequencies of coupled oscillations of two axially symmetric ring lattices with different types of resonators, which are characterised by different placement symmetry in free space. The obtained general analytical expressions for the frequencies of coupled oscillations are compared with the results of calculations obtained numerically, by solving linear systems of equations. A very good agreement between the solutions obtained by the two methods is demonstrated.
Conclusions. The developed theory is the basis for the design of many devices of the optical wavelength range, which are built on the basis of the use of a large number of dielectric resonators of various types. The obtained new analytical expressions for calculating coupled oscillations of dielectric resonators allow building new more efficient models of scattering for optimization of various optical communication devices.
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