Coupled Oscillations of Rectangular Lattices of Dielectric Resonators (Analytical Solutions)
DOI:
https://doi.org/10.20535/RADAP.2025.101.%25pKeywords:
dielectric resonator, coupled oscillations, rectangular lattice, spectral theoryAbstract
The frequency spectra and amplitude distributions of the natural oscillation fields of the same shape and dimensions coupled dielectric resonator (DR) systems, located in one-, two-, and three-dimensional rectangular lattices, are considered. Neglecting the coupling between non-adjacent resonators, general analytical solutions are found for the linear homogeneous system of equations proposed earlier for describing natural oscillations of coupled DR systems. An algorithm based on perturbation theory for solving systems of equations of coupled oscillations of identical DRs is proposed, which allows reducing the solution to the calculation of determinants of tridiagonal and pentadiagonal matrices. It’s shown that parameters of coupled oscillations of rectangular structures are determined through simple distributions of amplitudes and frequencies characteristic of Bloch waves of quantum particles in a periodic potential. Using the derived general analytical formulas, the calculated frequencies are compared with the natural oscillation frequencies obtained by numerical methods. For the first time, a general analytical solution is found for the distribution of amplitudes and frequencies of a rectangular lattice with DR doubly degenerate types of natural oscillations. It’s shown that in the case of zero coupling between degenerate oscillations of different types, the obtained analytical formulas transform into expressions describing the oscillations of DRs in simple rectangular lattices. General conditions are formulated, under which the solution of the equations for coupled oscillations of DR systems can be found in analytical form. It’s shown that under the specified conditions, the distribution of the amplitudes of coupled oscillations of identical DR lattices with degenerate and non-degenerate types of eigenoscillations are interconnected to each other. In this case, a new method for calculating the amplitudes and frequencies of coupled oscillations of DR lattices with degenerate oscillations is proposed. The obtained formulas allow us to estimate in general terms the characteristics of the spectrum of eigenoscillations with an increasing in the number of resonators. Several examples demonstrate a very good coincidence of the found analytical and numerical results. The obtained theoretical conclusions significantly simplify the calculation and optimization of scattering parameters of various communication devices in the microwave, infrared and optical wavelength ranges, which are built based on the use of rectangular DR structures.
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