THEORY OF GUIDED WAVES IN THE INFINITE SYSTEMS OF COUPLED DIELECTRIC RESONATORS
DOI:
https://doi.org/10.20535/2411-2976.22025.81-96Keywords:
dielectric resonator, eigen oscillations, lattice, coupled resonator transmission line, waveguide, perturbation theory, delay lineAbstract
Background. One of the promising elements of optical and quantum communication systems is various delay lines built on the high-quality dielectric resonators (DRs). These lines typically comprise a substantial number of elements, making the optimisation of their parameters quite challenging. The theory of DRs serves as a foundation for comprehending, calculating, and optimising the parameters of delay lines and other devices, facilitating a considerable reduction in the computational resources that typically require the use of powerful computers.
Objective. The study aims to derive analytical expressions for the electromagnetic parameters of diverse optical waveguides, composed of numerous types of DRs, to utilise them as transmission lines for optical communication systems. To address this issue, an infinite linear system of equations has been derived based on the perturbation theory applied to Maxwell's equations, which connects the complex amplitudes, wave numbers and the resonator frequencies.
Methods. To derive solutions for the analytical expressions, perturbation theory and the theory of infinite linear equations are employed. The outcome is a set of new general analytical formulae that describe the dispersion curves of lattices made up of an infinite number of various types of DRs.
Results. A theory of wave propagation in systems of interconnected one-, two-, and three-dimensional lattices of DRs extended infinitely in one or more directions has been developed. New analytical expressions for the dispersion characteristic of eigenwaves, delay times, and distributions of complex amplitudes of resonators, without any limitations on their quantity, have been derived. By utilising perturbation theory, a novel analytical model has been developed that describes the eigenwaves of three-dimensional lattices composed of identical ring structures of DRs. General analytical solutions for frequency dependencies and amplitudes for one-, two-, and three-dimensional lattices with varying arrangements of resonators have been identified.
Conclusions. The developed theory serves as the foundation for the analysis and design of many devices operating within the optical wavelength spectrum, constructed upon an infinite variety of distinct types of DRs. The obtained new analytical expressions for calculating optical waveguide parameters, based on coupled oscillations of DRs, enable the development of innovative and more efficient mathematical models for various optical communication devices.
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