A Perturbation Theory for Natural Frequencies of Two Co-Directional Dielectric Cylinders

This article examines the problem of finding normal modes in a system of two co-directional dielectric cylinders (waveguides). If the distance between the cylinders is large, their wavefields do not overlap and they can be considered independently. On the other hand, if the distance between the cylinders is small, the mode interaction becomes noticeable. This leads to a shift in the natural frequencies of the system compared with the frequencies of the noninteracting cylinders. This shift can be successfully determined using perturbation theory. Since the natural frequencies for a single cylinder are doubly degenerate due to axial symmetry, each natural frequency splits into two. Explicit expressions for the shift in natural frequency caused by the presence of the second cylinder are given, containing integrals of known functions. The dependence of the natural frequency shift on the geometrical and physical parameters of the second waveguide is calculated using the proposed approach. According to the theoretical estimates confirmed by numerical calculations, the frequency splitting is much smaller than its shift. The law of decrease in frequency change with the change in distance between cylinders is analyzed and analytical expressions for the distance corresponding to the maximum splitting of frequencies are derived. The results can be readily applied for modeling and optimization of various devices based on electromagnetic principles, such as antennas, waveguides, microdisk laser systems, transmission lines, and others.