  A rigorous analytical approach is applied to solve the eigenvalue problem for
a pair of circular dielectric cylinders with complex permittivity. This
approach relies on field expansion in terms of two sets of orthogonal azimuthal
modes, which are coupled due to finite distance between the cylinders. We
investigate the ability of a gain-dielectric cylinder operated in the
fundamental TM mode to compensate material losses of a larger cylinder operated
in the higher-order radial TM mode. To achieve such a loss compensation
phenomenon, a simple design strategy is developed. It is shown that this
phenomenon can be achieved for a certain distance between the cylinders, which
is associated with the exceptional point of the system. For smaller distances,
the adverse impact of high-order azimuthal (hybrid) modes are found to be
essential. The results obtained are validated against full-wave simulations.
