  A general picture of photonic eigenmodes of dielectric rings with a
rectangular cross section is presented, which is fundamentally different from
those of whispering gallery modes of the disc. The optical spectrum of a
rectangular dielectric ring consists of an infinite set of individual
galleries, each starting with broad transverse radial or axial Fabry-P\'erot
resonances due to two pairs of side faces. Each gallery continues with a set of
equidistant longitudinal modes with exponentially increasing quality factors.
Transverse radial and axial modes exhibit strict periodicity of line contours
in the Fano-Lorentz-Fano-Lorentz... sequence due to the periodicity of
cylindrical harmonics. Theoretical and experimental results are in perfect
agreement.
