The recently suggested self-coupling microfiber coil optical resonator (COR) is a simplest functional element for the future microfiber-based photonics. It could be created by wrapping a microfiber on a dielectric rod with smaller refractive index. It is feasible that COR, which is produced from a drawn optical microfiber, will not suffer from surface roughness as e.g. the lithographically fabricated 2D microrings. Therefore, COR may have extremely small losses and generate high-Q resonances. In this paper, the theoretical study of the basic electromagnetic properties of the uniform self-coupling COR with N turns is presented. The eigenmodes of COR, which are qualitatively different from the modes of the known types of resonators, exist for the discrete values of the dimensionless coupling parameter K=κS 〉 ½, where κ is the coupling coefficient between adjacent turns and S is the length of a turn. The spatial variation of the mode amplitudes does not have the wavelength scale oscillations and has no correlation with the period of COR, S. For certain series of K, the free spectrum range of COR is independent of the number of turns N and COR behaves similar to a single ring resonator. For N→∞, the microfiber coil optical waveguide (COW) has a simple dispersion relation implying the absence of stop bands. The value K = ½ corresponds to the crossover between two regimes of propagation: with and without zeroing of the group velocity. At the crossover, the dispersion relation of COW has inflexion points wherein the group velocity and the inversed group velocity dispersion simultaneously become zero.