Finesse of ring resonators

We deduce in this paper the exact solutions for the calculation of full width at half maximum (FWHM) and finesse F of ring resonators in which light is coupled from a bus to a ring. These solutions are shown to be simple in math. Furthermore, we deduce the electric field relationships for the case if light is generated within the ring and out-coupled to the bus and obtain the expression for the calculations of FWHM and finesse for this configuration. Our method is shown to produce more accurate results than those with the widely used approximation method, in particular, if the ring resonator works in low quality-factor regime.