Transmission properties of periodically sparse patterned microring resonators.

By using the transfer matrix method, we analyze the optical properties of periodically sparse patterned microring resonators in the resonator-waveguide system. The dispersion relation of a periodically sparse patterned microring is investigated theoretically. Two kinds of modes supported by the periodically sparse patterned microring resonators, traveling modes and stationary modes, are found. We also derive the reflectivity of the microring resonator side-coupled to a waveguide and find that just the stationary modes lead to total reflections. Traveling modes do not reflect, which can be used to extend free spectral range. All the transmission properties are confirmed by the finite-difference time-domain method numerically.