Matrix analysis of 2-D microresonator lattice optical filters

We present a transfer matrix analysis of a two-dimensional (2-D) filter to study its frequency response functions. The (M×N) array consists of N independent columns of microring resonators side-coupled to two channel bus waveguides, with equal spacing between columns and each column consisting of M coupled resonators. We show that such a general 2-D lattice network of lossless and symmetric resonators can approximate an ideal bandpass filter characterized by a flat-top box-like amplitude response without out-of-band sidelobes, and a linear phase response. The bandwidth is determined by the coupling factor between resonators. The 2-D periodic structure exhibits nonoverlapping photonic bandgaps arising from the complementary transmission properties of the row and column arrays. The row array behaves as a distributed feedback grating giving rise to narrow bandgaps corresponding to the flat reflection passbands of the filter with out-of-band sidelobes. The column array, on the other hand, acts as a high-order coupled-cavities filter with broad bandgaps that overlap with the sidelobe regions, thereby effectively suppressing the sidelobes. The phase response is linear except near the band edges, where enhanced group delay limits the usable bandwidth of the filter to about 80%. The minimum size of the array required is about 3×10, but is ultimately limited by waveguide loss.