A hybrid filter of Bragg grating and ring resonator

A theoretical model is presented for a hybrid resonator of Bragg grating on a ring, coupled to a linear waveguide. An explicit expression of the transfer matrix is obtained for the Bragg grating with a loss in each repeat unit of the refractive index variation. Then, the transmission and reflection spectra through the coupled linear waveguide are calculated. When the ring has an even number of repeat units, the Bragg reflection occurs at one of ring resonances, capable of producing a deep and wide stop band. We identify the relationship between the grating parameter and the coupling coefficient that optimizes the filter performance. The circumference of the hybrid resonator can be as short as 1/100 of the length of the linear Bragg grating that gives similar stop-band characteristics.