Filter Response of Nonuniform Almost-Periodic Structures

The filter response of nonuniform, almost-periodic structures, such as corrugated optical waveguides, is investigated theoretically. The filter process, leading to reflection of a band of frequencies near the Bragg frequency, is treated as a contradirectional coupled-wave interaction and shown to obey a Riccati differential equation. The nonuniformity of the structure is represented by a tapering in the coupling strength (e.g., the depth of the corrugation) and by a chirp in the period of the structure. For small reflectivities, the filter response is a Fourier transform of the taper function. For large reflectivities, the Riccati equation was evaluated numerically and plots are given for the response of filters with linear and quadratic tapers and with linear and quadratic chirps.