A systematic approach to the analysis of almost-periodic vertical-grating-assisted codirectional couplers with nonuniform duty ratios is presented. The Poisson sum formula is used to expand rigorously a nonuniform rectangular grating into a quasi-Fourier series, and local grating parameters such as period, width, and duty ratio can be defined unambiguously in such a procedure. On the basis of this expansion the coupled-local-mode formulation is the most natural extension for the analysis of such a nonuniform grating-assisted codirectional coupler filter. By transformation of the coupled-local-mode equations into the Zakharov–Shabat system the Gel’fand–Levintan–Marchenko inverse-scattering method is then used to synthesize special grating-assisted codirectional coupler filters. The design is illustrated by two typical examples: One is a third-order Butterworth filter, and the other is a linear filter.

© 2000 Optical Society of America

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