Rectangular characteristic gratings for waveguide input and output coupling

Normal-incidence planar-optical waveguide-imbedded phase gratings of finite aperture width and length are analyzed with Svidzinskii's (Sov. J. Quantum Electron. 10, 1103 (1980)] two-dimensional Braggdiffraction theory. Svidzinskii's characteristic-grating equations are adapted for the rectangulargrating case, and an overlap integral is used to extend the theory to account for the mode structure of the waveguide. The combined theory is used to optimize the throughput of a system composed of an input grating coupler, a waveguide, and an output grating coupler for both the highly multimode (thickwaveguide) and the few-mode (thin-waveguide) cases.