Theory of Bragg diffraction by limited-aperture gratings in a planar optical waveguide

An analysis is made of Bragg diffraction of light waves in a planar optical waveguide by diffraction gratings of finite aperture and length used as waveguide couplers in integrated optics. In the asymptotic coupled-wave approximation, the complex amplitude of an inhomogeneous diffracted wave is described by a hyperbolic partial differential equation with boundary conditions determined by the profile of the grating and the incident optical beam. The interaction constant of the coupled waves is calculated and its dependence on the diffraction angle is determined for the electric and magnetic modes of a planar optical waveguide. Examples are given of calculations of the diffraction field in the near-field zone for gratings having the configurations of a characteristic parallelogram and trapezium.