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A Bragg waveguide consisting of multiple dielectric layers with alternating index of refraction becomes an excellent option to form electron accelerating structure powered by high power laser sources. It provides confinement of a synchronous speed-of-light mode with extremely low loss. However, laser field cannot be coupled into the structure collinearly with the electron beam. There are three requirements in designing input coupler for a Bragg electron accelerator: side coupling, selective mode excitation, and high coupling efficiency. We present a side-coupling scheme using a distributed grating-assisted coupler to inject the laser power into the waveguide. Side coupling is achieved by a grating with a period on the order of an optical wavelength. The phase matching condition results in resonance coupling thus providing selective mode excitation capability. The coupling efficiency is limited by profile matching between the outgoing beam and the incoming beam, which has normally a Gaussian profile. We demonstrate a nonuniform distributed grating structure generating an outgoing beam with a Gaussian profile, therefore, increasing the coupling efficiency.