Effect of higher-order diffraction on the interference formed by Bragg scattering for large size optical surfaces

With an increase in the size and shape complexity of optical surfaces, the periodic structure on the optical surface can have significant effects on the beam profile, which propagates in the optical system. This necessitates a quantitative analysis of the interference between diffracted beam modes formed on the propagating beam profile after the reflection on the optical surfaces with periodic structures. Herein, a generalized equation, which determines the period of the constructive interference formed by Bragg scattering, is presented for the only two diffracted beam modes given by the m^th and n^th order. Two theoretical calculations, one being an analytical calculation based on the interference equation, and the other being a numerical simulation based on Fourier optics, are conducted for all the diffracted beam modes to reflect the effects of higher-order diffraction. Furthermore, the simulation results based on the Fourier optics are much closer to real phenomena because the boundary conditions are exactly considered. As the aspect ratio of the periodic structure on the optical surface increases, the period of constructive interference decreases, and its intensity is remarkably increased from the two theoretical calculations. Our results will be extensively applied as a novel theoretical tool to analyze quantitatively the effects of periodic structures on the propagating beam profile by providing a deep understanding about the interference formed by Bragg scattering.