Analytical derivation of far-field intensity distribution and grating equation for trapezoidal transmission grating

Previously, we proposed a binary trapezoidal transmission grating with the properties of quasi single-order diffraction. Numerical simulations based on the scalar diffraction theory and a visible experiment have been carried out to verify the diffraction properties in far-field. However, this work didnt give the analytic expression of the far-field intensity distribution and grating equation of the novel two-dimensional grating. Therefore, it is impossible to estimate the intensity distribution and position on the screen of every diffraction order quickly when the grating is applied to the actual spectral measurement, and at present these could only be obtained by the numerical simulation based on the theory of scalar diffraction slowly. To solve the above problem and use this novel grating in ray-tracing calculations, the analytic theory must be improved to describe the intensity distribution and grating equation of every diffraction order. In this paper, we give the analytical expression of the intensity distribution and the grating equation satisfies both transmission type and reflection type cases by using the method of analytical derivation. Finally, comparisons are made between the results of calculation by our analytical expression and simulation by the scalar diffraction module of the X-Lab to prove the validity of the analytical theory.