Use of Fourier coefficients in multilayer synthesis computations

Abstract

The objective of our synthesis technique is to find the refractive indices ni and the thicknesses ei of the layers in a nonabsorbing multilayer stack with prescribed properties. By introducing the constraint that each layer should have an optical thickness niei = ki λ0/4, where ki is an integer and λ0 a simple parameter as small as we wish, we have developed a calculation method for the optical properties, expressed as R/T, which is particularly advantageous. The form of the solution is a series of cosine terms. Given the reflectance R0 and the transmittance T0 of the filter to be synthesized, its R0/T0 curve is expanded in a Fourier series over a given frequency interval. The integers ki and the refractive indices of the layers are then adjusted until the coefficients of the cosine terms approximate those of the Fourier series of the filter being synthesized. The technique is demonstrated in the synthesis of a beamsplitter and a dichroic mirror.