Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses

Abstract

In this work, we present a comparative analysis of different numerical methods to obtain the focusing properties of the zone plates based on Fibonacci and Cantor sequences. The Fresnel approximation was solved numerically in order to obtain the axial irradiance provided by these diffractive lenses. Two different methods were applied. The first one is based on numerical integration, specifically the Simpson integration method and the two-dimensional Gaussian quadrature. The second consisted in the implementation of the Fast Fourier Transform in both one and two dimensions. The axial irradiance of the lenses, the relative error with respect to the analytical solution, and the calculation time required by each method are analyzed and compared. From this analysis it was concluded that the Gauss method presents the best balance between accuracy and computation time. This analysis could be useful to decide the most convenient numerical method to be used for the study of more complex diffractive structures.