A simple approach to the suppression of the Gibbs phenomenon in diffractive numerical calculations

Abstract

The Gibbs phenomenon is a well-known effect that is produced at discontinuities of a function represented by the Fourier expansion when it is truncated to perform numerical calculations. This phenomenon appears because it is not possible to fit a discontinuous function as the summation of continuous functions, such as it is done with the Fourier expansion. Only considering infinite terms of the summation, the Fourier expansion fits the real signal. From a general point of view, it will affect to the final results since the representation of the signal does not include higher frequencies. It is true that the higher is the truncation, the better are the results, but an error is always committed. The Gibbs phenomenon has been studied in electric signal and diffractive optics, where the Fourier expansion is commonly used. In this work, we drop complex mathematics to show the effect of the Gibbs phenomenon on the near field propagation of diffraction gratings (self-imaging phenomenon) and also possible implementations of some corrections which allow diminishing the analytical or numerical errors in comparison with less accurate approaches. Anyway, the conclusions of this work would be applicable to other numerically solved diffractive problems which include sharp edges apertures. Simulations are compared with experiments giving interesting results.