Analysis of the $\pi$ phase-shifts obtained in the Fourier spectra of phase gratings and grids by using two-window grating interferometry

Abstract

As it is very well known, the Fourier coefficients of only-phase gratings are Bessel functions of the first kind of integer order. Because the values of these real-valued functions oscillate around zero, they can adopt negative values, thereby producing phase shifts of $\pi$ between some diffraction orders. To better understand the practical implications of this effect, in this work the phase shifts in the Fourier spectra of several phase gratings and grids are numerically found in order to compare them with experimental phase shifts. These experimental shifts induce changes in the modulation of the interference fringes formed by grating (or grid) interferometers when placing two windows at object plane of the system. Measurements of the changes in the interferograms are presented.