Self-images location of amplitude/phase binary gratings

We analyze the near-field behavior of binary amplitude/phase diffraction gratings, which modulate at the same time the amplitude and phase of the incident light beam. As it is expected, the distance between two consecutive self-images of the grating depends only on the period of the grating and the wavelength of the illumination. However, the location of the self-images depends on the specific properties of the grating. In this work, we analyze the location of the self-images in terms of the Fourier coefficients of the grating, obtaining analytical expressions. This analysis can be useful in applications in which the position of the self-images must be at certain fixed distances from the grating. Finally, an experimental and numerical verification of the proposed theory is performed.