Theoretical analysis and experimental verification of six-step spatial phase-shifting shearing interferometry by double gratings

In order to measure the phase projection in moiré tomography, in this paper we present a new spatial phase-shifting shearing interferometry that consists only of a crossed grating and a linear grating. With it, six phase-shifted interferograms can be acquired simultaneously. The intensity distributions of these six interferograms are derived based on the scalar diffraction theory. Analytical results indicate that phase shifts are symmetrically and consistently distributed in the vertical and horizontal directions. Moreover, phase-shift values depend primarily on the parameters of the optical structure. And a six-step phase-shifting algorithm with arbitrary phase-shift values in two perpendicular directions is proposed to extract the phase information. The phase retrieval results of the spherical wave have verified the validity of the proposed method. Finally, an experiment with a plane incident wave is performed to measure the first-order derivative of the phase projection of a propane flame.