Phase retrieval in moiré volume computed tomography based on spatial phase shifting by triple-crossed gratings

Due to its advantages of nonintrusiveness, wide measurement range, and insensitivity to variation, Moiré deflectometry is a powerful tool for quantitative measurement of a flow field's physical parameters such as density and temperature. Moiré volume computed tomography (MVCT), combining the moiré deflectometry and volume optical computed tomography (VOCT), can realize real three-dimensional parameters reconstruction, in which the radial derivatives extraction of the projected phase is of great importance. In this paper, a spatial phase-shifting-interferometry-based MVCT system was proposed to extract the radial shearing phase distribution. The system is simple and compact, and consists of three crossed gratings and lenses, with no wave plates or polarizers introduced. Via using a 4-f system, the optical path was shortened, and four spatial phase-shifting grid moiré projections can be obtained simultaneously. Each grid interferogram was filtered and separated into two linear interferograms in two orthogonal directions. Moreover, a two-step spatial phase-shifting algorithm was applied to obtain the first-order derivative phase in two mutually perpendicular directions, respectively. Simulations were implemented to verify the feasibility and accuracy of the proposed phase retrieval method. The measured results for the radial first-order partial derivative of the phase projection of a propane flame in the experimental VOCT system are presented.