Measurement of three-dimensional displacements by radial shearing interferometer

Moire patterns formed by overlapping two circular gratings of slightly different pitches have been extensively used for measuring the two-dimensional (2D) and three-dimensional (3D) displacements. However, in the existing applications, Moire patterns are analyzed based on geometric superposition, by which the 3D displacements cannot be instantaneously or simultaneously measured with a high accuracy. In this paper, radial shearing interferometry with double circular gratings of slightly different pitches is presented to realize the simultaneous measurement of 3D displacements. The measurement is based on the principle that Moire patterns produced by radial shearing interferometry are determined not only by the 2D in-plane displacements, but also by the out-of-plane displacement that brings about a phase shift between Moire patterns of +1 and -1 diffraction orders. First, the production mechanism of Moire patterns by radial shearing interferometry is studied based on the scalar diffraction theory and the intensity distribution of Moire fringes of +1 and -1 orders is derived to establish the exact analytic relations between Moire patterns and 3D displacements. Second, on the basis of spectrum characteristics of circular grating, a semicircular ring filter is proposed for spatial filtering to realize the simultaneous imaging of Moire fringes of +1 and -1 orders. Then, the algorithm to quantitatively extract 3D displacements from Moire patterns is proposed and demonstrated by numerical simulation. In the algorithm, Moire patterns in the rectangular coordinate system are transformed into the polar coordinate system and skeletons are extracted to determine the feature points of the bright fringes. The in-plane displacements can be solved by feature points of +1 or -1 diffraction order, and the out-of-plane displacement can be computed by the feature points of +1 and -1 diffraction orders in the same bright fringe. Finally, experimental results prove that the maximum absolute error and mean error for in-plane displacements are 4.8 x 10(-3) mm and 2.0 x 10(-4) mm respectively, and 0.25 mm and 8.6 x 10(-3) mm for out-of-plane displacement. In conclusion, by using the Moire patterns of +1 and -1 diffraction orders imaged by radial shearing interferometer with double circular gratings of slightly different pitches, the 3D displacement can be simultaneously measured. The method has the advantages of simple device, high measurement accuracy, noncontact and instantaneous measurement, which provides an important guidance for practically measuring the 3D displacements.