Analysis of Interferograms with Arbitrary Phase Shifts Based on 3D-Trajectory Analysis

Optical interferometry is widely used in many branches of science and technology. The major problem of optical interferometry is to calculate the phase of the fringe pattern (interferogram) at each of its points. Recently, algorithms based on phase shifting have been used to solve this problem. The trajectory approach to the analysis of interferograms is known, in which the elliptical trajectory of interference signals (Lissajous figure) is transformed to a circular one. However, this approach requires finding the coefficients of the elliptic equation and its invariants with high accuracy, which presents certain technical difficulties. Also, this approach has a significant disadvantage - when stretching the ellipse to a circle, noise amplification occurs. A new approach is proposed that eliminates these difficulties. The approach is based on trajectory analysis of interference signals, which transforms the elliptical trajectory of interference patterns to a circular trajectory by projecting the points of the elliptical section of a 3D spatial shape onto the plane of the circular section. This transformation converts the elliptic trajectory into a circular one, which makes it possible to perform phase recovery using known formulas that do not explicitly contain the values of the phase shifts. The accuracy of phase calculation of different algorithms (algorithm based on the phase recovery formula for three fringe patterns, existent trajectory analysis algorithm and the proposed trajectory analysis algorithm) was compared. The experimental research was carried out on the synthesized interferograms with different values of phase shifts inaccuracy. The experimental data show that the phase recovery error for the first two algorithms is proportional to the phase shifts inaccuracy. Proposed algorithm has no such dependence. This allows to use the phase recovery formula for three fringe patterns with arbitrary phase shifts.