Study of linear phase shift algorithms and application to deflectometry

Abstract

For a number of optical methods, including deflectometry, information is encoded in fringe patterns. To extract the phase data from intensity values, a number of phase shift algorithms have been designed. In deflectometry, the nonlinearity of the fringe display implies that the displayed fringe pattern presents a harmonic content even if the input pattern is perfectly sinusoidal. The propagation of these harmonics through phase shift algorithms creates parasitic fringe patterns, reminiscent of the initial fringe pattern on the estimated phase. This phenomenon, known as print-through, has been identified as a serious performance limitation.In this paper, we revisit Surrel’s work on harmonic insensitive phase shift algorithms and demonstrate that the class of Discrete Fourier Transform (DFT) phase shift algorithms he defines is very appropriate for the field of deflectometry. We show how to choose the most suitable one depending on the application by performing a complete modeling of the harmonic print-through phenomenon for these DFT algorithms and studying the error propagation for shot noise and temporal perturbations. In a deflectometry context, we demonstrate by means of simulations that carefully chosen DFT algorithms can simultaneously be robust to print-through and perform better with respect to noise than the state of the art nonlinear phase shift algorithms.Lastly, by comparing experimental mirror shape measurements of the matrix of the secondary mirror of the European Extremely Large Telescope made on the one hand by DFT deflectometry and on the other hand by phase shift interferometry, we demonstrate that the use of DFT algorithms can substantially improve the high spatial frequency measurement capabilities of a deflectometry setup, without the need for a calibration of the display’s nonlinearities.