Fourier analysis of two-stage phase-shifting algorithms

Abstract

Differential phase-shifting algorithms (DPSAs) and sum phase-shifting algorithms (SPSAs) recover directly the phase difference and the phase sum, respectively, encoded in two patterns. These algorithms can be obtained, for instance, by an appropriate combination of phase-shifting algorithms (PSAs), which makes unnecessary the previous calculation and subtraction or addition of each individual optical phase by means of conventional PSAs. A filtering process in the frequency domain is presented that allows us to obtain in a simple and elegant manner a qualitative characterization with a Fourier description of the two-stage phase-shifting evaluation that reveals possible phase shifter miscalibration errors and unexpected harmonics in the signal.

Compensation of the two-stage phase-shifting algorithms in the presence of detuning and harmonics

“…The design and analysis in the Fourier domain of two-stage phase-shifting (TSPS) algorithms induce an interesting field of study that has been considered in some publications [1][2][3][4][5]. Interferometric applications described in [6][7][8] require adequate phase-shifting equations that can compensate error sources as deviated shifts due to linear (detuning) [9,10] or nonlinear miscalibration [11,12], the presence of harmonics in the intensity patterns [13,14], or the presence of noise, etc.…”

“…In the calculation of phase sums or phase differences, appropriate linear combinations of the interferograms can minimize, in part, the effect that some of these error sources can produce. In this paper, we are mainly interested in compensating the presence of detuning and harmonics for the two sets of interferograms that are required to obtain the phase sum 2ϕ Δϕ or the phase difference Δϕ, in the context of the work reported in [3]. Differential phase-shifting algorithms are susceptible to systematic errors, such as phase miscalibration or the presence of higher-order harmonics in the fringe profile, which have been recently demonstrated [4,5].…”

“…(3), the indexes k and p of the sums were removed to simplify notation. In [3], algorithms for the tangent of the phase difference Δϕ and the tangent of the phase sum 2ϕ Δϕ are described in Eqs. (5) and (6); however, for consistence we only are going to analyze the algorithm for the phase difference Δϕ.…”

The Monte Carlo method as a tool for statistical characterisation of differential and additive phase shifting algorithms

“…The knowledge of their sensitivities to these errors is fundamental to choose the best algorithm in each particular application. It is for this reason that we have considered of interest the characterization of the TSPSAs by means of numerical simulation, error linearization or an analysis of the process in the Fourier space (see our previous works [17][18][19][20][21]). The aim of the present paper is to show that the Monte Carlo method (MCM) [22] can be a practical alternative to analyze the error sensitivity of algorithms in which any approximation cannot be introduced, especially when it is not possible a linearization of the model or it is too complex.…”

Adjustment of the Parallelism of two Mirrors for Wide Angle Divided Mirror Michelson Wind Imaging Interferometer

“…The phaseshifting interferometry is realized by causing different additional phase in different time or space, and record the corresponding interferogram respectively, then calculate the phase from several interferograms to measure the parallelism [13][14][15] . Many methods have been developed as phaseshifting [16][17][18][19] …”