Phase-stepping algorithms for synchronous demodulation of nonlinear phase-shifted fringes

Standard phase-stepping algorithms (PSAs) estimate the measuring phase of linear carrier temporal-fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using feedback, closed-loop, optical phase-shifting devices. On the other hand, open-loop, phase-shifting devices, usually give fringe patterns with nonlinear phase-shifts. The Fourier spectrum of linear-carrier fringes is composed by Dirac deltas only. In contrast, nonlinear phase-shifted fringes are wideband, spread-spectrum signals. It is well known that using linear-phase reference PSA to demodulate nonlinear phase-shifted fringes, one obtains an spurious-piston. The problem with this spurious-piston, is that it may wrongly be taken as a real optical thickness. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with open-loop, nonlinear phase-shifted interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to demodulate nonlinear phase-shifted wideband fringes.