Phase space analysis of two-wavelength interferometry

Multiple wavelength phase shifting interferometry is widely used to extend the unambiguous range (UR) beyond that of a single wavelength. Towards this end, many algorithms have been developed to calculate the optical path difference (OPD) from the phase measurements of multiple wavelengths. These algorithms fail when phase error exceeds a specific threshold. In this paper, we examine this failure condition. We introduce a "phase-space" view of multi-wavelength algorithms and demonstrate how this view may be used to understand an algorithm's robustness to phase measurement error. In particular, we show that the robustness of the synthetic wavelength algorithm deteriorates near the edges of its UR. We show that the robustness of de Groot's extended range algorithm [Appl. Opt. 33, 5948 (1994)] depends on both wavelength and OPD in a non-trivial manner. Further, we demonstrate that the algorithm developed by Houairi & Cassaing (HC) [J. Opt. Soc. Am. 26, 2503 (2009)] results in uniform robustness across the entire UR. Finally, we explore the effect that wavelength error has on the robustness of the HC algorithm.