Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping

Since the early 1970s, optical two-wavelength phase-metrology (TWPM) has been used in a wide variety of experimental set ups. In TWPM one may compute the phase-sum and the phase-difference of two close phase measurements. Early TWPM optically computed the phase difference and phase sum by double exposure holography. However soon after, TWPM became almost synonymous to calculating the phase-difference only. The more sensitive phase-sum was largely forgotten. The standard application for phase-difference TWPM is to extend the phase measurement depth without phase-unwrapping for discontinuous phase-objects. This phase-difference, while non-wrapped, decreases however the signal-to-noise ratio (SNR) of the estimated phase. On the other hand, the phase-sum increases the phase sensitivity, and the SNR of the estimated phase. In spite of these two great advantages, the use of the phase-sum in TWPM has been almost ignored. In this paper we review and set the stage for digital TWPM for super-sensitive phase-sum estimation. This is coupled with two-sensitivity phase-unwrapping to obtain extended-range super-sensitive fringe-projection profilometry estimations. Here we mathematically prove, and experimentally show that using the phase-sum one obtains a huge increase in SNR with respect to using the phase-difference alone. The pioneer works on double exposure TWPM holography that uses the phase-difference and phase-sum are also properly acknowledged. Finally, two experimental results from fringe-projection profilometry that clearly show the huge SNR gain of the phase-sum, with respect to the phase-difference is now mathematically well established.