Towards a general theory for MxN pixelated carrier interferometry

Pixelated phase-mask (PPM) interferometers have become an industry standard for instantaneous phase-shifting interferometry. In commercially available PPM interferometers, an array with 2x2 unit-cells is used, which codify up-to 4 phase-steps within a single PPM interferogram. Recently we have shown that such 2x2 unit-cell arrays allows a harmonic rejection as good as the 4-step leastsquares phase-shifting algorithm (LS-PSA); this harmonics rejection is relatively-low and may not be enough to correctly demodulate some severely intensity distorted fringe patterns. In previous works we have proposed a new PPM with a 3x3 unit-cell to improve the harmonics rejection of the 2x2 array. With this new 3x3 unit-cell one is able to reject as many harmonics as with a 9-step LS-PSA¹⁰. In this paper we are extending the analysis of MxN unit-cell synchronous demodulation of PPM. The new results allow us to answer some important open questions about the method: for a given configuration, which harmonics cannot be rejected and why? Why, prior to low-pass filtering, we observe multiple copies of the interferogram's spectrum and what does this imply? We believe these preliminary results are important contributions towards a formulation of a general theory MxN unit-cell pixelated carrier interferometry.