General methods for generating phase-shifting interferometry algorithms

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Abstract
Two completely independent systematic approaches for designing algorithms are presented. One approach uses recursion rules to generate a new algorithm from an old one, only with an insensitivity to more error sources. The other approach uses a least-squares method to optimize the noise performance of an algorithm while constraining it to a desired set of properties. These properties might include insensitivity to detector nonlinearities as high as a certain power, insensitivity to linearly varying laser power, and insensitivity to some order to the piezoelectric transducer voltage ramp with the wrong slope. A noise figure of merit that is valid for any algorithm is also derived. This is crucial for evaluating algorithms and is what is maximized in the least-squares method. This noise figure of merit is a certain average over the phase because in general the noise sensitivity depends on it. It is valid for both quantization noise and photon noise. The equations that must be satisfied for an algorithm to be insensitive to various error sources are derived. A multivariate Taylor-series expansion in the distortions is used, and the time-varying background and signal amplitudes are expanded in Taylor series in time. Many new algorithms and families of algorithms are derived.