General methods for generating phase-shifting interferometry algorithms

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.

The general theory of phase shifting algorithms

“…Application of the F&K spectral analysis to PSI algorithm synthesis may be seen in [2]. In 1996 Surrel [3] developed an algebraic approach to analyze PSI algorithms based on what he called the characteristic polynomial associated to the quadrature filter.…”

Planet Formation Instrument for the Thirty Meter Telescope

“…oe] The RMS error for a two-sided pyramid wave-front sensor is RMS_PYR_2S=1/SQRT(2*num_phot) in the limit of no read noise. The four-bin interferometric wave-front sensor has an RMS error of RMS_4bin = 1/SQRT(num_phot/2) in this limit [Phillion 1997]. The RMS error for a Shack-Hartmann wave-front sensor may be expressed as RMS_SH~5.2*SQRT(1+(d/r o ) 2 )/SQRT(num_phot) [Bloemhof 2004 oe].…”

Construction and testing of wavefront reference sources for interferometry of ultra-precise imaging systems

“…At each step we recorded an interferogram consisting of a 1K by 1K frame of 12-bit pixels. We used a 12-bucket algorithm 5 to reduce these 12 frames to the measurement wave's amplitude and phase present at the plane of the CCD. Typically, we immediately repeated this process 8 times, averaging the complex phasors to form a single phase/amplitude map, which we stored.…”