<title>Fringe analysis in phase-shifting interferometers suppressing spatially nonuniform phase modulation, beam amplitude modulation, and nonlinearity of modulator</title>

Abstract

Abstract In a phase-shifting interferometer, spatial non-uniformity of the phase modulation happens in such a case where anaspherical mirror is compared to the corresponding aspherical standard surface which is translated along the optical axisby a piezo electric transducer. The amount of phase shift is different from position to position across the observingaperture depending on the direction cosine of the testing surface. In another case, when the reference optical flat istranslated by two or more piezo-electric transducers, we cannot ensure that the reference surface moves strictly parallelto the optical axis. When these transducers have different sensitivities, the phase modulation is no longer spatiallyuniform and varies across the observing aperture.Many phase measuring algorithms reported to date cannot compensate for the spatial nonuniformity if there is anonlinear phase shift. It is shown that even-order nonlinear sensitivity of the phase modulator causes a significant errorsin measured phase when there is a spatial nonuniformity in the phase shift. However, if we add a new symmetry to thesampling functions of the phase measuring algorithm we can suppress the lowest order of these errors. The newalgorithms need at least one more image frame to acquire the symmetry. The lowest-order algorithm compensating forquadratic spatially non-uniform phase modulation consists of six frames.Since the new algorithm acquires a new symmetry, it usually suffers from a larger random noise than the conventionalalgorithms. The random noise can be decreased if we added more sampling frames to the algorithm. The samplingamplitudes are determined by a Lagrangian method. The resultant algorithms show that the random noise decreasesinversely proportional to the square-root of the number of samples.