In phase-shifting interferometry spatial nonuniformity of the phase shift gives a significant error in the evaluated phase when the phase shift is nonlinear. However, current error-compensating algorithms can counteract the spatial nonuniformity only in linear miscalibrations of the phase shift. We describe an errorexpansion method to construct phase-shifting algorithms that can compensate for nonlinear and spatially nonuniform phase shifts. The condition for eliminating the effect of nonlinear and spatially nonuniform phase shifts is given as a set of linear equations of the sampling amplitudes. As examples, three new algorithms (six-sample, eight-sample, and nine-sample algorithms) are given to show the method of compensation for a quadratic and spatially nonuniform phase shift.

In phase measurement systems that use phase-shifting techniques, phase errors that are due to nonsinusoidal waveforms can be minimized by applying synchronous phase-shifting algorithms with more than four samples. However, when the phase-shift calibration is inaccurate, these algorithms cannot eliminate the effects of nonsinusoidal characteristics. It is shown that, when a number of samples beyond one period of a waveform such as a fringe pattern are taken, phase errors that are due to the harmonic components of the waveform can be eliminated, even when there exists a constant error in the phase-shift interval. A general procedure for constructing phase-shifting algorithms that eliminate these errors is derived. It is shown that 2j 1 3 samples are necessary for the elimination of the effects of higher harmonic components up to the j th order. As examples, three algorithms are derived, in which the effects of harmonic components of low orders can be eliminated in the presence of a constant error in the phase-shift interval.

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In phase-shifting interferometry, many algorithms have been reported that suppress systematic errors caused by, e.g., nonlinear motion of the phase shifter and nonsinusoidal signal waveform. However, when a phase-shifting algorithm is designed to compensate for the systematic phase-shift errors, it becomes more susceptible to random noise and gives larger random errors in the measured phase. The susceptibility of phase-shifting algorithms to random noise is analyzed with respect to their immunity to phase-shift errors and harmonic components of the signal. It is shown that for the most common group of error-compensating algorithms for nonlinear phase shift, both random errors and the effect of high-order harmonic components of the signal cannot be minimized simultaneously. It is also shown that if an algorithm is designed to have extended immunity to nonlinear phase shift, simultaneous minimization becomes possible.

Testing for flatness of an optical parallel plate in a Fizeau interferometer suffers from problems caused by multiple-beam interference noise. Each internal-reflection component can be separated from the signal by its modulation frequency in a wavelength-scanned interferometer; however, the frequency depends on the thickness and the refractive-index dispersion of the test plate and on the nonlinearity of the scanning source. With a new 19-sample wavelength-scanning algorithm we demonstrate the elimination of the reflection noise, the effect of the dispersion up to the second order of the reflectance of the test plate, and as the nonlinearity of the source. The algorithm permits large tolerance in the air-gap distance, thus making it somewhat independent of the thickness of the test plate. The minimum residual reflection noise with this algorithm for testing a glass plate is approximately lambda/600. Experimental results show that the front surface of the test plate was measured within 1 nm rms of its true shape over a 230-mm-diameter aperture.