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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The manufacture and testing of high-precision optical surfaces for the Laser Interferometer Gravitational Wave Observatory is described. Through the use of carefully shaped polishing laps made of a nondeformable polymer material coated on a rigid base, surfaces 250 mm in diameter with radii of curvature between 7 and 15 km were polished to an accuracy of several hundred meters in the curvature and with low values of waviness and microroughness. Metrology instrumentation used to measure the optical finish included a large-aperture digital interferometer calibrated to nanometer-level accuracy for measurements of curvature, astigmatism, and waviness and an interference microscope for measurements of microroughness. The power spectra of the data from both instruments were in good agreement.