Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces

The spherical geometry leads to phase-step nonuniformity because of the large off-axis angle, when testing high-numerical-aperture spherical surfaces using a phase-shifting interferometer. The calculated errors induced by phase-step nonuniformity for phase-shifting algorithms are derived as two components: one is the ripple error with the frequency twice that of fringes like that caused by phase-shift miscalibration; the other is the modified defocus error as a combination of defocus and spherical aberrations. A 13-frame interferogram-collecting sequence is captured once and separated into two 9-frame interferogram-calculation sequences; a 9-frame phase-shifting algorithm and an improved wavefront difference technique are implemented for calibration. The 9-frame phase-shifting algorithm is selected from mounts of algorithms for its insensitive characteristic to the ripple error in the presence of phase-step nonuniformity and phase-shift miscalibration. The modified defocus error is eliminated by least-squares fitting through processing the difference wavefront of two measured results from interferogram-calculation sequences. Both numerical simulation and experiments with a commercial Fizeau interferometer have been carried out, and the results indicate that the sub-nanometer-accuracy is achieved by the proposed calibration method.

► We derived the phase errors induced by phase-step nonuniformity into ripple error and modified defocus error. ► Improved wavefront difference techniques aiming at modified defocus error are implemented easily for instruments. ► Sub-nanometer accuracy can be achieved for testing of high-numerical-aperture spherical surfaces.