Fast pixel difference algorithm for determining piston step between optical mirror segments

With the rise of extremely large telescopes such as ELT, TMT, and GMT, the precision of mirror segment alignment will become critical to maintaining the resolution of the full aperture. A Shack-Hartmann wavefront sensor can be used to determine the tip/tilt of individual segments, but since it measures the displacement of a focal spot from the center, it is blind to any piston step (?) between segments. However, if a lenslet is centered over the gap between two mirror segments, the image results in an interference pattern from those segments. The interference pattern is highly dependent on the piston step between the two segments and is mathematically expressed by a modified sinc function. This leaves three main parameters that can be used to identify ?; the curve shape, the peak intensity, and the primary peak position. Curve fitting and correlation algorithms have previously been used to recover ? from the curve shape. However, they run significantly slower than a centroiding algorithm used for the position measurement of spots in the Shack-Hartmann wavefront sensor. For a system that is stable to segment piston, this is not an issue. But for a system like the GMT, where piston will need to be corrected almost continuously, an algorithm that is of a similar speed to centroiding could enable piston sensing to be integrated into a traditional Shack-Hartmann wavefront sensor (SHWFS). Also, the number of pixels needed to sample the interference spot has been much greater than the number for regular spot sampling in a SHWFS. This paper presents the derivation, simulation results, and an optical bench demonstration of a fast alternative piston-sensing algorithm; called the pixel difference algorithm, that is capable of utilizing a limited number of pixels across the interference spot (nominally 7). The pixel difference algorithm is based on the primary peak intensity parameter and is significantly faster than the curve fitting algorithm; by approximately a factor of 22 both in the simulation and in the optical bench demonstration. The simulation showed this speed comes with a trade-off; the method is more susceptible to noise and has approximately twice the error of the curve fitting algorithm at signal-to-noise ratios (SNR)<10. However, at high SNR<22, the pixel difference method has comparable, or better, accuracy than the curve fitting algorithm. The primary peak position parameter was also investigated but was found to have the same challenges as the pixel difference method, and so was not pursued in this paper. The pixel difference method presents an opportunity to use a Shack-Hartmann wavefront sensor for mirror segment alignment in an environment where ? needs to be corrected on a fast timescale.