<title>Phasing the Keck Telescope with out-of-focus images in the infrared</title>

We describe a method for phasing segmented optics which makes use of a novel variation of the established technique of curvature sensing. In traditional curvature sensing, the difference between inside-of-focus and outside-of-focus images provides a direct measure of the curvature of a relatively smooth wavefront. We illustrate how this approach can be extended to enable one to measure and correct the discontinuous wavefronts associated with segmented mirrors. A detailed algorithm, based not on curvature measurement, but on correlation of the `difference image' with theoretical images or templates, is presented. In a series of tests of this `Phase Discontinuity Sensing' or PDS algorithm at the Keck 1 Telescope, at a wavelength of 3.3 microns, the RMS piston error (averaged over the 36 primary mirror segments) was repeatedly reduced from about 230 nm to 40 nm or less. Furthermore the PDS phasing solution was shown to be consistent with our previous `phasing camera' results (to within 66 nm RMS), providing strong independent confirmation of this earlier approach.