Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm

In a previous paper, we described a successful technique, the broadband algorithm, for phasing the primary mirror segments of the Keck telescopes to an accuracy of 30 nm. Here we describe a complementary narrow-band algorithm. Although it has a limited dynamic range, it is much faster than the broadband algorithm and can achieve an unprecedented phasing accuracy of approximately 6 nm. Cross checks between these two independent techniques validate both methods to a high degree of confidence. Both algorithms converge to the edge-minimizing configuration of the segmented primary mirror, which is not the same as the overall wave-front-error-minimizing configuration, but we demonstrate that this distinction disappears as the segment aberrations are reduced to zero.