Telescope alignment from sparsely sampled wavefront measurements over pupil subapertures

We present a simple formalism that has proven useful in on-axis alignment of two-element telescopes when wavefront information is available from only a limited region (here two noncontiguous subapertures) of the pupil. Misalignments cause predictable full-aperture aberrations, which in turn cause predictable tip/tilt modes in the subapertures. For the most useful case in which secondary mirror tilts are independently constrained by optical monitoring, the four subaperture tip/tilt modes provide enough information to solve for the state of misalignment uniquely. A practically important and intuitively appealing simplification of this inversion occurs if the tip/tilts of the two subapertures are first transformed into a new basis consisting of differential and common-mode tilts in each of the x and y directions. Then the matrices interpreting subaperture modes as full-aperture aberrations and those in turn as mechanical misalignments become diagonal, so the mechanical adjustment required to align each degree of freedom is just a constant sensitivity multiplying one of the measured differential or common-mode tilt basis modes. Knowing that this simplification occurs allows rapid empirical calibration of sensitivities in the lab and then deterministic alignment, simply and transparently, with no need for ray tracing to model the optical effects of the adjustments at each step of the alignment.