Wave front sensorless adaptive optics, modal wave front sensing, and sphere packings

We investigate the properties of a class of adaptive optics systems that do not employ a wave front sensor but rather optimise a photodetector signal by appropriate control of an adaptive element. Such wave front control methods have already been implemented in various applications. It is often convenient to represent a wave front aberration by the superposition of several aberration modes, for example, using the set of Zernike polynomials. In many practical situations the total aberration can be accurately represented by a small number of such modes. It is shown that the design of wave front sensor-less adaptive optics systems based upon Zernike modes is related to the mathematical problem of sphere packing. This involves the arrangement of spheres in multiple dimensions, where the coordinate for each dimension corresponds to a Zernike mode amplitude. This observation permits optimisation of the systems providing considerable increases in efficiency over schemes that take no account of the geometries involved. We combine this approach with modal wave front sensing to provide efficient, direct measurement of Zernike aberration modes.