Wave front sensor-less adaptive optics: a model-based approach using sphere packings

Certain adaptive optics systems do not employ a wave front sensor but rather maximise a photodetector signal by appropriate control of an adaptive element. The maximisation procedure must be optimised if the system is to work efficiently. Such optimisation is often implemented empirically, but further insight can be obtained by using an appropriate mathematical model. In many practical systems aberrations can be accurately represented by a small number of modes of an orthogonal basis, such as the Zernike polynomials. By heuristic reasoning we develop a model for the operation of such systems and demonstrate a link with the geometrical problems of sphere packings and coverings. This approach aids the optimisation of control algorithms and is illustrated by application to direct search and hill climbing algorithms. We develop an efficient scheme using a direct maximisation calculation that permits the measurement of N Zernike modes with only N +1 intensity measurements.