Phasing optical phased arrays using an exact solution for adaptive optics

The pixelized nature of optical phased arrays (OPAs) or spatial light modulators such as liquid crystal devices (LCDs) or micro electromechanical devices (MEMs) allows us to treat these devices as segmented active optical elements. The individual OPA elements, for example, the LCD pixels, only control phase i.e. the piston degree of freedom parallel to the optical axis and not the tip and tilt or phase gradients. We assume in this paper that the OPAs operate in a 2-dimensional mode where the phase profile varies in both transverse directions. Wavefront sensor components of an adaptive optical system generally provide wavefront slope measurements (tip and tilt) and not piston. Thus, wavefront sensor measurements cannot be directly applied to OPAs. Since the wavefront slope measurements do not form a curl-free vector field, the phase reconstruction from the wavefront slopes must be optimized. The optimized phase reconstruction can be directly applied to the OPA device and therefore represents a control solution. We have found an exact analytic solution for the optimal phase reconstruction given a set of discrete gradient measurements.