Branch point problem in adaptive optics

It is shown that when branch points are present in the phase of a turbulence-distorted optical field, the ability of an adaptive optics system that utilizes a least mean square error type of wave-front reconstructor to sense all of the turbulence-induced phase perturbations is limited. There is a portion of the turbulence-induced phase perturbation, which portion we refer to as the hidden phase, that such a least mean square error type of wave-front reconstructor will, in effect, ignore. It is shown that the presence of branch points indicates that the measured phase-difference vector field cannot be considered to be simply the gradient of some scalar potential—the phase function—but is in part also the curl of a vector potential function. A solution is developed for this vector potential, and from this a simple closed-form solution for the hidden phase is developed. Sample numerical results are presented showing the nature of the hidden phase. Suggestions are provided for a branch-point-tolerant wave-front reconstructor based on use of the closed-form solution for the hidden phase.