Analysis of the Three-Dimensional Focal Positioning Capability of Adaptive Optic Elements

There are many situations where adaptive optic elements are useful for beam steering. When the beam is passed through a lens, the resulting focus may be scanned within a three-dimensional volume. In this article, we introduce expressions to describe the maximum distance to which a focus may be steered using a deformable mirror with a continuous phase distribution, limited either in its gradient or amplitude. Additionally for liquid crystal spatial light modulators (SLMs), an analytic approach is developed to describe the intensity reduction of a steered focus due to phase wrapping and pixellation. By calculating the total area on the SLM taken up by phase wrapping, we can estimate the proportion of light that is not directed to the focus. Thus, we derive an expression for the 3-D volume within which we may steer a focus given a permissible reduction in intensity. The results compare favorably with experiment.