Risley prisms as a conformal transformation device. I. Complex and graphic analyses of mapping images conformality

Mapping conformality and the singularities in the control law of the Risley prisms for target tracking are two internally related topics well suited for an investigation in one study. This is because an intrinsic ability of the Risley prisms is to produce 2D scan patterns in a polar frame by means of a combination of two 1D motions of the two prisms when their orientations are specified by two independent variables expressible as the x and y coordinates of a single point in a rectangular frame. In transformation of points between the rectangular and the polar frames, a singularity occurs inevitably at the pole where r = 0 in the polar frame. For a better understanding of the two topics and their relationship, this study presents an in-depth analysis, starting from the first-order approximation, and then the far-field condition is considered and finally is the rigorous theory developed for the Risley-prism-based beam steering system under the basis of a consideration of the effect of the thickness of the Risley prisms on the accuracy of beam steering. The rigorous theory provides correct results of mapping conformality in both the near and the far zones of the beam steering field.