Shape recovery for rotating surfaces: A tutorial

One of the key elements of the metrology system for the rotating stage used in the Reflective Electron Beam Lithography (REBL) instrument is a set of laser interferometers that measures the distance to a rotating cylindrical ring shaped mirror. This metrology arrangement allows the lateral x-y motion of the stage to be measured independent from the other four degrees of motion. One source of error in this measurement is the non-roundness of the shape of the mirror. By using redundant interferometry measurements, the shape of the mirror may be computed and its effects can be removed from the lateral motion estimate. This tutorial gives an overview of the mathematics of the shape recovery algorithm used in REBL as well as a guide for some of the practical issues that arise in shape estimation and stage design, not the least of which is the angular placement of the sensors.