High-Performance R-Theta Stage Motion Systems For Semiconductor Tools

Introduction: Whales & Natural Coordinates 

From the ocean-view decks of the Monterey Bay Aquarium, at a distance from the rocky shoreline outcrop, the fingerprint-like patterns of Humpback Whale tail flukes suddenly appear. To an observer on the deck, a hurried and excited description would soon follow. It’s this simple description that masks much deeper concepts that trace historical roots to Newton, symmetry arguments, and today can be found in cutting edge semiconductor process tools.

From Descartes to Newton 

Any description to a fellow human on where to find the whale would likely use information based on the polar coordinate system. Something like “at your 3 o’clock, about 500 feet out” would be a simple and effective instruction. It’s the curious commonality of description among observers that reveals a meaningful deeper non-intuitive truth. The whale is some radial distance off the coast (relative to observer) at some angle referenced against direct line of sight.  It’s clear the description isn’t based on the more common Cartesian coordinate system (using X and Y to locate the whale) but on a polar coordinate system using R (radial distance from a pole) and Theta (rotation angle relative to reference).  The polar coordinate system, as it would come to be known, was first applied in such a way by Isaac Newton in the late 17^th century.