Defining the measurand in radius of curvature measurements

Traceable radius of curvature measurements are critical for precision optics manufacturing. An optical bench measurement of radius is very repeatable and is the preferred method for low-uncertainty applications. On an optical bench, the displacement of the optic is measured as it is moved between the cat's eye and confocal positions, each identified using a figure measuring interferometer. This distance is nominally the radius of curvature. Traceability requires connection to a basic unit (the meter, here) in addition to a defensible uncertainty analysis. The identification and proper propagation of all uncertainty sources in this measurement is challenging. In this paper we report on a new mathematical definition of the radius measurand that is a single function that depends on all uncertainty sources, such as error motions, alignment uncertainty, displacement gauge uncertainty, etc. The method is based on a homogeneous transformation matrix (HTM) formalism, intrinsically defines an unbiased estimate for radius, and provides a single mathematical expression for uncertainty propagation through a Taylor-series expansion.