Conic Surfaces and Transformations for X-Ray Beamline Optics Modeling

TL;DR

This paper provides a unified mathematical framework for conic-shaped optical surfaces used in x-ray beamline optics, facilitating modeling, fabrication, and analysis of mirror systems.

Contribution

It introduces consistent equations and transformations for conic surfaces in grazing incidence optics, aiding ray tracing and optical surface analysis.

Findings

Unified equations for paraboloids, ellipsoids, hyperboloids

Transformations for off-axis mirror surfaces

Applications in ray tracing and misalignment studies

Abstract

Optical surfaces represented by second-degree polynomials (quadratic or conics) are ubiquitous in optics. We revisit the equations of the conic shapes in the context of grazing incidence optics, gathering together the curves commonly used in x-ray instruments and synchrotron beamlines. We present the equations for paraboloids, ellipsoids, and hyperboloids in a common and consistent notation. We develop the transformations from centered systems that are commonly used to describe conics and their axes of symmetry, to local coordinate systems centered on the off-axis mirror surfaces. The equations presented are directly applicable to ray tracing, fabrication, and metrology calculations. They can also be used to study misalignments, movement tolerances, and aberrations of optical surfaces.