Open AccessDerivation of closed‐form ellipsoidal X‐ray mirror shapes from Fermat's principle
Author(s) -
Goldberg Kenneth A.
Publication year - 2022
Publication title -
journal of synchrotron radiation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.172
H-Index - 99
ISSN - 1600-5775
DOI - 10.1107/s1600577522005793
Subject(s) - ellipsoid , optics , physics , surface (topology) , tangent , coordinate system , geometry , curvature , geometrical optics , plane (geometry) , mathematics , astronomy
Ellipsoidal and plane‐elliptical surfaces are widely used as reflective, point‐to‐point focusing elements in many optical systems, including X‐ray optics. Here the classical optical path function approach of Fermat is applied to derive a closed‐form expression for these surfaces that are uniquely described by the object and image distances and the angle of incidence at a point on a mirror surface. A compact description facilitates design, modeling, fabrication, and testing to arbitrary accuracy. Congruent surfaces in two useful coordinate systems — a system centered on the ellipsoid's axes of symmetry and a mirror‐centered or `vertex' system with the surface tangent to the xy plane at the mirror's center — are presented. Expressions for the local slope and radii of curvature are derived from the result, and the first several terms of the Maclauren series expansion are provided about the mirror center.