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Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle

Authors :
Kenneth Goldberg
Source :
Journal of Synchrotron Radiation. 29:991-996
Publication Year :
2022
Publisher :
International Union of Crystallography (IUCr), 2022.

Abstract

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.

Details

ISSN :
16005775
Volume :
29
Database :
OpenAIRE
Journal :
Journal of Synchrotron Radiation
Accession number :
edsair.doi.dedup.....dcb306a09fc4d19d793e03fe9a64e676
Full Text :
https://doi.org/10.1107/s1600577522005793