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

Goldberg, Kenneth A.

doi:10.1107/s1600577522005793

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

Journal Article · Mon Jun 20 00:00:00 EDT 2022 · Journal of Synchrotron Radiation (Online)

DOI:https://doi.org/10.1107/s1600577522005793· OSTI ID:1894090

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Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Advanced Light Source (ALS)

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.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1894090

Journal Information:: Journal of Synchrotron Radiation (Online), Journal Name: Journal of Synchrotron Radiation (Online) Journal Issue: 4 Vol. 29; ISSN 1600-5775

Publisher:: International Union of CrystallographyCopyright Statement

Country of Publication:: United States

Language:: English

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73 NUCLEAR PHYSICS AND RADIATION PHYSICS
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Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle

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References (14)

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