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

References (14)

Hard X-ray nano-focusing with Montel mirror optics Liu, Wenjun; Ice, Gene E.; Assoufid, Lahsen Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 649, Issue 1 https://doi.org/10.1016/j.nima.2010.11.080	journal	September 2011
Ellipsoidal mirror for two-dimensional 100-nm focusing in hard X-ray region Yumoto, Hirokatsu; Koyama, Takahisa; Matsuyama, Satoshi Scientific Reports, Vol. 7, Issue 1 https://doi.org/10.1038/s41598-017-16468-1	journal	November 2017
Mirror profile optimization for nano-focusing KB mirror Zhang, Lin; Baker, Robert; Barrett, Ray AIP Conference Proceedings https://doi.org/10.1063/1.3463335	conference	January 2010
Ex situ metrology and data analysis for optimization of beamline performance of aspherical pre-shaped x-ray mirrors at the advanced light source Yashchuk, Valeriy V.; Lacey, Ian; Gevorkyan, Gevork S. Review of Scientific Instruments, Vol. 90, Issue 2 https://doi.org/10.1063/1.5057441	journal	February 2019
Soft x-ray nanobeam formed by an ellipsoidal mirror Takeo, Yoko; Suzuki, Akihiro; Motoyama, Hiroto Applied Physics Letters, Vol. 116, Issue 12 https://doi.org/10.1063/1.5144932	journal	March 2020
CXVII. The total reflexion of X-rays Compton, Arthur H. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 45, Issue 270 https://doi.org/10.1080/14786442308634208	journal	June 1923
Theory and practice of elliptically bent x-ray mirrors Howells, Malcolm R. Optical Engineering, Vol. 39, Issue 10 https://doi.org/10.1117/1.1289879	journal	October 2000
Derivation of aberration coefficients for single-element plane-symmetric reflecting systems using Mathematica McKinney, Wayne R.; Palmer, Christopher A. SPIE Proceedings https://doi.org/10.1117/12.292730	conference	November 1997
New schemes in the adjustment of bendable elliptical mirrors using a long trace profiler Rah, Seung Y.; Locklin, Scott C.; Irick, Steven C. SPIE Proceedings https://doi.org/10.1117/12.295552	conference	November 1997
Second metrology round-robin of APS, ESRF and SPring-8 laboratories of elliptical and spherical hard-x-ray mirrors Rommeveaux, A.; Assoufid, L.; Ohashi, H. Optical Engineering + Applications, SPIE Proceedings https://doi.org/10.1117/12.736171	conference	September 2007
Design optimization of bendable x-ray mirrors McKinney, Wayne R.; Yashchuk, Valeriy V.; Goldberg, Kenneth A. SPIE Optical Engineering + Applications, SPIE Proceedings https://doi.org/10.1117/12.894175	conference	September 2011
Fermat's Principle and Geometrical Optics Shaw, J. H. American Journal of Physics, Vol. 33, Issue 1 https://doi.org/10.1119/1.1971227	journal	January 1965
Formation of Optical Images by X-Rays Kirkpatrick, Paul; Baez, A. V. Journal of the Optical Society of America, Vol. 38, Issue 9 https://doi.org/10.1364/JOSA.38.000766	journal	January 1948
WHO? DISCOVERED CODDINGTON'S Equations? Kingslake, Rudolf Optics and Photonics News, Vol. 5, Issue 8 https://doi.org/10.1364/OPN.5.8.000020	journal	January 1994

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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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