Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle
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 de...
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| Format: | Article |
| Language: | English |
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International Union of Crystallography
2022-07-01
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| Series: | Journal of Synchrotron Radiation |
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| Online Access: | http://scripts.iucr.org/cgi-bin/paper?S1600577522005793 |
| _version_ | 1818116236309954560 |
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| author | Kenneth A. Goldberg |
| author_facet | Kenneth A. Goldberg |
| author_sort | Kenneth A. Goldberg |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-11T04:19:18Z |
| format | Article |
| id | doaj.art-f225e5badcce430ba094d378cdc37425 |
| institution | Directory Open Access Journal |
| issn | 1600-5775 |
| language | English |
| last_indexed | 2024-12-11T04:19:18Z |
| publishDate | 2022-07-01 |
| publisher | International Union of Crystallography |
| record_format | Article |
| series | Journal of Synchrotron Radiation |
| spelling | doaj.art-f225e5badcce430ba094d378cdc374252022-12-22T01:21:10ZengInternational Union of CrystallographyJournal of Synchrotron Radiation1600-57752022-07-0129499199610.1107/S1600577522005793yi5120Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principleKenneth A. Goldberg0Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USAEllipsoidal 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.http://scripts.iucr.org/cgi-bin/paper?S1600577522005793x-raymirrorellipsoidellipsefermatseriesfocusing |
| spellingShingle | Kenneth A. Goldberg Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle Journal of Synchrotron Radiation x-ray mirror ellipsoid ellipse fermat series focusing |
| title | Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle |
| title_full | Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle |
| title_fullStr | Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle |
| title_full_unstemmed | Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle |
| title_short | Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle |
| title_sort | derivation of closed form ellipsoidal x ray mirror shapes from fermat s principle |
| topic | x-ray mirror ellipsoid ellipse fermat series focusing |
| url | http://scripts.iucr.org/cgi-bin/paper?S1600577522005793 |
| work_keys_str_mv | AT kennethagoldberg derivationofclosedformellipsoidalxraymirrorshapesfromfermatsprinciple |