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- Title
Off-axis representation of hyperbolic mirror shapes for X-ray beamlines.
- Authors
Goldberg, Kenneth A.; del Rio, Manuel Sanchez
- Abstract
Mirror-centered, closed-form expressions for hyperbolic surfaces used in X-ray beamlines have been derived. Hyperbolic mirrors create a virtual focus or source point and can be used to lengthen or shorten the effective focal distance of a compound optical system. The derivations here express off-axis segments of a hyperbolic surface in terms of the real and virtual focal distances and the incident glancing angle at the center of the mirror. Conventional mathematical expressions of hyperbolic shapes describe the surfaces in Cartesian or polar coordinates centered on an axis of symmetry, necessitating cumbersome rotation and translation to mirror-centered coordinates. The representation presented here, with zero slope and the origin at the central point, is most convenient for modeling, metrology, aberration correction, and general surface analysis of offaxis configurations. The direct derivation avoids the need for nested coordinate transforms. A series expansion provides a helpful approximation; the coefficients of the implicit equation are also provided.
- Subjects
X-rays; CARTESIAN coordinates; MIRRORS; SURFACE analysis
- Publication
Journal of Synchrotron Radiation, 2023, Vol 30, Issue 3, p514
- ISSN
0909-0495
- Publication type
Article
- DOI
10.1107/S1600577523001492