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 off-axis 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.
Funder
Director, Office of Science, Office of Basic Energy Sciences of the US Department of Energy
Publisher
International Union of Crystallography (IUCr)
Subject
Instrumentation,Nuclear and High Energy Physics,Radiation
Reference28 articles.
1. Properties And Performance Of Grazing Incidence Reflectors
2. Chuang, Y.-D., Anderson, C., Benk, M., Goldberg, K., Voronov, D., Warwick, T., Yashchuk, V. & Padmore, H. A. (2016). AIP Conf. Proc. 1741, 050011.
3. Solar X-Ray Image Obtained Using Grazing-Incidence Optics.
4. Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle
5. Analytic descriptions of parabolic X-ray mirrors