Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA.
ESRF, 71 Avenue des Martyrs, 38000 Grenoble, France.
J Synchrotron Radiat. 2023 May 1;30(Pt 3):514-518. doi: 10.1107/S1600577523001492. Epub 2023 Mar 10.
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.
已推导出用于 X 射线束线的双曲曲面的以镜心为中心的闭式表达式。双曲镜可形成虚拟焦点或源点，并可用于延长或缩短复合光学系统的有效焦距。这里的推导以镜面中心处的真实和虚拟焦距以及入射掠射角的形式表示了离轴段的双曲曲面。描述双曲形状的传统数学表达式以对称轴为中心在笛卡尔或极坐标中描述曲面，这需要繁琐的旋转和平移到以镜心为中心的坐标。此处给出的表示形式，斜率为零且原点位于中心点，对于离轴配置的建模、计量学、像差校正和一般表面分析最为方便。直接推导避免了嵌套坐标变换的需要。级数展开提供了有用的近似值；隐式方程的系数也提供了。
