Goldberg Kenneth A
Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA.
J Synchrotron Radiat. 2022 Jul 1;29(Pt 4):991-996. doi: 10.1107/S1600577522005793. Epub 2022 Jun 20.
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.
椭球面和平面椭圆面作为反射式点对点聚焦元件在包括X射线光学在内的许多光学系统中得到广泛应用。在此，应用费马经典的光程函数方法来推导这些表面的封闭形式表达式，这些表面由物距、像距以及镜面上某点的入射角唯一描述。紧凑的描述便于进行任意精度的设计、建模、制造和测试。给出了两个有用坐标系中的全等表面——一个以椭球对称轴为中心的系统，以及一个以镜心为中心或“顶点”系统，该系统的表面在镜心处与xy平面相切。从结果中推导出局部斜率和曲率半径的表达式，并给出了关于镜心的麦克劳林级数展开的前几项。
