{"title":"Mirror-centered representation of a focusing hyperbolic mirror for X-ray beamlines.","authors":"Jean Pierre Torras","doi":"10.1107/S1600577524009603","DOIUrl":null,"url":null,"abstract":"<p><p>Conic sections are commonly used in reflective X-ray optics. Hyperbolic mirrors can focus a converging light source and are frequently paired with elliptical or parabolic mirrors in Wolter type configurations. This paper derives the closed-form expression for a mirror-centered hyperbolic shape, with zero-slope at the origin. Combined with the slope and curvature, such an expression facilitates metrology, manufacturing and mirror-bending calculations. Previous works consider ellipses, parabolas, magnifying hyperbolas or employ lengthy approximations. Here, the exact shape function is given in terms of the mirror incidence angle and the source and image distances.</p>","PeriodicalId":48729,"journal":{"name":"Journal of Synchrotron Radiation","volume":" ","pages":"1464-1468"},"PeriodicalIF":2.5000,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11542650/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Synchrotron Radiation","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1107/S1600577524009603","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/10/29 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Conic sections are commonly used in reflective X-ray optics. Hyperbolic mirrors can focus a converging light source and are frequently paired with elliptical or parabolic mirrors in Wolter type configurations. This paper derives the closed-form expression for a mirror-centered hyperbolic shape, with zero-slope at the origin. Combined with the slope and curvature, such an expression facilitates metrology, manufacturing and mirror-bending calculations. Previous works consider ellipses, parabolas, magnifying hyperbolas or employ lengthy approximations. Here, the exact shape function is given in terms of the mirror incidence angle and the source and image distances.
圆锥形截面常用于反射 X 射线光学。双曲面反射镜可以聚焦会聚光源，经常与椭圆或抛物面反射镜搭配使用，形成 Wolter 型配置。本文推导了以镜面为中心的双曲面形状的闭式表达式，其原点处的斜率为零。结合斜率和曲率，该表达式有助于计量、制造和镜面弯曲计算。之前的研究考虑了椭圆、抛物线、放大双曲线或采用了冗长的近似值。在这里，精确的形状函数是根据镜面入射角和镜源与镜像的距离给出的。
期刊介绍：
Synchrotron radiation research is rapidly expanding with many new sources of radiation being created globally. Synchrotron radiation plays a leading role in pure science and in emerging technologies. The Journal of Synchrotron Radiation provides comprehensive coverage of the entire field of synchrotron radiation and free-electron laser research including instrumentation, theory, computing and scientific applications in areas such as biology, nanoscience and materials science. Rapid publication ensures an up-to-date information resource for scientists and engineers in the field.
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