Thick diffractive optical elements offer a promising way to achieve focusing or imaging at a resolution approaching 1 nm for X-ray wavelengths shorter than about 0.1 nm. Efficient focusing requires that these are fabricated with structures that vary in period and orientation so that rays obey Bragg's law over the entire lens aperture and give rise to constructive interference at the focus. Here the analysis method of ray-tracing of thick diffractive optical elements is applied to such lenses to optimise their designs and to investigate their operating and manufacturing tolerances. Expressions are provided of the fourth-order series expansions of the wavefront aberrations and transmissions of both axi-symmetric lenses and pairs of crossed lenses that each focuses in only one dimension like a cylindrical lens. We find that aplanatic zone-plate designs, whereby aberrations are corrected over a large field of view, can be achieved by axi-symmetric lenses but not the crossed lenses. We investigate the performance of 1 nm-resolution lenses with focal lengths of about 1 mm and show their fields of view are mainly limited by the acceptance angle of Bragg diffraction, and that aberrations can limit the performance of lenses with longer focal lengths. We apply the ray-tracing formalism for a tolerancing analysis of imperfect lenses and examine some strategies for the correction of their aberrations.

中文翻译：

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用于纳米聚焦的 X 射线多层劳厄透镜的射线迹分析

对于短于约 0.1 nm 的 X 射线波长，厚衍射光学元件提供了一种以接近 1 nm 的分辨率实现聚焦或成像的有前途的方法。有效的聚焦要求这些结构具有不同的周期和方向，以便光线在整个镜头孔径上遵循布拉格定律，并在焦点处产生相长干涉。此处将厚衍射光学元件的光线追踪分析方法应用于此类透镜，以优化其设计并研究其操作和制造公差。提供了轴对称透镜和像柱面透镜一样仅在一个维度上聚焦的成对交叉透镜的波前像差和透射率的四阶级数展开式的表达式。我们发现消球差波带片设计可以通过轴对称透镜而不是交叉透镜来实现，从而在大视场内校正像差。我们研究了焦距约为 1 mm 的 1 nm 分辨率镜头的性能，并表明它们的视场主要受布拉格衍射的接受角限制，并且像差会限制具有较长焦距的镜头的性能。我们将光线追踪形式应用于不完美镜片的公差分析，并检查一些用于校正其像差的策略。我们研究了焦距约为 1 mm 的 1 nm 分辨率镜头的性能，并表明它们的视野主要受布拉格衍射的接受角的限制，并且像差会限制具有较长焦距的镜头的性能。我们将光线追踪形式应用于不完美镜片的公差分析，并检查一些用于校正其像差的策略。我们研究了焦距约为 1 mm 的 1 nm 分辨率镜头的性能，并表明它们的视场主要受布拉格衍射的接受角限制，并且像差会限制具有较长焦距的镜头的性能。我们将光线追踪形式应用于不完美镜片的公差分析，并检查一些用于校正其像差的策略。