Abstract
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.
Highlights
Thick diffractive optical elements for X-ray wavelengths, including so-called sputtersliced zone plates and multilayer Laue lenses (MLLs), are fabricated by depositing alternating layers of materials onto a substrate to achieve the required diameter or height of the lens, and slicing the lens from this structure [1,2,3]
At the wavelengths considered here—of about 0.1 nm and below—reasonable diffraction efficiency demands structures that are hundreds or thousands of times thicker than the layer period and tilted to fulfil the Bragg condition of diffraction [1]. Such diffractive optical elements can be used as an objective lenses to construct various kinds of Xray microscopes, including a transmission microscope where the lens forms a magnified image of an object on an area detector or a scanning transmission microscope where the lens creates a focused probe through which the object is scanned while mapping the transmission or emission of the sample
X-ray multilayer Laue lenses (MLLs) can be considered as thick diffractive optical elements, in which diffraction occurs as a volume effect
Summary
Thick diffractive optical elements for X-ray wavelengths, including so-called sputtersliced zone plates and multilayer Laue lenses (MLLs), are fabricated by depositing alternating layers of materials onto a substrate to achieve the required diameter or height of the lens, and slicing the lens from this structure [1,2,3]. The modelling of diffractive optical elements in complex optical systems has been well established, with the development of holographic optical elements to produce arbitrary wavefronts abetted by the rise of computational ray-tracing methods and software [15,16,17,18,19,20,21] Some of these principles have been recently rediscovered for the analysis of meta-lenses [22,23], and we use them here for a comprehensive study of MLLs for nanometer focusing. The symbols used in this paper are listed in table 1 for convenience
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