Aberration-free aspherical lens shape for shortening the focal distance of an already convergent beam.

John P Sutter,Lucia Alianelli

doi:10.1107/s1600577517011808

Abstract

The shapes of single lens surfaces capable of focusing divergent and collimated beams without aberration have already been calculated. However, nanofocusing compound refractive lenses (CRLs) require many consecutive lens surfaces. Here a theoretical example of an X-ray nanofocusing CRL with 48 consecutive surfaces is studied. The surfaces on the downstream end of this CRL accept X-rays that are already converging toward a focus, and refract them toward a new focal point that is closer to the surface. This case, so far missing from the literature, is treated here. The ideal surface for aberration-free focusing of a convergent incident beam is found by analytical computation and by ray tracing to be one sheet of a Cartesian oval. An `X-ray approximation' of the Cartesian oval is worked out for the case of small change in index of refraction across the lens surface. The paraxial approximation of this surface is described. These results will assist the development of large-aperture CRLs for nanofocusing.

Highlights

Compound refractive lenses (CRLs) have been used to focus X-ray beams since Snigirev et al (1996) demonstrated that the extremely weak refraction of X-rays by a single lens surface could be reinforced by lining up a series of lenses
Because the absorption of X-rays in the lens material is generally significant, it becomes critical to design the CRL with the shortest length possible for the given focal length in order to minimize the thickness of the refractive material through which the X-rays must pass
At the end of this treatment, the focal spot profiles calculated by ray tracing for the compound refractive lens (CRL) of Table 1 will be compared with the diffraction broadening that inevitably results from the limited aperture

Summary

Introduction

Compound refractive lenses (CRLs) have been used to focus X-ray beams since Snigirev et al (1996) demonstrated that the extremely weak refraction of X-rays by a single lens surface could be reinforced by lining up a series of lenses. In this paper it will be shown that finite numerical precision can cause errors in the calculation of the Cartesian oval when the change in refractive index across the lens surface becomes very small, as is usually the case with X-rays. It has not been made explicit in the literature when it is reasonable to approximate the ideal Cartesian oval with various conic sections (ellipses, hyperbolas or parabolas). At the end of this treatment, the focal spot profiles calculated by ray tracing for the compound refractive lens (CRL) of Table 1 will be compared with the diffraction broadening that inevitably results from the limited aperture. This is a circle of radius centred at ð0; ÀÞ, where

Closed-form solutions of ideal lens surface

À 2 ðu1

The ‘X-ray approximation’ to the ideal lens surface

The paraxial approximation to a conic section

Findings

Testing the paraxial approximation