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

Abstract

Edited by M. Yabashi, RIKEN SPring-8 Center, JapanKeywords: X-ray; lens; oval; nanofocusing; aberration.Aberration-free aspherical lens shape for shortening the focal distance of an already convergent beam John P. Sutter* and Lucia Alianelli Diamond Light Source Ltd, Chilton, Didcot, Oxfordshire OX11 0DE, UK. *Correspondence e-mail: john.sutter@diamond.ac.ukThe 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.

Explicit Cartesian oval as a superconic surface for stigmatic imaging optical systems with real or virtual source or image

“…These particular cases are of special interest and are applied in different areas such as microscopy [10], astronomy [11] and machine vision [12], but recently it has been evidenced a practical interest in the Cartesian ovals and aspherical surfaces. Not limited to their applications, we find works as the design of catadioptric systems [13], the design of compound refractive lens systems (CRL) to focus X-rays [14,15], Cartesian ovals representations for illumination systems [16], the design of a immersion lens [17] and the manufacture of a high-resolution system to polish corrective optical surfaces with or without rotational symmetry [18]. In general, optical imaging systems are built using spherical surfaces, due to these surfaces being simpler to polish [19] and because they also present rigorously stigmatic points known as Young-Weierstrass points [20][21][22].…”

“…Additionally, we believe that an explicit expression could be used to design methods to formulate corrective surfaces that compensate aberrations in optical systems composed by RSS, when point object does not lie at the rigorously stigmatic point. Several works have addressed the explicit formulation of an RSS, based on Ferrari's method [30], both exactly [15,[31][32][33] or approximated, using series expansion [32] or using aspherical surfaces [34], some of them listed in table 1. The exact mathematical expression of RSS are emphasized in a real point image formation using a real point object, but the cases for virtual object or image have not received the same attention, but in [15], not limited to this configuration, are applied Cartesian surfaces to virtual point objects.…”

“…Several works have addressed the explicit formulation of an RSS, based on Ferrari's method [30], both exactly [15,[31][32][33] or approximated, using series expansion [32] or using aspherical surfaces [34], some of them listed in table 1. The exact mathematical expression of RSS are emphasized in a real point image formation using a real point object, but the cases for virtual object or image have not received the same attention, but in [15], not limited to this configuration, are applied Cartesian surfaces to virtual point objects. Due to the complexity of exact expressions for this kind of optical surfaces, the authors make use of third-party software as OSLO [34,37] to carry out the ray tracing.…”

Kinoform and saw-tooth X-ray refractive lenses development at SSRF

“…Since the breakthrough idea of compound refractive lenses (CRL) [1], people have proposed various designs and fabrication techniques for different applications. These interesting designs include (but are not limited to) parabolic lenses [2], multi-prism lenses [3], single element kinoform lens [4,5], compound kinoform lenses [6,7], oval lens [8], and axicon lens [9]. The refractive lenses are used for imaging [10], focusing [11], collimating [12], interfering [13], and other interesting applications [14].…”