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15 September 2016 Aberration-free x-ray lenses made of silicon
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Brilliant beams of hard x-rays, with geometrical cross-sections below 50×50 nm2, are a standard research tool for synchrotron users. With the advent of lower emittance sources, such as NSLSII, Petra III and Max IV, and planned upgraded lattices, such as APS-2, SPING8-II, ESRF II and DLS II, nanofocusing optics operating in transmission mode will become more competitive than they are currently. In general, they suffer from lower efficiency than reflective optics, however they often have easier set-up and alignment, combined with a smaller footprint. Fabrication and exploitation of ultra-short focal refractive lenses has not witnessed the same progress in the last decade as other optics, such as multilayer mirrors and multilayer Laue lenses. This paper reports on current status of high-resolution lithography for fabricating silicon lenses and on proposed designs for a new class of refractive lenses with zero aberrations and good efficiency. The new designs are created with geometrical parameters matching the spatial resolution achieved by modern lithography and silicon etch technology.



X-ray Nanoprobe instruments rely on focusing optics with very short focal lengths, typically of order 50 mm. The next frontier in Nanoprobe development is the achievement of smaller and smaller beam cross-sections, of order 10 nm, with sufficient flux for user’s experiments. Therefore a reduction of focusing optics focal length, to about 10 mm, will be necessary. The ultimate goal is single digit nanometer focusing, which has already been achieved with multilayer coated mirrors and multilayer Laue lenses. Such small beams are not yet widely available to hard X-ray beamline users. Use of such ultra-short focal optics is limited mainly because of the small apertures. Their use will become more convenient, on machines with lower emittance electron sources, and higher photon beam brilliance. Single-digit nanometer focusing refractive lenses have not been demonstrated yet. X-ray lens performance is hampered by absorption in the material, non-perfect fabrication results, and finally by properties of the material used, such as its non-crystalline form. The most recent silicon lens etch results are presented in the next section, as these give an important indication of what quality can be achieved with standard lithography and relatively advanced etch. We show the path towards achieving short focal performing lenses, using silicon. Despite its low efficiency at energy values of interest, silicon micro-structures are currently superior to other materials. We follow up the recent work on short focal aberration free lenses, with a practical recipe for ultra-short focal lens lenses, in the last section. A breakthrough in silicon lens fabrication will only happen in the frame of a strong collaboration between the synchrotron community and the silicon technology experts. Further developments in the field of nano-lithography methods [1] are necessary before the synchrotron community can finally enjoy use of ultra-short focal refractive X-ray lenses. A new class of nanofocusing X-ray lenses [3] will require use of lithographic techniques with spatial resolution better than 50 nm.



We have trialled fabrication of a silicon kinoform lens with overall spatial resolution just below 100 nm. Fabrication was made, using e-beam lithography and fast-switching Bosch™ deep reactive ion etching. The system employed was an Oxford Instruments PlasmaPro 100 Estrelas. Examples of the silicon kinoforms fabricated by this method are shown in the scanning electron microscopy (SEM) images in Figure 1, where the lenses have an elliptical curvature with a smallest radius at the apex R = 150 nm. The whole lens structures are several mm long and only the narrow ends of the elliptical nanofocusing surfaces are visible in these images. The smallest feature size (the thickness of lens sidewalls) with these lenses are t = 1, 2 and 4 μm. Etch angles of θ = 89.9° and scalloping amplitudes lower than 40 nm, were achieved, and shown in Figure 2.

Figure 1.

SEM images of planar silicon kinoform lens detail, with a radius at the apex of R = 150 nm


Figure 2.

SEM image of a 65 μm thick test structures etched with the planar silicon kinoform lenses in the etch optimisation process. Scalloping < 50 nm and etch angle θ = 89.9 deg.


The x-ray lens topography image, measured with monochromatic radiation with E = 12 keV, is shown in Figure 3. The lens was oriented to provide vertical focusing of the undulator source, with a focal distance of f = 30 mm. The data were collected by imaging the x-ray beam on a 5 μm thick Eu:LuAG scintillator. The image was relayed to a PCOEdge CMOS detector by magnifying visible light optics. The spatial resolution of the detector is of order 1 μm. The scintillator was placed at further distance from the lens focal plane, due to geometrical constraints. The line focus from this single-element elliptical lens, which is visible at the centre of the image, is therefore blurred.

Figure 3.

X-ray topographic images of a kinoform lens with focal length f = 30 mm, acquired on highly coherent synchrotron beamline I13-1 at Diamond Light Source. The lens is aligned to provide vertical focusing. The bright line at the centre is the focused radiation. The dark area on the left is the supporting wafer; the dark area around the focal line is the shade of the lens body. Refraction from borders of a pre-collimating large radius elliptical surface, visible in the SEM in 0, can be observed.


One drawback of silicon etch is that perfectly verticality is not reached. One consequence is the appearance of geometrical aberrations, as the shape of the refractive surface is not conserved along the lens height. Furthermore the thickness of the phase conservation kinoform features is not constant, therefore intensity and phase variations appear along the lens height. All experiments performed by this group in the past have shown variations in beam focused profile and intensity along the lens height [4]. These effects are visible in Figure 3, where the line focus intensity is not uniform, but rather has a constant variation with position. If the variation in thickness and curvature of the lens could be controlled and modelled, ways to design and align the lens could be found to minimise its effect. It is however not straightforward to assume that these imperfections can be completely controlled or compensated. A simplified approach can be used to model the effect of the non-zero etch angle φ=90° - θ. We assume that the ellipse semi-axes a and b change linearly along the z axis:


To a first approximation, these changes have an impact on effective focal length. The major contribution to its relative change is:




The change in focal length Δf in a silicon single element lens, integrated over a lens thickness D = 80 μm, and using a conservative value for the angle φ = 90 – θ = 0.1°, is a function of energy, like the lens depth of focus (DOF), which can be calculated as [5]:






The negative impact of etch angle on lens effective focal length can balanced by the somewhat large depth of focus for lenses with small numerical aperture. The etch angle φ ≠ 0 will potentially be an obstacle to effective and real improvement of lenses numerical aperture. This would be a necessary condition to achieving single-digit nanometer focusing. The total lens thickness D will be optimised to ensure that the total focal length variation is smaller than the depth of focus.

The other important parameter, minimum sidewall t, is determined by the aspect ratio that can be achieved during etching, currently around 50:1. Material thickness hugely affects the amount of absorption in the lens, and therefore its numerical aperture NA and diffraction limit s. These quantities are plotted in Figure 4. The kinoform lenses currently fabricated by our group, with minimum t = 1 and 2 μm, can reach NA = 1 mrad, which is the current best aperture possessed by refractive nano-focusing lenses. In order to improve performance, the desired sidewall thickness should be decreased. For instance a value of t = 0.4 μm provides an ambitious lens fabrication plan, bringing the numerical aperture to NA ~ 2 mrad. The diffraction limited spot size is therefore s < 30 nm for E > 10 keV (Figure 4). A realistic proposal for fabrication of next batch of silicon nanofocusing lenses would be using D = 30 μm, therefore aiming for an aspect ratio of 75:1.

Figure 4.

Diffraction limit s for kinoform silicon lenses (black curves) and numerical aperture NA (blue curves). The solid curves are calculated for t = 0.4 μm, the dashed lines for t = 1 μm and the dotted lines for t = 2 μm.




Efficiency and resolution of ideally fabricated kinoform lenses are limited by absorption, which is strongly affected by the sidewall thickness. The main technological issue to be solved is therefore achieving sub-μm sidewalls in silicon. The theoretical issue is to design arrays of lenses that introduce zero-aberrations. Arrays are necessary for ultra-short focal lengths due to the very small radii needed [6,7]. Aberration-free lenses are not a simple array of refractive surfaces with same shape, but rather arrays with decreasing radii, which make the X-ray beam progressively more convergent, in a controlled way [6].

in Figure 5(a) we show the ideal compound lens layout with its principal rays, as proposed in [3]. In Figure 5(b), we show the lens layout which would result by using Cartesian ovals to refocus the convergent beam from the first elliptical refractive surfaces, with each oval increasing the angle of any given paraxial ray from the optical axis. Using only elliptical surfaces or Cartesian ovals to design a lens without aberrations is possible as seen by comparing the designs in Figure 5(a) and (b). For reduced absorption, when a kinoform lens design is used, it is important to provide each kinoform step with a tilting angle equivalent to the angle of the rays to the optical axis. The compound kinoform lens design in Figure 5(c) is a simple elegant solution to the angle problem, as it inherently provides kinoform steps which are always parallel to the rays. Performance of a similar system, with focal length f = 21 mm, is summarized in Table 1. Fabrication of kinoform lenses for E > 15 keV will not be possible unless the resolution of lithographic patterning increases dramatically. This is due to the very small refractive index decrement and to the very small radius required at such energy values.

Figure 5.

Schematic representation of aberration-free x-ray lens composed of elliptical refractive surfaces (a). X-ray lens composed of ellipses Ovals of Descartes (b). Compound elliptical kinoform lens [3] (c). The principal rays are also shown. All geometrical rays refracted through the single elements are parallel to the kinoform steps. This is a necessary condition for phase preservation of the x-ray beam in the lens material. The principal rays are shown in red.


Table 1.

Performance of Si lens illustrated in Fig. 5(c), with focal length of 21 mm, using t = 0.4 μm.

Energy (KeV)Smallest Radius [nm]Effective Aperture [μm]NA [mrad]Diffraction limited spot size [nm]



We have offered a simple yet analytically correct design for planar nano-focusing refractive optics, and discussed current micro-fabrication technological limits. Silicon x-ray lenses are not currently available commercially; however the fabrication tools exist to make them a standard beamline optics component. Several fabrication methods are being trialled for achieving similar results in diamond, including moulding, laser cutting, and dry etching. Higher effective apertures and transmission values are possible with this desirable material, leading to smaller focused beams. This justifies the current trend in diamond micro-fabrication research for optics synchrotron applications.

A correct formulation of optical layout is a necessary condition to improve performance of x-ray lenses for high resolution synchrotron applications. Such lenses will rely on the use of refractive surfaces of elliptical, hyperbolic, or Cartesian oval form.


All lithography work prior to silicon etching was carried out in collaboration between Diamond Light Source and the Science and Technology Facilities Council. We wish to thank the people involved in this work: A. Malik, D. Jenkins and I. Loader. The x-ray image was acquired on beamline I13-1 at Diamond Light Source under proposal number MT9980. We are grateful to U. Wagner for setting up the beamline and assisting during beamtime.



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© (2016) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
L. Alianelli, I. Pape, J. P. Sutter, O. J. L. Fox, K. J. S. Sawhney, and K. Korwin-Mikke "Aberration-free x-ray lenses made of silicon", Proc. SPIE 9963, Advances in X-Ray/EUV Optics and Components XI, 99630S (15 September 2016);


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