1. Introduction

One of the key successes of synchrotron facilities is that they provide access to scientists from a wide range of disciplines in chemistry, life science and material science. Optics development has enabled in the past 20 years experiments with highly brilliant and coherent beams of hard x-rays. Hard x-ray nano-probes benefit studies of complex, inhomogeneous and disordered materials, with applications in microelectronics, material science, biology, geology, cultural heritage, environmental and earth science. The following (non-exhaustive) list of scientific applications of micro- and nano-focused beams justifies the fact that many large synchrotron facilities are investing in the development of focusing optics to be used in nano-probe beamlines where users will have access to hard x-ray beams of 10 – 30 nm in size: (i) nano-beams give local information, at the nano-scale rather than averaged over several nano-objects; therefore they are required in investigation of samples that are inhomogeneous or very small like whiskers and nano-wires; (ii) smaller and smaller beams are used in studies of adsorption onto particulate samples deposited on surfaces. In many cases some sample areas show well ordered and crystalline structures in small regions whilst other regions are disordered. It is particularly interesting to determine the structure of the ordered regions but also to be able to texture map the interface to observe the transition between the regions; (iii) imaging and diffraction with very hard x-rays are used for experiments in structural and process engineering. Stress, strain, deformation behaviour and phase transformations are analysed in samples of industrial interest using either highly collimated or highly focused beams; (iv) high resolution diffraction at medium x-ray energy for material science and magnetism can require sub-micron beams in studies of individual crystallites or domains. A technique that would benefit from an x-ray nano-probe is resonant diffraction applied to the study of anti-ferromagnetic domains, ferromagnetic domains and potentially orbital-ordering domains; (v) high resolution beams are required in diamond anvil cell experiments as minute sample areas (1-10 µm in dimension) are often used in extreme conditions studies; (vi) finally, focus quality evaluation would make kinoform optics, discussed in this paper, suitable for imaging and coherent diffraction experiments as they improve signal to noise ratio in experiments such as x-ray photon correlation spectroscopy and reflectometry [1,2].

In this paper we present recent work on kinoform lenses development, and suggest that they have reached sufficient maturity to be used as proper high flux nano-focusing devices.

It is understood that reflective and refractive optics with very short focal lengths can introduce geometrical aberrations when the curvature is not optimized for the shape of the incident wave front. The nano-focusing lenses described in this paper, in addition to the phase profile that minimises absorption, are designed with a double refractive surface, which improves the collimation of the beam prior to nano-focusing. The lenses were tested at the ID06 undulator beamline at the ESRF. The smallest focal length was f = 75 mm, and the measured focused spot full width at half maximum (fwhm) was s = 225 nm. The focal spot quality is satisfactory, and a peak-to-background ratio larger than 100 was measured.

2. Double kinoform lens system for nano-focusing

Kinoform lenses are semi-transparent refractive objects, in which the profile is optimised for a specific radiation wavelength and the transmission is maximised due to the very small amount of lens material [3]. The focal length of a lens with a single refractive surface, in the thin lens approximation, is f = R / δ where R is the radius of curvature at the apex of the surface, and δ is the refractive index real part decrement δ = Re (1 - n). The principal obstacles to the use of refractive optics to focus x-rays are the smallness of δ ~10⁻⁵ – 10⁻⁶, and the large absorption. Synchrotron focusing lenses have to be fabricated with very strong curvatures in materials with low Z number. A kinoform profile minimizes absorption. The profile is obtained by setting the length of the kinoform steps equal to multiples of$L=\lambda /\delta$, where λ is the x-ray wavelength. This ensures that each part of the kinoform lens which is illuminated by a collimated monochromatic wave, refracts the wave without introducing phase changes, apart from a factor multiple of 2π.

New generation kinoform focusing lenses were developed with improved design and fabrication [3,4] and micro-focusing of the hard x-ray beam was obtained with a lens with f = 150 mm [3]. In this paper we show that a lens with the same focal length, and slightly different design, will provide nano-focusing of the x-rays with good flux transmission over a large aperture. By decreasing the radius of the nano-focusing lens to R = 0.5 µm, rather than using refractive arrays, we have obtained an optics with focal length of f = 75 mm only.

It has been correctly reported that when a single refractive lens is used, the ideal surface to focus a planar wave to a point is elliptical [3,5]. However synchrotron radiation is not perfectly collimated, and the presence of thermal deformations of the crystal monochromators usually introduces further wave front distortions. For these reasons, strongly curved optical surfaces should follow a computer generated shape adjusted to the incoming wave front [6]. A simple solution to decrease aberrations and increase demagnification ratio by the nano-focusing optics, is illustrated in Fig. 1 .

 

Fig. 1 Simplified sketch of the nano-focusing system: lens A is designed for collimation, whilst lens B strongly demagnifies the beam. Geometrical aberrations are minimised by the use of two elliptical surfaces. The beam refracted by lens A and incident on lens B is highly collimated. Therefore it is highly parallel to the kinoform steps shown in the SEM in Fig. 2, and phase conservation across the optics is enhanced.

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This is a simple monolithic system where a pre-collimating lens is used to illuminate a nano-focusing lens. Two optical elements are fabricated onto the same chip, at reduced cost and simplified alignment. Better illumination of a nano-focusing lens is achieved, as the collimating lens modifies the incident wave front, increasing parallelism of the kinoform steps with the incident beam. This design allows better phase conservation across the optics, compared to previous kinoform lenses, and avoids possible refraction at the kinoform steps. The scanning electron microscope (SEM) image in Fig. 2 shows the resulting lens, fabricated using modern silicon dry etch technology. The aspect ratio of the lenses illustrated here is limited to about 50:1. As the material thickness, perpendicular to the lens optical axis, amounts to few micrometers only, the lens depth is limited as well. We have been able to fabricate x-ray lenses with maximum depth of 100 µm and are improving the hard mask process to increase this figure.

 

Fig. 2 Scanning electron microscope images showing details of nano-focusing kinoform lens optics. Both lenses are elliptical: the first has a focal length of 45 m therefore creates a collimated beam when placed at P = 45 m from the synchrotron source. The second lens has a focal length of 150 mm. The kinoform steps are parallel to the optical axis of the lenses. Therefore this system improves phase preservation across the lens aperture, when compared to kinoform lenses working in non-collimated beam.

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Optical design was accomplished using the Shadow ray-tracing code [7] which provides detailed simulation of geometrical focusing. The undulator beam divergence is α_undulator = 22 µrad (fwhm); the collimator reduces the divergence of the beam incident on the nano-focusing refractive lens to α < 1 µrad as illustrated in Fig. 3 .

 

Fig. 3 Simulation of beam divergence incident on the nano-focusing lens. The solid line is the undulator beam divergence (α_undulator = 22 µrad) incident on lens A in Fig. 1. The dotted line is the beam divergence after the collimator lens and incident on lens B (α < 1 µrad).

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The influence of a phase error introduced by the beam divergence can be estimated by calculating the phase of the x-rays across one or more kinoform steps, as a function of α. The difference in phase between two parallel rays traveling in air and in the material is:

When the first ray travels at an angle α the difference in phase is:

From Eq. (2) it follows that for very small values of α:

The deviation of Δφ from the ideal 2π value is in the range 10⁻¹-10⁻² for more divergent beams (bending magnet case) and 10⁻⁵-10⁻⁴ in the case of very collimated undulator beams. The pre-collimating lens removes these phase errors.

Design of kinoform lenses with shorter focal lengths compared to those discussed in this paper, is carried out with some extra attention to details such as: (i) arrays of kinoform optics have to be used [4]; (ii) refractive surfaces will have compensation of possible aberrations due to highly distorted front wave incident on lenses [6]; (iii) the kinoform steps should be tilted to follow the curvature of the wave refracted by the previous lens in the array.

3. Nano-focusing tests and lens efficiency

The kinoform lenses developed at Diamond Light Source have in the past been characterised on bending magnet and undulator beamlines [8–11]. The data in Figs. 4 -7 summarise results of measurements of vertically focused beam on the ID06 undulator source at the ESRF. A liquid nitrogen cooled Si-111 double crystal, fixed exit monochromator was used to adjust the x-ray energy. The lenses were mounted at distance P = 55 m from the source. Alignment was optimised using an imaging detector with micrometer spatial resolution, placed in the focal plane of the lenses. An image of the line focus showing lens dimensions and part of the direct unfocused beam outside the lens area is illustrated in Fig. 4.

 

Fig. 4 High resolution image of lens linear focus (f = 150 mm, E = 8 keV). The lens shadow is shown and part of the direct beam from the slits upstream the lens can be seen.

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Fig. 7 Measured transmission by the lenses with f = 150 mm and 75 mm at E = 8 keV. The data are the ratio between transmitted flux and incident flux versus lens aperture. The theoretical peak efficiency of the system is 75%.

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Beam size, depth of focus and lens transmission were measured scanning the edge of a metal wire in the focal plane. The plot in Fig. 5 is the signal from the wire, measured in transmission mode. The derivative representing that focus shape has a size s = 225 nm fwhm. Analysis of these data shows that there are two decades intensities between peak intensity and background. This is an indication of good quality of design and fabrication especially when considering that the measurement was made without additional slits or apertures between lens and diagnostic. The measured vertical source size on ID06 is S_undulator = 45 ± 5 µm fwhm [12] and the calculated focused beam size would be less than 200 nm for an effective aperture A ~100 µm. The larger measured beam size is caused by a number of possible unwanted effects, namely: scattering from the scalloping of the lens walls, which could mask a more pronounced peak; vibrations of wire or equipment; lens could be non perfectly aligned.

 

Fig. 5 Transmitted flux from knife edge scan in the focal plane of lens with f = 75 mm, E = 8 keV. The derivative represents the focused spot profile. The data shown are the raw data, without any correction for wire scan shape etc.

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The plot in Fig. 6 illustrates measured focal spot size s versus focusing distance. The solid lines represents the expected demagnified beam sizes, after convolution with the diffraction limited beam size, assuming lens effective apertures A = 100 µm, 200 µm or no diffraction limit at all. The plot indicates that the lenses developed up to now do not produce a diffraction limited spot size. It also shows that aperture is approximately 100 µm: whilst such acceptance value is not too competitive for micro-focusing optics, it would be rather attractive for shorter focal nano-focusing optics. For instance the reported apertures of current nano-focusing silicon planar CRL's and multilayer Laue lenses is in the range 30 - 50 µm [13–15].

 

Fig. 6 Focal spot size s (fwhm) versus focal distance. The experimental points summarise data from several measurements at the same beamline ID06 at the ESRF, with x-ray energy in the range E = [8–12] keV. The solid lines are calculated by convoluting the size of the geometrically demagnified beam with the size of the diffraction limited beam, assuming a lens effective aperture of A = 100 µm (black line), 200 µm (red line), with no diffraction at all (blue line).

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Kinoform lens efficiency is higher than the efficiency expected from planar silicon refractive lenses due to reduced absorption. At 8 keV x-ray energy the theoretical peak efficiency of this kinoform lens system is 75%. The measured transmitted flux is shown in Fig. 7: this was measured by recording the focused flux (without tails, i.e. between 10% and 90% of the maximum intensity in the knife edge scan) versus lens aperture, normalized to the incident flux. Measured peak efficiency is well above 50%. The flux gain G is the ratio of focused flux density to incident flux density. Lens efficiency varies across the aperture, however using the transmission data we obtain G > 200 for A = 200 µm for the lens with f = 150 mm. Transmission, gain and aperture of the new proposed kinoform lenses are larger than previously reported values for similar optics [14].

4. Discussion

Both lens efficiency and apertures described in this paper are comparable to or higher than the theoretical efficiency and apertures of high aspect ratio zone plates. This is a striking result when one considers that the lenses were fabricated in an old generation etcher. Lenses with better aspect ratio and reduced roughness or scalloping can be fabricated in modern fast cycle etchers. This implies that increased flux in the focused beam and shorter tails will be obtained with kinoform lenses provided that advanced silicon etchers become available to the synchrotron optics community. The improvements witnessed in recent years in the field of mirror and zone plates fabrication will be matched with refractive lenses when the most advanced silicon technologies are used.

This work shows that tightly focused beams can be obtained using moderate focal lengths. Refractive in-line focusing optics are easily kept and used in vacuum hence they do not require any window between optics and sample. Moreover kinoform lenses described in this paper do not produce unwanted diffraction or higher order focusing [16] and can be used as an add-on on many synchrotron experimental end-stations with the only requirement of suitable alignment stages.