Diamond refractive micro-lenses for full-field X-ray imaging and microscopy produced with ion beam lithography

: We demonstrate that ion-beam lithography can be applied to the fabrication of rotationally parabolic refractive diamond X-ray micro-lenses that are of interest to the ﬁeld of high-resolution X-ray focusing and microscopy. Three single half-lenses with curvature radii of 4.8 µm were produced and stacked to form a compound refractive lens, which provided diﬀraction-limited focusing of X-ray radiation at the P14 beamline of PETRA-III (DESY). As shown with SEM, the lenses are free of expressed low- and high-frequency shape modulations with a ﬁgure error of < 200 nm and surface roughness of 30 nm. Precise micro-manipulation and stacking of individual lenses are demonstrated, which opens up new opportunities for compact X-ray microscopy with nanometer resolution.


Introduction
The global trend towards the transition of modern accelerator X-ray sources to diffraction-limited synchrotrons (MAX IV, ESRF-EBS, PETRA-IV) and extremely brilliant Free Electron Lasers, provides great opportunities for coherent applications [1][2][3]. It also poses significant challenges to the development of optical elements adapted to these sources. Knowing that highly coherent X-ray radiation easily interacts on its way with various imperfections of optical elements and spoils transmitted wavefront with aberrations, X-ray optics have to preserve unique radiation properties and also be capable of performing such functions as beam transport, nano-focusing, phase-contrast imaging and microscopy. Speaking of refractive optics, compound refractive lenses (CRLs, [4]) have become one of the main tools at modern X-ray beamlines because of their reduced sensitivity to shape errors, overall ease-of-use and versatility [5]. Currently, beryllium parabolic lenses are the most common representatives of refractive optics, as beryllium has a low-attenuation coefficient combined with high refractive efficiency and well-developed manufacturing technology [6]. All this makes beryllium lenses easily accessible for a wide spectrum of X-ray applications in the energy range from 3 to 60 keV [7]. Beryllium CRLs have been used for beam conditioning being condensers, collimators, beam-shapers and higherharmonics suppressors [8][9][10][11][12]. They are also successfully applied in coherent diffraction, imaging and Fourier techniques [13][14][15][16][17][18]. Despite their many benefits for X-ray optics, beryllium is a polycrystalline material with a granular structure. Because of this, speckles are produced under coherent X-ray illumination of the lens, which limits the resolution in X-ray microscopy and related applications to 100 nm [19]. Therefore, there is a high demand for alternative optical materials that could be appropriate for the manufacturing of speckle-free X-ray lenses.
Considering an ideal X-ray optical material for refractive lenses, the most important parameter is refractive index n. This can be described by the formula n = 1 − δ + iβ, where δ and β are associated with the phase shift of the X-ray beam, and X-ray absorption in the lens material, respectively. The higher the δ/β ratio in the material, the better the optical performance of a lens. Hence, the choice of the material is limited to the elements with low atomic number (Be, B, C, Al, Si). The material should be further considered in the context of a refractive lens, where shape and optical resolution are other parameters to be taken into account.
If we refer to full-field microscopy, lenses should have the shape of paraboloid and achieve the diffraction-limited resolution that is defined as: where λ is the wavelength and NA is the numerical aperture of the lens. Clearly, as the resolution is inversely proportional to the numerical aperture, the latter needs to be maximal. As for refractive lenses, the NA can be expressed as: where A eff is an effective lens aperture, which can be smaller than the physical aperture due to the absorption in the lens material [20], and F is the focal length of a lens. The focal length is defined as where R is the radius of parabola apex and N is the number of double concave elements in the lens. Summarizing, the resolution can be written as: It is clear that a smaller radius of the parabola apex provides a shorter focal length, allowing to get a larger NA and resolution approaching the diffraction limit. Searching for and developing a suitable technological process for the production of highresolving refractive micro-lenses also becomes a complex task: it should be capable of repeatable production of lenses with the required shape and quality, out of a material with a high δ/ β ratio. In the context of the foregoing, additive manufacturing of polymer materials looked promising as all of the above requirements could be satisfied. Recently, it has been successfully applied to the production of double-concave parabolic refractive nano-lenses with small radii of curvature, R, of 5 µm [21]. Thirty individual nano-lenses were put together to form a CRL and tested in the full-field microscopy mode [22]. The resolution of 100 nm was experimentally achieved while 70 nm was expected. Despite the viability of the demonstrated approach, polymer lenses had several drawbacks: during the two-photon polymerization, which was used as a manufacturing method, the polymerization volume of material was elliptical due the elliptical 3D shape of the laser beam along the optical axis; in addition, the irradiation of polymer material with X-rays caused its continuous degradation. These factors altogether led to spherical aberrations and astigmatism, and this, in turn, limited the resolution. While the technological process of 3D-printing could be improved to correct the shape of the lens, the radiation damage under high-flux photon beams remains a systemic problem for polymers.
In contrast, manufacturing micro-lenses from a monocrystalline diamond is attractive. Unlike polymers, diamond can withstand high thermal loads from intense high-energy X-ray beams due to its high thermal conductivity and temperature stability. Moreover, the refractive index decrement of diamond, δ, is 2.7 times greater than for polymer materials. It means that 2.7 times fewer diamond lenses are required to achieve the same focal length, making the microscopy scheme with diamond lenses 2.7 times more compact. Fewer lenses reduce absorption in the CRL, which in turn generally increases effective aperture and resolution (Eq. (4)). Apart from that and unlike beryllium, monocrystalline diamond does not produce speckles in coherent X-ray beams, as it has no grains in its internal structure. The applicability of diamond to X-ray optical development was successfully demonstrated recently [23,24] where laser ablation was used for the fabrication of large-aperture rotationally parabolic X-ray lenses (A = 1 mm, R = 200 µm). It was shown that diamond lenses have great potential for beam-shaping and beam-conditioning applications in the high-heat flux beams of modern synchrotrons [25][26][27]. Despite successful implementation and testing, the laser ablation technique limits the quality of manufactured lenses, as the laser beam typically has an inhomogeneous shape and uneven spatial energy distribution. Consequently, lenses had micro-scale surface roughness and 5% deviation of the parabolic shape from the designed profile [26], which did not allow reaching a resolution better than 500 nm in microscopy mode. In addition, laser ablation does not allow manufacturing lenses with radii smaller than 50 µm. The aforementioned facts limited the use of the described diamond lenses for high-resolution X-ray microscopy.
To fabricate speckle-free lenses with small radii that would be suitable for high-resolution X-ray focusing and microscopy, the alternative method of diamond processing -direct-write ion beam lithography (IBL). Ion beam lithography offers high-resolution patterning on three-dimensional surfaces and is based on the interaction of a fine-focused ion beam (down to 5 nm in diameter) with a sample. The incidence angle, beam energy and dosage determine the amount of milled material while high-precision mechanics controls beam position, which allows one to etch complex patterns. The applicability of the IBL to precise optical manufacturing was demonstrated with the example of cylindrical and parabolic micro-lenses made of glass or silicon [28,29], while diamond was also used to create photonic structures [30][31][32]. Recently, IBL has been successfully applied to produce X-ray optical elements, such as binary and kinoform Fresnel zone plates [33,34]. In this paper, we introduce rotationally parabolic refractive X-ray micro-lenses that were fabricated with IBL from a monocrystalline diamond.

Fabrication of a compound refractive lens
The maskless direct milling of diamond lenses was carried out using a Zeiss CrossBeam 540 FIB-SEM system, equipped with liquid gallium ion source. In-situ scanning electron microscopy (SEM) was used to control the shape and geometry of the produced samples. Rotationally parabolic diamond half-lenses were milled in a 40-µm thick single-crystal diamond plate (100). The radius of curvature of a single parabolic surface and physical aperture were specified to be 5 µm and 20 µm, respectively. The milling process of a single half-lens took ∼2.5 hours. SEM was used to evaluate the circularity of the lens aperture ( Fig. 1(a)).
To assess the shape and surface quality of the milled lens, it was cut in half with FIB. Figure 1(b) shows the corresponding SEM image. Then, 200 points were manually set along the cut on the image using ImageJ software, with the precision of 3 pix (0.2 µm) that is a combination of cut and mark errors. Fitting an ideal parabolic curve with a radius, R, of 4.8 ± 0.1 µm to the experimental data ( Fig. 1(d)) and subtracting them from each other, a maximal variation of 50 nm was measured. Taking into account the integral precision of the measurement, we assume the figure error of the lens to be 0.2 µm. Visual inspection of the surface with higher magnification ( Fig. 1(c)) showed the surface roughness to be around 30 nm (root mean square).
To produce a compound refractive lens (CRL 3 ) the following procedure has been used, which consisted of 4 steps: 1) First of all, the reference indicators on the surface of the diamond plate were made to provide accuracy for further alignment processes. Then, three single diamond half-lenses were milled and redeposited material was removed from the surface with K 2 Cr 2 O acid heated to 60°C. Centers of lenses were aligned with respect to the edge of the diamond plate so the distance between the centers was 100 µm.
2) At the second step, half-lenses were cut from the diamond plate by making an L-like precise incision with the cut width of 4 µm to form small 96 × 96 µm plates with half-lenses in the center. A narrow conjunction close to the edge of the plate has been left to hold the plate for further mechanical translation ( Fig. 2(a)).
3) The last step was to fasten the lenses and form a CRL-stack. An in-chamber high-precision MM3A-EM micromanipulator (Kleindiek Nanotechnic GmbH, Reutlingen, Germany) with a tungsten (W) tip was used to mount and transfer single half-lens plates for stacking. The small conjunction, which held each lens, was removed by milling using a Ga+ ion beam with 15 nA current ( Fig. 2(b)). SEM imaging was used to monitor the stacking process. Three lenses were transferred to the Si substrate, coaxially aligned with respect to each other (Figs. 2(c) and (d)) and connected in two points by redeposition to form a stack.
4) Afterwards, the CRL 3 was transferred and attached to the silicon substrate. The quality of the stack and interconnection of lenses are shown in Fig. 3.

Experimental setup
The optical performance of the CRL 3 was tested at the EMBL P14 beamline at PETRA III, DESY (Hamburg, Germany). This is an undulator beamline where a standard U29 undulator produces X-ray radiation with an effective source size of 35 × 355 µm 2 (full width at half maximum, FWHM) at 12 keV [3,35]. The vertical size of 35 µm was measured experimentally and is slightly larger than the nominal size of 13 µm given by the DESY technical design report. This difference can be attributed to a broadening of the X-ray beam caused by thermal deformation of the surface of the monochromator crystals and high-frequency vibrations (>80 Hz) induced by the cryogenic cooling of the monochromator crystals [36]. Using double crystal Si-<111 > monochromator (DCM), P14 can operate X-ray energies varying from 6 to 30 keV. For our tests, the 'unfocused' beamline configuration was used, where no additional X-ray optical elements were installed on the path of the beam (Fig. 4) before it reached the sample position. The photon energy of 12 keV was chosen for all tests in the present study. CRL 3 was installed at the sample position on the high-precision MD3 Kappa-diffractometer (ARINAX, Moirans, France) with a <100 nm sphere of confusion for the vertical and downward Ω-axis. Primary centering of the sample with respect to an X-ray beam was performed using an integrated high-resolution optical microscope (on-axis viewing system, [37])

X-ray imaging
At first, propagation-based phase-contrast X-ray imaging was [38] used to estimate the coaxiality of the CRL 3 and to align its optical axis strictly parallel to the X-ray beam for further focusing tests. Given a high degree of transversal coherence and homogeneous parallel-beam illumination at the P14 beamline [36], phase contrast highlighted interfaces between areas with different electron density inside the CRL 3 [39] and corresponding intensity modulations were registered with the downstream detector ( Fig. 5(a)). Further use of a phase retrieval procedure [40] allowed us to obtain a descriptive representation of the projected thickness of the sample at its exit surface (Fig. 5(b)). Although some artifacts of the single-distance CTF-reconstruction [41] can be seen at the edge of the square diamond plate, they did not affect the main region of interest around the aperture of the lens, giving an opportunity to estimate its circularity. As can be seen in the Fig. 5(c), the projected thickness of the aperture of the CRL 3 has a circular symmetry with a diameter of 19.5 ± 0.5 µm in both vertical and horizontal directions. This indicates the precise coaxial alignment of single lenses in the stack, which allows one to expect the absence of spherical aberrations in further optical tests.

X-ray focusing
The next step was to test CRL 3 with X-ray focusing. X-ray focusing allows to estimate the quality of the focal spot, measure focal distance, as well as profiles and caustics of the focused . The circular shape of the aperture is slightly blurred in the horizontal direction, which is a consequence of a smaller transverse coherence (10 times) due to the source size. Gray values at the calibration bar correspond to the projected sample thickness at zero distance after the CRL 3 . Vertical and horizontal diameters of the aperture (yellow arrows) were measured as the distance between minima of the gray values.
beam. All this information reveals the optical properties of the lens and indicates the presence (or absence) of aberrations. As already mentioned in the Eq. (1), the focal distance, F, of the CRL is proportional to lens radius, R. Therefore, measuring the focal distance in the focusing experiment allows one to calculate the lens radius and compare it with the designed value. In order to measure the focal distance, one should take into account the thin lens equation where L 1 and L 2 are source-to-lens and lens-to-detector distances, respectively. Thus, knowing the designed lens radius, R 0 , one should expect to achieve the beam waist at the distance L 2 0 . And conversely, experimentally measured L 2 exp allows to recalculate the experimental radius, R exp , and compare it to R 0 . Practically, to measure the L 2 exp , we scanned the area near the L 2 0 along the optical axis of the lens, recording images of the focused radiation with the detector. Henceforth it will be called a longitudinal scan in the paper. The step of this scan, x, corresponded to less than half of the depth of focus (DOF), which is calculated by the equation: where A is the aperture of the lens, (∆l) min is the diffraction limit (Eq. (1)) and λ is the wavelength. It is worth mentioning that DOF/2 defines the measurement error of the L 2 exp and consequently translates to the error of R exp .
To reduce the impact of monochromator vibrations, 3 similar images were acquired with an exposure time of 0.01 sec. at each detector position in the longitudinal scan. Then, every image was cross-sectioned to measure the FWHM of the beam profile both in vertical and horizontal directions. Experimental data were fitted with a sum of Gauss and constant function using the Python LmFit library. A profile with a minimal size (FWHM) of a beam was chosen between these three repeatable measurements. FWHM was measured with an overall accuracy of 0.2 µm. This number takes into account beam flickering, pixel and fit errors.
Plotting beam profiles for each sample-to-detector position in the longitudinal scan, one can also reconstruct a beam caustic -the envelope of light rays coming from the CRL 3 to the focus and after it. The caustic helps to see possible lens aberrations as they inevitably lead to distortions of the wavefront. Additionally, if CRL had no aberrations, the intensity distribution in focus should have a Gaussian shape in both vertical and horizontal directions. One should compare its size (FWHM) with the expected value S, which can be calculated as: where s is the size of the undulator source and PSF is the point spread function of the X-ray imaging system. In the present paper, focusing properties of the CRL 3 were tested at the photon energy of 12 keV. We should note that the CRL 3 was insignificantly absorbing X-rays at this energy so the effective aperture -the aperture of the lens which is limited by the absorption of lens material was ∼10 times larger than the physical one. Therefore, we refer to the physical aperture in all calculations of the present paper. It also allowed us to use the Rayleigh definition of the diffraction limit (with the classical multiplier of 1.22) in the Eq. (1). Table 1 summarizes all parameters that were calculated as discussed above.
The focus was expected to be observed at the distance L 2 0 = 32 cm so a longitudinal scan was performed near this position by moving a detector stage along the optical axis in a range of L 2 = 22.5-39.5 cm and with a step of x = 1 cm. The result of this scan ( Fig. 6(a)) demonstrated that the beam waist ( Fig. 6(b)) was located at L 2 exp = 30.5 ± 3.0 cm. This value corresponded to an effective curvature radius of the single half-lens of R exp = 4.6 ± 0.5 µm. The term 'effective' means that this value is an average radius for all 3 single half-lenses in the CRL. A minor  discrepancy between radii measured with SEM (R 0 = 4.8 µm) and X-ray focusing (R exp = 4.6 µm) can be caused by a relatively high measurement error in X-ray focusing tests (0.5 µm) due to the large depth of focus of the CRL 3 . We also admit, however, that it may be caused by the technological process itself and, while being beyond the scope of the present paper, further examinations should be considered. At the same time, R exp allows one to recalculate diffraction limit (∆ 1 ), depth of focus (DOF 1 ) and expected waist size (S theor ) to be 1.9 µm, 6 cm and 2.0 × 2.7 µm (VxH), respectively. Experimentally measured waist size (S exp ) was 2.2 ± 0.2 µm and 2.9 ± 0.2 µm (FWHM of the Gaussian fit) for vertical and horizontal directions, respectively (Fig. 6(c)). The measured values fully corresponded to the expected waist size within the measurement error. Caustics of the focusing X-ray beam (Fig. 6(d)) in both vertical and horizontal directions also indicated high homogeneity of the wavefront, which was confirmed by Gaussian cross-section profiles at every measured longitudinal position (L 2 ) downstream of the lens. To conclude, the absence of enlargement of the focal size, the absence of deviations from a Gaussian profile and symmetry in the intensity distribution of the focusing beam prove the absence of expressed lens aberrations.

Conclusion
In the present paper, we demonstrated that ion-beam lithography can be applied to micromachining of X-ray refractive lenses. With the help of IBL, the hardest of current materials -diamond -was milled and micro-scale diamond half-lenses were produced. Lenses had a rotationally parabolic profile with radii of parabola apexes of <5 µm. As has been confirmed with SEM, the profiles of produced lenses were free of expressed low-and high-frequency modulations: figure errors of fabricated lenses were <200 nm, while the surface roughness was estimated to be 30 nm. SEM has also indicated that the aperture of the lens has a circular shape which additionally confirms high positioning accuracy of the ion beam. Single lenses were stacked in the CRL 3 within one technological process with high alignment precision that has been further verified by acquired X-ray radiograms. The optical performance of the CRL 3 was successfully tested at a third-generation synchrotron, where the lenses provided diffraction-limited focusing of X-ray radiation and demonstrated intensity profiles with Gaussian distributions at every measured longitudinal position (along the optical axis) downstream of the CRL 3 .
As IBL gives a unique opportunity to precisely control the shape of diamond micro-lenses, one could easily vary the radii and profiles of lenses, tailoring the CRL stack to a specific application. Therefore, as the next step in its development, we aim to produce the CRL with bigger number and smaller radii of individual diamond micro-lenses, in order to increase the numerical aperture and thus the resolution. According to our calculations, the resolution of 40 nm can be achieved at the photon energy of 12 keV, using 20 diamond half-lenses with 50µm apertures and 3µm radii of the parabola apex. It is worth mentioning that the production of a larger amount of lenses would not necessarily lead to a linear increase in the overall manufacturing time as the technological process can be significantly optimized through the use of H 2 O-assisted FIB milling process [42]. This will not only speed the milling up by 2-2.5 times but also will reduce material redeposition which improves surface roughness. The foregoing facts can allow to produce large-aperture diamond lenses, which can be used at highly collimated high-flux X-ray sources (as Free Electron Lasers) for beam-shaping and beam conditioning applications.
Precise micro-manipulation and stacking of individual lenses also open new opportunities for compact X-ray objectives. The precise stacking which we demonstrated in the present paper can be further used to add entrance and exit pinholes to the CRL, thus limiting the working area of the lens and suppressing unnecessary radiation which is of importance for X-ray microscopy. Another interesting idea would be the creation of a so-called lab-on-a-chip, where a short-focal objective lens would be integrated with the sample of study and any other additional micro-optical elements.

FIB
The process of fabrication has been automated by SmartFIB software (Zeiss). The energy of an ion beam was 30 keV and ion current varied in the range from 50 pA to 100 nA. The volume of ∼1600 µm 3 was milled with Ga+ ion beam at a current of 3 nA.
The grayscale digital image (8-bit) was created, where pixel coordinates defined ion beam position in the orthogonal direction (x-y) and pixel gray intensity values set the beam dose (0-100%) in each pixel (Fig. 7). In our case, the equation describing a paraboloid of revolution was used to create the image: where x and y are pixel coordinates, R is the radius of the parabola apex and I corresponds to the pixel gray value after normalization. The software automatically controlled the local milling rate in each pixel and the amount of removed material. The continuous manufacturing was accompanied by the redeposition of the material (mainly amorphous carbon and implanted gallium) from the bottom of the lens to its upper parts [43]. To remove the redeposited material, the diamond plate was washed for 3 hours in K 2 Cr 2 O acid heated to 55°C (Fig. 8). The amount of redeposited carbon can be decreased with the optimization of the milling technology -using a gas precursor and increasing the number of passes of the ion beam at the surface. This will allow to skip the chemical washing without sacrificing surface quality and will also reduce the manufacturing time.

Image registration
We used an X-ray imaging system (Optique Peter, Lyon, France), consisting of a thin (8.4 µm) LSO:Tb scintillator (ESRF, Grenoble, France), an OLYMPUS UPlanFI 20-fold objective (OLYMPUS, Tokyo, Japan), a 45°mirror reflecting the image of the scintillator upwards, an OLYMPUS F180 mm tube lens and a PCO.edge 4.2 sCMOS-camera (PCO AG, Germany) with the resolution of 2048 × 2048 pixels and 1-100 fps acquisition frequency. Considering the impact of all the optical elements, an effective pixel size of 0.325 µm and a field of view of 666 × 666 µm 2 were achieved. The resolution of the imaging system is defined by a point spread function (PSF) of 2 pixels, which equals 650 nm. The distance between the sample and the scintillator is assumed as the sample-to-camera distance. The X-ray imaging system was installed at the translation detector stage (in-house design), which offers five degrees of freedom (vertical and horizontal translation, roll, pitch and yaw). The sample-to-detector distance is also adjustable between 10 cm and 3 m.
To ensure maximum contrast and maximum signal-to-noise ratio the images from the PCO camera were corrected by a flatfield in two steps. At first, we collected a set of thirty images without the sample registering slightly different illumination conditions due to the fluctuations of the X-ray beam. In the second step, we corrected each X-ray image with the sample by dividing it on the flatfield image with the highest similarity. For this operation, we used the similarity index (SSIM) implemented in the scikit-image Python-module as a metric [44].

Phase-contrast X-ray imaging and phase retrieval
For the experiment described in the present paper, X-ray radiograms were acquired in the near-field edge-enhancing regime at the sample-to-detector distance L 2 of 10.8 cm and at the photon energy of 12 keV. Flatfield-corrected images were further processed using a single-distance non-iterative holographic reconstruction procedure (the so-called CTF-approach) to obtain projected phase maps of the sample [45,46]. The CTF-approach was implemented in the Python-based program code (in-house development).
experiments. We would also like to show gratitude to Leonid Dubrovinsky from the University of Bayreuth for sharing his pearls of wisdom with us during the initialization of this project. Travel accommodation for X-ray tests at PETRA III, DESY was supported by the Russian Academic Excellence Project at the Immanuel Kant Baltic Federal University.

Disclosures
The authors declare no conflicts of interest.