X-ray zoom lens allows for energy scans in X-ray microscopy.

We introduce a new design and development of a compound refractive X-ray zoom lens for energy scans in X-ray microscopy. Energy scans are, in principle, equivalent to radial scans in the reciprocal space for X-ray diffraction. Thanks to the absence of sample or detector motions, energy scans are better suited for microscopy, which requires high stability. In addition, close to the absorption edge of an element, energy scans can yield chemical information when coupled with resonant effects in full field diffraction X-ray microscopy (FFDXM) or X-ray absorption near edge structure (XANES) microscopy. Here, we demonstrate the concept by using a customized compound refractive X-ray zoom lens for 11 keV near the Ge Kα-edge. The working distance and magnification were kept constant during the energy scans by adapting the lens composition on switchable zoom lens fingers. This alleviates the need to reposition the lens while changing the energy and makes quantitative analysis more convenient for FFDXM. The fabricated zoom lens was characterized and proven suitable for the proposed measurement.

and tilt information on an area larger than 100 µm x 100 µm at the sample surface with 200 nm resolution (line and space). By coupling with resonant scattering, information on the chemical composition of the anomalous element can be further obtained.
To achieve this, the microscopy setup has to be changed from a chromatic CRL to an objective with variable focal length, which can easily be included into the existing experimental setup. The recently introduced X-ray zoom lens [5,6] can keep a fixed working distance and constant magnification factor over a required energy range. This X-ray zoom lens consists of SU-8 lens elements fabricated by deep X-ray lithography [7][8][9] and positioned on bendable zoom lens fingers. The focal length can be adjusted by bending lens elements in or out of the beam path. This X-ray optics is a comparably cheap solution and is used with a customized lens layout in this new experimental setup.

Optical setup and X-ray zoom lens layout
For demonstration, the target energy of 11.103 keV was selected for the X-ray zoom lens. This energy corresponds to the Ge-Kα-edge found in isolated compositionally graded SiGe pillars, which were previously studied with the K-map technique at ID01 [4]. To correctly apply the scheme of multi-wavelength anomalous X-ray scattering [10], it is necessary to measure several data points close to the Ge-Kα-edge in energy steps of 5 eV and additionally two data points 50 eV far away from the edge. The customized X-ray zoom lens has one configuration for each of these energies to achieve FFDXM imaging with a fixed working distance and a constant magnification. The experimental setup for this demonstration is shown in Fig. 1. The sample distance sd is defined from the sample plane to the first edge of the first finger of the X-ray zoom lens at position z = 0. The z-direction is defined along the optical axis. The working distance wd is defined from the first edge of the first zoom lens finger of the X-ray zoom lens (z = 0) to the detector plane with wd = 6.5 m. The CRL's maximum length is LCRL = 61 mm. , with a sketch of optical path through the sample in Bragg-condition, objective (X-ray zoom lens), and detector (bottom). z = 0 is defined to be at the first edge of the first zoom lens finger. The sample distance sd is the distance from the sample plane to z = 0 and the working distance wd is the distance from the detector plane from z = 0.
The detector is an Andor Zyla 4.2 PLUS sCMOS camera with 2160 x 2560 active pixels and pixel size pCMOS = 6.5 µm. For quantitative analysis, it is essential that the magnification is kept constant for all photon energies. To achieve a desirable resolution of σmin = 400 nm per line and space, the magnification factor was set to M ~65: The factor four takes into account that four detector pixels are needed to resolve a pair of line and space as indicated by the Nyquist-Shannon sampling theorem [11]. With these requirements the sample distance sd and characteristics of the X-ray zoom lens can be calculated for the photon energies around the absorption edge at a fixed working distance wd = 6.5 m. Because the sample and the detector positions are fixed during the energy scan, the objective has to be adapted by changing its characteristics to keep the back focal point F' at a fixed position [see Fig. 2]. For a CRL made of identical and equidistant elements with overall length LCRL, radius of curvature in the apex of the parabola R, δ the decrement of the refractive index and N lens elements, the focal length f can be approximated by [12,13] CRL . 26 To achieve fixed working distance for very small energy step size of 5 eV, the new zoom lens was designed with lens elements having a slightly different radii R, resulting in non-equidistant lens elements like in Fig. 2. Consequently, the approximation (2) is here replaced by raytracing. The effective focal length fzl is defined from the back principle plane H' of the lens system to the back focal point F' [14]. The refractive index n of any material is slightly below one in the X-ray regime and in general denoted as: Here, β is the absorption coefficient and δ the decrement of the refractive index. For the X-ray zoom lenses, we used a SU-8 resist (type "mr-X-50" from micro resist technology GmbH, Berlin) and measured δ for photon energies EPh for a 400 µm thick resist ranging from 8.7 keV to 40 keV. The value of the absorption coefficient δ for the desirable energy range can be approximated by: with a tolerance below 1%. For EPh = 11.103 keV δ is 2.202·10 6 . The absorption coefficient β can be found in the literature [15]. The object distance so from the sample to the front principle plane is calculated by the image distance si from detector towards the back principle plane and the magnification factor M [see Fig. 2 The sample distance was determined as sd = 67.5 mm using Eq. (5) and the specific lens element positions on the zoom lens fingers [see Fig. 4]. The X-ray zoom lens was optimized in a way that the distances sd and wd stay constant when changing the photon energy. Therefore, the focal length of the zoom lens was adapted accordingly. Deep X-ray lithography allows for the manufacturing of 1D line focussing refractive lenses. A point focus requires a 2D-lens and therefore two line focus lenses are mounted perpendicular to each other rotated around the optical axis [see Fig. 3]. For small energy steps such as 5 eV, it is technically unfeasible to vary the number of lens elements in the beam. Instead, the fixed working distance is achieved by exchanging lens elements by elements with slightly different radii. Nine different zoom lens configurations have been realized in total, for FFDXM measurement at nine energies around the Ge Kα-edge [see Table 1]. Only configurations that are symmetrical in relation to the lens center of the zoom lens [see Fig. 4] were used to keep the position of the principle planes and hence the magnification M as constant as possible. The focal length varies slightly for each configuration, as it depends also on the length of the CRL LCRL [see Eq. (1)]. The latter becomes larger as lens elements far away from the center are switched in. The optimized zoom lens layout is shown in Fig. 4. All lens elements in the current zoom lens layout have an aperture of A = 98 µm and radii between R = 7.050 µm and 8.305 µm. The variation of the radius for neighboring lens elements is in average ΔR ~150 nm. Overall N = 66 1D lens elements are manufactured on 22 zoom lens fingers, respectively for the horizontal and vertical focusing half lens [see Fig. 4].
In each configuration six horizontal and vertical zoom lens fingers are placed in the beam. The total number of lens elements in the beam is thus N = 18 for each focusing direction. The average transmission for the configurations is (20.2 ± 0.2)%. The radii of horizontal and corresponding vertical lens elements are slightly different to compensate for the astigmatism originated from the shifted lens element positions. As a result, the back focal point F' is slightly different for horizontal and vertical direction. The configuration #5 for the photon energy at the Ge Kα-edge is shown in Fig. 4 (green highlighted). The selected six zoom lens fingers are v05, v10-v13, v18 for the vertical focusing half lens and h05, h10-h13, h18 for the horizontal focusing half lens. All other zoom lens fingers are kept out of the beam path for this configuration. Fig. 4. X-ray zoom lens layout for the energy scan with FFDXM at nine different energies around the Ge Kα-edge. There are 22 zoom lens fingers (shown as grey and green stripes), respectively, for the horizontal focusing lens elements (h01-h22) and the vertical focusing lens elements (v01-v22). In each configuration, six zoom lens fingers with overall N = 18 lens elements are positioned in the beam path, for vertical and horizontal direction, respectively. The configuration #5 for the exact photon energy at the Ge Kα-edge of EPh_GeK = 11.103 keV is highlighted in green. Table 1 shows all parameters of the optimized X-ray zoom lens for all nine energies (seven with 5 eV step size and two with 50 eV step size). "X" marks zoom lens fingers that are positioned in the beam and "o" marks zoom lens fingers that are moved out of the beam path [see Table 1, column 5]. Note that while the X-ray zoom lens can also be optimized to achieve a constant effective focal length fzl (i.e., the distance from the back principle plane H' to the back focal point F' [Fig. 2]), in the microscopy setup, the goal is to achieve a constant magnification factor M. To emphasize this effect, Table 2 shows exemplarily the effect on M when the configuration of the X-ray zoom lens is kept constant (e.g., config. #5). For an energy step of 50 eV, for instance, to get a sharp image in the detector plane, one would have to move the detector up to Δwd = 8.5 m.

Fabrication of customized X-ray zoom lens
The X-ray zoom lens consists of lens elements of SU-8 polymer resist, which is cross-linked during deep X-ray lithography exposure at a synchrotron source. The material is radiation stable up to a deposited dose of at least 2 MJ/cm 3 [16,17]. With deep X-ray lithography all refractive lens elements are fabricated well aligned in one step. The mandatory X-ray absorber mask is written with an electron beam writer with a spot size of 8 nm diameter [18]. The length of one lens element Llens along the optical axis is the sum of two parabolas and the web in between. With a web size of w = 7 µm and an aperture of A = 98 µm the length of a lens is calculated by: The smallest variation in the radius of curvature in this layout is ΔR = 8 nm. Therefore, the change in the length of lens elements with an average radius of curvature in the apex of parabola of R = 7.6 µm can be estimated by: with ΔR « R. It can be noted that the 8 nm writing spot size of the electron beam writer is equal to only ±2.4% of required change L1-L2 = 0.330 µm. This guaranties a precise and reliable fabrication of the X-ray zoom lens elements. Two silicon stripes, each containing 66 lens elements, were glued to a zoom lens substrate providing the bending fingers [see Fig. 3 and Fig. 5]. Each zoom lens substrate with the glued silicon stripe was diced to form the 22 zoom lens fingers as shown in the layout [see Fig. 4]. These bendable zoom lens fingers were preloaded with a stainless steel bar as deadlock and held the lens elements at their free end [see Fig. 5 left]. For a point focus X-ray zoom lens two zoom lens substrates, one for vertical and one for horizontal focusing direction, are mounted perpendicular to each other rotated around the optical axis [see Fig. 3 and Fig. 5 right]. The achieved angular tolerance was 90°±0.06° by laser triangulation. Details of the construction, mounting and precision of the lens elements as well as the electronic control of the X-ray zoom lens are explained elsewhere [5].

Optical characterization of X-ray zoom lens
At beamline ID01, ESRF the custom-designed X-ray zoom lens was characterized and tested. In Fig. 3 (middle and left) the radiograph at 11 keV of the aligned zoom lens is shown with the two crossed line focus lenses acting as a point focusing optics. The flat field, stopped down to 90 µm x 90 µm showed good homogeneity with only 12.5% intensity deviation for the average intensity. The quality of the fabricated lens was characterized with a series of ptychography scans on a Siemens-star test sample in transmission. With a 300 nm step size the sample is moved through the focused X-ray beam in an area of 10 µm x 10 µm. A far-field diffraction pattern was recorded with a MaxiPix 2x2 detector with 516 x 516 active pixels and pixel size of 55 µm at a distance of 102 cm. The illuminated object and the wave field probing the sample were reconstructed with PyNX.Ptycho [19]. In Fig. 6 is shown the beam profiles of the X-ray zoom lens in configuration #5 at 11.103 keV. On the left side the caustic focus profiles are shown along the beam path in horizontal [ Fig. 6(1a)] and vertical [ Fig. 6(1b)] direction. The observed difference in the focal distance was by design, in order to compensate for the position shift of one lens element between the vertical and horizontal lenses [see also Fig. 7]. On the right side [ Fig. 6(2)], a point focus was achieved as evidenced by the intensity distribution in the focal plane.
More ptychography measurements were carried out in different configurations and at different energies. In Table 3 are found the reconstructed focal distances f and focal spot sizes σ by ptychography compared to their theoretical values. All reconstructed focal distances f are defined as the distance from z = 0 to the back focal point F'; not to be confused with the effective focal length fZL as calculated in section 2 [ Table 1 and Table 2]. The focal spot sizes shown in Fig. 7 were calculated using wavefront propagation [19]. On the left of Fig. 7, it was demonstrated that the focal point, horizontal or vertical, remained indeed constant despite changing the energy by switching to the corresponding configurations (#4, #5, and #9). The simulated and measured focal distances f all showed the expected behavior regarding the astigmatism and required static focal point, with variations in the range of less than 0.1% [see Table 3, col. [3][4][5][6][7]. This validates the concept and functionality of the designed zoom lens. Focal spot sizes in the range of σ = 175 nm to 193 nm were obtained with an uncertainty of ~10%. The results are comparable to the calculated minimal focal spot sizes after taking into account the diffraction limit, source size and astigmatism [ Table 3, col. 8-9] [5].
In a second approach, the difference between using a fixed chromatic lens and an adaptive zoom lens at different energies is shown. Ptychography measurements were performed for configuration #5 at its intended photon energy and at 50 eV above [see Fig. 7 right]. A shift of focal point of 0.8 mm was observed by operating at 50 eV above its intended energy. The shift was further confirmed by the simulated data [ Table 4]. Such shift should be avoided as it introduces a change in the magnification M and blurs the FFDXM image. For the designed zoom lens, this was achieved by switching to the corresponding configuration #9. In a last approach, the designed X-ray zoom lens was used as imaging optics for the FFDXM setup in a transmission mode. A test pattern was positioned at distance sd = (67 ± 1) mm upstream of the lens. Two sets of slits with 0.3 mm opening upstream of the sample were used to define the aperture of the zoom lens. The detector was placed downstream of the zoom lens with wd = 6.5 m. The magnification was therefore M ~65. A series of images was taken to estimate the resolution of the zoom lens with different configurations and at different energies. In Fig. 8 the entire test structure was imaged with the X-ray zoom lens (FoV = 86 µm x 86 µm) in configuration #6 at EPh = 11.103 keV [ Fig. 8. top left]. A detailed view shows the smallest structures resolved [Fig. 8, top right]. At the bottom of Fig. 8 are shown three images taken with Fig. 8(a) configuration #6 at its intended energy 11.103 keV and Fig. 8(b) configuration #9 at its intended energy 11.148 keV. Both show equal imaging quality, the 150 nm lines are resolved. Figure 8(c) is taken in configuration #6, but at the energy 11.148 keV, thus out of focus. Compared to Fig. 8(b), Fig. 8(c) shows broadened stripes between the structured fields and significant edge effects, which indicates the out of focus position of the lens. The 100 nm lines are not resolvable as they are under-sampled with only two pixels per line and space in the detector plane. With the adaptive zoom lens, the measured change in the magnification (by comparing results from Figs. 8(a) and 8(b)) is 35% smaller compared to the expected change for a chromatic CRL (by comparing results from Figs. 8(a) and 8(c)). The images of the test pattern also confirmed that the desired resolution of σmin = 400 nm per line and space was reached for the current zoom lens. This microscopy setup was restricted by the detector resolution. Taking the the ptychography scan results with a measured focal spot size of σ ~180 nm into account, a resolution of 350 nm line and space is achievable with this X-ray zoom lens. Fig. 8. Test pattern with vertical and horizontal line structures of different sizes imaged by the X-ray zoom lens with a FoV of 86 µm x 86 µm in configuration #6 at EPh = 11.103 keV (top left). A more detailed view is shown (top right) for the smallest resolvable structures. On the bottom are shown a comparison between three images taken with (a) configuration #6 its intended energy and (c) 45 eV above, and (c) configuration #9 at its intended energy.

Conclusion
A customized compound refractive X-ray zoom lens has been designed and realized for energy scans in diffraction X-ray microscopy. The X-ray zoom lens, which consists of SU-8 lenses fabricated by deep X-ray lithography, has been tested and characterized at ID01, ESRF. The working distance and the magnification factor of this zoom lens, to be used as objective in the microscopy setup, are kept constant for nine energies around the Ge Kα-edge by switching to the corresponding configuration. Hence, there is no sample, objective or detector motion involved during energy scans, the images are kept sharp in microscopy mode and the magnification constant in shadow projection mode. The concept was validated by ptychography and full field microscopy. The ptychography showed that a fixed working distance was achieved at the intended energies with a measured focal spot sizes of σ = (180 ± 18) nm. Additionally, the imaging of line test structures showed an intended resolution of 400 nm line and space on a FoV of 86 µm x 86 µm. The resolution is assumed to be around 350 nm, but was restricted by the detector resolution. The result shows the potentials of such X-ray zoom lens for energy scans to eliminate sample movement in nano-focused, full field X-ray diffraction and absorption microscopy. The detailed description and analysis of the experiment with SiGe pillars at ID01 shall be described in a separate publication.