Hard X-ray multi-projection imaging for single-shot approaches
P. Villanueva-Perez, B. Pedrini, R. Mokso, P. Vagovic, V. Guzenko, S. Leake, P. R. Willmott, C. David, H. N. Chapman, M. Stampanoni
HHard X-ray multi-pro jection imaging forsingle-shot approaches
P. Villanueva-Perez , B. Pedrini , R. Mokso , P. Vagovic ,V. Guzenko , S. Leake , P. R. Willmott , C. David ,H. N. Chapman , and M. Stampanoni Paul Scherrer Institut, Villigen, Switzerland Center for Free-electron Laser Science (DESY), Hamburg, Germany Max IV Laboratory, Lund University, Lund, Sweden ESRF - The European synchrotron, Grenoble, France University of Hamburg, Hamburg, Germany Centre for Ultrafast Imaging, Hamburg, Germany Institute for Biomedical Engineering, UZH/ETH Z¨urich, Z¨urich, Switzerland * Corresponding author: P. Villanueva-Perez, [email protected]
Abstract
Obtaining 3D information from a single X-ray exposure at high-brilliance sources, such as X-ray free-electron lasers (XFELs) [1] ordiffraction-limited storage rings [2], allows the study of fast dynami-cal processes in their native environment. However, current X-ray 3Dmethodologies are either not compatible with single-shot approachesbecause they rely on multiple exposures, such as confocal microscopy [3,4] and tomography [5,6]; or they record a single projection per pulse [7]and are therefore restricted to approximately two-dimensional ob-jects [8]. Here we propose and verify experimentally a novel imag-ing approach named X-ray multi-projection imaging (XMPI), whichsimultaneously acquires several projections without rotating the sam-ple at significant tomographic angles. When implemented at high-brilliance sources it can provide volumetric information using a singlepulse. Moreover, XMPI at MHz repetition XFELs could allow a wayto record 3D movies of deterministic or stochastic natural processes a r X i v : . [ phy s i c s . op ti c s ] A ug n the micrometer to nanometer resolution range, and at time scalesfrom microseconds down to femtoseconds. Since their discovery, hard X-rays have been crucial in natural sciencesbecause of their penetration power and short wavelength, which allows high-resolution imaging of thick samples, even in native conditions. Among thecurrently used X-ray imaging techniques, phase-contrast methods enhancethe contrast sensitivity by exploiting the phase shift due to variations in theelectron density rather than the intensity attenuation characteristic of radio-graphic approaches [9]. Coherent techniques, which exploit phase contrast,are regarded as the most suitable to achieve high-resolution [10], in thatthey can address micrometer to nanometer scales. Because the high bril-liance is the key parameter for coherent techniques, their advent coincidedwith the realization of third generation synchrotron light sources. Novel X-ray sources with orders of magnitude higher brilliance, such as diffractionlimited storage rings [2] and X-ray free electron lasers (XFELs) [1, 11, 12],enlarge the spectrum of coherent applications, especially addressing shortertimescales [13, 14]. XFELs in particular provide ultraintense femtosecondpulses which can image samples before inducing any radiation damage [15].This concept, known as diffract before destroy, was demonstrated experi-mentally [16] by reconstructing an object from an X-ray pulse but before itCoulomb explodes. The resolution and contrast sensitivity are limited by thenumber of photons available in a single pulse, and not by the maximum toler-able dose which preserves a given resolution [17], as is the case for continuoussample illumination. As a consequence, any method that requires multipleexposures of the same sample, including three-dimensional (3D) techniquessuch as tomography [5, 6] and confocal microscopy [3], or any scanning tech-nique cannot be applied. Thus, XFEL applications aiming at 3D structuralinformation, which deliver high dose, either require imaging of several identi-cal copies of the object [18], or are restricted to retrieving partial informationfrom a single exposure [8], as desired in ankylography [7].Here we propose a scheme christened X-ray multi-projection imaging(XMPI), which provides 3D structural information via multiple 2D projec-tions at different tomographic angles acquired simultaneously from the sameobject. The key component of XMPI is a beam splitter that generates anumber of beams by Laue diffraction, which illuminate a sample simultane-ously from different angles. Each of these beams retains the correspondingprojection information. This idea was proposed in 1994 for the soft X-ray2egime [19] using a phase-grating splitter. In the hard X-ray regime un-der consideration, however, suitable gratings are unrealistic; for example,the grating pitch to achieve a deflection for the first diffracted order of 20 ◦ for 4 keV photons would be 8.5 ˚A, which is too small for presently-knownmanufacturing methods. In contrast, Laue diffracted beams are much moresuitable because the deflection angles reach tens of degrees, compatible withthe requirement for true tomographic projections. In a general case, theLaue condition can be achieved simultaneously for two different reflectionsby appropriately orienting the crystal [20]. This number can be increasedby exploiting symmetries of the crystal lattice, setting the X-ray energy tospecific values, and positioning the crystal so that several reflections sit si-multaneously on the Ewald sphere. Figure 1(a) illustrates the generationof eight deflected beams by the { } -family of Laue reflections of a face-centered-cubic crystal, such as diamond or silicon. The incoming beam direc-tion, defined by its momentum vector (cid:126)k , is set parallel to a high-symmetryaxis, corresponding to the (001)-reflection direction in the depicted example.All reflections related by a rotation around the symmetry axis, e.g. corre-sponding to { } -family, form identical angles π/ − θ with respect to theincoming beam direction and share the same reflection plane spacing d . TheLaue condition for the wavelength λ , λ = 2 d sin( θ ) , (1)is then fulfilled simultaneously by all eight planes, yielding eight diffractedbeams with a deflection angle of 2 θ . For a silicon crystal, the photon energythat sets the { } planes in the Laue condition is 12 .
56 keV (see Methods).Figure 1(b) provides experimental evidence for the simultaneous generationof the eight beams described above. The experiment was performed at theMaterial Science beamline [21] of the Swiss Light Source (SLS), using a siliconcrystal with the aforementioned arrangement. Figure 1(c) sketches the ar-rangement of the beam-splitter crystal and a sample positioned downstreamthe crystal in the overlap region of all eight diffracted beams. To ensure thesimultaneous illumination of a sample of size t by all the beams, the incom-ing beam diameter S and maximum distance from the sample center to theclosest face of the crystal L are constrained (see Methods).XMPI is a technique which can be applied to the near-field and far-fieldimaging regimes. In this work, we demonstrate that the different projectionsof an object are retrieved for both regimes with resolutions around 17 µ m and80 nm, respectively. 3he near-field imaging experiment was carried out at the TOMCATbeamline at SLS [22]. Propagation-based phase-contrast imaging was per-formed (see Methods) using the setup depicted in Fig. 2(a). The collimatedbeam at 12.56 keV illuminated a Si(001) splitter, mounted on a triple-axisgoniometer. Due to geometrical limitations of the experimental setup, thecrystal could not be oriented to hit simultaneously the eight reflections ofthe Si { } -family (Fig. 1(a)), but only the Si(131) and the Si(111) reflec-tions with deflections angles of 35 . ◦ and 18 . ◦ , respectively. A moth placeddirectly downstream of the splitter was illuminated simultaneously by thethree beams. Three near-field images shown in Fig. 2(b-d) were recorded bytranslating the detector to intercept each of the three beams. The forward-direction image exhibits lower noise because of the higher intensity. Theimage resolution of such images was estimated to be about 17 µ m by analyz-ing the edge profiles. The rotation axes that relate the direct-beam projection(b) with the two deflected beam projections (c) and (d) form the expectedangle of 11 . ◦ . The features of the moth head observed in the three imagesconcur with being projections of the same object along the directions givenby (001), (111), and (¯1¯31).The far-field imaging experiment at 12.56 keV was performed at the ID01beamline of the European Synchrotron Research Facility (ESRF) [23]. Weperformed coherent diffraction imaging (CDI) [24] (see Methods), a well-established technique at storage rings and XFELs [10], using the setup shownin Fig. 3(a) (see Methods). A Si(001) splitter was mounted on a small hexa-pod to adjust the orientation. The crystal was oriented such that the two { } -family diffracted beams accessible in the ID01 diffractometer geom-etry were seen simultaneously on a pixel detector. A gold nanostructure,exhibiting non-trivial 3D features (Fig. 3(b)) and grown on a silicon nitridemembrane (see Methods), was glued on the downstream surface of the crystal.As the coherent flux was not sufficient, the beam was focused to a size about S = 1 µ m at the crystal surface with a numerical aperture which matchedthe Darwin width of the Si(131) reflection [25, 26]. Unfortunately, S did notallow the simultaneous illumination of the sample by the different beams asdesired. However, this is not a limitation at sources with higher coherentflux such as XFELs and diffraction-limited synchrotrons. The sample wasthen translated transversely to produce diffraction patterns on the detectorpositioned at 2.37 m distance. The three recorded diffraction patterns areshown in the third column of panels (c-e) of Figure 3, along with correspond-ing simulations (first column) with same signal levels. The experimental4atterns from the diffracted beam clearly manifest larger background levelsdue to lower flux and backgrounds components enhanced by the crystal. TheCDI reconstructions from the experimental diffraction images, obtained byapplying phase-retrieval algorithms (see Methods), and the simulated projec-tions of the sample are shown in the fourth and second column of the samepanels as above. Their comparison confirms that the expected projectionshave been measured. The resolution of the reconstructions, established usingthe phase-retrieval transfer-function criterion [27], was 18 nm for the directbeam projection and 77 and 85 nm for the two skew projections. A 3D re-construction of the object using the three measured projections is depictedin Fig. 4. The reconstruction in yellow is compared to the simulated modelin semi-transparent red.In conclusion, we have validated experimentally XMPI that relies on asingle crystal as beam splitter to generate simultaneously tomographic pro-jections from a single exposure of a sample to the X-rays. XMPI circum-vents rotating the sample as for tomography, and represents a clear improve-ment with respect to pseudo-3D single-shot methods. We conceived XMPIas an imaging method for XFELs. In the diffract-before-destroy approach,essential to achieve sub-micrometer resolution from weakly scattering, non-reproducible objects, XMPI paves the way to 3D object reconstructions.Other applications appear however to be meaningful. If XFELs that offerpulse trains at MHz repetition rates, such as the European XFEL or the LinacCoherent Light Source after the planned upgrade, are operated at fluencesbelow the sample damage threshold, XMPI may enable to track 3D structuraldynamics of stochastic and deterministic [28] processes at the submicrosec-ond time scale. At synchrotron facilities, XMPI may find applications in thecase that a sample cannot be rotated due to the complexity of the sampleenvironment. Furthermore, at diffraction-limited sources, such as MAX-IVand future ones, the time resolution for structural dynamics investigationsmay be reduced well below the millisecond regime. We therefore anticipatethat dedicated XMPI instruments may be realized at operational and futurehard X-ray user facilities. 5 ethods Crystal beam splitters
We propose the use of face-centered-cubic crystals as beam splitters due totheir high degree of symmetry, i.e. they can potentially generate simulta-neously multiple deflected beams. Specifically, we focus on diamond andsilicon crystals. Diamond is a good candidate for X-ray free-electron lasersoptics due to its low X-ray absorption, damage threshold, and good heatconductivity. On the other hand, silicon is a crystal which can be producedinexpensively, and with high purity, low strain, and no defects. Table 1 re-ports the deflection angles for a few allowed silicon and diamond reflectionfamilies whose corresponding wavelengths are in the hard X-ray regime andwith practical deflection angles. The energy to set a family of planes in Braggcondition is given by E = hc d sin θ , (2)where E is the energy, h is the Planck constant, and c is the speed of lightin vacuum.Refl. Symm. direction Refl. mult. E Deflection angle 2 θ Si(131) (001) 8 12.56 keV 35.1 ◦ Si(331) (001) 8 13.70 keV 48.2 ◦ C(¯311) (011) 6 13.52 keV 50.5 ◦ C(1¯1¯1) (111) 3 9.03 keV 38.9 ◦ C(¯11¯3) (111) 6 11.04 keV 63.0 ◦ Table 1: Bragg-reflection families of diamond and silicon cubic crystal struc-tures suitable for multi-beam generation in the photon energy range ( E )between 2 and 14 keV, with a deflection angle between 20 ◦ to 65 ◦ . Simultaneous illumination geometry
In order to image the sample simultaneously by all the generated beams,two conditions have to be satisfied. First, we impose a geometric condition,which requires that all the beams fully illuminate the sample. Therefore, thesample has to be positioned downstream the crystal at a position L ≤ L g ,where the maximum geometrically allowed distance L g for a thin crystal is6onstrained by the beam diameter S , the maximum transverse dimension ofthe sample t , and the deflection angle 2 θ : Lg ≤ sin (2 θ ) (cid:32) S cos(2 θ ) − t (cid:33) . (3)Second, the direct and the deflected beams have different optical paths, thusthey do not illuminate exactly at the same time the sample. Thus given amaximum tolerable time delay (∆ t ) which ensures the immutability of thesample, the maximum tolerable distance between the crystal and the sampleis given by L t = c ∆ t cos (2 θ ) − , (4)where c is the speed of light. At XFELs, the maximum tolerable time delaybetween deflected and the direct beam is constrained by time interval betweenthe arrival of the imaging pulse and the observation of radiation damage. Forexample, for a biological sample like a Lysozyme crystal imaged by 3 × photons at 12 keV focused on a 100 ×
100 nm the aforementioned interval isbelow 10 fs [15] and the maximum distance is of the order of 10 µ m. Finally,the sample to crystal distance L is chosen to be smaller than the minimumof the geometrical L g and temporal L t constraints. Propagation-based phase contrast
Phase-contrast imaging techniques exploit the phase change of the exit wave-front after transmitting through a sample rather than the change of trans-mission due to absorption. Such techniques are specially useful to distinguishbetween two materials with similar transmission or transparent materials tothe probing radiation. The high penetration power of hard X-rays, speciallyfor low-Z materials, makes them a perfect radiation type to exploit phase-contrast techniques. In the context of this work, we exploit propagation-based phase contrast, i.e. we use the free-space propagation to observe thephase change of the exit wavefront as intensity variations in the recordedimages [29]. The phase information is retrieved from the intensity varia-tions using phase-retrieval algorithms. The main solutions to this inversionproblem in the near-field regime linearize either the transmissivity of thesample as exploited by the contrast-transfer function approach (CTF) [30]or the propagator as transport-of-intensity equations (TIE) [31] do. As the7resented images are acquired at propagation distances of the order of thedepth of focus of the used microscope, we can exploit TIE algorithms, suchas that presented in Ref. [32].
TOMCAT setup and data collection
The phase-contrast near-field experiments were performed at the TOMCATbeamline of the Swiss Light Source. The X-rays provided by a bendingmagnet source were monochromatized by a multilayer monochromator to12.6 keV with approximately a 2 % bandwidth. The natural divergence of thebending magnet X-ray beam at TOMCAT is about 2 mrad in the horizontaland 0.6 mrad in the vertical direction, which is larger than the Darwin widthof the used crystal. A 100 µ m thick Si(001) crystal was illuminated by a10 × beam after conforming it with three sets of slits. The crystalwas mounted on a triple-axis goniometer to generate simultaneously severalreflections. Behind the crystal a moth was positioned to be illuminated bythe different generated beams. At 10 cm from the moth the detector waspositioned to record the different phase-contrast images. For each of theimages the detector was translated in a plane perpendicular to the incomingbeam. The detector was an X-ray 1:1 (Optique Peter) microscope with ahigh efficiency scintillator, which converts X-rays to optical photons. Thecamera used was a pco.edge 4.2 CMOS detector with a pixel size of 6.5 µ mand 2048 × × ph/mm on the crystal beam splitter [33]. Around 63 % of thefluence on the crystal beam splitter contributed to form the direct-beamimage, while the contribution to the silicon (111) and (131) images was of4 % and 2 %, respectively. Coherent diffraction imaging
Coherent diffraction imaging (CDI) [24] is a lensless technique. The sampleis illuminated by plane waves, and the diffraction patterns are recorded inthe far-field regime. CDI provides a reconstruction of the complex X-raytransmissivity of the sample by means of phase-retrieval procedures basedon iterative transform algorithms [34, 35]. The achievable resolution is givenby the largest diffraction angle at which the intensity of the diffraction pat-tern exhibits sufficient signal-to-noise in order to reliably reconstruct thephase [29], so that the ultimate resolution limit is set by the wavelength of8he incident radiation. The reconstruction presented in Fig. 3(c), (d), and (e)are obtained after averaging 20 reconstructions, where each of them was ob-tained after 2800 iterations of shrink-wrap algorithm [36] and 1200 iterationof hybrid input-output [35] with β = 0 .
3D reconstruction
The 3D reconstruction was retrieved by using the filtered backprojectionalgorithm. As the experimental data recorded at ID01 was limited to onlythree projections, we have applied symmetry constraints. First, we applied afour-fold symmetry constraint around the beam direction, i.e. perpendicularto the view in Fig. 3(c). Second, we applied a mirror symmetry constraintaround a mirror plane defined perpendicular to the projection in Fig. 3(c).Once the 3D model was reconstructed, a histogram constraint vetoing theoutlaiers was applied. The 3D data visualization has been obtained usingParaView software [37].
ID01 setup and data collection
The CDI experiments were performed at the ID01 beamline at ESRF. Thebeam was monochromatized by a double crystal monochromator at 12.56 keVwith a ∼ × − bandwidth as required to efficiently generate eight beamsby a silicon crystal (Table 1). The coherent portion of the beam was focusedby compound refractive lenses to a focal spot of 1 µ m on the sample. Thesample was composed of a 100 µ m thick Si(001) crystal and 500 nm goldnanostructures attached to it. The sample was mounted on a hexapod stagecapable to precisely orient the sample to generate the eight deflected beams.The intensity of the deflected beams was 10 % less intense than the trans-mitted beam. Once the crystal was aligned, a SmarAct 3D piezo system wasused to position precisely the gold nanostructures to be illuminated by thedeflected beams. As the beam was tightly focused around the sample and thesample was not exactly positioned behind the crystal, we translated around50 µ m the sample from the position in the direct beam to the position ineach of the diffracted beams. The diffractometer was positioned on each ofthe three accessible beams to record the diffraction patterns. The detectorused was a Maxipix with 55 µ m pixel size and 512 ×
512 pixels. The Max-ipix was positioned 2.37 m from the sample on the diffractometer arm and9 vacuum pipe with a beamstop was installed between the sample and thedetector to reduce the air scattering. The direct-beam image was recordedwith a fluence of 1 . × ph/ µ m illuminating the crystal beam splitter,but only 38 % of that fluence illuminated the sample. The deflected-beamimages were recorded for both reflections with a fluence of 3 . × ph/ µ m with an efficiency of 4 %. Sample preparation for CDI
Gold nanoparticles for CDI experiments were manufactured on 250 nm thickSi N membranes by means of two steps: electron-beam lithography andelectroplating [38, 39]. First, a metal stack of Cr/Au/Cr (5nm/10nm/5nm)was evaporated on a silicon nitride membrane. Subsequently, a negativetone e-beam resist HSQ (FOX16, Dow Corning, 1:1 dilution with MIBK)was spin-coated at 3000 rpm, resulting in a film thickness of about 250 nm.In the first e-beam lithography step, an array of micro-rings were exposedusing Vistec EBPG5000Plus direct writing e-beam lithography system, op-erated at 100 kV accelerating voltage. After development of the exposedHSQ in NaOH buffered solution (MICROPOSIT 351, Rohm and Haas) andrinsing in deionized water, silicon dioxide discs with inner/outer diameter of250 nm/4 µ m, respectively, and the distance between the neighboring parti-cles of 100 µ m were defined at specified locations. For the following e-beamexposure, approximately 900 nm thick layer of positive tone resist (PMMA950k, 8% in anisole, MicroChem Corp.) was spin-coated on the membrane at4000 rpm and baked out at 175 ◦ C on a hotplate. By performing the secondoverlay exposure and developing the samples in IPA:DI H O (7:3) solution,structures with various manifold rotational symmetry, fitting into a circle of500 nm diameter, were created exactly above the SiO discs. This way, moldsfor electroplating gold nanoparticles with well-resolved 3D shape control weredefined. After a short Cl -based plasma etching step (required to remove theupper Cr layer), the mold was filled with gold during the electroplating stepto a height of 500 nm. After removing the PMMA layer and subsequentlyHSQ discs in acetone and BOE (buffered oxide etch), respectively, individualnanoparticles with complex 3D shape along the rotational symmetry axisanchored on the Si N membrane were fabricated, an example is shown inFig. 3(b). 10 cknowledgments We are grateful to P. B¨osecke, T. Celcer, Y. Chushkin, H. Djazouli, M.Gordan, M. Lange, D. Meister, P. Oberta, and T. Schulli, for their sup-port in setting up the infrastructure for the experiments, and M. Guizar-Sicairos, A. Irastorza-Landa, G. Lovric, T. White, O. Yefanov, the TOMCATgroup, and the coherent X-ray imaging group at CFEL for fruitful discus-sions and comments. Part of this work has been supported by the ERCgrant ERC-2012-StG 310005-PhaseX. We also acknowledge funding withinthe R¨ontgen-Angstr¨om-Cluster by the German Ministry for Education andResearch (BMBF) under grant number 05K18XXA and by the Swedish Re-search Council under grant number 2017-06719.
Author contributions
P.V.-P., B.P., R.M., and M.S. initially conceived the research project, pro-posed the experiments, and developed the main concept of XMPI. P.V.-P.and P.V. devised the beam splitter concept and imaging implementations forXFELs and diffraction limited synchrotrons. P.V.-P., B.P., R.M., and P.R.W.demonstrated experimentally the beam splitter at the MS beamline. P.V.-P., B.P., R.M., and M.S. performed the measurements at TOMCAT. P.V.-P.,B.P, R.M., and S.L. performed the experiments at ID01. V.G. and C.D. fab-ricated the 3D nanosamples for the CDI experiments at ID01. The projectwas supervised by M.S. and H.N.C. P.V.-P. and B.P. wrote the bulk of themanuscript. All authors contributed to the preparation of the manuscript.
Competing interests statement
The authors declare no competing interests.
Data Availability
The data that support the plots within this paper and other findings of thisstudy are available from the corresponding author upon reasonable request.11 eferences [1] P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek, A. Brach-mann, P. Bucksbaum, R. Coffee, F. J. Decker, Y. Ding, D. Dowell, S. Ed-strom, A. Fisher, J. Frisch, S. Gilevich, J. Hastings, G. Hays, P. Her-ing, Z. Huang, R. Iverson, H. Loos, M. Messerschmidt, A. Miahnahri,S. Moeller, H. D. Nuhn, G. Pile, D. Ratner, J. Rzepiela, D. Schultz,T. Smith, P. Stefan, H. Tompkins, J. Turner, J. Welch, W. White, J. Wu,G. Yocky, and J. Galayda. First lasing and operation of an ˚angstrom-wavelength free-electron laser.
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Microelectronic Engineering , 121:127–130, 2014.16igure 1: Beam splitter. (a) Illustration of the eight reflections in reciprocalspace of the { } -family of a face-centered cubic crystal. The dotted-dashedcurve represents the intersection between the Ewald sphere and the l=1 plane.The sample is positioned downstream the crystal to be illuminated by all thegenerated beams. (b) Image of the direct beam and of the eight diffractedbeams on a phosphor screen, generated from a single incoming beam travers-ing a 100 µ m thick Si(001) crystal perpendicularly to the (001) surface. (c)Representation of the requirement to the maximum distance L from the crys-tal beam splitter surface to the sample such that the latter is illuminated byboth the direct and the diffracted beams. The relevant parameters are thediameter of the direct beam S , the transverse extension of the sample t andthe deflection angle 2 θ of the diffracted beam.17igure 2: Near-field imaging experiment. (a) Experimental setup used at theTOMCAT beamline of the Swiss Light Source. (b-d) Phase contrast imagesin the near-field regime recorded with the area detector placed in the hori-zontal plane at deflection angles of (b) 0 ◦ (direct beam direction), (c) 18 . ◦ (diffracted beam from the Si(111) reflection), and (d) 35 . ◦ (diffracted beamfor the Si(311) reflection), respectively. The detection plane was perpen-dicular to the direct beam. The rotation axes and rotation directions withrespect to the projection in (a) are marked with dashed-red lines and blackarrows. The scale bar in (b) corresponds to 500 µ m and the two red-dashedlines illustrate the angle between the rotation axes.18igure 3: Far-field imaging experiment. (a) Experimental setup used atID01 beamline of the ESRF synchrotron. (b) SEM image of the gold nanos-tructure sample. (c-e) Data related to the direct beam (c) and to the twoaccessible projections (d,e). From left to right: simulated diffraction pattern,corresponding simulated object projection, experimental diffraction pattern,corresponding CDI reconstruction. The scale bars in the diffraction pat-terns and in the reconstructions correspond to 2 × − nm −1