Ptychography imaging of the phase vortices in the x-ray beam formed by nanofocusing lenses
D. Dzhigaev, U. Lorenz, R. Kurta, F. Seiboth, T. Stankevic, S. Mickevicius, A. Singer, A. Shabalin, O. Yefanov, M. N. Strikhanov, G. Falkenberg, C. G. Schroer, R. Feidenhans`l, I. A. Vartanyants
PPtychography imaging of the phase vortices in thex-ray beam formed by nanofocusing lenses
D Dzhigaev , , U Lorenz , ∗ , R Kurta , F Seiboth , T Stankevic ,S Mickevicius , A Singer , † , A Shabalin , O Yefanov ,M N Strikhanov , G Falkenberg , C G Schroer , R Feidenhans‘l andI A Vartanyants , Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany National Research Nuclear University, “MEPhI”, 115409 Moscow, Russia Institute for Structural Physics, Technische Universit¨at Dresden, D-01062 Dresden, Germany Niels Bohr Institute, University of Copenhagen, DK-1165 Copenhagen, Denmark Center for Free-Electron Laser Science CFEL, Notkestraße 85, D-22607 Hamburg, GermanyE-mail: [email protected]
Abstract.
We present the ptychography reconstruction of the x-ray beam formed bynanofocusing lenses (NFLs) containing a number of phase singularities (vortices) in the vicinityof the focal plane. As a test object Siemens star pattern was used with the finest features of50 nm for ptychography measurements. The extended ptychography iterative engine (ePIE)algorithm was applied to retrieve both complex illumination and object functions from the setof diffraction patterns. The reconstruction revealed the focus size of 91.4 ± ±
1. Introduction
Phase singularities are a common feature of different forms of waves and represent a fundamentaltopology property of wavefields [1]. In 1804 Young has described effects of interference fromdifferent types of obstacles in the path of the light beam [2]. When three or more waves interfere,points of zero intensity could appear. At these positions the phase is undefined (singular), and,in general, all phase values in the interval [0; 2 π ] occur around a vortex point, leading to acirculation of the optical energy. Phase singularities were discussed in the terms of dislocationsin wave trains in 1974 [3] and were first observed in the optics of visible light [4]. Ten years agophase vortices were observed in x-ray regime using spiral zone plate at 9-keV photon energy [5].Phase singularities could also appear after interaction with the crystal lattice dislocations [6, 7].One generally describes light as a plane wave, that is an electromagnetic field with a constantphase that extends infinitely and normally to the Poynting vector, ψ ( r )= ρ ( r ) exp( iϕ ( r )), where ρ ( r ) is the real amplitude and ϕ ( r ) is the phase at the position r . Vortices can be characterized ∗ Present address: Institut f¨ur Chemie, Universit¨at Potsdam, Karl-Liebknecht-Straße 24-25, D-14476 Potsdam,Germany † Present address: The University of California, San Diego, La Jolla, CA 92093, USA a r X i v : . [ phy s i c s . op ti c s ] N ov igure 1. Experimental setup. The incoming x-ray beam (red arrow) goes along the Z axisand is focused by a pair of perpendicularly positioned NFLs. The test sample (S) in the form ofthe Siemens star is mounted on the movable stage (M) and is illuminated at the positions of araster grid. Diffraction patterns are collected by the detector (D) 2.1 m downstream. Insets (a)and (b) show horizontal and vertical profiles of the intensity of the reconstructed probe functionacross the central maximum.by the integer number S (positive or negative) that is called a strength or topological chargeof the singularity and is determined by S=( / π ) (cid:72) C d ϕ , where C is any closed path aroundthe vortex point. Dynamics of the vortices include processes of nucleation, annihilation, andpropagation in three dimensions [8].Ptychography, first proposed by Hoppe in the field of electron microscopy more than 40 yearsago [9], became a well established x-ray microscopy technique during the last decade [10, 11].Development of the phase retrieval algorithms [12] made this approach especially useful for thebeam characterization with the use of the test patterns [13]. It allows to reconstruct complexillumination function (probe) and complex object function simultaneously. In this work we applyptychography technique to reconstruct the wavefield generated by nanofocusing lenses (NFLs)in hard x-ray regime.
2. Experiment
We performed an experiment at the nanoprobe end station of P06 beamline at PETRA IIIsynchrotron source at DESY [14]. The geometry of the experiment is shown in Fig. 1. Twoperpendicularly positioned NFLs based on parabolic compound refractive x-ray lenses (CRLs)were used to obtain a nano-sized focus of the incident x-ray beam with the 15.25 keV energy.These lenses were fabricated by electron beam lithography and deep trench reactive etching. Thelenses were produced from silicon, because they can be shaped accurately on the sub-micrometerscale [15]. The flux of the beam in the focus was 4 × photons/sec. Pilatus 300K hybrid-pixeldetector (Dectris, Switzerland) with the pixel size of the 172 × µm was used. Detectorwas positioned 2.1 m downstream from the sample. A tantalum test sample in the form ofthe Siemens star, fabricated by nanolithography was mounted on the movable sample stage andpositioned in the focal plane. The ptychography scan was performed on a Cartesian grid with 50 igure 2. Results of ptychography reconstruction at the sample plane. (a) Amplitude ofthe probe function. White arrows indicate two points of zero intensity corresponding to phasesingularities (see inset for an enlarged view). (b) Phase of the probe function. In the inset anenlarged view of two singularities with opposite directions is shown. (c) Amplitude of the objectfunction. (d) Phase of the object function. Smallest resolved lines are 50 nm in size. Color barin (a) and (c) show normalized values of the amplitude functions and in (b) and (d) values ofthe phase in radians.nm step size and 41 ×
41 scan positions, in the horizontal and vertical directions, perpendicularto the optical axis of the beam. The step size corresponds to 58% of the probe overlap [16]. Theacquisition time was 0.3 seconds per scan position.
3. Results and discussion
An extended ptychography iterative engine (ePIE) algorithm [17] was applied to determinethe complex probe (see Fig. 2 (a, b)) and complex object function (see Fig. 2 (c, d)).They were reconstructed from 1681 diffraction patterns. The field of view was about 2 × µm . Reconstruction procedure started from an initial guess of the probe function of a roundshape with the uniform positive value inside and zero value outside, and a uniform objectwith a constant transmissivity without the phase shift. The final result was obtained after 100iterations. The pixel size in this reconstruction is 6 nm. The smallest detail in the object pattern igure 3. Propagation of the wavefield. Three 2D cuts perpendicular to the beam propagationdirection are shown: 0.24 mm in front of the focal plane (F), at the focal plane, and 0.11 mmbehind the focal plane. At the first position the pair of vortices nucleate, at the last one theyannihilate.that was resolved is 20 nm in size (see Fig. 2 (c, d)).The size of the illumination spot on the sample was obtained by fitting of the intensitydistribution by a Gaussian function. It turned out to be 91.4 ± ± { } topological charges.The reconstructed complex wave field profile was propagated from 1 mm in front of the focusto 1 mm behind it in the frame of paraxial approximation. The region, where the pair of vorticesnucleate and annihilate, is shown in Fig. 3. In three dimensions the wave function ψ ( r ) vanishesat each point of the line, called nodal line. These lines are clearly seen in two dimensional cutsthrough the propagated wave field along the beam direction (see Fig. 3). The length of thevortex lines is 0.24 mm before and 0.11 mm after the focal plane, giving in total 0.35 mm.The presence of vortices in the illuminating wavefield may cause degradation of the qualityof the phase retrieval procedure. In this case, a number of corrections should be taken intoaccount during reconstruction as shown for high-resolution transmission electron microscopy(TEM) [18] and classical wavefields [19]. Nevertheless, the ptychography technique is verytolerant to imperfections of the probe and the features of the object much smaller than thebeam size can be reconstructed as demonstrated in this work.
4. Conclusions
We have obtained the ptychography reconstruction of the x-ray field focused by NFLs containinga number of phase singularities (vortices) in the vicinity of the focal plane. Siemens star patternwas used as a test object with the finest features of 50 nm. After inversion procedure with theePIE algorithm a complex wave field function was obtained. The reconstruction revealed theocus size (FWHM) of 91.4 ± ± ± Acknowledgments
We acknowledge fruitful discussions and support of this project by E. Weckert, and carefulreading of the manuscript by M. Sprung. This work was supported by the EU grant for theproject 280773 “Nanowires for solid state lighting” , the Virtual Institute VH-VI-403 of theHelmholtz Association and by BMBF Proposal 05K10CHG “Coherent Diffraction Imaging andScattering of Ultrashort Coherent Pulses with Matter” in the framework of the German-Russiancollaboration “Development and Use of Accelerator-Based Photon Sources”.
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