X-ray beam compression by tapered waveguides

We have fabricated linear tapered waveguide channels ﬁlled with air and imbedded in silicon for the hard x-ray regime, using a processing scheme involving e-beam lithography, reactive ion etch-ing, and wafer bonding. Beam compression in such channels is demonstrated by coupling a pre-focused undulator beam into the channels, and recording the exit ﬂux and far-ﬁeld diffraction patterns. We achieved a compressed beam with a spot size of 16.48 nm (horizontal) (cid:1) 14.6 nm (vertical) near the waveguide exit plane, as determined from the reconstructed near-ﬁeld distribution, at an exit ﬂux which is eight times higher than that of an equivalent straight channel. Simulations indicate that this gain could reach three to four orders of magnitude for longer channels with tapering in two directions. V C 2015 AIP Publishing LLC . ¼ 334 nm and d ¼ 73 nm, and the taper angle h ¼ 26 l rad. The simulation illustrates the rapid decay

We have fabricated linear tapered waveguide channels filled with air and imbedded in silicon for the hard x-ray regime, using a processing scheme involving e-beam lithography, reactive ion etching, and wafer bonding. Beam compression in such channels is demonstrated by coupling a pre-focused undulator beam into the channels, and recording the exit flux and far-field diffraction patterns. We achieved a compressed beam with a spot size of 16.48 nm (horizontal) Â 14.6 nm (vertical) near the waveguide exit plane, as determined from the reconstructed near-field distribution, at an exit flux which is eight times higher than that of an equivalent straight channel. Simulations indicate that this gain could reach three to four orders of magnitude for longer channels with tapering in two directions. V C 2015 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4921095] X-ray waveguides (WGs) can be used to generate quasipoint sources for hard x-ray imaging. 1 These kinds of x-ray optics have first been realized as planar layered systems [one-dimensional waveguide (1DWG)], [2][3][4][5] and later also as two-dimensional confined channels (2DWGs), 6,7 as required for applications in coherent imaging. In contrast to most other nano-focusing optics such as Kirkpatrick-Baez (KB) mirrors, [8][9][10][11] Fresnel zone plates (FZP), 12 or compound diffractive lenses, 13 2DWGs deliver nanoscale x-ray beams with controllable spatial coherence. The quasi-point source of the WG exit emits smooth wavefronts which are ideally suited for holographic imaging. 1 At the same time, the absorption in the cladding assures extremely low background illumination, without contributions from higher diffraction orders as in FZP focusing or the pronounced tails of KBoptics. Hence, they also enable nanoscale diffraction studies at significantly reduced background. 14,15 In addition to spatial and coherence filtering, WGs can also concentrate the beam, based on geometric tapering. 16,17 For present imaging applications, WGs without beam compression are used and have to be combined with high gain pre-focusing optics to reach the flux needed for imaging applications. 7 In a state of the art realization of this scheme, a KB-beam of about 200-300 nm cross-section was coupled into WGs with an exit size of d ¼ 50 À 70 nm, delivering a flux in the range 1 10 8 À 10 9 ph/s. The smallest WG beam diameters down to 10 nm have been achieved 18 by crossing two high transmission 1DWGs slices to form an effective 2DWG. This crossed 2DWG system is limited, however, for reasons of its intrinsic astigmatism, as well as the absorption in the carbon guiding layer of the WGs, which becomes important for lower photon energy. Furthermore, the planar thin film fabrication technology is not compatible with variations in shape function. To overcome these limits, we have developed an alternative fabrication scheme based on e-beam lithography, reactive ion etching, and wafer bonding, which gives buried air or vacuum channels in silicon. 19 In this work, we use this fabrication approach to realize WGs with a taper function, which concentrate and filter the beam at the same time. This can either be used to match the entrance width D to the pre-focusing optics and to funnel more photons to a smaller exit width d, or in future extensions to make pre-focusing optics in front of the WG obsolete, at least for some applications. An earlier experimental realization of a tapered waveguide (tWG) 16 was demonstrated by an air-gap 1DWG, 20 while tapered 2DWG is addressed in this work. To demonstrate the concept of tWG beam compression in lithographically defined channels, we couple an undulator beam focused by KB-mirrors to 300 nm into a tWG and guide it further to d ' 75 nm at the exit.
For the design of the present experiment, we have used finite difference simulations (FDSs) to study the beam propagation in tWGs, based on the parabolic wave equation. 21 In addition, the propagation in x-ray WGs with straight channels [20][21][22] has been studied previously, showing that beam compression and filtering can be combined in a tWG. 23 Both Bergemann et al. 24 and Kukhlevsky et al. 25 have studied linearly tapered x-ray WGs or capillaries by analytical and numerical calculations, addressing the minimum beam width at the exit; while Panknin et al. 23 have also included non-linear tapering profiles and Bukreeva et al. 26 have studied corrugated WGs. Fig. 1(a) shows the wave optical simulation of guided wave propagation in a one-dimensional linear tWG (vacuum in silicon), as obtained by FDSs. The WG length L ¼ 5 mm was chosen for all WGs to ensure sufficient absorption of radiative modes at the given photon energy E ¼ 13.8 keV, with a calculated transmission T si ¼ 5:25 Â 10 À7 , correspondingly assuring an extremely low background signal. The slope of the WG interface and the associated linear tapering geometry are limited by the critical angle of total internal reflection h c ' ffiffiffiffiffi 2d p ¼ 2:25 mrad corresponding to the index of refraction n si ¼ 1 À d þ ib, with d ¼ 2:554 Â 10 À6 and b ¼ 2:047 Â 10 À8 , and guiding layer with n vac ¼ 1. In the example shown in Fig. 1 of the wave intensity in bulk silicon and the much weaker absorption in the tapered guiding layer. With wðx; zÞ denoting the field inside the WG and w 0 the incident plane wave, the intensity enhancement can be defined as the maximum of the normalized intensity I max ¼ max fjwðx; zÞj 2 =jw 0 j 2 g, where I max ¼ 11 for the simulated WG in Fig. 1(a). This enhancement appears towards the end of the WG as a maximum with a flux density one order of magnitude higher than at the WG entrance. In this manner, beam compression by the tWG is clearly evidenced. In order to demonstrate this concept experimentally, we have fabricated a WG chip with many parallel tWGs, and finally diced to L ¼ 5 mm. 19,27 The optical characterization of the WGs was performed at the GINIX (Göttingen Instrument for Nano-Imaging with X-rays) at the P10 beamline at the PETRA III synchrotron facility in Hamburg (DESY). Fig. 1(b) shows a schematic of the experimental setup, along with electron micrographs of the entrance and exit plane of a representative WG (labeled as "C" below). The WG entrance had a width of 334 nm and a depth of 69 nm; the WG exit had a width of 73 nm and a depth of 51 nm. The WG entrance was aligned in the focal plane of the KB-mirror system. The far-field images were recorded by a single photon counting pixel detector (Pilatus 300 K, Dectris) with 172 Â 172 lm 2 pixel size, placed 5.1 m behind the WG. For 13.8 keV, a KB-flux of I KB ¼ 7:32 Â 10 10 photons/s was recorded, and a focal spot size of 295 nm (FWHM) in horizontal and 181 nm (FWHM) in vertical direction, respectively. The far-field intensity distribution of the WG labeled "C" is shown in Fig. 1(c), with 2:79 Â 10 10 photons/s impinging onto its entrance, for a 1s accumulation. The flux at the exit was measured to I C WG ¼ 2:69 Â 10 8 photons/s. Next, the angular acceptance x of the WG, defined as the width (FWHM, Gaussian fit) of the exit intensity as a function of angle of incidence, was determined by rotating the WG around the y-axis by an angle a and measuring the integrated far-field intensity, as shown as solid squares in Fig. 1(d). In addition, numerical simulations for WGs of different taper angles h, labeled by A, B, and D with h ¼ 55, 31, and 10.2 lrad, respectively, are depicted by lines. Label "S" stands for a straight channel reference. Both curves are normalized to the maximum intensity and are in very good agreement. As shown in Fig. 1(e), the angular acceptance decreases with the taper angle xðhÞ as expected from wave optical simulations (Table I). In a geometric optical framework, this is explained by the increase in the number of reflections n and in particular, by the correspondingly increasing internal reflection angles h n at the interfaces between guiding layer and cladding, see also the sketch in Fig. 3(b). The beam can only be guided in the z direction, as long as the internal angle is smaller than the critical angle h n h c , else it is absorbed in the cladding.
A precise analysis of the beam confinement due to the WG geometry is obtained by reconstructing the complex wave field using the measured far-field intensities in the nearfield regime right behind the WG exit using the error reduction (ER) algorithm. 18 The reconstructed near-field intensity distribution and the corresponding reconstruction of the farfield pattern are shown in Fig. 2. As usual in iterative phase retrieval methods, the "reconstructed far-field" is computed from the reconstructed near-field intensity and can be tested against the measured far-field (see Fig. 1(c)), in order to check consistency and convergence of the phase retrieval process. The ER algorithm was initialized with the auto-correlation of the near-field, obtained by an inverse Fourier transform of the recorded far-field data, shown in Fig. 1(c). The support constraint was implemented using a Gaussian function with a width (FWHM) eight times larger than the size of the WG exit, in the respective direction. Fig. 2(a) shows the reconstructed far-field intensity pattern of a tWG ("C") on logarithmic scale, which is found in very good agreement with the measured data, as shown in Fig. 1(c  I. Entrance and exit size of the guiding layer, with the corresponding taper angle h, and the angular acceptance x for the tWGs labeled as "A," "B," "C," and "D" along with a straight reference channel "S." source with a width of 16.48 nm (x-axis) and depth of 14.60 nm (y-axis); the results for the tapered ("C") and straight ("S") channels are also tabulated in Table II. The transmission of the WG defined as the ratio of the intensity impinging onto the WG and the intensity exiting the WG is calculated using

WG label
for the simulation T Sim and the experiment T Exp , where A WG and A KB are the cross section areas of the WG entrance and the KB beam size. The measured flux of the tWG ("C") is I C WG ¼ 2:69 Â 10 8 cps and of the straight reference WG ("S") I S WG ¼ 3:34 Â 10 7 cps. The simulated transmission T Sim ¼ 8:9% of the tWG is 9.27 times higher than the experimental transmission T Exp ¼ 0:96%, which is affected by channel roughness and internal defects. The gain G, defined as the ratio of the intensity of the tWG over the straight reference, is calculated by yielding to G Sim ¼ 8.05 for the numerical simulation, and G Exp ¼ 6.89 for the experiment. This demonstrates clearly that a linear tWG can efficiently compress the beam, even if real structure effects such as roughness and waviness deteriorate the experimental values for T. In fact, after obtaining the results shown in Figs. 1 and 2, further improvements in the fabrication protocol regarding, in particular, wafer bonding under inert gas and changes in the exposure protocol were implemented and led to significantly better controlled channel side walls, and correspondingly higher T as detailed in elsewhere. 27 In order to further optimize tWGs in view of I max , the entrance width can be enlarged to collect more photons from the incoming beam, while the exit width can be tailored to the favored size and divergence for a specific experiment. In this way, the improved WG imaging could even be implemented without high gain focusing optics, such as the present KB-system. Fig. 3(a) illustrates the intensity enhancement I max as a function of the taper angle h, achieved by compressing a beam to an exit size of d ¼ 10 nm over a channel length L ¼ 2.4 mm, simulated for vacuum in Si at E ¼ 11 keV. We chose d ¼ 10 nm in view of holographic imaging experiments which require a small source size. The I max ðhÞ curve first steeply increases, as more photons are collected, but reaches a maximum at h max ¼ 0:124h c , before declining again, when the internal reflection angle increases above the critical angle h n ! h c towards the end of the channel, see (b). For illustration, we trace a ray for h max shown as a red line in (b). From h n ¼ ð2n À 1Þh h c , we can infer that the channel supports n ¼ 4 internal reflections. 27 For higher numbers of reflections n ! 5; h n ! h c lead to a leaking into the cladding material, see blue arrow in (b). However, a fraction of the x-ray beam is still reflected and guided inside the tWG (red arrow). The  gray dashed line in (b), (e), and (f) indicates the position where the fifth reflection occurs, corresponding to a tWG beam with width of 16.34 nm. This is in good agreement with the visual inspection of the leakage based on the logarithmically scaled intensity distribution, see (e) and (f). Despite the fact that the reflected angle exceeds the critical angle towards the end of the channel, the WG still reaches I max ¼ 66.7, underlining the substantial compression. For the same parameters, but in a two-dimensional taper geometry, the simulation yields I max ¼ 4206, as shown in (f).
In summary, tWGs have been designed based on FDSs, fabricated and characterized by experiments with synchrotron radiation. A pre-focused monochromatic x-ray beam was coupled into the front side of a tWG with an entrance width matching the focal spot size. The beam coupled into the WG was further compressed by propagation in the guiding channel with linear taper, resulting in a gain of about one order of magnitude over a straight channel waveguide. Simulations for a two-dimensional cone geometry indicate that gain values G ' 10 4 could be reached by a WG tapered in two dimensions, given future improvements in realizing tapered profiles in both directions orthogonal to the optical axis.
We thank Mike Kanbach for helping WG fabrication, Anna-Lena Robisch for phase retrieval, and Michael Sprung for the support at the P10 beamline. We gratefully acknowledge the German Research Foundation (DFG) for funding through Grant No. SFB 755/C1, and appreciate the financial support from the Ministry of Science and Technology of China (NSC 102-2917-I-564-062).