Sub-15 nm beam conﬁnement by two crossed x-ray waveguides

Sub-15 nm beam confinement by twocrossed x-ray waveguides

S. P. Kruger1,2, K. Giewekemeyer1, S. Kalbfleisch1, M. Bartels1,H. Neubauer1, and T. Salditt1,∗

1Institut fur Rontgenphysik, Universitat Gottingen, Friedrich-Hund-Platz 1,37077 Gottingen, Germany

[email protected]∗[email protected]

Abstract: We have combined two high transmission planar x-raywaveguides glued onto each other in a crossed geometry to form aneffective quasi-point source. From measurements of the far-field diffractionpattern, the phase and amplitude of the near-field distribution is retrievedusing the error-reduction algorithm. In agreement with finite difference fieldsimulations (forward calculation), the reconstructed exit wave intensitydistribution (inverse calculation) exhibits a full width at half maximum(FWHM) below 15 nm in both dimensions. Finally, holographic imaging issuccessfully demonstrated for the crossed waveguide device by translationof a lithographic test structure through the waveguide beam.

© 2010 Optical Society of America

OCIS codes: (340.7440) X-ray imaging; (110.7440) X-ray imaging

References and links1. S. Di Fonzo, W. Jark, S. Lagomarsino, C. Giannini, L. De Caro, A. Cedola, and M. Muller, “Non-destructive

determination of local strain with 100-nanometre spatial resolution,” Nature 403, 638–640 (2000).2. S. Eisebitt, J. Luning, W. F. Schlotter, M. Lorgen, O. Hellwig, W. Eberhardt, and J. Stohr, “Lensless imaging of

magnetic nanostructures by X-ray spectro-holography,” Nature 432, 885–888 (2004).3. C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-Based Off-Axis Holography with Hard X-Rays,” Phys. Rev.

Lett. 97, 254801 (2006).4. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray

fields,” Nat Phys 2, 101–104 (2006).5. C. Bergeman, H. Keymeulen, and J. F. van der Veen, “Focusing X-Ray Beams to Nanometer Dimensions,” Phys.

Rev. Lett. 91, 204801 (2003).6. A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N.

Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, B. Weckert, S. Schoder, M. Burghammer, and C.G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96,091102–3 (2010).

7. O. Hignette, P. Cloetens, W.-K. Lee, W. Ludwig, and G. Rostaing, “Hard X-ray microscopy with reflectingmirrors status and perspectives of the ESRF technology,” J. Phys. IV France 104, 231–234 (2003).

8. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatialresolution better than 15 nm,” Nature 435, 1210–1213 (2005).

9. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K.Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nmbarrier in hard-X-ray focusing,” Nat Phys 6, 122–125 (2010).

10. H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G.B. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. 92,221114 (2008).

11. W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, “Demonstration of 12 nm Resolution Fresnel ZonePlate Lens based Soft X-ray Microscopy,” Opt. Express 17, 17669–17677 (2009).

#123298 - $15.00 USD Received 25 Jan 2010; revised 18 Mar 2010; accepted 27 Apr 2010; published 8 Jun 2010(C) 2010 OSA 21 June 2010 / Vol. 18, No. 13 / OPTICS EXPRESS 13492

12. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L.Vincze, A. van der Hart, and M. Kuchler, “Hard x-ray nanoprobe based on refractive x-ray lenses,” Appl. Phys.Lett. 87, 124103–3 (2005).

13. T. Salditt, S. P. Kruger, C. Fuhse, and C. Bahtz, “High-Transmission Planar X-Ray Waveguides,” Phys. Rev. Lett.100, 184801–4 (2008).

14. F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, “Two-Dimensional X-ray Waveguides and PointSources,” Science 297, 230 (2002).

15. L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S.Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77,081408 (2008).

16. I. A. Vartanyants and A. Singer, “Analysis of Coherence Properties of 3-rd Generation Synchrotron Sources andFree-Electron Lasers,” (2009).

17. Strictly speaking, only a mono-modal waveguide acts as a perfect coherence filter. However, numerical simula-tions of the coupling process show that even for a waveguide with three modes, coherence is already significantlyfiltered (M. Osterhoff et al., unpublished).

18. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).19. C. Fuhse and T. Salditt, “Finite-difference field calculations for one-dimensionally confined X-ray waveguides,”

Physica B: Condensed Matter 357, 57–60 (2005).20. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-

contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).21. C. Fuhse, “X-ray waveguides and waveguide-based lensless imaging,” Ph.D. thesis, University of Gottingen

(2006).22. A minor difference with respect to one of the four parameters was the following: The detector area used for the

reconstruction shown in Fig. 4 was 256×241 pixels, for the simulation we have used a square area of 248×248pixels with the same pixel size as in the experiment.

23. L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in x-ray coherentFresnel projection-geometry diffraction,” Appl. Phys. Lett. 90, 041105 (1982).

1. Introduction

X-ray waveguides (WG) can be used to filter short wavelength radiation at nanoscale dimen-sions, replacing the function of macroscopic slits and pinholes used in conventional x-ray ex-periments. Waveguides can thus provide localized and highly coherent beams for diffractionstudies at significantly reduced sample volume [1], as well as for coherent x-ray imaging andholography [2, 3, 4]. Depending on the materials employed for the guiding and cladding layers,waveguides are in principle capable to deliver beams with two-dimensional cross sections downto about d ' 10 nm [5], below the values currently achieved by focusing optics such as com-pound refractive lenses, mirrors and Fresnel zone plates [6, 7, 8]. At x-ray energies of 20 keV,even higher beam confinement in one dimension is demonstrated by multilayer mirrors [9] andat energies up to 10 keV sub-20 nm is demonstrated by Laue lenses [10] with an efficiency of∼ 30%. The efficiency of Fresnel zone plates decreases strongly at higher photon energies dueto the limits in aspect ratios which can be fabricated, values on the order of 1% are reported[11]. Focusing with compound refractive lenses are more efficient. Flux density gains of ∼ 104

have been reached [12]. For optimized high transmission waveguide design [13], simulationtransmission can reach values above 90%, if the waveguide is illuminated coherently, i.e. by aplane wave. Furthermore, the coherence properties and cross section of the beam are decoupledfrom the primary source. And finally, over-illumination and stray radiation, often accompany-ing other forms of x-ray focusing (with far-field optics), is efficiently blocked by the claddingand cap layers, since the radiation in the near-field is confined to ' d .

To optimize the transmission and to minimize absorption losses, we have recently introduceda two-component cladding [13]. An appropriate interlayer was placed between the guiding coreand the high absorption cladding, resulting in significantly enhanced transmission. This wasdemonstrated with planar one-dimensional waveguides (1DWG). Contrarily, the vast major-ity of applications would need two-dimensional waveguides, demonstrated for the first time in[14], with however impractically low efficiencies. The main challenge is thus in fabrication of


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Fig. 1. (a) Schematic of the two crossed waveguide. (b) Profiles of the real and the imag-inary parts of the index of refraction n = 1− δ + iβ , calculated for a photon energyE=17.5 keV. Transmission of the guided modes in the C guiding layer is enhanced bythe high δ but relatively low β of Mo. (c) The scanning electron microscopy (SEM) image(magnification 52.85 kx) shows the Mo/C/Mo layers encompassed by highly absorbingGe and a In52Sn48 alloy which acts as the bond material to an additional Ge cap wafer. (d)The thicknesses of the guiding layer and the interlayers are clearly identified in the highresolution SEM image (magnification 200 kx).

two-dimensionally confining waveguides (2DWG) with attractive specifications. In this workwe have combined two high transmission 1DWG slices glued onto each other in a crossedgeometry to form an effective two-dimensional quasi-point source for holographic imaging.Important advantages of this scheme are the compatibility with a wide range of thin layer depo-sition techniques, geometric parameters and material choices. Compared to channel waveguidesprepared by electron lithography, smaller guiding layers and more complex layer systems be-come amenable. In contrast to the previously reported serial arrangement of two crossed 1DWG[15], the present device is much more compact, so that the horizontal and vertical focal planesnearly coincide.

2. Waveguide design and experimental methods

Figure 1 shows the schematics of device design and fabrication. An optical film layer sequenceGe/Mo[di=30 nm]/C[d=35 nm]/Mo[di=30 nm] was deposited on 3 mm thick Ge single crystal


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substrates (Incoatec GmbH, Germany). The interlayer thickness di=30 nm is designed to en-compass the evanescent wave component of the propagating mode. A second so-called capwafer (Ge, 440 µm thickness) was bonded onto the WG wafer by an alloying process toblock the beam areas not impinging onto the waveguide entrance. Bonding was achieved by anIn52Sn48 alloy (GPS Technologies GmbH, indalloy number 1E) ’sandwiched’ between the Nifaces of the WG and cap wafers, under a pressure of p=67 mbar and heated up to T =250◦C un-der vacuum conditions (sub-1 mbar). The resulting ’sandwich’ sample was cut by a dicing saw(DISCO DAD 321) to the desired lengths l1 = 400 µm (1DWG-1) and l2 = 207 µm (1DWG-2), used as the horizontal and vertical components of the crossed two-dimensional waveguide(c2DWG), respectively. The cutting process led to smearing of material at the entrance and exitfaces. Therefore the waveguide slices were further treated using Focused Ion Beam (FIB) pol-ishing (FEI, Nova 600 Nanolab). The FIB process also enables to correct the waveguide lengthup to sub-1 µm. Figure 1(c) shows the exit sides after FIB polishing, exhibiting the 35 nmthick guiding layer, the cladding, the interlayers and the bonding alloy. The index profile ofthe two-component waveguide is shown in Fig. 1(b) for the photon energy E=17.5 keV used inthe experiment and for optical constants corresponding to ideal (bulk) electron densities. The Clayer embedded in the high δMo = 5.82× 10−6 Mo cladding forms a relatively deep potentialwell. At the same time, a relatively low βMo = 1.01×10−7 value of Mo reduces the absorptionin the (interlayer) cladding and hence enables an increased transmission T . Note that at thisenergy, the low electron density C layer with βC = 2.77×10−10 contributes less than 2% to theeffective absorption µe f f . In other words C ’acts’ essentially like a vacuum guiding layer.

The experiment was performed at the ID22NI undulator beamline of the third generation syn-chrotron facility ESRF , Grenoble. The beamline was operated in the so-called pink mode (nocrystal monochromators) at a photon energy of E=17.5 keV, using the intrinsic monochromatic-ity of the undulators and the bandpass of the multilayer Kirkpatrick-Baez (KB) mirror system.The KB focal spot size was Dhorz = 129 nm (FWHM) in the horizontal and Dvert = 166 nm(FWHM) in the vertical direction, respectively. The c2DWG was aligned in terms of threetranslations and two rotations in the focal plane of the KB. The total flux exiting the waveguidewas 6.4× 108 cps measured by a single photon counting diode. The corresponding transmis-sion of the c2DWG T = 0.052 is significantly lower than the value of Tsim = 0.904 obtained bysimulation.


What are the reasons for this large discrepancy in theoretical and experimental transmis-sion ? We first consider the effect of partial coherence. Note that the simulation leading to thetheoretical value assumes a coherent plane wave impinging on the waveguide, while the actualwave front in the KB focus may not be well described by an idealized plane wave. In fact, fromthe theory of coherence propagation (Gaussian shell model) [16] we can calculate the degreeof coherence in front of the KB. In the vertical direction, the source size σvert = 30 µm, thesource divergence σ ′vert = 5 µrad and the 63 m distance from the undulator yield a degree ofcoherence of 0.18 at the source. Cutting the beam size by the 400 µm vertical slit size in frontof the KB, the degree of coherence of the beam illuminating the KB increases to 0.33. In thehorizontal direction, on the other hand, a virtual source realized by a 10 µm slit at a distanceof 27 m downstream from the undulator, provides a nearly completely coherent beam over the190 µm horizontal KB slit size. Thus in front of the KB, as in its focus, about 33% of theflux is coherent. Since the waveguide essentially accepts only the coherent flux [17], a factorof three in the discrepancy can thus be attributed to partial coherence. The remaining factorof about 5-6 must be due to other factor(s). The most likely reason is the finite depth of fo-cus (DOF), which must be compared to the thickness of the waveguide slices. If, for example,the first vertically oriented slice (1DWG-1) was exactly in the focus, the entrance of the sec-ond slice (1DWG-2) would already be displaced by 400 µm, corresponding to the thicknessof 1DWG-1. We estimate the depth of focus by DOF ≤ 2zR, where zR = kσ2

KB is the Rayleighlength, k the wavenumber, and σKB the lateral width of the focus, to DOFvert = 440 µm andDOFhorz = 266 µm, for the two directions, respectively. Since the DOF is likely to be smallerfor a partially coherent beam (see coherence factor above), this may very well account for the5-6 fold intensity ratio not explained by the coherence argument. Note that the equality in theexpression for the DOF holds only in the limit of full coherence. Finally we stress that exper-imental and theoretical transmission values were found to be in good agreement for the givenwaveguides when illuminated by unfocused parallel beams [13].

For demonstration of holography, in the next step a high resolution chart (NTT-AT, Japan,model # ATN/XRESO-50HC) consisting of a 500 nm thick nanostructured tantalum layer ona Ru/SiC/SiN membrane was placed in the beam at a distance z1 = 4.48 mm downstreamfrom the c2DWG, as determined by an on-axis optical microscope. At 17.5 keV, the expectedphase shift of a 500 nm Ta pattern is 0.40 rad, and the transmission is 0.93. A low noise directphoton counting pixel detector (Medipix, ESRF) with a pixel size of 55 µm and 256×256 pixelswas used to image the in-line hologram at a distance z1 + z2 = 3.09 m from the waveguide(positioned in the KB focal plane), as sketched in Fig. 2.

3. Simulations and experimental results

Figure 3(a) shows the measured far-field pattern of the c2DWG as a function of the two recip-rocal space coordinates qx and qy after combination of 15 accumulations (exposure time 2 seach) with the detector shifted in the xy-plane to increase the field of view. A relatively uniformand flat intensity distribution in the center is framed by a characteristic arrangement of verticaland horizontal fringes. We attribute the fringes to interference of the wave ψxy guided by both1DWG slices, with the wave components ψx ty and ψy tx. The latter terms denote the wavesguided only by one of the 1DWG slices and attenuated by the other, with simultaneous diffrac-tion from its planar interfaces. Figure 3(b) shows a two-dimensional representation of the simu-lated 1DWG-1 and 1DWG-2 far-field, obtained by multiplication of the respective simulationsof 1DWG slices. The simulation is based on a finite-difference (FD) algorithm to obtain thesimulated electromagnetic field distribution of the propagating modes inside the 1DWGs andthe near-field distributions. The far-fields are obtained by a fast Fourier transformation (FFT) ofthe near-field distribution. The simulation does not account for the simultaneous diffraction at


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Fig. 3. (a) and (b) Measured and simulated far-field diffraction and pattern of the c2DWG,the intensity is encoded logarithmically in the colormap (I [cps] and I [arb. units]). (c)Simulated electromagnetic field intensity inside the 1DWG at 17.5 keV within a rangeof 221-261 µm in propagation direction z. (d) A Fourier transformation with respect to zof the simulated electromagnetic field in the 1DWG showing the guided modes. (e) Fieldintensities which correspond to the dashed lines in (c) illustrating 3-mode propagation.(f) FWHM of the simulated near-field distribution (top) and far-field distribution (bottom)as a function of the waveguide length l. (g) Autocorrelation of the measured far-field. (h)Reconstructed near-field intensity obtained from the error-reduction algorithm (see text). (i)and (j) Reconstructed intensity along with the simulated near-field intensity of the 1DWG-2and 1DWG-1, respectively.


the planar interfaces of the 1DWGs but illustrates the form of the far-field around the maximumintensity. Note that only the combined thickness of the two crossed slices l1 + l2 is thick enoughto completely block the beam, while a single slice exhibits measurable transmitted photon flux,facilitating alignment of the c2DWG. At the same time, the finite 1DWG contributions do notimpede holographic imaging, as shown below.

To corroborate this and to further characterize the near-field distribution in amplitude andphase, we have adapted an inverse scattering approach, where the near-field is reconstructediteratively from the measured far-field pattern by use of the error-reduction algorithm (ER)[18]. By application of a support constraint in the exit plane of the c2DWG with a cross sectionof 150× 150 nm2 (smoothed by an error function), the near-field intensity in a virtual planebehind the c2DWG can be retrieved iteratively in intensity and phase. Figure 3(h) shows thereconstructed near-field intensity, obtained after Ni = 1000 iterations and an initial guess of aGaussian amplitude with FWHM=35 nm. The reconstruction for Ni = 1000 did not show anysignificant differences with respect to shorter and longer runs, e.g. Ni = 10 or Ni = 10000 ,underlining the rapid convergence. The ER reconstruction always yields a flat exit wavefront(no curvature). The reconstructed near-field must thus be associated with a virtual plane whichcan be considered as the effective confocal plane of the c2DWG. The high beam confinementis also in agreement with the autocorrelation function of the field at the exit surface of thec2DWG, calculated as the modulus of the FFT applied to the measured far-field intensity. Thebeam confinement in both directions due to the 2DWG effect is clearly evidenced by the centermaximum, visible in Fig. 3(g). The nearly isotropic shape indicates that the c2DWG-sourcecan be described as quasi point-like. The full width of the autocorrelation function (FWHM)obtained by Gaussian fits was 14.2 nm and 17.9 nm for the vertical and horizontal direction,respectively.

Next, we have compared the reconstructed near-field distribution to finite-difference (FD)simulations of the 1DWG slices. A WG with a 35 nm C guiding layer supports three modes,leading to a periodically alternating field distribution by interference of the modes (mode beat-ing), as shown in Fig. 3(c) for a closeup of a 17.5 keV simulation [19]. A Fourier transformationof the field with respect to the propagation direction z decomposes the simulated electromag-netic field into the guided modes which are shown in Fig. 3(d). ∆k corresponds to the differ-ence between the respective propagation constants βm and the wavenumber k in free space. InFig. 3(d) only the ψ0 and the ψ2 modes are visible, the ψ1 mode is not exited by a plane waveimpinging on the waveguide at normal incidence due to symmetry. The observed differenceβ0− β2 = 2.28× 10−4 nm−1 is in excellent agreement with the analytical result. Due to theperiodically alternating field, a corresponding oscillating confinement of the fields dependingon the propagation distance is obtained, as illustrated by dashed lines in Fig. 3(c), and the cor-responding near-field profiles in Fig. 3(e). Thus, the exit wave field will depend on the exactlength of the WG slice. The FWHM (full width at half maximum) of the simulated near-fieldintensity ∆x [nm] and the corresponding far-field intensity ∆q [A−1] as a function of the wave-guide length l are plotted in Fig. 3(f). Finally, Fig. 3(i) and (j) show the comparison of the FDsimulation and the ER reconstructions for the vertical and horizontal direction, respectively.The width (FWHM) of the Gaussian fit to the near-field intensity distributions obtained by ERreconstruction is 9.2 nm and 9.6 nm, compared to 12.5 nm and 13.6 nm of the FD simulation,for the vertical and horizontal direction, respectively.

After characterization of the near- and far-field patterns, the Siemens star was used as awell defined test structure with controlled increase of spatial frequencies from the outer tothe inner regions, to demonstrate holographic imaging with the c2DWG. The Siemens starwas mapped by translation in the xy-plane at a defocus position z1 = 4.48 mm, correspondingto a beam size of 6.72 µm (intensity FWHM) at the sample. A mesh of 15×15 scan points


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was recorded. For holographic reconstruction, the projection geometry used here was mappedonto parallel beam propagation by a variable transformation [20, ?]. Given the distance z1 be-tween source and sample and z2 between sample and detector, parallel beam reconstructionby Fresnel-Kirchhoff back-propagation of the recorded intensity can be applied using the ef-fective defocus (propagation) z = z1z2/(z1 + z2). At the same time the hologram is magnifiedcorresponding to the geometrical projection by a factor of M = (z1 + z2)/z1. Figure 4 showsthe image reconstruction after summing up the 15×15 scan points in real space and a line scanthrough the phase distribution of the reconstructed image near the center of the Siemens star.Each hologram was reconstructed from the intensity in the center of the far-field, correspondingto |qx|, |qy| ≤ 0.0035 A−1. The raw images were regridded by a factor of 2 for the image recon-struction and by a factor of 8 in the line scan. The image resolution was determined from a fit ofphase step to an error function yielding a width of 70 nm (FWHM). Note that the magnificationM = 690 for the given defocus corresponds to an image pixel size of 80 nm (before regridding).

The spatial resolution of reconstructed holographic images obtained with the present setup isinfluenced by several factors, which limit the maximum accessible resolution for a given sampleon different levels. On the most fundamental level the resolution is limited by the highest anglewith respect to the optical axis, at which diffracted photons can be collected, i.e. the numericalaperture of the diffracted light cone. Depending on the total fluence incident on the sample, thesample scattering strength and the diameter of the waveguide exit wave the diffracted light conecan be larger or smaller than the waveguide exit cone.

On a less fundamental level, the resolution can be limited even further by the geometry ofthe experiment, i.e. the numerical aperture of the detector, and the geometric magnificationfactor. Due to steric constraints in the positioning stages, higher magnifications M (smallerz1) as well as higher numerical apertures of the detector and thus a possibly higher resolutioncould not be tested. In order to estimate the resolution range that can be achieved with the


Fig. 5. (a) Simulated normalized hologram of a lines-and-spaces (LS) pattern with the samescattering contrast as the sample used in the real experiment and a total photon count on thedetector 104 times higher than collected in the experiment per one second. Further simu-lation parameters are given in the main text. (b) Holographic reconstruction correspondingto subfigure (a). The LS pattern has a half-period of 13.5 nm. (c) Holographic reconstruc-tion corresponding to the hologram shown in subfigure (a), however now simulated witha total number of photons 104 times lower. Scale bars indicate 3 mm (a) and 50 nm (b/c).Colorbars indicate normalized intensity (a) and phase in rad (b/c).

present setup, i.e. leaving the detector area and pixel size, the sample scattering strength andthe photon wavelength constant, we have simulated and reconstructed holograms with thesefour parameters identical to those in the experiment [22].

In the simulated experiment, the geometry was optimized to a high spatial resolution, i.e.a pattern consisting of lines and spaces with a half period of 13.5 nm was placed 74.7 µmdownstream of the waveguide exit plane. The detector received nearly the full WG exit cone,which was modelled as a Gaussian beam with a waist full width at half maximum (FWHM)of 10 nm, at a distance of 1.37 m downstream of the WG exit plane and accumulated a totalnumber of 6.4 · 1012 photons, which could have been collected in about 104 s ≈ 2.8 h in thereal experiment, during which a total number of 6.4 · 108 photons were exiting the waveguideper second (see above). The resulting simulated average fluence on the sample was thus 1.2 ·1013 ph/µm2 with a geometrical magnification factor of M = 18394. The normalized hologramresulting from this simulation is presented in Fig. 5(a) with the corresponding holographicreconstruction shown in Fig. 5(b), indicating that line pairs with a half period of 13.5 nm areclearly resolved, with minor artifacts due to the direct holographic inversion of the data, whichcan be further improved by iterative methods. Figure 5(c) shows a holographic reconstructionfrom a simulated dataset obtained with the same set of parameters as before, except a totalphoton number of 6.4 · 108, corresponding to the photon count accumulated in one second onthe detector area in the real experiment. Even here the line pairs are still visible, indicating ahigh robustness of the holographic reconstruction with respect to strong noise.

4. Conclusion

In summary, we have demonstrated the formation of a two-dimensional waveguide quasi-pointsource by combination of two crossed planar waveguide structures, each of a 35 nm guidinglayer, in a compact geometry, with horizontal and vertical focus coinciding within 207 µm.A total flux in the waveguide beam of 6.4× 108 cps was achieved by focusing a KB beaminto the front face of the waveguide. Note that KB focusing and x-ray waveguide optics areboth essentially non-dispersive, so that broad bandpass (pink beam) experiments can be used.In the next step, advanced reconstruction schemes taking into account the phase fronts of theempty beam beyond the point source approximation, may lead to an increased resolution and


an improved image quality. The use of support constraints and corresponding reconstructionalgorithms, which are well established in coherent diffractive imaging, could also be combinedwith holographic reconstruction, as shown in [23], and applied to the current imaging scheme.

Acknowledgements

We thank the ID22 team for experimental support. We acknowledge financial support byDeutsche Forschungsgemeinschaft through SFB755 Nanoscale Photonic Imaging and the Ger-man Ministry of Education and Research under Grant No. 05KS7MGA.


Sub-15 nm beam conﬁnement by two crossed x-ray waveguides

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Transcript of Sub-15 nm beam conﬁnement by two crossed x-ray waveguides