We have combined two high transmission planar x-ray waveguides glued onto each other in a crossed geometry to form an effective quasi-point source. From measurements of the far-field diffraction pattern, the phase and amplitude of the near-field distribution is retrieved using the error-reduction algorithm. In agreement with finite difference field simulations (forward calculation), the reconstructed exit wave intensity distribution (inverse calculation) exhibits a full width at half maximum (FWHM) below 15 nm in both dimensions. Finally, holographic imaging is successfully demonstrated for the crossed waveguide device by translation of a lithographic test structure through the waveguide beam.
©2010 Optical Society of America
X-ray waveguides (WG) can be used to filter short wavelength radiation at nanoscale dimensions, replacing the function of macroscopic slits and pinholes used in conventional x-ray experiments. Waveguides can thus provide localized and highly coherent beams for diffraction studies at significantly reduced sample volume , as well as for coherent x-ray imaging and holography [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 down to about d ≃ 10 nm , below the values currently achieved by focusing optics such as compound 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  and at energies up to 10 keV sub-20 nm is demonstrated by Laue lenses  with an efficiency of ~ 30%. The efficiency of Fresnel zone plates decreases strongly at higher photon energies due to the limits in aspect ratios which can be fabricated, values on the order of 1% are reported . Focusing with compound refractive lenses are more efficient. Flux density gains of ~ 104 have been reached . For optimized high transmission waveguide design , simulation transmission can reach values above 90%, if the waveguide is illuminated coherently, i.e. by a plane wave. Furthermore, the coherence properties and cross section of the beam are decoupled from the primary source. And finally, over-illumination and stray radiation, often accompanying other forms of x-ray focusing (with far-field optics), is efficiently blocked by the cladding and 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 introduced a two-component cladding . An appropriate interlayer was placed between the guiding core and the high absorption cladding, resulting in significantly enhanced transmission. This was demonstrated with planar one-dimensional waveguides (1DWG). Contrarily, the vast majority of applications would need two-dimensional waveguides, demonstrated for the first time in , with however impractically low efficiencies. The main challenge is thus in fabrication of two-dimensionally confining waveguides (2DWG) with attractive specifications. In this work we have combined two high transmission 1DWG slices glued onto each other in a crossed geometry 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 deposition techniques, geometric parameters and material choices. Compared to channel waveguides prepared by electron lithography, smaller guiding layers and more complex layer systems become amenable. In contrast to the previously reported serial arrangement of two crossed 1DWG , the present device is much more compact, so that the horizontal and vertical focal planes nearly coincide.
2. Waveguide design and experimental methods
Figure 1 shows the schematics of device design and fabrication. An optical film layer sequence Ge/Mo[di=30 nm]/C[d=35 nm]/Mo[di=30 nm] was deposited on 3 mm thick Ge single crystal substrates (Incoatec GmbH, Germany). The interlayer thickness di=30 nm is designed to encompass the evanescent wave component of the propagating mode. A second so-called cap wafer (Ge, 440 µm thickness) was bonded onto the WG wafer by an alloying process to block the beam areas not impinging onto the waveguide entrance. Bonding was achieved by an In52Sn48 alloy (GPS Technologies GmbH, indalloy number 1E) ‘sandwiched’ between the Ni faces of the WG and cap wafers, under a pressure of p=67 mbar and heated up to T=250°C under vacuum conditions (sub-1 mbar). The resulting ‘sandwich’ sample was cut by a dicing saw (DISCO DAD 321) to the desired lengths l 1 = 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 exit faces. Therefore the waveguide slices were further treated using Focused Ion Beam (FIB) polishing (FEI, Nova 600 Nanolab). The FIB process also enables to correct the waveguide length up to sub-1 µm. Figure 1(c) shows the exit sides after FIB polishing, exhibiting the 35 nm thick guiding layer, the cladding, the interlayers and the bonding alloy. The index profile of the two-component waveguide is shown in Fig. 1(b) for the photon energy E=17.5 keV used in the experiment and for optical constants corresponding to ideal (bulk) electron densities. The C layer embedded in the high δMo = 5.82 × 10−6 Mo cladding forms a relatively deep potential well. At the same time, a relatively low βMo = 1.01 × 10−7 value of Mo reduces the absorption in the (interlayer) cladding and hence enables an increased transmission T. Note that at this energy, the low electron density C layer with βC = 2.77 × 10−10 contributes less than 2% to the effective absorption µeff. In other words C ‘acts’ essentially like a vacuum guiding layer.
The experiment was performed at the ID22NI undulator beamline of the third generation synchrotron facility ESRF, Grenoble. The beamline was operated in the so-called pink mode (no crystal monochromators) at a photon energy of E=17.5 keV, using the intrinsic monochromaticity 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 three translations and two rotations in the focal plane of the KB. The total flux exiting the waveguide was 6.4 × 108 cps measured by a single photon counting diode. The corresponding transmission of the c2DWG T = 0.052 is significantly lower than the value of Tsim = 0.904 obtained by simulation.
What are the reasons for this large discrepancy in theoretical and experimental transmission? We first consider the effect of partial coherence. Note that the simulation leading to the theoretical value assumes a coherent plane wave impinging on the waveguide, while the actual wave front in the KB focus may not be well described by an idealized plane wave. In fact, from the theory of coherence propagation (Gaussian shell model)  we can calculate the degree of coherence in front of the KB. In the vertical direction, the source size σvert = 30 µm, the source divergence σ′vert = 5 µrad and the 63 m distance from the undulator yield a degree of coherence of 0.18 at the source. Cutting the beam size by the 400 µm vertical slit size in front of the KB, the degree of coherence of the beam illuminating the KB increases to 0.33. In the horizontal direction, on the other hand, a virtual source realized by a 10 µm slit at a distance of 27 m downstream from the undulator, provides a nearly completely coherent beam over the 190 µm horizontal KB slit size. Thus in front of the KB, as in its focus, about 33% of the flux is coherent. Since the waveguide essentially accepts only the coherent flux , a factor of three in the discrepancy can thus be attributed to partial coherence. The remaining factor of about 5-6 must be due to other factor(s). The most likely reason is the finite depth of focus (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 second slice (1DWG-2) would already be displaced by 400 µm, corresponding to the thickness of 1DWG-1. We estimate the depth of focus by DOF ≤ 2zR, where zR = kσ2KB is the Rayleigh length, k the wavenumber, and σKB the lateral width of the focus, to DOFvert = 440 µm and DOFhorz = 266 µm, for the two directions, respectively. Since the DOF is likely to be smaller for a partially coherent beam (see coherence factor above), this may very well account for the 5-6 fold intensity ratio not explained by the coherence argument. Note that the equality in the expression for the DOF holds only in the limit of full coherence. Finally we stress that experimental and theoretical transmission values were found to be in good agreement for the given waveguides when illuminated by unfocused parallel beams .
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 on a Ru/SiC/SiN membrane was placed in the beam at a distance z 1 = 4.48 mm downstream from the c2DWG, as determined by an on-axis optical microscope. At 17.5 keV, the expected phase shift of a 500 nm Ta pattern is 0.40 rad, and the transmission is 0.93. A low noise direct photon counting pixel detector (Medipix, ESRF) with a pixel size of 55 µm and 256 × 256 pixels was used to image the in-line hologram at a distance z 1 + z 2 = 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 reciprocal space coordinates qx and qy after combination of 15 accumulations (exposure time 2 s each) with the detector shifted in the xy-plane to increase the field of view. A relatively uniform and flat intensity distribution in the center is framed by a characteristic arrangement of vertical and horizontal fringes. We attribute the fringes to interference of the wave ψxy guided by both 1DWG slices, with the wave components ψx ty and ψy tx. The latter terms denote the waves guided only by one of the 1DWG slices and attenuated by the other, with simultaneous diffraction from its planar interfaces. Figure 3(b) shows a two-dimensional representation of the simulated 1DWG-1 and 1DWG-2 far-field, obtained by multiplication of the respective simulations of 1DWG slices. The simulation is based on a finite-difference (FD) algorithm to obtain the simulated electromagnetic field distribution of the propagating modes inside the 1DWGs and the near-field distributions. The far-fields are obtained by a fast Fourier transformation (FFT) of the near-field distribution. The simulation does not account for the simultaneous diffraction at the planar interfaces of the 1DWGs but illustrates the form of the far-field around the maximum intensity. Note that only the combined thickness of the two crossed slices l 1 + l 2 is thick enough to 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 not impede holographic imaging, as shown below.
To corroborate this and to further characterize the near-field distribution in amplitude and phase, we have adapted an inverse scattering approach, where the near-field is reconstructed iteratively from the measured far-field pattern by use of the error-reduction algorithm (ER) . By application of a support constraint in the exit plane of the c2DWG with a cross section of 150 × 150 nm2 (smoothed by an error function), the near-field intensity in a virtual plane behind the c2DWG can be retrieved iteratively in intensity and phase. Figure 3(h) shows the reconstructed near-field intensity, obtained after Ni = 1000 iterations and an initial guess of a Gaussian amplitude with FWHM=35 nm. The reconstruction for Ni = 1000 did not show any significant 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 which can be considered as the effective confocal plane of the c2DWG. The high beam confinement is also in agreement with the autocorrelation function of the field at the exit surface of the c2DWG, calculated as the modulus of the FFT applied to the measured far-field intensity. The beam confinement in both directions due to the 2DWG effect is clearly evidenced by the center maximum, visible in Fig. 3(g). The nearly isotropic shape indicates that the c2DWG-source can 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 beating), as shown in Fig. 3(c) for a closeup of a 17.5 keV simulation . A Fourier transformation of the field with respect to the propagation direction z decomposes the simulated electromagnetic field into the guided modes which are shown in Fig. 3(d). Δk corresponds to the difference between the respective propagation constants βm and the wavenumber k in free space. In Fig. 3(d) only the ψ0 and the ψ 2 modes are visible, the ψ 1 mode is not exited by a plane wave impinging on the waveguide at normal incidence due to symmetry. The observed difference β 0 − β 2 = 2.28 × 10−4 nm−1 is in excellent agreement with the analytical result. Due to the periodically alternating field, a corresponding oscillating confinement of the fields depending on the propagation distance is obtained, as illustrated by dashed lines in Fig. 3(c), and the corresponding near-field profiles in Fig. 3(e). Thus, the exit wave field will depend on the exact length of the WG slice. The FWHM (full width at half maximum) of the simulated near-field intensity Δx [nm] and the corresponding far-field intensity Δq [Å−1] as a function of the waveguide length l are plotted in Fig. 3(f). Finally, Fig. 3(i) and (j) show the comparison of the FD simulation 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 ER reconstruction 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 a well defined test structure with controlled increase of spatial frequencies from the outer to the inner regions, to demonstrate holographic imaging with the c2DWG. The Siemens star was mapped by translation in the xy-plane at a defocus position z 1 = 4.48 mm, corresponding to a beam size of 6.72 µm (intensity FWHM) at the sample. A mesh of 15 × 15 scan points was recorded. For holographic reconstruction, the projection geometry used here was mapped onto parallel beam propagation by a variable transformation [20, ?]. Given the distance z 1 between source and sample and z 2 between sample and detector, parallel beam reconstruction by Fresnel-Kirchhoff back-propagation of the recorded intensity can be applied using the effective defocus (propagation) z = z 1 z 2/(z 1 + z 2). At the same time the hologram is magnified corresponding to the geometrical projection by a factor of M = (z 1 + z 2)/z 1. Figure 4 shows the image reconstruction after summing up the 15 × 15 scan points in real space and a line scan through 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, corresponding to |qx|,|qy| ≤ 0.0035 Å−1. The raw images were regridded by a factor of 2 for the image reconstruction and by a factor of 8 in the line scan. The image resolution was determined from a fit of phase step to an error function yielding a width of 70 nm (FWHM). Note that the magnification M = 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 is influenced by several factors, which limit the maximum accessible resolution for a given sample on different levels. On the most fundamental level the resolution is limited by the highest angle with respect to the optical axis, at which diffracted photons can be collected, i.e. the numerical aperture of the diffracted light cone. Depending on the total fluence incident on the sample, the sample scattering strength and the diameter of the waveguide exit wave the diffracted light cone can be larger or smaller than the waveguide exit cone.
On a less fundamental level, the resolution can be limited even further by the geometry of the experiment, i.e. the numerical aperture of the detector, and the geometric magnification factor. Due to steric constraints in the positioning stages, higher magnifications M (smaller z 1) as well as higher numerical apertures of the detector and thus a possibly higher resolution could not be tested. In order to estimate the resolution range that can be achieved with the present setup, i.e. leaving the detector area and pixel size, the sample scattering strength and the photon wavelength constant, we have simulated and reconstructed holograms with these four parameters identical to those in the experiment .
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 µm downstream 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 total number of 6.4 · 1012 photons, which could have been collected in about 104 s ≈ 2.8 h in the real experiment, during which a total number of 6.4 · 108 photons were exiting the waveguide per 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 hologram resulting from this simulation is presented in Fig. 5(a) with the corresponding holographic reconstruction shown in Fig. 5(b), indicating that line pairs with a half period of 13.5 nm are clearly resolved, with minor artifacts due to the direct holographic inversion of the data, which can be further improved by iterative methods. Figure 5(c) shows a holographic reconstruction from a simulated dataset obtained with the same set of parameters as before, except a total photon number of 6.4 · 108, corresponding to the photon count accumulated in one second on the detector area in the real experiment. Even here the line pairs are still visible, indicating a high robustness of the holographic reconstruction with respect to strong noise.
In summary, we have demonstrated the formation of a two-dimensional waveguide quasi-point source by combination of two crossed planar waveguide structures, each of a 35 nm guiding layer, 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 beam into the front face of the waveguide. Note that KB focusing and x-ray waveguide optics are both 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 the empty beam beyond the point source approximation, may lead to an increased resolution and an improved image quality. The use of support constraints and corresponding reconstruction algorithms, which are well established in coherent diffractive imaging, could also be combined with holographic reconstruction, as shown in , and applied to the current imaging scheme.
We thank the ID22 team for experimental support. We acknowledge financial support by Deutsche Forschungsgemeinschaft through SFB755 Nanoscale Photonic Imaging and the German Ministry of Education and Research under Grant No. 05KS7MGA.
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