1. Introduction

The high degree of spatial coherence required by modern lensless x-ray imaging techniques calls for suitable methods to quantify wavefront distortion and spatial coherence [1–3]. The mutual intensity function Γ or the complex degree of coherence needs to be quantified to control the coherence of the wavefront. To this end, powerful methods based on interferometry have been developed [4–7]. Most of these have been used almost exclusively on macroscopic scales, e.g. for collimated quasi-parallel beams. However, the most demanding high resolution imaging experiments today require x-ray beams focused to nanoscale focal spot sizes. In this work, we extend the methods of coherence characterization by the Talbot effect to nano-focused beams with highly curved divergent wavefronts behind the focus. We extract the equal-time mutual intensity function j(z,d) describing the phase correlations between two points separated by a distance d in a plane perpendicular to the optical axis a distance z > 0 behind the focus. We show that by use of an x-ray waveguide the partial coherent wavefront of a Kirkpatrick-Baez (KB) mirror system can be filtered to yield nearly fully coherent exit waves behind the waveguide at the relevant sample distances.

Based on coherent x-ray beams, lensless x-ray imaging offers the potential to overcome the resolution barrier associated with x-ray lens fabrication, in particular in the hard x-ray range. Coherent x-ray imaging can be grouped into two classes: (A) Coherent diffraction imaging (CDI) where the diffraction pattern is recorded in the (Fraunhofer) far-field [8], and (B) propagation imaging in projection geometry, based on the measurement of the (Fresnel) near-field [9–11]. For (A) quasi-plane wave illumination, and for (B) quasi-point source illumination is desired, but for both cases, coherent wavefronts are essential. While the need for efficient nano-focusing is obvious in projection propagation imaging, it is also essential for high resolution CDI, requiring high photon flux densities in the sample plane, and hence in most cases high gain focusing optics on the incidence side.

In the limiting case of full coherence, reconstruction of both the intensity and phase distribution of a nano-focused x-ray beam, in or around the focus position is achieved by iterative reconstruction algorithms and the experimentally accessible far-field intensity distribution. Reconstruction algorithms can be based on support constraints from simple error reduction [12] to more elaborate schemes [13, 14], or on the overlap constraints defining the class of so-called ptychographic algorithms [15]. In its advanced variants [16, 17], ptychography is capable to reconstruct the unknown complex-valued illumination function probed by scanning along with an generally also unknown object through the beam. This approach has been used to characterize nanoscale wavefronts [18, 19]. However, ptychographic wavefront reconstruction is based on the assumption of a fully coherent beam, and thus fails or is compromised in the case of partial coherence, at least in current implementations. We therefore turn to Talbot interferometry for the quantification of spatial coherence.

We have performed the experiments with the KB mirror system and setup for propagation imaging installed at the coherence beamline P10 of the new storage ring PETRA III at Hasylab, DESY [20]. Two diagnostic tools have been used for beam characterization: (i) an x-ray waveguide (WG), and (ii) a set of nanoscale gratings exhibiting the Talbot effect. Aside from diagnostic purposes, insertion of the WG can also be used to reduce the beam diameter from typical values of 200 nm (KB) to values below 15 nm [21]. The WG provides highly divergent and coherent exit beams for phase contrast propagation imaging [11, 22]. Importantly, the coherence properties and cross-section of the exit beam are decoupled from the primary source. According to the reduction in the number of modes, the coherence of the exit wave is significantly enhanced. In the first part of this work, before considering the coherence filtering effects, we simply use the small diameter D of the WG guiding layer as an ’ultra-narrow slit’ to scan the beam around the KB focus. In the last section, the characteristic mode structure of waveguides is exploited, using the waveguide as a coherence filter. Waveguide mode structure and transmission is calculated using finite-difference (FD) simulations [23–25], based on the parabolic wave equation [26]. The paper is organized as follows. After this introduction, the experimental setup and parameters are presented, followed by a section on the focal intensity distribution. Section 4 contains the Talbot experiment and analysis. Section 5 studies the effect of partial coherence on the propagation image of a simple test pattern, allowing for the comparison to an analytical expression. Section 6 presents a brief example of coherence filtering by a waveguide, before the paper closes with a summary and conclusions.

2. Experimental setup and parameters

Figure 1 shows the schematic of the experiment carried out at the holographic imaging end-station of the coherence beamline P10 of PETRA III. The source in low β configuration consists of a 5 m long undulator with a period of 29 mm, with a source size of 36μm × 6μm (1σ, horz. × vert.). The beamline was operated in monochromatic mode using a Si(111) double crystal monochromator, positioned at 35 m behind the source. At a distance of 87.7m the x-ray beam is focused by two elliptically shaped mirrors (fixed shape) in Kirkpatrick-Baez (KB) geometry, contained in a vaccuum vessel. The mirrors are configured to a fixed elliptical shape, corresponding to an incidence angle of α = 4.00 mrad and α = 4.05 mrad, as well as a focal length of 305 mm and 200 mm, for the vertically (v) focusing mirror (WinlightX, France) and the horizontally (h) focusing mirror (JTEC Corporation, Japan), respectively. The mirrors consist of Pd-covered silicon (v-mirror) and Pd-covered silica (h-mirror), respectively. For the h-mirror, which was polished by elastic emission machining [27], the maximum deviation from the ideal elliptical shape (uncoated) was independently measured to 3.1 nm (peak-to-valley) [20]. The entrance slits in front of the KB-system were set equal or larger than the geometric acceptance of the mirrors of 0.4 mm, i.e. the full mirror length was illuminated by the beam.

 

Fig. 1 Experimental setup: A parallel hard x-ray wave front is focused by the KB mirror system. A Talbot grating is illuminated at a (defocus) distance z ₁ behind the KB focal plane, and the magnified projection Talbot image is recorded by an area detector positioned in the far-field at a distance z ₁ + z ₂. The schematic shows the example of a grating period a = 1 μm with 500 nm lines and spaces (l&s) recorded by a pixel detector with 172 μm (Pilatus, Dectris), while smaller grating periods corresponding to 200 nm l&s, and 50 nm l&s required a CCD (LCX, RoperScientific) with smaller pixel size of 20μm.

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The KB near-field intensity distribution was characterized by scanning x-ray waveguides through the KB focus, using a miniaturized fully motorized goniometer with optical encoders (Attocube), with three translations in xyz, and two rotations along two directions orthogonal to the optical axis. Alignment of the waveguide as well as the Talbot gratings was facilitated by use of two on-axis optical microscopes whose focal planes coincide with the KB. The two microscopes are directed to the focus with a ’downstream’ and an ’upstream view’, respectively. For x-ray beam path, the ’downstream view’ microscope (Optique Peter) has a hole in the deflecting mirror, while the ’upstream view’ microscope observing the sample at a fixed working distance of 32 mm, has a drilled objective (Bruker AXS), providing in-situ inspection. The sample stage is equipped with an air-bearing rotation (Micos) for ultra-high precision turns needed for nano-tomography. On top of the rotation, a group of xyz piezos (Physik Instrumente) is used for aligning the sample in the axis of rotation. Additional xyz stages (Micos) below the rotation are used for aligning the rotation axis in the x-ray beam and for distance variation between the waveguide and the sample. The detectors were placed at a distance of z ₁ + z ₂ = 5.29m behind the sample, see Fig. 1. Along with a direct-illumination CCD (LCX, Princeton Instruments) used primarily for the data analyzed here, a scintillator CCD (SCX, Princeton Instruments) and a single photon counting pixel detector (Pilatus 300K, Dectris) was available, and used to cross-check the far-field signal and/or to determine the integral flux.

Waveguides: The beam is coupled into the waveguide by the so-called front-coupling scheme [22]. Here we use two new waveguide types. (A) Lithographic channel waveguides fabricated by wafer bonding (bonded-2DWG), yielding air filled channels [11, 20], which can be cut to a length according to the requirements of the photon energy E. An array of (well separated) waveguide channels is first exposed into an e-beam resist layer on a Si wafer by electron beam lithography. After development this resist layer acts as an etching mask for the reactive ion etching (RIE) of the waveguide channels. After the removal of the etching mask, a second (cap) wafer is bonded onto the first (structured) wafer, forming the waveguide chip. The entrances and exits on the cleaved edges of the waveguide chip are then polished by Focused Ion Beam (FIB) polishing (FEI, Nova 600 Nanolab). The cross-section of the channels can be varied, typically in the range between 30 nm×20 nm and approx. 120 nm×60 nm, depending on the specific application of the experiment. (B) For smaller waveguide dimensions, an arrangement of crossed high-transmission planar x-ray waveguides is used similar to [28]. These waveguides are made of a transmission optimized sequence of sputtered thin films, with amorphous C as the guiding layer [21,29], even for very small layer thicknesses. For the specific guide used here, an optical film layer sequence Ge/Mo[d _i=30 nm]/C[D=35 nm]/Mo[d _i=30 nm]/Ge was deposited on a 3 mm thick Ge single crystal substrate (Incoatec GmbH, Germany). 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. The resulting ‘sandwich’ sample was cut by a dicing saw to the desired lengths l = 200 μm, and cleaned by FIB.

Simulation of beam propagation and coherence: In order to simulate the focusing properties and near-field intensities of the KB-focus, wave-optical propagation was carried out numerically. The undulator source at 7.9 keV with the nominal source size as given above was discretized into a set of independent emitters, with Fresnel-Kirchhoff integrals used to calculate the propagation from source to mirror, as well as from mirror to the focus. To this end, the measured height profile of the mirror is taken into account. Partial coherence is modeled by averaging stochastic realizations emitted by the virtual point-sources. The equal time complex degree of coherence $j_{1,2}\hspace{0.17em}=\hspace{0.17em}〈u_1{u}_2^{*}〉/\hspace{0.17em}\sqrt{I_1{I}_2}$ between two points (1,2) with the associated complex-valued wave amplitudes u _1,2 and intensities I _1,2 was computed from a random phase superposition u(x) = Σ_n w _n c _n u _n(x), with equally distributed) phases and (real valued) weight factors w _n adjusted to the Gaussian envelope of the source. For each basis field u _n, free space propagation and reflection was carried out, and an ensemble average of 8000 stochastic realizations of random field distributions was evaluated in the ’detection’ plane of interest, modeling the time-averaging measurement process. The reflection process was modeled by the Fresnel equations, applied locally to a discretized elliptically shaped curve (in 1+1 dimension) including the measured surface deviation from the perfect ellipse, using 32000 data points for integration, randomly distributed on the curve (surface). Details of this method will be described elsewhere (Osterhoff and Salditt, in preparation). Results for the near-field intensity distribution around the focal point are shown in Fig. 2(e) for the case of the horizontally focusing mirror (JTEC) on logarithmic scale, simulated for the experimental photon energy, horizontal source size, and distances, as given above.

 

Fig. 2 Beam width (FWHM) along the (a) horizontal (x) and (b) vertical (y) direction, as a function of z along the optical axis, as measured by scanning the waveguide through the focus, along with the Fresnel-Kirchhoff simulations (solid black line) and a fit to a Gaussian beam profile (solid red line), convolved with a residual width (see text). (c) Schematic of the waveguide scan. (d) Focus intensity profile along the vertical direction in the focal plane after iterative alignment, along with a Lorentzian fit (solid red line). Side minima and maxima are observed around the central focus. Two-dimensional (e) simulated and (f) experimental intensity distribution in the focal region (vertical plane), after logarithmic color encoding (see colorbar).

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Grating and test structures: For coherence measurements by the Talbot effect, a high resolution chart (NTT-AT, Japan, model # ATN/XRESO-50HC) consisting of a 500 nm thick nano-structured tantalum layer on a Ru/SiC/SiN membrane was placed in the beam at a distance z ₁ downstream from the KB focus, as determined by the on-axis optical microscope. For this study, we used three different gratings on this test structure, namely 500 nm lines and spaces, 200 nm lines and spaces, and 50 nm lines and spaces, respectively. Note that for example 500 nm lines and spaces (l&s), means 500 nm lines followed by 500 nm spaces, i.e. the given value is half the grating period. At 7.9 keV, the calculated phase shift of a 500 nm Ta pattern is ϕ = 0.830 rad, and the intensity transmission is T = 0.871. In addition to the grating measurements, the hologram of a double slit test structure was analyzed, based on analytical forward calculations and least square fitting. The double slit test structure was fabricated by FIB etching (FEI, Nova 600 Nanolab). First, a thin Au layer was deposited by e-beam deposition on a 200 nm SiN foil (Silson), coated with a small 5 nm Ti adhesion layer. The Au film thickness was determined to 35 nm by profilometry (Veeco Dektak 6M). The Au was then etched away in a double slit pattern, with a slit length of 4 μm, a width w = 1.6 μm and a distance between the two slits of 6 μm. After fabrication, the geometric layout was checked by e-beam microscopy.

3. Intensity distribution of the KB focus

Figures 2(a) and 2(b) show the results of the x-ray waveguide scans through the focal plane of the KB to characterize the near-field intensity distribution. By scanning one-dimensional (planar) waveguides through the focal spot after careful angular alignment, they are used as a direct probe for the beam width, see the schematic in (c). The thickness of the guiding layer was D = 35 nm in the horizontal and D = 30 nm in the vertical direction. The lateral and vertical profile of the focal intensity distribution can then be analyzed as a function of z at and around the focal plane, to determine the spot sizes and the field of depth. In (a,b), the beam cross section (FWHM) are plotted as a function of z after batch fitting of scans along the y (vertical) and z (horizontal) directions for each z to a Gaussian peak profile. The over-estimated error bar values drawn correspond to ±50 nm, and may considered an upper bounds for the uncertainties, including systematic errors in parameter initialization during the batch fit. Single profiles in particular in the focal plane have been analyzed by hand with individual initialization of parameters, and fitting range, resulting in much smaller errors. Along with the experimental results (circles), the values determined from the Fresnel-Kirchhoff simulations (solid black line) and an empirical fit to a Gaussian beam profile (solid red line) are shown. The simulations of beam propagation take into account the geometrical parameters (undulator source size, distances) and measured height profiles of the mirrors, as shown in Fig. 2(e), for the case of the vertical direction. The resulting focal distribution is comparable to the experimentally measured 2D intensity distribution, as shown in (f), obtained from a series of waveguide scans (similar to those shown in (a,b)), after normalization to the peak area for each scan. The simulation and the experimental results are in qualitative agreement, as far as beam width and depth of focus is concerned, but show differences concerning the exact lineshapes and side maxima. The functional form of the empirical fit is based on a convolution of a diffraction limited beam with a Gaussian (FWHM value ζ) taking into account finite source size and all other spurious beam broadening effects

4. Coherence measured by Talbot effect

Following [1, 30], the normalized mutual coherence function also termed complex degree of coherence describes the correlation between two wave field u probed at two points (1) and (2), and at times t and t +τ, respectively, and can be written for stationary fields as

Before turning to the experimental procedure, let us briefly consider the expected spatial coherence properties based on the simulations. Figure 3 shows the results for the relevant field behind the KB focus in the horizontal plane, i.e. (a) a two-dimensional plot of the |j(z,y)|, as well as selected curves |j(y)| for z const., with the corresponding Gaussian fits. Finally, (c) shows the linear scaling of the full width at half maximum (FWHM) of the curves |j(y)| as a function of z, in other words the lateral coherence length ξ _FWHM. The linear scaling of the coherence length and the Gaussian functional form, support the use of a Gaussian Shell-model (GSM) for the coherence properties, as put forward in [3]. In the simulation, the actual undulator source was assumed to be completely incoherent, but a finite coherence ‘builds up by free space propagation. According to the GSM, the cross-spectral density W (or for the quasi-monochromatic the mutual intensity J) for any two points in a plane orthogonal to the optical axis separated by a distance $d\hspace{0.17em}=\hspace{0.17em}\sqrt{x_1^2\hspace{0.17em}-\hspace{0.17em}{x}_2^2}$ can be written as

 

Fig. 3 Simulation results. (a) Degree of coherence |j(x,z)| in the horizontal (xz) plane, evaluated for point (1) on the optical axis (0,z), and point (2) at (x,z). (b) The degree of coherence |j| as a function of x for selected constant values of z, along with a fit (red solid line) to a Gaussian. (c) The full width at half maximum ξ _FWHM of the Gaussian fits as a function of z shows linear scaling (solid red line), defining the (full) opening angle of the coherent beam cone ξ _FWHM /z ≃ 0.733 mrad, which is smaller than the 2 mrad (FWHM) angle of the full beam in the horizontal direction. In the vertical plane, simulations predict full coherence, as a result of the significantly smaller vertical undulator source size.

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We now turn to the Talbot experiment. Figure 4 gives illustrative examples of the Talbot effect in the magnification setup and the data analysis scheme for a grating with 200 nm lines and spaces (l&s), aligned vertically, i.e. in the case of probing the coherence properties in the horizontal plane. As the sample is scanned in z ₁, the visibility of the fringes oscillates between weak or no contrast as shown in (a) to high contrast as shown in (b). After vertical averaging over a region of interest (ROI), the spectral density of the profile is computed. The integrated intensity of the first order (|f _{m =1}|) is then evaluated as a function of z ₁. Since the magnification M also changes with z ₁, the position of the m = 1 moves linearly in the PSD curve. An automated peak search function was used to single out the m = 1 peaks. When z ₁ corresponds to a Talbot minimum, this procedure did not work well for the 200 nm l&s grating. In a refined procedure, the movement of the peak was analyzed by linear regression to predict the peak position, followed by intensity readout, see also Fig. 5(b), comparing the two procedures of simple peak search (open symbols) and of peak search based on regression (solid symbols). According to Talbot theory (Eq. (3) in [4]) the first order Fourier coefficient of the intensity is

 

Fig. 4 (a) Intensity distribution (linear gray scale) of the 200nm lines and spaces (l&s) at z ₁ = 8.4 mm recorded with the CCD detector, after empty beam correction (scale bar: 0.8 mm in the detector coordinates, corresponding to 1.3μm in the sample plane. (b) same as in (a) but for z ₁ = 8.5 mm, where high contrast is observed. (c) Intensity profile obtained by vertical averaging of (b) and subtraction of the mean. (d) Modulus of the Fourier transform of the profile (spectral density, PSD) as a function of inverse pixel, showing the strong m = 1 order.

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Fig. 5 (a) The magnitude of the first (m = 1) order Fourier component |f _{m =1}| of the 200 nm l&s grating (solid symbols), as a function of the defocus distance z ₁, computed by Fourier analysis of the recorded Fresnel patterns, along with a least-square fit to Eq. (5) (solid line). The intensity oscillates periodically as the grating position is scanned, with a period of the expected Talbot distance Z. (b) The same for the 500 nm l&s grating, with (solid symbols) and without (open symbols) a refined image analysis procedure (see text). As expected, the amplitude of the |f _{m =1}| oscillation is higher for the 500 nm l&s grating than for the 200 nm l&s grating.

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The experimental values of the m = 1 Fourier coefficient extracted from the Talbot scans are fitted to the equation

Figure 5(a) and 5(b) show the measured Talbot curves (symbols) along with the least-square fits (solid lines), for the 200 nm l&s / a = 400 nm grating, and for the 500nm l&s/ a = 1μm grating. The magnitude of the first order (m = 1) maximum in the power spectral density (PSD), as computed by Fourier analysis of the recorded Fresnel diffraction pattern, is plotted as a function of the defocus distance z ₁. The fitted Talbot distances for the two gratings of Z ₂₀₀ = 1.020 μm, and Z ₅₀₀ = 6.306 μm correspond well with the expected Talbot distances a ² /λ, within the experimental uncertainties. A small error δz ₁ of the nominal defocus z ₁ was allowed as a free parameter in the least-square fit. The resulting value δz ₁ = −0.62 mm and δz ₁ = −0.44 mm for the two gratings, respectively, show that the position of the test pattern as determined by the optical microscope was probably not exact. The focal plane of the optical microscope had initially been adjusted to the KB focus, but due to realignment after injection the focal plane of the KB can vary slightly from one ring filling to the next. Finally, the third and last free parameter was the fitted |j|-value associated with the amplitudes of the Talbot curves, which were evaluated to j ₂₀₀ = 0.415 ± 0.003, and j ₅₀₀ = 0.927 ± 0.007, respectively. This measured coefficient |j(z ₁ , d)| corresponds to the degree of coherence in the sample plane z ₁ for any two points separated by a lateral distance d = λz _eff /a. During the scan, z ₁ and d both increase linearly. For z ₁ ≪ z ₂, z ₁ ≃ z _eff, and a constant value |j| = const. is measured along the line forming an angle λ/a with respect to the optical axis. Note that the data sets could of course be fitted with models parameterizing a variation of j along the scan, but the amplitude of the Talbot effect proves constant on visual inspection, and the rather good agreement of the data with the three parameters motivates the conclusion that |j(z ₁ ,λz ₁ /a)| is constant. This conclusion is plausible, since the coherence length ξ (z ₁) increases linearly with z ₁ due to propagation. The reduced mean square sum is χ ² = 4.0 and χ ² = 0.51, for the two fits. The overall agreement is quite satisfactory of a three parameter fit, and precludes models with more parameters.

Next, the same gratings with l&s oriented horizontally were used in analogous scans to determine the |j| values in the vertical (yz)-plane (data not shown). Least-square analysis resulted in j ₂₀₀ = 0.473 ± 0.009, and j ₅₀₀ = 0.948 ± 0.007, for the 200 nm l&s and the 500 nm l&s grating, respectively. The |j| values in the vertical plane are therefore only slightly larger than in the horizontal plane. One would have expected a higher degree of coherence based on the smaller vertical (primary) source size, which leads to a nominally almost fully coherent illumination of the vertically focusing mirror. However, the presence of the double crystal monochromator in the vertical plane was observed to introduce vibrations, leading to a decease in coherence. This effect does not seem to be as systematic as the finite horizontal source size. Thus, apart from the reduced amplitude of the Talbot scans, which already disprove full coherence, the Talbot self image quality seems somewhat inferior and more irregular. In the Talbot scans, |f _{m =1} | never decreased to zero, and the quality of the fits was worse than in the horizontal direction. In any case, the vertical coherence length ξ _x(z ₁) was thus also smaller than the beam diameter.

Figure 6 compiles the results on |j(z ₁ , d)| in the xz-plane and illustrates the geometry of the probing scans. (a) shows the two experimental j values corresponding to the two gratings, as a function of normalized lateral distance X = (x/z). A one-parameter fit to a Gaussian of unit amplitude exp(–X ²/2Σ²) yields a normalized coherence length Σ = 0.30±0.02, or correspondingly a Full Width at Half Maximum (FWHM) ${\xi }_{FWHM}\hspace{0.17em}=\hspace{0.17em}2\sqrt{2ln(2)}\mathrm{\Sigma }z\hspace{0.17em}=\hspace{0.17em}0.71\hspace{0.17em}\cdot \hspace{0.17em}{10}^{-3}\hspace{0.17em}z$. The corresponding values for the vertical yz plane are Σ = 0.33 ± 0.02, and ξ _FWHM = 0.78 · 10^–3 z, respectively. Along with the experimental values, the resulting |j(X)| curve of the coherence simulations is shown (thin black line), as calculated for the plane z ₁ = 10 mm, scaled to the X = x/z coordinate, showing good agreement. In (b), The function |j(z ₁ , x)| = exp(–(x/z ₁)²/2Σ²) is shown as a contour plot, with selected contour levels. The two oblique lines (white dotted lines on both sides of the optical axis ) in the (xz)-plane indicate the measurement points probed by the two defocus scans. The solid white lines indicate the divergence of the KB-beam in the vertical (xz) plane. Finally, the lateral coherence length ξ _FWHM(z ₁) is plotted in (c), for the experimental values (open circles) with errors corresponding to the Σ fit, along with the numerical coherence simulations (solid red squares), and the analytical results for an equivalent fully incoherent source at distance z ₁. The only parameter in this expression is the ’effective’ horizontal source size, which best approximates the experimental results when set to s = 194.6 nm (FWHM). In other words, all observations at the respective distances probed experimentally are equivalent to the coherence properties of an incoherent source with the above cross section. The corresponding treatment for the vertical plane yields an effective vertical source size of s = 179.0 nm (FWHM). In conlcusion, the coherent opening angle of the beam was determined to 0.71 mrad in the horizontal and 0.78 mrad in the vertical plane, which divided by the total beam opening angle, 1.97 mrad and 1.15 mrad, fixes the coherence parameter q = 0.36, and q = 0.68, for the two planes, respectively. Throughout this treatment we have assumed that the focusing, propagation and coherence properties factorize in the two orthogonal directions. The results are in excellent agreement in the horizontal plane, but deviate from expectation in the vertical plane, for which q ≥ 1 was expected, since the vertical mirror should have been illuminated coherently due to the smaller vertical undulator source.

 

Fig. 6 (a) Degree of coherence |j(z ₁ , d)| in the xz-plane as a function of normalized lateral distance X = (x/z), with the two experimental j values corresponding to the two gratings (open squares), along with a one-parameter fit to a Gaussian of unit amplitude exp(–X ²/(2Σ²)), yielding a normalized coherence length Σ = 0.30± 0.02, or correspondingly a FWHM value o ${\xi }_{FWHM}\hspace{0.17em}=\hspace{0.17em}2\sqrt{2ln(2)}\mathrm{\Sigma }z\hspace{0.17em}=\hspace{0.17em}0.707\hspace{0.17em}\cdot \hspace{0.17em}{10}^{-3}\hspace{0.17em}z$. Along with the experimental values, the degree of coherence | j| from the simulations is shown (thin black line), as evaluated for the plane z ₁ = 10 mm, scaled to the X = x/z coordinate. The j values corresponding to the vertical direction (filled triangles) are also included for comparison. A corresponding Gaussian fit yields Σ = 0.33 ± 0.02 (not shown). (b) The function |j(z ₁ , x)| = exp(–(x/z ₁)²/(2Σ²)) as determined for the horizontal plane, shown as a contour plot, with selected contour levels, along with the two oblique lines (white dotted lines) in the (xz)-plane probed by the two defocus scans, and the edge of the beam (solid white lines) indicating the divergence of the KB-beam, here for the horizontal (xz) plane. (c) The lateral coherence length ξ _FWHM(z ₁) corresponding to the experimental values (open circles) (with errors from the Σ fit), and the numerical simulations (solid red squares), as well as the coherence length corresponding to a fully incoherent source of size s, observed at distance z ₁. The fitted slope corresponds to s = 194.6 nm (FWHM), see text.

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5. Holographic imaging of a double slit

Next, we will deduce information on the lateral coherence by analyzing the hologram of a well-defined test structure. The idea is to model the measured intensity profile by an analytical expression derived for full coherence, followed by Gaussian convolution to account for partial coherence. The convolution width should then be related to the coherence length ξ. The use of a simple double slit (DS) pattern was motivated by the fact that one can easily derive an analytical function for the Fresnel diffraction pattern, for arbitrary complex-valued transmission functions. For the present photon energy and film thickness, the sample primarily interacts by phase contrast. E = 7.9keV, a structure in 35 nm thick Au layer primarily interacts by phase contrast with an expected phase shift of ϕ = 0.069 rad and a transmission of T = 0.986. As described above, the DS had a width w = 1.6μm and a distance between the two slits of 2l = 6μm, adapted to the expected coherence length. To avoid the complications associated with adaptation of the phase reconstruction algorithms to partially coherent wave fields, we here take a forward-scattering based approach, i.e. we analyze the hologram, based on a simple analytical equation which can easily be derived for the DS pattern as a function of the detector coordinate x _D,

 

Fig. 7 (a) Near-field propagation image (linear gray scale) of a double slit pattern with predominant phase contrast after correction of dark current and empty beam (flat field). The DS is illuminated by the partially coherent and divergent KB beam. Scale bar denotes 0.8 mm in the detector plane, corresponding to 3.025μm in the sample plane. (b) Horizontal intensity trace showing the characteristic Fresnel oscillations (dark circles) with least-square fit (red line), see text.

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How to understand the convolution in simple terms, based on partial coherence corresponding to the finite source size? Taking a simple geometric relation for the expected width (FWHM) of the convolution function ζ = Δs z ₂ /z ₁ [32], and the source size Δs ≃ 200 nm (FWHM), a FWHM value of 2.64 pixel is expected, substantially smaller then the fit result of 4.03 pixel. One may be tempted to readily attribute the discrepancy to all other smearing effects aside from coherence, such as vibrations between beam and sample and/or detector point spread (PSF) function. However, the PSF of the direct illumination CCD was measured, confirming essentially the 1 pixel value expected for a CCD with negligible cross-talk. Vibration amplitudes on the order of 200 nm are also unlikely. Instead, investigating the error returned for each parameter by the least-square fit, we see that the lower bound of the confidence interval defined by the increase of χ ² → χ ² +1 is ζ ≃ 1.4, and that the expected value would thus still be within the confidence interval. Accordingly, the smearing of the Fresnel diffraction pattern is not in disagreement with the experimental coherence length or source size. At the same time, we also see that imaging a test structure at fixed defocus is not sensitive enough to determine |j|. At the same time, we can conclude that imaging under given parameters and conditions of partial coherence is well described by the ideal propagation contrast followed by an semi-empirical Gaussian convolution, and that imaging by the KB beam alone without further coherence filtering already gives quantitatively tractable results.

6. Coherence filtering by a waveguide

Next, the Talbot scans were continued, but now with a waveguide inserted in the KB focal plane for further reduction of the beam size by an order of magnitude, down to below 20 nm, in order to demonstrate the associated gain in spatial coherence. The sputtered thin film system Ge/Mo[30 nm]/C[35 nm]/Mo[30 nm]/Ge with a waveguide length of 200μm was used as described above. Simulation and previous experiments have shown that very small near-field beam cross sections (FWHM) in the range of 10–15 nm can be obtained in theses waveguides by multi-modal interference and damping of higher orders [21]. Figure 8(a) shows the schematic of the setup, and (b) a numerical calculation of the field propagation in and directly behind the waveguide, based on finite difference equations [23]. Since the grating is a one-dimensional structure, one-dimensional beam confinement was sufficient and thus the horizontal WG (for vertical beam confinement) was used in the Talbot scan. The grating defocus distance was first calibrated by the optical microscope. At its reference position, the fully motorized microscope has a focal spot coinciding with the KB mirror. Next, the z ₁ value with respect to the waveguide exit plane was determined in the x-ray beam by vertical translations of the grating, yielding a magnification of M = 8997 and correspondingly a defocus of z ₁ = 0.5880 mm, in perfect agreement with the thickness of the waveguide. Figure 8(c) shows the far-field intensity pattern of the 50 nm l&s grating at 1 mm defocus with respect to the KB focus. The intensity matrix was divided first by the empty waveguide far-field matrix, and then divided by the average horizontal (vector) intensity profile (mean over all rows). Finally, (d) shows the result of the Talbot scan (solid circles) with a least-square fit to Eq. (5) (solid line). Given the significantly reduced Talbot replication period Z _T = a ² /λ, the curve is not sampled finely enough, but a least-square fit is nevertheless possible, if the Z _T is kept constant, or fixed within a small range of values. The resulting amplitude of the Talbot curve is significantly increased with respect to the results without waveguide. The amplitude fit parameter of 0.445 directly illustrates the gain in coherence, since within error it equals the prefactor in Eq. (5), with c = 0.440 calculated for the given optical constants of the grating and photon energy. Accordingly, this indicates a degree of coherence |j| ≃ 1 over the measured range, in line with the coherence simulations, which predict a fully coherent waveguide exit beam.

 

Fig. 8 (a) Schematic of the waveguide Talbot setup. (b) Numerical simulation of beam propagation in the waveguide, assuming a plane incidence wave. Transmission, beam confinement and the coupling of the field into and out of the waveguide can be studied. Due to damping of the higher order modes, the field in the exit plane is highly confined, with a width (FWHM) smaller than the guiding layer D, resulting in a broad diffracted emission cone used to illuminate the grating. (c) Illustrative example of the Talbot image as measured with the detector at the same position as above, with the 50 nm l&s grating at a defocus value z ₁ = 0.5880 mm behind the exit plane of the waveguide. The vertical scale bar represents 1.8 mm in the detector plane, corresponding to 200 nm in the sample plane. (d) Talbot scan yielding the magnitude of the first (m = 1) order Fourier component of the 50 nm l&s grating (circles), as determined from Fourier analysis of the recorded Fresnel patterns, along with a least-square fit to Eq. (5) (solid line), as a function of the defocus distance (here with respect to KB focus).

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7. Summary and conclusion

The spatial coherence of an x-ray beam emanating from a nano-focused partially coherent secondary source of 200 nm cross section has been characterized. We have first mapped out the focal intensity distribution by scanning x-ray waveguides through the beam. At a storage ring current of 60 mA and a photon energy of E = 7.9 keV, the monochromatic flux (Si(111) double crystal monochromator) in the focal spot was 2.13 · 10¹¹cps. To characterize the lateral coherence in particular of the defocused beam behind the mirror focus, grating interferometry measurements as well as numerical wave propagation were employed. Defocus values have been chosen in view of propagation imaging. This study is part of a long term goal to relax coherence requirements in imaging applications, i.e. to develop advanced reconstruction algorithms which take a (measured) finite degree of coherence into account [1], and enable imaging at optimized flux for tomography and dynamical imaging. To this end, the magnified propagation image of a test pattern was analyzed, and fitted by an analytical expression convolved with a Gaussian smearing function which takes into account the effects of partial coherence. This example shows that imaging under partial coherence can in this case be well described by the ideal propagation contrast followed by a semi-empirical Gaussian convolution, and that imaging by the KB beam alone without further coherence filtering already gives quantitatively tractable results.

A more complete treatment of the lateral coherence in two orthogonal directions to the beam was achieved by grating interferometry using gratings of 500 nm, 200 nm, and 50 nm lines and spaces, respectively, at defocus distances in the range of 1 mm to 15 mm, relevant for magnified projection imaging. In this range the coherence length ξ _x,y was essentially indistinguishable from that of an incoherent secondary source of the same size and distance. In other words the effect of partial coherence simply changes the ratio of beam diameter and coherence length, but not the coherence length in absolute value. This is true for defocus distances larger than the Rayleigh length, in line with simple wave optical coherence propagation, both in analytical approximation as in a more complete numerical treatment. The agreement in the horizontal directions with the numerical results was excellent. In the vertical direction, the degree of coherence was very similar to the horizontal values, in contrast to expectation based on the smaller vertical (primary) source size. The small source size (σ = 6μm) in the vertical direction should have led to a nearly fully coherent vertical beam, over the acceptance of the KB mirror. This was, however, not the case, and the discrepancy can be attributed to the double nitrogen cooled crystal monochromator which is known to introduce vibrations, and to compromise coherence. Possible improvements are now under close investigation at PETRA III.

In conclusion, as the coherence of the x-ray optical system is an essential prerequisite for coherent imaging, one needs suitable diagnostic tools and procedures for the case of propagation imaging and for coherent diffraction imaging alike. Extending previous studies carried out in unfocused or moderately focused synchrotron beams, the present work has addressed the case of highly focused radiation by Talbot interferometry, which was previously treated only in the limit of full coherence, e.g. by ptychographic wavefield reconstruction. We expect that for future applications of full field propagation imaging, the coherence requirements can be relaxed, if properly measured. If this is achieved, a much wider range of partially coherent optical systems can be used, to the advantage of higher flux and lower accumulation times. However, for applications needing more stringent coherence requirements and high aperture divergent beams, further optical coherence (and spatial) filtering elements are needed. To this end, we have inserted an x-ray waveguides in the KB focus, leading to an increase in spatial coherence, as evidenced by the increased amplitude of the oscillatory grating visibility, in the Talbot scans.