1. Introduction

The diffraction-limited storage ring at the MAX IV Laboratory is the first of the 4th generation synchrotrons to come into operation [1,2]. Of the beamlines currently in service, the hard X-ray nanoprobe (NanoMAX) is the first to take full advantage of the low-emittance source. It has been in operation with one experimental station since 2017, providing its user community with a focused and coherent X-ray beam in the energy range 5-28 keV [3]. The current experimental station is designed for scanning diffraction and coherent diffraction imaging, and arguably provides the highest-quality coherent hard X-ray beam available today, enabling novel experiments in fields such as semiconductor engineering, catalysis, the geosciences, and biology.

Coherent X-ray beams are conveniently characterized using forward scattering ptychography, a form of imaging where an extended sample is scanned through the incident beam across a series of overlapping positions [4]. The resulting diffraction patterns are then processed through iterative phase retrieval to produce phase- and absorption contrast images. While ptychography is mainly an imaging technique, it also provides the complex wavefront of the incident illumination in the sample plane as a by-product, making it a useful tool for beam diagnostics [5,6]. Indeed, ptychography is used routinely to set up and align the nano-focusing mirror system at NanoMAX.

The ptychography technique has also been extended to cope with partially coherent illumination [7]. The forward model for each diffraction pattern is then expanded into an intensity sum of a number of mutually incoherent modes. Probe mode decomposition can absorb several kinds of disturbance of the coherent propagation aside from partial coherence in the beam, for example sample vibration, motion blurring, or detector point-spread. Assuming that partial coherence dominates, this extension of the ptychographic model can be used to characterize the tranverse coherence properties of the beam itself.

In this study, we first characterize the X-ray beam focus created at the beamline’s experimental station and its energy dependence. We then use mode decomposition to indirectly measure the coherent flux of the focused X-ray beam. By varying the parameters of the optical system, we explore the experimental conditions for full transverse coherence at the NanoMAX endstation and observe the emergence of partial coherence in the beam.

2. Optical system

Focused beams at NanoMAX are produced by a set of Kirkpatrick-Baez (KB) mirrors. The KB system consists of two Pt-coated, fixed curvature elliptical mirrors (JTEC, Japan) whose mutual alignment is controlled by piezoelectric actuators. Their arrangement is schematically illustrated in Fig. 1(a). The mirror pair produces a symmetrically focused square beam with numerical aperture

 

Fig. 1. A) Schematic optical layout, not to scale, where the rays represent the maximum acceptance of each mirror. Acceptances and focal lengths are $h_1 = 378$ µm, $h_2 = 225$ µm, $f_1 = 310$ mm, and $f_2 = 180$ mm, resulting in the beam divergence $\alpha = 1.2 $ mrad. Note that the actual mirrors are centered on the optical axis. B) Intensity profile incident on the SSA, together with an illustration of possible openings. The innermost rectangle shows the slit opening ($s_{x}^{\mathrm {coh}}$, $s_{y}^{\mathrm {coh}}$) predicted to produce coherent illumination of the KB mirrors at 10 keV (see below and Table 1).

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Table 1. SSA openings that nominally produce coherent illumination of the KB mirrors, together with diffraction-limited spot sizes (FWHM) for the NanoMAX KB mirror system.

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The beamline achieves a coherent wave field at the experimental station by means of spatial filtering through a secondary source aperture (SSA) situated at a distance $d = 46.7$ m upstream of the mirrors. The monochromatic beam forms a 1:1 image of the undulator source in the plane of the SSA (see Fig. 1(b)). The complex degree of transverse coherence is defined as

Equation (3) allows setting a criterion for what can be considered a fully coherent beam, and choosing the SSA opening in the horizontal and vertical directions ($s_x$, $s_y$) accordingly. A reasonable choice might be to limit the arguments of the $ {\textrm{sinc}} $ factors in Eq. (3) to $\tfrac {\pi }{4}$, which limits each factor to 0.90. Physically, this means that the degree of coherence between the center of the beam and the edges of each of the mirrors is 0.90, and that between the beam center and corner of the mirror pair is $0.90^2=0.81$. The mirror edges are located at around $\theta _x =h_2/2d$ and $\theta _y=h_1/2d$. The following expressions for slit openings should therefore provide a nominally fully coherent illumination across the mirrors.

3. Experimental and numerical methods

Ptychography was performed by scanning a 1 µm thick patterned tungsten film through the beam over an area of about 2.5 µm × 2.5 µm. The sample was placed either 400 µm or 800 µm downstream of the stigmatic X-ray focus. A Pilatus 100k detector (DECTRIS, Switzerland) was placed 3.873 m away from the focal plane to record diffraction patterns in the far field. A He-filled flight tube with kapton windows was placed between sample and detector to reduce absorption and air scattering. All ptychography data were acquired by software-controlled step scanning of a three-axis piezoelectric stage in a spiral pattern (see Fig. 2). For each energy, ptychography scans were collected for a range of slit openings $s_x$ and $s_y$ at the aspect ratio defined in Eq. (5). To avoid exceeding the linear range of the detector, double-polished Si attenuators were introduced as needed. An ion chamber located after the KB mirrors was used to independently measure the incoming photon flux, which was then corrected for the absorbance of the Si attenuators. During the measurements, the 3 GeV synchrotron stored an electron beam of around 250 mA, which provides a photon flux at the SSA on the order of $10^{12}$ per second.

 

Fig. 2. Energy dependence of the properties of the focused beam. A) Example ptychographic scan at 10 keV, showing the complex retrieved probe (left) and object (right) on the same scale. B) The complex wavefronts retrieved at different energies propagated to the X-ray focus, together with horizontal intensity profiles. Reconstructions performed using one probe mode only. Colorbar: Hue and saturation represent phase and amplitude, repectively.

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Ptychographic phase retrieval was done using Ptypy [9]. Reconstructions consisted of 300 iterations of the Difference Map algorithm [10], followed by 300 iterations of Maximum Likelihood refinement [11]. Ten probe modes were used for multimode phase retrieval and orthogonalized afterwards, and their intensities scaled to the ion chamber measurements. Retrieved probes were propagated numerically using the near-field propagator from the Ptypy library. To find the focal plane the probe was propagated across the range −1000 µm to 1000 µm in 5 µm steps. The three-dimensional intensities were then squared and summed across the plane perpendicular to the propagation direction, and the position of maximum sum-squared intensity was considered the position of the focal plane.

4. Energy dependence of the X-ray focus

Figure 2(a) shows an example ptychography reconstruction with the complex probe and retrieved object on the same scale. Also shown are the scanning positions at which diffraction data were collected. The probe is a divergent wavefront and the object mostly shows phase contrast features. Data such as those in Fig. 2(a) were collected for the photon energies 6, 8, 10, 14, 18, and 22 keV, with the SSA set at or just below the limit for a fully coherent beam, Eq. (4). They were reconstructed with one probe mode and therefore assume full coherence. For each dataset the retrieved probe was numerically propagated until the focus was found. Complex wavefronts in the focal plane are shown in Fig. 2(b) together with horizontal intensity profiles. The beam shape in focus is characteristic of the diffraction pattern from a square aperture, such as defined by the acceptance of the KB mirrors, and described by the $ {\textrm{sinc}} $ function in Eq. (6).

The beam sizes at the focal position measured from the propagated ptychography reconstructions are in striking agreement with the predictions for the diffraction limited beam (Eq. (8)). This agreement shows that, within the sensitivity of the ptychographic technique, there are no sources of broadening of the focal spot beyond what is dictated by the diffraction limit. Specifically, the focused X-ray beam is not aberrated to an extent detectable with ptychography. Direct measurements of the NanoMAX KB focus in the horizontal direction recently showed similar results with beam energies around 18 keV [12].

5. Estimated coherent flux

A key number for many scientific applications is the coherent flux available at an instrument. The full coherent flux is a measure of the number of photons per second that can be harnessed to build up a coherent diffraction pattern. In the context of ptychography, it can be defined as the flux achievable in the strongest reconstructed probe mode. As outlined above, transversely coherent illumination of the KB mirrors relies on the SSA admitting only a small part of the undulator cone, which results in a tradeoff between beam intensity and degree of coherence. By performing multimode ptychography reconstructions on data collected at different SSA openings, the coherent flux at the endstation can be estimated.

Figure 3 shows the results of a large number of ptychography scans and reconstructions, collected as described above. The plots on the left show the absolute intensity of the strongest orthogonal coherent mode, while the plots on the right show the relative intensity of the first mode compared to the full beam, both at varying SSA openings. As expected, the power of the first mode initially increases as the slits are opened, and then saturates. This is an operational measure of the full coherent flux, previously used for partially coherent X-ray beams at other sources [7]. At moderate beam energies of 6-10 keV, the full coherent flux of the KB setup can be seen to exceed $10^{11}$ photons per second.

 

Fig. 3. Coherent flux estimation using probe mode decomposition. The left-hand side shows the absolute full coherent flux estimation, while the right-hand side shows the onset of partial coherence as the secondary source is made larger. The vertical lines mark the slit settings ($s_x^{\mathrm {coh}}$, $s_y^{\mathrm {coh}}$) prescribed by Eq. (4). Both vertical axes are on a linear scale.

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Another experimentally important quantity is the X-ray flux that can be delivered in a “single mode” beam that can be considered fully coherent. For example, while some coherent imaging techniques such as 2-dimensional ptychography in forward scattering geometry are robust to partial coherence effects, techniques such as Coherent Diffraction Imaging (CDI) [13] or 3D ptychography in the Bragg geometry [14] are more sensitive to the beam quality and require a high degree of coherence in the illumination. As can be seen on the right-hand side of Fig. 3, the portion of the beam contained in the strongest coherent mode is independent of the SSA setting for small openings. There are clear plateaus before the curves turn downwards at energy-dependent thresholds. The figures show the slit settings ($s_x^{\mathrm {coh}}$, $s_y^{\mathrm {coh}}$) as vertical lines, and the data show that these estimated settings are on the coherence plateau, but close to where the curve turns downwards. Opening the SSA beyond these values therefore causes partial coherence to appear. This confirms that the choice made in Eq. (4) ensures a beam that can be considered fully coherent.

It must be noted that the mode decomposition approach provides a conservative estimate of the degree of coherence of the beam. As mentioned above, many effects beyond true partial coherence contribute to additional probe modes. Thus, for the higher energies, the fraction of the intensity in the strongest mode plateaus at values below unity. The independence of slit opening on the plateaus is taken as indication that the beam itself is fully coherent to the extent observable by ptychography.

6. Conclusion

From the above ptychographic analysis, we conclude that the nanofocused beam at NanoMAX can be made diffraction limited, and that beams which are fully coherent in an operational sense are achieved using the recommended slit settings. Figure 4 summarizes these results. Furthermore, if partial coherence is experimentally acceptable, the full coherent flux for beam energies between 6 and 10 keV exceeds $10^{11}$ photons per second, a number which will further increase as the storage ring current approaches its nominal value of 500 mA.

 

Fig. 4. Summary of the results. The left-hand figure shows the full coherent flux of the beam, together with the flux of a “single mode” beam that can be considered fully coherent. The right-hand plot shows the beam size in the focal plane compared to the diffraction limit.

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