Focusing X-ray and extreme ultraviolet (XUV) radiation to small focal spots is desired in many experiments to maximize the fluence on the sample or to improve the spatial resolution in imaging experiments. For this reason, in recent years, a lot of effort has been spent on producing high-quality X-ray optics to generate small spot sizes of only tens of nanometers [1,2]. However, measuring the amplitude and phase of nano-focused XUV and X-ray beams is challenging. Hartmann wavefront sensors, which are similar to those in the visible, are also used in the XUV range [3,4], and even more sophisticated methods were developed to measure the spectrally resolved wavefront of discrete harmonics [5]. However, Hartmann wavefront sensors are limited in the accuracy by the micrometer-scale pixel size of the detectors and, due to the large mask size, they have to be placed in the converging beam between the focusing element and the focal spot, which hinders practical implementation. Other techniques, such as lateral shearing interferometry [6], provide full spectral characterization, but rely on complicated and demanding experimental setups and are not single-shot.

To characterize small focal spots, multi-shot techniques, such as scanning coherent diffractive imaging (CDI) (ptychography [7,8]), are routinely used [9–16]. Since ptychography relies on the assumption that the illuminating beam is constant over time, it struggles with shot-to-shot fluctuations, which makes it difficult to reliably characterize the beam profile of, e.g., free-electron lasers (FELs) [17]. Hence, a single-shot method for beam characterization is highly desired for real-time and shot-to-shot measurements.

Here we present a single-shot wavefront sensor, which measures the amplitude and phase of a focused beam directly in the focal plane. An amplitude-only mask, consisting of an array of nano-holes is placed directly in the focus of a beam. The far-field diffraction pattern of the transmission of the beam through the mask is recorded with a detector. Finally, a phase retrieval algorithm is used to reconstruct the electric field of the beam, right after transmission through the mask, from the diffraction pattern [18]. Since the mask is binary, the illuminating beam can be interpolated from the retrieved electric field at the pinhole positions (for illustration, see Fig. 1). In contrast to earlier approaches, our method does not rely on any a priori knowledge about the focusing element [19], and the wavefront is measured directly in the focal plane [20]. Further, a modulator between the mask and detector is not needed [21–23], and arbitrary, extended beams can be investigated.

To prove the feasibility of our method, we conducted a proof-of-principle experiment using a high-order harmonic generation (HHG) beam centered at 13.5 nm driven by a 75 W high-power femtosecond fiber-laser generating a photon flux of approximately $5 \times {10^9}{\rm ph}/{\rm s}/{\rm eV}$ at 13.5 nm. (For a detailed description of the XUV source, see [24].) We used three multilayer mirrors to select a relative bandwidth of ${\Delta}\lambda /\lambda = 0.015$ and to focus the beam to ${\sim} 3\; \unicode{x00B5}{\rm m}$ diameter (FWHM). The prototype wavefront sensor consists of a simple, binary transmission mask, which can be sufficiently characterized by an electron microscope. The mask consists of $10 \times 10$ holes which were cut in a 50 nm ${\rm Si}_3{\rm N}_4$ membrane coated with 150 nm of copper using a focused ion beam [Fig. 2(a)]. Therefore, the XUV beam was sampled with a total of 100 pixels. Each hole has a diameter of 300 nm, and the pitch was chosen to be 600 nm which results in an overall mask size of 6 µm. The overall transmission of the mask is 20%, but can be increased by enlarging the size of the 300 nm holes.

 

Fig. 1. Schematic representation of our wavefront sensor. A coherent (XUV) beam is focused onto an amplitude-only mask which consists of an array of holes. The diffraction pattern is recorded by a detector, and the diffracted beam is reconstructed with a phase retrieval algorithm.

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Fig. 2. (a) Scanning electron microscope image of the amplitude-only mask. The distance between two holes is 600 nm, and each hole has a diameter of 300 nm. The top right quadrant has a slight offset of 125 nm to account for the twin image problem. The scale bar corresponds to 2 µm. (b) Measured diffraction pattern in logarithmic scale using a photon energy of 92 eV. The scale bar corresponds to $5\;{{\unicode{x00B5}{\rm m}}^{- 1}}$.

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A well-known problem of CDI is the so-called twin image problem, which states that an object ${O}({{x},{y}})$ and its twin (${O^*}({- {x}, - {y}})$, complex conjugated and rotated by 180°) result in the same diffraction pattern [25]. Hence, for a centrosymmetric support, the reconstruction of a beam will lead to two possible solutions, which are rotated by 180°. To avoid this problem, we added a slight offset of 125 nm to the top right quadrant. The offset breaks the symmetry and enables us to identify the true orientation of the beam without ambiguity.

A pinhole with a diameter of 2.7 µm was placed in the focus of the 92 eV HHG beam to create a stable and well-defined illumination. The mask-based wavefront sensor was placed approximately 500 µm behind the pinhole, and the far-field diffraction pattern of the 92 eV beam was recorded 33 mm behind the wavefront sensor with an XUV camera (Andor Ikon-L). The diffraction pattern was recorded with an exposure time of 1 s [see Fig. 2(b)]. The pixels of the CCD were binned 2 by 2 to an effective pixel size of $27\; {\unicode{x00B5}{\rm m}} \times 27\; {\unicode{x00B5}{\rm m}}$. The diffraction pattern shown in Fig. 2(b) contains $512 \times 512\;{\rm pixels}$ and covers a numerical aperture (NA) of 0.2, which results in a real-space pixel size of 33 nm. Hence, nine pixels in diameter represent a single sub-aperture of the wavefront sensor mask. The amplitude and phase of the transmitted beam were reconstructed with an iterative phase retrieval algorithm, which propagates between the detector plane, where the modulus constraint is applied, and the sample plane, where the support constraint is applied. Since the exact geometry of the binary mask is known from a scanning electron microscope image [Fig. 2(a)], the support is fixed and fed to the reconstruction algorithm. Thus, the reconstruction is stable and converges quickly. For the presented results here we used a total of five cycles, where each cycle consists of 80 iterations of the relaxed averaged alternating reflections algorithm [26] with a beta parameter of 0.8 and 20 iterations using the error-reduction algorithm [18].

The whole reconstruction took less than 20 s on a standard computer. Note that the iterative phase retrieval algorithm could be replaced by an artificial neural network [27] to speed up the numerical phase reconstruction and reduce the phase retrieval time to milliseconds. The reconstructed amplitude and phase are depicted in Figs. 3(a) and 3(c). To provide a more intuitive representation of the beam profile, the amplitude and phase were interpolated from the reconstructed pixels. For this purpose, the phase was averaged over the reconstructed pixels of a sub-aperture and then cubically interpolated. The resulting interpolated amplitude and phase are shown in Figs. 3(b) and 3(d), respectively. The phase values in the corresponding pinholes were averaged and a lineout was taken along a vertical and horizontal pinhole row in the center [indicated by a black dotted line in Fig. 3(c)]. The measured phases and the corresponding interpolation are shown for the vertical and horizontal direction in Figs. 3(e) and 3(f) after a global phase value was subtracted. To quantify the precision of our method, we acquired another nine diffraction patterns with the same settings. The beam was reconstructed and from all reconstructions, a global phase and tip/tilt, which are due to imperfect centering of the diffraction pattern on the detector, were removed. A vertical and horizontal lineout similar to the dotted lines in Fig. 3(c) was taken for each of the 10 measurements, and the standard deviation is calculated and added as an error bar in Figs. 3(e) and 3(f). The maximum standard deviation is calculated to be 80 mrad, which corresponds to a precision better than $\lambda /75$.

 

Fig. 3. (a) Reconstructed amplitude of the wavefront sensor. (b) Interpolation of the reconstructed amplitude shown in (a). (c) Reconstructed phase. (d) Interpolation of the reconstructed phase shown in (c). The scale bar in (a) corresponds to 2 µm. (e) Vertical lineout from (c). (f) Horizontal lineout of (c). Error bar in (e) and (f) corresponding to the standard deviation calculated from 10 independent measurements.

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In the next step, we scanned the beam profile and wavefront behind the pinhole in the propagation direction ($ z $ direction). For this purpose, we moved the wavefront sensor in 500 µm steps towards the detector and acquired diffraction patterns for three planes (${z} = 500$, 1000, and 1500 µm). The vertical and horizontal lineouts of the interpolated phase are shown in Fig. 3. It appears that the radius of curvature of the wavefront increases with increasing distance from the pinhole. To verify the measurement, we simulated the expected wavefront behind the pinhole by assuming a circular aperture with a diameter of 2.7 µm illuminated with a flat wavefront and a constant amplitude, which are reasonable assumptions, since the pinhole was placed in the focus of a larger beam. The diffraction pattern in the respective planes was calculated by numerically propagating the beam transmitted through the pinhole with a near-field propagator (angular spectrum of plane waves [28]). The corresponding phase profiles are shown in Fig. 4 and match remarkably well with the measured wavefronts. To quantify the accuracy, we calculated the RMS value of the difference between the measured and simulated wavefront. For all three planes, an accuracy of better than $\lambda /70$ was identified.

 

Fig. 4. Vertical and horizontal lineouts of the reconstructed phase for 500 [(a) and (b)], 1000 [(c) and (d)], and 1500 µm [(e) and (f)] behind the pinhole. The blue curves show the expected wavefront from a 2.7 µm pinhole illuminated with a flat wavefront.

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In the present experiment, the field of view was limited by the size of the mask to $6\; {\unicode{x00B5}{\rm m}} \times 6\; {\unicode{x00B5}{\rm m}}$. In principle, larger masks can be fabricated, and the maximum field of view is only limited by sufficient sampling of the diffraction pattern. In contrast, the spatial resolution only depends on the pitch (size) of the sub-apertures and the measured maximum diffraction angle (NA). Today, structure sizes well below 10 nm can be fabricated using helium ion beam lithography [29]. Measuring at very large diffraction angles and thus with NAs $\gt0.5$ has already been demonstrated in previous CDI experiments [30–32]. Thus, the scaling of the presented method to sub-10 nm resolution for soft X-ray applications appears feasible. Further, the maximum phase curvature, which can be determined by our method, is limited by phase wrapping. A phase jump from two adjacent sub-apertures larger than $\pi$ can no longer be uniquely assigned. Therefore, the maximum measurable phase curvature of our experiment is given by $\pi /600\;{\rm nm} $. The maximal measurable phase curvature can be increased by using smaller sub-apertures.

In the next step, we investigated the required minimum photon number on the sensor. Therefore, we acquired diffraction patterns with varying total detected photon numbers (exposure times) and estimated the validity by calculating the RMS compared to the measurement with the highest detected photon number ($6.3 \times {10^5}$ photons). The reconstructed phase along the vertical and horizontal axis (Fig. 5) shows that starting from $0.6 \times {10^5}$ detected photons (which corresponded in our experiment to an exposure time of 300 ms), wavefronts can be accurately (${\rm RMS} {\text -} {\rm error} = 89\;{\rm mrad} $) reconstructed. The reconstruction with $0.2 \times {10^5}$ photons still showed the rough wavefront curvature, but deviations to the expected wavefront increased significantly (${\rm RMS} = 235\;{\rm mrad} $). Consequently, considering the transmission of the mask of 20%, the wavefront can be accurately reconstructed with only $3 \times {10^5}$ photons incident on the mask. This analysis shows that our method is suitable to measure single-shot wavefronts of X-ray FELs (XFELs) (usually ${\gt}{10^{11}}$ photons per pulse [33]) or HHG beamlines (up to ${10^{11}}$ photons per pulse [34]), even if only a fraction of the beam is split for online analysis.

 

Fig. 5. (a) Lineout of the phase along the vertical and horizontal direction for varying detected photon numbers (exposure times). Reasonable results were obtained with only $0.6 \times {10^5}$ photons, which corresponded in our experiment to 300 ms exposure time.

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In summary, we presented a novel method to characterize the wavefront of focused, coherent beams. It relies on a simple binary mask that can be easily incorporated into existing setups at, e.g., HHG, synchrotron, or XFEL beamlines. Note that the presented concept is not restricted to the XUV range. It can be adapted to any region in the electromagnetic spectrum, as long as a suitable absorbing material exists. Although fabrication of suitable masks for the hard X-ray range will be difficult due to longer absorption lengths, we believe that suitable masks for photon energies up to 1 keV, where reasonably absorbing materials (e.g., Au) still exist, are possible. The method is photon efficient and fast, and requires only the acquisition of a single diffraction pattern. Thus, it is ideally suited for real-time beam characterization, the alignment of complex optical systems, or feedback to adaptive optics. Moreover, it can be employed for the characterization of single shots, e.g., from FELs or HHG sources, and scaling towards sub-10 nm resolution appears feasible.