Single-shot transverse coherence measurements with Young's double pinholes at FLASH2

We measured the transverse coherence at FLASH2, a variable gap undulator line at the FLASH free-electron laser user facility at DESY in Hamburg. We demonstrate, theoretically and experimentally, a revised version of Young’s double pinhole approach to perform single-shot, repeatable and non-invasive transverse coherence measurements. At beamline FL24 of FLASH2, the transverse coherence of pulses was systematically characterized at wavelengths of 8, 13.5 and 18 nm for different FEL source settings. We determine degrees of coherence of 57% to 87% in the vertical and horizontal direction, respectively. These measurements can facilitate the planning of novel, coherence-based experiments at the FLASH facility.

In a YDP setup for FEL characterization, double pinholes were usually placed in the focus of the optics, which enabled planar illumination of each pinhole. Under this condition, the far-field interference pattern in an observation plane is only due to the optical path difference of light passing through pinholes. Thus, a straightforward analytical solution was given in [31] to extract the degree of transverse coherence from the measured interference pattern as a function of the pinhole separation. However, a typical focus on the micrometer scale made it impractical to perform a wide scan over the different pinhole separations. Additionally, the highly intense focused pulses destroyed the pinhole sets, requiring replacements after every single shot.
In this paper, we demonstrate an alternative approach, by illuminating a pinhole plate with a highly divergent beam downstream of the focus. This out-of-focus setup ensures a non-invasive, non-destructive illumination of the pinholes and allows a wide scan through the beam width, leading to precise transverse coherent measurements. However, the transition from planar to divergent illumination causes a significant change of the phase over each pinhole, which incorporates the beamʼs curvature into the interference pattern for a wide range of pinhole separations. The complexity of YDP experiments with a divergent illumination was Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. mentioned and partially investigated in [32]. In this work, we demonstrate two methods to determine the degree of transverse coherence of FELs with an out-of-focus YDP setup.
We performed a single-shot YDP experiment at the FLASH2 beamline [33], an additional variable-gap undulator line operational since 2016. The FLASH2 source generates radiation in the soft x-ray regime and operates as a SASE FEL which statistically behaves as a chaotic source. Due to the stochastic fluctuations, FLASH2 may not exhibit the same degree of transverse coherence as a fully coherent laser-like source, such as the seeded XUV FERMI FEL-2 source [34].
We present a general analytical solution to calculate the transverse coherence of out-of-focus FEL pulses and, additionally, we introduce the deconvolution method and its capabilities to extract the degree of transverse coherence. Utilizing both methods, we systematically characterized the degree of coherence for different FEL machine settings and different wavelengths of operation at FLASH2. Finally, we compare the analytical approach with the deconvolution method.

Experimental setup
Double pinholes (DPH) were illuminated near and downstream of a focus produced by Kirkpatrick-Baez (KB) focusing optics installed at beamline FL24 of FLASH2. Equipped with bendable mirrors, the focal length of the KB was set to 2 m while being inspected with a Hartmann wave front sensor (WFS) [35] (figure 1(a)). After the focus was established at the foreseen position, the end of the beamline was changed to replace the WFS with an XUV detector, positioned approximately5.8 m from the position of the pinhole plate which was itself approximately1 m downstream of the focus ( figure 1(b)).
The pinhole plate consists of two sets of ten double pinhole pairs, oriented vertically and horizontally, respectively. The separation between the DPHs was ranging from m 50 m to 1557μm in several steps (see figure 2). The pinholes were manufactured using laser drilling with a diameter of approximately m 10 m. The YDP plate was illuminated using various FEL source settings such as different number of active undulators, electron bunch charge and three different wavelengths of 8 nm, 13.5 nm and 18 nm. The FEL machine settings are listed in table 1.
Another plate with a phosphor coating and apertures for each pinhole pair was attached to the pinhole plate to allow a direct characterization of the FEL beam size with an optical camera by positioning the beam on a blank space between two pinhole pairs. The images were recorded using a Basler acA1300-30 gm camera under 45°a ngle. The KB mirrors were adjusted such that the beam diameter was covering the maximum available pinhole separation. The second-moment width D4σ in horizontal and vertical direction was determined for each direct beam measurement based on ISO 11 146 method of variances [36]. Assuming a Gaussian intensity profile, the Full Width at Half Maximum (FWHM) is given as D 4σ·2.35/4. For each FEL source setting the averaged beam widths are shown in figure 3 and also given later in table 2.
For each pinhole pair consecutive single-shot interference patterns were recorded using a Princeton Instruments PIXIS1024B XUV camera. Background images were collected by positioning the beam on the phosphor plate between two pinhole pairs. Single-shot information of pulse energies was provided by beam position monitoring [37,38] installed in the beamline. Therefore, for each recorded single-shot a corresponding background image with the same energy could be determined. This allowed an accurate shot-to-shot analysis of the interference patterns. 3. Transverse coherence determination 3.1. Analytical fit Figure 4 sketches the geometry of the YDP experiment at FLASH2. The focal plane of the KB optics is at z 0 =0, and the pinhole pair is located downstream at position z 1 . The electric field E s z , can be   written as follows, We consider the free-space, paraxial propagation of the light through each pinhole to the measurement plane at z 2 to be determined by where d and D are the pinhole separation and diameter respectively, D =z z z 2 1 is the relative distance between the two planes of propagation, and = identifies the effective propagation distance due to the divergent illumination of the pinholes.
We express the interference of the two fields at z 2 as where γ is the degree of transverse coherence. We numerically simulated equation (3) as a fit function with z 1 , D and d as the adjustable free parameters. We note that due to the pointing instability of the FLASH source a fixed z 1 did not lead to a convergence of the fits. For a stable source of radiation (such as seeded FELs) the variation in z 1 can probably be ignored. Additionally, an imbalanced illumination of pinholes was considered. Thus, equation (3) was fitted to all measured data by using the experimental values as the initialization of free parameters for the fits. A similar approach has been discussed in [39].
We call a heuristic measure that has a strong influence on the interference pattern interpretation, the dimensionless Fresnel number With F=1 we can omit the spherical phase term in equation (2) inside the integrand and apply the Fraunhofer diffraction formula [30,31]. An analytical derivation of equation (3) under the Fraunhofer  approximation was given in [29] where the pinholes were illuminated with approximately planar waves at the focus of an x-ray focusing optics. In equation (4), for a given z M and λ, F=1 is satisfied for the small pinhole separations and equation (2) relies on the Fraunhofer approximation. On the other hand, for F values close to or larger than unity, the phase curvature of illumination is then sampled by the two pinholes and the assumption of a planar illumination is not held any longer. This can be understood by expanding the spherical phase factor as ( ) ( ). The first term of the expansion, within the integrand, results in the Fraunhofer formula and the second term involves the phase curvature over the pinhole width.
We chose a wide range of pinhole separations, spanning different values of F, to escalate the fit accuracy. Relying solely on the Fraunhofer approximation, we could only scan less than three pinhole separations over the out-of-focus beam whereas equation (3) enabled us to interpret data from a much wider range of pinhole separations.
In figure 5, we have shown three cases of interference pattern analysis where F1. The shown data were chosen to display the significant fluctuations of the FEL intensity. We discovered that for each separation the maximum of γ corresponds to the single shots illuminating both pinholes equally.

Deconvolution method
It has been shown by [40,41] that a partially coherent intensity I pc can be expressed as the convolution of a fully coherent intensity I fc and the Fourier transformation of the transverse coherence function γ, Therefore, we realized that the blind deconvolution approach (equation (5)) can be an alternative solution to resolve γ. We note that the blind deconvolution is a non-convex method and highly sensitive to the noise level. Thus, a proper background subtraction for each single shot is demanded. Here, in contrast to the previous section, we obtain the transverse coherence function explicitly without a need to fit.
For each measured intensity the transverse coherence extraction was performed with the algorithm shown in figure 6. The general steps in the algorithm are summarized as follows:  as a 2D-Gaussian:  We reconstruct the measured interference patterns by convolving both retrieved elements of the right hand side of equation (5). After scanning through various vertical and horizontal widths of g F{ }the optimal values were determined by the minimal χ 2 -distance. In figure 7 we have displayed the monitoring of the algorithm's accuracy. When the numerical convolution (green solid line) reaches the measured partially coherent intensity (blue solid line) the algorithm stops. The positivity constraint was applied to the coherent intensity (red solid line) to avoid non-physical solution.  We note that within the algorithm no symmetry constraint was imposed to equation (6). Due to different transverse electron bunch compressions at the FEL machine, an asymmetrical Gaussian distribution for γ was expected.
Also, within the measured datasets, we observed that the intensity distributions, though they transversely fluctuate, can be approximately described by a Gaussian distribution. Theoretically, Vartanyants et al [43] have shown that the Gaussian-Shell model would suffice to analyze the radiation at the FLASH facility. Therefore, we concluded, through the measured statistics, that the Gaussian-Shell model can adequately represent the radiation of the mentioned FEL source setting. Thus, for the deconvolution method, we define the degree of transverse coherence based on the aforementioned presumption of describing FLASH pulses in the Gaussian-Shell model. We notice that further studies with advanced methods such as ptychography [44] or modal-based diffraction imaging [45] would provide a more accurate representation of the FLASH2 pulses.
The size of a Gaussian beam can be characterized by the rms width [36] s B,hor in horizontal direction (s B,ver in vertical direction): With the knowledge of both, transverse coherence length and beam size, the degree of transverse coherence z hor can be determined. In the framework of the Gaussian-Shell model [46]

Results and discussion
Broadly, both methods, namely analytical fits of the interference profiles and the deconvolution method, were used to determine the transverse coherence of the pulses with different FEL machine settings and pinhole separations. Due to the nature of the SASE process, an intrinsic shot-to-shot variation of intensity and position was observed. Thus, despite of a controlled alignment of the double-pinhole plate on the optical axis, unequal illumination of the pinholes was inevitable. The influence of unequal illumination on the determination of the transverse coherence has been studied in [47]. Addressing this, the highest value of γ for each pinhole separation through the measured dataset was taken. An illustration is shown in figure 8 for ( ) . Observing more fluctuation for certain ratios can be attributed to an imperfect alignment of the pinholes from the optical axis (beam propagation axis). In fact, due to the rather small misalignment ( figure 8(a)), to the left from the optical axis, more statistics have been measured for the ratio of < 1 ( ) . Also, for a different pinhole pair a misalignment to the right resulted in observing relatively more fluctuations in the right half of figure 8(b). The chaotic nature of SASE operating FELs and shot-to-shot fluctuations also have to be considered. As discussed in [48], during the FEL amplification process the degree of transverse coherence improves until single-mode saturation is reached. A post-saturation condition (corresponding to the 12 undulators here) could result in growth of higher-order radiation modes, which degrades the transverse coherence. The degree of transverse coherence γ is plotted against the different pinhole separations d for all used FEL machine settings in figure 9.
It was found that for the pinhole separations larger than m 1020 m the analytical fitting model might not converge to a unique solution with reliable free-parameters of fitting. In that specific range, the deconvolution algorithm outperforms significantly the analytical fitting, and yields unique and reproducible solutions. Additionally, for pinhole separations less than m 250 m, we realized that the deconvolution method was trapped in a local minimum instead of reaching to the global minimum. Since in that length span the pulses have a very high degree of transverse coherence close to unity, deconvolving the measured profiles as stated in equation (5) fails due to the negligible contribution of the transverse coherence function. In summary, for a wide scan of pinhole separations the two proposed methods complement, to precisely measure the transverse coherence of pulses.
The coherence length x ver,hor ( ) is calculated as the rms width of γ(d). Using equation (8), we determine degrees of transverse coherence z ver,hor ( ) in vertical and horizontal direction of 57% to 71% at a wavelength of 8 nm, 63 to 83% at 13.5 nm and 78% to 87% at 18 nm (see table 2). The given values are obtained through a wide scan covering pinhole separation widths from m 107 m to m 1335 m. Practically, both methods yield comparable results to previously measured values at the FLASH1 facility. An exception was seen where the values for longer wavelengths are slightly smaller than expected.

Conclusion
We performed a single-shot YDP experiment to determine the transverse coherence of the FLASH2 facility, at the FL24 beamline. Due to the intense focus of the KB optics at FL24, the double-pinhole was installed far downstream of the focus, resulting in a divergent illumination of the pinhole pair. Incorporating a varying phase curvature over each pinhole, we introduced and demonstrated two reliable approaches to characterize the transverse coherence properties of an FEL source.
Our measurements, for three different wavelengths of 8 nm, 13.5 nm and 18 nm, involving an in-depth statistical analysis yielded a high degree of transverse coherence of approx. 60% to 90% with some variations for different wavelengths and FEL settings. The positional instability of the FEL pulses resulted in unequal irradiation of the pinholes. Its influence on the determined transverse coherence length was studied and accommodated into the data analysis. Our analysis shows that the FLASH2 source delivers highly coherent pulses as it was conceived for this facility. Further foreseen work on approaches such as mixed-state ptychography and speckle tracking will entail studying the partial coherence in detail. We believe that the presented approaches can be applied as a fast, on-line diagnostic tool at beamlines demanding the knowledge of transverse coherence.