## 6 Soft X-Rays: Virtual Spectrometer
Typical single-pulse spectral distributions reconstructed with VS are presented in Figure 6d where the blue dashed line is the actual grating spectra that has not been smeared out via convolution. Although individual spikes cannot be reconstructed by the VS, the main features are clearly seen with intermediate resolution. For very low pulse energies, increasing the operating gas pressure is limited due to HV sparking. When selecting the target gas, the main criteria becomes to maximize the photoionization cross-section. Nonetheless, a poor signal-to-noise ratio makes data  
Figure 6: (**a**) TOF spectra with sxPES for Ne 1s from one single pulse registered on several detectors with 2 GSs\({}^{-1}\). (**b**) Same spectra calibrated to photon energy. (**c**) Same photon pulse with VS (red line). Four spikes are clearly resolved that cannot be discerned from the raw data. The uncertainty band, that comes from the noise, corresponds to a 68 % confidence level. Comparison with GS convolved with the estimated instrumental linebroadening (green dashed line) demonstrates the validity of the method. (**d**) Examples of typical single-pulse spectral distributions for Ne 1s collected with the VS and compared with non-convolved GS validation data (blue dashed line). (**e**) Examples at the challenging condition of very low pulse energies with Xe 3d target around \(\hbar\omega=1390\,\mathrm{eV}\). interpretation difficult and thus motivates the use of machine learning. Figure 6e shows three VS single-shot spectra at around 1390 eV acquired with Xe 3d (\(\sigma=0.61\) Mb) [12]. The spin-orbit splitting 3d\({}_{3/2}\) (E\({}_{B}=689.0\) eV, \(\Gamma=490\) meV) and 3d\({}_{5/2}\) (E\({}_{B}=676.4\) eV, \(\Gamma=510\) meV) [15] is sufficiently large to avoid overlap at the typical spectral bandwidth \(<12\) eV. The flight-tube retardation voltages were selected to include both spin-orbit lines, thus demonstrating that the VS by default takes advantage of all available information. Despite the low pulse energies (\(<\)60 uJ) that lead to a poor signal-to-noise ratio in the raw data, and high kinetic energy (\(>\)700 eV) that leads to lower instrumental resolution, the reconstruction is still reliable when compared with the validation spectrum (blue dashed line). Under these conditions, we estimate a resolution of 2.3 eV full width half maximum for the reconstructed spectra. Another benefit of the VS is the situation when Auger-Meitner lines overlap the photoelectron spectra. This is, for example, the case around 770 eV when nitrogen 1s (E\({}_{B}=410.0\) eV overlaps with the nitrogen KVV band at around 360 eV. Normally, photon diagnostics is prevented under these conditions, but the VS automatically removes these features as part of the background subtraction. The virtual spectrometer has the limitation that it assumes that the PES and XGM configurations remain stable throughout the measurements, which imposes a boundary on the ranges of photon energies to be measured before refitting the model by re-inserting the GS. Such a limitation could be removed by establishing a large database of PES and GS conditions and interpolating between them as necessary. This planned development requires a large data-taking campaign covering several photon energies, beam intensities, and related PES configurations. While such a task would be time-consuming, it is technically feasible. More complex regression models may be used to further enhance the capabilities of the system in such conditions. Furthermore, considering that the PES resolving power varies non-linearly over the spectral bandwidth, a future operation mode with VS that we are exploring is to optimize each of the flight-tube retardation voltages to cover a separate photon energy region, thus improving the resolution further. Finally, we plan to implement the VS also for hard X-rays where the training would be performed with hxPES and the HIREX spectrometer [7].