# Resonant X-ray Emission Spectroscopy with a SASE Beam
## 3 Results and Discussion  
The RXES planes were reconstructed by means of Equation (2) for different numbers \(k\) of the measured incident and emission X-ray spectra. The measured data and the Fe K\(\beta\) RXES planes reconstructed for four different numbers of pulses are shown in Figure 1. For the analysis, we considered 1200, 6000, 12,000, 24,000, and 60,000 incidence pulses that, at 120 Hz repetition rate, correspond to the total acquisition time of 10, 50, 100, 200, and 500 s, respectively. In Figure 1\({}^{\prime}\)s left column, the measured incident SASE pulses' spectra are plotted. As shown, the spectra jitter around the Fe K-edge binding energy, but the average over a greater number of pulses (red line) takes on a Gaussian shape. In the central column, the measured Fe K\(\beta\) XES spectra are presented. Because of the XFEL beam jitter, certain pulses contained mainly photons with energy smaller than the Fe K-edge binding energy, while others delivered some or many photons with energy above the ionization threshold. As a result, some incident SASE pulses did not induce any Fe K\(\beta\) emission, while others induced some or strong emission, hence the plots' ragged character. The gaps appearing periodically along the vertical axis (pulse ordering number) in the Fe K\(\beta\) XES column result from the variation of photon energy controlled with a Vernier element, periodically disabling the Fe K\(\beta\) emission [18]. In Figure 1\({}^{\prime}\)s right column, the reconstructed Fe K\(\beta\) XES planes are presented. The obtained RXES planes provide detailed information on the occupied and unoccupied electronic states in the sample. For the increasing number of the processed pulses, the reconstructed RXES plane becomes less grainy, showing the electronic structure with better precision.  
To quantify the noisiness \(n(k)\) of the reconstructed RXES planes \(I_{k}(E_{1},E_{2})\) with the number of pulses \(k\), incidence energy \(E_{1}\), and emission energy \(E_{2}\), the following steps were taken. First, a residual \(r(k,E_{1},E_{2})\) of the reconstructed 3D RXES plane was calculated with the plane reconstructed for \(k_{\max}=60,000\) XFEL pulses taken as the reference plane:  
\[r(k,E_{1},E_{2})=I_{k}(E_{1},E_{2})-I_{k_{\max}}(E_{1},E_{2}). \tag{3}\]  
Second, the absolute value of residual function was integrated over the entire incident, and emission energy ranges covered in the experiment, which yielded a single number for each residual function. Finally, the obtained numbers were divided by the reference plane's integral, providing a relative deviation of the reconstructed plane from the reference one:  
\[n(k)=\frac{\int|r(k,E_{1},E_{2})|\mathrm{d}E_{1}\mathrm{d}E_{2}}{\int\,I_{k_{ \max}}(E_{1},E_{2})\mathrm{d}E_{1}\mathrm{d}E_{2}}. \tag{4}\]  
The reconstructed RXES planes were processed in the described way, and the result, i.e., an estimate of the RXES planes' noisiness as a function of the number of SASE pulses, is shown in Figure 2. As expected, the noisiness decreases monotonically for the increasing number of pulses processed in the reconstruction and converges well with a \(\sim 1/\sqrt{k}\) function (in blue) representing the expected error dependence. The plotted data can be used in the determination of the number of XFEL pulses needed to obtain the demanded accuracy of the RXES planes' reconstruction. In the case studied in this work, at least a few thousand pulses are required to maintain the statistical significance below 10%. For the very low number of pulses required, one should reckon with the potential necessity of reducing the number of points in the incident spectra (through either binning or cutting out a part of the spectrum) to fulfill the vital reconstruction condition \(k>m\).  
Figure 1: The measured incident SASE and Fe K\(\beta\) XES spectra (**left** and **central** column) and the Fe K\(\beta\) RXES planes (**right** column) reconstructed for: 1200, 6000, 12,000, and 24,000 pulses (rows 1–4).