## 4 Discussion
### Time-Dependent Response  
To model the photoinduced dynamics of the superlattice reflection we fit the data with a model which assumes two distinct physical processes govern the observed changes. Just after pump excitation, we observe a fast, step-like drop in the superlattice intensity, limited by the time resolution. At later times, a slower and more gradual decay arises, lasting several tens of nanoseconds. Based on these observations, we propose a two-component fit function of the form  
\[y(t)= \begin{cases}1&t<0\\ 1-A_{1}-A_{2}(1-e^{-t/\tau})&t\geq 0\end{cases} . \tag{1}\]  
where \(A_{1}\), \(A_{2}\) and \(\tau\) are fit parameters, representing amplitudes and a timescale, respectively. The model \(y(t)\) is further convolved with a gaussian response function with a FWHM of 70 ps to account for the finite time resolution imposed by the X-ray pulse duration. The resulting model curves fit \(I(t)/I_{0}\) fairly well, as seen in Figures 2a and 3a. The fit parameters are summarized in Table 1.  
We first discuss the fast component of the response, represented by the fit parameter \(A_{1}\). \(A_{1}\) increases linearly with fluence up to 3.5 mJ/cm\({}^{2}\) (Figure 2b) and decreases with probe penetration depth. The nearly linear increase in the intensity reduction with fluence, which is similar to the observations reported by [PERSON] et al. for commensurate superlattice reflections in the electronically driven photoinduced phase transition in Pr\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\)[5]. This fast change in the intensity of the superlattice peak is strongly suggestive of an electron-phonon driven effect, where the excitation of states across the bandgap modifies the coupling that stabilizes the incommensurate modulation, leading to a drop of the amplitude of the structural modulation. The saturation of the pump-probe effect at high fluences (Figure 2b) is likely an effect of the spatial inhomogeneity in the pumped region; since the pump penetration depth is limited to 20 nm and the field penetration depth of the X-rays is in all cases larger than 70 nm, the pump does not initially excite the entire probed volume of the sample. The slower process, represented by \(A_{2}\) and \(\tau\), leads to complete suppression of the superlattice for the lowest incident angle value at 3.5 mJ/cm\({}^{2}\) fluence. Rocking curve scans at several time delays confirm that the integrated intensity truly vanishes and does not simply escape detection by a change in momentum transfer. As shown in Figure 2c the magnitude of the slower, second intensity reduction also increases with fluence, but saturates at already at low fluences of around 2 mJ/cm\({}^{2}\). In all cases the amplitude of the second response, \(A_{2}\), is larger than the fast response, \(A_{1}\). The value of relaxation time \(\tau\) varies strongly with both fluence and X-ray incidence angle. The changing value of \(\tau\) with probing depth suggests strongly that transport effects are relevant. These relaxation times, \(\tau\), versus  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fluence** & **Probing Depth** & \(A_{1}\) & \(A_{2}\) & \(\tau\) \\
**[m]/cm\({}^{2}\)** & **[nm]** & **[-]** & **[-]** & **[ns]** \\ \hline
3.5 & 71 & 0.30 \(\pm\) 0.01 & 0.64 \(\pm\) 0.01 & 8.25 \(\pm\) 0.94 \\
3.5 & 130 & 0.33 \(\pm\)  
the effective X-ray field penetration depth, \(d\), is shown in Figure 3c. The data fits well to a line of the form \(d=v\tau+d_{0}\), where \(v=3.7\pm 0.2\) m/s and \(d_{0}=37\pm 2\) nm. This velocity is significantly lower than the speed of sound \(v_{s}=2.4\cdot 10^{3}\) m/s [41]. Figure 3c indicates a relatively slow propagation of the transformed phase into the material. The limiting factor determining the speed of the phase transformation could be heat transport. A model describing a purely diffusive heat transport has been attempted but was insufficient to appropriately reproduce the timescale and shape of the measured response, especially when considering different pulse energies and probing depths.