# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments
## 4 Case Example: Analysis of a Femtosecond Pump-Probe VMI Experiment
Due to a possible different number of photons involved, the temporal resolution \(\tau_{xc,2}\) can be a priori different from the other contributions. The complete mathematical description can be expressed as follows:  
\[C_{2}(r,\alpha,t)=T_{2}(t)\times A_{2}(\alpha)\times R_{2}(r), \tag{18}\]  
with  
\[T_{2}(t)=\exp\left(-4\ln 2\left(\frac{t^{2}}{\tau_{xc,2}^{2}}\right)\right) \otimes\left[H(t-t_{2})\times\left(e^{-\frac{(t-t_{2})}{\tau_{d,2}}}\right) \right], \tag{19}\]  
\[A_{2}(\alpha)=1+\beta_{2,2}P_{2}(\cos\alpha)+\beta_{4,2}P_{4}(\cos\alpha), \tag{20}\]  
and  
\[R_{2}(r)=H(r_{2}-r)\exp\left(-4\ln 2\left(\frac{(r-r_{2})^{2}}{\sigma_{2,a}^{2 }}\right)\right)+H(r-r_{2})\exp\left(-4\ln 2\left(\frac{(r-r_{2})^{2}}{ \sigma_{2,b}^{2}}\right)\right). \tag{21}\]  
The second minor contribution, \(C_{3}\), has more localized velocities around the region of the contribution of interest (around \(C_{1}\)). Moreover, it has a temporal expression similar to \(C_{1}\). Additionally, its anisotropy requires the consideration of both \(\beta_{2}\) and \(\beta_{4}\) constant terms. This last contribution is modeled by the following equations:  
\[C_{3}(r,\alpha,t)=T_{3}(t)\times A_{3}(\alpha)\times R_{3}(r), \tag{22}\]  
with  
\[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]0  
\[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]1  
and  
\[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]2  
In what follows, the specific results of the 3D fitting procedure applied to the mentioned case example are presented. Figure 4 shows two typical images, out of the sequence of 80 images recorded, fitted at short (120 fs) and asymptotic (10 ps) delay times. In order to remove the standard 3-4 pixel region of noise produced by the Basex inversion around the central column of the images [28], the fit has been carried out four pixels away from the central axis. Those points are thus removed from the fitted and measured images shown in \[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]3 Additionally, some of the mathematical functions used in the fitting procedure present a singularity at the center of the images. In order to avoid the singularities, a 10 pixel circle was set to zero in the fitted and measured matrices. The elimination of these areas of the images does not represent any problem in the analysis, since all the relevant contributions are far away from the center. The advantage of the 3D fitting procedure is that the anisotropy is considered. In the present case, the main contribution (ring) starts with a pronounced anisotropy (of perpendicular character), with the ion signal concentrated at the equator of the images, and later the ring evolves to become more isotropic. The anisotropy parameters of Equation (16) obtained from the fit for this main contribution are \(\beta_{2,1}=-0.50\pm 0.03\), \(\Delta\beta_{2,1}=-0.51\pm 0.03\) and \(\tau_{\beta_{2,1}}=1.26\pm 0.13\) ps. Therefore, the fitted anisotropy at \(t=t_{1}\) is \(\beta_{2,1}(t=t_{1})=-1.01\pm 0.06\), which is characteristic of a purely perpendicular transition. The two secondary contributions are visible in the experimental images shown in \[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]4 The first can be clearly seen at the center of the image shown in Figure 4a; the second is appreciatedat the vicinity of the main ring in Figure 4d. These secondary contributions, together with the main contribution, can be distinguished in a more clear way in the velocity distributions shown in Figure 5 at two delay times.