# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments
## 4 Case Example: Analysis of a Femtosecond Pump-Probe VMI Experiment
The time evolution of the velocity distribution can be well represented by angular integration of the images, as shown in \[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]5  
\[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]6 (**a,d**) experimental Abel inverted velocity map imaging (VMI) images recorded for the \(\Gamma(^{2}P_{1/2})\) fragment at 120 fs and 10 ps delay times, respectively; (**b,e**) corresponding fitted reconstructed images; (**c,f**) are the residuals corresponding to the experimental image subtracted from the fit. \[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]7 \(\mathbb{I}^{\star}(^{2}P_{1/2})\) fragment velocity distributions (in pixels) at (**a**) 120 fs and (**b**) 10 ps delay times. The experimental data (**-**), the fitted total (**-**) and the fitted different contributions (main (**-**) and secondary (**-** - and **-**)) are shown. The main contribution (ring) has been fitted in velocity (pixels) using the sum of two Gaussian functions, which have the same width (\(\sigma_{1,a}=\sigma_{1,b}=5.27\pm 0.04\) pixel) as indicated in Ref. [38]. The central position has been found to be at \(r_{1,a}=69.9\pm 0.5\) px for the main peak and at \(r_{1,b}=65.7\pm 0.5\) px for the minor shoulder with an amplitude \(\epsilon=0.147\pm 0.002\). As was indicated above, the main peak and the shoulder are assigned to dissociation channels yielding CH\({}_{3}(\
u=0)\) and CH\({}_{3}(\
u_{1}=1)\), respectively. The temporal evolution of the secondary contribution \(C_{2}\) appearing in the center of the images has been fitted using a \(\tau_{xx,2}=400\pm 50\) fs Gaussian convolution of a \(\tau_{d,2}=1.7\pm 0.1\) ps exponential decay. The other secondary contribution, \(C_{3}\), has a time constant of \(\tau_{d,3}=1.48\pm 0.05\) ps. The temporal behavior of the contribution of interest \(C_{1}\) isolated from the secondary contributions is shown in Figure 7 as a 1D transient. The data points shown in this figure have been obtained through subtraction of the fitted secondary contributions from the experimental measurement. Then, the resulting signal has been fully integrated angularly and radially integrated over the region of interest. An exponential-type growth (Equation (15)) with a time constant of \(\tau_{d,1}=1.54\pm 0.04\) ps for this particular case has been found from the fit. Several (from 6 to around 30) independent sets of measurements across the full temporal range were acquired. Each time resolved-VMI measurement is independently fitted, keeping the same mathematical function. Each Levenberg-Marquardt fit provides fitting parameters and their corresponding standard deviations. A statistical study of the parameters that may not change with the scan, such as lifetimes of a given state, is hence performed. For such parameters, the extracted value is defined by the center of mass of the parameters found by each fitting procedure weighted by the inverse of its own standard deviation. Its respective error is  
\[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]8 2D representation corresponding to the angular integration of the \(\mathbf{I}^{\bullet}(^{2}P_{1/2})\) fragment image sequence as a function of the delay time. (**a**) experimental measurement; (**b**) the corresponding fitting; and (**c**) the residuals corresponding to the experiment subtracted from the fit. \[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]9 1D transient for the amplitude of the main contribution as a function of time. (\(\circ\)) experimental data and (\(\dasheddot\)) fit. See text for more details. hence the weighted dispersion of the individual values. The time constant thus obtained has been found to be \(1.5\pm 0.1\) ps, as shown in Ref. [38].