# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments
## 4 Case Example: Analysis of a Femtosecond Pump-Probe VMI Experiment
The observed final anisotropy, measured through angular distributions of the fragments, can differ from the limiting cases of parallel (\(\beta_{2}=2\)) or perpendicular (\(\beta_{2}=-1\)) anisotropy, mainly because rotation of the molecule prior to dissociation tends to blur the angular preference. This is especially relevant in the case of an indirect dissociation process if the time scales are similar or slower than molecular rotation. Inspection of a few experimental images of the sequence shows that the anisotropy does evolve in this way [38], i.e., images taken at earlier delay times show a more pronounced anisotropy than those taken at later times. The present example, therefore, provides a test of the fitting procedure where one of the angular parameters is a function of time. The functional form of the loss of anisotropy with time is not known. An exponentially decreasing modulus of the \(\beta_{2}\) parameter is proposed as:  
\[\beta_{2}(t)=\beta_{2_{0}}+\Delta\beta_{2}e^{-\frac{(t-t_{0})}{\beta_{2}}}, \tag{13}\]  
where \(\beta_{2_{0}}\) is the asymptotic anisotropy parameter, \(\beta_{2_{0}}+\Delta\beta_{2}\) is the amplitude of the anisotropy parameter at \(t=t_{0}\) and \(\tau_{\beta_{2}}\) is the relaxation time. Some theoretical work has been dedicated to define the final value of \(\beta_{2_{0}}\) for a dissociation process that is either parallel or perpendicular, considering the dissociation time [44; 45]. Those theoretical models are based on the rotation of the excited wave packet prior to dissociation. Although no predictions are available on the relaxation time, it can be estimated to be in the picosecond time scale. The complete 3D function \(C_{1}\) that was employed to describe the main contribution (anisotropic ring) on the \(B\)-band photodissociation of CH\({}_{3}\)I can be written as follows:  
\[C_{1}(r,\alpha,t)=T_{1}(t)\times A_{1}(\alpha,t)\times R_{1}(r), \tag{14}\]  
where  
\[T_{1}(t)=\exp\left(-4\ln 2\left(\frac{t^{2}}{\tau_{xx,1}^{2}}\right)\right) \otimes\left[H(t-t_{1})\times\left(1-e^{-\frac{(t-t_{1})}{\tau_{xx,1}}}\right) \right], \tag{15}\]  
\[A_{1}(\alpha,t)=1+\left(\beta_{2_{0},1}+\Delta\beta_{2_{1}}e^{-\frac{(t-t_{1} )}{\beta_{2,1}}}\right)P_{2}(\cos\alpha), \tag{16}\]  
and  
\[R_{1}(r)=\exp\left(-4\ln 2\left(\frac{(r-r_{1,\alpha})^{2}}{\sigma_{1,\alpha} ^{2}}\right)\right)+\epsilon_{1}\exp\left(-4\ln 2\left(\frac{(r-r_{1,b})^{2}}{ \sigma_{1,b}^{2}}\right)\right), \tag{17}\]  
where two Gaussian functions describe the radial contributions corresponding to CH\({}_{3}\) fragments in the two vibrational states \(\
u=0\) and \(\
u_{1}=1\), and with \(\epsilon\) representing the fraction of the population of CH\({}_{3}\) in \(\
u_{1}=1\). Femtosecond pump-probe VMI images contain, in general, some other low intensity secondary contributions, which can arise from different mechanisms, as, for example, absorption of two pump photons, relaxation of the molecule over another state which has lower absorption probability, detection by non-resonant ionization or background gas contamination. The secondary contributions have to be considered in the fitting procedure if they are overlapped with the contribution or contributions of interest. In fact, a good reproduction of the secondary signals allows the isolation of the contribution of interest from the rest by subtraction of the previously fitted secondary signals. If these secondary contributions are neglected, then an error is generated in the fit with a magnitude corresponding to the sum of the secondary signals in the spatial overlap region. In the present case example [38], the secondary contributions can be grouped into two classes. The first, C\({}_{2}\), appears in a broad range of velocities modeled by an asymmetric Gaussian function,i.e., with different widths (\(\sigma_{2,a}\) and \(\sigma_{2,b}\)) at each side of the maximum. It presents a \(\beta_{2}\) and \(\beta_{4}\) anisotropy. This contribution appears quickly at short delay times and then exponentially decreases with a constant time \(\tau_{d,2}\).