# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments
## 2 Construction of a Sequence of Velocity Map Images and Description of Its Time Evolution
Multichannel processes lead to images \(I\) with a collection of contributions \(C_{i}\) like the one described above. In the case that the different contributions do not interfere with each other, the experimental signal can be described by the following sum:  
\[I=\sum_{i}a_{i}C_{i}, \tag{3}\]  
Figure 1: 2D representation of (**a**) the radial distribution given by Equation (1) with parameters \(r_{c}=\)40 px and \(\sigma_{c}\)=3 px; (**b**) the angular distribution given by Equation (2) using \(\beta_{2}=2\) and \(\beta_{2n>2}=0\); (**c**) the product of the radial and angular distributions shown in (**a,b**), and (**d**) the corresponding Abel projection of (**c**). where \(a_{i}\) is the amplitude of each contribution. In the case that interferences occur, Equation (3) would include additional interference terms. Let us now consider the particular case of a channel for which the shape of the velocity and angular distributions does not change with time. In that case, the temporal evolution \(T(t)\) of the channel is represented just by an amplitude factor of the contribution that changes with time. For instance, in the case of a contribution which appears from the depopulation of an excited state (described by an exponential-type growth of the amplitude factor), the temporal behavior can be modeled as follows:  
\[T(t)=\int_{-\infty}^{\infty}\exp\left(-4\ln 2\frac{(t-x)^{2}}{\tau_{xx}^{2}} \right)\times\left[H(x-t_{0})\times\left(1-\exp\left(-\frac{x-t_{0}}{\tau_{d} }\right)\right)\right]dx, \tag{4}\]  
or in a more symbolic form,  
\[T(t)=I_{xc}(t)\otimes\left[H(t-t_{0})\times\left(1-e^{-\frac{t-t_{0}}{\tau_{d} }}\right)\right], \tag{5}\]  
where \(H(t)\) is the Heaviside step function (for which \(H(t<0)=0\) and \(H(t>0)=1\)) and the temporal resolution corresponds to the cross-correlation between the two laser pulses \(I_{xc}(t)\), modeled by a Gaussian function with a full width at half maximum of \(\tau_{xc}\), which is convoluted by the dynamics of the population of the state. The variable \(t_{0}\) represents the central pump-probe delay time for which the contribution appears, and \(\tau_{d}\) is the time constant of the exponential. Therefore, the complete 3D description of a contribution \(C_{i}(r,\alpha,t)\) can be written as  
\[C_{i}(r,\alpha,t)=R_{i}(r)A_{i}(\alpha)T_{i}(t). \tag{6}\]  
This contribution is illustrated in Figure 2.  
In the particular case described above, time appears completely decorrelated from the other dimensions. However, in a more general case, some parameters of the radial or angular distributions can change with time. For example, the anisotropy can relax, the central position of the ring can shift or its width can vary. Hence, the contribution can be written in a more general way as:  
\[C_{i}(r,\alpha,t)=R_{i}(r,t)A_{i}(\alpha,t)T_{i}(t), \tag{7}\]  
Figure 2: Time evolution of a contribution with a temporal behavior described by Equation (5) with \(\tau_{d}=1\) ps, \(t_{0}=250\) fs and where \(I_{xc}(t)\) is a Gaussian function with a full-width-at-half-maximum (FWHM) of \(\tau_{xc}=300\) fs. The images shown below correspond to the contribution depicted in Figure 1 at selected delay times. The images labelled (**a**–**d**) correspond to the time delays depicted in the transient shown in the top panel. where now both \(R\) and \(A\) become functions that depend on time. The \(T_{i}(t)\) function could be contained in the functional form of \(R_{i}\) or \(A_{i}\), but we have chosen to keep it explicitly written in Equation (7) for convenience. This formalism can be generalized in a straightforward manner to any image scan as a function of an arbitrary observable, such as wavelength or temperature, for example, thus extending the dimensionality of the problem. In some cases, the study of the dependence of the velocity distribution with time can be sufficient to extract all the relevant information of the pump-probe VMI experiment, with no additional insight to be gained from the anisotropy.