# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments
## 2 Construction of a Sequence of Velocity Map Images and Description of Its Time Evolution  
The first demonstration of charged-particle imaging applied to reaction dynamics was made by [PERSON] and [PERSON]. In their 1987 work [1], they showed how it is possible to record, at once, the entire spatial distribution of fragments originating from a photodissociation event. In this way, within the resolution limits, it is possible to directly measure the angular and velocity distributions of the products of a chemical reaction. The technique of ion/photoelectron imaging made a giant leap a decade later with the discovery by [PERSON] and [PERSON] of the technique that is known as velocity map imaging [2], where through the use of open-lens electrodes with appropriate voltages, it is possible to work in a configuration where products of a photodissociation event with the same initial velocity vector are imaged onto the same position in the detector. This implies that the observed images are in fact 2D projections onto the plane of the detector of a velocity distribution on a spherical surface. Inversion techniques, such as the inverse Abel transform [4; 28; 29; 30; 31], need to be implemented in order to extract the true distribution from the 2D projection. Further developments of this technique led to the discovery of slice imaging for ions [32; 33; 34], where the extraction and detection conditions are such that only the central slice of the distribution on the sphere is detected, eliminating the need for inversion procedures that invariably introduce additional noise and required the existence of cylindrical symmetry in the interaction. A typical image acquired in this type of experiment, either raw (through slice imaging), or, equivalently, mathematically inverted (through velocity map imaging), contains, in general, a set of \"contributions\" in the form of isotropic or anisotropic rings, by which we mean each of the possible processes or channels associated with a given ion or photoelectron. Typically, a \"channel\" is characterized by ions or photoelectrons with a given velocity (or center-of-mass kinetic energy) distribution, which, on the image, can be modeled, for instance, by a radial Gaussian function of the form  
\[R(r)=\exp\left[-4\ln 2\left(\frac{r-r_{c}}{\sigma_{c}}\right)^{2}\right], \tag{1}\]where \(r_{c}\) is the radial distance from the center of the image and \(\sigma_{c}\) is the full-width-at-half-maximum (FWHM) of the contribution. Figure 1a shows an example of such a contribution in the form of an isotropic ring with parameters \(r_{c}=40\) and \(\sigma_{c}=3\) in units of pixels (px) of the CCD camera, which are proportional to the speed of the charged species. The angular distribution (anisotropy) of charged particles for a given radius provides additional information on the nature of the channel. In the case of inversion symmetry, the anisotropy \(A\) can be written as  
\[A(\alpha)\propto 1+\sum_{n}\beta_{2n}P_{2n}(\cos\alpha), \tag{2}\]  
where \(\alpha\) is the angle between the polarization axis of the electric field and the considered direction and \(n\) takes maximum values of 1 and 2 for one-photon and two-photon processes, respectively. Legendre polynomials, \(P_{2n}(\cos\alpha)\), represent a complete angular basis set, which has the advantage that only few terms in Equation (2) are generally sufficient to describe the ring anisotropy. A 2D plot of this function using \(\beta_{2}=2\) and \(\beta_{2n>2}=0\) is shown in Figure 1b. A channel can be fully described by the product of the velocity and angular distributions, \(R(r)A(\alpha)\). Figure 1c shows the result of the product of the 2D radial and angular representations shown in Figure 1a,b. The corresponding raw velocity map image can be simulated by applying the Abel projection [4] and the result is shown in Figure 1d. This simulation of a VMI image has been obtained assuming cylindrical symmetry on the 3D distribution, for which Figure 1c is the central slice.