# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments

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## Abstract

**:**

**2010**, 133, 234313). We show that the multidimensional approach is essential for the analysis of complex images that contain several overlapping contributions where reduced dimensionality analyses cannot provide a reliable description of the features present in the image sequence. This methodology can be generalized to many types of multidimensional data analysis.

## 1. Introduction

## 2. Construction of a Sequence of Velocity Map Images and Description of Its Time Evolution

## 3. Numerical Fitting Procedure

## 4. Case Example: Analysis of a Femtosecond Pump-Probe VMI Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**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**).

**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 ${I}_{xc}\left(t\right)$ 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.

**Figure 3.**(

**a**) 2D representation of the signal shown in Figure 2 angularly integrated as a function of radius and time; (

**b**) velocity distribution (in pixels of the CCD camera) extracted from (

**a**) at a fixed delay time of 5 ps.

**Figure 4.**(

**a**,

**d**) experimental Abel inverted velocity map imaging (VMI) images recorded for the I*(${}^{2}{P}_{1/2}$) 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.

**Figure 5.**I*(${}^{2}{P}_{1/2}$) 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

**$\xb7-$**)) are shown.

**Figure 6.**2D representation corresponding to the angular integration of the I*(${}^{2}{P}_{1/2}$) 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.

**Figure 7.**1D transient for the amplitude of the main contribution as a function of time. (○) experimental data and (

**-**) fit. See text for more details.

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**MDPI and ACS Style**

Loriot, V.; Bañares, L.; De Nalda, R. Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments. *Appl. Sci.* **2018**, *8*, 1227.
https://doi.org/10.3390/app8081227

**AMA Style**

Loriot V, Bañares L, De Nalda R. Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments. *Applied Sciences*. 2018; 8(8):1227.
https://doi.org/10.3390/app8081227

**Chicago/Turabian Style**

Loriot, Vincent, Luis Bañares, and Rebeca De Nalda. 2018. "Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments" *Applied Sciences* 8, no. 8: 1227.
https://doi.org/10.3390/app8081227