# Distortion of Thomson Parabolic-Like Proton Patterns Due to Electromagnetic Interference
## 3 Processing Antenna Signals and TP Snapshots  
The obtained data by both the TP and the antenna were processed by a MATLAB script specifically written for this purpose. First, the parabolic curve (1) that overlapped the track of protons was plotted in the TP snapshot. In such a way, one can read the intensity of pixels associated with the parabolic curve, and plot this intensity as a function of the horizontal position \(x\) with respect to the origin of the parabola (i.e., to the zero-deflection point). Then, the velocity of ions \(v\) was derived for every point of the parabola (i.e., for each pixel) from the general equation for magnetic deflection [9]:  
\[v=\frac{qBL_{/B}}{xm_{0}}\Big{(}\frac{L_{fB}}{2}+L_{rB}\Big{)}, \tag{2}\]  
where \(x\) stands for the distance between the origin of the parabolic track and the actual position of an ion in the detector plane due to its deflection in the magnetic field. By knowing the distance between target and recording system, the corresponding proton time of flight (TOF) was calculated. Therefore, the dependence of the pixel intensity (along  
Figure 2: **(a)** Experimental setup at PALS laser facility showing position of the TP spectrometer and plane antenna. **(b)** Typical TP snapshot showing the parabolic track of protons distorted due to the electromagnetic pulse (EMP) in the Cartesian coordinate system. the parabolic curve) on the TOF of protons was derived for each point of the parabolic track as can be seen in Figure 3. This calculation shows which part of the EMP signal influenced the TP, because the antenna (recording the EMP signal) was positioned directly on the TP electrodes. As Figure 3 shows, the signal recorded by the antenna between 0 and 243 ns (the edge of the detector) corresponds to the entire parabolic track of protons from zero-deflection point up to the edge of MCP. Since the proton signal occurred only between 120 and 243 ns in this particular measurement, only the part of the EMP signal between 120 and 243 ns was considered in further processing. This part of the signal is highlighted with red in Figure 4, showing the measured antenna signal. The EMP that affected the TP before the fastest protons (in our case, before 120 ns) can also be included in analysis. This would basically also predict the distortion of the parabolic track in the region where no protons were detected in this experiment. Nevertheless, analysis was mainly focused on the region between 120 and 243 ns because the results could be compared with the measured parabolic track of protons. Figure 4: Detected EMP signal (blue curve). Its part highlighted with red corresponds to the TOF derived from the parabolic track of protons in Figure 3 from 120 to 243 ns; (inset) 9 samples into which the signal corresponding to TOF was divided. Figure 3: Intensity of pixels along the parabolic track of protons as a function of time of flight (TOF) derived from the velocity corresponding to each point of the parabolic curve. Subplot shows the TP snapshot in grayscale with the parabolic curve used for reading pixel intensity. The edge of the MCP screen is clearly visible. The TOF corresponding to the edge was taken as a point where the EMP no longer influenced the proton trajectories. The red part of the recorded EMP signal (i.e., the part corresponding to detected protons) in Figure 4 was divided into 9 samples. The length of each sample must be sufficient to retrieve its frequency spectrum using fast Fourier transform (FFT). This condition together with the length of the red part naturally led to the number of samples (in this case, N = 9). After applying the FFT on each time sample, we found 3 highest peaks corresponding to the 3 main frequencies \(f_{i}^{1},f_{i}^{2},f_{i}^{3}\) included within the i-th time sample. In addition, phases \(\phi_{i}^{1},\phi_{i}^{2},\phi_{i}^{3}\) and amplitudes \(A_{i}^{1},A_{i}^{2},A_{i}^{3}\) were obtained.