# Distortion of Thomson Parabolic-Like Proton Patterns Due to Electromagnetic Interference
## 3 Processing Antenna Signals and TP Snapshots
Lastly, this physical quantities (frequencies, phases, and amplitudes) were used to calculate the modulated electric field \(E_{m}\) in the model developed in MATLAB. In addition, frequencies \(f_{i}^{1},f_{i}^{2},f_{i}^{3}\), which were found in time sample \(t_{i}\), contributed to the modulation of the electric field only within time sample \(t_{i}\), and their contribution was cancelled by setting \(t_{i}=0\) outside this particular time sample. In such a way, the track of protons in time sample \(t_{i}\) was distorted only by the electromagnetic waves that affected the TP while these protons were traversing the electric field. The other bunch of protons (e.g., related to time sample \(t_{i+1}\)) were influenced by waves that were interacting with the TP within time sample \(t_{i+1}\). The modulation of electric field \(E_{m}\) can be thus expressed as a function of time corresponding to the TOF of protons. The modulation itself is considered to be a sum of sine waves because the sources of modulations are electromagnetic waves generated during laser-target interaction. Assuming the linear transfer of modulation from EMP to electric field, sinusoidal modulations with main frequencies (and corresponding phases and amplitudes) were added to the constant term of the electric field. The relation for final electric field \(E(t)\) in the model may be written as  
\[E(t)=E_{o}+K\cdot E_{m}(t,f_{i=1,\dots,9}^{i=1,\dots,3},\varphi_{i=1,\dots,9}^{ i=1,\dots,3})=E_{o}+K\cdot\sum_{i=1}^{9}\sum_{j=1}^{3}A_{i}^{j}\cdot\sin(2\pi f _{i}^{j}t_{i}+\varphi_{i}^{j}), \tag{3}\]  
where \(E_{0}\) is the electric field applied to the electrodes, and \(K\) is the constant that may be changed in order to amplify the influence of modulation. This constant was less important, since we only investigated the similarities of the frequency. Resulting electric field \(E(t)\) was no longer constant and led to the distortion of parabolic tracks on the recording system. In particular, the electric field (3) was substituted in Equation (1), which is plotted in the TP snapshot. Additionally, the captured track of protons was extracted from the snapshot by finding the maximum of intensity in a neighborhood of the parabolic track of protons. Eventually, both the extracted parabolic and the modulate track could be plotted. Both curves are shown in Figure 5 with the TP snapshot converted into grayscale in the Cartesian coordinate system. Figure 5: (**a**) Extracted parabolic track of protons as measured. (**b**) Modulated parabola plotted in MATLAB by introducing a varying electric field (3) into Equation (1). We now demonstrate that the signal detected by the antenna was the same as the one that influenced the TP electrodes. The generated EMP on the target could propagate through the vacuum pipe connecting the TP spectrometer to the interaction chamber. Therefore, the cutoff frequency of the waveguide with a circular cross-section (radius of 50 mm) was analytically derived in order to understand whether the EMP could affect the TP electrodes from the inside. The cutoff frequency of the TE\({}_{11}\) mode (the fundamental transverse electric mode of circular waveguide) was calculated as \(\approx\)1.75 GHz, which means that any electromagnetic wave of lower frequency could not propagate through the vacuum pipe towards the spectrometer. Thanks to the EMP characterization already performed at PALS, the main portion of the EMP frequency spectrum could be found below 1.5 GHz [7; 10]. Hence, the antenna placed on TP electrodes recorded the EMP causing the distortion of parabolic tracks.