Figure 1

Geometry of a high energy particle shower. The top figure shows a side view ($r-z$ plane) of the charge excess profile. The shower front propagates at velocity $v$ and is modeled as a thin pancake $\delta (z^{\prime }-v{t}^{\prime })$. The charge evolution of the shower front traces out the longitudinal profile $Q(z^{\prime })$ shown here as an asymmetric distribution. The lateral spread of the shower due to Coulomb scattering, shown in (blue) arrows, is modeled by a velocity vector $\mathbf{v}=\mathbf{v}(r^{\prime },{\varphi }^{\prime })$ that is mostly directed along the shower axis with small radial components. The scatter results in observation angles that differ from the nominal angle ${\theta }_z$ relative to the shower axis and will lead to an asymmetric pulse. The radiative portion of the electric field lies along the vector $\mathbf{p}$ which is the orthogonal projection of $\widehat{\mathbf{z}}$ along the direction $\widehat{\mathbf{u}}$ from the source to the observer. The bottom panel shows a frontal view ($r-\varphi$ plane) of the shower. The lateral charge excess distribution $f(r^{\prime })$ is represented in (blue) shading.Reuse & Permissions

Figure 2

The vector potential from electromagnetic showers in homogeneous ice (density $\rho =0.924\mathrm{g}{\mathrm{cm}}^{-3}$ and refractive index $n=1.78$, ${\theta }_C\sim 55.8°$) observed at the Cherenkov angle for various energies from the ZHS simulation. The functional behavior is identical except for an overall scaling factor that is directly proportional to the shower energy. The width is determined by the lateral distribution of the shower while the asymmetry is mainly due to the radial spread of particle tracks; both are the result of the Coulomb scattering in the medium.Reuse & Permissions

Figure 3

Results of the ZHS Monte Carlo (full lines) compared to our calculation (dashed lines) Eq. (29) for the radiation due to a $3\times {10}^{18}\mathrm{eV}$ electromagnetic shower observed at $\theta ={\theta }_C-0.3°$ in the far-field in ice. The method is applied to the charge excess longitudinal profile obtained from the same simulation. Top panel: Vector potential in the time-domain with 10 ps time sampling (corresponding to a sampling frequency of 100 GHz) as obtained in ZHS simulations. This is compared to the calculation presented in this work. The longitudinal charge profile is shown as the 1D model presented in 29 (lighter lines) where the depth and charge are linearly rescaled to give the observer time and vector potential. Middle panel: Electric field in the time-domain comparing our method to ZHS results. Bottom panel: The electric field amplitude spectrum obtained from the ZHS simulation and the Fourier-transform amplitudes of the electric field obtained from the calculation presented in this work. Note that the discrepancy between the time-domain electric field of the ZHS simulations and our results are due to the incoherent radiation at high frequencies.Reuse & Permissions

Figure 4

Sketch illustrating the polarization dependence with time. As the shower evolves the observer sees the radiation contributions at different angles corresponding to different source positions $z^{\prime }$. This means that the observer will see the polarization vector changing as a function of time $t$.Reuse & Permissions

Figure 5

The Fresnel region vector potential for an electron-induced 100 PeV shower in ice from the ZHS Monte Carlo (dashed lines) compared to the results of Eq. (22) (solid line beneath the dashed lines). The longitudinal profile of the excess charge is convolved with the form factor $F_p$. The observers are located at $(x,y,z)$ where $z$ lies along the shower axis and $y=0$. The vector potentials have been arbitrarily shifted in time by multiples of 50 ns for clarity. Pulses from left to right associated to observation points with decreasing z. The spiked vector potential in the middle (blue) corresponds to an observation where the peak of the longitudinal shower profile is near the Cherenkov angle. The vector potentials to the right of it (green and red) are due to observations above the Cherenkov angle while the vector potentials to the left (magenta and yellow) are due to observations below the Cherenkov angle. Note the inverted order of the primary and secondary peaks.Reuse & Permissions

Figure 6

Illustration of the vector potential characteristics due to the apparent motion of the charge distribution for an observer seeing the region around shower maximum with angles close to the Cherenkov angle in ice ($n=1.78$). In the bottom plot we show the longitudinal charge excess profile following a Greisen parameterization. The center plot shows the observer time as a function of the source time for a shower exceeding the speed of light in a medium. The solid (red) portion, corresponding to the minimum of the source and observer time relation, contributes significantly to the radiation near the Cherenkov peak. The resulting vector potential is shown in the right hand side. The Cherenkov radiation spike corresponds to the compressed mapping of the charge excess distribution to the observer time.Reuse & Permissions

Figure 7

Apparent motion analysis of the charge distribution for the vector potentials shown in Fig. 5. Bottom panel: Longitudinal profile of the excess negative charge in the 100 PeV electron-induced shower. Middle panel: Observer time $t$ vs source position $z^{\prime }$ for the same observers as in Fig. 5. Top panel: Angle with which each observer sees the different positions $z^{\prime }$ in the shower axis. Lines from top to bottom associated to observation points with increasing z. The Cherenkov angle (${\theta }_C=55.8°$ in ice) is indicated with a long-dashed horizontal line. See text for a description of the mapping of these relations to the features of the vector potentials shown in Fig. 5.Reuse & Permissions