## 2 Theoretical Background  
The fundamental mechanism of emission considered here is the Transition Radiation (TR) effect as generated when a relativistic charged particle crosses the boundary between two media of different dielectric properties, e.g., a vacuum-metal screen interface [21]. TR originates from the time varying currents induced, on the target surface, by the electromagnetic field of the traveling charges, due to optical inhomogeneities in the space. Assuming a perfectly conducting flat screen with rectangular \(h\times l\) dimensions, the spectral and spatial intensity distribution from a single electron \(\frac{d^{2}I_{sp}}{d\omega d\Omega}\) is well described by the virtual-photon method [22]:  
\[\frac{d^{2}I_{sp}}{d\omega d\Omega}=\frac{1}{4\pi\epsilon_{0}}\frac{e^{2} \omega^{2}}{4\pi^{2}\epsilon}\left(|E_{x}|^{2}+|E_{y}|^{2}\right), \tag{1}\]  
where \(E_{x(y)}\) is the horizontal (vertical) components of the emitted field:  
\[ \begin{split} E_{x(y)}(k_{x},k_{y})=&\frac{4 ek}{2 \pi^{2}\epsilon\textit{R}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dx_{x} d\textit{x}_{y}\frac{\kappa_{x(y)}}{\kappa_{x}^{2}+\kappa_{y}^{2}+\alpha^{2}} \frac{sin(\frac{1}{2}(k_{x}+\kappa_{x})}{k_{x}-\kappa_{x}}\\ &\left\{\frac{sin[\frac{1}{2}(k_{y}-\kappa_{y})]}{k_{y}-\kappa_{ y}}-\frac{sin[\frac{b}{2}(k_{y}-\kappa_{y})]}{k_{y}-\kappa_{y}}\right\},\end{split}  \tag{2}\]with \(\alpha=\frac{\omega}{\beta C\tau}\), \(R\) is the distance between the center of screen and the observation point, while \(k_{x}\) and \(k_{y}\) are the projections of the light wavevector \(\vec{k}\) on the \(x\)-\(y\) plane. When considering a bunch of spatially distributed \(N\) charged particles, each particle emits TR on the screen with the same intensity, while the relative phase difference depends on the position within the bunch and on the radiation emission direction. The total emitted intensity can be written as  
\[\frac{d^{2}I}{d\omega d\Omega}=\frac{d^{2}I_{sp}}{d\omega d\Omega}[N+N(N-1)F( \omega)] \tag{3}\]  
where \(F(\omega)\) is the longitudinal form factor of the bunch which is the squared of Fourier transform of the longitudinal charge distribution. The total emitted energy \(\frac{d^{2}I}{d\omega d\Omega}\) shows two different regimes: (i) at wavelengths \(\lambda=2\pi c/\omega\) smaller than the longitudinal electron bunch size, typically tens or few hundreds of microns, \(F(\lambda)\) tends to 0 and the spectrum is dominated by incoherent emission (whose intensity is proportional to the number of particles \(N\) in the bunch); (ii) at longer wavelengths, with respect to the bunch length, emission starts to be coherent (corresponding to an intensity proportional to \(N^{2}\)) because each electron emits in phase and the amplitudes add constructively: \(F(\lambda)\) tends to 1. For sub-ps electron beams the coherent regime occurs at THz range. From this point of view, the \(N^{2}\) dependency of coherent radiation results in a huge intensity gain with respect to most existing THz sources. For a single gaussian-shaped electron beam, shown in Figure 1a, the calculated spectrum combining Equations (2) and (3) is depicted in Figure 1b by a dotted red line. As one can observe, a single electron bunch with a 100 fs (RMS) pulse duration generates a broadband THz spectrum up to 6 THz. However, the emitted spectrum suffers of a low frequency cut-off due to finite screen dimension [22], which occurs when natural extent of the particle field \(\gamma\lambda/2\pi\) exceeds the screen size. In particular, as observed in Figure 1b, for a \(30\times 30\) mm the low frequency cut-off is around 0.3 THz. If a longitudinally modulated electron beam, called comb beam, with THz repetition rate is transported to the TR screen, narrowband emission occurs at discrete frequencies multiple to the inverse of the pulse train inter-distance.