## 3 Betatron Radiation Sources  
Plasma wakefield acceleration has also been demonstrated to be a promising platform for producing secondary radiation [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. The compactness of plasma wakefield accelerators is correlated with the shortness of the accelerating buckets so that only very short bunches of particles can be accelerated down to a few femtoseconds [27; 28]. This means that the radiation pulses secondarily emitted by these particles will also be short, delivering high peak power. The fact that plasma wakefield accelerators produce for ultra-relativistic electrons also makes possible the emission of intense bursts of light with extremely interesting angular and spectral properties, e.g., the high-energy photon emission within narrow solid angles. Here, we focus on incoherent radiation, which is the case for the radiation emitted in plasma wakefield accelerators by beams undergoing betatron oscillations [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. The radiation field far from the source is found as solution of the Maxwell equations [19]:  
\[\vec{H}(\vec{R},\omega)=\frac{\omega}{4\pi c}\frac{e^{i\omega\vec{R}}}{R^{2}} \vec{R}\times\widetilde{\vec{j}}(\vec{k},\omega) \tag{8}\]  
where the observation vector is \(\vec{R}=R\{\cos\varphi\sin\theta,\sin\varphi\sin\theta,\cos\theta\}\), with \(\theta\) the polar angle and \(\varphi\) the azimuth. The Fourier-Laplace transform of the current density \(\vec{j}(\vec{r},t)=q\delta(\vec{r}-\vec{r}_{e})\) associated with a single electron, whose motion is described by \(\vec{r}_{e}=\vec{r}_{e}(t)\), evaluated in the spatial-frequency/time domain, is:  
\[\vec{j}(\vec{k},t)=q\frac{d\vec{r}_{e}}{dt}e^{-i\vec{k}\cdot\vec{r}_{e}} \tag{9}\]  
The trajectory of a generic electron injected in the plasma accelerator is given by [33]:  
\[\vec{r}_{e}=\left\{\left(\frac{\gamma_{0}}{\gamma(t)}\right)^{\frac{1}{4}}x_{ \beta}\sin\left(\omega_{\beta}t+\psi_{x}+\varphi_{0}\right)\right.\left(\frac{ \gamma_{0}}{\gamma(t)}\right)^{\frac{1}{4}}y_{\beta}\sin\left(\omega_{\beta}t +\psi_{y}+\varphi_{0}\right)\right.\int_{0}^{t}dtc\sqrt{1-\frac{1}{\gamma^{2} (t)}-\frac{v_{x}^{2}+v_{y}^{2}}{c^{2}}} \tag{10}\]  
where the velocity of the particle is \(\vec{v}_{e}=\{v_{x},z_{y},v_{z}\}=d\vec{r}_{e}/dt\) and the transverse coordinates (i.e., the first two components of \(\vec{r}_{e}\), since we have chosen \(z\) as beam axis) represent the so-called betatron oscillations occurring at frequency \(\omega_{\beta}=\omega_{p}/\sqrt{2\gamma(t)}\). The Lorentz factor of the particle gaining energy from the plasma wakefield is \(\gamma(t)\), while \(\gamma_{0}\) is the initial value for the same particle at the injection plane. The phase shift, related to initial injection condition for the electron, is for the two planes:  
\[\psi_{x,y}=\arctan\left[\left(\frac{\gamma_{0}}{\gamma(t)}\right)^{\frac{1}{4} }\frac{\{x_{\beta},y_{\beta}\}\omega_{\beta}}{c\vartheta_{x,y}}\right] \tag{11}\]  
Figure 7: **(Left)**: Example of electron beam energy spectrum on the diagnostic scintillator placed after the bending magnet, plotted in vertical-divergence/MeV units **(Right)**: shot-to-shot fluctuation of the beam energy and rms energy spread; the uncertainty is determined by standard deviation of the measurements. where \(\theta_{x,y}=v_{x,y}/v_{z}\) is the injection angle of the electron into the plasma wakefield. With writing \(\{x_{\beta},y_{\beta}\}\) we mean \"either \(x_{\beta}\) or \(y_{\beta}\), according whether the \(x\) or the \(y\) direction is considered, respectively\". The phase \(\phi_{0}\) is instead the phase retardation with respect to a reference particle, which accounts for the fact that not all particles are injected at the same time.