## 3 Betatron Radiation Sources
All the above equations make a self-consistent set to calculate the number of photons \(dN\) emitted in the frequency bandwidth \(d\omega\), through the Lienard-Wiecher formula [34] starting from the magnetic field at Equation (8):  
\[\frac{dN}{d\omega}=\frac{q^{2}\omega}{16\pi^{3}\varepsilon_{0}\hbar c^{3}} \int d\Omega dx_{\beta}dy_{\beta}d\theta_{x}d\theta_{y}d\varphi_{0}d\gamma_{0} \rho_{beam}\Bigg{|}\frac{\vec{R}}{R}\times\int_{t+\frac{\phi_{0}}{\omega_{ \beta}}}^{L/c}dt\vec{v}(t)e^{i\omega t-\vec{k}\cdot\vec{\tau}_{e}}\Bigg{|}^{2} \tag{12}\]  
where \(\rho_{beam}=\rho_{beam}(x_{\beta},y_{\beta},d\theta_{x},d\theta_{y},\phi_{0}, \gamma_{0})\) is the trace space density of the particle beam at the injection plane. The lower limit of the time integral must depend on \(\phi_{0}\), since in injection schemes as the self-injection the particles can be continuously injected all through the laser propagation in the plasma, so that they can have very different acceleration lengths (this is source of energy spread at the end of the acceleration process). Despite Equation (12) not being possible to be solved analytically, in general, asymptotic limits can be guessed for particles such that \(\gamma v_{x,y}/c>>1\). In that case, \(v_{x,y}\exp\left(i\omega t-\vec{k}\cdot\vec{\tau}_{e}\right)\propto K_{2/3}( \omega/\omega_{c})\) for radiation emitted on axis. The modified Bessel function \(K_{n}(x)\) of the second kind presents a parameter called critical energy \(\omega_{c}\), determining the shape of the betatron spectrum. For synchrotron radiation, the same parameter divides the spectrum in two regions, each carrying half of the total energy irradiated. Choosing a description of the betatron spectrum as a synchrotron-like spectrum, we expect for high-frequencies (\(\omega>>\omega_{c}\)) that \(dN/d\omega\propto e^{-\omega/\omega_{c}}\). Figure 8 shows experimental data of measured betatron spectra at VEGA 2. The critical energies reported in the legend \(E_{c}=\hbar\omega_{c}\) have been fitted with the exponential function introduced above. The curves have been represented in order of consecutive shots, where a higher critical energy does not necessarily correspond to a higher number of emitted photons due to the fluctuations of other experimental parameters. The most critical parameter determining instability for critical energy is the electron energy, since critical energy scales non-linearly with it. In the following, we provide the scaling of critical energy upon all the experimental parameters. Experimentally, a shot-to-shot fluctuation of \(E_{c}\) up to 40% has been observed. An expression for this parameter can be found [28,33]:  
\[E_{c}=\frac{3}{2}\hbar\gamma^{3}(t)\bigg{(}\frac{\gamma_{0}}{\gamma(t)}\bigg{)} ^{\frac{1}{4}}\frac{\{x_{\beta},y_{\beta}\}\omega_{\beta}^{2}}{c}\propto\gamma (t)^{\frac{2}{4}}n_{e} \tag{13}\]  
Figure 8: Measurements of betatron radiation spectra. For “Betatron Photon Number”, \(\sim(dN/d\omega)\times\Delta\omega\) with \(\Delta\omega=0.13\,keV\), which the resolution of the diagnostic apparatus. A fluctuation of the mean energy \(\sim\)10%, such as that shown by the data of Figure 7, together with an energy spread of 13%, is translated already in a critical energy fluctuation of \(\sim\)40%. Adding the measured density fluctuation of 5% would not change the result much; therefore, we believe that the main sources of critical energy instability were the energy gain and the energy spread of the electron beam. Moreover, in the above reasoning, we neglected fluctuations of the betatron oscillation amplitudes, which apparently are also less important than the energy/spread fluctuations, since the latter can already explain the observed shot-to-shot variation of the critical energy of the betatron spectrum. The mean critical energy for the shots in Figure 8 was 0.78 keV.