## 3 Betatron Radiation Sources
We consider a quadratic energy gain, as in [33], and averaging Equation (13) on the acceleration length we can infer an effective value (one for both directions) for the betatron oscillation amplitude equal to \(\sim\)0.2 \(\upmu\)m, where in Equation (13) we have replaced \(E_{c}\) with the measured mean value of 0.78 keV as well as replacing \(\gamma(L/c)\simeq 468\) given the results in Figure 7. Figure 8 also shows a simulation performed using analytical formulas of synchrotron-like betatron radiation, including electron acceleration [33]. The parameters used for the simulation have been: rms betatron oscillation amplitude equal to \(\sim\)0.2 \(\upmu\)m (gaussian beam), electron final Lorentz factor at \(z=L=1.7\) mm equal to 468, and finally the laser and plasma parameters used in the experiment. The effect of the electron energy spectrum and of the radial distribution of the beam has been obtained by summing incoherently the betatron radiation emission from a statistical sample of 10,000 electrons, representing the total charge of 373 pC. The simulation is reasonably in agreement with the results, even if the synchrotron-like approximation seems to overestimate the low-energy photons. The mean number of measured photons per shot in the range \(>\)2 keV was \(\sim\)\(4\times 10^{5}\), with a fluctuation of the 50 %. The detector response (a CCD X-ray camera operated in single photon counting) for lower photon energies was not reliable so it was neglected, due to thermal noise. The used camera was a Great Eyes 1024 256, with an image area of 26.6 mm \(\times\) 6.7 mm, a pixel size of 26 \(\upmu\)m \(\times\) 26 \(\upmu\)m, operated at a pixel readout frequency of 500 kHz and a temperature of the sensor equal to \(-20^{\circ}\). The algorithm used to retrieve the photon spectrum has been based on the determination of isolated illuminated pixels on the sensor matrix and in the construction of the histogram of their values. This algorithm can lead to a slight underestimation of photons, in general, but this was not the case in our experiment, as no significant contiguous events were detected on the sensor. The process has been implemented after background subtraction and considering only photons above the thermal noise threshold. Moreover, a deconvolution with the transmission function of 2 \(\upmu\)m thick Al window in front of the sensor (to reflect spurious laser light preventing the sensor damage) has been performed. If simulated within a synchrotron-like approximation, as in shown in Figure 8, the number of measured photons would correspond to 1% of the total photon number, which is \(\lesssim\)10\({}^{8}\). The missing photons, constituting the majority of emitted radiation, have not been detected as they fell below 2 \(keV\). The scaling for the total photon number can be expressed as [33]:  
\[N_{\beta}=\left(\frac{\gamma_{0}}{\gamma(t)}\right)^{\frac{1}{2}}\frac{2\pi \alpha Q\{x_{\beta},y_{\beta}\}\omega_{p}^{2}L}{18 ec^{2}}\propto Qn_{e}L\gamma^ {-\frac{1}{2}} \tag{14}\]  
Considering the fluctuations of the quantities upon which the betatron photon number scales, we obtain an overall fluctuation \(\sim\)45 %, which explains rather well the observed results (still neglecting the fluctuations on the betatron oscillation amplitudes). Once evaluated for the parameters of our experiment, Equation (14) yields a photon number \(\lesssim\)10\({}^{8}\) for a betatron amplitude \(\sim\)0.2 \(\upmu\)m, which is in agreement with our previous estimation based on a simulation of a synchrotron-like spectrum (see Figure 8).