## EOS Detector  
The diagnostics based on EOS enable time-resolved measurements of relativistic electron charge, mean energy, and temporal length [40; 41]. The EOS diagnostics are based on a 500 \(\upmu\)m thick ZnTe crystal. By adopting the spatial encoding technique [42] and the probe laser intersecting the crystal at \(\theta_{i}\approx 28^{\circ}\) incidence angle, the emitted electron's temporal profile is mapped onto the probe laser transverse profile. The temporal resolution of our experimental setup is about 100 fs with a time window of about 10 ps. The Coulombian electric field encoded in the EOS-probe signal is retrieved adopting a polarizing optics placed after the ZnTe crystal, as shown in Figure 10. The polarizer is oriented perpendicularly (90\({}^{\circ}\)) to the input laser polarization, resulting in no light transmission when there is no EO-induced phase retardation. This setup facilitates the conversion of probe polarization modulation into intensity modulations, with the diagnostic signal easily detectable using a CCD camera. Figure 11 shows the electric field encoding process. The CCD camera records the signal generated by the total overlap, namely temporal and spatial, of the probe laser and the locally induced birefringence by the interaction yield electric field. The bunch generates an electric field, which induces the electro-optic effect, proportional to its charge. The latter can be evaluated from the intensity of the signal, thus the EOS diagnostics act as a temporally resolved charge measurement with a resolution on the order of femtosecond. In addition, in the EOS signal width is encoded with the temporal length of the bunch as shown in Figure 11 and described in Ref. [20]. Furthermore, our EOS diagnostics can work as a time-of-flight monitor [42; 43], providing energy measurements resolved in time (see Figure 12). It should be highlighted that these diagnostics can be applied on electron bunches produced both in LWFA and PWFA experimental campaigns. Figure 10: The main beam of the FLAME laser is focused on a stainless steel target, producing a beam of relativistic electrons and electromagnetic pulses. These interaction outputs cause a local birefringence in a ZnTe crystal, enabling a linearly polarized laser passing through the crystal to probe this effect and, in turn, retrieve the yield properties. Adapted from Ref. [19]. Interferometry plays a crucial role in investigating gas and plasma densities in plasma-based acceleration experiments. At the FLAME test facility, a Mach-Zehnder interferometer coupled with a probe beam is adopted for a plasma density diagnostics, as depicted in Figure 13. Within the interaction chamber, the probe beam is divided into two identical beams. One passes through the gas or plasma, while the other, acting as the reference, travels through a vacuum path of the same length. Finally, a second beam splitter merges the two beams, and the resulting interference fringes are captured by a CCD camera (Basler scA640-70 gm, Basler AG, Ahrensburg, Germany). Additionally, a lens is included in the setup to allow for adjustments in image magnification. Figure 11: Spatial encoding process description for the electron bunches propagating a few mm below the EOS crystal along the y-axis (orange arrow): (**left**) The Coulomb field locally induces birefringence in the crystal. (**center**) As the electron bunch propagates along the y-axis, in turn, its electric field moves accordingly through the crystal so the local birefringence shifts downward. (**right**) The synchronous probe laser traversing the crystal rotates its polarization in the overlapping zone, meaning the resulting signal is generated in this region where the birefringence and the probe laser are coinciding both temporally and spatially. Adapted from Ref. [20]. Figure 12: Calibration of diagnostics in relation to time-of-flight (TOF) measurements. Each point represents a single step in the delay line (3 fs).