Direct visualization of relativistic Coulomb field in the near and far field ranges

Coherent emission coming from relativistic charged bunches is of great interest in a wide range of user-oriented applications and high-resolution diagnostics. The complete characterization of such emission is therefore important in view of a complete understanding of its potential. Here we present a complete temporally-resolved characterization of the radiation emitted by ultra-short relativistic electron bunches using a temporal diagnostic based on electro-optical sampling with a few tens fs of temporal resolution. We have characterized, for the first time to our knowledge, the evolution of the radiation (in THz range) both in amplitude and direction of propagation by varying the detection (i.e. the observer) position from the near to the far field (FF) range. Results show that in the near-field regime the emitted radiation propagates collinearly with the electron beam; while, approaching the FF regime, the radiation behaves as the classical Cherenkov radiation.

In the last few decades, great interest has been shown in coherent radiation produced by ultra-short electron bunches with femtosecond durations. The radiation is emitted as an ultra-short pulse in the millimeter and sub-millimeter (THz) range and can be adopted in a wide range of scientific areas. It can be employed to directly probe the microscopic structure of matter [1,2] or as a diagnostic of the bunch itself, for instance, to retrieve its longitudinal charge profile [3][4][5][6].
The usual picture to describe the radiation emission process is to start from the electromagnetic wave generated by a single charged particle, as formulated by Lienard and Wiechert. According to this theory, the emitted electromagnetic field is radiated when a charged particle undergoes acceleration [7], meaning a variation of its velocity due to an external field, or crosses the boundary between two different media [8]. This approach can be used to describe all the main formation mechanisms for synchrotron, transition, diffraction and Vavilov-Cherenkov radiations (VCRs) [9][10][11][12][13][14]. The radiation field is composed of two terms [7,15], (i) the near-field (NF), falling as the square of the distance and well-known as the Coulomb field for a particle at rest and (ii) the far-field (FF), scaling as the inverse of the distance for accelerated charges. In many works the FF term (∝ r −1 ) is the only one that is considered since the observer is usually considered to be far from the radiation formation region. But when the detector is located near the emitting bunch, the NF term (∝ r −2 ) should definitely be included since its effect becomes dominant.
In this work, we present experimental results showing the temporally-resolved characterization of the THz radiation emitted by an ultra-short relativistic electron bunch. The detection of the radiation is performed using a temporal diagnostic based on electro-optical sampling (EOS) [16]. The experiment has been carried out at the SPARC_LAB test-facility [17] by employing the electron bunches produced by the high-brightness SPARC photo-injector [18,19]. We have characterized, for the first time to our knowledge, the evolution of the THz radiation features, namely its amplitude and duration, in the near-to the FF observation regimes [20]. We observe, in the FF limit, that radiation moves along the Cherenkov angle, Figure 1. Setup of the experiment. The horizontally polarized IR laser crosses the crystal with 30 • incidence angle. The electron beam travels near the crystal, inducing the electro-optic effect that rotates the laser polarization. The position of the electron beam with respect to the crystal is adjusted with a magnetic steerer located upstream. The modulation of the laser polarization is then converted into an intensity modulation by means of a polarizer whose axis is orthogonal to the initial laser polarization. The modulated laser transverse profile is then imaged onto a CCD camera. The inset shows the location of the NF and FF areas. typical of the VCR; on the contrary, in the NF limit, the pattern of the radiation is dominated by NF effects and leads to a radiation front propagating collinearly with the radiating electron bunch. The results have been compared with previous theoretical models [21], showing an excellent agreement.
The experimental setup is shown in figure 1. An ultra-short electron bunch is produced by the SPARC_LAB photo-injector [22], consisting of a radio-frequency (RF) gun providing a 120 MV m −1 peak electric field followed by three accelerating sections. The bunches are generated by sending UV pulses, whose shape and duration can be tailored according to the experimental task, directly on the metallic Cu cathode [23] and are longitudinally compressed by the photo-injector down to a few tens of femtosecond durations [24]. The beam diagnostics consist of an EOS station able to monitor the bunch features in a non-intercepting and single-shot way. The EOS is based on the spatial encoding technique [25] and its temporal resolution is estimated to be of the order of 50 fs [26]. The beam diagnostics is completed by an RF-deflector and a magnetic spectrometer allowing to characterize the time and energy profiles of the beam in correspondence of a cerium-doped yttrium aluminum garnet (Ce:YAG) screen located on a 14 • beamline [27]. Beam current monitors installed along the machine allow us to measure the beam charge at different locations.
The SPARC photo-injector is capable to produce different beam configurations according to the experimental campaigns. For the purposes of this work, we tuned the longitudinal compressor to produce 200 pC bunches with ≈150 fs temporal duration. The beam energy was varied in the range 37-96 MeV. The EOS diagnostics consists of 200 µm thick ZnTe and 100 µm thick GaP crystals. The electron beam propagates at distances in the range of 0.15-2.5 mm from the side of the EOS crystals, as shown in figure 1. The EOS relies on a ≈100 fs probe IR laser, directly split from the photo-cathode laser system, ensuring a jitter-free synchronization. The laser impinges the crystal at 30 • incidence angle, providing an effective temporal window of ≈10 ps. The EOS setup is completed by a lens (f = 30 cm) installed downstream of the crystal and used to image the crystal surface on the CCD camera where the EOS signals are actually recorded. In such a way we can assume the crystal itself as the effective position of the observer. The temporal overlap of the radiation produced by the emitting electrons and probe laser in correspondence with the EOS crystal is obtained by a 3 fs resolution delay-line on the probe path.
To figure out how the detection is performed, figure 2 shows three recorded EOS snapshots related to an electron bunch with a temporal duration of 150 fs and an energy of 96 MeV at several positions R with respect to the EOS crystal (i.e. the observation point of view). The detected EOS signals are able to give information both on the electron bunch properties and on its irradiated field. The electric field (Σ e ) and the temporal duration (τ e ) of the emitting electron bunches can be retrieved from the signal amplitude and width [28], respectively. Concurrently, the experimental traces allow us to fully characterize the radiation, namely its amplitude and propagation angle, both depending on the observation regime (near-or FF, according to the well-known Lienard-Wiechert theory). When moving toward the FF limit (i.e. growing distance R from crystal), it is noticeable the appearance of an additional signal. When the distance is as large as R − →3 mm, the additional signal related to the FF signal increases its amplitude. On the contrary, the NF signal amplitude is depleting.   2 also shows tat the two signals appear delayed in time and this corresponds to two non co-propagating beams along the EOS crystal. It is straightforward to determine the propagation angle of the FF signal, considering the distance between the two signals (L s , namely the distance between the two peaks shown in figure 2) and the crystal thickness (L c ) As mentioned before, we have adopted both high and low energy electron bunches to show the features of the radiation in the near and FF limits. Adopting equation (1), figure 3 was sorted out. It shows the experimental results regarding the effective propagation angle as a function of the observation distance for the four different bunch energies. In the case E = 96 MeV and the distance from R = 0.15 mm to R − → 0.7 mm, we get θ < 7 • , indicating a radiation front propagating almost parallel to the bunch propagation direction. On the contrary, for the same energy but the distance from R ∼ 0.7 mm to R − → 3 mm, we get θ ∼ 70 • , close to the classical Cherenkov angle value θ c = arccos (1/nβ 0 ) ∼ 72 • . Moreover, it is noticeable that the NF range decreases (FF range increases) decreasing the electron bunch energy (E = 71 − → 37 MeV) according to the theory developed in [21]. This non-collinear behavior of the radiation is consistent for all the beam energies and by using two electro-optic crystals (i.e. different indexes of refraction) with different thicknesses as shown in figure 3. Indeed, considering the retrieved properties of the electron bunches, the radiation wavelength λ THz = 30 µm, the indexes of refraction of the ZnTe and GaP crystals in our THz range n ZnTe ∼3 and n GaP ∼3, we calculated the effective propagation angle of the emitted radiation as [21] θ th (R) = arccos wherep r,z are the average radial and longitudinal momenta of the electromagnetic field. It is worth to notice that here we use a definition of angle which is complementary to the one in [21], since this choice is more in line with standard notation for the Cherenkov angle. The momenta are given byp r,z = N r,zhkr,z , where N r,z is the number of photons andhk r,z their average momentum.
Equation (2) converges to the classical Cherenkov angle in the FF limit, as mentioned before, with the effective Cherenkov angle increasing when decreasing the electron bunch energy. The analytical calculations are also reported in figure 4 showing an excellent agreement with the experimental data reported in figure 3.
Finally, to fully characterize the THz radiation, we studied the amplitude of the NF component, the experimental behavior of the FF amplitude component and of the temporal duration of both signals as a function of the R parameter with respect to the energy of the emitting relativistic electron bunches. The space-charge field for a gaussian beam can be written as follows where σ is the electron beam radius. Adopting equation (3), we were able to fit the experimental data regarding the NF amplitude of the emitted THz field as a function of the observation distance for the four different bunch energies, as shown in figure 5, (Up). The fit parameters are reported in table 1.
Considering the retrieved fit parameter A NF0 , we measure that the radiation field is independent of the electron bunch energy in our experimental conditions. In figure 5, (bottom), the experimental behavior of the FF radiation amplitude is shown. A linear growth is retrieved in the spatial range of interest. Finally, the temporal duration of both signals was retrieved as a function of the observation distance. As shown in figure 6, the duration of the NF signal is constant (∼150 fs) within the studied range, meanwhile, the FF signal duration grows almost linearly from ∼160 fs to ∼250 fs at the furthest point. In the NF, the duration of the signal corresponds to the one of the electron bunch because the diagnostic system is not cutting any spectral bandwidth for R ≲ γβλ/2π. Meanwhile, going towards the furthest position R ≳ γβλ/2π, the NF signal is depleted and the FF signal is enhanced and stretched because of the cutting of the high frequencies in the spectral bandwidth. In our experimental campaign, γλ/2π is in the mm range. It is noticeable, as   shown in figure 6, that the maximum temporal duration is reached sooner by the lower energy electron bunches because they cut high frequencies for lower distances.
In conclusion, we have presented direct temporally-resolved sub-picosecond resolution measurements of the radiation emitted by relativistic electron bunches. According to the electron energy (37-96 MeV range) and the distance (0.15-2.5 mm range) between the electron bunch and the EOS diagnostic, the radiation can be considered in the NF of the VCR, i.e. where the field is similar to a point charge field propagating collinearly to the emitting bunch, or in the FF, where the radiation propagates at the well-known Cherenkov angle arccos(1/β 0 n). Analytic calculations were performed to validate the behavior experimentally retrieved. The results provide a more complete picture of the radiation process emission, underlying its formation mechanism.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).