We summarize the experimental results published so far. Details can be found in the respective publications (or here).

The general setup for experiments without electron bunch (SM experiments only) is shown on the figure.

Self-modulation (SM) is the result of perieodic focuaing and defocusing of p⁺ along the long bunch. The effect thus manifests itself on time reolved images of the proton bunch as the formation of a train of dense regions made of focused p⁺, the microbunches, separated by regions of defocused p⁺. We use a streak camera with a time resolution on the order of 1ps to obtain the time reolved images of the bunch.

The camera yields the image of a thing slice (x,t) (labelled "Space" and "Time" on figures) of the bunch about its propagation axis. The figure [AWAKE Collaboration, Phys. Rev. Lett. 122, 054802 (2019), arXiv:1809.04478] shows single event images of the p⁺ bunch with various plasma densities, at the same time as corresponding discrete Fourier transform DFT) spectra of the density profiles shown in green. All beam images show SM at times later than the relativistic ionization front (shown by the red line) show formation of a micro-bunch train. Figure c) shows that a long train is formed.
The modulation frequency is given by the frequency of the wakefields, which itself is close to the plasma electron frequency: f_pe~n_e0^1/2. The above figure [AWAKE Collaboration, Phys. Rev. Lett. 122, 054802 (2019), arXiv:1809.04478] shows that when changing the electron plasma density through the rubidium vapor density over a large range (~10), the measured frequency has the expected values and dependency (n_e0^1/2=n_Rb^1/2).

In essence, SM is an instability, i.e., its outcome varies from event to event. Previous results already showeed that SM can be seeded by uing the relativistic ionization front (RIF) the laser pulse creates and having it propagate within the p⁺ bunch. Systematic measurements showed that one can observe the transition between the non-reproducible instability (SMI) and the reproducible seeded SM (RIF-SSM) regimes [F. Batsch et al., Phys. Rev. Lett. 126, 164802 (2021), arXiv:2012.09676]. We observe this transition when transverse wakefields exceed 2.8 to 4.0GV/m of amplitude, which occurs with the RIF is placed <∽ 2σ_z ahead from bunch center. This difference between the two regimes is clearly visible on the figure, images of consecutive bunch trains. Later measurement indicate that this transition occurs at the same RIF position for various bunch parameters and plasma densities [J. Pucek, in preparation]. Measurement of tining or pghase variations show rms values close to 0.27 in the SMI regime, cosnistent with unifor ditribution of the events over (at least) one period of the wakefields and <0.07, indicating a much smaller variation. This rms is limited by the ability to determine the bunch timing or phase from single picosecond scale images. As a reminder, small variations in timing of the bunch train, and thus of wakefields it drives, is essential for external injection of electrons in the accelerating and focusing phase of the wakefields.

We succesfully demonstarted seeding and reproducibility of the SM process using a relativistic ionization front (RIF) [F. Batsch et al., Phys. Rev. Lett. 126, 164802 (2021), arXiv:2012.09676]. This seeding mechanism leaves the part of the bunch ahead of the RIF and traveling in vapor (no wakefields) not modulated, as it exits the seld-modulator plasma. The long length of the acceletor plasma required to reach large energy gain requires the plasma to be pre-formed (no RIF). Therefore, the front of the p⁺ bunch could experience SMI in the second plasma. SMI wakefields could interfere with those driven by the bunch train and disturb the acceletaion process. Therefore, we studied the seeding of the SM process with a short electron bunch traveling ahead of the long p⁺ bunch in the plasma [P. Muggli et al., 2020 J. Phys.: Conf. Ser. 1596 012066 (2020), arXiv:2002.02189]. This seeding method leads to SM of the entire p⁺ bunch and thus avoid the possible issue mentioned above. We use the electron bunch that was used for acceleration in previous experiments. We also use the same analysis method as for RIF-seeding [fabian] to characterize the reproducibility of the SM process.
Experimental results show that the electron bunch indeed seeds the SM process [L. Verra et al., AWAKE Collaboration, Phys. Rev. Lett. 129, 024802 (2022), arXiv.2203.13752]. The figure shows two time-resolved images of the p⁺ bunch obtained with the electron bunch delayed by 6.7ps with respect with each other. These images are the sum of at least ten succesive events each and thus demontarte that SM is reproducible since the bunch train pattern is clearly visible. In bot cases the RMS od the timing/phase variarion is comparable to that measured with RIF-seeding, i.e., <0.08. Moreover, the shift in time between the two patterns (∽7ps) is equal to the delay between the electron bunch for the two cases, demonstarting the direct link between seed electron bunch and bunch train pattern.
The SM process grows from seed wakefields that depend on the electron bunch parameyers, with a growth rate that depends on the p⁺ bunch parameters (at a given plasma density). This seeding method thus allow for independent control of the two parameters, through, for example, the charge of the two bunches. The figure shows the external envelope of the p⁺ bunch charge distribution observed when varying (a) the electron and (b) the p⁺ bunch charge. It is clear in both cases that the envelope is wider, and thus the growth of SM larger when we increase the charge of either bunches. This confirms the theroretical expectation that seed wakefields and growth rate increas with the charge of the bunches. This seeding method thus allows for control of the SM process and thus of the amplitude of the wakefields. This control could be used to fine tune the acceletaor parameters.

The SM instability (SMI) is a convective instability, which means that its phase velocity (that of the wakefields) is different from that of the bunch. Calculations and numerical simulations show that it is slower than that of the p⁺s. The phase velocity of the wavefields can be influenced by imposing a density gradient along the plasma. The vapor souce allows for a (essentially) linear gradient to be imposed. In the experiments, the density at the plasma entrace is kept constant.
One expects that with a velocity of wakefields different from (slower than) that of the p⁺s, p⁺s can find themselves in the focusing or defocusing phase at different positions as they propagate along the plasma. In particular, p⁺s later along the bunch will see the sign of the transverse wakefields to change more often and they are thus more likely to exit the wakefields and less likely to appear im micro-bunches. Time-resolved images (Figure (b)) [F. Braunmueller et al., Phys. Rev. Lett. 125, 264801 (2020), arXiv:2007.14894 [physics.plasm-ph]] suggests that later micro-bunches contain less charge than earlier ones. One can thus expect that a density gradient that makes the phaae velocity of wakefields more similar(disimilar) to that of the p⁺s would increase (decrase) the charge per micro-bunch (see figures in publication) and the train length. Figure (a) ((c)) shows that a positive(negative) density gradient making the phase velocity of the wakefields faster(slower) does indeed make the bunch longer(shorter) than in the constant density case (b). This indicates that the phase velocity of the wakefields (in a constant density plasma) is indeed slower than that of the p⁺s.

These results conform another fundamental property of self-modulation.

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There is no direct measuremnt of the amplitude of the wakefields. However, their amplitude can be estimated from the radial size of the bunch halo as measured on time integrated image of the transverse proton diatribution. [M. Turner et al., Phys. Rev. Lett. 122, 054801 (2019), arXiv:1809.01191]
The maximum position they reach at the screen is proportional to the transverse momentum they acquired, itself proportional to integral of the amplitude of transverse wakefields they experienced while inside the wakefield (before leaving them and traveling balistically to the screen. The estimate of the amplitude of the tranverse wakefields can be compared to the amplitude of the seed wakefields calculated from the bunch density at the RIF position.

It was established (see above) that the frequency or period of the modulation of the p⁺ is given by the plasma electron density and frequency: f_mod~f_pe~n_e0^1/2. When a (linear) plasma density gradient is imposed along the bunch propagation axis, the plasma density changes and so should the modulation frequency [F. Braunmueller et al., Phys. Rev. Lett. 125, 264801 (2020), arXiv:2007.14894 [physics.plasm-ph]]. The figure shows that f_mod (measured after the plasma, is indeed lower/higher with a negative/positive density gradient. However, it does not follow f_pe expected from (z=10m):
f_{pe exit}=f_mod=f_pe(1+gz/100)^1/2. [P.I. Morales Guzmán et al., Phys. Rev. Accel. Beams 24, 101301 (2021), arXiv:2107.11369 [physics.plasm-ph]] The modulation frequency is measured using two independent methods: the DFT of the charge density of the bunch near is axis (as above) from streak camera, time-resolved images (f_streak)and the frequncy of coherent transition radiation (CTR) emitted by the bunch train f_CTR. The agreement between the two measurements is excelent.

As mentioned above p₊ reaching larger radii at the various screens left the wakefields earlier along the plasma, where SM developed. With a plasma density gradient, their distribution might cary information about the bunch modulation, and thus the plasma frequency at these earlier locations.

This is visible from the DFT analysis of either a narrow/wide transverse region of the bunch on the time-resolved images, as shown on the figure [P.I. Morales Guzmán et al., Phys. Rev. Accel. Beams 24, 101301 (2021), arXiv:2107.11369 [physics.plasm-ph]]. The narrow window result, including only the micro-bunch train show a modulation frequency essentially equal to the plasma a frequency at the plasma exit (with negative density gradient). The variation in frequency with ensity gradient is smaller with positive gradient because the micro-bunch train is longer and thus supports a lower frequency bandwidth than the shorter train. The result with the wide transverse region shows essentially no frequency variation for negative density gradient, and the same as that of the narrow window for positive density gradient. Again, as the micro-bunch train is short for g<0, the signal on the image is dominated by protons that have left wakefields early along the plasma, which has a density that does not change with g. With g>0, the modulation frequency changes more than with g<0 because the modulation frequency is initially higher than the plasma frequency, because of the additional resoring force on plasma electrons exerted by the continuous positive charge of the proton bunch. We observe these dependencies from experimental and numerical simulation images.

The radial dependency of the modulation frequency is confirmed by analysis of narrow radial regions, as shown on the figure: [P.I. Morales Guzmán et al., Phys. Rev. Accel. Beams 24, 101301 (2021), arXiv:2107.11369 [physics.plasm-ph]] higher frequencies near the axis and lower frequencies away from the axis, in the g<0. These radial differences in modulation frequency give "D" or "C" shapes to the protron charge distribution.