The self-modulation (SM) instability transforms a long charged particle bunch traveling in plasma into a train of microbunches that resonantly drives large-amplitude wakefields. We present the first determination of the saturation length of SM using experimental and numerical results. The saturation length is the distance over which wakefields reach their maximum amplitude along the plasma. By varying the plasma length and measuring the radius of the transverse distribution of the bunch, we find that the saturation length of SM decreases with plasma density and initial field amplitude, e.g., when seeding. The saturation length is a fundamental parameter of the instability, and these results are key for understanding SM and designing plasma wakefield accelerators driven by long bunches, such as AWAKE, or by long laser pulses for radiation production.
Introduction -Instabilities play a central role in plasma physics (see, e.g., Ref. [1]). In particular, understanding their development and saturation is essential. A fundamental parameter characterizing many instabilities is the saturation length. An instability can grow in space from an initially small (linear) amplitude. Growth leads to nonlinear effects, and at the saturation length the amplitude reaches a large and essentially constant value, lim-ited by these effects. In general, the effect sought after is maximum at saturation, and its event-to-event variations are also smaller than during growth.
For example, in free electron lasers (FELs), a particle bunch modulates to emit light with exponentially growing intensity. The saturation length, L sat , can be determined by measuring the radiation power as a function of undulator length (see e.g., Ref. [2]). Determining L sat is crucial, as it sets the minimum undulator length required for the FEL to reach maximum output power.
In plasma, an analogous microbunching process occurs in the self-modulation (SM) instability of a relativistic charged particle bunch. SM develops when the duration of the bunch σ t is much longer than the plasma electron period τ pe [3]. It has been observed with electron [4][5][6] and proton [7] bunches. SM also occurs with long laser pulses propagating in plasmas [8][9][10] and has been observed in experiments [11]. Despite these observations, the saturation length of SM has not yet been measured experimentally.
The initial transverse wakefields driven by the long bunch have small amplitude and periodically modulate the bunch density with period τ pe . Since the modulated bunch drives higher-amplitude wakefields, a feedback loop between density modulation and wakefields is set up. This leads to growth of SM along the plasma and along the bunch. The growth saturates when the bunch is fully modulated, i.e., when the resulting train of microbunches resonantly excites wakefields to the highest amplitudes.
Unlike in FELs, where radiation power can be measured directly, wakefield amplitudes are difficult to measure [12]. It is therefore necessary to rely on indirect effects of the wakefields to determine the saturation length of SM.
Because SM is a transverse process, the formation of the microbunch train leads to that of a halo of defocused particles surrounding the train. This halo formation is both the cause (modulation of the bunch density) and a consequence (effect of transverse wakefields) of the growth of the wakefields along the plasma [13]. One can expect the formation process to slow significantly once the microbunch train is fully formed, and when saturation of SM is reached, at L sat . This key parameter can therefore be experimentally determined by characterizing the evolution of the halo of defocused particles along the plasma [14].
In this Letter, we determine for the first time, with experimental and numerical simulation results, the saturation length of the SM process. By varying the plasma length and measuring the radius of the halo at a screen downstream of the plasma, we identify the distance over which this radius increases and saturates as a function of experimental parameters. We find that the saturation length of the radius decreases with increasing plasma electron density, and that seeding the SM process makes this length shorter than in the non-seeded case. Numerical simulation results indicate that the saturation length of the halo radius provides a good estimate for that of SM. There is excellent agreement between numerical simulation and experimental results for the saturation length of the halo radius.
Experimental Setup -Measurements were performed in the context of AWAKE (Advanced WAKefield Experiment) at CERN [15]. A schematic layout of the experiment is shown on Fig. 1.
A 400 GeV proton bunch from the Super Proton Synchrotron, with a population N b ∼ = 3 × 10 11 particles, is focused to an rms transverse size σ r ∼ = 200 µm at the entrance of the plasma. The bunch has a normalized emittance ϵ N ∼ = 2.2 mm-mrad, and an rms duration σ t ∼ = 170 ps.
A short laser pulse with FWHM duration 120 fs and energy ∼100 mJ co-propagates with the proton bunch. It forms a relativistic ionization front (RIF), and ionizes a rubidium vapor (RbI → RbII), resulting in a plasma electron density n pe that can be varied in the range ∼ = (1 -10) × 10 14 cm -3 [16]. This range of densities corresponds to plasma periods much shorter than the bunch duration (τ pe = 2π meϵ0 npee 2 ∼ = (11 -3) ps ≪ σ t ∼ = 170 ps), enabling the development of SM. When placed within the bunch, the RIF defines the onset of the beam-plasma interaction, and thus the timing and amplitude of the initial wakefields that can seed SM [17].
The laser pulse propagation can be stopped at different locations along the 10.3 m-long vapor column by inserting a thin (200 µm) aluminum foil (laser blocks on Fig. 1). The plasma length (L p ) over which th
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