In **Particle-driven Wakefield Accelerators (PWFA)** the plasma wave is excited by the space charge forces of the driving electron bunch that displace the plasma electrons creating high gradient accelerating fields between the local depleted and enriched plasma electron densities. In that way the driving electron pulse can transfer a large fraction of its kinetic energy to a subsequent bunch (witness bunch) placed at a proper distance.

The **accelerating field** scaling law

$\large{E_{acc}\left[ MV/m\right] = 244 \frac{N_b}{2 \times 10^{10}}\left( \frac{600}{\sigma_z[\mu m]}\right)^2}$

shows that **high charge** ($N_b$) and **short bunch length** ($\sigma_z$) play an essential role in achieving ultra-high gradients. For example with a beam of $N_b = 2 \times 10^{10}$ particles (corresponding to $\sim 3.2 \, nC$) and $\sigma_z= 20\, \mu m$, accelerating gradients of the order of $100 \, GV/m$ can be driven in a plasma with a **density** of $10^{17} \, cm^{-3}$. Moreover since the dependence on the bunch length is inverse quadratic, more challenging accelerating gradients of the order of $1 \, TV/m$ could be obtained with shorter bunches: $\sigma_z = 0.8\, \mu m$ and $N_b = 10^8$ (corresponding to $\sim 20 \, pC$), in a plasma with a density of $7 \times 10^{19} \, cm^{-3}$.

There are several possible PWFA scenarios when operating such experiments, mainly depending on the working regime (linear or nonlinear). The** linear regime**, where the bunch density is very small compared to the plasma, has the advantage that the plasma **oscillation** may be resonantly driven, but has the notable drawback that the **focusing is nonlinear** in radial coordinate and the same dependence would apply for the provided acceleration. In the nonlinear regime, called **blow-out regime** [1], the beam density is much greater than the plasma density (nb >> np) and the beam channel is completely rarefied of plasma electrons. Under these conditions the plasma **response is very nonlinear**, the plasma wavelength **λ _{p}**

**depends**on the wakefield

**amplitude**and large

**wave-breaking spikes**are excited at the end of the first plasma oscillation.

To overcome these limitations, the beam driven experiment at SPARC LAB, named “COMB”, will operate in the so-called **quasi-nonlinear regime** [2]. It requires operations where one uses beam with relatively low charge and longitudinal and transverse beam size smaller than a plasma wavelength, σ_{r}, σ_{z} << λ_{p} ~ 200 µm. In this case, the beam **density** may **exceed** that of the **plasma**, producing blowout (strongly non linear regime) [3], but due to the small total charge, producing a **disturbance** that behaves in many ways as **linear**, having frequency essentially that of linear plasma oscillations.

The reduction of the bunch transverse dimensions to match the desired beam density requires a proper **focusing system**. Such a task cannot be accomplished with normal-conducting electromagnetic solenoids or quadrupoles because the provided field gradients would be too weak. In this context the state of the art is represented by **permanent-magnet quadrupoles (PMQ)** [4]. So far, field gradients of about 560 T/m have been reported when using small-bore PMQs manufactured with neodymium–iron–boride (NdFeB) materials. Another promising way to focus the beam is to use an active **plasma lens **[5], where the discharge that ionize the plasma act as the focusing element. The field gradients available by plasma lenses are in the order of kT/m, thus requiring even less space than PMQ to focus the beam. More information are available on the plasma lens page.

In a **PWFA** driven by a **single** electron **bunch**, the peak **accelerating field** is, in principle, limited to **twice** the value o the peak decelerating field within the bunch (transformer ratio R=2). Therefore the maximum possible energy gain for a trailing bunch is less than twice the incoming energy. Several methods has been proposed **to increase** the accelerating field. A very promising method is the so called **ramped bunch train** and consists of using a train of $N_T$ equidistant bunches wherein the charge increases along the train producing an accelerating field resulting in a transformer ratio proportional to the number of driving bunches. For this application, it is essential to create trains of high-brightness femtosecond long microbunches with stable and adjustable length, charge, and spacing.

A lot of efforts are now ongoing worldwide to produce the required **bunch train** configurations. The method we propose to achieve the required bunch train quality is based on the so called Laser Comb Technique [6] that we have proposed some year ago and which has been tested with the SPARC photoinjector [7]. In this injector operating mode, the photocathode is illuminated by a** comb-like laser pulse** in order to produce a train of sub-picosecond high-charge density electron bunches within the same RF gun accelerating bucket. Downstream of the gun exit, the work done by the space charge force produces a linear **energy chirp** along each pulse, which can be exploited **to compress** the initial charge profile with an RF accelerating structure, operating in the velocity bunching mode [8].

With such a train of $N_T$ bunches a **resonant** excitation of plasma waves can be performed with a convenient scaling with $N_T$ in the **accelerating field**:

$\large{E_{acc}\left[ MV/m\right] = 244 \frac{N_b}{2 \times 10^{10}}\left( \frac{600}{\sigma_z[\mu m]}\right)^2 \times N_T^2}$

For example train of 4 bunches with 16 pC/bunch separated by one plasma wavelength (160 µm), propagating in a plasma of density 3 10^{22 }particles/m^{3} can generate an accelerating field in excess of **3 GV/m**.

Useless to mention that electron pulse trains with some hundreds pC charge, a sub-picosecond length and a repetition rate of some terahertz can be useful also to drive pump and probe or multi-color free-electron laser (FEL) experiments, generation of narrow-band terahertz radiation [9] and to drive Dielectric Wake Field Acceleration (DWFA) experiments [10].

**Publication highlights**

- “Laser comb with velocity bunching: Preliminary results at SPARC”, M. Ferrario et al.,
*Nucl. Instrum. Meth. Phys. Res. A*vol. 637 (2011) pag. S43, doi: 10.1016/j.nima.2010.02.018 - “Beam manipulation with velocity bunching for PWFA applications”, R. Pompili et al.,
*Nucl. Instrum. Meth. Phys. Res. A*vol. 829 (2016) pag. 17, doi: 10.1016/j.nima.2016.01.061 - “Focusing of High-Brightness Electron Beams with Active-Plasma Lenses”, R. Pompili et al.,
*Phys. Rev. Lett.*vol. 121 (2018) pag. 174801, doi: 10.1103/PhysRevLett.121.174801 - “Energy spread minimization in a beam-driven plasma wakefield accelerator”, R. Pompili et al.,
*Nature Phys.*vol. 17 (2021) pag 499, doi: 10.1038/s41567-020-01116-9 - “First emittance measurement of the beam-driven plasma wakefield accelerated electron beam”, V. Shpakov et al.,
*Phys. Rev. Accel. Beams*vol. 24 (2021) pag. 051301, doi: 10.1103/PhysRevAccelBeams.24.051301 **“Free-electron lasing with compact beam-driven plasma wakefield accelerator”, R. Pompili et al.,***Nature*vol. 605 (2022) pag 659, doi: 10.1038/s41586-022-04589-1- “Stable Operation of a Free-Electron Laser Driven by a Plasma Accelerator”, M. Galletti et al.,
*Phys. Rev. Lett.*vol. 129 (2022) pag 234801, doi:10.1103/PhysRevLett.129.234801

Plasma experiments