• bubble regime
• electron acceleration
• ionization injection
• laser wakefield accelerator
• plasma

In a free electron medium, such as a plasma, a coherent displacement of electrons from their equilibrium position generates an unbalanced electric charge which produces an electrostatic field. This field, if friction effects are small, pulls back the electrons which start to oscillate around their equilibrium position at the frequency ωp, known as plasma frequency [1,2] which depends on the electron density in the medium.
In a plasma the oscillations can have a phase velocity vp, originating electrostatic plasma waves. These electrostatic waves are longitudinal, have always frequency ωp, if the plasma can be assumed cold, and have a maximum possible amplitude (the so-called wavebreaking limit [3,4]).
But they can have any value of vp: the key idea in exciting an electrostatic wave in cold plasmas by mean of a driver is that the value of vp can be set by the velocity of the chosen driver, because the frequency of oscillations is always ω = ωp while the wavenumber k is not determined[5].

In plasma based accelerators plasma waves are excited in the wake of an electron beam (Plasma
Wake-Field Accelerator (PWFA)) or of a laser pulse (Laser Wake-Field Accelerator (LWFA)), can have
large amplitudes and phase velocities close to c. They can sustain, also in linear, classical conditions, accelerating gradients[2] up to 100 GV/m. In PWFA an electron beam driver loses energy to plasma and excites wakewaves, while in LWFA the key role in the excitation of a wakewaves is played by the laser ponderomotive push, which can be seen as the force due to the radiation pressure of the laser light[2,6].

A longitudinal plasma wave with phase velocity vp can be seen as a sequence of cavities[5] of length vpTp/2, dephased by Tp/2 where Tp = 2π/ωp is the plasma oscillation period; an injected electron with velocity ve = vp will take a time Tp/2 to cross the cavity and will always see the electrostatic field in the same direction, getting a net acceleration. The accelerated electrons need to remain in phase with the wave. This implies that, because the electrons cannot exceed the velocity of light, plasma wave must have a phase velocity vp ≲ c.
Nowadays LWFAs are capable to produce accelerated electrons with energy from hundreds of MeV up to GeV within length scales from millimeters to centimeters [7,8,9].

A crucial aspect in a plasma based accelerator is how electrons are injected into plasma. From the
choice of the injection scheme can depend the quality of the final obtained beam in terms of gained
energy, carried charge, divergence and energy spread of the energy distribution.way to internally inject electrons in a plasma wave is via ionization [10,11,12,13], achieved by ionizing deep bound electrons from a high atomic number gas at phase inside the laser field.

During this thesis work a numerical study, performed via the Fourier-Bessel Particle in Cell code[14], of a possible laser-driven wakefield accelerating scheme was conducted, involving the excitation of a plasma bubble structure, with high amplitude accelerating field and strong focusing fields[15], in a Helium plasma. These allowed the acceleration and focusing of an internally injected electron bunch, obtained by mean of ionization of deep bounded electrons of Nitrogen confined in a strip 400μm long, short compared to the total length of the system which is 2 mm. This obtained bunch carries a charge of 115 pC and has a peak current of about 24 kA; at the end of the acceleration section it has been accelerated up to 195 MeV with an energy spread of 6%.

A preliminary parametric analysis was performed to find the right laser-plasma parameters to be
used in the settings of the simulations, following three main guidelines[2,15]:

• matching the laser pulse length to plasma wavelength to have an efficient wakefield generation[1]
• matching the transverse dimension of the plasma bubble to laser spot size to generate a well
defined bubble in the laser wake
• considering the ratio between the laser power P and and a critical power Pc[2], which sets a limit to trigger the guiding of the laser inside the plasma, without triggering any wave breaking.

An ”artificial” dephasing between the injected bunch and the rear was set up with a lengthening
of the plasma wavelength caused by a smooth density ramping of the acceleration section with a
linear gradient. This must be smooth enough to not trigger electron self-injection from a density
transition[16,17].

The bunch is asymmetric with greater extension and particle momentum spread in the x-direction
than in the y-direction, mostly due to the linear polarization of the laser, which also set a great spreadof the particles momentum in the transverse x-direction because of off-laser-peak extraction.
A study of the downramp section[18] was made to limit the growth of the emittance of the bunch
at the exit of plasma, with reduction of momentum spread and a controlled expansion of bunch size:
this is obtained with a density profiling which assures an adiabatic reduction of the focusing force.

Comparison with a ”free-drift” case, where plasma density rapidly falls, was made, showing that a
controlled expansion and a reducing of the momenta spread are possible without any growth of the
emittance. Besides, in the initial part of the downramp a further acceleration of the bunch is possible.

Finally the numerical solution of a model based on the invariant envelope equation 19 for bunch
size evolution is compared to the data obtained by the particle-in-cell (PIC) simulation performed.
It is shown a quite good agreement in x-direction, while in y-direction seems to be a sort of phase
difference between this solution and simulation data: this must be further studied. This comparison
shows that the model can be used to predict the main trend for some of the quantities of interest
of a particle bunch during its acceleration without involving an accurate PIC simulation: with this
model is possible to reduce the problem of the evolution, given some initial conditions and knowing the accelerating and focusing fields, of bunch properties to a first order ODE problem and can be useful for preliminary studies in accelerating schemes, saving computational time and data storage resources.

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