Analytical theory of blowout regime in radially inhomogeneous plasmas
Lately a lot of attention has been given to plasma acceleration methods [1, 2] in which an intense laser pulse or a bunch of charged particles is used to excite a plasma wakefield whose large longitudinal field is used for acceleration. Such methods provide acceleration rates orders of magnitude higher than conventional methods. So far electron bunches with the energy up to 4.2 GeV at the acceleration distance of 9 cm have been observed in experiments  for laser-wakefield acceleration (LWFA), while for plasma-wakefield acceleration (PWFA) the possibility of energy doubling from 42 GeV to 85 GeV at a distance of approximately one meter has been demonstrated .
One of the most promising regimes of plasma acceleration is the so-called “bubble” or “blow-out” regime in which electrons behind the driver are almost completely expelled and a spherical plasma cavity free of electrons is formed . This cavity is usually called “bubble” On its border a thin electron sheath consisting of the expelled electrons is created. The cavity itself travels with near-luminous velocity through the plasma. The longitudinal electric field in it is uniform is the transversal direction, while the focusing force acting on the accelerated electrons is linear in radius and uniform along the cavity. In spite of the fact that this regime provides large acceleration gradients, obtaining electron bunches with low emittance, low energy spread, and high stability is a challenging task. One of methods of improving the bunch quality is using deep (or hollow) channels in plasma. The lack of ion column at the axis for such kind of channels significantly reduces the focusing force acting on the electrons, while, for the case of LWFA, the parameters of channel also provide additional freedom for balancing between the laser depletion and dephasing lengths. The possibility of obtaining electron bunches with the energy of 7.5 GeV and the energy spread of only 0.3% in a plasma with a deep channel has been shown in numerical simulations . Due to the complexity of the strongly nonlinear “bubble” regime of the plasma wakefield it is commonly studied using 3D particle-in-cell (PIC) simulations. But its theoretical description is also of considerable interest. However, all recent theories address only the case of homogeneous plasmas, which cannot be applied to the case of plasmas with channels.
Here we develop a generalized theory analytically describing the bubble envelope for plasmas with arbitrary cylindrically symmetric plasma profiles . This theory is based on the theory for homogeneous plasma by Lu et al. .
Model of a strongly-nonlinear wakefield
To develop a general model we assume that a laser pulse or a relativistic electron bunch is propagating through the plasma along the axis \(z\) and is exciting a wakefield in the strongly-nonlinear regime in it. The plasma is described by its radially symmetric ion density profile \(\rho_i(r)\). As the structure of the wakefield slowly changes during its propagation through the plasma, the quasi-stationary approximation, under which all values depend only on the difference \(\xi = t - z\) between the time and longitudinal coordinate, is also assumed.
In our theory the following phenomenological model of a bubble is used: inside the plasma cavity there is no plasma electrons left, outside it the plasma is non-perturbed, while on its border, which is described by a function \(r_b(\xi)\), there is a thin electron sheath of small constant width \(\Delta\). Using this model the following wakefield potential can be obtained inside the bubble:
\(\Psi(r,\xi) = \Psi_0(\xi) - \int_0^r S_i(r')r'dr'\),
where \(S_i (r) = \int_0^r \rho_i(r') r' dr'\) .
The wakefield potential is defined as \(\Psi = \varphi - A_z\), where \(\varphi\) is the electric potential and \(A_z\) is the longitudinal component of the vector-potential (the Lorenz gauge is also assumed). Using this expression for the potential as well the expression for the radial component of the vector-potential \(A_r\), it is possible to calculate the fields inside the bubble and therefore to describe the electron motion.
As the bubble sheath consists of electrons, its inner border is also an electron trajectory, which allows to find an equation for the bubble's envelope:
\(A(r_b) r''_b + B(r_b) r'_b + C(r_b) = \lambda(\xi)/r_b\).
The source \(\lambda(\xi)\) in the RHS of this equation depends only on the radial coordintate. When the width of the electron sheath is small and the bubble is large enough, the coefficients of this equation depend only on the plasma profile \(\rho_i(r)\). The comparison of analytical solutions found by using this equation to the results of 3D PIC simulations (Fig. 1) shows that the model correctly describes both the shape of the bubble and the longitudinal electric field in it.
FIG. 1. Electron density distributions and corresponding longitudinal electric field profiles on the axis of the bubbles as observed in PIC simulations for different channel radii \(r_c\). Analytical solutions for the bubbles’ shapes and the fields in them are shown with the dashed lines.
Homogeneous accelerating field by the adjustment of the accelerated bunch profile
The model also allows to describe plasma cavities with arbitrary electron bunches. In order to improve the quality of the accelerated bunch it is necessary to have homogeneous accelerated field in the whole volume of the bunch. Transverse homogeneity of this field is always present in the scope of this model, therefore it is required only to obtain longitudinal homogeneity. It can be shown that this field cannot be homogeneous in a non-loaded bubble (i.e. a bubble without an accelerated bunch). However, the accelerated bunch itself affects the shape of the bubble, therefore we may try obtaining homogeneous longitudinal field by ajusting the profile of the bunch \(\lambda(\xi)\).
It turns out that it is possible to do for an arbitrary plasma profile. The resulting electron bunch profile is close to trapezoidal in the trasverse direction. In order to verify that such electron bunch can provide homogeneous longitudinal electric field, we have carried out 3D PIC simulations (Fig. 2). The results of these simulations have been compared to the predictions of our theory. It is indeed visible that the accelerated bunch deforms the shape of the bubble in such a way that the longitudinal field is uniform along the bubble.
FIG. 2. Electron density distributions and corresponding longitudinal electric field profiles on the axis of the bubbles for an accelerated electron
bunch with the necessary profile. Analytical solutions are shown with the dashed lines.
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