Transition from wakefield generation to soliton formation

It is well known that when a short laser pulse propagates in an underdense plasma, it induces longitudinal plasma oscillations at the plasma frequency after the pulse, typically referred to as the wakefield. However, for plasma densities approaching the critical density, wakefield generation is suppressed, and instead the EM-pulse (electromagnetic pulse) undergoes nonlinear self-modulation. In this article we have studied the transition from the wakefield generation to formation of quasi-solitons as the plasma density is increased. For this purpose we have applied a one-dimensional relativistic cold fluid model, which has also been compared with particle-in-cell simulations. A key result is that the energy loss of the EM-pulse due to wakefield generation has its maximum for a plasma density of the order 10% of the critical density, but that wakefield generation is sharply suppressed when the density is increased further.

Figure 1

The spatial profiles of longitudinal field are plotted at $t=159$ for the case when a 5 cycle laser pulse with $a_0=0.01$ interacts with a plasma having density $0.001n_c$ (a) and $0.1n_c$ (b). The corresponding spatial profiles of the transverse field (EM driver) are also presented in (c) [profile on right (left) is for $0.001n_c$ ($0.1n_c$)].

Figure 2

The spatial profiles of the longitudinal field (left panel) are plotted at $t=749$ for the cases when a 5 cycle laser pulse with $a_0=0.01$ (a), $0.1$ (b), and 0.3 (c) interacts with a plasma having density $0.1n_c$. The spatial profiles of the transverse field with $a_0=0.01$ (d), 0.1 (e), and 0.3 (f) are also presented in right panel for $t=109$ (left), 409 (center), and 749 (right).

Figure 3

Fluid simulation: The wakefield for different plasma densities 0.001 $n_c$ (a), $0.04n_c$ (b), $0.4n_c$ (c), and $0.6n_c$ (d) are presented as calculated at $t=263$. The laser parameters are $a_0=0.3$, 5 cycle (FWHM) Gaussian linearly polarized laser pulse, incident on the plasma slab of $275\lambda$.

Figure 4

Spatiotemporal evolution of electrostatic (a), electromagnetic (b) fields with 5 cycle laser with amplitude $a_0=0.3$ and $n_e=0.6n_c$. Enlarged version of the solitonic structures is also presented separately (c). In all panels, field profiles are illustrated from left to right with ascending temporal order.

Figure 5

The wakefield energy is calculated as ${\int }_{z_0}^{z_1}{\left|E_z\right|}^{2}dz$. Here, $z_0$ and $z_1$ are $20\lambda$ and $10\lambda$ behind the peak of the laser at $t=263$ for different plasma densities.

Figure 6

A comparison of the group velocity ${\upsilon }_g$, the line fitted by Eq. (19), and the propagation velocity ${\upsilon }_p$, the dots computed from the position of the central peak of the EM field, as a function of plasma density. Here, the case of a 5 cycle EM pulse with $a_0=0.3$ is considered.

Figure 7

PIC simulation: The wakefields for different plasma densities $0.001n_c$ (a), $0.04n_c$ (b), $0.4n_c$ (c), and $0.6n_c$ (d) are presented as calculated at $t=265$. The laser parameters are $a_0=0.3$, 5 cycle (FWHM) Gaussian linearly polarized laser pulse, incident on the plasma slab of $275\lambda$.