Two-dimensional $s$-polarized solitary waves in relativistic plasmas. I. The fluid plasma model

The properties of two-dimensional linearly $s$-polarized solitary waves are investigated by fluid-Maxwell equations and particle-in-cell (PIC) simulations. These self-trapped electromagnetic waves appear during laser-plasma interactions, and they have a dominant electric field component $E_z$, normal to the plane of the wave, that oscillates at a frequency below the electron plasma frequency ${\omega }_{pe}$. A set of equations that describe the waves are derived from the plasma fluid model in the case of cold or warm plasma and then solved numerically. The main features, including the maximum value of the vector potential amplitude, the total energy, the width, and the cavitation radius are presented as a function of the frequency. The amplitude of the vector potential increases monotonically as the frequency of the wave decreases, whereas the width reaches a minimum value at a frequency of the order of $0.82{\omega }_{pe}$. The results are compared with a set of PIC simulations where the solitary waves are excited by a high-intensity laser pulse.

Figure 1

Panels (a) and (b) show the electron density normalized with the unperturbed density $n_0$ at time 612 ${\omega }_{pe}^{-1}$. Panels (c)–(d) and (e)–(f) display at the same instant the electric and the magnetic fields in units of $m_{e}c{\omega }_{pe}/e$ and $m_e{\omega }_{pe}/e,$ respectively. The lengths are given in units of $c/{\omega }_{pe}$, and in panels (c) and (e) the arrows indicate the directions of the fields and the color their modulus.

Figure 2

Panels (a) and (b) show the time evolution of $E_{\perp }=\sqrt{E_x^2+E_y^2\phantom{{}^{tm}}}$ and $E_z$ (in units of $m_{e}c{\omega }_{pe}/e$). Panels (c) and (d) show $B_{\perp }=\sqrt{B_x^2+B_y^2\phantom{{}^{tm}}}$ and $B_z$ (in units of $m_e{\omega }_{pe}/e$). The fields are measured at the position $(x,y)=(127,1)c/{\omega }_{pe}$, respectively, and time is given in units of ${\omega }_{pe}^{-1}$.

Figure 3

Solitary wave with $\Omega =0.8$ and $\alpha =0.01$. Panels (a), (b), (c), and (d) show the amplitude of the vector potential, the electron density, the azimuthal magnetic field $B_{\theta }$, and the electric field normal to the plane of the wave $E_z$ in units of $m_e{c}^2/e$, $n_0$, $m_e{\omega }_{pe}/e$, and $m_{e}c{\omega }_{pe}/e$, respectively. The variable $\tau$ is the time normalized with the period of the wave, and the radius $r$ is expressed in $c/{\omega }_{pe}$ units.

Figure 4

Panels (a) and (b) show the temporal profiles of $a$ at $r=0$ and the energy evolution of the solitary wave shown in Fig. 3 ($\Omega =0.8$). Panels (c) and (d) correspond to a wave with $\Omega =0.7$. $U$ and $E_{\text{kin}}$ are the electromagnetic and electron kinetic energies per unit of length, and they are given in $m_e^2{c}^2/{\mu }_0{e}^2$ units.

Figure 5

Some properties of the solitary waves of the type shown in Fig. 3 vs the frequency. Panel (a), (b), (c), and (d) correspond to the maximum of the vector potential amplitude, the width of the wave at $a_{\text{max}}/2$, the total energy, and the cavitation radius in the case of a cold plasma.

Figure 6

Amplitude-frequency relationship for solitary waves computed with the fluid model (solid and dashed lines) and excited in PIC simulations. Red circles correspond to our simulations and black stars to the results of Ref. [2].

Figure 7

A second type of solitary wave with $\Omega =0.8$ and $\alpha =0.01$. Panels (a), (b), (c), and (d) show the amplitude of the vector potential, the electron density, the azimuthal magnetic field $B_{\theta }$, and the electric field normal to the plane of the wave $E_z$.