Analytic theory of relativistic self-focusing for a Gaussian light beam entering a plasma: Renormalization-group approach

Using the renormalization-group approach, we consider an analytic theory describing the formation of a self-focusing structure of a laser beam in a plasma with relativistic nonlinearity for a given radial intensity distribution at the entrance and derive approximate analytic solutions. We study three stationary self-focused waveguide propagation modes with respect to controlling laser-plasma parameters for a Gaussian radial intensity distribution at the plasma boundary. The proposed theory specifies the domains and their boundaries on the plane of the controlling parameters where (1) self-trapping, (2) self-focusing on the axis, and (3) tubular self-focusing solutions occur. We review the concept of the critical power and show that it must be correlated to the form of the entering light pulse and its value corresponding to the minimum power that admits self-channeling can be significantly lower than the widely used value $17({\omega }^2/{\omega }_{\mathrm{pe}}^2)$ GW.

Figure 1

The boundaries of the domains I, II, III, and IV on the parameter plane $\{i_0,\rho \}$ for different types of solutions (8). These boundaries are defined by the relations $\rho ={\rho }_{\mathrm{axis}}\left(i_0\right)$ (curve a, boundary between I and II), $\rho ={\rho }_{\mathrm{off}}\left(i_0\right)$ (curve b, boundary between II and III), and $\rho ={\rho }_{\mathrm{cav}}\left(i_0\right)$ (curve c, boundary between IV and all the others, I, II, and III).

Figure 2

The spatial distribution of the eikonal derivative $v$ (left image) and the square of the amplitude of the electric field of the light beam $w$ (right image) for different values of the longitudinal coordinate $z$ along the beam axis for $i_0=0.01$ and $\rho =13.96$: the different curves correspond to the transition from $z=0$ to $0.6;0.8;0.9;1.0;1.2$. An increase of $z$ is accompanied by a clearly discernible decrease of $w$ in the region of small $r\to 0$.

Figure 3

The spatial distribution of the eikonal derivative $v$ (left image) and the square of the amplitude $w$ of the electric field of the light beam (right image) for different values of the longitudinal coordinate $z$ along the beam axis for $i_0=0.1$ and $\rho =3.99$: the different curves correspond to the transition from $z=0$ to $0.6;0.8;0.9;1.0;1.2$. An increase of $z$ is accompanied by a clearly discernible decrease of $w$ in the region of small $r\to 0$.

Figure 4

The spatial distribution of the electron density $N_e$ corresponding to the intensity distribution of the laser beam for the plasma and beam parameters in Figs. 2 (left image) and 3 (right image).

Figure 5

The spatial distribution of the eikonal derivative $v$ (left image) and the square of the amplitude $w$ of the electric field of the light beam (right image) for different values of the longitudinal coordinate $z$ along the beam axis for $i_0=0.2$ and $\rho =5$: the value of the longitudinal coordinate of the axial singularity for the chosen parameter values $i_0=0.2$ and $\rho =5$, given by condition (15), is equal to $z_{\mathrm{axis}}=0.82$. The different curves correspond to the transition from $z=0$ to $0.6z_{\mathrm{axis}};0.7z_{\mathrm{axis}};0.8z_{\mathrm{axis}};0.9z_{\mathrm{axis}};0.95z_{\mathrm{axis}}$ from the top down for curves in the left image and from the bottom up in the right image (for small $r\to 0$).

Figure 6

The spatial distribution of the eikonal derivative $v$ (left image) and the square of the amplitude $w$ of the electric field of the light beam (right image) for different values of the longitudinal coordinate $z$ along the beam axis for $i_0=5$ and $\rho =3$: the values of the longitudinal coordinates of the on- and off-axis singularities for the chosen parameter values are $z_{\mathrm{axis}}\approx 0.564$ and $z_{\mathrm{br}}\approx 0.454$. The different curves correspond to the transition from $z=0$ to $0.6z_{\mathrm{br}};0.7z_{\mathrm{br}};0.8z_{\mathrm{br}};0.85z_{\mathrm{br}};0.92z_{\mathrm{br}}$ from the top down for curves in the left image and from the bottom up in the right image (for small $r\to 0$).

Figure 7

The spatial distribution of the electron density $N_e$ corresponding to the intensity distribution of the laser beam for the plasma and beam parameters in Figs. 5 (left image) and 6 (right image). The effect of electron cavitation in the axial zone in the left image and in the off-axis zone in the right image is clearly visible.

Figure 8

The dependencies of $z_{\mathrm{axis}}$ (left image) and $p_{\mathrm{w}}$ (right image) on $i_0$ for different $\rho =5$, 4, 3, 2, and 1.2. The dashed curves in both images limit the range of $i_0$ for each given $\rho$ due to the condition of no initial cavitation ${\rho }_{\mathrm{cav}}=\rho$.