# Propagation Characteristics of High-Power Vortex Laguerre-Gaussian Laser Beams in Plasma
## 4 Simulation Results  
The propagation characteristics of the \(x\)-polarized high-power LG\({}_{0}^{1}\) laser beam in plasma is first modeled and simulated. The laser beam used is as the one depicted in Figure 1, where the beam waist \(w_{0}=3\lambda_{\rm L}\) with the wavelength \(\lambda_{\rm L}=351\)nm in the ultraviolet region. The nominal electric field of the laser beam is \(E_{0}=2.55\times 10^{10}\)V/m, which is feasible and practicable in modern giant laser facilities [1]. The total three-dimensional computational region for the following FDTD simulations is \(12\lambda_{\rm L}\times 12\lambda_{\rm L}\times 28\lambda_{\rm L}\) in the \(x\), \(y\), and \(z\) directions and is surrounded by perfectly matched layers (PMLs). A cubic Yee cell with size \(\Delta x=\Delta y=\Delta z=\lambda_{\rm L}/20\) is applied for the space discretization. The initial electron density of simulated plasma is spatially homogeneous with \(n_{\rm e0}=3.25\times 10^{21}\)cm\({}^{-3}\) and the collision frequency is \(v_{\rm c0}=0.05\omega_{\rm p0}\). As the interaction process is simulated in a relatively short time \(t=3000\Delta t=3000\Delta x/2c=8.78\times 10^{-14}\)s for a total of 3000 time steps, a constant electron temperature \(T_{\rm e}=1\times 10^{4}\)K is also considered. Figure 3a,b show the simulation results for the longitudinal views of the redistributed light field magnitude of high-power doughnut-shaped LG\({}_{0}^{1}\) laser beam on the \(xOz\) and \(yOz\) planes, respectively, when it propagates in plasma. Figure 4a-f show the cross-sectional views of light field magnitude of LG\({}_{0}^{1}\) laser beam at several propagation distances \(z=4\lambda_{\rm L}\), \(8\lambda_{\rm L}\), \(12\lambda_{\rm L}\), \(16\lambda_{\rm L}\), \(20\lambda_{\rm L}\) and \(24\lambda_{\rm L}\) from the beam waist. The dynamic interaction process between the LG\({}_{0}^{1}\) laser beam and plasma are vividly recorded in Videos S1 and S2. See Supplementary Videos S1 and S2 for details. Secondly, the propagation characteristics of the \(x\)-polarized high-power LG\({}_{1}^{2}\) laser beam in plasma is modeled and simulated. The investigated laser beam is as the one depicted in Figure 2 and the parameters for wavelength and light-field intensity are the same as those of the LG\({}_{0}^{1}\) laser beam. The parameters for FDTD simulations and plasma parameters are also the same as the first numerical example. Figure 5a,b show the simulation results for the longitudinal views of the redistributed light field magnitude of high-power double-ring-shaped LG\({}_{1}^{2}\) laser beam on the \(xOz\) and \(yOz\) planes, respectively, when it is propagating in plasma. Figure 6a-f show the cross-sectional views of light field magnitude of LG\({}_{1}^{2}\) laser beam at several specified propagation distances \(z=4\lambda_{\rm L}\), \(8\lambda_{\rm L}\), \(12\lambda_{\rm L}\), \(16\lambda_{\rm L}\), \(20\lambda_{\rm L}\) and \(24\lambda_{\rm L}\) from the beam waist. The dynamic interaction process between the LG\({}_{1}^{2}\) laser beam and plasma are vividly recorded in Videos S3 and S4. See Supplementary Videos S3 and S4 for details. Figure 3: Simulation results for the propagation of the high-power doughnut-shaped LG\({}_{0}^{1}\) laser beam in plasma (see Video S1). The longitudinal views of the light-field magnitude of laser beam normalized to \(E_{0}\) and shown in logarithmic scale (**a**) on the \(xOz\) plane and (**b**) on the \(yOz\) plane. Figure 4: The cross-sectional views of the light-field magnitude profiles of the high-power doughnut-shaped \(\mathrm{LG}_{0}^{1}\) laser beam in plasma at the several specified propagation distances (**a**) \(z=4\lambda_{\mathrm{L}}\), (**b**) \(z=8\lambda_{\mathrm{L}}\) (**c**) \(z=12\lambda_{\mathrm{L}}\), (**d**) \(z=16\lambda_{\mathrm{L}}\) (**e**) \(z=20\lambda_{\mathrm{L}}\) and (**f**) \(z=24\lambda_{\mathrm{L}}\) from the beam waist (see Video S2).