# Propagation Characteristics of High-Power Vortex Laguerre-Gaussian Laser Beams in Plasma
## 2 Theoretical Background
However, when the light intensity is extremely high, the spatially-inhomogeneous light beam exerts a significant nonlinear Lorentz force, called the ponderomotive force, on the charged particles in the  
Figure 1: The field magnitude and phase distributions of the doughnut-shaped Laguerre-Gaussian vortex laser beam of \(\mathrm{LC}_{1}^{0}\) mode in free space: (**a**) The longitudinal view of the normalized field magnitude distribution \(E_{\mathrm{m}}(r,z)/E_{0}\); (**b**) the cross-sectional view of the normalized field magnitude distribution \(E_{\mathrm{m}}(r,z=0)/E_{0}\) on the plane of beam waist; (**c**) the cross-sectional view of principal argument distribution \(\Phi(r,z=0)\) of the complex field of laser beam on the plane of beam waist. Figure 2: The field magnitude and phase distributions of the double-ring-shaped Laguerre-Gaussian vortex laser beam of \(\mathrm{LG}_{1}^{2}\) mode in free space: (**a**) The longitudinal view of the normalized field magnitude distribution \(E_{\mathrm{m}}(r,z)/E_{0}\); (**b**) the cross-sectional view of the normalized field magnitude distribution \(E_{\mathrm{m}}(r,z=0)/E_{0}\) on the plane of beam waist; (**c**) the cross-sectional view of principal argument distribution \(\Phi(r,z=0)\) of the complex field of laser beam on the plane of beam waist. plasma. Since the ponderomotive force scales with the inverse of particle mass, the ponderomotive effect on ions is generally negligible with respect to that of electrons. Thus, the electron density in plasma is manifestly changed by such a force until the balance is restored with the plasma pressure gradient force. According to the theory of ponderomotive nonlinearity, the redistributed electron density is exponentially dependent on the square of light-field magnitude \(E_{\rm m}^{2}(r,z)\) and is given by [20]:  
\[n_{\rm e}(r,z)=n_{\rm e0}\exp\left[-\frac{e^{2}}{4m_{\rm e}k_{\rm B}T_{\rm e} \omega_{\rm L}^{2}}E_{\rm m}^{2}(r,z)\right]=n_{\rm e0}\exp\left[-\alpha E_{\rm m }^{2}(r,z)\right] \tag{4}\]  
where \(n_{\rm e0}(r,z)\) is the initial spatial distribution of electron density before the presence of laser beam, \(E_{\rm m}^{2}(r,z)\) is the square of light-field magnitude, \(T_{\rm e}\) is the electron temperature of plasma in unit of kelvin, \(k_{\rm B}\) is [PERSON]'s constant, \(\omega_{\rm L}\) is the angular frequency of laser beam, and the coefficient \(\alpha=e^{2}/(4m_{\rm e}k_{\rm B}T_{\rm e}\omega_{\rm L}^{2})\) is defined for the sake of conciseness of following equations. On the other hand, the electromagnetic properties of plasma are usually described by the famous Drude model of permittivity:  
\[\varepsilon(\omega)=\varepsilon_{0}\left(1-\frac{\omega_{\rm p}^{2}}{\omega^{ 2}-\rmi\omega v_{\rm c}}\right) \tag{5}\]  
where \(\varepsilon_{0}\) is permittivity of free space, \(\omega_{\rm p}\) is the plasma frequency and \(v_{\rm c}\) is the collision frequency. Theoretical derivations [7; 11] demonstrated that \(\omega_{\rm p}^{2}\) and \(v_{\rm c}\) both are proportional to the electron density \(n_{\rm e}(r,z)\). Therefore, for high-power laser beam incidence, both the plasma frequency and collision frequency are modulated by the square of field magnitude as follows:  
\[\omega_{\rm p}^{2}(r,z)=\omega_{\rm p0}^{2}\exp\left[-\alpha E_{\rm m}^{2}(r,z)\right] \tag{6}\]  
and  
\[v_{\rm c}(r,z)=\
u_{\rm c0}\exp\left[-\alpha E_{\rm m}^{2}(r,z)\right] \tag{7}\]  
where \(\omega_{\rm p0}=e\sqrt{n_{\rm e0}/\varepsilon_{0}m_{\rm e}}\) and \(\
u_{\rm c0}\) are the initial plasma frequency and collision frequency before the incidence of laser beam. Thus, the modified Drude model of permittivity for describing the interaction of high-power laser beams with plasma is given by  
\[\varepsilon(r,z)=\varepsilon_{0}\left\{1-\frac{\omega_{\rm p0}^{2}\exp\left[- \alpha E_{\rm m}^{2}(r,z)\right]}{\omega^{2}-\rmi\omega v_{\rm c0}\exp\left[- \alpha E_{\rm m}^{2}(r,z)\right]}\right\} \tag{8}\]  
As apparent in Equation (8), such a physical model of permittivity for plasma is related to the light intensity of laser beam.