# Propagation Characteristics of High-Power Vortex Laguerre-Gaussian Laser Beams in Plasma
## 2 Theoretical Background  
In physical optics, the complex-form expression of the electric field of a Laguerre-Gaussian beam propagating in the free space under cylindrical coordinates \((r,\ \phi,\ z)\) is given by:  
\[E_{lp}(r,z)=E_{0}\frac{w_{0}}{w(z)}\left(\frac{\sqrt{2}r}{w(z)}\right)^{|l|}L_{p }^{|l|}\left(\frac{2r^{2}}{w^{2}(z)}\right)\exp\left[-\mathrm{i}\left(\frac{kr^ {2}}{2q(z)}+kz+l\phi-\varphi_{pl}(z)\right)\right] \tag{1}\]  
where \(E_{0}\) is the nominal magnitude of light field of laser beam, \(k=2\pi/\lambda_{\mathrm{L}}\) is the wave number of laser beam with wavelength \(\lambda_{\mathrm{L}}\), \(w_{0}\) is the radius of beam waist, \(w(z)=w_{0}[1+(z/z_{\mathrm{R}})^{2}]^{1/2}\) is the beam size at distance \(z\) from the beam waist with Rayleigh range \(z_{\mathrm{R}}=\pi w_{0}^{2}/\lambda\), \(q(z)=z+\mathrm{i}z_{\mathrm{R}}\) is the complex beam parameter, \(\varphi_{pl}(z)=(2p+|l|+1)\tan^{-1}(z/z_{\mathrm{R}})\) is the Gouy phase shift, and \(L_{p}^{l}(\cdot)\) are the associated Laguerre polynomials. Herewith, each Laguerre-Gaussian beam of \(\mathrm{LG}_{p}^{l}\) mode is specified by two mode indices, the angular mode index \(l\ (l=0,\ \pm 1,\ \pm 2,\ \dots)\) and the transverse radial mode index \(p\ (p=0,\ 1,\ 2,\ \dots)\). It is noted that a Laguerre-Gaussian beam degenerates to the fundamental Gaussian beam when \(p=l=0\). In this work, we focus on the category of vortex Laguerre-Gaussian laser beams with \(l\
eq 0\). To maintain simplicity without losing generalizability, we investigate the propagation of two typical low-order Laguerre-Gaussian beams in plasma. The first one is the doughnut-shaped \(\mathrm{LG}_{0}^{1}\) laser beam with electric field written as:  
\[E_{10}(r,z)=\frac{E_{0}\sqrt{2}w_{0}r}{w^{2}(z)}\exp\left[-\mathrm{i}\left( \frac{kr^{2}}{2q(z)}+kz+\phi-2\tan^{-1}(\frac{z}{z_{\mathrm{R}}})\right)\right] \tag{2}\]  
Figure 1 shows the field magnitude and phase distributions of the \(\mathrm{LG}_{0}^{1}\) laser beam propagating in free space, where \(E_{m}(r,z)=\left|E_{lp}(r,z)\right|\) and \(\Phi(r,z)=\arg\left\{E_{lp}(r,z)\right\}\) are the magnitude and principal argument of the complex electric field \(E_{lp}(r,z)\) of laser beam given by Equation (1), respectively. The size of beam waist is \(w_{0}=3\lambda_{\mathrm{L}}\) as depicted in Figure 1, where \(\lambda_{\mathrm{L}}\) is the wavelength of laser beam. The distinct doughnut-shaped feature makes it ideal for applications in many areas of optics, such as optical trapping [17] and optical tweezers [18]. The second laser beam is the double-ring-shaped Laguerre-Gaussian vortex laser beam of \(\mathrm{LG}_{1}^{2}\) mode with its electric field given by  
\[E_{21}(r,z)=E_{0}\frac{2w_{0}r^{2}}{w^{3}(z)}\left(3-\frac{2r^{2}}{w^{2}(z)} \right)\exp\left[-\mathrm{i}\left(\frac{kr^{2}}{2q(z)}+kz+2\phi-5\tan^{-1}( \frac{z}{z_{\mathrm{R}}})\right)\right], \tag{3}\]  
according to the associated Laguerre polynomial \(L_{1}^{2}(x)=3-x\). Figure 2 shows the magnitude and phase distributions of an electric field of the Laguerre-Gaussian laser beam of \(\mathrm{LG}_{1}^{2}\) mode propagating in free space. Here, a smaller size of beam waist \(w_{0}=2\lambda_{\mathrm{L}}\) is applied, which implies a larger diffraction effect as we can see from Figure 2a. The cross-sectional view of field magnitude distribution \(E_{\mathrm{m}}(r,z=0)/E_{0}\) in Figure 2b shows that the laser beam is double-ring-shaped. The dynamic interaction of laser beams with plasma is complex and dependent on the level of laser power [19]. When the intensity of the incident laser beam is low, the charged particles such as ions and electrons in plasma driven by the light fields are oscillating around their nearly-fixed oscillation centers. Therefore, the spatial distribution of the electron density keeps nearly unchanged.