Product-Gauss integration method in the numerical investigation of relativistic self-focusing critical power of intense laser beams with LG10 and LG20 modes in plasma

In this paper, the critical power of relativistic self-focusing laser beams with LG(10) and LG(20) modes in plasma is calculated numerically. An envelope equation for the laser spot size is obtained. This equation contains a defocusing term and a focusing term. The first one is due to diffraction spreading and the second one is due to relativistic oscillations of the plasma electrons. To identify the critical power, the envelope equation is discussed in terms of an effective potential. Numerical methods are used to solve the governing equations. We use Product-Gauss integration method to obtain numerical results. To certify the numerical methods, the known analytical solution of the relativistic self-focusing critical power for TEM00 mode is compared with our numerical results. Our numerical results and known analytical results are compatible. The relativistic self-focusing critical powers for LG(10) and LG(20) laser beams are about 6.5 and 18 times those for a TEM00 laser beam. Our numerical method agrees with the previously published analytical solutions.