Laguerre-Gaussian beams have better stability in atmospheric propagation due to their unique vortex characteristics. However, the coherence of the transmitted beams will be significantly reduced under the influence of turbulent media. Furthermore, the generation of the Kerr effect can also affect the light transmission characteristics with high power output. We simulate the Kerr effect in the propagation of high-power partially coherent vortex beam and compare the influence of the Kerr effect with different coherence lengths and topological charges. The results indicate that the Kerr effect can help maintain the ring structure of light intensity, and increasing the topological charge can effectively alleviate the decay of vortex structures caused by the coherence decrease.

During analyzing the influence of the partial coherence of laser beams, the coherent mode decomposition method is adopted because the phase change cannot be directly added to the cross-spectral density function of the partially coherent beams. The modal expansion of the cross-spectral density of Gaussian Schell-model beams can be regarded as the incoherent superposition of a set of coherent sources, which is characterized by a Laguerre polynomial. Then, phase modulation is loaded onto the coherent light of each mode. When Kerr phase modulation is applied to coherent light, the split-step Fourier method can be employed to solve the problem. The main idea is to evenly divide the light transmission path into segments, with each segment considering linear and nonlinear effects. The influence of the Kerr effect on light wave transmission can be calculated by a model similar to a multi-layer phase screen. In calculation, linear transmission should be separated from nonlinear effects. Firstly, the diffraction effect of linear transmission at each distance, and the nonlinear effect on corresponding transmission distance need to be considered. The phase modulation caused by the Kerr effect is superimposed on the amplitude within the distance. The cross-spectral density function of partially coherent beams under the influence of the Kerr effect can be obtained by calculating the propagation of coherent beams of each mode, and then the sum is weighted.

When the laser power is too small to simulate the Kerr effect, the hollow characteristics of the light intensity distribution of partially coherent Laguerre-Gaussian beams should disappear after transmission [Fig. 2 (a)]. With the increasing laser power, the Kerr effect will cause some coherent Laguerre-Gaussian modes to converge, which results in a rapid increase in light intensity [Fig. 3 (b)], and the overall vortex ring structure of the synthesized light intensity will be restored [Fig. 3 (a)]. We analyze the influence of coherence length on the Kerr effect, as shown in Fig. 4. With the rising coherence length, the diffraction effect weakens. The modulation effect of the Kerr effect on light intensity will increase with the concentration ratio of the intensity distribution. Additionally, it will result in the rapidly rising peak light intensity. Furthermore, the ratio of peak light intensity to central light intensity continues to increase, and the ring structure of light intensity distribution gradually becomes apparent. The most interesting result is that the disappearance of light intensity ring characteristics under low coherence can be improved by adjusting the topological charge number of vortex beams. As shown in Fig. 5, the increase in topological charge number brings about the rising radius of the light spot. Meanwhile, the central part of the light intensity significantly decreases compared to the surrounding ring area. The vortex ring structure is well restored, and better coherence leads to more obvious improvement brought by increasing the topological charge.

Based on the nonlinear Schrodinger equation, the propagation characteristics of partially coherent Laguerre-Gaussian beams are simulated by the split-step Fourier method and the coherent mode decomposition method. The results indicate that the intensity convergence effect of the Kerr effect can effectively compensate for the degradation of vortex characteristics caused by beam diffraction. Thus, the ring structure of partially coherent vortex beams can be maintained. In addition, changing the topological charge of the beam can also affect the intensity distribution structure of vortex light. When the ring structure decays due to the low coherence degree of the beam, it can be compensated by increasing the topological charge. When the coherence length is 1 mm, raising the topological charge to 3 can well alleviate the decay of the vortex ring structure.