Self-focusing of high-intensity beams with grid structures

TL;DR

This work tackles self-focusing of high-power beams in Kerr media by introducing a 2D grid of beamlets that exploit inter-beamlet nonlinear interactions to redistribute power and delay collapse. Using a non-paraxial, scalar propagation model and split-step Fourier integration for a grid of Gaussian beamlets in fused silica at $\lambda=800$ nm, the authors quantify how the optimal lattice spacing modifies the effective critical power. They reveal a dimension-dependent, multi-stage coalescence where the grid progressively forms smaller grids and eventually a single beam, achieving up to about $15\%$ higher stable transmitted power than the sum of independent beamlets ($N^2P_0$) at an optimal spacing $D_{\text{optimal}} \approx 2.54 r_0$. The study also derives scaling relations showing $D_{\text{optimal}}/r_0$ is largely independent of $r_0$ and depends on the propagation distance relative to the Rayleigh range, with $\beta_{\max}$ approaching $\sim1.15$ for larger grids. These insights suggest a practical beam-shaping approach to enhance high-power beam stability and reduce material damage in advanced optical and directed-energy systems.

Abstract

Laser beams with high optical power propagating in a Kerr medium can undergo self-focusing when their power exceeds a critical power determined by the optical properties of the medium. The highly concentrated energy close to the in the region of the self-focus can lead to other nonlinear phenomena and cause significant irreversible damage to the material. We propose a transverse grid beam structure that effectively suppresses self-focusing even in the absence of other competing effects through the redistribution of optical power by inter-beamlet nonlinear interaction. We find that a beam with a $N \times N$ grid structure with optimized lattice spacing undergoes a dimension-dependent multi-stage self-focusing. We also identify specific grid layouts that can increase the total transmitted power beyond that permitted by the critical level of self-focusing for each beamlet. Lastly, we derive a general numerical relation between the optimal grid lattice spacing and the size of beamlets. Our results could potentially inform the use of beam shaping to prevent damage to optical components in high-powered and directed-energy applications.

Figures (5)

• Figure 1: Intensity profile at the entrance surface of an $N \times N$ grid beam ($N = 3$). The spatial coordinates are scaled to the Rayleigh range $(z_R)$ of each Gaussian beamlet.
• Figure 2: The relation between the power enhancement factor $\beta$ and the lattice constant $D$ for a $4\times4$ grid beam with the maximum propagation distance $z_{\rm max} = 4z_{\rm R}$ and a beamlet radius of $r_0 = 5\lambda$. The enhancement of the threshold power required for self-focusing becomes most pronounced at $D = 2.54r_0$. Note that the four segments of the curve correspond to the four coalescence stages of a $4\times4$ grid beam. The blue dashed line ($\beta = 1$) indicates the asymptotic case where each beamlet propagates independently as the interaction among beamlets are negligible due to a large lattice constant.
• Figure 3: (a) Demonstration of a coalescing process of $4\times 4$ grid beam into a $3\times 3$ grid. Here $P_{\rm input} = 11.0P_0, D = 2.10r_0$. (b) Peak intensity and (c) transmitted power over the propagation distance in the optimal $4\times 4$ grid configuration where $D=2.54r_0$. Under this configuration, the $4\times 4$ grid beam does not collapse before $4z_R$ and can transmit approximately 16.7% more power beyond the total critical power ($16P_0$).
• Figure 4: (a) The relation between the threshold power, $P_{\rm max}$, and the lattice spacing, $D$, for grid beams with an even dimension. (b) The relation between the threshold power, $P_{\rm max}$, and the lattice spacing, $D$, for grid beams with an odd dimension. The total propagation distance is kept fixed at $z_{\rm max} = 4z_{\rm R}$. Notice the significant increase of $P_{\rm max}$ due to the inter-beamlet interaction at the optimal lattice spacing, $D_{\text{optimal}}$. At $D_{\text{optimal}}$, the collapse of each individual beamlet is delayed because the self-focusing is maximally compensated by the net effect of diffraction and the nonlinear refractive index change induced by surrounding beamlets.
• Figure 5: The maximum power enhancement factor $(\beta_{\rm max})$ remains insensitive to the change of the beamlet radius $(r_0)$ for different $N \times N (3 \leq N \leq 9)$ grid-beams at $z_{\rm max} = 4z_{R}$. An anomalous dependence on $r_0$ is observed when the beamlets are organized into a 2-by-2 grid.