With the widespread application of femtosecond laser pulses in micromachining, ophthalmic surgery, and biophotonics, existing computational models for laser-plasma evolution often neglect the crucial back-reaction of the plasma on the light field. Meanwhile, comprehensive research on the effects of numerical aperture (NA) on plasma evolution remains limited. This study proposes a nonlinear transient coupling model, aiming to elucidate the plasma generation mechanisms under different NA conditions and to analyze the key physical processes of laser-induced water breakdown and subsequent plasma evolution, thereby deepening theoretical understanding.

The theoretical framework of the model was grounded in Maxwell’s equations and incorporated the transient and nonlinear effects arising from plasma formation. The evolution of the free electron density (FED) was described by a global rate equation, accounting for multiphoton ionization (MPI), avalanche ionization (AI), recombination, and diffusion, with MPI being primarily responsible for initiating AI. This model was specifically applicable to interaction scenarios involving moderate pulse energies and sub-picosecond pulse durations, under which electron solvation effects could be minimized. The back-reaction of the plasma on the light field was quantified through changes in the spatial current density, which in turn affected the total relative permittivity. For numerical solution, a two-dimensional axisymmetric model was constructed using the finite element method (FEM) to simulate the propagation of an incident Gaussian laser pulse along the x-axis. Standard material parameters (Table 1) and a diffraction-limited Gaussian beam profile (Fig.1) were employed in the calculations, and the coupled equations were numerically solved to track the transient evolution of the light field and plasma characteristics in real time.

The results revealed that ionization mechanisms and plasma evolution exhibited significant differences under different numerical apertures. The optical breakdown threshold was closely correlated with the laser intensity required to reach the critical free electron density. When the plasma frequency exceeded the light frequency, optical shielding occurred, leading to an FED plateau or decline. The temporal evolution curves of the maximum FED (Fig.2) showed that the plasma core migrated from the medium boundary toward the beam waist, peaking around 106.5 fs for a 100 fs pulse, after which it gradually decreased (Fig.3). NA was confirmed to be the key parameter governing plasma morphology (Fig.4, Fig.5). At low NA (0.7), a relatively large, regular elliptical plasma was formed. At NA = 1.0, the plasma’s elliptical structure began to lose regularity, indicating that the interaction between the plasma and the light field had a significant effect on morphology. At high NA (1.2), the initial plasma morphology significantly deviated from regularity, forming a “bifurcated protruding tail structure” opposite to the laser propagation direction. This irregularity is a direct consequence of the severe distortion of the laser field by the high-density plasma (Fig. 5c). Analysis of the absorption coefficient (Fig.6) revealed a bell-shaped distribution across all NA conditions, with a clear correlation between NA and peak absorption. The integral area of these curves reflected the effective interaction volume. High NA corresponded to small-volume strong interaction, while low NA corresponded to large-volume weak interaction, the latter correlating with a larger heat-affected zone.

The proposed nonlinear transient coupling model overcomes the limitations of existing models by treating the laser light field and plasma as a coupled whole, and is employed to perform simulation studies on laser-induced water breakdown and plasma evolution under different NAs. These findings provide a critical theoretical basis for optimizing laser parameters and enhancing system performance, and can guide researchers in selecting an appropriate NA by comprehensively considering factors such as precision, nonlinear side effects, or desired plasma size.