At the nanoscale, it is often difficult to achieve absolute flatness and certain curvature on the deposited substrate surface. An Ag/SiO₂ core/shell nanostructure model deposited on a curved substrate has been designed to address the near-field enhancement of nanoparticles under femtosecond laser action. The electromagnetic field, two-temperature model, and plasma physical field are coupled, with femtosecond laser breakdown mediated by this nanostructure carried out.

We employ the radio-frequency module, electromagnetic wave module, and frequency domain interface of COMSOL. COMSOL is multiphysics simulation software that enables the modeling and simulation of coupled physical phenomena. The electromagnetic wave propagation in different media and structures is modeled by the electromagnetic waves module, and Maxwell's equations involved in this module are solved. The two-temperature model is adopted to simulate the evolution of lattice temperature within nanoparticles. This model likely considers the separate temperatures of the electrons and the lattice, accounting for heat diffusion and energy transfer processes during femtosecond laser pulse irradiation. Meanwhile, the two-temperature model is coupled with the electromagnetic model to account for resistive losses during the interaction of the femtosecond laser pulse with nanostructures. Plasma rate equations derived from the Keldysh theory are solved to describe multiphoton ionization, avalanche ionization, diffusion, and recombination losses. These equations likely capture the complex dynamics of the plasma formed around nanoparticles due to the laser irradiation. Additionally, we calculate the dynamics of free-electron plasma density around nanoparticles by solving the plasma rate equations, which provide insights into the behavior of the plasma, including its formation, expansion, and recombination processes. The plasma dynamics model is coupled with the electromagnetic model by considering parameters such as the electric field value and changes in the dielectric function of water due to free-electron plasma formation. This ensures that the effects of plasma on electromagnetic wave propagation are accurately incorporated into the simulation. The gridded model is shown in Fig. 1, where the outer shell coating layer and deposition substrate of silver nanoparticles are silica, surrounded by water. A free tetrahedral mesh is used in the silver nanospheres, deposited substrates, and their neighborhoods, where the maximum mesh size of the silver nanospheres is 1/10 of their diameter, and the maximum mesh size of the deposited substrates and their neighborhoods is 1/20 of the laser incident wavelength. The ideal matching layer (PML) adopted for solving electromagnetic physical fields is a swept grid. We integrate multiple models and computational techniques to comprehensively analyze the interaction of femtosecond laser pulses with nanostructures, with both electromagnetic and thermal effects, and the formation and dynamics of plasma considered. Finally, this approach provides a detailed understanding of the complex physical processes involved in these interactions.

Parametric scanning calculations are performed for deposition substrates with different curvatures, and the maximum relative electric field enhancement of the silver nanoparticles trimer is 12.7 times when R=140 nm and θ=20° (Fig. 7). Meanwhile, under R=190 nm and θ=20°, the maximum relative electric field enhancement of silver nanoparticles trimer is 4.37 times (Fig. 8), which shows that when θ remains unchanged, the electric field decreases when the curvature radius increases. The maximum relative electric field enhancement of silver nanoparticles trimer is 5.01 times when R=140 nm and θ=25° (Fig. 9), and it is 5.01 times when R=140 nm and θ=25°. The electric field enhancement is 3.59 times (Fig. 10), which can be observed by comparing with Figs. 7 and 8 respectively, where the electric field decreases with the increasing θ. Considering the influence of the angle θ on the near-field enhancement separately (Fig. 12), the maximum relative electric field enhancement at θ=15° is 8.25 times, and the maximum electric field position is located at the nanoparticle edge. The relative electric field enhancement at θ=20° is shown in Fig. 13, and the maximum relative electric field enhancement is 4.17 times. The relative electric field enhancement at θ=25° is presented in Fig. 14, and the maximum relative electric field enhancement is 3.5 times. As the angle θ between the nanoparticles increases, the maximum relative electric field enhancement decreases rapidly, which is caused by the distance increase between nanoparticles. When the angle θ rises from 15° to 25°, the strong mutual coupling between silver nanoparticles weakens, and the strong electric field regions shown in red evolve from overlapping each other in Fig. 12 to Fig. 13 and move away from each other in Fig. 14. In particular, this trend becomes more obvious as the spacing between nanoparticles increases. For comparison, the relative electric field enhancement is calculated when the substrate changes to a planar substrate, which means R=200 nm remains unchanged, the center distance of two adjacent nanoparticles is 53.59 nm, and its maximum relative electric field enhancement is 7.2 times (Fig. 10). Compared with Fig. 12 where the base is curved, the maximum relative electric field enhancement is weakened. Further analysis indicates that the center distance of two adjacent nanoparticles in Fig. 12, or the center arc length of two adjacent nanoparticles is 52.36 nm, which is slightly smaller than 53.59 nm in Fig. 15. The results demonstrate that the distance between nanoparticles determines the intensity of electric field enhancement, and the magnitude of relative electric field enhancement is highly sensitive to the distance between nanoparticles.

The results show that the near-field enhancement effect is highly sensitive to the spatial (planar/curved surface) distance between silver nanoparticles. The closer distance leads to a stronger near-field enhancement effect. Meanwhile, the bending curvature of the silver nanoparticle substrate has a greater influence on near-field enhancement, which in turn affects the breakdown energy of the femtosecond laser to the medium. Under the mediation of this new nanostructure, when the environmental medium water is ionized and broken down, the core of the silver nanoparticles does not reach the melting point and can theoretically remain intact. Finally, great significance is provided for the reuse of silver nanoparticles in practice.