Fig. 1. A schematic of the 3D simulations for bulk fused silica (a), and the coating of the multi-layer quarter-wave mirror with a fused protective layer (b). The normally incident few-cycle laser pulse (c) is driven from the minimum y boundary into a 0.1 μm vacuum region before the target. The regions are represented in yellow and the in blue. A cross-section of the multi-layer mirror in the plane is shown in (d) (the dimension is not to scale). The thickness of the top layer is 270.90 nm and then there are alternating layers of (98.95 nm) and (135.45 nm).

Fig. 2. The electron density at the center of the x–z plane (averaged over six cells in x and y) along y at 20 fs for a series of PIC simulations at various laser fluences with a 7 fs duration pulse interacting with bulk . The critical plasma density and electron instability density from Ref. [72] are labeled with dashed lines. The latter predicts damage around .

Fig. 3. The peak energy density for the series of PIC simulations at various fluences with a 7 fs duration pulse interacting with bulk fused silica. The experiment^[16] being modeled observed damage around and ablation^[21] at . The shaded horizontal line/bands indicate approximate energy density thresholds for melting, boiling and dissociation. The shaded vertical bands represent uncertainty in experimental damage and ablation thresholds. The simulations including collisions have a higher predicted energy density. Including both our Keldysh photoionization model and collisional effects shows agreement between the expected damage fluence and the dissociation energy.

Fig. 4. Excited electron energy distributions in a region of the target at the center of the interaction for a pulse. A simulation with just field ionization is shown in (a) and a simulation with field ionization and collisions is shown in (b). A fit with the given temperature is shown. We observe the nonthermal nature, especially for early times and for simulations without collisions.

Fig. 5. The electron density at the center of the x–z plane along y at 20 fs for a series of PIC simulations at various fluences with a 7 fs duration pulse interacting with a multi-layer mirror. The yellow area is and blue area is . Different critical electron densities are labeled in the figure with dashed lines.

Fig. 6. The peak electron energy density at the center of the x–z plane along y at 20 fs for a series of PIC simulations at various fluences with a 7 fs duration pulse interacting with a multi-layer mirror. The yellow area is fused and blue area is fused . The boiling and melting energy densities are at about and , respectively, indicated as yellow dashed lines, and the boiling and melting energy densities are at about and , respectively, indicated as blue dashed lines.

Fig. 7. The (a) energy density and (b) electron density at the center of x on the y–z plane at 20 fs and with a 7 fs duration pulse interacting with a multi-layer mirror. The mirror surface starts at y = 0, and the dashed red lines are the interfaces between layers. The white areas have no excited electrons. (c) The maximum accumulated normalized intensity over the entire simulation.

Fig. 8. The energy histogram for the electrons in the first layer (a) and layer (b) at 6, 10, 24 fs for the simulation with distribution fitted. The black curve is the Maxwell–Boltzmann distribution given the average energy at the stable stage around 24 fs.

Fig. 9. The and electron energy spatial distribution for a fluence of is shown on the left-hand side at 24 fs. The highest energy particles are found in the center of the first layer and at the interfaces between layers. Particle trajectories for random 2% of electrons with low (below average) final energy demonstrate the ionization dynamics in the first layer up to 16 fs (after the pulse has left). Ionization near the surface generally occurs at later times, as indicated by the color of the tracks.

Fig. 10. Percent error for 10–500 terms in the infinite sum in Equation (3) for at varying laser intensities. The error is calculated with the result for 1000 terms as the ‘actual’ value. Our simulation framework uses 500 terms, which shows a negligible difference compared to 1000 terms for these intensities. The errors for (not shown) are of similar magnitude to those for .