## 4 Discussion
### LIPSS Formation
In Figure 10, simulations and experimental results are shown after a double pulse of \(E_{p}\) = 125 mJ/cm\({}^{2}\) irradiated (001) TiO\({}_{2}\) for two delays, \(t_{delay}\) = 4 ps and \(t_{delay}\) = 10 ps. An interesting point regarding the two values of the pulse separation is that they are situated before and past an optimum pulse separation value (\(t_{delay}\)\(\sim\)6 ps) for which the depth is critical to lead to a transition of structures from ripples (i.e., subwavelength structures oriented perpendicularly to the laser polarization direction) to grooves (i.e., suprawavelength structures that are aligned with laser polarization direction); therefore it is important to evaluate the impact of small and large pulse separation. Due to the fact that for a smaller pulse separation (\(t_{delay}\) = 4 ps), damage related  
Figure 8: Surface structures on (001) rutile TiO\({}_{2}\) obtained after laser irradiation (248 nm, 450 ps, \(NP\) = 100) with total measured fluence of 250 mJ/cm\({}^{2}\) and delay between pulses of 0 ps (**a**), 4 ps (**b**), 10 ps (**c**) and 30 ps (**d**). Doubled–ended arrow indicates laser polarisation direction. temperatures will be developed into smaller regions, a shallower profile will be produced compared to that for \(t_{delay}=10\) ps. Hence, the condition for promoting the production of grooves will be satisfied more easily at longer delays. Simulation results show rippled structures (Figure 10a) for \(NP=80\) for the above conditions at \(t_{delay}=4\) ps of periodicity \(\sim\)220 nm and grooves (Figure 10b) for \(NP=80\) at \(t_{delay}=10\) ps of periodicity \(\sim\)1.8 \(\upmu\)m. Experimental results on (001) TiO\({}_{2}\) rutile surface that validate the simulations are illustrated in Figure 8b,c, which show the production of ripples and grooves, respectively, for \(NP=100\). Interestingly, irradiation of (100) rutile TiO\({}_{2}\) surface does not lead to the formation of grooves [46]. Instead of grooves, a featureless flat area replaces the ripples when the delay between laser pluses increases (see Appendix C). The formation of this unusual structure is not yet explained by simulations. This emphasises that crystalline orientation of the irradiated surfaces is a key parameter in the formation processes of suprawavelength periodic structures under femtosecond laser irradiation. The multiscale model used in this work described the physical processes that account for the formation of LSFL on TiO\({}_{2}\) while a transition from ripples to grooves was also derived. It is evident that more accurate conclusions will be drawn if more appropriately developed experimental (for example time-resolved) protocols are also introduced to evaluate ultrafast dynamics and relevant parameter valuessuch as electron-electron and electron-hole scattering frequencies, electron-phonon relaxation times, and reflectivity changes. It is noted that although the theoretical predictions appear to agree with the evolution of the surface structures, simulation results manifested a significant ablation that takes place for the fluence values used in the experiment which to some extent deviates from experimental observations. This can also be attributed to the choice of various parameters such as the electron effective mass that affects the dielectric constant and eventually the optical properties of the material and energy absorption; this is a critical issue to both the induced carrier density and thermal response of the material. Values for \(m_{opt}^{*}\) used in the current work and at low wavelengths might need to be revised appropriately through a more rigorous theoretical approach and suitable experimental  
Figure 10: Simulation results of periodic surface profiles for \(t_{delay}=4\) ps (**a**) and \(t_{delay}=10\) ps (**b**). Results are illustrated in a quadrant. Results are for (001) TiO\({}_{2}\). Double-ended arrow indicates laser polarisation direction. Figure 9: SPP wavelength _vs. NP_. (\(t_{delay}=4\) ps, \(E_{p}=125\) mJ/cm\({}^{2}\) for (100) and (001) TiO\({}_{2}\)). protocols.