## 2 Theoretical Model
### LSFL Formation  
Among the physical processes that are aimed to account for the formation of LSFL structures, the most dominant scenarios are based on: (a) the interference of excited Surface Plasmon Polariton waves (SPP) with the incident beam [22; 30; 34; 73], (b) the interference of the incident laser beam with induced surface scattered waves [40] or (c) the role of near field effects [39]. A key ingredient, though, for the excitation of SPP or surface scattered waves is the presence of an initial corrugation (for example, surface defects [22]). In regard to the first scenario that will be adopted in the current work, the interference of the incident and SPP leads to the formation of periodic ripples (LSFL) that are oriented _perpendicularly_ to the electric field of the laser beam [18; 21; 22; 31; 32; 33; 74; 75; 76; 77; 78; 79; 80]; this is generated by a combination of multiscale processes such as the spatially periodic energy distribution (as a result of the interference of the incident beam and a surface wave), an induced characteristic spatial modulation of the lattice system through electron-phonon scattering and effects related to fluid dynamics (for conditions that allow the material to undergo a phase transition) leading eventually to a rippled profile when the resolidification process ends [22; 78; 79; 81]. To describe the excitation of SPP, dispersion conditions are derived through the elaboration of the boundary conditions (continuity of the electric and magnetic fields at the interface between a metallic and dielectric material) (\(\varepsilon_{d}=1\)) for a flat surface (number of pulses, \(NP=1\)). Therefore, a requirement for a semiconductor to obey the above relation and conditions [22; 82] is that \(Re(\varepsilon^{\prime})<-1\) and the computed SPP wavelength \(\Lambda_{\mathrm{SPP}}\) is given by  
\[\Lambda_{\mathrm{SPP}}=\frac{\Lambda_{L}}{k_{e}\sqrt{\left(\frac{\varepsilon^ {\prime}\varepsilon_{d}}{\varepsilon^{\prime}+\varepsilon_{d}}\right)}} \tag{8}\]  
The condition \(Re(\varepsilon^{\prime})<-1\) and Equations (6) and (8) can be used to derive the range of values of the excited carrier densities that lead to SPP excitation. According to the above condition, only carrier densities larger than \(\sim\)13 \(\times\) 10\({}^{22}\) cm\({}^{-3}\) (for (100)) are capable to lead to the excitation of SPP (Figure 1a) for nearly flat profiles. Although, Equation (8) can be used to calculate the SPP for nearly flat surfaces, there exists a discrepancy between the experimentally measured ripple periodicities and predictions due to the above expression. This is due to the fact that in order to observe periodic profiles experimentally, a sufficient number of pulses are required. On the other hand, repetitive irradiation is accompanied with a continuously varying depth of the periodic profile which results from a shift of the SPP wavelength to smaller values [21; 78; 83; 84] that leads to SPP frequencies that deviate from the value resulting from Equation (8). On the other hand, for nearly flat profiles, it is evident that due to the different dielectric parameters and optical masses of TiO\({}_{2}\) for (001 and (100), a larger range of \(\Lambda_{\mathrm{SPP}}\) values for (100) is predicted (\(\sim\)205 nm) compared to the computed value (\(\sim\)232 nm) for (100). The minimum values of \(\Lambda_{\mathrm{SPP}}\) occur for \(\sim\)27 \(\times\) 10\({}^{22}\) cm\({}^{-3}\) (001) and \(\sim\)44 \(\times\) 10\({}^{22}\) cm\({}^{-3}\) (100), respectively. To describe the formation of LSFL with scenario (b), the electron density values and surface corrugation are used to determine the orientation and the periodicity of LIPSS. To correlate \(N_{c}\) with the formation of laser-induced surface structures, the inhomogeneous energy deposition into the irradiated material is computed through the computation of the product \(\eta(\vec{\underline{K}},\ \overrightarrow{k_{i}})\times\left|\underline{ \left(\vec{\underline{K}}\right)}\right|\)[40].