## 2 Theoretical Model
### Fluid Transport. Ripples and Grooves  
To model the surface modification of TiO\({}_{2}\) upon irradiation of material with repetitive pulses, the laser fluence is assumed to be sufficiently high that results in a phase transformation. The melting temperature of the material (\(T_{melting}=2110\) K [86]) is considered as the threshold at which phase transformation occurs. Following a slow resolidification process, a non-flat relief is induced on the  
Figure 1: (a) Surface plasmon wavelength as a function of the excited carrier densities for (001) and (100) crystal orientation. Prediction of structure periodicity based on [PERSON]’s theory for (b) \(44\times 10^{22}\) cm\({}^{-3}\) (100) and (c) \(27\times 10^{22}\) cm\({}^{-3}\) (001). Inset shows the profile of the efficacy factor along \(K_{x}\) (_white_ dashed line). The _black_ double headed arrow indicates the direction of the polarisation vector. material. Moreover, the fluence values used for the simulations and experiments in the current work are sufficiently high to cause ablation. To simulate mass ablation, it is assumed that it occurs if the material is heated above a critical temperature (\(\sim\)\(T_{L}\)\(>\)\(T_{L}^{buling}\) where \(T_{L}^{buling}\)\(\sim\)3245 K for TiO\({}_{2}\)). It is noted that the choice of the critical temperature \(T_{critical}\) above which mass removal occurs has been investigated in previous studies. According to the discussion in reference [87], a typical value of \(T_{critical}\) is 1-2 times higher than the boiling temperature. In other studies, it was noted that a solid material which is heated with ultrashort pulsed lasers at sufficiently high fluences undergoes a phase transition to a superheated liquid whose temperature reaches values \(\sim\)0.90 \(\times\)\(T_{\sigma}\) (\(T_{\sigma}\) stands for the thermodynamic critical point [88]) and evaporation due to dynamics of the Knudsen layer (adjacent to the liquid-vapor interface [22; 88]). To argue the choice of the ablation temperature threshold, it would be useful to know an estimate of \(T_{\sigma}\), however, to the best of our knowledge, a relevant value for \(T_{\sigma}\) for TiO\({}_{2}\) does not exist in the literature (except from a very high value for Ti (\(\sim\)15,500 K [89])). Based on the above discussion, for the sake of simplicity, the boiling temperature was selected to represent the threshold for mass removal. To describe the dynamics of an uncompressible fluid a Navier-Stokes equation (NSE) is used [83]  
\[\rho_{0}\Bigg{(}\frac{\partial\overrightarrow{u}}{\partial t}+\overrightarrow {u}\cdot\overrightarrow{\
abla}\overrightarrow{u}\Bigg{)}=\overrightarrow {\
abla}\Bigg{(}-P+\mu\bigg{(}\overrightarrow{\
abla}\overrightarrow{u} \bigg{)}+\mu\bigg{(}\overrightarrow{\
abla}\overrightarrow{u}\bigg{)}^{T} \Bigg{)} \tag{9}\]  
The boundary conditions imposed at the liquid free surface of the material are the following: \(\frac{\partial u}{\partial Z}=-\sigma/\mu\frac{\partial T_{L}}{\partial X}\) and \(\frac{\partial v}{\partial Z}=-\sigma/\mu\frac{\partial T_{L}}{\partial Y}\), where (\(u\),\(v\),\(w\)) are the components of \(\overrightarrow{u}\) in Cartesian coordinates. \(\rho_{0}\) (\(\rho_{0}=3.8375-2.8\times 10^{-4}\)\(T_{L}\) gr/cm\({}^{3}\)) [90] and \(\mu\) stand for the density and viscosity of molten TiO\({}_{2}\), while \(P\) and \(\overrightarrow{u}\) stand for the pressure and velocity of the fluid, respectively. On the other hand, \(\sigma\) corresponds to the surface tension of the material (see [9], for a more detailed description of the fluid dynamics module; similarly, temperature dependent values for \(\mu\), \(\sigma\) are used [91]). It is noted that in this work, Cartesian coordinates are used and (\(x\),\(y\),\(z\)) converts to (\(X\),\(Y\),\(Z\)) assuming a profile change with increasing \(NP\). In Equation (9), superscript \(T\) denotes the transpose of the vector \(\overrightarrow{\
abla}\overrightarrow{u}\).