## 4 Discussion
### LIPSS Formation  
The above investigation provided a detailed description of the ultrafast dynamics after the heating of rutile TiO\({}_{2}\) of different crystal orientation with ultrashort double laser pulses. As noted in the introductory section, a question of paramount importance is how the pulse separation and the consideration of irradiation of rutile TiO\({}_{2}\) (001) and (100) influences the surface morphological features (i.e., types of structures and periodicities). The images in Figure 8 show the TiO\({}_{2}\) (001) surface after irradiation with two pulses of (peak) fluence (for each pulse) \(E_{exp}\) = 125 mJ/cm\({}^{2}\) and different temporal delays between laser pulses. According to results in a previous report [46], this fluence corresponds to the LSFL domain of morphological surface maps (MSM). In agreement, ripples were formed at a small delay of \(t_{delay}\)\(\leq\) 4 ps (Figure 8a,b). However, at a longer delay of \(\sim\)10 ps (Figure 8c), suprawavelength grooves were formed; they coarsened at even longer delay of \(\sim\)30 ps (Figure 8d) and disappeared at \(\sim\)60 ps, being replaced by spike-like columns (not shown). To relate experimental data with theoretical results, it is important to estimate the dependence of the produced carrier densities with the induced surface structures as an increase in the irradiation dose (i.e., increase in \(NP\)) modifies the energy absorption and characteristics of the morphological features. For the investigation of the LIPSS formation, between the two proposed mechanisms stated in the previous section based on [PERSON]'s theory and the excitation of SPP, the former suffers from fundamental inconsistencies: more specifically, it is already known that one of the limitations of the efficacy factor-based theory is the neglect of the so-called 'feedback mechanism' which is very important to calculate the evolution of the periodicity of the induced periodic structures [34; 40]. On the other hand, the SPP-based approach is used to compute the periodicities considering the produced carrier density values following the surface morphology that is induced for every pulse. In contrast to more precise electrodynamics simulations based on Finite Difference Finite Domain Schemes (FDTD) used in previous reports to correlate the induced LIPSS periodicities with increasing _NP_[21; 24; 39; 84; 98], an alternative and approximating methodology was employed, in this work, to relate the SP wavelength with the produced maximum depth of the corrugated profile [78; 79] (i.e., which is directly linked with _NP_). The methodology is based on the spatial distribution of the electric field on a corrugated surface of particular periodicity and height and how continuity of the electromagnetic fields influences the features of the associated SPP. Results for SPP wavelength as a function of _NP_ are illustrated in Figure 9 for \(E_{p}\) = 125 mJ/cm\({}^{2}\) for (100) and (001) TiO\({}_{2}\), respectively (\(t_{delay}\) = 4 ps). As mentioned in the previous section, the decrease in SPP wavelength with the increase in the irradiation dose _NP_ is related to the expected shift of SPP wavelength to smaller values as the profile becomes deeper [21; 78]. Furthermore, according to Figure 7a-h, the larger carrier densities (and thermal response) produced for the orientation (001) lead to a more rapidly attained deeper profile with respect to that for (100). Hence, the curve that shows the decrease in the ripple periodicity as a function of _NP_ should be steeper for (001) while SPP wavelengths also are expected to expand to a larger range of values (Figure 9). Furthermore, simulation results predict a substantially smaller SPP wavelength for (001) orientation which should be also projected on the periodicity of the induced ripples. Apart from the comparison of the LIPSS periodicities following irradiation with the same fluence, pulse separation (\(t_{delay}\) = 4 ps), and _NP_, it is also very important to evaluate potential morphological variations for different pulse separation.