## 2 Theoretical Model
### Ultrafast Dynamics
Based on previous reports [50], scattering rates for electron-hole and electron-electron collisions scale as \(N_{c}\) and \(N_{c}^{1/3}\), respectively, yielding approximate value \(\tau_{c}\)-1 fs for TiO\({}_{2}\) regardless of the orientation. This is the value that was used in this work. \(E_{g}\) stands for the energy band gap of rutile TiO\({}_{2}\) (\(\sim\)3 eV) [46]. The'source term' that appears in the first equation in Equation (1) [22; 56; 57] is given by  
\[S(\overrightarrow{r},t)=(\alpha_{SPA}+\alpha_{FCA})I(\overrightarrow{r},t)+ \beta_{TPA}I^{2}(\overrightarrow{r},t)-\frac{\partial N_{c}}{\partial t} \Big{(}E_{g}+3k_{B}T_{c}\Big{)} \tag{3}\]  
In Equation (1), it is evident that carrier transport and carrier heat diffusion have been ignored. Although more rigorous expressions have been presented in previous works in which the contribution of those effects have been incorporated, reports showed that estimation of damage thresholds are not influenced significantly if those terms are ignored [56; 57]. Similarly, for the sake of simplicity, it is assumed that the energy band gap does not vary significantly within the pulse. The laser intensity \(I\Big{(}\overrightarrow{r},t\Big{)}\) in Equations (1)-(3) is obtained by considering the propagation loss due to one-photon, two-photon and free carrier absorption, respectively [22; 26; 30; 36; 57; 65],  
\[\frac{\partial I(\overrightarrow{r},t)}{\partial z}=-(\alpha_{SPA}+\alpha_{ FCA})I(\overrightarrow{r},t)-\beta_{TPA}I^{2}(\overrightarrow{r},t) \tag{4}\]  
for a laser beam that is Gaussian, both temporally and spatially, the transmitted laser intensity on a _flat_ surface for double pulses is provided by the following expression  
\[I(\mathrm{x},\mathrm{y},\mathrm{z}=0,t)=\frac{\sqrt{\mathrm{ln}2}E_{p}(1-R( \mathrm{z}=0,t))}{\sqrt{\pi}\tau_{p}}e^{-\frac{(2\frac{1}{2}\omega^{2}+\gamma 2)}{E_{0}^{2}}}\Bigg{[}e^{-4\ln 2(\frac{t-3 \tau_{p}}{\tau_{p}})^{2}}+e^{-4\ln 2(\frac{t-3\tau_{p}-t_{delay}}{\tau_{p}})^{2}} \Bigg{]} \tag{5}\]  
where \(2E_{p}\) is the total fluence of the laser beam and \(\tau_{p}\) is the pulse duration (i.e., full width at half maximum), \(R_{0}\) is the irradiation spot-radius (distance from the centre at which the intensity drops to \(\
icefrac{{1}}{{e^{2}}}\) of the maximum intensity, and \(R\) is the reflectivity while irradiation under normal incidence was assumed. The value of \(3\tau_{p}\) in the exponent in Equation (5) has been chosen to ensure that the intensity switches off at \(t\) = 0. The objective of the current work is to illustrate the impact of temporarily separated pulses, therefore, Equation (5) provides the total intensity of a double pulse of pulse delay equal to \(t_{delay}\). In Equation (5), Cartesian coordinates have been used. The calculation of the free carrier absorption coefficient and the reflectivity are derived from the dielectric parameter of the material (assuming also corrections due to band and state filling [66]), \(\varepsilon^{\prime}\)  
\[\varepsilon^{\prime}=1+(\varepsilon_{un}-1)\Big{(}1-\frac{N_{c}}{N_{v}}\Big{)} -\frac{e^{2}N_{c}}{\varepsilon_{0}\omega_{L}!}\frac{1}{m_{opt}^{*}\big{(}1+i \frac{1}{\omega_{L}\tau_{c}}\big{)}} \tag{6}\]  
where \(N_{v}\) stands for the valence band carrier density (\(\sim\)\(5\times 10^{23}\) cm\({}^{-3}\) for TiO\({}_{2}\)) and \(\varepsilon_{un}\) is the dielectric parameter of the unexcited material at \(\lambda_{L}\) = 248 nm. In previous reports, the values of \(\varepsilon_{un}\)[67; 68; 69; 70; 71; 72] and the optical mass \(m_{opt}^{*}\)[47; 48; 50; 51; 52; 53; 54] were measured and estimated for both (001) and (100), respectively. As noted above, the dielectric parameter is expected to affect the energy absorption and therefore precise parameter values for \(m_{opt}^{*}\) and \(\varepsilon_{un}\) are required. Based on the aforementioned works, the values\(\varepsilon_{un}^{(100)}=1.3+7.61i\), \(\varepsilon_{un}^{(001)}=3.09+6.95i\), \(m_{opt}^{\star(100)}=6m_{e}\), \(m_{opt}^{\star(001)}=2m_{e}\) were used in the current study.