## 4 Supercontinuum Generation
### Numerical Simulations  
In order to confirm the qualitative insights into the nonlinear dynamics occurring during the spectral broadening process, we have performed simulations based on the generalized nonlinear Schrodinger equation (GNLSE) governing the evolution of the complex envelope of the electrical field \(A\)(z,\(t\)) [2; 12; 48]:  
\[\frac{\partial A}{\partial z}=-\frac{\alpha}{2}A+\sum_{k\geq 2}i^{k+1}\frac{ \beta_{k}}{k!}\frac{\partial^{k}A}{\partial z^{k}}+i\gamma\bigg{(}1+\tau_{ shock}\frac{\partial}{\partial t}\bigg{)}\bigg{(}A(z,t)\int_{-\infty}^{+ \infty}R(tt)\times|A(z,t-tt)|^{2}dt\bigg{)} \tag{1}\]  
where \(z\) and \(t\) are the distance and the temporal coordinates respectively. \(\alpha\) is the linear loss, \(\gamma\) is the usual nonlinear coefficient, \(R(t)\) is the response function that takes into account both instantaneous electronic and delayed Raman contributions defined from the parameters given in Ref. [24]. \(\beta_{k}\)'s are the dispersion coefficients associated with the Taylor series expansion of the propagation constant \(\beta\) around the central frequency. Note that we do not include terms based on the two-photon absorption (\(\alpha_{2}\)) and three photon absorption (\(\alpha_{3}\)) since titanium dioxide presents negligible nonlinear absorption in the spectral region of interest. The main limitations of the above GNLSE is related to the fact that visible light obtained through frequency tripling of the input pulse cannot be modelled [49; 50] and that it assumes a pure single-mode propagation. However, note that the aim of these numerical simulations is to qualitatively reproduce the observed SC generation. Indeed, several parameters of the TiO\({}_{2}\) waveguide such as its dispersive properties or the exact value of the nonlinear coefficient have not been experimentally assessed yet and would deserve a devoted study that is well beyond the scope of the present letter. The exact physical features of the optical pulse entering into the waveguide are not accurately known but it can be considered as an initial 90-fs transformed-limited hyperbolic-secant pulse centred at 1.64 \(\upmu\)m (as shown in Figure 2c). For the qualitative discussion, our numerical simulations used a nonlinear index value (\(n_{2}=0.16\times 10^{-18}\) m\({}^{2}\)/W) and a Raman coefficient inferred from the literature [24], which leads to a nonlinear parameter about 1.2 W\({}^{-1}\).m\({}^{-1}\).  
In the presence of higher-order dispersion, a soliton sheds away energy into radiative dispersive waves (DWs). The frequencies of those DWs can be determined by the following phase matching condition [47]:  
\[\sum_{k\geq 2}\frac{\beta_{k}(\omega_{sol})}{k!}(\omega-\omega_{sol})^{k}=(1- f_{R})\gamma P_{sol} \tag{2}\]  
where \(\omega_{sol}\) is the angular frequency of the soliton and \(P_{sol}\) its peak power. The calculated phase matching curves are shown in Figure 5a. These curves help to understand the results of the simulations summarized in Figure 5b which have to be compared with spectra recorded and depicted in Figure 4. In particular, the resulting supercontinuum is bounded by two dispersive waves located around 1.2 and 2.3 \(\upmu\)m. Note that the SC bandwidth can be controlled by a suitable engineering of the waveguide dispersion. Here the main experimental SC features and evolution as a function of input energy are well predicted from the GNLSE simulations. In particular, the soliton frequency-shift towards 1.9 \(\upmu\)m wavelength as well as the position and the blue-shift of the dispersive waves are indeed well reproduced. The optical signal to noise ratio being much higher in our simulations compared to our experiments, we find spectral features in Figure 5 that can be only hardly observed in the experimental results. Nevertheless, we conclude that our approach based on the GNLSE captures most of the dominant spectral properties of the SC generated in our TiO\({}_{2}\) waveguides.