## 4 Supercontinuum Generation  
### Experimental Results  
Figure 3 summarizes the typical spectral properties of the SC generated in the 1.5-\(\upmu\)m wide TiO\({}_{2}\) waveguide. The input peak power of the fs-pulses injected into the waveguide was estimated to almost 1.3 kW (corresponding to a soliton number N = 15 [2,43]) but given the experimental difficulty to evaluate precisely the energy coupled into the fundamental mode, this value remains a rough estimate. By means of a careful adjustment of the input polarization aligned to the fundamental TE mode which is expected to be in anomalous dispersion regime at the pump wavelength, an octave-spanning supercontinuum spectrum is then generated, with a central part ranging from 1050 nm up to 1910 nm (\(-\)20 dB spectral width in Figure 3a). Compared to the pioneer experiments of spectral broadening leading to tens of nanometres around 1550 nm reported by [PERSON] et al. [24], the present results represent a significant improvement by much more than one order of magnitude. A third harmonic signal is also generated into the waveguide [45] so that visible light is also clearly observed by the naked eye in the early stage of propagation as can be seen from the photography taken from the top of the waveguide (Figure 3b,c). From these pictures, we can also notice that green light is generated first, followed by yellow, orange and then red, which is consistent with the progressive frequency shift of the femtosecond pulse during its propagation within the nonlinear waveguide. The visible light emitted along the waveguide represents therefore an indirect but practical way to track the longitudinal evolution of the central wavelength of the Raman-shifted soliton. However, the level of losses of TiO\({}_{2}\) in the visible region remains slightly too high and, given the sensitivity of our spectral analysis devices, has prevented the detection of the visible part of the generated light at the output of the 2-cm-long waveguide. Figure 4a shows the dispersion curve of the waveguide obtained from finite element simulations and highlights the significant contribution of the waveguide to the overall dispersion that strongly differs from the properties of the TiO\({}_{2}\) material. The TE fundamental mode is expected to be anomalous around the pumping wavelength, therefore enabling a soliton-driven SC dynamics [3,46]. Moreover, we confirm here that the waveguide presents two zero dispersion wavelengths that ultimately limit the spectral expansion of our supercontinuum [47]. Note also that the TM mode lies in the normal dispersion regime at the pump wavelength and its considerable dispersion value makes that only a negligible broadening can be expected (not presented here because similar to the input pulse). This point therefore further explains the high polarization sensitivity of the input conditions. In order to better describe the SC dynamics, other spectra in the infrared part of the spectrum are plotted in Figure 4b as a function of the estimated coupled energy into the waveguide. As expected, the input pulse undergoes a strong and rapid self-phase modulation process with increased input energy, then followed by the emission of a dispersive wave at short wavelengths located in normal dispersion regime. Indeed, due to the presence of two zero-dispersion wavelengths, short- and long-wavelength SC edges are determined through phase-matched dispersive wave generation, namely below 1.3 \(\upmu\)m and beyond 2.1 \(\upmu\)m. However, note that the possible residual contribution of spurious double-order mode in the recorded spectra of Figure 4b in the 2200-2400 nm wavelength range prevent us to assert the generation of a dispersive wave in this spectral region. This is the reason why this wavelength area has been shaded in Figure 4b to avoid any misleading interpretation. Simultaneously, we observe the Raman self-frequency shift of one soliton, emitted from the initial pulse breakup, until 1.84 \(\upmu\)m for the maximum input energy.