## 3 Numerical Results
Figure 5 shows the variation of the average SC spectrum obtained in our tellurite fiber for a core diameter ranging from 3 to 4 \(\upmu\)m. Here the fiber length was fixed to 12 cm, and the input peak power of 200-fs pulses at 2300 nm was set to 3 kW. \[\left|g_{12}^{(1)}(\omega)\right|=\left|\frac{\left\langle\widetilde{A}_{i}^{* }(\omega)\widetilde{A}_{j}(\omega)\right\rangle_{i\
eq j}}{\left\langle\left| \widetilde{A}(\omega)\right|^{2}\right\rangle}\right| \tag{2}\]  
One can mainly distinguish three configurations, namely, (i) the pumping in the normal dispersion regime (\(\phi<3.3\)\(\upmu\)m), which rather leads to narrow spectral broadenings, (ii) the pumping in the anomalous dispersion regime between two close ZDWs (\(3.3<\phi<3.6\)\(\upmu\)m), which results in the broadest SC spectra, and (iii) the pumping in the anomalous dispersion regime when the second ZDW is far from the pump or does not exist (\(\phi>3.6\)\(\upmu\)m). In the first case (i), SC generation is driven by self-phase modulation (SPM) and optical wave-breaking, thus the spectrum remains nearly symmetric around the pump wavelength [33] and stable to input noise, which guarantees a full SC coherence, as shown in Figure 5b. These are typical features of SC generation in ANDi fibers even if the input soliton number (corresponding to input pulse and fiber parameters) is well above 10 [31] (see Figure 5c), for increasing core diameters in this regime, the group-velocity dispersion approaches zero. For the second case (ii), the spectral broadening reaches maximum bandwidth possible and results from well-known soliton dynamics and Raman soliton-self-frequency shift [26]. Note the formation of dispersive waves (DWs) in both regions of normal dispersion, in the near-IR below 2000 nm and in the mid-IR beyond 3000-3500 nm, induced by the presence of two close ZDWs. However, we observe the degradation of SC coherence in particular in the wavelength range of DWs (note that the soliton number is around 10). In the third case (iii), spectral broadening dynamics are similar, but without mid-IR dispersive wave since the second ZDW is far-detuned. Consequently, the SC bandwidth is reduced for increasing fiber core diameters, in particular due to the larger anomalous dispersion that restrains soliton dynamics. Note that the soliton number decreases well below 10, thus resulting in the recovery of the full degree of SC coherence. Figure 5: \(Cont.\)  
To gain further insight into the limiting pumping parameters on SC coherence, we performed similar numerical simulations for the same pump wavelength but with distinct input peak powers, namely 5 kW (see Figure 6) and 1 kW (see Figure 7). In Figure 6, we recover almost the same results as those described for Figure 5 with increased spectral broadenings, except that the degradation of SC coherence now occurs even in the third SC pumping configuration with a single ZDW. We also clearly note that the corresponding soliton number is always above 10. It is worth mentioning that the generated SC becomes completely incoherent in the second configuration when pumping between two ZDWs. Now when looking at Figure 7 with a lower input peak power, the full SC coherence is preserved regardless of the fiber dispersion properties (i.e., the fiber core diameter). The empirical rule for SC design is again verified since the calculated soliton number is below 10 for pumping in the anomalous dispersion regime. We also check that the SC spectrum can still extend in the mid-IR until 4.5 \(\upmu\)m with such a moderate input power. Figure 5: (**a**) Evolution of the average supercontinuum (SC) spectrum calculated from an ensemble of 100 simulations for each fiber core diameter of the step-index tellurite fiber (black dashed lines indicate the position of zero-dispersion wavelengths (ZDWs)). The fiber length was fixed to 12 cm, and the input peak power of 200-fs pulses at 2300 nm was set to 3 kW (equivalent to 600-p] pulse energy, see Section 4).