## 3 Numerical Results  
We first performed numerical simulations of the nonlinear pulse propagation in step-index tellurite fibers with different core diameters, more particularly by choosing pumping parameters from the laser characteristics used in the later experimental work (200-fs Gaussian pulses in the wavelength range 2100-2400 nm). Our modelling is based on the generalized nonlinear Schrodinger equation that includes the full dispersion curve (\(\hat{\beta}\)), both instantaneous Kerr and delayed Raman nonlinear responses, and the dispersion of nonlinearity (\(\tau_{\mathrm{s}}\)) [26]. This equation governs the evolution of the complex envelope of the electrical field \(A(z,t)\) as follows:  
\[\frac{\partial A}{\partial z}=-\frac{\mathrm{\AA}}{2}A+\hat{D}A+i\gamma\Big{(} 1+\tau_{\mathrm{s}}\frac{\partial}{\partial t}\Big{)}\Big{(}A(z,t)\int_{- \infty}^{+\infty}[(1-f_{R})\delta(t^{\prime})+f_{R}h_{R}(t^{\prime})]\times|A (z,t-t^{\prime})|^{2}dt\Big{)} \tag{1}\]  
The nonlinear Kerr coefficient \(\gamma\) is calculated by means of the previously simulated effective mode area and the following nonlinear refractive index \(n_{2}=3.8\times 10^{-19}\) m\({}^{2}\).W\({}^{-1}\)[28]. For the Raman response function, we used an intermediate-broadening model using convolutions of Lorentzians and Gaussians adapted from spontaneous Raman scattering spectra and estimated Raman gain coefficient given in Reference [29]. The contribution of the delayed Raman response was found to be \(f_{R}=0.25\). Our simulations also include the fiber losses (\(\mathrm{\AA}\)) in the 1-4 \(\upmu\)m spectral range measured in Figure 2 and extrapolated above 4 \(\upmu\)m (i.e., the multi-phonon absorption edge). The dependence of SC stability on the input pulse characteristics and the fiber parameters has been extensively studied in the last decade [30; 31]. It was found in the anomalous dispersion regime that the competition of coherent (soliton fission) and incoherent (modulation instability) spectral broadening mechanisms governs the SC coherence properties. It is then reasonable to consider a pump pulse soliton number of \(N<10\) as a general rule to maintain high coherence (thus corresponding, in practice, to pump pulse durations well below the picosecond). Recall that \(N\) is defined as follows: \(N=\big{(}\gamma P_{0}T_{0}^{2}/|\beta_{2}|\big{)}^{1/2}\), where \(P_{0}\) is the peak power, \(T_{0}\) is the half-width (at \(1/e\)-intensity point) of the input pulse, and \(\beta_{2}\) is the group velocity dispersion at the pump wavelength. In contrast, the use of all-normal dispersion fibers allows highly coherent SC to be designed for initial soliton numbers up to \(N\sim 600\) and pump pulse durations of up to 1.5 ps. In addition to spectral broadening, in the following, we also analyze the associated coherence properties as a function of fiber parameters for the available pump conditions. Figure 4: (**a**) Refractive index curves of both TZNF core and TZNG cladding glasses. (**b**) and (**c**) Wavelength-dependent curves of dispersion parameter D and effective mode area A\({}_{\mathrm{eff}}\) of the fundamental guided mode in distinct step-index tellurite fibers with respective core diameters \(\phi=3\), 3.25, 3.5, 3.75, and 4 \(\upmu\)m.  
The sensitivity of SC fluctuations to input noise was then studied, as described in Reference [26]. We specifically performed 100 simulations with different random input noise imposed on the input pulse for each fiber. The ensemble of output SC fields allows the calculation of the complex degree of first-order coherence (given by Equation (2)) defined at each wavelength in the SC, typically used to characterize the shot-to-shot stability of SC sources. The angular brackets in Equation (2) denote averaging over an ensemble of independent SC realizations [26; 32].