## 3 Linear and Nonlinear Characterizations  
### Linear Characterizations  
Linear transmission spectra of the fabricated SRN ridge waveguides with lengths of 5.5 mm, 13 mm and 21 mm were measured from 1860 nm to 1980 nm. Then the cut-back method was used to extract the propagation losses and the coupling losses to a tapered lensed fiber for waveguides with different \(w\) and \(h\). In order to reduce the accidental error, six identical waveguides were fabricated and measured for each waveguide dimension configuration. To serve as an example, Figure 2a,b exhibits the propagation and coupling losses with respect to the wavelength for the fabricated SRN ridge waveguides with \(h\) = 380 nm and \(w\) = 950 nm, and \(h\) = 640 nm and \(w\) = 1320 nm, respectively. Due to the Fabry-Perot interference effect between the two facets of a waveguide, the nonperfect uniformity of the six identical waveguides and the measurement inaccuracy, there were statistical errors for the extracted losses, as shown in the figures. Nevertheless, the propagation losses of the fabricated SRN ridge waveguides were around 2 dB/cm in the investigated wavelength range. Meanwhile, the coupling losses were around 4 dB/facet. Results on the other waveguide dimensions are summarized in Table 1 in the end of this section. The propagation loss \(\alpha\) of the fabricated SRN ridge waveguide includes linear absorption loss \(\alpha_{1}\) and scattering loss \(\alpha_{2}\). \(\alpha_{1}\) can be derived from the imaginary part of the effective RI, \(N_{eff,r}\) of the SRN ridge waveguide by considering the absorption of the SRN material and expressed as [41]  
\[\alpha_{1}=\frac{2\omega}{c}\text{Im}(N_{eff,r}) \tag{1}\]  
Figure 1: The fabricated silicon-rich nitride (SRN) ridge waveguide. (**a**) Schematic diagram of the structure. (**b**) Scanning electron microscopy image. (**c**) Simulated fundamental transverse-electric (TE\({}_{0}\)) mode profile at 1950 nm of the waveguide with a width of 1.32 \(\upmu\)m and a rib height of 640 nm. Figure 2: The measured propagation loss and coupling loss with respect to the wavelength for the fabricated silicon-rich nitride ridge waveguides with (**a**) \(h\) = 380 nm and \(w\) = 950 nm and (**b**) \(h\) = 640 nm and \(w\) = 1320.  
where \(\omega\) is the optical angular frequency and \(c\) is the speed of light in vacuum. The scattering loss \(\alpha_{2}\) depends on the sidewall roughness, half the waveguide width \(w/2\), RI steps from the SRN core to the surroundings, and scaling factor of a ridge waveguide to a channel waveguide (i.e., \(h=H\)). The sidewall roughness is characterized by the correlation length \(L_{c}\) and the mean squared error \(\sigma^{2}\) deviated from the sidewall surface. Thus, the scattering loss \(\alpha_{2}\) can be expressed as Equation (2) [42]  
\[\alpha_{2}=4.343\frac{\sigma^{2}}{\sqrt{2}k_{0}(w/2)^{4}N_{eff,r}}g\cdot f\cdot s \tag{2}\]  
where the unit of \(\alpha_{2}\) is dB/m, \(k_{0}\) is the wave vector in vacuum, and function \(g\) is completely determined by the dimensions of the SRN channel waveguide and \(f\) is relevant to \(L_{c}\) and RI steps. Details for calculating \(g\) and \(f\) can be found in [PERSON] and [PERSON]'s work for analyzing a planar waveguide's scattering loss using the exponential autocorrelation function [43]. The scaling factor \(s\) can be calculated as [42]  
\[s=\frac{\delta N_{eff,r}/\delta d}{\delta N_{eff,c}/\delta d} \tag{3}\]  
where \(N_{eff,c}\) is the effective RI of the SRN channel waveguide and \(d\) is half the waveguide width \(w/2\). A larger \(h\) and a smaller \(w\) mean more SRN being etched and more light fields overlapping the rough sidewalls and thus a larger scattering loss. In contrast, for the SRN ridge waveguide with a small \(h\) and a large \(w\), almost all the light is confined inside the SRN core and, thus, \(\alpha\) is dominantly determined by the absorption loss \(\alpha_{1}\).