## 4 Optimal Design for the Maximal Nonlinear Energy Efficiency  
For waveguides without nonlinear losses, we can use a figure of merit (FOM) to evaluate the nonlinear energy efficiency of the waveguide, this is the ratio of the waveguide nonlinear parameter to the linear loss, expressed as [46]  
\[\text{FOM}=\frac{\gamma}{\alpha} \tag{11}\]  
For the proposed SRN ridge waveguide, both \(\gamma\) and \(\alpha\) are dependent on the width and rib height. While a deeper etching and a moderately smaller width can give a stronger confinement and hence a larger nonlinearity, this also leaves more light overlapping the rough sidewalls and therefore gives a larger linear propagation loss. Thus, the FOM can be used to fairly evaluate the overall nonlinear efficiency of the SRN ridge waveguide with different dimensions. We have calculated the nonlinear \begin{table}
\begin{tabular}{c c c c c} \hline  & \begin{tabular}{c} **Measured \(\alpha\)** \\ **(dB/cm)** \\ \end{tabular} & \begin{tabular}{c} **Calculated \(\alpha\)** \\ **(dB/cm)** \\ \end{tabular} & \
\begin{tabular}{c} \(\gamma\) \\ **(W\({}^{-1}\) m\({}^{-1}\))** \\ \end{tabular} &
\begin{tabular}{c} **\(\mathbf{n_{2}}\)** \\ **(m\({}^{2}\)/W)** \\ \end{tabular} \\ \hline \(h=230\) nm, \(w=840\) nm & \(1.7\pm 1.6\) & \(1.8\) & \(3.74\) & \(1.77\times 10^{-18}\) \\ \hline \(h=230\) nm, \(w=1360\) nm & \(2.1\pm 1.0\) & \(1.9\) & \(2.90\) & \(1.30\times 10^{-18}\) \\ \hline \(h=380\) nm, \(w=950\) nm & \(1.7\pm 1.0\) & \(2.1\) & \(2.79\) & \(0.87\times 10^{-18}\) \\ \hline \(h=380\) nm, \(w=1350\) nm & \(2.3\pm 1.0\) & \(2.1\) & \(2.49\) & \(0.80\times 10^{-18}\) \\ \hline \(h=640\) nm, \(w=800\) nm & \(3.3\pm 1.4\) & \(3.3\) & \(5.98\) & \(1.24\times 10^{-18}\) \\ \hline \(h=640\) nm, \(w=1320\) nm & \(2.7\pm 0.9\) & \(2.6\) & \(4.62\) & \(1.13\times 10^{-18}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of the linear and nonlinear properties of the fabricated silicon-rich nitride ridge waveguides with various rib heights (\(h\)) and widths (\(w\)) at 1950 nm. Figure 5: Measured output four-wave mixing spectra (**a,b**) and the wavelength conversion efficiencies with respect to the coupled pump powers (**c,d**) of the fabricated silicon-rich nitride ridge waveguides with a length of 21 mm. (**a**) and (**c**) are for the waveguide with \(h=\)380 nm and \(w=\) 950 nm. (**b**) and (**d**) are for the waveguide with \(h=\) 640 nm and \(w=\) 1320 nm. parameter, linear loss and FOM of the SRN ridge waveguides for many combinations of \(w\) and \(h\) by knowing the linear and nonlinear RIs of SRN and the fabrication quality (i.e., \(L_{\text{c}}\) and \(\sigma\)) and shown them in Figure 6a-c, respectively. As expectations, \(\gamma\) and \(\alpha\) generally have the similar varying trend as the waveguide width or etch depth changes. This finally yielded a maximal value of 0.0804 W\({}^{-1}\) of the FOM at the rib height \(h\) = 700 nm and width \(w\) = 1100 nm. Here, \(\gamma\) an \(\alpha\) were calculated to be 5.50 W\({}^{-1}\) m\({}^{-1}\) and 2.97 dB/cm, respectively. We have also calculated the nonlinear performances of the FWM wavelength conversion for the designed SRN ridge waveguide with the maximal FOM (\(h\) = 700 nm, \(w\) = 1100 nm). Figure 7a shows the CEs as the waveguide length increases, when the input pump wavelength/power and input signal wavelength/power were set to be 1950 nm/500 mW and 1951 nm/1 mW, respectively. Here, CE is defined as the ratio of the output idler power to the input signal power. The nonlinear effect for generating new idler photons dominated the power of the idler for a short waveguide while for a long waveguide, the linear loss took charge. Thus, there was an optimal length for maximal CE and, here in Figure 7a, a length of 14.6 mm was obtained for a maximal CE of -36.2 dB. It is worth noting that, to obtain a higher CE, one can further increase the pump power or use a resonating structure to enhance the nonlinear interactions between light and the SRN ridge waveguide.