## 3 Linear and Nonlinear Characterizations
### Nonlinear Characterizations
They can be calculated as [45]  
\[A_{eff}(\omega_{m})=\frac{\left|\iint\Re{\left[\overset{\rightarrow}{e}(x,y, \omega_{m})\times\overset{\rightarrow}{h}^{*}(x,y,\omega_{m})\right]\cdot \hat{e}_{z}dxdy}\right|^{2}}{\iint\left\{\Re{\left[\overset{\rightarrow}{e}(x,y, \omega_{m})\times\overset{\rightarrow}{h}^{*}(x,y,\omega_{m})\right]\cdot \hat{e}_{z}}\right\}^{2}dxdy} \tag{8}\]  
\[\overline{n}_{2}(\omega_{m})=\frac{\varepsilon_{0}}{\mu_{0}}\frac{\iint n_{2} (x,y,\omega_{m})n^{2}(x,y,\omega_{m})\left|\overset{\rightarrow}{e}(x,y, \omega_{m})\right|^{2}\left|\overset{\rightarrow}{e}^{*}(x,y,\omega_{m}) \right|^{2}dxdy}{\iint\left\{\Re{\left[\overset{\rightarrow}{e}(x,y,\omega_ {m})\times\overset{\rightarrow}{h}^{*}(x,y,\omega_{m})\right]\cdot\hat{e}_{z }}\right\}^{2}dxdy} \tag{9}\]  
where \(n(x,y,\omega_{m})\) and \(n_{2}(x,y,\omega_{m})\) are the linear and nonlinear RIs of the material at position \((x,y)\) at the frequency \(\omega_{m}\), respectively. \(\overset{\rightarrow}{e}(x,y,\omega_{m})\) and \(\overset{\rightarrow}{h}(x,y,\omega_{m})\) are the electric and magnetic field distributions on the waveguide transverse plane. \(\hat{e}_{z}\) is the unit vector along the propagation direction. \(\varepsilon_{0}\) and \(\mu_{0}\) are the dielectric constant and the permeability constant, respectively. Finally, the conversion efficiency \(\eta\) (in the unit of dB) is defined as the ratio of the output idler power to the output signal power, that is  
\[\eta(\text{dB})=-10\text{lg}\frac{\left|A_{I}(L)\right|^{2}}{\left|A_{S}(L) \right|^{2}} \tag{10}\]  
where \(L\) is the physical length of the waveguide. We have measured the output FWM spectra of 21-mm-long SRN ridge waveguides with different waveguide dimensions under various coupled pump powers and extracted the conversion efficiencies (CEs), when the signal wavelength and the incident signal power were fixed at 1953 nm and 10 mW, respectively. For example, Figure 5a,b shows the measured FWM spectra of the waveguide with \(h\) = 380 nm and \(w\) = 950 nm under a coupled pump power of 61.5 mW, and \(h\) = 640 nm and \(w\) = 1320 under a coupled pump power of 52.4 mW, respectively. CEs of -53.2 dB and -51.1 dB can be extracted from the spectra. It should be noted that there was already some idler light generated in the incident fiber. We have normalized the output idler powers to that coming from the fiber and found little difference on the CEs between the normalizations before and after. Figure 5c,d shows the measured and normalized CEs with respect to the coupled pump power for the waveguides with \(h\) = 380 nm and \(w\) = 950 nm, and \(h\) = 640 nm and \(w\) = 1320 nm, respectively. By solving the equations from (4) to (10), we can fit the measured CEs versus the coupled pump power and the nonlinear parameter \(\gamma\) can be extracted to be 2.79 W\({}^{-1}\) m\({}^{-1}\) for the waveguide with \(h\) = 380 nm and \(w\) = 950 nm and 4.62 W\({}^{-1}\) m\({}^{-1}\) for the waveguide with \(h\) = 640 nm and \(w\) = 1320, respectively. With the extracted \(\gamma\) value, the nonlinear index \(n_{2}\) of the SRN material can also be calculated by using Equation (7). Here, we assume the whole nonlinear effect was contributed by the SRN material since the silica surroundings have a nonlinear RI orders lower than that of the SRN material [45]. We have summarized the linear and nonlinear properties of the fabricated SRN ridge waveguides with different waveguide dimensions in Table 1. While there were variations for these measured or extracted values, the fabricated waveguides generally exhibited a linear loss of \(\sim\) 2 dB/cm and a nonlinear index of \(\sim\)1.13 \(\times\) 10\({}^{-18}\) m\({}^{2}\)/W. These values are indeed consistent with that from literatures where the SRN material and waveguide were measured at 1550 nm [30].