## 3 Conversion Efficiency  
### Method  
The conversion efficiency \(\eta\) is defined as the power ratio between the generated SH and the pump, both estimated inside the bus waveguide:  
\[\eta=\left|\frac{b_{1,SH}}{a_{1,f}}\right|^{2} \tag{1}\]  
where \(b_{1,SH}\) is the electric field amplitude of the SH mode in the waveguide after the microring and (since the fundamental mode is \(Hz\) instead of \(Ez\)) \(a_{1,f}\) is the magnetic field amplitude of the fundamental mode in the waveguide before the microring (Figure 1a). \(\eta\) gives a useful insight into which geometry should be used for SHG among the numerous possibilities previously determined.  
\(\eta\) expression is derived from [23]:  
\[\eta=4\pi^{2}P_{in}|K|^{2}\frac{\alpha_{SH}^{2}(1-t_{SH}^{2})}{1+\alpha_{SH}^ {2}t_{SH}^{2}-2\alpha_{SH}t_{SH}\cos(\varphi_{SH})}\times\left(\frac{\alpha_{ f}^{2}(1-t_{f}^{2})}{1+\alpha_{f}^{2}t_{f}^{2}-2\alpha_{f}t_{f}\cos(\varphi_{f})} \right)^{2} \tag{2}\]  
where \(P_{in}\) is the pump power inside the bus waveguide; \(t\) and \(\alpha\) are, respectively, the transmission coefficient of the coupling for the amplitude and the transmission coefficient after one round-trip inside the microring. They are computed from the coupling quality factor \(Q^{cpl}\), and the intrinsic one \(Q^{int}\), by \(Q^{i}=\pi m\frac{\sqrt{x}}{1-x}\) where \(x=\alpha\) if \(i=int\) and \(x=t\) if \(i=cpl\)[31].  
\(K\) is the overlap between the fundamental and the SH modes, in the case of a phase mismatch \(\Delta m=m_{SH}^{\prime}-2m_{f}^{\prime}\), where \(m_{SH}^{\prime}\) and \(m_{f}^{\prime}\) are pseudo-azimuthal numbers computed by linear interpolation  
Figure 2: Fundamental wavelengths and geometries fulfilling the requirements for the SHG in a Si microring cladded by Si\({}_{3}\)N\({}_{4}\) for \(p_{f}=1\), \(p_{SH}=3\), a Hz fundamental mode, and a Ez SH mode. (**a**) For a fixed thickness \(e=280\) nm and three different widths \(w\). The points correspond to different fundamental azimuthal numbers \(m_{f}\). (**b**) For a fixed width \(w=800\) nm and different thicknesses represented by the different colored points. The three curves correspond to different \(m_{f}\) values.  
corresponding to non-resonant wavelengths. The pseudo-azimuthal numbers also allow to evaluate the phase: \(\varphi=-2m^{\prime}\pi\). According to [23] the overlap reads:  
\[K=K_{2+}\times\exp[i(\Delta m+2)\pi]\times\text{sinc}[(\Delta m+2)\pi]+K_{2-} \times\exp[i(\Delta m-2)\pi]\times\text{sinc}[(\Delta m-2)\pi] \tag{3}\]  
where \(K_{2\pm}\) is the field overlap in the case of the phase matching:  
\[K_{2\pm}=\frac{i}{4m_{f}^{2}}\sqrt{\frac{2\mu_{0}\omega_{SH}}{m_{SH}\mathcal{N }_{SH}}}\frac{1}{m_{f}\mathcal{N}_{f}}(-\chi_{zxx}^{(2)}\pm 2i\chi_{zxy}^{(2)}+\chi_{ zyy}^{(2)})\iint_{fring}rE_{z}^{SH*}\left(\frac{m_{f}H_{z}^{f}}{r}\pm\frac{ \partial H_{z}^{f}}{\partial r}\right)^{2}drdz \tag{4}\]  
where \(\chi_{zxx}^{(2)}=\chi_{zyy}^{(2)}=0\) and \(\chi_{zxy}^{(2)}\) is factorized out of the integral due to the assumption of a uniform value. We assume in \(\chi_{zxy}^{(2)}=1\) pm/V (which is close to the value given in [18] or in [22]). \(\mathcal{N}_{SH}\) is a normalization constant given by:  
\[\mathcal{N}_{SH}=\iint\frac{|E_{z}|^{2}}{r}dS, \tag{5}\]  
whereas \(\mathcal{N}_{f}\) is given by Equation (5) where \(E_{z}\) is replaced by \(H_{z}\).  
In the following, we assume a critical coupling condition between the bus-waveguide and the microring, with the Q-factors equal to \(10^{4}\) and \(P_{in}=1\) mW. Simulations were done in the undepleted pump approximation. The assumption of a constant Q-factor is quite crude. However the main goal of this paper is to demonstrate the influence on fabrication defects on the SHG in a model system. More refined modeling for a fabricated microresonator to-be should clearly consider the geometrical dependence of the Q-factor.