## 4 Robust Geometries against Fabrication Errors
### Application  
The relationship between the internal radius and the thickness values, which fulfill the SHG condition, has an interesting consequence. As shown in Figure 5, the function \(R_{in}=f(e)\) has a flat maximum around \(e=330\) nm, so the required microring geometry to obtain a SHG at these thicknesses has a range of radii values instead of a fixed radius value. From a practical point of view, this means that we can tolerate variations of \(\pm 10\) nm in the thickness about the selected value of 330 nm or of the radius about the selected values of 60.755 \(\upmu\)m without compromise on the generation efficiency. Therefore the proposed geometry is robust with respect to fabrication tolerances. Keeping in mind the general observation made in Section 4.2, the same optimization can be done for other materials. The choice of the proper geometrical parameters for the resonator and the proper pump and SH wavelengths determine the same kind of robust geometry as the one here reported.  
In order to illustrate the interest of the robust geometry presented before, we have computed the conversion efficiency for a silicon microring (\(m_{f}=400\)) for two different internal radii: (i) \(R_{in}=60.755\)\(\upmu\)m which corresponds to the robust geometry determined from Figure 5, and (ii) \(R_{in}=57.208\)\(\upmu\)m which corresponds to the SHG point for \(e=440\) nm. To simulate a fabrication error, the thickness is swept around the nominal value, \(e=330\) nm in the first case and \(e=440\) nm in the second one. For each new thickness, the fundamental wavelength is adjusted around 2.3 \(\upmu\)m in order to keep the pump resonant and the fundamental mode at the azimuthal number \(m_{f}=400\). On the other side, the SH is not resonant (except for \(e=330\) nm or 440 nm); thus, a pseudo-azimuthal number \(m_{SH}^{\prime}\) is interpolated between the closest resonant wavelengths in order to compute \(\eta\). The results are shown in Figure 6a.  
In both cases, \(\eta_{max}\approx 1.7\times 10^{-4}\) when the thickness has the optimum value, which yields the optimum microring radius. For the robust geometry (blue line), the SHG remains efficiently, even when the thickness differs by 10 nm from the optimum value, whereas for the other geometry (orange line), \(\eta\) decreases by at least two orders of magnitude when the thickness differs from the optimum value. In the first case, \(\eta\) is larger than \(10^{-4}\) on a thickness range spanning more than 15 nm, while the range is as narrow as 1 nm in the second case. This result confirms what was found in the previous section: the \(R_{in}=f(e)\) plots allow one to find geometries which can accommodate fabrication errors. Our result is different from what was observed by [23] for GaAs, since here we used resonant conditions while neither the fundamental nor the SH modes were resonant in the other article.  
Figure 5: Internal radii values that fulfill the criteria of SHG with high efficiency as a function of the waveguide thickness. We assumed a silicon microring of width \(w=800\) nm and a fundamental azimuthal number \(m_{f}=400\). The colors are the same as Figure 2b to ease the comparison between the two plots.  
Figure 6b shows the same kind of study when the internal radius is swept while the thickness is fixed to (i) \(e=330\) nm in the case of the robust geometry and (ii) \(e=440\) nm in the other case. We can see that a fabrication deviation of the radius is not as critical as the one for the thickness, since in both cases at least a \(45\) nm-deviation on \(R_{in}\) is allowed to keep the conversion efficiency larger than \(10^{-4}\).  
Finally, we performed the same optimization for the most promising geometry for SHG, i.e., for small radii (Figure 3a), and with \(m_{f}=100\) to use \(\lambda_{f}>2.2\) um (Figure 2a). The robust geometry corresponds to \(e=320\) nm, \(R_{in}=14.275\) um and leads to \(\eta\approx 10^{-3}\) for \(P_{in}=1\) mW which is only one order of magnitude smaller than what is theoretically expected in GaAs [23], for instance.