# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
## 4 Results and Discussion
### Effect of Rotation in RCSS  
Rotating the RCSS microresonator in the coupling region can remove the restrictions imposed by the requirement of a minimal phase mismatch between the resonator mode and the waveguide mode in these resonator systems, as well as its dependency on the interaction length between the resonator and feed waveguide. The preliminary results of [PERSON] et al. [36] showed that the quality factor of these resonators can be improved by increasing the rotation angle (0-45\({}^{\circ}\)) of the resonator with respect to the feed waveguide around the coupling point. The analysis also established an approach for achieving different quality factors with the same microresonator just by changing its orientation with respect to the feed waveguide; in some cases, the quality factor was significant. All RCSS microresonator cases discussed in Section 4.2 and Figure 8 have now been rotated by 45 degrees with respect to the feed waveguide. The corresponding values of \(\alpha\)  
Figure 9: Comparison between the results obtained from the theoretical modeling, the FEA method, and optical experiments for (i) the RCSS microresonator with the optimal radius of curvature \(R_{c}=15\)\(\upmu\)m (green) and (ii) the circular microring resonator (radius 63.66 \(\upmu\)m), keeping the optical path length at 400 \(\upmu\)m (red), TE mode. and \(\beta\) as well as \(\epsilon\) and \(Q\) are computed and plotted, as shown in Figure 10. Comparing Figures 8-10, it can be observed that both \(\alpha\) and \(\beta\) are close to unity in all cases at a 45-degree rotation, with a radius from \(R_{c}\) = 5 \(\upmu\)m to 55 \(\upmu\)m; in contrast, in the case of no-rotation, the values of \(\alpha\) and \(\beta\) varied significantly between zero and unity. This results in a higher quality factor for the rotated RCSS at all curvature radii from 0 to 55 \(\upmu\)m (\(\sim\)10\({}^{5}\)) and a consistent lower transmission dip (\(\epsilon\) between 0 and 0.5, or \(r_{e}\)\(\sim\) 5 dB to 13 dB). However, as the radius of curvature further increases to 60 \(\upmu\)m and 63.66 \(\upmu\)m (circular ring), the \(\alpha\) and \(\beta\) slightly decrease due to an increase in transmission and propagation losses at every point along the circumference of the resonator, leading to a decrease in the quality factor by an order of magnitude (from \(\sim\)10\({}^{5}\) to \(\sim\)10\({}^{4}\)). As the radius of curvature increases, the length of the straight portions of the RCSS microresonator decreases, eventually transforming it into a microring resonator. The straight portions of the RCSS microresonator minimize the transmission and propagation losses but introduce the dependency on the interaction length between the waveguide and the resonator. By rotating the RCSS microresonator by 45\({}^{\circ}\) around the coupling point, this dependency and risk of phase mismatch are eliminated (similar to a circular ring). Therefore, the advantage of minimizing losses due to linear wave propagation along the straight sides is retained, which results in a better quality factor (\(\sim\)10\({}^{5}\)) in comparison to conventional microring resonators (\(\sim\)10\({}^{4}\)). Additionally, the computed values of \(\alpha\), \(\beta\), \(\epsilon\), and \(Q\) for the RCSS microresonator (\(R_{c}\) = 50 \(\upmu\)m), with varying angles of rotation \(\theta\) in relation to the feed waveguide around the coupling point are plotted in Figure 11. When the RCSS is rotated from 0 to 45 degrees, the coupling region changes from straight to curved, and mixed in between. Consistent with the findings [36], as \(\theta\) increases from 0 to 45 degrees and the dependency on the interaction length between the RCSS resonator and the feed waveguide decreases, both \(\alpha\) and \(\beta\) steadily increase to reach a value close to unity, with the difference between them continuously decreasing, thus leading to a consistent and significant improvement in the quality factor, which underscores the importance of coupling.