# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
## 4 Results and Discussion
### Effect of the Radius of Curvature in RCSS  
From the results of the numerical simulation, the optical coupling parameters \(\alpha\) and \(\beta\) of the coupling model presented in this study can be computed. For the specific case of RCSS microresonators [27], the computed values of \(\alpha\) and \(\beta\) as well as \(\epsilon\) and \(Q\) are shown in Figure 8. As the radius of curvature at the corners is increased, with the total optical path length kept constant at 400 \(\upmu\)m, the propagation \(\alpha\) value changes significantly, ranging between 0.07 and 0.98. Propagation losses are a cumulative effect of bend losses, transition losses, and intrinsic transmission losses, which fluctuate with the changing radius of the bends, even when the total optical path length remains the same. The quality factor \(Q\) also varies considerably. The transmission \(\beta\) varies less significantly and stays between 0.83 and 0.99. Due to the constant optical path length, the free spectral range remained the same (\(\sim\)1.1 nm \(\pm\) 0.1 nm) for all cases presented in this study.  
The optimal radius of curvature \(R_{\rm c}\) of the RCSS of [PERSON] and [PERSON] [27] is 15 \(\upmu\)m, where the quality factor (\(\sim\)10\({}^{5}\)) is the highest (the resonance peak is the narrowest) and the throughput port transmissivity shows a very significant dip at resonance (i.e., \(Q\) is the highest, \(\epsilon\) is low, and the extinction ratio is high). These spectral and quality characteristics are desirable in bio-sensing and optical switching applications. This is because both \(\alpha\) and \(\beta\) are high in value (close to unity) and the difference between them is small, allowing the resonance signal at the throughput port to have both high quality (narrow linewidth, \(\Delta\lambda_{FWHM}\)) and a significant transmissivity dip (\(r_{e}\sim\) 6 dB). In a similar manner, the least  
Figure 8: Optical parameters for a waveguide coupled with RCSS microresonator, TE mode, the total optical path length is 400 \(\upmu\)m.  
desirable radius of curvature is 40 \(\upmu\)m, where the extremely small value of \(\alpha\) coupled with the lowest value of \(\beta\) leads to a negligible transmissivity dip (\(r_{e}\sim\) 0.2 dB) and an extremely small quality factor (\(\sim\)10\({}^{3}\)), making the resonance spectral signal indistinguishable from non-resonance regions.  
Figure 9 shows the accuracy of the two-parameter transmissivity equation, Equations (10) and (12), with respect to the full spectral signal simulated by the EMFD and the optical experiments for both an RCSS and a ring microresonator. It can be seen that the results from the coupling model and from the EMFD FEA numerical simulation results are virtually identical over the entire free spectral range. These results are also validated by the experimental data, as shown in the figure. Since this is the most important spectral range for applications of resonant microstructures, the coupling model is powerful in describing the optical resonance response. Results from the theoretical model, FEA analysis, and experiments confirm that the round-cornered square-shaped microresonator at an optimum radius of curvature (\(\sim\)15 \(\upmu\)m) provides a quality factor (\(\sim\)10\({}^{5}\)) that is better than that of the conventional microring resonator (\(\sim\)10\({}^{4}\)), which is highly desirable for biosensing and other similar applications. Note that the extinction ratio of the conventional ring is 13 dB and that of the RCSS is 6 dB.