# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
## 4 Results and Discussion
### Effects of Propagation (\(\alpha\)) and Transmission (\(\beta\))  
The effects of coupling transmission on the transmission characteristics of the RCSS in order to find an optimal set of parameters are investigated. \(\beta\) has been increased from 0.6 to 1, while other parameters are kept unchanged. A general parametric response that illustrates the wavelength spectra near resonance is presented in Figure 4 for different values of the resonator propagation value \(\alpha\), and the coupling region transmission value \(\beta\). As the resonator becomes lossier (i.e., the value of resonator propagation \(\alpha\) reduces), the resonance peak becomes broader and the quality factor decreases. Losses in the RCSS resonators have previously been identified to be a result of (i) radial radiation losses at the bends, (ii) the transition losses in the regions where bends and straight segments connect, and (iii) the intrinsic propagation losses per unit length along with the entire travel distance in the microresonator [27]. From Figure 4, it can be seen that in addition to the intrinsic losses in the resonator above, the characteristics of the coupling also significantly influence the spectral response of the coupled waveguide-resonator system. The dip in the transmissivity at the throughput port at resonance is a function of the difference between \(\alpha\) and \(\beta\) and is maximized, i.e., the transmissivity ratio \(\epsilon\) is at a minimum (=0) when the values of \(\alpha\) and \(\beta\) are the same (see Equations (11) and (12)). Thus, as shown in Figure 5, to get the maximum spectral transmissivity dip (minimum \(\epsilon\)) at the throughput port at resonance, the intrinsic propagation property \(\alpha\) of the resonator should be closest to the coupling transmission \(\beta\), which is a function of the design of the coupling region. Figure 3: Schematic configuration of the experimental setup. For a given propagation value \(\alpha\), higher values of the coupling transmission \(\beta\) yield narrower resonance peaks and therefore higher quality factors. This phenomenon is captured in Figure 6 where the finesse \(\mathcal{F}\) is shown to span several orders of magnitude, which is the same trend for the quality factor \(Q\) as per Equation (25). The quality factor is a function of the product of \(\alpha\) and \(\beta\), which is unlike the dependency of throughput port transmissivity on the difference of \(\alpha\) and \(\beta\). The quality factor \(Q\) increases monotonically as a function of \(\alpha\beta\). Thus, to obtain the largest throughput port transmissivity dip, i.e., the lowest \(\tau/\tau_{\max}\) and the highest extinction ratio, accompanied by the highest quality factor \(Q\), the values of \(\alpha\) and \(\beta\) should be close to unity and as close to each other as possible. This would correspond to a lossless or near-lossless resonator, with a coupling design  
Figure 4: Effect of transmission (\(\beta\)) and propagation (\(\alpha\)) on optical response around the resonant wavelength (\(\lambda_{0}\)). Figure 5: Variation of the transmission ratio \(\epsilon\) (Equation (21)) at the resonant wavelength (\(\lambda_{0}\)) as propagation (\(\alpha\)) and transmission (\(\beta\)) parameters change for an RCSS microresonator; \(\epsilon=0\) (ideal) when \(\alpha=\beta\). that gives the highest coupling transmission. Note that the higher the transmission \(\beta\), the lower the coupling \(c\) is. However, the value of \(\beta\) can never be unity since that would imply \(c=0\), which is only theoretically possible for an isolated microresonator unaffected optically by the waveguide, i.e., not coupled with the waveguide (see Equation (14)). An isolated lossless microresonator would theoretically possess a flat spectral response, except for a delta-function change at resonance, and it is the design of the coupling and its characteristics that introduce the broadened spectral response.