# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
## 5 Conclusions  
We applied a coupling model to an on-chip structure consisting of a round-cornered square-shaped (ROSS) resonator as an optical filter. The theoretical relationships established in this study indicate that in addition to intrinsic losses in the resonator, the characteristics of the coupling also significantly influence the resonant spectral response of the coupled waveguide-resonator system. The quality factor and finesse of the microresonators were shown to be functions of both transmission (\(\beta\)) and resonator propagation (\(\alpha\)) values. Theoretical results provided a good understanding of the influencing parameters, and a simple expression that uses only \(\alpha\) and \(\beta\) to represent the entire spectral response was demonstrated.  
It was seen that in order to obtain a high-quality factor and the greatest transmission signal dip at resonance, \(\alpha\) and \(\beta\) should be similar in value and as close to unity as possible. Furthermore, these approaches were used to confirm that the quality factor and transmission dip at resonance improve significantly with the rotation of the ROSS resonator in relation to the waveguide around the coupling point. Additionally, the study also experimentally, theoretically, and numerically validated that the ROSS microresonator, at an optimum radius of curvature and an optimum angle with respect to the input waveguide, provides a significantly better resonance response than the conventional microring resonator. The present analysis can assist in improving the coupling characteristics and optical performance of resonators of polygonal, circular, or any other geometries.  
The present study establishes that round-cornered square-shaped (ROSS) microresonators can be competitive alternatives to circular ring microresonators, and that with the proper selection of the radius of curvature and the angle of rotation with respect to the  
Figure 12: ROSS (\(R_{c}\) = 50 \(\upmu\)m) with 45\({}^{\circ}\) rotation in relation to the feed waveguide around the coupling point. (**left**) Hybrid meshing of the ROSS for FEA analysis; (**right**) SEM image of the fabricated ROSS for optical experiments.  
Figure 13: Comparison of the optical response of the on-chip ROSS device with different angles of rotations: (**left**) FEA method, (**right**) optical experiments, and theoretical modeling. Here, \(R_{c}\) = 50 \(\upmu\)m, and the total optical path length is 400 \(\upmu\)m, TE mode.  
input waveguide, the RCSS can deliver a better quality factor as compared to a circular ring microresonator.  
Conceptualization, P.P. and S.K.; investigation, B.P.; methodology, P.P. and N.S.S.M.; supervision, S.K.; validation, B.P. and M.R.; visualization, N.S.S.M.; writing--original draft, P.P. and N.S.S.M.; writing--review and editing, P.P., N.S.S.M., B.P., M.R. and S.K. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by the research grant, ADEC Award for Research Excellence (AARE-0073) from the Abu Dhabi Education Council.  
Not applicable.  
The data that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.