# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
###### Abstract  
An on-chip structure consisting of a round-cornered square-shaped (RCSS) resonator as an optical filter is studied via optical experiments, analytical modeling, and numerical techniques. A general coupling model is shown to accurately represent the entire spectral response; the model also provides an understanding of the influence of geometrical and coupling parameters on the resonance characteristics of the RCSS microresonators. The selection of an optimum radius of curvature for rounding off the sharp corners of square microresonators can provide higher quality factors than that of conventional circular resonators. The rotation of the RCSS at the central corner coupling point is also shown to improve the quality factor and remove the minimal phase mismatch requirement and dependency on interaction length. The model results are validated with an electromagnetic finite domain analysis (EMFD) and optical experiments, for which an RCSS on a silicon-on-insulator platform is fabricated. It is shown that the optical performance characteristics (quality factor, transmission ratio, and extinction ratio) of the microresonators are very sensitive to the coupling parameters and must be carefully considered in addition to geometrical length, rotation, and curvature effects. Due to the change in coupling introduced by rotation, the quality factor of the round-cornered square-shaped microresonator can be significantly larger than a circular ring with the same size, thereby establishing RCSS as a competitive alternative to circular microresonators. microring resonator; fiber waveguides; couplers; round-cornered square-shaped microring; on-chip structure; propagation; scattering; losses +
Footnote †: journal: Applied Physics Letters B  
1 Mechanical Engineering Department, New York University Tandon School of Engineering,
by manipulating fabrication procedures to reduce scattering, absorption, and intrinsic losses, researchers have demonstrated an ultra-sensitive resonator with a quality factor of up to 422 million [9]. In another study, the coupling loss was lowered by tailoring the directional coupler excitation to the resonator, which experimentally achieved a quality factor of 81 million on the \(Si_{3}N_{4}\) optical resonator platform [10]. [PERSON] et al. [11] studied a 1D photonic crystal ring resonator and demonstrated a quality factor of more than \(10^{10}\) at 1.55 um. In this realm, ring resonators and their variants such as racetrack configurations have good performance parameters and find various applications such as in optical networks and as sensors, for which they have been extensively studied in theory and for practical applications [8; 12; 13; 14; 15; 16; 17]. For racetrack resonators, it was found that the radius of curvature and the length of the coupling region play an essential role in the total optical loss, where a larger radius of curvature reduces the bending loss while increasing the scattering loss. In order to minimize overall loss and improve the quality factor, optimal values for the radius of curvature and the length of the straight coupling section were obtained [18]. However, concerns about short interaction lengths between the circular portion of such resonators and the straight waveguides [19; 20] have led researchers to consider polygonal shapes [21; 22]. In the absence of sharp corners, the mode changes adiabatically between the radial mode in the curved sections and the normal mode in the straight section [19; 23]. In order to address the issues above and to obtain high optical length interaction, a lot of efforts have been made, including the proposition of various structures as well as the use of novel materials. As an example, polygonal microresonators have been introduced due to their ability to provide long interaction lengths and ease the air-gap spacing tolerance [24; 25; 26]. A longer interaction length and an optimal gap can create a strong modal field overlap of the resonator and the waveguide, which is essential for a strong coupling.