# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
## 2 Coupling Model and Analysis  
In 1969, [PERSON] [29] presented one of the preliminary coupling models for an integrated optical ring resonator. Since then, waveguide-resonator coupling has been studied for various types of geometries [11; 12; 30; 31; 32]. Based on the coupled-mode theory, a mathematical model was developed to analyze the effect of curvature on waveguide modes and to simulate the transmission spectra of the 1D photonic crystal ring resonator [11]. Using the scattering theory, [PERSON] and co-workers modeled the coupling and showed that the reflection and transmission properties of the waveguide are primarily influenced by the waveguide-resonator coupling, symmetry and degeneracy of the resonant modes, mode resonant frequency, and cavity-loss [12; 30]. Several researchers were able to achieve optimum quality factor and extinction ratio [31; 32] just by tuning the coupling parameters between the resonator and the input waveguide. In the present study, a simple form of an optical filter with a round-cornered square-shaped (RCSS) resonator is used and shown in Figure 1. The optical source from the left-hand side of the structure is coupled with the bus waveguide and excites the electromagnetic wave in the RCSS resonator with a narrow-band resonance, leading to the light filtering. A sharp dip exists in the resonance transmission spectra ; hence, a good quality filtering wavelength is obtained in the transmission spectrum. The complex output fields \(O\) and \(o\) are related to the complex input fields \(I\) and \(i\) through the self-coupling and cross-coupling coefficients \(T\) and \(t\), and \(C\) and \(c\), respectively. The optical coupling can be expressed as follows [33]:  
\[\left[ \begin{array}{c}O\\ o\end{array} \right]=\left[ \begin{array}{cc}T&c\\ C&t\end{array} \right]\left[ \begin{array}{c}I\\ i\end{array} \right] \tag{1}\]  
All the variables in Equation (1) are complex values. Since \(o\) is coupled back with the input \(i\), we have:  
\[i=po \tag{2}\]  
where \(p\) is the propagation along the resonator. From the equation above, we obtain the following:  
\[\frac{O}{I}=\frac{T-(Tt-Cc)p}{1-tp} \tag{3}\]  
Tracking the energy in the waveguide and the ring resonator gives:  
\[ \begin{array}{l}\left|O\right|^{2}=\left|TI\right|^{2}+\left|ci\right|^{2} \\ \left|o\right|^{2}=\left|CI\right|^{2}+\left|ti\right|^{2}\end{array}  \tag{4}\]  
Figure 1: Schematic of a planar waveguide-coupled RCSS microresonator channel. or  
\[\left|\mathcal{O}\right|^{2}+\left|\varrho\right|^{2}=\underbrace{\left(\left|T \right|^{2}+\left|\mathcal{C}\right|^{2}\right)}_{l^{2}}\left|I\right|^{2}+ \underbrace{\left(\left|c\right|^{2}+\left|t\right|^{2}\right)}_{l^{2}}\left|i \right|^{2} \tag{5}\]  
where \(L^{2}\) and \(l^{2}\) are the loss variables of the waveguide and the resonator, respectively. The left side of Equation (5) can also be expanded by using Equation (1), and using complex conjugates to compute absolute values yields the following:  
\[\left. \begin{array}{l}Tc^{*}+Ct^{*}=0\\ T^{*}c+C^{*}t=0\end{array} \right\}\rightarrow\frac{\left|T\right|^{2}}{\left| t\right|^{2}}=\frac{\left|\mathcal{C}\right|^{2}}{\left|c\right|^{2}} \tag{6}\]  
where the variables with an asterisk \(*\) denote complex conjugates. With the help of the set of equations in Equation (6), Equation (3) can be rewritten as:  
\[\frac{O}{I}=\frac{t^{*}-l^{2}p}{1-tp}\frac{T}{t^{*}} \tag{7}\]  
To simplify the result further, the phases \(\phi_{p}\) and \(\phi_{t}\) are introduced as:  
\[ \begin{array}{l}p=\overline{p}e^{\phi_{p}}\\ t=\overline{t}e^{\phi_{t}}\end{array}  \tag{8}\]  
where \(\overline{p}\) and \(\overline{t}\) are magnitudes of \(p\) and \(t\), respectively. By defining the scaled resonator propagation value and the coupling region transmission value as \(\alpha=\overline{p}l\) and \(\beta=\overline{t}/l\), respectively, we get:  
\[\frac{O}{I}=\frac{\beta-\alpha e^{\phi_{p}}}{1-\alpha\beta e^{\phi_{p}}}\frac {T}{\beta} \tag{9}\]  
where \(\phi=\phi_{t}+\phi_{p}\).