# Coupling and Optical Analysis of a Round-Cornered Square-Shaped Microresonator
## 2 Coupling Model and Analysis
The energy transmission ratio (transmissivity) to throughput port \(\tau\) using Equation (9) can be expressed as follows:  
\[\tau=\left|\frac{O}{I}\right|^{2}=\frac{OO^{*}}{II^{*}}=\frac{\alpha^{2}+ \beta^{2}-2\alpha\beta\cos\phi}{1+\alpha^{2}\beta^{2}-2\alpha\beta\cos\phi} \left|\frac{T}{\beta}\right|^{2} \tag{10}\]  
In a coupling condition when resonance occurs, minimum energy is transmitted through the throughput port, that is:  
\[\tau_{\text{min}}=\left.\tau\right|_{\phi=0}=\frac{\alpha^{2}+\beta^{2}-2 \alpha\beta}{1+\alpha^{2}\beta^{2}-2\alpha\beta}\left|\frac{T}{\beta}\right|^ {2}=\frac{\left(\alpha-\beta\right)^{2}}{\left(1-\alpha\beta\right)^{2}} \left|\frac{T}{\beta}\right|^{2} \tag{11}\]  
Similarly, the energy transmitted to the throughput port is maximum when:  
\[\tau_{\text{max}}=\left.\tau\right|_{\phi=\pi}=\frac{\alpha^{2}+\beta^{2}+2 \alpha\beta}{1+\alpha^{2}\beta^{2}+2\alpha\beta}\left|\frac{T}{\beta}\right|^ {2}=\frac{\left(\alpha+\beta\right)^{2}}{\left(1+\alpha\beta\right)^{2}} \left|\frac{T}{\beta}\right|^{2} \tag{12}\]  
The value of the \(\tau_{min}\) becomes zero if \(\alpha=\beta\), that is:  
\[\overline{p}l=\frac{\overline{t}}{l}\rightarrow\overline{p}l^{2}=\overline{t} \tag{13}\]  
If \(\alpha=\beta=1\), \(\tau_{min}\) would become undefined, as seen in Equation (11). However, this would yield the following:  
\[\overline{p}l=\frac{\overline{t}}{\overline{l}}=1\rightarrow\overline{t}=l \ \,\ \ \overline{p}=\frac{1}{l} \tag{14}\]Since \(\vec{t}^{2}+\vec{c}^{2}=l^{2}\), we would get \(\vec{c}^{2}=0\) if \(\beta=1\), which means that there would be no coupling. Thus, \(\alpha=\beta=1\) is not physically possible. The transmission at full-width at half-maximum (FWHM) in this configuration is given by:  
\[\tau_{FWHM}=\tau_{\max}-\frac{\tau_{\max}-\tau_{\min}}{2}=\frac{\tau_{\max}+ \tau_{\min}}{2} \tag{15}\]  
Substituting the \(\tau_{min}\) and \(\tau_{max}\) from Equations (11) and (12) into Equation (15), we get:  
\[\tau_{FWHM}=\frac{1}{2}\Bigg{(}\frac{\left(\alpha+\beta\right)^{2}}{\left(1+ \alpha\beta\right)^{2}}+\frac{\left(\alpha-\beta\right)^{2}}{\left(1-\alpha \beta\right)^{2}}\Bigg{)}\Bigg{|}\frac{T}{\beta}\Bigg{|}^{2} \tag{16}\]  
\(\tau_{FWHM}\) can also be found in Equation (10):  
\[\tau_{FWHM}=\frac{\alpha^{2}+\beta^{2}-2\alpha\beta\cos\phi_{FWHM}}{1+ \alpha^{2}\beta^{2}-2\alpha\beta\cos\phi_{FWHM}}\left|\frac{T}{\beta}\right| ^{2} \tag{17}\]  
Comparing Equations (16) and (17) and simplifying the expression, \(\cos\phi_{FWHM}\) is expressed as:  
\[\cos\phi_{FWHM}=\frac{2\alpha\beta}{1+\alpha^{2}\beta^{2}}=\frac{2\vec{t} \overline{p}}{1+\vec{t}^{2}\overline{p}^{2}} \tag{18}\]  
Defining \(\gamma\) as the ratio of the \(\Delta\lambda_{FWHM}\) (spectral full-width at half-maximum, which is the wavelength spacing at the half level of the resonance intensity peak) to the \(\Delta\lambda_{{}_{FSR}}\) (the free spectral range or the wavelength spacing between two successive optical intensity maxima or minima), together with the help of Equation (18), we have the following:  
\[\tau=\left|\frac{O}{I}\right|^{2}=\frac{OO^{*}}{II^{*}}=\frac{\alpha^{2}+ \beta^{2}-2\alpha\beta\cos\phi}{1+\alpha^{2}\beta^{2}-2\alpha\beta\cos\phi} \left|\frac{T}{\beta}\right|^{2} \tag{10}\]0  
where \(\mathcal{F}\) is the finesse, a quality measure of the microring, which is the ratio between the FSR and the FWHM. High finesse devices have a small FWHM, and a strong intensity build-up in the ring when in resonance is thus a measure of the sharpness of resonances relative to their spacing [15; 34]. For simplicity, we introduce \(X=\alpha\beta\) and \(\epsilon=\tau_{\min}/\tau_{\max}\).