## 2 Proposed Architecture and Operation Principles
It is usually expressed by the normalized group delay as [24]  
\[\frac{\tau_{g}}{\tau_{0}}=\frac{\partial\tan^{-1}\frac{Im(H)}{Re(H)}}{ \partial\phi_{1}}, \tag{5}\]  
Figure 1: Schematic picture for the structure of the proposed integrated optical delay line based on racetrack resonator with balanced Mach–Zehnder interferometer. where \(\tau_{0}\) is the delay time for one round - trip of the ring. From Equations (4) and (5), the group delay can be tuned by the two phases \(\phi_{1}\) and \(\phi_{2}\). Equation (4) can also be written in a form similar to the formula for a simple all-pass ring (APR) resonator. By treating the MZI as a tunable coupler, the output and input of the coupler can be described by a generalized matrix as  
\[\left[ \begin{array}{c}E_{out}\\ E_{3}\end{array} \right]=M_{c}M_{p}M_{c}\left[ \begin{array}{c}E_{in}\\ E_{2}\end{array} \right]=\left[ \begin{array}{cc}r&\kappa\\ \kappa&r^{\prime}\end{array} \right]\left[ \begin{array}{c}E_{in}\\ E_{2}\end{array} \right], \tag{6}\]  
where \(r\) and \(r^{\prime}\) are the transmission coefficients of the MZI at the trough ports and \(k\) is the overall coupling coefficient of the MZI, which can be written as  
\[r=a_{0}\left[(1-k)-ke^{i\phi_{2}}\right]\equiv\rho e^{i\psi}, \tag{7}\]  
\[r^{\prime}=a_{0}\left[-k+(1-k)e^{i\phi_{2}}\right]\equiv\rho e^{i\psi^{\prime }}, \tag{8}\]  
\[\kappa=a_{0}\left[-i\sqrt{(1-k)}k\left(1+e^{i\phi_{2}}\right)\right]. \tag{9}\]  
Then, the transfer function can be written as  
\[H=\frac{r+\left(\kappa^{2}-rr^{\prime}\right)a_{1}e^{i\phi_{1}}}{1-r^{\prime} a_{1}e^{i\phi_{1}}}=\frac{r-a^{\prime}e^{i\phi_{1}}}{1-r^{\prime}a_{1}e^{i\phi_{1}}}, \tag{10}\]  
where  
\[a^{\prime}=a_{0}^{2}a_{1}e^{i\phi_{2}}\equiv\sigma e^{i\phi_{2}}, \tag{11}\]  
with \(\sigma=a_{0}^{2}a_{1}\). Equation (10) is of a similar form as the transfer function of a simple APR resonator [31]. It is equivalent to Equation (4) by substituting all the parameters of Equations (7)-(11) into Equation (10). For a simple APR with a lossless coupler, \(r=r^{\prime}=\sqrt{1-k}\) and \(\kappa=-i\sqrt{k}\), the transfer function is reduced to a simpler form:  
\[H=\frac{r-a_{1}e^{i\phi_{1}}}{1-ra_{1}e^{i\phi_{1}}}. \tag{12}\]  
It is well known that, for given ring parameters of a simple APR, the group delay is maximal at the resonant condition where \(\phi_{1}\) is an integer multiple of \(2\pi\). Under this resonant condition, the light is mostly localized in the ring. For a racetrack resonator with tunable coupling, the denominator of Equation (10) can be rewritten as \(1-\rho a_{1}e^{i(\phi_{1}+\psi^{\prime})}\), so the resonant condition for this tunable racetrack resonator occurs at \(\phi_{1}+\psi^{\prime}=2m\pi\), where \(m\) can be an arbitrary positive integer. In addition to the phase tuning \(\phi_{1}\) on the racetrack path, the phase of the tunable coupler, given in Equations (7)-(9), will also change the resonant condition. The normalized group delay for a racetrack resonator with tunable coupling can be derived from Equations (5) and (10) as  
\[\frac{\tau_{g}}{\tau_{0}}=\frac{\sigma^{2}-\rho\sigma\cos(\phi_{1}+\phi_{2}- \psi)}{\rho^{2}+\sigma^{2}-2\rho\sigma\cos(\phi_{1}+\phi_{2}-\psi)}+\frac{\rho a _{1}\cos(\phi_{1}+\psi^{\prime})-\rho^{2}a_{1}^{2}}{1+\rho^{2}a_{1}^{2}-2\rho a _{1}\cos(\phi_{1}+\psi^{\prime})}. \tag{13}\]  
From Equation (13), the group delay is a function of both \(\phi_{1}\) and \(\phi_{2}\). Therefore, the tuning of \(\phi_{2}\) for tuning the group delay will change the resonant wavelength where the peak group delay is located. Equation (13) is obtained by neglecting the contribution of group delay caused by the MZI. This assumption is reasonable for a MZI with balanced arms. In order to achieve a large group delay with a single-stage racetrack resonator, it is desired to operate the device at the peak group delay condition, i.e., the resonant condition.