## 2 Proposed Architecture and Operation Principles  
Figure 1 illustrates a schematic of the proposed tunable ODL structure. The circuit implements a combination of racetrack resonator, balanced Mach-Zehnder interferometer (MZI), and two tuning heaters. The balanced MZI with a heater at its lower arm is to tune the coupling coefficient for the light to couple into or out of the racetrack loop from the straight waveguide. It is well known that the maximal group delay occurs at the resonant condition, and the coupling coefficient determines the quality factor of the loop and, thus, the delay time [31]. However, the tuning of the coupling coefficient also gives rise to a phase change that will shift the resonant wavelength. In order to achieve a large change in the group delay with a single racetrack resonator, the operation wavelength needs to be aligned to the resonant peak. Therefore, the phase change while tuning the coupling coefficient has to be compensated with a phase change in the racetrack section by another thermal heater. This thermal heater can also be used for correcting the error in the resonant wavelength due to the manufacturing fluctuation. An MZI consisting of two 3 dB multimode interference (MMI) couplers operating at the wavelength of 1550 nm is used as the equivalent tunable coupler for the racetrack resonator, while the racetrack resonator itself is constructed by connecting one output port of the MZI to one of its input ports. The lengths of the two arms of the MZI are equal. A thermal heater (Heater 2) is placed above the lower arm of the MZI to produce a phase change. The change in the phase relationship between the light traveling through the two arms by the thermo-optic effect can change the optical power coupled into the racetrack resonator [24; 32] and, thus, achieve tunability of the coupling efficiency. Another thermal heater (Heater 1) is placed above the racetrack waveguide to control the resonant wavelength position. The device characteristics can be easily formulated in matrix forms [21; 24]. The coupler at the input/output end of the MZI can be written as  
\[M_{c}=\left[ \begin{array}{cc}\sqrt{1-k}&-i\sqrt{k}\\ -i\sqrt{k}&\sqrt{1-k}\end{array} \right], \tag{1}\]  
where \(k\) is the coupling coefficient of the coupler. The matrix for the two paths between the two couplers is  
\[M_{p}=\left[ \begin{array}{cc}a_{0}e^{i\theta_{0}}&0\\ 0&a_{0}e^{i(\theta_{0}+\phi_{2})}\end{array} \right], \tag{2}\]  
where \(\phi_{2}\) and \(\theta_{0}\) are the phases of the path with and without tuning, respectively, and \(a_{0}\) is the propagation loss for the two arms in the MZI. The output and input light can be related to the matrix multiplication \(M_{c}M_{p}M_{c}\) as  
\[\left[ \begin{array}{c}E_{out}\\ E_{3}\end{array} \right]=M_{c}M_{p}M_{c}\!\!\left[ \begin{array}{c}E_{in}\\ E_{2}\end{array} \right]=e^{i\theta_{0}}\!\!\left[ \begin{array}{cc}a_{0}\!\! \left[1-k-ke^{i\phi_{2}}\right]&-ia_{0}\sqrt{k(1-k)}\!\!\left[1+e^{i\phi_{2}} \right]\\ -ia_{0}\sqrt{k(1-k)}\!\!\left[1+e^{i\phi_{2}}\right]&a_{0}\!\!\left[-k+(1-k)e^ {i\phi_{2}}\right]\end{array} \right]\left[ \begin{array}{c}E_{in}\\ E_{2}\end{array} \right]. \tag{3}\]  
Using Equation (3), as well as adding the phase shift, \(\phi_{1}\), and the propagation loss, \(a_{1}\), caused by the racetrack path, the transfer function of this tunable ODL can be derived and expressed by  
\[H=\frac{E_{out}}{E_{in}}=\frac{-a_{0}k+a_{0}(1-k)e^{i\phi_{2}}-a_{0}^{2}a_{1}e ^{i(\phi_{1}+\phi_{2})}}{1-a_{0}a_{1}(1-k)e^{i\phi_{1}}+a_{0}a_{1}ke^{i(\phi_ {1}+\phi_{2})}}. \tag{4}\]  
Equation (4) is obtained by describing the racetrack path with \(E_{2}=E_{3}a_{1}e^{i\phi_{1}}\), where \(E_{2}\) and \(E_{3}\) are defined in Figure 1. The group delay can thus be obtained from the derivative of the phase of \(H(\omega)\) to the optical angular frequency, \(\omega\).