## 3 Device Structure and Simulation Results  
The feasibility of using this type of structure to tune the group delay has been experimentally demonstrated using silica waveguides of \(3\times 3\)\(\upmu\)m\({}^{2}\) core size [24]. The incorporation of a thermal phase shifter in a SOI-based photonic circuit has been reported and proved to be consistent with simulation results [33; 34; 35; 36; 37; 38]. We demonstrate here the operation of such a push-pull tunable delay line with device simulators. To demonstrate the device performance as close to the real device as possible, the simulation needs to account for multiple physical domains and the interactions of thermal, electrical, and optical parameters. The temperature profile around the waveguide region by injecting electric current into the heater stripe needs to be obtained in order to calculate the change in the refractive index. The change in the effective refractive index for light passing through the waveguide is then calculated by finding the light field distribution with an optical waveguide mode solver. Then, the tunable delay characteristics can be obtained by using a photonic circuit simulator that accounts for the refractive index change from thermal tuning.  
The tunable ODLs are designed on the SOI platform with a silicon waveguide layer of 220 nm in thickness and a 2 \(\upmu\)m thick silicon dioxide layer between the silicon layer and the silicon substrate. The silicon waveguide layer is buried with silicon dioxide. This platform is commercially available for MPW services. The heating metal layer is located 0.6 \(\upmu\)m above the waveguide layer. The whole structure is covered with a thick silicon dioxide layer. All the waveguides in an ODL are etched into strip waveguides of 0.45 \(\upmu\)m in width. The length of the two MZI arms is the same with \(L_{2}=173.832\)\(\mu m\), and the circumference of the racetrack resonator is \(L_{1}=336.664\)\(\mu m\). The length and width of the heaters on the racetrack resonator and the lower MZI arm are the same, i.e., 100 and 1.2 \(\upmu\)m, respectively. The length of \(L_{1}\) and \(L_{2}\) can be chosen to meet the practical application requirements such as the free spectral range and the range of delay time. From Equation (16), the delay time can be larger for a longer \(L_{1}\), which gives a larger \(\tau_{0}\). Here, \(L_{1}\) and \(L_{2}\) are just chosen as examples for device simulation. \(L_{2}\) needs to accommodate the bending waveguides of the MZI and the heater length for phase shifting. The waveguide loss is assumed to be _1 dB/cm_.  
The group delay can be tuned in a continuous manner by properly varying the input power of the heater. By using the Lumerical DEVICES HEAT transport solver [39], we first simulate the temperature profile around the waveguide with different input powers. The thermal crosstalk to the neighbor components is included. This result is then imported into MODE Solution where the optical waveguide modes are calculated. The overlapping of the optical mode and the temperature profile is then analyzed to obtain the changes in the effective refractive index and phase as a function of the input power of the heaters [39]. These thermally induced changes on the refractive index and phase are then imported to the Lumerical INTERCONNECT module to calculate the group delay for each set of input powers to the heaters. We use a uniform heat source on the heater to model the heat generated in the heater and keep the bottom of the substrate fixed at room temperature. The top surface of the simulation region is bounded by air. A temperature monitor is placed around the waveguide region, which will automatically store the temperature map into an output file for each input power to the heaters.