## 3 Device Structure and Simulation Results
### Thermal Heating Simulation  
Figure 3a shows the temperature profile around the simulation region, and the temperature varies from 300 to 408 K for an input heater power of 30 mW to each heater. Figure 3b, d illustrates the temperature profile around the lower arm of the MZI waveguide region and the racetrack resonator waveguide region, respectively. As expected, the temperature profile around the two heaters is almost identical and the thermal heating to the waveguides beneath the heaters can be clearly seen. On the other hand, Figure 3c illustrates the temperature profile around the upper arm of the MZI waveguide region, where the temperature remains unchanged. This result implies that the heating in this structure is very localized, and no thermal crosstalk happens. In our design, thermal crosstalk is avoided by keeping a safe spacing between the two arms of the MZI waveguides.  
To characterize the optical response, including the changes in the effective refractive index and phase as a function of the input heater power, the temperature map from the previous heat transport simulation needs to be imported into the finite difference eigenmode (FDE) solver of MODE Solutions. An index perturbation grid attribute is used to convert the temperature map into a change in refractive index of the material based on a user-specified thermal-optic coefficient. In the simulation, the thermal coefficients, (dn/dT), of \(1.855\times 10^{-4}\) and \(1.317\times 10^{-5}\) (K\({}^{-1}\)) are used for silicon and silicon dioxide, respectively [40]. The temperature data files are loaded into the temperature grid attribute of the Lumerical MODE Solution FDE solver to calculate the effective refractive index \(\mathrm{n_{eff}}\) and the change in effective refractive index \(\mathrm{\Delta n_{eff}}\) of the TE mode for different input powers of the heaters. Figure 4a shows the change in the effective index for the racetrack waveguide and lower waveguide of the MZI. The phase shift in \(\pi\) can be obtained with less than 30 mW of heating power. Figure 4b shows that the effective index change of the upper MZI waveguide (w/o heater) is three orders of magnitude smaller than that of the heated waveguide. Therefore, the heating power that comes from the two heaters does not affect the waveguide of the upper arm of the MZI. That is, no thermal crosstalk occurs in our design. The change in the effective refractive index for the upper arm racetrack waveguide and lower arm of the balanced Mach-Zehnder interferometer (BMZI) waveguide as a function of the input power of the heater is shown in Figure 4a, and its accuracy level of the change in refractive index is within six decimal digits.  
Figure 4: Change in effective refractive index as a function of input power of heater (**a**) for lower arm of MZI and for racetrack waveguide and (**b**) for upper arm of balanced Mach–Zehnder interferometer (BMZI).  
Figure 3: Temperature profile (**a**) around the simulation region, (**b**) around the lower arm of the Mach–Zehnder interferometer (MZI) waveguide region, (**c**) around the upper arm of the MZI waveguide region, and (**d**) around the racetrack waveguide region.