## 2 Materials and Methods
### OFC Generation
These parameters can easily integrate the proposed devices with other photonic components to form a compact, low-cost optical comb generator. To identify the paths of generation, a description of the parametric space \((\Delta,|S|^{2})\) is required, as proposed in [42]. This method uses the Lugiato-Lefever Equation (LLE) to perform various simulations and verify the final state of the generated spectrum. We choose this approach because it allows us to observe in detail the propagation of the electromagnetic field in the microresonator as we vary the input laser parameters. The LLE equation is defined as  
Figure 3: Photonic structure for OFC generation. Figure 2: MRR structure. \[t_{R}\frac{\partial E(t,\tau)}{\partial t}=\left[-\alpha-i\delta_{0}-iL\frac{\beta_{ 2}}{2}\bigg{(}\frac{\partial^{2}}{\partial^{2}\tau^{2}}\bigg{)}+i\gamma L|E|^{2 }\right]E+\sqrt{\theta}E_{\text{in}} \tag{1}\]  
where \(E(t,\varphi)\) is the complex envelope of the total intracavity field, \(t\) is the time variable, \(\tau\) is the transverse coordinate, \(t_{R}\) is the round-trip time, \(\delta_{0}\) is the phase detuning, \(L\) is the cavity length, and \(E_{\text{in}}\) is the pump field. To solve this equation numerically, we use the split-step Fourier method. It is important to emphasize that the pump power and detuning are normalized (see Equations (2) and (3)) to compare the results with those of other methods in the literature [42], where  
\[\Delta=\frac{\delta_{0}}{\alpha} \tag{2}\]  
\[S=E_{\text{in}}\sqrt{\frac{\gamma L\theta}{\alpha^{3}}} \tag{3}\]  
To conduct the simulations, we initiate the field within the cavity in the frequency domain \(E(\omega)\) using complex Gaussian noise with a standard deviation of \(\sigma_{\text{noise}}=10^{-9}\left[\text{W}^{1/2}\right]\). Subsequently, we set the initial state \((\Delta_{1},\left|S_{1}\right|^{2})\) and maintain this state for a duration of \(1.5\,\upmu\)m. After this, we transition to a final state \((\Delta_{2},\left|S_{2}\right|^{2})\) and we hold this state for another \(1.5\,\upmu\)s. We repeat this process for different final states and reconstruct the parameter space as proposed in [42]. A single soliton or dissipative Kerr soliton (DKS) is chosen as the final state because the frequency comb contains the largest number of optical carriers, which is essential for our method. For the 200 GHz MRR, we set \((\Delta_{1}=0,\left|S_{1}\right|^{2}=6)\), and, for the 50 GHz MRR, we set \((\Delta_{1}=0,\left|S_{1}\right|^{2}=4)\). We generate a stable OFC by varying the parameters \((\Delta,\left|S\right|^{2})\) using a path obtained from the parameter space. We select a detuning value of \(\Delta=0\) as the starting point, and the parameters are varied until reaching a final point in the region of DKS in the lower-right part of the parameter space \((\Delta,\left|S\right|^{2})\). This choice is justified because this comb type exhibits more significant wavelengths with high coherence and stability. Figure 4b shows the path in blue and orange. We establish a non-linear power variation to avoid the chaotic region, as mentioned in [42]. We verify the probability of generating a DKS with the established paths. We carry out this process because the paths do not always produce the same type of OFC, and we need to adopt a deterministic approach. We perform multiple simulations using the LLE equation with the established path to verify the probability of DKS generation. During each simulation, we count the number of pulses appearing in the microresonator at the end of the process. If only one pulse occurs, DKS generation is considered successful. We repeat this process for each simulation and generate a histogram that shows the number of pulses obtained during 200 repetitions. This result allows for the evaluation of the proposed path's suitability or the need to offer alternative means to achieve the goal of deterministic  
Figure 4: (**a**) Parametric space as a function of peak numbers \((\Delta,\left|S\right|^{2})\) for an MRR with an R (\(\upmu\)m).