# On-Demand Waveguide-Integrated Microlaser-on-Silicon
## 2 Results and Discussion
### Numerical Simulations  
Three-dimensional (3D) finite-difference time-domain (FDTD) simulations were performed to obtain an in-depth understanding of light coupling. Figure 4a depicts the index profiles of the top and side views of the simulated structure. For a direct comparison with the study, we used structural parameters similar to those of the fabricated device, as shown in Figure 2d. As in the experiment, the InGaAsP microdisk was slightly misaligned by 700 nm with respect to the center of the Si-post and was in the longitudinal center of the 100-\(\mu\)m-long Si WG. To excite the resonant WGMs, an electric dipole with a wavelength of 1501 nm was introduced in the microdisk. Along the WG, we set a series of monitoring planes perpendicular to the axis of the WG ((i), Figure 4a), which included the cross-section of the Si WG on the SiO\({}_{2}\) substrate ((ii), Figure 4c). Then the total power emitted from the microdisk (Out\({}_{\rm disk}\)) and the transmitted power passing through the monitor planes (\(T_{\rm WG}\)) were calculated. Furthermore, we introduced additional monitoring planes at positions 270 and 570 nm above the microdisk and WG, respectively, to calculate the vertical component of the emitted powers from the microdisk (Top\({}_{\rm disk}\)) and WG ends (Top\({}_{\rm WG}\)) ((ii), Figure 4a). Figure 4b depicts the calculated electric field intensity distributions at the center of the microdisk (i) and Si WG (ii) (white dotted lines of the inset in Figure 4a), exhibiting a strongly confined WGM within the microdisk and coupled light propagation through the WG ((ii), Figure 4b). To estimate the light-coupling efficiency (\(\eta_{c}\)), we plotted the normalized transmitted power (\(T_{\rm WG}\)/Out\({}_{\rm disk}\)) as a function of the monitoring position ((i), Figure 4c). Then, the data obtained from positions ranging from 3-48 \(\upmu\)m were linear-fitted to exclude the power directly from the microdisk. The extrapolated value on the \(y\)-axis represents the coupling efficiency (\(\eta_{c}\)), which was estimated to be approximately 44.52%. In addition, we calculated the \(y\)-component of the electric fields of the propagating TE-like WG mode at a 40 \(\upmu\)m distant detection position. The field profile revealed that the light  
Figure 4: Numerical simulation of light coupling and propagation. (**a**) Index profiles of the simulated structures. Top-(i), and side-view (ii) of the Si WG-integrated microdisk. The diameter and thickness of the microdisk (Si-post) were 4.51 \(\upmu\)m (1.55 \(\upmu\)m) and 260 nm (300 nm), respectively. The width, thickness, and length of Si WG were 470 nm, 260 nm, and 100 \(\upmu\)m, respectively. Out\({}_{\rm disk}\) and \(T_{\rm WG}\) indicate the total power emitted from the microdisks and the transmitted power through the WG. Top\({}_{\rm disk}\) and Top\({}_{\rm WG}\) indicate the vertical component of emitted power from the microdisk and of scattered power at WG end, respectively. Scale bars: 10 \(\upmu\)m and 1 \(\upmu\)m for (i) and inset of (ii). (**b**) Calculated electric field intensity distributions at the center of microdisk (i) and WG (ii) (white dotted lines of inset of (a)). (**c**) Light coupling analysis. (i) Normalized transmitted power (\(T_{\rm WG}\)/Out\({}_{\rm disk}\)) as a function of distance from the center of WG. (ii) Schematic of WG cross section. Scale bar: 200 nm. (iii) \(E_{y}\)-component of electric fields of the fundamental propagating WG mode. (**d**) Comparison of simulation and experiment: Top\({}_{\rm WG}\)/Top\({}_{\rm disk}\) (simulation), versus \(I_{\rm WG}\)/\(I_{\rm disk}\) (experiment). from the microdisk was coupled to the fundamental WG mode ((iii), Figure 4c). Finally, we calculated the normalized power in the vertical direction (Top\({}_{\text{WG}}\)/Top\({}_{\text{disk}}\)) at both ends and directly compared them with those from the experiment (\(I_{\text{WG}}\)/\(I_{\text{disk}}\)), as shown in Figure 4d.