## 3 Coupling Performance and Integrated Optical Modulator  
Figure 2 reports on the optical coupling performance based on different parameters of the proposed amorphous GeSi taper. To verify the potential of achieving a relatively homogenous performance with a modern, large-scale manufacturing of the Si wafer, we studied the variation in optical coupling based on taper tip width (Figure 2a), taper end width (Figure 2b), taper length (Figure 2c), amorphous Ge\({}_{0.83}\)Si\({}_{0.17}\) refractive index (Figure 2d), and the relative positions between the taper and the Ge-based FKE optical modulator (Figure 2e,f). The coupling loss between the 1 \(\upmu\)m wide, 1 \(\upmu\)m high Si\({}_{3}\)N\({}_{4}\) waveguide and the 200 nm thick, 590 nm wide Ge-based FKE layer could be as low as 1.2 dB. As in Figure 2a, a relatively-consistent coupling loss of 1.2 dB was maintained as long as the taper tip width did not become larger than 50 nm. Optical coupling loss became slightly better as the taper tip width became smaller than 10 nm, which could be attributed to a better optical mode matching with the Si\({}_{3}\)N\({}_{4}\) waveguide at the tip of the taper. We selected the taper tip width of 30 nm, which gave a good compromise between optical coupling performance and fabrication tolerance. For the taper end width, as in Figure 2b, 1.2 dB coupling loss was sustained for the width variation of \(\pm\)\(\leq\)10 nm, which was comparable with that of taper tip width. For the length of the taper, optimized coupling performance  
Figure 1: The fundamental optical mode of (**a**) 1 \(\upmu\)m high and 1 \(\upmu\)m wide Si\({}_{3}\)N\({}_{4}\) waveguide, (**b**) amorphous-GeSi lateral taper, and (**c**) 200 nm high and 590 nm wide Ge-based Franz–Keldysh effect (FKE) optical modulator. (**d**) Schematic view of the proposed coupling structure. (**e**) Top view of optical propagation from the Si\({}_{3}\)N\({}_{4}\) waveguide via amorphous Ge\({}_{0.83}\)Si\({}_{0.17}\) taper to the Ge-based FKE optical modulator. of 1.2 dB was achieved with a 15 \(\upmu\)m long taper, and only significant fabrication variation of larger than \(\pm\)0.5 \(\upmu\)m (500 nm) began to affect the optimized coupling loss value, as shown Figure 2c. As a variation in the refractive index of deposited amorphous GeSi was reasonably expected, Figure 2d investigates its effect on the coupling performance. It would have taken a refractive index variation of more than \(\pm\)0.5% to cause a noticeable increase in coupling loss, which was within the capability of modern deposition tools. For the variation in the relative position between the amorphous Ge\({}_{0.83}\)Si\({}_{0.17}\) taper and the Ge-based FKE optical modulator, as reported in Figure 2e,f, the optimized coupling performance was maintained as long as the misalignment of vertical and horizontal positions were within \(\pm\)10 nm. As a result, it was certain that the proposed taper structure could be realized with homogenous performance with respect to the capability of advanced microelectronic process technology [35; 36; 37]. It should be noted that we focused the conversation on the ability to maintain optimized optical coupling, as each dB increase in optical loss necessitated a correspondingly higher laser optical power, therefore, the ability to maintain low optical loss was crucial for low-power optical interconnection. Figure 2: Optical coupling variation with respect to (**a**) taper tip width, (**b**) taper end width, (**c**) taper length, (**d**) amorphous Ge\({}_{0.83}\)Si\({}_{0.17}\) refractive index, and the relative (**e**) vertical and (**f**) horizontal positions between the taper and the Ge-based FKE optical modulator.