## 1 Introduction  
Photonic-electronic convergence on silicon (Si) has the strong potential to provide high-volume, low-cost, and energy-efficient solutions to the increasingly-high bandwidth demands for data center interconnects and long-haul telecommunication networks [1; 2; 3; 4]. In this regard, Ge-on-Si photonic devices are promisingly being developed in waveguide configurations for optical modulation, detection, and emission around the C-band telecommunication wavelength of 1.55 \(\upmu\)m [5]. Accordingly, efficient integration between Ge-based photonic devices and crystalline Si on insulator (SOI) waveguides are the subject of intense research in order to enable on-chip integration between various photonic devices on SOI substrates. Ge-based optical modulators and photodetectors integrated with SOI waveguide were developed with very impressive high-speed performance at 1.55 \(\upmu\)m [6; 7; 8; 9; 10; 11; 12; 13; 14], using electro-absorption based on the Franz-Keldysh effect (FKE), which is intrinsically fast [15]. Nevertheless, to integrate photonics with complementary metal-oxide semiconductor (CMOS) electronics and take full advantage of large-scale manufacturing, it is also of importance to have the capability to monolithically develop photonic circuitry on bulk Si substrate [16; 17; 18], which could be a significant challenge for SOI-based devices. Moreover, in order to further accommodate a rapid increase in data traffic, the adoption of wavelength multiplexing is considered critical. In these regards, Si\({}_{3}\)N\({}_{4}\) waveguide could be regarded as a promising alternative suitable for wavelength division multiplexing implementation in photonicintegrated circuits on bulk Si substrate [19; 20; 21; 22; 23]. Due to the relatively-low thermo-optics coefficient of Si\({}_{3}\)N\({}_{4}\) and a moderate refractive index contrast between Si\({}_{3}\)N\({}_{4}\) and SiO\({}_{2}\), Si\({}_{3}\)N\({}_{4}\) was proposed to be advantageous for wavelength multiplexing in terms of polarization independence, phase error due to fabrication imperfections, temperature stability, and wideband operation for telecommunications applications [23; 24; 25]. Additionally, a Si\({}_{3}\)N\({}_{4}\) refractive index \(\sim\)2 is sufficiently high to realize compact photonic circuits with SiO\({}_{2}\) cladding (n \(\sim\) 1.45) [20; 23]. Moreover, silicon nitride-based waveguides can possibly be deposited directly onto bulk Si substrate at a relatively low temperature [23; 26; 27]. Nevertheless, comparing to SOI waveguides, efficient optical coupling between Si\({}_{3}\)N\({}_{4}\) waveguide and Ge-on-Si photonic components could be considered significantly more challenging due to an even larger refractive index difference between Si\({}_{3}\)N\({}_{4}\) and Ge. In this paper, we propose and present theoretical analysis on the use of a compact amorphous GeSi lateral taper (15 \(\upmu\)m in length) to facilitate the optical coupling between a Si\({}_{3}\)N\({}_{4}\) waveguide (refractive index of \(\sim\)2), and a low-voltage Ge-based FKE optical modulator (refractive index of \(\sim\)4.3) using 3D Finite-Difference Time-Domain (3D-FDTD) simulation (Lumerical Inc. Vancouver, BC, Canada) at the optical wavelength of 1.55 \(\upmu\)m. As there was no need for a crystalline layer for its deposition, the amorphous material was grown on SiO\({}_{2}\) layers, providing design flexibility to obtain the most suitable configuration of optimized optical coupling; moreover, it was also possible to develop the structure on bulk Si substrate. Coupling structure in a lateral direction was used in order to be preferable for quasi-transverse-electric (quasi-TE) polarization, which was typically employed for Ge-based Franz-Keldysh optical modulators in waveguide configuration [6; 7; 8; 9].