## 2 Materials and Methods
### Filter Design  
In order to obtain a proper filter design, we started with a simple mathematical description of an ideal ring resonator transfer function, comparing it with the filter requirements that are reported in Table 1. It is worth noting that the requirements of the adjacent and non-adjacent suppression of channels are relatively relaxed, as the structure of next-generation passive optical network units (ONUs) often has the optical signal passing through two filters. On the other hand, this introduces the requirement of an extremely low filter insertion loss (IL). By analyzing the filter-cascade transfer function, we determined that a cascade of at least two resonators is required to satisfy the requirements of a Si-photonic-based WDM NG-PON, and that the bus-to-ring (\(k_{B}\)) and ring-to-ring (\(k_{R}\)) field-coupling-coefficients, should be set to 0.26 and 0.05, respectively. Following this, we performed a series of 3D-FDTD simulations, to identify the geometry of waveguides and coupling regions that could provide the desired coupling coefficients. All simulations were performed using the commercially available \"FDTD Solutions\" (by Lumerical Solutions Inc., Vancouver, BC V6E 2M6, Canada), by considering the silicon waveguide and resonator structures to be fully surrounded by SiO\({}_{2}\) cladding layers, and by working on TE light polarization. Furthermore, minimum feature sizes and other manufacturing constraints were provided by the fabrication procedures at the CMOS-compatible Si-photonic foundry of CEA-Leti. The principal limits imposed by the fabrication and overall integration requirements are:  
* the minimum gap between adjacent waveguides was set to 200 nm, in order to allow a full-etch of the region using standard lithographic techniques;
* the bus waveguide had to be 500 nm wide, so as to match the width of the waveguides used for the other components of the final chip;
* the height of the waveguides could be 220 nm (corresponding to the height of the Si layer present on the SOI wafers), or could be reduced to 100 nm, as a fabrication step with a 120-nm etch depth was already present in the process flow, but no other height-value could be used for the filter's design;
* it was important to realize a fabrication-tolerant design, as the fabrication process was based on standard lithographic techniques and was not exploiting the e-beam high-resolution, as often done in scientific research where performance is the main challenge and production-price is not an issue. Analysis immediately identified that standard (500 \(\times\) 220 nm, width \(\times\) height) Si-waveguides could not achieve the required bus-ring coupling coefficient, while maintaining a waveguide-to-waveguide gap \(>\) 200 nm. However, by taking advantage of the fact that the TE optical mode propagating in height-reduced waveguides more deeply extends outside of the Si-core (see the top part of Figure 1), it is possible to increase the waveguide coupling coefficients. This creates the possibility of two different solutions, which were investigated separately--(i) the use of race-track resonators with standard waveguide; and (ii) the use of reduced-height waveguides with a curved coupling region. In both cases, the coupling region is increased, which provides a higher coupling efficiency. In addition, for the second case (curved coupling region), we expect a reduction of propagation losses [19], and a simultaneous increase of the coupling coefficients, as the optical mode further expands outside of the waveguide-core region. For racetrack-based filters, the length of the straight section was initially determined through numerical simulations, so as to obtain the required coupling (while keeping the gap between the bus waveguide and the resonator equal to 200 nm). The curvature radius was then calculated, based on