## 2 Device Design and Structure
For that, adiabatic tapers with over 95% transmission have been designed. This component is crucial to reduce insertion losses in the widely tunable laser and to minimize back-reflections in the SOA booster. A DVS-benzocyclobutene (BCB) passivation cladding has been used to increase the lateral optical confinement, hence avoiding significant mode leakage towards the Si slab while preventing lateral current leakage under high current injection [7]. Such geometry allows for tighter mode confinement in the III-V waveguide compared to other designs with much wider III-V mesas, which helps reducing the laser's threshold current. The ring-based Vernier filter has been designed in a 310 nm-thick SOI platform using strip waveguides. The inset below in Figure 2a shows an optical mode cross-section simulation of the fundamental TE-like mode. As seen, the optical mode is tightly confined in the waveguide, which allows designing compact photonic circuits with small bend radius. The Vernier filter is composed by two racetrack resonators coupled to a bus waveguide, each of them with a roundtrip length of 164.6 \(\upmu\)m and 153.1 \(\upmu\)m, respectively. Such parameters provide a theoretical, Vernier free spectral range (FSR) of 50 nm (see the grey horizontal arrow in Figure 2a). A careful optimization of the self-coupling coefficient (r) between the racetrack resonators and the bus waveguide was carried out to ensure single frequency operation of the widely tunable laser with a low insertion loss penalty. Too low r values would induce negligible insertion loss in the laser cavity, but at the price of a mitigated mode filtering that would derive in multimode unstable operation. To avoid that, the Vernier frequency response should have a sufficiently high side mode suppression ratio (SMSR) between its central lobe with maximum transmission and the adjacent transmission lobes with lower transmission (see the grey vertical arrow at the left-hand side of Figure 2a). In our case, we have defined a Vernier SMSR\({}_{\text{v}}\) of 6 dB. Moreover, the mode selectivity at the central lobe also plays an important role in rejecting adjacent longitudinal close-in modes. The upper inset of Figure 2a provides a zoom-in view of the Vernier's spectral response around its maximum of transmission (orange trace) overlaid with the simulated longitudinal cavity mode spectrum, assuming an enhanced optical length at resonance (blue trace). As seen, the ability to filter out longitudinal close-in modes can also be evaluated with an SMSR\({}_{\text{FP}}\) parameter, which is around 5 dB in our widely tunable laser (see the vertical pale red arrow at the inset of Figure 2a). This value can be adjusted by either varying the FSR of the longitudinal cavity modes (for instance, by modifying the effective optical length of the cavity) or by narrowing (or widening) the half width at half maximum (HWHM) of the maximum transmission lobe. For the current design presented in this work, we have used a HWHM of 10.3 GHz and a longitudinal mode FSR\({}_{\text{FP}}\) of 13.3 GHz for a wavelength of 1549 nm. Thus, the lasing wavelength of the widely tunable laser is defined by the spectral positioning of such maximum transmission lobes, which is governed by the optical phase of the Vernier filter. In our design, we have used a couple of thermal heaters placed on top of each racetrack resonator to finely control such optical phases by means of the thermo-optic effect. Further details on the design of Vernier filters can be found in [22]. The optical feedback in the widely tunable laser cavity has been performed with two Bragg mirrors etched in a Si rib waveguide with a core height of 310 nm. Inset of Figure 2b shows the fundamental TE-like mode profile. The Si waveguide has a rib width of 2 \(\upmu\)m, a slab width of 10 \(\upmu\)m and rib height of 143 nm. Total internal reflection is obtained through a highly reflective Bragg mirror that is composed by 39 periods, has an etching depth of 40 nm, a duty cycle of 50% and a period of 276 nm.