# Effect of Process Parameters on Mode Conversion in Submicron Tapered Silicon Ridge Waveguides
## 2 Structure and Analysis
### Taper Design for Efficient Mode Conversion
#### 2.2.1 (0.6. H) Ridge Waveguide (Ridge Height > Slab Height)  
We start our analysis with a deeply etched ridge waveguide. The ridge height is 300 nm (0.6 H), corresponding to a slab height of 200 nm. Figure 4 shows mode conversion efficiencies when the TM\({}_{0}\) mode is launched and coupled to the TE\({}_{1}\) mode. The TM\({}_{0}\) mode propagates along the adiabatic taper with a start and end widths of W\({}_{1}\) = 1 \(\upmu\)m and W\({}_{2}\) = 2 \(\upmu\)m, respectively. The simulations are performed for angled sidewalls and different top cladding conditions. From Figure 4, one can realize very highly efficient tapers for TM\({}_{0}\) to TE\({}_{1}\) mode conversion with \(\sim\)100% conversion efficiency by appropriately selecting L\({}_{\rm tp}\). The L\({}_{\rm tp}\) required for mode conversion is observed to be strongly dependent on the process asymmetries related to the angled sidewalls and cladding material.  
Figure 2: The calculated \(\Delta\)n\({}_{\rm eff}\) (n\({}_{\rm eff}\) (mode)-n\({}_{\rm eff}\)(slab)) for the eigen modes of SOI ridge waveguide with different etch depths along with hybridized mode profiles. (**a**) h = 300 nm (0.6 H); (**b**) h = 250 nm (0.5 H); (**c**) h = 200 nm (0.4 H). The total height of silicon is H = 500 nm and simulations are done at 1550 nm wavelength. Here the sidewall angle \(\theta=0^{0}\) and n\({}_{\rm cl}\) = 1.445 (oxide cladding).  
Under an ideal rectangular waveguide with symmetric cladding (SiO\({}_{2}\)), a taper length of 1820 \(\upmu\)m is required for the complete mode conversion. However, when angled sidewalls are accounted for, the size is reduced to 1600 \(\upmu\)m. This enhanced mode hybridization corresponds to a 12% reduction in taper length. Alternatively, when an air top cladding and sidewall angle are simultaneously present, the length for 100% mode conversion is reduced to 1110 \(\upmu\)m. This corresponds to a total reduction of 39%, which is very significant compared to an ideal case.  
It is important to note that mode conversion for SiO\({}_{2}\)-cladding waveguide can be significantly reduced by choosing ultra-short taper lengths. From Figure 4, the optimum length is 10 \(\upmu\)m, for which the conversion efficiency is very low. For this taper length, the TM\({}_{0}\) mode preserving ability is highest, i.e., 98%, see FDTD (Finite-Difference Time-Domain) simulation in Figure 4b). Since it is not possible to avoid angled sidewalls, it must be considered when designing low-loss TM tapers. Figure 4c shows the transmission efficiency assuming a sidewall angle of 8\({}^{0}\). The drop-in efficiency due to angled sidewalls is only 1%, which is acceptable.  
Figure 5 shows FDTD simulation along an adiabatic taper of length 200 \(\upmu\)m when the launched mode is TM\({}_{0}\). Figure 5a,b are shown for SiO\({}_{2}\) cladding with 0\({}^{0}\) and 8\({}^{0}\) sidewall angles, respectively, while Figure 5c is shown for angled air-cladding waveguide. As predicted, the conversion efficiency increases with process asymmetries of the angled sidewalls and air-top cladding.  
Figure 4: (**a**) The mode conversion efficiency (\(\eta\)) as the function of taper length (L\({}_{\text{tp}}\)) when TM\({}_{0}\) mode is launched. (**b**) Light propagation with \(\theta=0^{0}\) and n\({}_{\text{cl}}=1.445\). (**c**) Light propagation with \(\theta=8^{0}\) and n\({}_{\text{cl}}=1.445\). The total height of silicon is H = 500 nm, ridge height (h) = 0.6 H and simulations are done at 1550 nm wavelength.  
Figure 3: Schematic of regular adiabatic taper.