# Effect of Process Parameters on Mode Conversion in Submicron Tapered Silicon Ridge Waveguides
## 3 Discussion  
The insertion loss (IL) of a photonic device is defined as:  
\[\text{IL}=\ -10\log\Bigl{(}\frac{\text{P}_{\text{out}}}{\text{P}_{\text{in}}} \Bigr{)} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Silicon Height H = 500 nm**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Ridge Height (0.6H)**} & \multicolumn{2}{c}{**Ridge Height (0.5 H)**} & \multicolumn{2}{c}{**Ridge Height (0.4 H)**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Mode Converter**} & \multicolumn{2}{c}{**TM Preserving**} & \multicolumn{2}{c}{**Mode Converter**} & \multicolumn{2}{c}{**TM Preserving**} \\  & \multicolumn{2}{c}{**(Ltp)**} & \multicolumn{2}{c}{**(Ltp = 10 \(\upmu\)m)**} & \multicolumn{2}{c}{**(Ltp)**} & \multicolumn{2}{c}{**(Ltp = 10 \(\upmu\)m)**} & \multicolumn{2}{c}{**(Ltp)**} & \multicolumn{2}{c}{**(Ltp = 10 \(\upmu\)m)**} \\ \hline \(\theta=0^{0}\) & \multirow{2}{*}{1821 \(\upmu\)m} & \multirow{2}{*}{98\%} & \multirow{2}{*}{985 \(\upmu\)m} & \multirow{2}{*}{96\%} & \multirow{2}{*}{440 \(\upmu\)m} & \multirow{2}{*}{85\%} \\ \(n_{\text{cl}}=1.445\) & & & & & \\ \hline \(\theta=8^{0}\) & \multirow{2}{*}{1600 \(\upmu\)m} & \multirow{2}{*}{97\%} & \multirow{2}{*}{750 \(\upmu\)m} & \multirow{2}{*}{94.5\%} & \multirow{2}{*}{376 \(\upmu\)m} & \multirow{2}{*}{83\%} \\ \(n_{\text{cl}}=1.445\) & & & & & \\ \hline \(\theta=8^{0}\), \(n_{\text{cl}}=1\) & 1110 \(\upmu\)m & 96\% & 655 \(\upmu\)m & 94\% & 277 \(\upmu\)m & 80\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results Summary.  
where \(\rm{P_{in}}\) indicates input optical power and \(\rm{P_{out}}\) indicates output optical power. In addition to surface roughness, the main loss mechanism in these devices is the presence and excitation of unwanted optical modes. In the case of TM to \(\rm{TE_{1}}\) mode converter, the percentage of TM mode in the output power contributes towards loss. Therefore, Equation (2) can be modified as:  
\[\rm{IL}\ (\rm{Mode}\ \rm{Convertor})=\ -10\log\biggl{(}\frac{\rm{P_{out}}( \rm{TE_{1}})}{\rm{P_{in}}(\rm{TM_{0}})}\biggr{)} \tag{3}\]  
Similarly, for a TM taper, the excitation of unwanted \(\rm{TE_{1}}\) mode is the main source of the loss. The insertion loss of the TM preserving taper can be defined as:  
\[\rm{IL}\ (\rm{TM}\ \rm{preserving}\ \rm{Taper})=\ -10\log\biggl{(}\frac{\rm{P_{out}}( \rm{TM_{0}})}{\rm{P_{in}}(\rm{TM_{0}})}\biggr{)} \tag{4}\]  
Therefore, the best low-loss configuration for the mode conversion is the air cladding with a partial etch of 0.4 H (0.176 dB), whereas for the TM preserving taper is the case of symmetric cladding with a ridge height of 0.6 H (0.084 dB). Here we have assumed an rms sidewall roughness of 3 nm.  
Mode converters based on mode hybridization in adiabatic tapers are highly tolerant to fabrication tolerance. This is because irrespective of fabrication deviation in width, the mode hybridization width is always found along the taper. However, if the input width (\(\rm{W_{1}}\)) is too close to the mode hybridization width (\(\rm{W_{h}}\)), the device is more prone to fabrication variations. Along with width variation, there is also a possibility of thickness variation and changes in the sidewall angle. We study the effect of thickness and changes in sidewall angle (see Figure 10). From this figure, it can be seen that the decrease in the ridge height slightly improves the mode conversion efficiency. Similarly, the increase in sidewall angle had a similar effect on the efficiency. However, fabrication variation does not cause much difference in the performance.