# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 2 Materials and Methods
\[\epsilon_{\text{gc}}\ =\epsilon_{\text{r}}+\frac{\partial[\widetilde{\omega}_{c} \epsilon_{\text{r}}(\mathbf{r},\widetilde{\omega}_{c})]}{\partial\widetilde{ \omega}_{c}} \tag{15}\]  
In \(V_{\textit{eff},5}\), the surface integration term in Equation (10) is replaced by the integration of energy density in the perfectly matched layer (PML) in the simulation domain. As \(\mathbf{E}(\mathbf{r})\) exponentially diverges in space when \(\,|\,\mathbf{r}\,|\,\rightarrow\infty\), a leaky mode cannot be readily normalized with energy consideration based merely on the conjugated form of the electric field inner product \(\mathbf{E}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})^{*}\), which is the modulus square of the electric field used in \(V_{\textit{eff},1,2,3}\). To include the impact of absorption and/or radiation loss, which is embedded in Im(\(\mathbf{E}\)), the unconjugated form \(\mathbf{E}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})\) needs to be used for leaky cavities. The formulae of \(V_{\textit{eff},A}\) and \(V_{\textit{eff},5}\) are both based on this energy consideration of the unconjugated form \(\mathbf{E}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})\), therefore, theoretically, these last two effective modal volume expression yield more accurate \(V_{\textit{eff}}\) value for cavities with significant amount of absorption and/or radiation loss. Comparing \(V_{\textit{eff},A}\) and \(V_{\textit{eff},5}\), \(V_{\textit{eff},4}\) requires a surface integration of electric energy density over the boundary of a large simulation domain, which may constrain its applicability for users with limited physical memory of their computation tool. The physical memory requirement for eigenfrequency study in COMSOL grows linearly with the increase in degree of freedom and meshing elements; therefore, there exists an upper limit for simulation domain size. On the other hand, \(V_{\textit{eff},5}\) requires the knowledge of the exact permittivity and permeability distributions in the PML, which may not be possible in all commercial software [30]. Nevertheless, the normalization condition of \(V_{\textit{eff},5}\) can be easily extended for dispersive and magnetic materials. To simulate these three different cavities, eigenfrequency study is performed with the Wave Optics module in the commercial finite element method (FEM) software COMSOL. The cavities are enclosed with spherical shape background material and perfectly match layer (PML). The detailed meshing conditions are listed in Table 2, where \(\lambda_{L}\) is 1.26 \(\upmu\)m, corresponding to the lower edge of InGaAsP quantum well gain spectrum at room temperature; \(n_{\text{InGaAsP}}\) and \(n_{\text{air}}\) are refractive indices of InGaAsP and air, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Fine** & **Normal** \\ \hline \multirow{2}{*}{PML} & Type & Sweep & Sweep \\ \cline{2-4}  & Number of layers & 10 & 5 \\ \hline \multirow{4}{*}{Cavity} & Type & Free tetrahedral (extremely fine) & Free tetrahedral (fine) \\ \cline{2-4}  & Maximum element & \(\lambda_{L}/(6\cdot n_{\text{InGaAsP}})\) & \(\lambda_{L}/(4\cdot n_{\text{InGaAsP}})\) \\ \hline \multirow{2}{*}{Background} & Type & Free tetrahedral (normal) & Free tetrahedral (normal) \\ \cline{2-4}  & Maximum element & \(\lambda_{L}/(6\cdot n_{\text{air}})\) & \(\lambda_{L}/(4\cdot n_{\text{air}})\) \\ \hline \multirow{2}{*}{Metal-dielectric boundary} & Type & Free tetrahedral (extremely fine) & Free tetrahedral (extra fine) \\ \cline{2-4}  & Maximum element & 10 & 20 \\ \hline \end{tabular}
\end{table} 
Table 2: PML and meshing conditions in eigenfrequency study.