# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 2 Materials and Methods  
In this section, we present five expressions for \(V_{\textit{eff}}\) from the literature. The first expression, \(V_{\textit{eff},1}\), was presented for use in a photonic crystal cavity with an air gap and containing only dielectric materials [17]. It is:  
\[V_{\textit{eff},1} = \frac{\sum_{\alpha}\int_{V_{\alpha}}d^{3}\textbf{r}_{\alpha} \left(\varepsilon_{\alpha}(\textbf{r}_{\alpha})|E_{\alpha}(\textbf{r}_{ \alpha})|^{2}\right)}{\varepsilon(\textbf{r}_{\max})\text{max}(|E(\textbf{r}_ {\max})|)^{2}} \tag{6}\]  
The denominator of Equation (6) includes the maximum of the electric field, where \(\textbf{r}_{\max}\) denotes the position of electric field antinode. Additionally, the permittivity of the denominator equals that of the material in which the maximum electric field antinode is located. The second expression, \(V_{\textit{eff},2}\) was used in the analysis of metallic nanocube antennas [18]. It defines the effective volume as the ratio of total electrical energy to the peak value of electrical energy density. Using the earlier introduced notation:  
\[ \begin{array}{ll}V_{\textit{eff},2}&=\frac{\Xi_{r}}{\max(U_{c})}\\ &=\frac{\sum_{\alpha}\int_{V_{\alpha}}d^{3}\textbf{r}_{\alpha}\left(\varepsilon _{\alpha}(\textbf{r}_{\alpha})|E_{\alpha}(\textbf{r}_{\alpha})|^{2}\right)}{ \text{max}\left(\varepsilon(\textbf{r})|E(\textbf{r})|^{2}\right)}\end{array}  \tag{7}\]  
If the peak energy density is located in the active region, then Equation (7) leads to the reference expression for \(\Gamma\), i.e., Equation (3), after division by \(V_{a}\). If the peak electric field lies in the active region, then Equations (6) and (7) yield the same result. However, in some geometries, the peak electric field could reside in a low index region, such as an air gap, while the peak energy density lies in the high index active region, in which case Equations (6) and (7) will yield different results. The third effective volume expression, \(V_{\textit{eff},3}\), was used in the analysis of spontaneous emission enhancement in a dielectric pillar microcavity [19] and a metallo-dielectric cavity [22]. It is:  
\[V_{\textit{eff},3} = \frac{\sum_{\alpha}\int_{V_{\alpha}}d^{3}\textbf{r}_{\alpha} \left(n_{\alpha}(\textbf{r}_{\alpha})^{2}|\textbf{f}_{\alpha}(\textbf{r}_{ \alpha})|^{2}\right)}{n_{\alpha}^{2}} \tag{8}\]  
where **f** is the normalized electric field such that \(\textbf{1}\textbf{f}(\textbf{r})\textbf{1}=1\) at the field antinode. While Equation (8) appears to differ from Equations (6) and (7) in that the electric field is absent from the denominator, the normalization effectively supplies the electric field to both the numerator and denominator. If all materials in the cavity are lossless dielectric and the peak electric field resides in the active region, then \(V_{\textit{eff},3}\) yields identical results to \(V_{\textit{eff},2}\) and \(V_{\textit{eff},1}\). However the use of the active region refractive index in the denominator of Equation (8), regardless of the location of peak field or energy, generally leads to values of \(V_{\textit{eff},3}\) different from \(V_{\textit{eff},1}\) and \(V_{\textit{eff},2}\). where \(V_{q}\) is defined as:  
\[V_{q}\equiv\frac{\sum_{\alpha}\left(\int_{V_{\alpha}}\ d^{3}\mathbf{r}_{\alpha} \epsilon_{gc,\alpha}(\mathbf{r}_{\alpha})\mathbf{E}_{\alpha}(\mathbf{r}_{\alpha })^{2}\right)+\int_{V_{PML}}d^{3}\mathbf{r}\epsilon_{gc}(\mathbf{r})\mathbf{E} (\mathbf{r})^{2}}{\epsilon_{gc}(\mathbf{r}_{\text{max}})\text{max}(\mathbf{E}( \mathbf{r}_{\text{max}}))^{2}} \tag{14}\]  
For \(V_{\textit{eff},5}\), \(\epsilon_{\text{r}}\) is replaced by \(\epsilon_{\text{gc}}\), where 'gc' stands for a complex group permittivity [21].