# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 4 Discussion  
Based on the results, the type of cavity clearly affects the robustness of the various expressions of \(V_{\mathit{eff}}\) found in the literature. To quantify the stability of a \(V_{\mathit{eff}}\) expression, we use the maximum percentage difference, defined as:  
\[\Delta_{(\#)}\ =\left(\max\frac{\left|V_{\mathit{eff,(\#)}}^{m}-V_{\mathit{ eff,(\#)}}^{n}\right|}{V_{\mathit{eff,(\#)}}^{m}}\right)\times 100\% \tag{16}\]  
where m and n refer to different simulation conditions. All the evaluation results are tabulated in Table 3 for fair comparison of the five effective modal volume expressions.  
If the cavity supports a well-confined photonic mode, the electric field maximum resides in the active region and the five expressions yield nearly identical results if numerical error can be eliminated. This is confirmed with the metallo-dielectric cavity, where we observed that \(\Delta 1=\Delta 2=\Delta 3\approx\Delta 4\approx\Delta 5\) < 0.21% as the simulation domain size increases. This can be considered a negligible change.  
For cavities supporting plasmonic modes, the proximity of the cavity resonance to the metal plasma frequency plays an essential role in determining the efficacy of the \(V_{\mathit{eff}}\) expressions. For the Ag-coaxial cavity under fine meshing, \(\Delta 2<2.02\%\), suggesting it is the most robust expression. However, for the AZO cavity, \(V_{\mathit{eff,1}}\), \(V_{\mathit{eff,2}}\), and \(V_{\mathit{eff,3}}\) all diverge nearly at the same rate, while \(V_{\mathit{eff,4,5}}\) converge but oscillate around certain values. Therefore, based on their robustness performance, for confined mode \(V_{\mathit{eff,2}}\) is suggested; for leaky mode, \(V_{\mathit{eff,4,5}}\) are suggested to obtain a more precise effective modal volume value. In addition, the choice of the correct effective modal volume formula should rely on a solid physical understanding of where the light-matter interaction of interest is happening.  
Figure 5: (**a**) Effective modal volume of the coaxial cavity using AZO as a metal for various air domain size \(d_{\text{air}}\). (**b**) Absolute value of the real part of the relative permittivity \(\left|\varepsilon_{\text{r}}\right|\) (solid curves) and attenuation coefficient \(\alpha\) (dashed curves) of Ag (gray curves) and AZO (red curves) as functions of vacuum wavelength.  
In addition, notice that all five \(V_{\mathit{eff}}\) formulae shown here only calculate the electric energy in the physical domain (including cavity and background material). However, for metallic cavities, especially at room temperature in the near-IR regime, the significant absorption loss of plasmonic or hybrid modes will result in unequal electric and magnetic energy in the cavity. For example, in the Ag-coaxial cavity, the electric energy is usually two to five times greater than the magnetic energy in the same material region. Therefore, calculating only the electric energy in metallic cavities should be understood as an approximation, and the \(V_{\mathit{eff}}\) formulae should be modified for plasmonic modes if greater accuracy is desired [15; 16; 21].