# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 3 Results
### Coaxial Resonator with Ag  
A cross section of the Ag-cladded coaxial cavity is shown in Figure 2a, where \(\mid\)E(**r**)\(\mid\) of the transverse electro-magnetic-like (TEM-like) mode is plotted. The cavity is enclosed by a spherical simulation space shown in Figure 2b. Despite the fact that the optical mode is fairly well confined to the active region of the cavity, there is notably more leakage into the air plug compared to the metallo-dielectric cavity. As a result, the electric field antinode is located in the air plug rather than the active region of the cavity. The quality factor of this plasmonic mode in the coaxial cavity is  
Figure 1: (**a**) Closeup of the transverse electric (TE) TE\({}_{012}\) mode in the metallo-dielectric cavity (lengths expressed in nm) (**b**) Entire simulation space for the metallo–dielectric cavity (**c**) \(V_{\mathit{eff}}\) as a function of \(d_{\mathrm{air}}\) evaluated for the various expressions. All of the traces overlap. (\(Q\) = 2700.8 at 2.0169e14 Hz (\(\lambda_{0}\) = 1486.4 nm).)on the order of 100, which is about an order of magnitude lower than the photonic mode in the metallo-dielectric cavity.  
The evaluated effective modal volumes of the Ag-cladded coaxial cavity are shown in Figure 3, for various simulation conditions, which include different meshing finesse and simulation domain sizes. As the electric field maxima and electric field energy density maxima located in the air region and active region respectively, \(V_{\mathit{eff,3}}\) has the largest denominator among all five \(V_{\mathit{eff}}\) expressions. Therefore, \(V_{\mathit{eff,3}}\) exhibits the smallest values. As \(V_{\mathit{eff,2}}\) takes the electric field energy density in its denominator, this \(V_{\mathit{eff}}\) expression is less sensitive to the change of simulation domain size, and is the most stable one. Among the other three expressions, as the field quickly vanishes at the simulation boundary, \(V_{\mathit{eff,1}}\), \(V_{\mathit{eff,4}}\) and \(V_{\mathit{eff,5}}\) all show converging values with less than 30% variation (\(V_{\mathit{eff,1}}\approx 1.29\)\(V_{\mathit{eff,4}}\) and \(1.27\)\(V_{\mathit{eff,5}}\)) for all simulation conditions. From Figure 2a, it is apparent that the maximum electric field resides in the air region, which leads to a denominator for \(V_{\mathit{eff,1}}\), \(V_{\mathit{eff,4}}\) and \(V_{\mathit{eff,5}}\) that is significantly smaller than the denominators of both \(V_{\mathit{eff,2}}\) and \(V_{\mathit{eff,3}}\). Hence, the values of \(V_{\mathit{eff,1}}\), \(V_{\mathit{eff,4}}\) and \(V_{\mathit{eff,5}}\) are significantly greater than those of \(V_{\mathit{eff,2}}\) and \(V_{\mathit{eff,3}}\). Due to this same reason, \(V_{\mathit{eff,1}}\), \(V_{\mathit{eff,4}}\) and \(V_{\mathit{eff,5}}\) are more sensitive to the change of simulation domain size compared to \(V_{\mathit{eff,2}}\). Comparing Figures 3a and 3b, which present results for fine and normal meshing, respectively, we observe a rather negligible change in the evaluated expression of \(V_{\mathit{eff,2}}\). However, it is important to note that because \(V_{\mathit{eff,1}}\), \(V_{\mathit{eff,3}}\), \(V_{\mathit{eff,4}}\)\(V_{\mathit{eff,5}}\) are normalized with respect to the maximum electric field rather than to the maximum electric field energy density as in the case of \(V_{\mathit{eff,2}}\), if this maximum lies in the air domain, then the mesh size could play an important role. For example, this singularity of the field produced by the corners of the Ag core can be suppressed by reducing the mesh size [33]. Additionally, the accuracy of the simulation can be further improved by introducing the idea of subpixel smoothing of the isotropic dielectric function [34].  
Figure 2: (**a**) Closeup of the TEM-like mode of the coaxial cavity with Ag as cladding material (lengths expressed in nm). (**b**) Side view of the entire spherical simulation space for the coaxial cavity. (\(Q=79.074\) at 2.0542e14 Hz (\(\lambda_{0}=1459.4\) nm)).