# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 1 Introduction
The latter is calculated as:  
\[\varepsilon_{\mathrm{g}}=\frac{\partial[\omega\varepsilon_{R}(\mathbf{r}, \omega)\ ]}{\partial\omega} \tag{5}\]  
Use of Equation (4) in place of simply \(\varepsilon_{r}\) accounts for the physical effects of dispersion and usually ensures mathematically that the all the terms in the denominator of Equation (3) will be positive because in most cases \(\varepsilon_{\mathrm{g}}+\varepsilon_{R}>0\). For example, use of the Drude model for the complex permittivity of gold (Au) leads to \(\varepsilon_{\mathrm{g}}+\varepsilon_{R}>0\)[12]. However, use of experimental data [13] for the permittivity of Ag may lead to \(\varepsilon_{\mathrm{g}}+\varepsilon_{R}<0\), such that \(\varepsilon_{\mathrm{avg}}<0\). In this case, use of \(\varepsilon_{r}\) in Equation (3) may be replaced directly by Equation (5), which will always be positive [14; 15; 16]. Despite the circumvention of the failure of Equation (3) via Equations (4) and (5), a number of different expressions for the effective modal volume have appeared in the literature [17; 18; 19; 20; 21]. It is the purpose of this paper to assess the utility of these expressions for the design and analysis of nanoscale photonic and plasmonic cavities. For concreteness, we evaluate the expressions for near-infrared (IR) resonant cavities. In Table 1, we list all three different nanoscale cavities we have evaluated, and categorize them according to the optical mode they support. Therefore, we provide a practical guideline on how to implement each effective modal expression. It should be understood, however, that our approach could be applied to cavities supporting modes at other frequencies as well. The paper is organized as follows. In Section 2 we expand upon the introduction and review the various expressions for \(V_{\mathit{eff}}\) found in the literature. Next, in Section 3 we apply these expressions to several canonical cavity geometries, including a metallo-dielectric cavity that supports photonic modes [22] and a coaxial cavity that supports plasmonic modes [6]. In all cases, indium gallium arsenide phosphide (InGaAsP) serves as the active region material, with alloy composition chosen such that emission occurs in the wavelength range of 1260 nm \(<\lambda_{0}<\) 1590 nm. For the coaxial geometry, we study both Ag and aluminum-doped zinc oxide (AZO) as the cladding materials, whereas only Ag is used for the metallo-dielectric geometry. The plasma frequency of Ag and AZO lie in the \begin{table}
\begin{tabular}{c c c c} \hline  & **Metallo–Dielectric** & **Metallic Coaxial Cavity** & **Metallic Coaxial Cavity** \\  & **Cavity with Ag** & **with AZO** & **with Ag** \\ \hline Photonic Mode & x & x & \\ \hline Plasmonic Mode & & & x \\ \hline Confined Mode & x & & x \\ \hline Leaky Mode & & x & \\ \hline \end{tabular}
\end{table} 
Table 1: Mode features of investigated nanoscale cavities. (AZO: aluminum-doped zinc oxide. )ultraviolet and near-IR wavelengths, respectively, with the latter being tunable depending on its alloy composition [23]. Therefore, we compare cases when the active material emission frequency and the cavity resonance frequency are far from and close to the plasma frequency of the constituent metal. Room temperature conditions are assumed for all calculations and values of permittivity are taken from the literature [23; 24; 25]. It should be emphasized that in calculating \(V_{\textit{eff}}\), we include both the real and imaginary parts of the metal permittivity to account for realistic absorption losses. However, we use only the real part of the InGaAsP permittivity because we are primarily interested in the degree to which electromagnetic energy may be stored in the active region. When considering absorption or amplification of energy in the active region, the numerator of Equation (3) can be used with the substitution of \(\varepsilon_{I}\) for \(\varepsilon_{r}\), which has been theoretically proved and verified by numerical simulation, respectively [26; 27].