# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 1 Introduction
\begin{array}{ll}\Gamma&=\frac{\Xi_{a}}{\Xi_{a}}\\ &=\frac{\int_{V_{a}}d^{3}\mathbf{r}_{\mathrm{n}}U_{\mathrm{e}}(\mathbf{r})}{ \int_{V}d^{3}\mathbf{r}U_{\mathrm{e}}(\mathbf{r})}\\ &=\frac{\int_{V_{a}}d^{3}\mathbf{r}_{\mathrm{n}}\left(\mathbf{D}(\mathbf{r}) \cdot\mathbf{E}(\mathbf{r})^{*}\right)}{\int_{V}d^{3}\mathbf{r}\left(\mathbf{ D}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})^{*}\right)}\\ &=\frac{\int_{V_{a}}d^{3}\mathbf{r}_{\mathrm{n}}\left(\varepsilon_{0}\ell_{ \mathrm{r}}(\mathbf{r})|\mathbf{E}(\mathbf{r})|^{2}\right)}{\int_{V}d^{3} \mathbf{r}\left(\varepsilon_{0}\ell_{\mathrm{r}}(\mathbf{r})|\mathbf{E}( \mathbf{r})|^{2}\right)}\\ &=\frac{\int_{V_{a}}d^{3}\mathbf{r}_{\mathrm{n}}\left(\varepsilon_{0}\ell_{ \mathrm{r}}(\mathbf{r})|\mathbf{E}(\mathbf{r})|^{2}\right)}{\sum_{a}\int_{V_{a}} d^{3}\mathbf{r}_{\mathrm{n}}\left(\varepsilon_{0}\ell_{\mathrm{r}},A(\mathbf{r}_{ \mathrm{n}})|\mathbf{E}_{\mathrm{n}}(\mathbf{r}_{\mathrm{n}})|^{2}\right)} \end{array}  \tag{3}\]  
where \(\mathbf{D}\) and \(\mathbf{E}\) are the displacement and electric field vectors, respectively (\({}^{*}\) denotes the conjugated form of the electric field, therefore, the inner product would be the norm of the electric field), \(\varepsilon_{0}\) is the permittivity in vacuum, and \(\varepsilon_{r}=\varepsilon_{R}+\mathrm{i}\varepsilon_{I}\) is the, generally, complex valued relative permittivity, with the R and I subscripts denoting the real and imaginary part, respectively. In the last equality, \(\alpha\) represents each unique material domain in the simulation over which volume integration must be executed, with the total energy being the sum over these domains. For the study presented in the proceeding sections, we will consider only non-magnetic materials and therefore use Equation (3) as our reference point. For sake of clarity, we define a material as dielectric or metallic at a given wavelength when \(\varepsilon_{R}>0\) and \(\varepsilon_{R}<0\), respectively. Hence, the same \"material\" can behave as both dielectric and metal if the simulation includes dispersion over the range of wavelengths in which \(\varepsilon_{R}\) changes sign. Also, we define a material as absorptive at a given wavelength when \(\varepsilon_{I}<0\) (the negative sign is used to be in accordance with the expression in our simulation software, COMSOL Multiphysics(r) 5.3 (Burlington, MA, USA)). As most dielectric materials show very weak dispersion and absorption property in the optical regime, if all materials are dielectric, the evaluation of Equation (3) is straightforward, as all of the terms are positive. If some materials are metallic, but the mode is confined primarily to a dielectric region, Equation (3) remains useful because the denominator remains positive and the small amount of energy contribution from the metallic region introduces only negligible errors in the calculation. However, for cavities supporting surface plasmon polariton (SPP) modes or localized surface plasmon resonance (LSPR) modes where the electric field is strongly confined at the metal-dielectric interface, the amount of energy stored and dissipated by metal is no longer negligible. Equation (3) becomes invalid because the resulting negative energy has no physical meaning. To circumvent this problem, the real part of the permittivity in Equation (3) can be replaced by an average permittivity, defined as [12]. As \(\omega\gg\gamma\) (damping constant) in most of the optical applications, both \(\varepsilon_{\mathrm{g}}\) and \(\left(\varepsilon_{\mathrm{g}}(\omega)+\varepsilon_{R}(\omega)\right)/2\) are positive. \[\varepsilon_{\mathrm{avg}}\ (\mathbf{r},\omega)=\frac{\left(\varepsilon_{ \mathrm{g}}(\mathbf{r},\omega)+\varepsilon_{R}(\mathbf{r},\omega)\right)}{2} \tag{4}\]  
where \(\omega\) is the angular frequency, related to the vacuum wavelength via the vacuum speed of light, \(c\), by \(\omega=2\pi c/\lambda_{0}\), and \(\varepsilon_{\mathrm{g}}\) is the group permittivity.