# Effective Modal Volume in Nanoscale Photonic and Plasmonic Near-Infrared Resonant Cavities
## 1 Introduction  
Semiconductor nanolasers serve simultaneously as a platform for studying intriguing fundamental physics and as a technologically relevant solution for next generation photonic integrated circuits [1; 2]. A major drive in the study and application of nanoscale cavities rests in the Purcell effect, which describes the inverse relation between cavity mode volume and the rate of spontaneous emission [3; 4]. Smaller cavities generally permit higher rates of spontaneous emission and a greater ratio of spontaneous emission channeled into the lasing mode, which, in principle, provides increased modulation bandwidths and lower lasing thresholds [5; 6]. In many nanoscale structures, however, the physical volume of the cavity does not reflect the volume of the cavity mode. Therefore, the effective modal volume, \(V_{\mathit{eff}}\)--a unitless quantity normalized with respect to \((\lambda_{0}/n_{a})^{3}\), was introduced to account for this discrepancy [7]. A complementary expression, the confinement factor, \(\Gamma\), may be defined to express the ratio of the volume of the active region, \(V_{a}\), to \(V_{\mathit{eff}}\), namely [8]. \[\Gamma\equiv\frac{V_{a}}{V_{eff}} \tag{1}\]In purely dielectric cavities with a large physical volume with respect to the cubic wavelength, \(\Gamma\approx 1\) because the mode is completely confined to the active region. As the active region volume approaches the diffraction limit (approximately given by \((\lambda_{0}/(2n_{a}))^{3}\), where \(\lambda_{0}\) and \(n_{a}\) are the vacuum wavelength and the real part of the active region refractive index, respectively), the cavity mode becomes less confined and \(\Gamma<1\). Eventually, when \(V_{a}/(\lambda_{0}/(2n_{a}))^{3}<<1\), modes are no longer supported in the purely dielectric cavity of which the physical size is below the diffraction limit in all three dimensions. To continue reducing cavity volume, the incorporation of metals becomes inevitable [9]. The plasmonic modes supported by metal-dielectric interfaces in such cavities exhibit considerably different modal profiles from their dielectric counterparts. Namely a substantial portion of modal energy rests at the metal-dielectric interface rather than in the center of a purely dielectric region. Calculation of \(V_{\mathit{eff}}\) in three-dimensional (3D) cavities generally requires use of numerical methods due to the geometric and material complexity of the cavities. Physically, the cavity mode is a spatial distribution of electromagnetic energy [10]. Therefore, the calculation essentially requires a comparison of integration of the electromagnetic energy density over the active region domain, \(U_{a}\), to the integration of the energy density over the entire simulation domain, \(U\). If the simulation domain consists only of dielectric materials, then this calculation is straightforward, and can be expressed as:  
\[ \begin{array}{ll}\Gamma&=\frac{\Xi_{a}}{\Xi_{a}+\Xi_{m}}\\ &=\frac{\int_{V_{a}}d^{3}\mathbf{e}_{\mathrm{n}}U_{\mathrm{e}}(\mathbf{r})+ \int_{V_{a}}d^{3}\mathbf{r}_{\mathrm{n}}U_{\mathrm{m}}(\mathbf{r})}{\int_{V}d ^{3}\mathbf{r}U_{\mathrm{e}}(\mathbf{r})+\int_{V}d^{3}\mathbf{r}U_{\mathrm{m} }(\mathbf{r})}\end{array}  \tag{2}\]  
where \(\Xi\) is the energy, \(\mathbf{r}\) is the position vector, and the e, m, and a subscripts refer to electrical energy, magnetic energy, and active region respectively. For a simulation of non-absorbing and non-dispersive materials, the stored electrical and magnetic energy are identical [11]. Hence, Equation (2) may be simplified to its more commonly seen form of:  
\[