## 2 Dyadic Green's Function for the Electric Field  
### Free-Space Dyadic Green's Function  
The goal of this section is to introduce the necessary tools to study the polarizability of a nanoparticle in the presence of substrate. In particular, we will introduce the electric field dyadic Green's function, which allows us to solve the wave equation for the electric field in the presence of source currents. Although the material in this section is relatively well-known, we present it here in some detail both for the sake of completeness and to fix notations used throughout the paper. In frequency space, the dyadic Green's function, \(\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\), allows us to express the total electric field, \(\mathbf{E}(\mathbf{r},\omega)\), in the presence of a source current, \(\mathbf{j}_{f}(\mathbf{r},\omega)\), as  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+i\omega\mu_{ n}\mu_{0}\int d^{3}\mathbf{r}^{\prime}\overleftrightarrow{G_{0}}(\mathbf{r}, \mathbf{r}^{\prime},\omega)\cdot\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega), \tag{1}\]  
where \(\mathbf{E}_{0}(\mathbf{r},\omega)\) is the electric field in the absence of the source current, \(\mu_{0}\) is vacuum's permeability, and \(\mu_{n}\) is the relative permeability of the medium where the source current is embedded in. If the source current is due to a point dipole located at \(\mathbf{r}=\mathbf{r}_{0}\), we have that \(\mathbf{j}_{f}(\mathbf{r},\omega)=-i\omega\mathbf{d}_{0}\delta\left(\mathbf{r }-\mathbf{r}_{0}\right)\), where \(\mathbf{d}_{0}\) is the electric dipole moment. In this case, Equation (1) reduces to (for \(\mathbf{r}\
eq\mathbf{r}_{0}\))  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]  
In order to determine the dyadic Green's function we will follow a method originally described in [24]. The first step to determining \(\overleftrightarrow{G_{0}}\) in this approach is noticing that in the presence of a source current, the wave equation for the electric field can be written as an inhomogeneous Helmholtz equation (see Appendix A for a derivation) as follows:  
\[-\
abla^{2}\mathbf{E}(\mathbf{r},\omega)-k_{n}^{2}\mathbf{E}(\mathbf{r},\omega )=i\omega\mu_{n}\mu_{0}\left[\mathbf{j}_{f}(\mathbf{r},\omega)+\frac{1}{k_{n} ^{2}}\
abla\left(\
abla\cdot\mathbf{j}_{f}(\mathbf{r},\omega)\right)\right], \tag{3}\]  
where we have introduce the quantity \(k_{n}=\omega/v_{n}\), where \(v_{n}=1/\sqrt{\epsilon_{0}\epsilon_{n}\mu_{0}\mu_{n}}\) is the speed of light in the medium, \(\epsilon_{0}\) is the vacuum's permittivity, and \(\epsilon_{n}\) is the medium relative dielectric constant. The general solution of the Helmholtz equation can be written as (see Appendix B)  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+i\omega\mu_{ n}\mu_{0}\int_{V_{g}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{0}\left( \mathbf{r},\mathbf{r}^{\prime},\omega\right)\left[\overleftrightarrow{I}+ \frac{1}{k_{n}^{2}}\
abla^{\prime}\
abla^{\prime}\right]\mathbf{j}_{f}( \mathbf{r}^{\prime},\omega), \tag{4}\]  
where  
\[g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)=\frac{e^{ik_{n}| \mathbf{r}-\mathbf{r}^{\prime}|}}{4\pi\left|\mathbf{r}-\mathbf{r}^{\prime} \right|}, \tag{5}\]  
is the Green's function for the scalar Helmholtz equation [1; 24; 25] and \(\int_{V_{g}(\mathbf{r})}\) represents integration in the principal value sense, where an infinitesimal volume, \(V_{\delta}(\mathbf{r})\), enclosing the point \(\mathbf{r}^{\prime}=\mathbf{r}\) is excluded. We have written \(\
abla^{\prime}\
abla^{\prime}\equiv\
abla^{\prime}\otimes\
abla^{\prime}\) with \(\otimes\) denoting the tensor product and the prime indicates that the derivative acts on \(\mathbf{r}^{\prime}\) variables.