## 2 Dyadic Green's Function for the Electric Field
### Polarizability of a Quantum Emitter in Proximity to a Planar Interface  
If the nanoparticle is situated in the vicinity of an interface, it is also possible to write an equation of the Lippmann-Schwinger type for the electric field similar to Equation (39). The only difference is that in order to take into account the interface, the free-space dyadic Green's function must be replaced by the one which incorporates the reflection from the substrate, \(\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\to \overleftrightarrow{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)= \overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}^{\prime},\omega)+ \overleftrightarrow{G_{r}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\). Likewise, the external field \(\mathbf{E}_{0}(\mathbf{r},\omega)\) must be replaced by a solution of the electric field wave equation in the presence of the substrate, \(\mathbf{E}_{0}(\mathbf{r},\omega)\to\mathbf{E}_{\text{ext}}(\mathbf{r},\omega) =\mathbf{E}_{0}(\mathbf{r},\omega)+\mathbf{E}_{r}(\mathbf{r},\omega)\), where \(\mathbf{E}_{0}(\mathbf{r},\omega)\) is the incident field and \(\mathbf{E}_{r}(\mathbf{r},\omega)\) is the reflected field. Therefore, the Lippmann-Schwinger equation for the electric field taking into account the substrate is given by  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{\text{ext}}(\mathbf{r},\omega)+ \omega^{2}\mu_{1}\mu_{0}\int_{V}d^{3}\mathbf{r}^{\prime}\left(\epsilon_{\text {np}}(\omega)-\epsilon_{1}\right)\overleftrightarrow{G}(\mathbf{r},\mathbf{ r}^{\prime},\omega)\cdot\mathbf{E}(\mathbf{r},\omega). \tag{47}\]  
We now proceed in the same fashion as before, assuming \(k_{1}R\ll 1\), approximating \(\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}(\mathbf{r}_{0},\omega)\) as constant inside the nanoparticle, and keeping only the dominant contributions from \(\overleftrightarrow{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)\) in the limit \(|\mathbf{r}-\mathbf{r}^{\prime}|\to 0\). Therefore, we write [22]  
\[\overleftrightarrow{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)\simeq-\frac{1}{ 3k_{1}^{2}}\overleftrightarrow{I}\delta(\mathbf{r}-\mathbf{r}^{\prime})+i \frac{k_{1}}{6\pi}\overleftrightarrow{I}+\overleftrightarrow{G}_{r}( \mathbf{r}_{0},\mathbf{r}_{0},\omega), \tag{48}\]  
where we have used the fact that \(\overleftrightarrow{G_{r}}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)\) is regular. Introducing Equation (48) into Equation (47), and approximating \(\mathbf{E}(\mathbf{r},\omega)\simeq\mathbf{E}(\mathbf{r}_{0},\omega)\), we can solve for \(\mathbf{E}(\mathbf{r}_{0},\omega)\) as a function of \(\mathbf{E}_{\text{ext}}(\mathbf{r}_{0},\omega)\), obtaining  
\[\mathbf{E}(\mathbf{r}_{0},\omega)=\frac{3\epsilon_{1}}{\epsilon_{\text{np}}( \omega)+2\epsilon_{1}}\left[\overleftrightarrow{I}-\alpha_{\text{CM}}\omega^{ 2}\mu_{1}\mu_{0}\left(\frac{ik_{1}}{6\pi}\overleftrightarrow{I}+ \overleftrightarrow{G}_{r}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)\right) \right]^{-1}\cdot\mathbf{E}_{\text{ext}}(\mathbf{r}_{0},\omega). \tag{49}\]  
The electric dipole moment is thus given by  
\[\mathbf{d}=V\epsilon_{0}\left(\epsilon_{\text{np}}(\omega)-\epsilon_{1}\right) \mathbf{E}(\mathbf{r}_{0},\omega)=\overleftrightarrow{\alpha}_{\text{eff}} \cdot\mathbf{E}_{\text{ext}}(\mathbf{r}_{0},\omega), \tag{50}\]  
with the effective polarizability defined as  
\[\overleftrightarrow{\alpha}_{\text{eff}}=\alpha_{\text{CM}}\left[ \overleftrightarrow{I}-\mu_{1}\mu_{0}\omega^{2}a_{\text{CM}}\left(iS \overleftrightarrow{G}_{0}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)+ \overleftrightarrow{G}_{r}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)\right)^{-1} \right]^{-1}. \tag{51}\]  
This equation can be expressed in terms of the free-space polarizability Equation (45) as  
\[\overleftrightarrow{\alpha}_{\text{eff}}=\overleftrightarrow{\alpha}_{0}\left[ \overleftrightarrow{I}-\mu_{1}\mu_{0}\omega^{2}\overleftrightarrow{G}_{r}( \mathbf{r}_{0},\mathbf{r}_{0},\omega)\cdot\overleftrightarrow{\alpha}_{0} \right]^{-1}. \tag{52}\]  
Equation (51) has been derived in the literature before, following a similar argumentation [6; 21; 22].