## 2 Dyadic Green's Function for the Electric Field
### Polarizability of a Quantum Emitter in a Homogeneous Medium  
The polarizability of a nanoparticle, \(\overleftrightarrow{\alpha}\), treated as a point object, relates the electric dipole moment, \(\mathbf{d}\), that is induced in the nanoparticle to the value of the externally applied electric field, \(\mathbf{E}_{\mathrm{ext}}\left(\tau_{0}\right)\), at the nanoparticle's position, \(\tau_{0}\), via  
\[\mathbf{d}=\overleftrightarrow{\alpha}\cdot\mathbf{E}_{\mathrm{ext}}\left( \tau_{0}\right). \tag{37}\]  
Note that \(\mathbf{E}_{\mathrm{ext}}\left(\tau_{0}\right)\) does not include self-field effects, that is, the electric field generated by the nanoparticle itself when subjected to \(\mathbf{E}_{\mathrm{ext}}\left(\tau_{0}\right)\). Let us consider a homogeneous medium characterized by \(\epsilon_{1}\) and \(\mu_{1}\), in which a nanoparticle with dielectric function \(\epsilon_{\mathrm{np}}(\omega)\) lives. Then, the electric field obeys Equation (1) with the free current due to the nanoparticle polarization (excluding the current due to the polarization density of the homogeneous medium) being written as  
\[\mathbf{j}_{f}(\mathbf{r},\omega)=-i\omega\left[\mathbf{P}_{\mathrm{np}}( \mathbf{r},\omega)-\mathbf{P}_{1}(\omega)\right]=-i\omega\epsilon_{0}\left( \epsilon_{\mathrm{np}}(\omega)-\epsilon_{1}\right)\mathbf{E}(\mathbf{r}, \omega), \tag{38}\]  
where we have used the usual linear constitutive relation \(\mathbf{P}_{n}(\omega,\mathbf{r})=\epsilon_{0}\left(\epsilon_{n}-1\right) \mathbf{E}(\mathbf{r},\omega)\). \(\mathbf{P}_{\mathrm{np}}(\mathbf{r},\omega)\) is the polarization density due to the nanoparticle, \(\mathbf{P}_{1}(\omega)\) is the polarization density due to the homogeneous  
Figure 2: Representation of the primary field (\(\mathbf{E}_{0}^{\pm}\)), emitted by a point dipole (represented by the golden ball with an arrow), and the reflected (\(\mathbf{E}_{r}\)) and the transmitted (\(\mathbf{E}_{t}\)) field due to the presence of the interface at \(z=0\). The \(\pm\) sign in \(\mathbf{E}_{0}^{\pm}\) indicates whether the field is emitted along the positive/negative \(z\) direction. medium, and \(\mathbf{E}(\mathbf{r})\) is the total electric field in the nanoparticle. Therefore, from Equation (1), the electric field obeys a Lippmann-Schwinger equation [44]  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{\mathrm{ext}}(\mathbf{r},\omega)+ \omega^{2}\mu_{1}\mu_{0}\int_{V}d\mathbf{r}^{\prime}\left(\epsilon_{\mathrm{np} }(\omega)-\epsilon_{1}\right)\overset{\longleftrightarrow}{C_{0}}(\mathbf{ r},\mathbf{r}^{\prime},\omega)\cdot\mathbf{E}(\mathbf{r},\omega) \tag{39}\]  
where \(\mathbf{E}_{\mathrm{ext}}(\mathbf{r},\omega)\) is a solution of the wave equation in the homogeneous medium, and \(V\) is the volume of the nanoparticle. We want to solve for the electric field inside the nanoparticle. We will follow the approximate approach of [22]. We shall assume a spherical nanoparticle, with radius \(R\), and assume that \(k_{n}R\ll 1\). This allows us to approximate \(\mathbf{E}(\mathbf{r},\omega)\) as being constant inside the nanoparticle and to take the limit \(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\to 0\) for \(\overset{\longleftrightarrow}{C_{0}}(\mathbf{r}-\mathbf{r}^{\prime},\omega)\). Taking into account Equations (12)-(14), in the limit \(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\to 0\) we can write the regular part (excluding the Dirac \(\delta\)-function term) of the free-space dyadic [PERSON]'s function as  
\[\overset{\longleftrightarrow}{C_{0}}^{\mathrm{reg}}\left(\mathbf{r},\mathbf{ r}^{\prime},\omega\right)\simeq\frac{1}{6\pi}\frac{1}{\left|\mathbf{R}\right|} \overset{\longleftrightarrow}{I}+i\frac{k_{1}}{6\pi}\overset{ \longleftrightarrow}{I}, \tag{40}\]  
where \(\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}\).