## 2 Dyadic Green's Function for the Electric Field
### Free-Space Dyadic Green's Function
This allows us to write  
\[\int_{\backslash V_{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{0}\left( \mathbf{r},\mathbf{r}^{\prime},\omega\right)\
abla^{\prime}\left(\
abla^{ \prime}\cdot\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega)\right)=\int_{\backslash V _{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}\
abla^{\prime}\
abla^{\prime}g_ {0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)j_{f}^{k}(\mathbf{r}^{ \prime},\omega)-\frac{1}{k_{n}^{2}}\overleftrightarrow{L}_{V_{\delta}}\cdot \mathbf{j}_{f}(\mathbf{r},\omega)\]  
where the dyadic \(\overleftrightarrow{L}_{V_{\delta}}\) is defined as [24]  
\[\overleftrightarrow{L}_{V_{\delta}}=\int_{\partial V_{\delta}(\mathbf{r})} \frac{d^{2}\mathbf{r}^{\prime}}{4\pi}\frac{(\mathbf{r}^{\prime}-\mathbf{r}) \otimes\mathbf{n}^{\prime}}{\left|\mathbf{r}^{\prime}-\mathbf{r}\right|^{3}}, \tag{9}\]  
which can be interpreted as a depolarization term. Therefore, Equation (4) can be written in the form of Equation (1) with the dyadic Green's function, \(\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\), given by  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]0  
where \(\mathrm{P.V. }_{\cdot V_{\delta}}\) indicates that the small volume \(V_{\delta}\) centered at \(\mathbf{r}^{\prime}=\mathbf{r}\) is to be excluded. Notice that \(\overleftrightarrow{L}_{V_{\delta}}\) depends on the shape of the chosen excluded volume [24]. For a sphere, it is straightforward to show that \(\overleftrightarrow{L}_{\mathrm{Sphere}_{\delta}}=\overleftrightarrow{I}/3\). In this case, the free-space dyadic Green's function in real space can be written as the sum of four terms [26; 27]:  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]1respectively, the far-, intermediate-, near- and self-field terms, which are written as  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]2 \[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]3 (13) \[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]4 (14) \[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]5  
where the terms \(\overleftarrow{G_{0}}^{\text{FF}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\), \(\overleftarrow{G_{0}}^{\text{IF}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\) and \(\overleftarrow{G_{0}}^{\text{NF}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\) are to be understood in the principal value sense, and we have introduced the definitions \(\hat{\mathbf{R}}=(\mathbf{r}-\mathbf{r}^{\prime})\) / \(|\mathbf{r}-\mathbf{r}^{\prime}|\) and \(\hat{\mathbf{R}}\hat{\mathbf{R}}=\hat{\mathbf{R}}\otimes\hat{\mathbf{R}}\). We point out that in more standard derivations of the dyadic Green's function it is easy to miss the depolarization term Equation (9) [1; 28], or fail to recognize its dependence on the shape of the excluded volume [29; 30]. It will be shown in Section 3 that this term is crucial to correctly capture self-field contributions to a particle polarizability. The dependence of depolarization term on the shape of the volume is also important when dealing with non-spherical particles. An alternative derivation of the depolarization term Equation (9) based on the vector potential can be found in [31].