## 2 Dyadic Green's Function for the Electric Field
### Free-Space Dyadic Green's Function
The Helmholtz Green's function is the solution of  
\[\left[-\
abla^{2}-k_{n}^{2}\right]g_{0}\left(\mathbf{r},\mathbf{r}^{\prime}, \omega\right)=\delta(\mathbf{r}-\mathbf{r}^{\prime}) \tag{6}\]  
in a way that is clarified in Appendix B. Notice that \(g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\) is integrable, and therefore, the exclusion of the volume \(V_{g}(\mathbf{r})\) is not usually emphasized. However, it will be important when determining the behaviour of \(\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\) when \(\mathbf{r}=\mathbf{r}^{\prime}\). Although Equation (4) already allows us to compute the electric field as a function of the current, it is useful to obtain an alternative expression which does not involve derivatives of the current. Such an expression can be obtained by carefully performing integration byparts. It must the noticed that due to the excluded volume surrounding \(\mathbf{r}^{\prime}=\mathbf{r}\), boundary terms are generated when doing this. In particular we have that  
\[\int_{\backslash V_{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{ 0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\
abla^{\prime}\left( \
abla^{\prime}\cdot\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega)\right)=-\int_{ \partial V_{\delta}(\mathbf{r})}d^{2}\mathbf{r}^{\prime}\mathbf{n}^{\prime} \left[g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\left(\
abla^{ \prime}\cdot\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega)\right)\right]\\ -\int_{\backslash V_{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime }\
abla^{\prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\left( \
abla^{\prime}\cdot\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega)\right), \tag{7}\]  
where \(\mathbf{n}^{\prime}\) is a outward-pointing unit vector, normal to the surface \(\partial V_{\delta}(\mathbf{r})\) of the enclosing volume \(V_{\delta}(\mathbf{r})\). In the limit of infinitesimal excluded volume, the first term of the above equation vanishes, since the element of area scales as \(d^{2}\mathbf{r}^{\prime}\sim\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{2}\), while \(g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\sim 1/\left|\mathbf{r}- \mathbf{r}^{\prime}\right|\). For the second term, we perform integration by parts once again. For clarity we explicitly write the tensorial components in a Cartesian basis, with repeated induces being summed over, and obtain  
\[-\int_{\backslash V_{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime }\partial_{i}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right) \partial_{k}^{\prime}j_{f}^{k}(\mathbf{r}^{\prime},\omega)=\int_{\partial V_{ \delta}(\mathbf{r})}d^{2}\mathbf{r}^{\prime}n_{k}^{\prime}\left[\partial_{i}^{ \prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)j_{f}^{k}( \mathbf{r}^{\prime},\omega)\right]\\ +\int_{\backslash V_{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime }\left[\partial_{i}^{\prime}\partial_{k}^{\prime}g_{0}\left(\mathbf{r}, \mathbf{r}^{\prime},\omega\right)j_{f}^{k}(\mathbf{r}^{\prime},\omega)\right]. \tag{8}\]  
Now the boundary term is finite. In the limit of an infinitesimal volume, we take \(\mathbf{r}^{\prime}\rightarrow\mathbf{r}\), such that \(j_{f}^{k}(\mathbf{r}^{\prime},\omega)\rightarrow j_{f}^{k}(\mathbf{r},\omega)\) and use the \(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\to 0\) limit of \(\partial_{i}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\) Equation (A26).