## 4 Extension of the Formalism When the Quantum Emitter Has Both an Electric and a Magnetic Dipole
### Free-Space Electric, Magnetic and Mixed Green's Functions  
Our starting points are the inhomogeneous Helmholtz equations for the electric and the magnetic fields (in fact the magnetic field induction \(\mathbf{B}(\mathbf{r},\omega)\)) in the presence of a source current density (see Appendix A for the derivation):  
\[-\
abla^{2}\mathbf{E}(\mathbf{r},\omega)-k_{n}^{2}\mathbf{E}( \mathbf{r},\omega) = i\omega\mu_{n}\mu_{0}\left[\mathbf{j}_{f}(\mathbf{r},\omega)+ \frac{1}{k_{n}^{2}}\
abla\left(\
abla\cdot\mathbf{j}_{f}(\mathbf{r},\omega) \right)\right] \tag{76}\] \[-\
abla^{2}\mathbf{B}(\mathbf{r},\omega)-k_{n}^{2}\mathbf{B}( \mathbf{r},\omega) = \mu_{n}\mu_{0}\
abla\times\mathbf{j}_{f}(\mathbf{r},\omega). \tag{77}\]  
As before, we can write the solution for the inhomogeneous Helmholtz equations as  
\[\mathbf{E}(\mathbf{r},\omega) = \mathbf{E}_{0}(\mathbf{r},\omega)+i\omega\mu_{n}\mu_{0}\int_{V_{ \delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r}^ {\prime},\omega\right)\left(\overleftrightarrow{I}+\frac{1}{k_{n}^{2}}\
abla ^{\prime}\
abla^{\prime}\right)\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega) \tag{78}\] \[\mathbf{B}(\mathbf{r},\omega) = \mathbf{B}_{0}(\mathbf{r},\omega)+\mu_{n}\mu_{0}\int_{V_{\delta} (\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime },\omega\right)\
abla^{\prime}\times\mathbf{j}_{f}(\mathbf{r}^{\prime},\omega). \tag{79}\]  
In the same spirit of Equation (38), we write the current in terms of polarization, \(\mathbf{P}_{f}\), and magnetization, \(\mathbf{M}_{f}\), densities as  
\[\mathbf{j}_{t}(\mathbf{r},\omega)=-i\omega\mathbf{P}_{f}(\mathbf{r},\omega)+ \
abla\times\mathbf{M}_{f}(\mathbf{r},\omega). \tag{80}\]  
Inserting the latter result into Equations (78) and (79) we obtain  
\[\mathbf{E}(\mathbf{r},\omega) = \mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu_{n}\mu_{0}\int_{V_ {\delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r} ^{\prime},\omega\right)\left(\overleftrightarrow{I}+\frac{1}{k_{n}^{2}} \
abla^{\prime}\
abla^{\prime}\right)\mathbf{P}_{f}(\mathbf{r}^{\prime},\omega) \tag{81}\] \[+ i\omega\mu_{n}\mu_{0}\int_{V_{\delta}(\mathbf{r})}d^{3}\mathbf{ r}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\
abla^{ \prime}\times\mathbf{M}_{f}(\mathbf{r}^{\prime},\omega),\] \[\mathbf{B}(\mathbf{r},\omega) = \mathbf{B}_{0}(\mathbf{r},\omega)-i\omega\mu_{n}\mu_{0}\int_{V_{ \delta}(\mathbf{r})}d^{3}\mathbf{r}^{\prime}g_{0}\left(\mathbf{r},\mathbf{r} ^{\prime},\omega\right)\
abla\times\mathbf{P}_{f}(\mathbf{r}^{\prime},\omega)\] (82) \[+ \mu_{n}\mu_{0}\int_{V_{\delta}(\mathbf{r})}d^{3}\mathbf{r}^{ \prime}g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\left(-\
abla^{ \prime 2}+\
abla^{\prime}\
abla^{\prime}\right)\mathbf{M}_{f}(\mathbf{r}^{\prime },\omega),\]  
where we have used the fact that \(\
abla^{\prime}\cdot\left(\
abla^{\prime}\times\mathbf{M}_{f}(\mathbf{r}^{ \prime},\omega)\right)=0\) and \(\
abla^{\prime}\times\
abla^{\prime}\times\mathbf{M}_{f}(\mathbf{r}^{\prime}, \omega)=\
abla^{\prime}\left(\
abla^{\prime}\cdot\mathbf{M}_{f}(\mathbf{r}^{ \prime},\omega)\right)-\
abla^{\prime 2}\mathbf{M}_{f}(\mathbf{r}^{\prime},\omega)\). We now proceed as in Section 2.1, using integration by parts, while taking into account the boundary terms due to the excluded volume \(V_{\delta}\) enclosing the point \(\mathbf{r}^{\prime}=\mathbf{r}\), in the same form that we have already dealt with for the electric field Green's function. The crossed terms relating the magnetization to the electric field and the polarization to the magnetic field only involve one derivative of the Helmholtz Green's function and therefore the generated boundary term vanishes in the limit of infinitesimal excluded volume.