## 4 Extension of the Formalism When the Quantum Emitter Has Both an Electric and a Magnetic Dipole
### Weyl's or Angular Spectrum Representation of Magnetic and Mixed Green's Functions  
Now we will see what is the Weyl's (or angular spectrum) representation of the magnetic and mixed Green's functions. The magnetic Green's function is almost the same as the electric Green's function, the only difference being the different additional \(\overleftrightarrow{I}\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)/k_{ n}^{2}\) self-field term, which is isotropic and independent of the chosen excluded volume. Therefore, we can write  
\[\overleftrightarrow{G}_{0}^{MM}\left(\mathbf{p}_{\parallel},z,z^{\prime}, \omega\right)=\hat{e}_{s}\hat{e}_{s}\frac{i}{2\beta_{n}}e^{i\beta_{n}|z-z^{ \prime}|}+\hat{e}_{p,n}^{\pm}\hat{e}_{p,n}^{\pm}\frac{i}{2\beta_{n}}e^{i\beta _{n}|z-z^{\prime}|}+\frac{1}{k_{n}^{2}}\left(\overleftrightarrow{I}-\hat{e}_ {z}\hat{e}_{z}\right)\delta\left(z-z^{\prime}\right). \tag{94}\]  
We point out that the demagnetization term \(\left(\overleftrightarrow{I}-\hat{e}_{z}\hat{e}_{z}\right)\) was previously obtained in [38]. As for the mixed Green's function, their Weyl's representation can be obtained by making the replacements: \(g_{0}\left(\mathbf{r},\mathbf{r}^{\prime},\omega\right)\to g_{0}\left( \mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\), \(\left(\partial_{x},\partial_{y}\right)\to i\mathbf{p}_{\parallel}\) and \(\partial_{z}\to\pm i\beta_{n}\) for \(z\gtrless z^{\prime}\). Therefore, we obtain  
\[\overleftrightarrow{G}_{0}^{EM}\left(\mathbf{p}_{\parallel},z,z^{\prime}, \omega\right)=\overleftrightarrow{G}_{0}^{ME}\left(\mathbf{p}_{\parallel},z,z ^{\prime},\omega\right)=\frac{1}{k_{n}}\left[ \begin{array}{ccc}0&\sigma\beta_{ n}&-p_{y}\\ -\sigma\beta_{n}&0&p_{x}\\ p_{y}&-p_{x}&0\end{array} \right]\frac{i}{2\beta_{n}}e^{i\beta_{n}|z-z^{\prime}|}, \tag{95}\]  
where \(\sigma=\pm 1\) for \(z\gtrless z^{\prime}\). As for the electric and the magnetic Green's functions, the mixed Green's functions in the Weyl representation can also be written in terms of the \(s\)- and \(p\)-polarization vectors. It is straightforward to verify that  
\[\frac{1}{k_{n}}\left[ \begin{array}{ccc}0&\sigma\beta_{n}&-p_{y}\\ -\sigma\beta_{n}&0&p_{x}\\ p_{y}&-p_{x}&0\end{array} \right]=\hat{e}_{p,n}^{\pm}\hat{e}_{s}-\hat{e}_{s} \hat{e}_{p,n}^{\pm}, \tag{96}\]  
which allows us to write  
\[\overleftrightarrow{G}_{0}^{EM}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega \right)=\overleftrightarrow{G}_{0}^{ME}\left(\mathbf{p}_{\parallel},z,z^{ \prime},\omega\right)=\left[\hat{e}_{p,n}^{\pm}\hat{e}_{s}-\hat{e}_{s}\hat{e}_{p,n}^{\pm}\right]\frac{i}{2\beta_{n}}e^{i\beta_{n}|z-z^{\prime}|}. \tag{97}\]  
This representation is useful, as it allows for a simple interpretation of the emitted fields generated by the electric and magnetic dipoles in terms of \(s\)- and \(p\)-polarized electromagnetic waves. If we are interested in the problem of scattering at a planar interface between two dielectric media with \(e_{1}\) for \(z>0\), and \(e_{2}\) for \(z<0\), we can construct reflected and transmitted Green's functions expressed in terms of reflection and transmission coefficients, as done previously for \(\overleftrightarrow{G_{0}}^{EE}\left(\mathbf{p}_{\parallel},z,z^{\prime}, \omega\right)\). However, some care must be taken in what the polarization vectors mean in the Green's function, considering that the polarization of an electromagnetic field is usually defined by the polarization of the \(\mathbf{E}\) field. The quantity \(\overleftrightarrow{G_{0}}^{EM}\left(\mathbf{p}_{\parallel},z,z^{\prime}, \omega\right)\) gives us the electric field generated by a point magnetic dipole located at \(z^{\prime}\).