## 2 Dyadic Green's Function for the Electric Field
### Free-Space Dyadic Green's Function
Doing this yields  
\[\mathbf{E}(\mathbf{p}_{\parallel},z,\omega)=\mathbf{E}_{0}(\mathbf{p}_{ \parallel},z,\omega)+i\omega\mu_{n}\mu_{0}\int dz^{\prime}\overleftrightarrow{ G_{0}}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\mathbf{j}_{f}( \mathbf{p}_{\parallel},z^{\prime},\omega), \tag{22}\]  
with \(\overleftrightarrow{G_{0}}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\) being the dyadic Green's function in [PERSON]'s representation  
\[\overleftrightarrow{G_{0}}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega \right)=\text{P.V.}\left[\overleftrightarrow{I}-\frac{1}{k_{n}^{2}}\mathbf{ p}_{n}^{\pm}\mathbf{p}_{n}^{\pm}\right]\frac{i}{2\beta_{n}}e^{i\beta_{n}|z-z^{ \prime}|}-\frac{1}{k_{n}^{2}}\hat{e}_{z}\hat{e}_{z}\hat{\delta}\left(z-z^{ \prime}\right), \tag{23}\]  
where we have introduced \(\mathbf{p}_{n}^{\pm}=\mathbf{p}_{\parallel}\pm\beta_{n}\hat{e}_{z}\), with the \(\pm\) sign applying for \(z\gtrless z^{\prime}\). The last term in the above equation is the depolarization term, that arises from the boundary contributions when performing integration by parts, due to the exclusion of an infinitesimal line element around \(z^{\prime}=z\) in \(f\). The principal value in the first term indicates that a small region around \(z^{\prime}=z\) is to be excluded. We also notice that this depolarization term could also have been obtained from the general depolarization dyadic in real space, Equation (9), if we choose as excluded volume an infinite slab located at \(-\delta<z<\delta\) (with \(\delta\to 0\)). For this excluded volume, we would obtain \(\overleftrightarrow{L}_{\text{Slab}_{i}}=\hat{e}_{z}\hat{e}_{z}\). It is possible to write Equation (23) in a more meaningful way by introducing the \(s\)- and \(p\)-polarization vectors. The \(s\)-polarization vector lies in the \(xy-\)plane and is therefore written as [33]  
\[\hat{e}_{s}=\frac{\mathbf{p}_{\parallel}\times\hat{e}_{z}}{p_{\parallel}}. \tag{24}\]  
On the other hand, the \(p\)-polarization vector is orthogonal to \(\mathbf{p}_{n}^{\pm}\) and \(\hat{e}_{s}\), and therefore we write it as [33]  
\[\hat{e}_{p,n}^{\pm}=\frac{\hat{e}_{s}\times\mathbf{p}_{n}^{\pm}}{k_{n}}=\frac{ p_{\parallel}}{k_{n}}\hat{e}_{z}+\frac{\beta_{n}}{k_{n}}\frac{\mathbf{p}_{ \parallel}}{p_{\parallel}}, \tag{25}\]  
where \(\hat{e}_{p,n}^{\pm}\) is the \(p\)-polarization vector for a field propagating in the positive/negative \(z-\)direction. With these definitions we have that \(\overleftrightarrow{I}-\mathbf{p}_{n}^{\pm}\mathbf{p}_{n}^{\pm}/k_{n}^{2}= \hat{e}_{s}\hat{e}_{s}+\hat{e}_{p,n}^{\pm}\hat{e}_{p,n}^{\pm}\) and, therefore, we can write Equation (23) as [34; 35]  
\[\overleftrightarrow{G_{0}}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega \right)=\hat{e}_{s}\hat{e}_{s}\frac{i}{2\beta_{n}}e^{i\beta_{n}|z-z^{\prime}| }+\hat{e}_{p,n}^{\pm}\hat{e}_{p,n}^{\pm}\frac{i}{2\beta_{n}}e^{i\beta_{n}|z-z^ {\prime}|}-\frac{1}{k_{n}^{2}}\hat{e}_{z}\hat{e}_{z}\hat{\delta}\left(z-z^{ \prime}\right), \tag{26}\]  
with the first and the second terms corresponding to the \(s\)- and \(p\)-polarization components of the free-space dyadic Green's function, respectively. A different derivation of previous two equations has been given in the literature previously [36; 37; 38]. The same decomposition has been used in the study of an emitter's life-time near a graphene sheet [39; 40] and in the context of the calculation of the electric field of a dipole near graphene [41].