## 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
Therefore, the reflected and transmitted Green's functions are constructed in the same way as for \(\overleftrightarrow{G_{0}}^{EE}\left(\mathbf{p}_{\parallel},z,z_{0},\omega\right)\), and for \(z_{0}>0\) we obtain  
\[\overleftrightarrow{G_{r}}^{EM}\left(\mathbf{p}_{\parallel},z,z_{0},\omega \right) = r_{p}\frac{i}{2\beta_{1}}\delta_{p,1}^{+}\delta_{s}e^{i\beta_{ 1}\left(z+z_{0}\right)}-r_{s}\frac{i}{2\beta_{1}}\delta_{s}\delta_{p,1}^{-}e ^{i\beta_{1}\left(z+z_{0}\right)}, \tag{98}\] \[\overleftrightarrow{G_{r}}^{EM}\left(\mathbf{p}_{\parallel},z,z_{0 },\omega\right) = t_{p}\frac{i}{2\beta_{1}}\delta_{p,2}^{-}\delta_{s}e^{-i\beta_{ 2}z}e^{i\beta_{1}z_{0}}-t_{s}\frac{i}{2\beta_{1}}\delta_{s}\delta_{p,1}^{-}e^{ -i\beta_{2}z}e^{i\beta_{1}z_{0}}. \tag{99}\]  
For the magnetic Green's functions, \(\overleftrightarrow{G}^{MM}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\) and \(\overleftrightarrow{G}^{ME}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\), we must take into account that these describe a field \(\mathbf{B}\) generated by, respectively, a point magnetic and electric dipole. For electric and magnetic dipoles, \(\mathbf{d}_{0}\) and \(\mathbf{m}_{0}\), located at \(z_{0}\), the primary magnetic field emitted for \(z_{0}>z>0\) is given by  
\[\mathbf{B}_{0}\left(\mathbf{p}_{\parallel},z,\omega\right) = \mu_{1}\mu_{0}k_{n}^{2}\overleftrightarrow{G}_{0}^{MM}\left( \mathbf{p}_{\parallel}z,z_{0},\omega\right)\cdot\mathbf{m}_{0}-\omega\mu_{1} \mu_{0}k_{n}\overleftrightarrow{G}_{0}^{ME}\left(\mathbf{p}_{\parallel}z,z_{0 },\omega\right)\cdot\mathbf{d}_{0} \tag{100}\] \[= B_{0,s}e^{i\beta_{1}\left|z-z_{0}\right|}\delta_{s}+B_{0,p}e^{i \beta_{1}\left|z-z_{0}\right|}\delta_{p,1}^{-},\]  
with  
\[B_{0,s} = \mu_{n}\mu_{0}k_{1}^{2}\frac{i}{2\beta_{1}}\left(\hat{e}_{s} \cdot\mathbf{m}_{0}\right)+\omega\mu_{1}\mu_{0}k_{1}\frac{i}{2\beta_{n}}\left( \hat{e}_{p,1}^{-}\cdot\mathbf{d}_{0}\right), \tag{101}\] \[B_{0,p} = \mu_{n}\mu_{0}k_{1}^{2}\frac{i}{2\beta_{1}}\left(\hat{e}_{p,1}^{- }\cdot\mathbf{m}_{0}\right)-\omega\mu_{1}\mu_{0}k_{1}\frac{i}{2\beta_{n}}\left( \hat{e}_{s}\cdot\mathbf{d}_{0}\right). \tag{102}\]  
The corresponding electric field can be obtained from [PERSON]'s equations as \(\mathbf{E}_{0}(\mathbf{p}_{\parallel},z,\omega)=-\omega^{-1}\mathbf{p}_{n}^{\pm} \times\mathbf{B}_{0}(\mathbf{p}_{\parallel},z,\omega)\). More explicitly (for \(z>0\)) we have for the primary field  
\[\mathbf{E}_{0}\left(\mathbf{p}_{\parallel},z,\omega\right)=v_{1}B_{0,s}e^{i \beta_{1}\left|z-z_{0}\right|}\delta_{p,1}^{-}-v_{1}B_{0,p}e^{i\beta_{1}\left| z-z_{0}\right|}\delta_{s}. \tag{103}\]  
This primary electric field is scattered by the interface at \(z=0\), giving origin to a reflected field for \(z>0\), which reads  
\[\mathbf{E}_{r}\left(\mathbf{p}_{\parallel},z>0,\omega\right)=r_{p}v_{1}B_{0,s}e ^{i\beta_{1}\left(z+z_{0}\right)}\delta_{p,1}^{+}-r_{s}v_{1}B_{0,p}e^{i\beta_{ 1}\left(z+z_{0}\right)}\delta_{s}, \tag{104}\]  
and to a transmitted field for \(z<0\)  
\[\mathbf{E}_{t}\left(\mathbf{p}_{\parallel},z<0,\omega\right)=t_{p}v_{1}B_{0,s}e ^{-i\beta_{2}z}e^{i\beta_{1}z_{0}}\delta_{p,2}^{-}-t_{s}v_{1}B_{0,p}e^{-i\beta_ {2}z}e^{i\beta_{1}z_{0}}\delta_{s}. \tag{105}\]  
The corresponding magnetic fields can be obtaining using [PERSON]'s law applied to Equations (104) and (105). For example: if \(\mathbf{E}=E_{0}\hat{e}_{p,n}\) then \(i\omega\mathbf{B}=\
abla\times\mathbf{E}=i\mathbf{p}_{n}^{-}\times\hat{e}_{p,n} ^{-}E_{0}=ik_{n}\hat{e}_{s}E_{0}\), where \(k_{n}=\left|p_{n}^{-}\right|\) and \(E_{0}\) is the amplitude of the \(s\)-component of the field.