## Appendix A Derivation of the Wave Equation  
Let us start revising the basics of electromagnetic theory writing [PERSON]'s equations for a homogeneous medium of relative dielectric permittivity \(\epsilon_{n}\) and relative permeability \(\mu_{n}\):  
\[\
abla\times\mathbf{E}(\mathbf{r},t) = -\frac{\partial\mathbf{B}(\mathbf{r},t)}{\partial t}, \tag{10}\] \[\
abla\times\mathbf{H}(\mathbf{r},t) = \frac{\partial\mathbf{D}(\mathbf{r},t)}{\partial t}+\mathbf{j}_{ f}(\mathbf{r},t),\] (11) \[\
abla\cdot\mathbf{D}(\mathbf{r},t) = \rho_{f}(\mathbf{r},t),\] (12) \[\
abla\cdot\mathbf{B}(\mathbf{r},t) = 0, \tag{13}\]  
The free current density \(\mathbf{j}_{f}(\mathbf{r},t)\) (current per unit volume) and the free charge density \(\rho_{f}(\mathbf{r},t)\) (charge per unit volume) are linked via the continuity equation:  
\[\
abla\cdot\mathbf{j}_{f}(\mathbf{r},t)+\frac{\partial\rho_{f}(\mathbf{r},t)} {\partial t}=0. \tag{14}\]  
By free, we mean those currents that are not already taken into account by the polarization an magnetization densities included in electric displacement, \(\mathbf{D}(\mathbf{r},t)\), and in the magnetic strength field, \(\mathbf{H}(\mathbf{r},t)\). The connection between the displacement and electric fields, and between the magnetic induction and magnetic strength fields is given by (for linear media)  
\[\mathbf{D}(\mathbf{r},t) = \epsilon_{n}\epsilon_{0}\mathbf{E}(\mathbf{r},t), \tag{15}\] \[\mathbf{H}(\mathbf{r},t) = \mu_{n}^{-1}\mu_{0}^{-1}\mathbf{B}(\mathbf{r},t), \tag{16}\]where \(\epsilon_{n}\) and \(\mu_{n}\) are the medium relative permittivity and permeability, respectively. Taking the curl of Equation (A1) and using Equation (A2) we obtain the wave equation for the electric field  
\[\
abla\times\
abla\times\mathbf{E}(\mathbf{r},t)+\frac{1}{v_{n}^{2}}\frac{ \partial^{2}\mathbf{E}(\mathbf{r},t)}{\partial t^{2}}=-\mu_{n}\mu_{0}\frac{ \partial\mathbf{j}_{f}(\mathbf{r},t)}{\partial t}\] (A8)  
where \(v_{n}=\sqrt{1/(\mu_{n}\mu_{0}\epsilon_{n}e_{0})}\) is the speed of light in the medium. Let us now consider harmonic fields with a time dependence \(e^{-i\omega t}\). In this case, the wave equation reads  
\[\
abla\times\
abla\times\mathbf{E}(\mathbf{r},\omega)-\frac{\omega^{2}}{v_{n} ^{2}}\mathbf{E}(\mathbf{r},\omega)=i\omega\mu_{n}\mu_{0}\mathbf{j}_{f}( \mathbf{r},\omega).\] (A9)  
Taking the curl of Equation (A2) we find a wave equation for the magnetic induction  
\[\
abla\times\
abla\times\mathbf{B}(\mathbf{r},t)+\frac{1}{v_{n}^{2}}\frac{ \partial^{2}\mathbf{B}(\mathbf{r},t)}{\partial t^{2}}=\mu_{n}\mu_{0}\
abla \times\mathbf{j}_{f}(\mathbf{r},t).\] (A10)  
Considering a harmonic time dependence of the fields and of the current it follows that  
\[\
abla\times\mathbf{H}(\mathbf{r},t) = \frac{\partial\mathbf{D}(\mathbf{r},t)}{\partial t}+\mathbf{j}_{ f}(\mathbf{r},t),\]0 (A11)  
It is possible to rewrite Equations (A9) and (A11) as inhomogeneous Helmholtz equations. In order to do so, we make use of the identity \(\
abla\times\
abla\times\mathbf{v}=\
abla\left(\
abla\cdot\mathbf{v}\right)- \
abla^{2}\mathbf{v}\) and write Equations (A9) and (A11) as  
\[\
abla\times\mathbf{H}(\mathbf{r},t) = \frac{\partial\mathbf{D}(\mathbf{r},t)}{\partial t}+\mathbf{j}_{ f}(\mathbf{r},t),\]1 (A12) \[\
abla\times\mathbf{H}(\mathbf{r},t) = \frac{\partial\mathbf{D}(\mathbf{r},t)}{\partial t}+\mathbf{j}_{ f}(\mathbf{r},t),\]2 (A13)  
Next, we use Equation (A4) to write \(\
abla\cdot\mathbf{B}(\mathbf{r},\omega)=0\), and Equations (A2) and (A6) to write \(\
abla\cdot\mathbf{E}(\mathbf{r},\omega)=\epsilon_{n}^{-1}\epsilon_{0}^{-1} \rho_{f}(\mathbf{r},\omega)\). Using the continuity Equation (A5), the free charge density can be written in terms of the free current density, as \(\rho_{f}(\mathbf{r},\omega)=\
abla\cdot\mathbf{j}_{f}(\mathbf{r},\omega)/(i \omega)\).