## 2 Dyadic Green's Function for the Electric Field
### Free-Space Dyadic Green's Function
Weyl's or Angular Spectrum Representation of the Dyadic Green's Function: A Useful Formulation for Interfaces  
Although Equation (10) can be used directly to evaluate \(\overleftarrow{G_{0}}(\mathbf{r},\mathbf{r}^{\prime},\omega)\), for many applications such formulation might not be the most useful. In the the case of scattering by planar interfaces it is useful to make a (two-dimensional) Fourier transform of the fields in the coordinates parallel to the interface. This representation of the fields and of the Green's function is generally referred to as Weyl's or angular spectrum representation. In this representation, the electric field is written as  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]6  
where \(\mathbf{p}_{\parallel}\) is the in-plane wave-vector and \(\boldsymbol{\rho}=(x,y)\) are in-plane coordinates. In this representation Equation (4) becomes  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]7  
where \(\mathbf{j}_{f}(\mathbf{p}_{\parallel},z,\omega)\) is the Weyl representation of the current density, defined in analogous way to Equation (16), \(\mathcal{D}^{\prime}=\mathbf{p}_{\parallel}-i\tilde{e}z_{0}^{\prime}\), \(\mathcal{D}^{\prime}\mathcal{D}^{\prime}\equiv\mathcal{D}^{\prime}\otimes \mathcal{D}^{\prime}\), \(f\) represents the principal value integral in one dimension, excluding the point \(z^{\prime}=z\), and \(g_{0}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\) is the Helmholtz Green's function in the Weyl representation, defined such that  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]8  
The function \(g_{0}(\mathbf{p}_{\parallel},z,z^{\prime},\omega)\) can be easily obtained from the components of the three-dimensional Fourier transform of the [PERSON]'s function, \(g_{0}\left(\mathbf{p},\omega\right)=\left(p_{\parallel}^{2}+p_{z}^{2}-k_{n}^{ 2}\right)^{-1}\), as  
\[\mathbf{E}(\mathbf{r},\omega)=\mathbf{E}_{0}(\mathbf{r},\omega)+\omega^{2}\mu _{n}\mu_{0}\leftr\overleftrightarrow{G_{0}}(\mathbf{r},\mathbf{r}_{0},\omega) \cdot\mathbf{d}_{0}.\right. \tag{2}\]9This integral can be easily performed by contour integration yielding  
\[-\
abla^{2}\mathbf{E}(\mathbf{r},\omega)-k_{n}^{2}\mathbf{E}(\mathbf{r},\omega )=i\omega\mu_{n}\mu_{0}\left[\mathbf{j}_{f}(\mathbf{r},\omega)+\frac{1}{k_{n} ^{2}}\
abla\left(\
abla\cdot\mathbf{j}_{f}(\mathbf{r},\omega)\right)\right], \tag{3}\]0  
where \(\beta_{n}\) is defined as  
\[\beta_{n}= \begin{cases}\sqrt{k_{n}^{2}-p_{\parallel}^{2}},&k_{n}^{2}>p_{ \parallel}^{2}\\ i\sqrt{p_{\parallel}^{2}-k_{n}^{2}},&k_{n}^{2}<p_{\parallel}^{2}\end{cases} . \tag{21}\]  
Clearly, Equation (20) is written is terms of both propagating and evanescent waves [32]. Similarly to what we have done in the previous section, we can rewrite Equation (17) by moving the derivatives \(\partial_{z}^{\prime}\) from \(\mathbf{j}_{f}(\mathbf{p}_{\parallel},z^{\prime},\omega)\) to \(g_{0}\left(\mathbf{p}_{\parallel},z,z^{\prime},\omega\right)\).