solution of can be written as a superposition of incident, transmitted and reflected waves. For the electric field, we can write these in the form of angular-spectrum expansions [@Tsukerman-Markel14]:

\[eq:Efine\] $$\begin{aligned}
\label{eq:Ei}
&& {\mathbf{e}}_i({\mathbf{r}}) = \int {\mathbf{s}}_i(k_x, k_y) e^{i\left(k_x x + k_y y +
    k_z z \right) } dk_x dk_y \ ,    \\
\label{eq:Et}
&& {\mathbf{e}}_t({\mathbf{r}}) = \int {\mathbf{s}}_t(k_x, k_y)
e^{i\left(k_x x + k_y y + k_z z \right)} d k_x d k_y \ , \ \ z > L \ ,    \\
\label{eq:Er}
&& {\mathbf{e}}_r({\mathbf{r}}) = \int {\mathbf{s}}_r(k_x, k_y) e^{i \left(k_x x + k_y y - k_z
    z \right)} d k_x d k_y \ , \ \ z < 0 \ ,\end{aligned}$$

where $$\label{eq:disp_vac}
   k_z = \sqrt{k_0^2 - k_x^2 - k_y^2} \ ,$$ and the square root branch is defined by the condition $0 \leq \arg k_z < \pi$. Expressions for the magnetic field are obtained from by using the second Maxwell equation in . In , ${\mathbf{s}}_i(k_x,k_y)$, ${\mathbf{s}}_t(k_x,k_y)$ and ${\mathbf{s}}_r(k_x,k_y)$ are the angular spectra of the incident, transmitted and reflected fields. Waves included in these expansions can be evanescent or propagating. For propagating waves, $k_x^2 + k_y^2 < k_0^2$, otherwise the waves are evanescent.

Everywhere in space, the 