The reflection coefficients are related to the incident and reflected portions of the electric field.
\begin{equation} r_{\parallel}=\dfrac{E_{\mathrm{r}\parallel}}{E_{\mathrm{i}\parallel}}\\ \end{equation} \begin{equation} r_{\perp}=\dfrac{E_{\mathrm{r}\perp}}{E_{\mathrm{i}\perp}}\\ \end{equation}The portions of the electric field can be found from the boundary condition to Maxwell’s equations for the electric and magnetic fields (\citet{modest2003}, Eqs. 2.66 and 2.67).
\begin{equation} \mathbf{E}_{c1}\times\mathbf{\hat{n}}=\mathbf{E}_{c2}\times\mathbf{\hat{n}}\\ \end{equation} \begin{equation} \mathbf{H}_{c1}\times\mathbf{\hat{n}}=\mathbf{H}_{c2}\times\mathbf{\hat{n}}\\ \end{equation}Complex representation of the electric field, broken into parallel and perpendicular components (\citet{modest2003}, Eq. 2.46)
\begin{equation} \mathbf{E}_{c}=\mathbf{E}_{0}e^{-2\pi i\left(\mathbf{w}^{\prime}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}\cdot\mathbf{r}\right)},\quad\mathbf{E}_{0}=E_{\parallel}\mathbf{\hat{e}}_{\parallel}+E_{\perp}\mathbf{\hat{e}}_{\perp}\\ \end{equation}Considering incident, reflected, and transmitted electric fields at a point on the boundary, Eqs. \ref{eq:eboundary} and \ref{eq:hboundary} become:
\begin{equation} \mathbf{E}_{c\mathrm{i}}\times\mathbf{\hat{n}}+\mathbf{E}_{c\mathrm{r}}\times\mathbf{\hat{n}}=\mathbf{E}_{c\mathrm{t}}\times\mathbf{\hat{n}}\\ \end{equation} \begin{equation} \mathbf{E}_{0\mathrm{i}}e^{-2\pi i\left(\mathbf{w}^{\prime}_{\mathrm{i}}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}_{\mathrm{i}}\cdot\mathbf{r}\right)}\times\mathbf{\hat{n}}+\mathbf{E}_{0\mathrm{r}}e^{-2\pi i\left(\mathbf{w}^{\prime}_{\mathrm{r}}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}_{\mathrm{r}}\cdot\mathbf{r}\right)}\times\mathbf{\hat{n}}=\mathbf{E}_{0\mathrm{t}}e^{-2\pi i\left(\mathbf{w}^{\prime}_{\mathrm{t}}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}_{\mathrm{t}}\cdot\mathbf{r}\right)}\times\mathbf{\hat{n}}\\ \end{equation}and
\begin{equation} \mathbf{H}_{c\mathrm{i}}\times\mathbf{\hat{n}}+\mathbf{H}_{c\mathrm{r}}\times\mathbf{\hat{n}}=\mathbf{H}_{c\mathrm{t}}\times\mathbf{\hat{n}}\\ \end{equation} \begin{equation} \mathbf{H}_{0\mathrm{i}}e^{-2\pi i\left(\mathbf{w}^{\prime}_{\mathrm{i}}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}_{\mathrm{i}}\cdot\mathbf{r}\right)}\times\mathbf{\hat{n}}+\mathbf{H}_{0\mathrm{r}}e^{-2\pi i\left(\mathbf{w}^{\prime}_{\mathrm{r}}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}_{\mathrm{r}}\cdot\mathbf{r}\right)}\times\mathbf{\hat{n}}=\mathbf{H}_{0\mathrm{t}}e^{-2\pi i\left(\mathbf{w}^{\prime}_{\mathrm{t}}\cdot\mathbf{r}-i\mathbf{w}^{\prime\prime}_{\mathrm{t}}\cdot\mathbf{r}\right)}\times\mathbf{\hat{n}}\\ \end{equation}With the coordinate origin on the boundary, at the point \(\mathbf{r}=0\), Eqs. \ref{eq:eboundary} and \ref{eq:hboundary} become
\begin{equation} \left(\mathbf{E}_{0\mathrm{i}}+\mathbf{E}_{0\mathrm{r}}\right)\times\mathbf{\hat{n}}=\mathbf{E}_{0\mathrm{t}}\times\mathbf{\hat{n}}\\ \end{equation} \begin{equation} \left(\mathbf{H}_{0\mathrm{i}}+\mathbf{H}_{0\mathrm{r}}\right)\times\mathbf{\hat{n}}=\mathbf{H}_{0\mathrm{t}}\times\mathbf{\hat{n}}\\ \end{equation}These equations can then be broken into parallel and perpendicular components.
\begin{equation} \left(E_{\parallel\mathrm{i}}+E_{\parallel\mathrm{r}}\right)\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}+\left(E_{\perp\mathrm{i}}+E_{\perp\mathrm{r}}\right)\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}=\left(E_{\parallel\mathrm{t}}\right)\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}+\left(E_{\perp\mathrm{t}}\right)\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}\\ \end{equation} \begin{equation} \left(H_{\parallel\mathrm{i}}+H_{\parallel\mathrm{r}}\right)\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}+\left(H_{\perp\mathrm{i}}+H_{\perp\mathrm{r}}\right)\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}=\left(H_{\parallel\mathrm{t}}\right)\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}+\left(H_{\perp\mathrm{t}}\right)\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}\\ \end{equation}Unit vectors are related by the following:
\begin{equation} \mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}=-\mathbf{\hat{e}}_{\perp}\cos\theta\\ \end{equation} \begin{equation} \mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}=\mathbf{\hat{t}}\\ \end{equation}So, the boundary equations become
\begin{equation} \left(E_{\parallel\mathrm{i}}+E_{\parallel\mathrm{r}}\right)\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{1}\right)+\left(E_{\perp\mathrm{i}}+E_{\perp\mathrm{r}}\right)\mathbf{\hat{t}}=\left(E_{\parallel\mathrm{t}}\right)\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{2}\right)+\left(E_{\perp\mathrm{t}}\right)\mathbf{\hat{t}}\\ \end{equation} \begin{equation} \left(H_{\parallel\mathrm{i}}+H_{\parallel\mathrm{r}}\right)\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{1}\right)+\left(H_{\perp\mathrm{i}}+H_{\perp\mathrm{r}}\right)\mathbf{\hat{t}}=\left(H_{\parallel\mathrm{t}}\right)\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{2}\right)+\left(H_{\perp\mathrm{t}}\right)\mathbf{\hat{t}}\\ \end{equation}which results in 4 equations (\citet{modest2003}, Eq. 2.81-84).
\begin{equation} \left(E_{\parallel\mathrm{i}}+E_{\parallel\mathrm{r}}\right)\cos\theta_{1}=E_{\parallel\mathrm{t}}\cos\theta_{2}\\ \end{equation} \begin{equation} E_{\perp\mathrm{i}}+E_{\perp\mathrm{r}}=E_{\perp\mathrm{t}}\\ \end{equation} \begin{equation} \left(H_{\parallel\mathrm{i}}+H_{\parallel\mathrm{r}}\right)\cos\theta_{1}=H_{\parallel\mathrm{t}}\cos\theta_{2}\\ \end{equation} \begin{equation} H_{\perp\mathrm{i}}+H_{\perp\mathrm{r}}=H_{\perp\mathrm{t}}\\ \end{equation}The magnetic field can be eliminated by relating it to the electric field
\begin{equation} \mathbf{H}_{0}=\pm\dfrac{m}{c_{0}\mu}\left(E_{\parallel}\mathbf{\hat{e}}_{\perp}-E_{\perp}\mathbf{\hat{e}}_{\parallel}\right)\\ \end{equation}where the upper sign applies to incident and transmitted waves, and the lower sign applies to reflected waves. With this definition, the boundary condition for the magnetic field can be rewritten as
\begin{equation} \left(\mathbf{H}_{0\mathrm{i}}+\mathbf{H}_{0\mathrm{r}}\right)\times\mathbf{\hat{n}}=\mathbf{H}_{0\mathrm{t}}\times\mathbf{\hat{n}}\\ \end{equation} \begin{equation} \left(\dfrac{m_{1}}{c_{0}\mu_{1}}\left(E_{\parallel\mathrm{i}}\mathbf{\hat{e}}_{\perp}-E_{\perp\mathrm{i}}\mathbf{\hat{e}}_{\parallel}\right)-\dfrac{m_{1}}{c_{0}\mu_{1}}\left(E_{\parallel\mathrm{r}}\mathbf{\hat{e}}_{\perp}-E_{\perp\mathrm{r}}\mathbf{\hat{e}}_{\parallel}\right)\right)\times\mathbf{\hat{n}}=\dfrac{m_{2}}{c_{0}\mu_{2}}\left(E_{\parallel\mathrm{t}}\mathbf{\hat{e}}_{\perp}-E_{\perp\mathrm{t}}\mathbf{\hat{e}}_{\parallel}\right)\times\mathbf{\hat{n}}\\ \end{equation}Assuming the magnetic permeability \(\mu\) is the same in both media and distributing the cross products.
\begin{equation} m_{1}\left(E_{\parallel\mathrm{i}}\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}-E_{\perp\mathrm{i}}\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}\right)-m_{1}\left(E_{\parallel\mathrm{r}}\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}-E_{\perp\mathrm{r}}\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}\right)=m_{2}\left(E_{\parallel\mathrm{t}}\mathbf{\hat{e}}_{\perp}\times\mathbf{\hat{n}}-E_{\perp\mathrm{t}}\mathbf{\hat{e}}_{\parallel}\times\mathbf{\hat{n}}\right)\\ \end{equation} \begin{equation} m_{1}E_{\parallel\mathrm{i}}\mathbf{\hat{t}}-m_{1}E_{\perp\mathrm{i}}\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{1}\right)-m_{1}E_{\parallel\mathrm{r}}\mathbf{\hat{t}}+m_{1}E_{\perp\mathrm{r}}\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{1}\right)=m_{2}E_{\parallel\mathrm{t}}\mathbf{\hat{t}}-m_{2}E_{\perp\mathrm{t}}\left(-\mathbf{\hat{e}}_{\perp}\cos\theta_{2}\right)\\ \end{equation}Collecting components gives the following two equations.
\begin{equation} m_{1}\left(E_{\parallel\mathrm{i}}-E_{\parallel\mathrm{r}}\right)=m_{2}E_{\parallel\mathrm{t}}\\ \end{equation} \begin{equation} m_{1}\cos\theta_{1}\left(E_{\perp\mathrm{i}}-E_{\perp\mathrm{r}}\right)=m_{2}\cos\theta_{2}E_{\perp\mathrm{t}}\\ \end{equation}These two equations are equivalent to Eqs. 2.86 and 2.87 in \citet{modest2003} without the assumption of non-absorbing media. There is now a system of four equations in terms of the electric fields, \(E_{\parallel\mathrm{i}}\), \(E_{\perp\mathrm{i}}\), \(E_{\parallel\mathrm{r}}\), \(E_{\perp\mathrm{r}}\), \(E_{\parallel\mathrm{t}}\), and \(E_{\perp\mathrm{t}}\), the material properties, \(m_{1}\) and \(m_{2}\), and the incidence and reflected angles, \(\theta_{1}\) and \(\theta_{2}\).
The problem can be simplified by assuming \(m_{2}=n_{2}\) if the algebra proves too difficult.
The problem is now sufficiently specified that the reflection coefficients can be expressed in terms of \(m_{1}\), \(m_{2}\), \(\theta_{1}\), and \(\theta_{2}\). From the generalized Snell’ls law in the above section, \(\theta_{2}\) can be expressed in terms of \(m_{1}\), \(m_{2}\), and \(\theta_{1}\).