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  • Generalized Snell’s law and Fresnel equations

    Goal: find the reflectivity of an electromagnetic wave incident on an interface between two absorbing media (or, if that proves too challenging, on an interface between an absorbing and a non-absorbing medium with the wave originating in the absorbing medium)

    To accomplish this goal, first a generalized Snell’s law for an interface between two absorbing media must be found to relate the incident angle with the transmission angle. Then, the reflection coefficients must be found to determine the reflectivity.

    Generalized Snell’s Law

    To find a generalized Snell’s law, start with the requirement of continuity of the tangential component for the transmitted wave vector (Modest (2003), Eq. 2.104):

    \begin{equation} \label{eq:1}w^{\prime}_{\mathrm{i}}\sin\left(\theta_{1}\right)=w^{\prime}_{\mathrm{t}}\sin\left(\theta_{2}\right)\\ \end{equation}

    Complex wave vectors \(\mathbf{w}=w^{\prime}-iw^{\prime\prime}\) and complex refractive indices \(m=n-ik\) are related by the definition of complex refractive index (Modest (2003), Eq. 2.31):

    \begin{equation} \label{eq:refractindex}\mathbf{w}\cdot\mathbf{w}=\eta_{0}^{2}m^{2}\\ \end{equation}

    Expanding both sides of Eq. \ref{eq:refractindex}

    \begin{equation} \mathbf{w}\cdot\mathbf{w}=w^{\prime 2}-w^{\prime\prime 2}-2iw^{\prime}w^{\prime\prime}\\ \end{equation} \begin{equation} \eta_{0}^{2}m^{2}=\eta_{0}^{2}\left(n^{2}-k^{2}-2ink\right)\\ \end{equation}

    Equating the real and imaginary parts of each side gives:

    \begin{equation} w^{\prime 2}-w^{\prime\prime 2}=\eta_{0}^{2}\left(n^{2}-k^{2}\right)\\ \end{equation} \begin{equation} w^{\prime}w^{\prime\prime}=\eta_{0}^{2}\left(nk\right)\\ \end{equation}

    The final two equations can be applied to both the medium 1 where the wave originated and medium 2 where the wave is transmitted.

    \begin{equation} \label{eq:2}w^{\prime 2}_{\mathrm{i}}-w^{\prime\prime 2}_{\mathrm{i}}=\eta_{0}^{2}\left(n_{1}^{2}-k_{1}^{2}\right)\\ \end{equation} \begin{equation} \label{eq:3}w^{\prime}_{\mathrm{i}}w^{\prime\prime}_{\mathrm{i}}=\eta_{0}^{2}\left(n_{1}k_{1}\right)\\ \end{equation} \begin{equation} \label{eq:4}w^{\prime 2}_{\mathrm{t}}-w^{\prime\prime 2}_{\mathrm{t}}=\eta_{0}^{2}\left(n_{2}^{2}-k_{2}^{2}\right)\\ \end{equation} \begin{equation} \label{eq:5}w^{\prime}_{\mathrm{t}}w^{\prime\prime}_{\mathrm{t}}=\eta_{0}^{2}\left(n_{2}k_{2}\right)\\ \end{equation}

    Equations \ref{eq:1}, \ref{eq:2}, \ref{eq:3}, \ref{eq:4}, and \ref{eq:5} constitute a system of five equations with the following five unknowns: \(\theta_{2},w_{\mathrm{i}}^{\prime},w_{\mathrm{i}}^{\prime\prime},w_{\mathrm{t}}^{\prime},w_{\mathrm{t}}^{\prime\prime}\). \(\theta_{2}\) can therefore be solved for in terms of \(m_{1}\) and \(m_{2}\).

    If the algebra for that proves too complicated, the equations can be simplifed if medium 2 is assumed to be non-absorbing. Equations \ref{eq:4} and \ref{eq:5} reduce to

    \begin{equation} w_{\mathrm{i}}^{\prime}=\eta_{0}n_{1}\\ \end{equation} \begin{equation} w_{\mathrm{i}}^{\prime\prime}=0\\ \end{equation}

    Reflection Coefficients

    \label{eq:hboundary}

    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 (Modest (2003), 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 (Modest (2003), 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 equ