Pre-Quantum Electrodynamics

• Gr 9.3.2

Boundary between two media: \(x-y\) plane. Plane wave of frequency \(\omega\) travelling in \(z\) direction and polarized in \(x\) direction approaches surface: \[ {\boldsymbol E}_I (z,t) = E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol x}, \hspace{1cm} {\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y} \label{Gr(9.75)} \] (in which we have used \(B_0 = \frac{1}{v} E_0\), EBmpw, adapted for the light velocity in this medium).

Reflected wave: \[ {\boldsymbol E}_R (z,t) = E_{0_R} e^{i (-k_1 z - \omega t)} \hat{\boldsymbol x}, \hspace{1cm} {\boldsymbol B}_R (z,t) = \frac{-1}{v_1} E_{0_R} e^{i (-k_1 z - \omega t)} \hat{\boldsymbol y} \label{Gr(9.76)} \] (minus sign is conventional here, but handy for inverting Poynting vector direction).

For the transmitted wave, we put

\begin{align*} {\boldsymbol E}_T (z,t) &= E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \\ {\boldsymbol B}_T (z,t) &= \frac{1}{v_2} E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol y}. \end{align*}

Our boundary is by choice of coordinate system at \(z = 0\). Our setup calls for solving the boundary conditions disc_nfc with \({\boldsymbol E}_I + {\boldsymbol E}_R\) and \({\boldsymbol B}_I + {\boldsymbol B}_R\) on one side, and \({\boldsymbol E}_T\) and \({\boldsymbol B}_T\) on the other.

At normal incidence, there are no perpendicular components of the fields relative to the surface, so disc_nfc (i) and (ii) are obeyed. (iii) means that \[ E_{0_I} + E_{0_R} = E_{0_T} \label{Gr(9.78)} \] while (iv) means that \[ \frac{1}{\mu_1} \frac{1}{v_1} (E_{0_I} - E_{0_R}) = \frac{1}{\mu_2} \frac{1}{v_2} E_{0_T}, \label{Gr(9.79)} \] which can be rewritten as \[ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm} \beta \equiv \frac{\mu_1 v_1}{\mu_2 v_2} = \frac{\mu_1 n_2}{\mu_2 n_1}. \label{Gr(9.80)} \]

Solving these coupled equations, we can write outgoing amplitudes in terms of incident ones:

\[ E_{0_R} = \frac{1 - \beta}{1 + \beta} E_{0_I}, \hspace{10mm} E_{0_T} = \frac{2}{1 + \beta} E_{0_I}. \tag{ERT}\label{ERT} \]

The intensity of reflected and transmitted waves is the power per unit area transported by the wave. In vacuum, this was \[ I \equiv \langle S \rangle = \frac{1}{2} c \varepsilon_0 E_0^2. \label{Gr(9.63)} \] Here, this becomes \[ I = \frac{1}{2} v \varepsilon E_0^2. \label{Gr(9.73)} \] The fraction of incident intensity in the reflected intensity is \[ R \equiv \frac{I_R}{I_I} = \frac{E^2_{0_R}}{E^2_{0_I}} = \left( \frac{1 - \beta}{1 + \beta} \right)^2 \label{Gr(9.86)} \] while the transmitted intensity is

\begin{equation} T \equiv \frac{I_T}{I_I} = \frac{v_2 \varepsilon_2}{v_1 \varepsilon_1} \frac{E^2_{0_T}}{E^2_{0_I}} = \sqrt{\frac{\mu_1 \varepsilon_2}{\mu_2 \varepsilon_1}} \frac{E^2_{0_T}}{E^2_{0_I}} = \frac{4 \beta}{(1 + \beta)^2}. \end{equation}

We thus have that the reflection and transmission coefficients satisfy \[ R + T = 1. \label{Gr(9.88)} \]