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Reflection and Refraction at a
Plane Interface

Suppose a plane wave is incident upon a plane surface that is an interface between two materials, one with $\mu ,\epsilon $ and the other with $\mu ',\epsilon '$.

Figure 9.1: Geometry for reflection and refraction at a plane interface between two media, one with permittivity/permeability $\mu ,\epsilon $, one with permittivity/permeability $\mu ',\epsilon '$.
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In order to derive an algebraic relationship between the intensities of the incoming wave, the reflected wave, and the refracted wave, we must begin by defining the algebraic form of each of these waves in terms of the wave numbers. The reflected wave and incident wave do not leave the first medium and hence retain speed $v = 1/\sqrt{\mu\epsilon}$, $\mu$, $\epsilon $ and $k = k'' = \omega \sqrt{\mu\epsilon} = \omega/v$. The refracted wave changes to speed $v' = 1/\sqrt{\mu'\epsilon'}$, $\mu'$, $k' = \omega \sqrt{\mu'\epsilon'} = \omega/v'$.

Note that the frequency of the waves is the same in both media as a kinematic constraint! Why is that?

This yields the following forms for the various waves:

Incident Wave


$\displaystyle \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$E$}_0 e^{i(\mbox{\scriptsize\boldmath$k$}\cdot\mbox{\scriptsize\boldmath$x$} - \omega t)}$ (9.53)
$\displaystyle \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle \sqrt{\mu\epsilon} \frac{\mbox{\boldmath$k$} \times \mbox{\boldmath$E$}}{k}$ (9.54)

Refracted Wave


$\displaystyle \mbox{\boldmath$E'$}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$E'$}_0 e^{i(\mbox{\scriptsize\boldmath$k'$}\cdot\mbox{\scriptsize\boldmath$x$} - \omega t)}$ (9.55)
$\displaystyle \mbox{\boldmath$B'$}$ $\textstyle =$ $\displaystyle \sqrt{\mu'\epsilon'} \frac{\mbox{\boldmath$k'$} \times \mbox{\boldmath$E'$}}{k'}$ (9.56)

Reflected Wave


$\displaystyle \mbox{\boldmath$E''$}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$E''$}_0 e^{i(\mbox{\scriptsize\boldmath$k$}''\cdot\mbox{\scriptsize\boldmath$x$} - \omega t)}$ (9.57)
$\displaystyle \mbox{\boldmath$B''$}$ $\textstyle =$ $\displaystyle \sqrt{\mu \epsilon} \frac{\mbox{\boldmath$k$}'' \times \mbox{\boldmath$E''$}}{k}$ (9.58)

Our goal is to completely understand how to compute the reflected and refracted wave from the incident wave. This is done by matching the wave across the boundary interface. There are two aspects of this matching - a static or kinematic matching of the waveform itself and a dynamic matching associated with the (changing) polarization in the medium. These two kinds of matching lead to two distinct and well-known results.



Subsections
next up previous contents
Next: Kinematics and Snell's Law Up: Plane Waves Previous: Polarization of Plane Waves   Contents
Robert G. Brown 2013-01-04