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 $\mu',\epsilon'$.

Incident Wave

$$\mbox{\boldmath$E$} = \mbox{\boldmath$E$}_0 e^{i(\mbox{\scriptsize\boldmath$k$}\cdot\mbox{\scriptsize\boldmath$x$} - \omega t)}$$ (9.48)

$$\mbox{\boldmath$B$} = \sqrt{\mu\epsilon} \frac{\mbox{\boldmath$k$} \times \mbox{\boldmath$E$}}{k}$$ (9.49)

Refracted Wave

$$\mbox{\boldmath$E'$} = \mbox{\boldmath$E'$}_0 e^{i(\mbox{\scriptsize\boldmath$k'$}\cdot\mbox{\scriptsize\boldmath$x$} - \omega t)}$$ (9.50)

$$\mbox{\boldmath$B'$} = \sqrt{\mu'\epsilon'} \frac{\mbox{\boldmath$k'$} \times \mbox{\boldmath$E'$}}{k'}$$ (9.51)

Reflected Wave

$$\mbox{\boldmath$E''$} = \mbox{\boldmath$E''$}_0 e^{i(\mbox{\scriptsize\boldmath$k$}\cdot\mbox{\scriptsize\boldmath$x$} - \omega t)}$$ (9.52)

$$\mbox{\boldmath$B''$} = \sqrt{\mu \epsilon} \frac{\mbox{\boldmath$k$} \times \mbox{\boldmath$E''$}}{k'}$$ (9.53)

where the reflected wave and incident wave do not leave the first medium and hence retain speed $v = 1/\sqrt{\mu \epsilon}$, $\mu$, $\epsilon$ and $k = \omega \sqrt{\mu\epsilon} = \omega/v$. The refracted wave changes to speed $v' = 1/\sqrt{\mu'\epsilon'}$, $\mu'$, $k' = \omega \sqrt{\mu'\epsilon'} = \omega/v'$.

[Note that the frequency $\omega$ of the wave is the same in both media! Ask yourself why this must be so as a kinematic constraint...]

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.