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Kinematics and Snell's Law

The phase factors of all three waves must be equal on the actual boundary itself, hence:

\begin{displaymath}
(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$})_{z=0} = (\mbo...
...z=0} =
(\mbox{\boldmath$k''$}\cdot\mbox{\boldmath$x$})_{z=0}
\end{displaymath} (9.54)

as a kinematic constraint for the wave to be consistent. That is, this has nothing to do with ``physics'' per se, it is just a mathematical requirement for the wave description to work. Consequently it is generally covered even in kiddy-physics classes, where one can derive Snell's law just from pictures of incident waves and triangles and a knowledge of the wavelength shift associated with the speed shift with a fixed frequency wave.

At $z = 0$, the three $\mbox{\boldmath$k$}$'s must lie in a plane and we obtain:

$\displaystyle k \sin{\theta_{\rm incident}}$ $\textstyle =$ $\displaystyle k'\sin{\theta_{\rm refracted}} = k
\sin{\theta_{\rm reflected}}$  
$\displaystyle n \sin{\theta_{\rm incident}}$ $\textstyle =$ $\displaystyle n'\sin{\theta_{\rm refracted}} = n
\sin{\theta_{\rm reflected}}$ (9.55)

which is both Snell's Law and the Law of Reflection, where we use $k = \omega/v = n \omega/c$ to put it in terms of the index of refraction, defined by $v = c/n$. Note that we cancel $\omega/c$, using the fact that the frequency is the same in both media.


next up previous contents
Next: Dynamics and Reflection/Refraction Up: Reflection and Refraction at Previous: Reflection and Refraction at   Contents
Robert G. Brown 2007-12-28