Kinematics and Snell's Law

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

$$(\Vec{k}\cdot\Vec{x})_{z=0} = (\Vec{k'}\cdot\Vec{x})_{z=0} = (\Vec{k''}\cdot\Vec{x})_{z=0}$$ (11.55)

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 $\Vec{k}$ 's must lie in a plane. The angles of incidence $\theta_i$ , reflection $\theta_l$ , and refraction $\theta_r$ add to the angles in the dot product to make $\pi/2$ , so the cosine in the dot product becomes the sine of these angles and we obtain: k (&thetas#theta;_i) & = & k'(&thetas#theta;_r) = k (&thetas#theta;_l)
n (&thetas#theta;_i) & = & n'(&thetas#theta;_r) = n (&thetas#theta;_l) which is both Snell's Law and the Law of Reflection, obtained in one fell swoop.

Note well that we used $k = \omega/v = n \omega/c = k''$ and $k' = \omega/v' = n' \omega/c$ to put it in terms of the index of refraction, defined by $v = c/n$ and $v' = c/n'$ . Then we cancel $\omega/c$ , using the fact that the frequency is the same in both media.

Snell's Law and the Law of Reflection are thus seen to be kinematic relations that are the result of the requirement of phase continuity on the plane interface - a ``wavefront'' of the $k$ (or $k''$ ) wave must be the same as the wavefront of the $k'$ wave.