Kinematics and Snell's Law

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

$$(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$})_{z=0} = (\mbo... ...z=0} = (\mbox{\boldmath$k''$}\cdot\mbox{\boldmath$x$})_{z=0}$$ (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:

$$k \sin{\theta_{\rm incident}} = k'\sin{\theta_{\rm refracted}} = k \sin{\theta_{\rm reflected}}$$

$$n \sin{\theta_{\rm incident}} = n'\sin{\theta_{\rm refracted}} = n \sin{\theta_{\rm reflected}}$$ (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.