Reflection and refraction of light
The subject of this chapter is the reflection and refraction of light—or electromagnetic waves in general—at surfaces. We have already discussed the laws of reflection and refraction in Chapters 26 and 33 of Volume I. Here’s what we found out there: (Earlier, we used $i$ for the incident angle and $r$ for the refracted angle. Since we can’t use $r$ for both “refracted” and “reflected” angles, we are now using $\theta_i={}$incident angle, $\theta_r={}$reflected angle, and $\theta_t={}$transmitted angle.) Our earlier discussion is really about as far as anyone would normally need to go with the subject, but we are going to do it all over again a different way. Why? One reason is that we assumed before that the indexes were real (no absorption in the materials). But another reason is that you should know how to deal with what happens to waves at surfaces from the point of view of Maxwell’s equations. We’ll get the same answers as before, but now from a straightforward solution of the wave problem, rather than by some clever arguments.  We want to emphasize that the amplitude of a surface reflection is not a property of the material, as is the index of refraction. It is a “surface property,” one that depends precisely on how the surface is made. A thin layer of extraneous junk on the surface between two materials of indices $n_1$ and $n_2$ will usually change the reflection. (There are all kinds of possibilities of interference here—like the colors of oil films. Suitable thickness can even reduce the reflected amplitude to zero for a given frequency; that’s how coated lenses are made.) The formulas we will derive are correct only if the change of index is sudden—within a distance very small compared with one wavelength. For light, the wavelength is about $5000$ Å, so by a “smooth” surface we mean one in which the conditions change in going a distance of only a few atoms (or a few angstroms). Our equations will work for light for highly polished surfaces. In general, if the index changes gradually over a distance of several wavelengths, there is very little reflection at all.