A Physical Explanation of Refraction

  One can trace Snell's law of refraction to the boundary conditions acting on the electromagnetic wave equations for the electromagnetic fields. To show this we can assume that the incident, reflected and refracted rays are simple plane waves.

In what follows let ψ represent an arbitrary electric or magnetic field component. In the first medium the field is a sum of the incident and reflected fields and there is only one field in the second medium. The relation between the angular frequency, wavelength and phase velocity, ω=kv_ph=kc/n allows us to express the wave vectors for each medium in terms of the magnitude, k=nω/c, of the corresponding wave vector in a vacuum. We assume the plane of incidence is the x,y-plane with horizontal axis x and vertical axis y. Along the boundary between the media y=0. For reflection we know that the reflected angle is equal to the angle of incidence so we can set β equal to α. This allows us to simplify the field equation for the first medium slightly. If a field is continuous along the boundary then its derivative will also be the same for both media.

Equating these functions we get two equations for the wave amplitudes A_k and the second equation yields Snell's law after eliminating the common factors.

So the continuity of the fields appears to be the physical cause for the "broken" or refracted light path. The wave vectors do not depend on the values for the incident fields but we can expect the amplitudes of the reflected and refracted rays to depend on them.