The Huygens-Fresnel Principle

In order to proceed with the discussion we have to define two terms. A wave front is the surface of constant phase. In a plane wave these are planes and in a spherical wave these are spheres. A ray travels perpendicular to the fronts.

Huygens postulated that as a wave propagates through a medium each point on the advancing wavefront acts as a new point source of the wave. This is correct physics for the water waves but not for light waves. However the Helmholtz equation for diffraction of EM waves gives a solution identical to that give by Huygens' principle.

Look at the figure which shows a wavefront AB coming to a surface and is reflected creating the front CD. The point A hits the surface first. The point B hits a time v t v t later. During that time a spherical wave is emitted from A and travels a distance v t v t . In fact this happens for every point along the wavefront. The next figure attempts to show how a number of waves line up along the line DC and that this is perpendicular to the line AD.

From this we see that sin θ i = v t A C sin θ i = v t A C and sin θ r = v t A C sin θ r = v t A C so θ i = θ r θ i = θ r

For refraction a similar thing happens. See figure (geometric optics / Huygens refraction.vsd )

In this case the velocities are different in the two media and so one obtains: sin θ i = v i t A C sin θ i = v i t A C and sin θ t = v t t A C sin θ t = v t t A C which then can be rearranged sin θ i v i t = sin θ t v t t sin θ i v i t = sin θ t v t t or rearranging some more sin θ i sin θ t = v i t v t t sin θ i sin θ t = v i t v t t or sin θ i sin θ t = n t n i sin θ i sin θ t = n t n i finally n t sin θ t = n i sin θ i n t sin θ t = n i sin θ i which is Snell's law. Now note that normally one uses rays, in which case the angles are measured w.r.t. the normal to the surface.