Geometric Derivation of Snell's Refraction Angle Law

This animation shows how Snell's laws of the refraction angle develops without the usual tangential component phase matching at the dielectric interface. In this anination, n is the index of refraction . It relies on the fact that both normal and tangential component of the wave velocity are reduced by the factor 1/n at the transparent interface. Thus the refracted wavefront orientation can be computed by iterating both components of the velocity. This is done here by defining points on the incident wavefront and integrating the positions of by using the equations xi(t)+=vxi(t)dt and yi(t)+=vyi(t)dt. In addition to the iteration, the animation shows a purely geometric proof (Exploded Detail) of Snell's law. This proof relies on the fact that the points on a wavefront have to be transverse to their wavefront propagation direction. The proof computes the time difference dt between arrival at the interface of point 1 and the next adjacent point, point 2. This time difference is the reason that the wavefront orientation and therefore the propagation direction changes. The two side by side triangles share a common hypotenuse. This allows relating the index medium internal ray length (see arrows) to the external ray length and, from this, Snell's law is derived. With respect to the interface normal, the incident angle is a and the refracted angle is b. As always, the learner is welcome to hit F12 in Windows to view the Javascript code that I used.