Reflection and Refraction A short overview
Plane wave A plane wave can be written as follows: Here A represent the E or B fields, q=i,r,t and j=x,y,z So this is a representation of the waves that is valid i all three cases, i.e. the incoming, the reflected and the transmitted wave
Boundary conditions The most importent condition at the boundary is the existence of continuity This gives us that the tangential component of the argument in the wave-equations have to be equal, i.e. the waves are in the plane of incidence
So we have the following: With an arbitary point, r, on the surface that are spanned by the xy-plane in z=0, we can write the equation on scalar form: This can also be written as a cross-product:
If we use the ralation We can see that, and since the incomming and the reflected waves propagates in the same medium, we know, where k 0 is the wavenumber in vacuum and n 1 is the refractive index in medium 1. This gives us
In medium 2, one get the following: This is Snell’s law. The more general way of writing this is the vector form:
Generalisation of the laws One just have to look on the general case close enough, so one can approximate the wave as plane and the interface as a tangent plane through the point in question. This can be done locally for each point on the surface.
Brewster’s law If the wave is comming in at a angle such that then we get from Snell’s law This will result in a polarisation of the reflected wave, because the E vector of the reflected wave won’t have a component in the plane of incidence. From the picture on the next page one can see
Total reflection If the incomming wave has an angle that corresponds to a refraction index of Then we get the refracted wave to propagate along the interface If the incomming angle is larger than the critical angle then we have total reflection, but the will exist a field in medium 2, but it do not transport energy
All reflected waves will be polarised, and the degree of the polarisation is determined by the refractive index and the incomming angle