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 vt later. During that time a spherical wave is emitted from A and travels a distance vt . 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.

Figure 7.1. 

Figure 7.2. 

From this we see that and so θ_i = θ_r

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

Figure 7.3. 

In this case the velocities are different in the two media and so one obtains: and which then can be rearranged or rearranging some more or finally n_t sin θ_t = n_i sin θ_iwhich 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.

Fermat postulated that rays of light follow the path that takes the least time. This is a very profound idea! There is something very deep in it. It also gives the experimentally observed results! Lets apply it to reflection and see what results:

Figure 7.4. 

We want to find the length (which is the same as time times the speed) AEB. To do this we construct a fake point B' which is on the other side of the surface the same perpendicular distance from the surface such that the line BB' is a perpendicular to the surface. Then clearly the length AEB equals the length AEB'. So which point on the surface gives the shortest path to B, the one that gives the shortest path to B' and that clearly lies on the straight line AB'. I have labeled this point C.

Now clearly θ_r = θ_r^′ and also θ_r^′ = θ_i so we get θ_r = θ_i

Now lets apply Fermat's principle to refraction. Look at the next figure:

Figure 7.5. 

We want the shortest time from A to B. Clearly that is To find the minimum we want to solve for x such that Thus which is obviously or Snell's law n_t sin θ_t = n_i sin θ_i

If light travels via many different media then the time is or we can rewrite this as The quantity is the optical path length ( OPL ) . For a continuously varying medium then the summation becomes (for light traveling from S to P ) OPL = ∫_S^Pn ( s ) ⅆ s Fermat's principle could be restated that we minimize the OPL In fact this is inadequate, for example one can construct an example where the optical path length is not the minimum.(See for example figure 4.37 in the book "Optics" by Hecht (Fourth Edition).The correct statement of Fermat's principle is that there is a stationary point in the optical path length. (Ie. its derivative is zero).

We want to understand with Electromagnetism what happens at a surface. From Maxwell's equations we can understand what happens to the components of the and fields: First lets look at the field using Gauss' law. Recall

Figure 7.6. 

Consider the diagram, the field on the incident side is . On the transmission side, the field is . We can collapse the cylinder down so that it is a pancake with an infinitely small height. When we do this there are no field lines through the side of the cylinder. Thus there is only a flux through the top and the bottom of the cylinder and we have; I have set ∫ ρ ⅆ V = 0 since we will only consider cases without free charges. So we have ε_iE_{i ⊥} + ε_iE_{r ⊥} = ε_tE_{t ⊥} if is a unit vector normal to the surface this can be written

Similarly Gauss' law of Magnetism gives B_{i ⊥} + B_{r ⊥} = B_{t ⊥} or

Amperes law can also be applied to an interface.Then

Figure 7.7. 

(note that in this case is perpendicular to the page)

Now we will not consider cases with surface currents. Also we can shrink the vertical ends of the loop so that the area of the box is 0 so that . Thus we get at a surfaceor

Similarly we can use Faraday's law and play the same game with the edges to get E_{i ∥} + E_{r ∥} = E_{t ∥} or (notice ε does not appear)

In summary we have derived what happens to the and fields at the interface between two media:

Consider an electromagnetic wave impinging upon an interface: Where describes the incoming wave, the reflected wave, and the transmitted wave. At the interface (ie. at points where the vector points to the plane of the interface), all the waves must be in phase with each other. This means that the frequencies must all be equal and there can be no arbitrary phase between the waves. The net result of this is that we must have (for an interface passing through the origin): from which we get k_i sin θ_i = k_r sin θ_r .

It is important to note now that we are doing this at the interface. We have chosen a coordinate system so that the interface is at y = 0 and contains the origin. This implies that the vector is lying in the plane of the interface at the point where we say that the above is true.

Finally, since the incident and reflected waves are in the same medium we must have k_i = k_r and thus θ_i = θ_rAlso, we get that all line in a plane (because defines a plane). We also have and following the same arguments find that all line in a plane and that k_i sin θ_i = k_t sin θ_t . Now we know that ω_i = ω_t so we can multiply both sides by c / ω_i and get n_i sin θ_i = n_t sin θ_t