Last post, I talked about the fastest route between two mediums, and how light seems to follow that route when travelling from one medium to another. From what I just summarized, one would think light did that just because it is in a rush to get somewhere or something, but it is instead due to the wave nature of light. In classical physics, one can model how light reflects and refracts by applying something called Huygens Principle. Basically, a light wave at its edge forms a spherical wave at every point. Those spherical waves combine to form a single smooth wave. In the case of refraction, it looks like this:

Interestingly, the following model gives you the same math: Instead of the waves making more waves, let’s imagine instead that there are a chain of people holding hands and walking on the beach at constant speed. They suddenly reach the ocean, but at an angle. The person who reaches it first then bend the path, and then go forward. Otherwise, if the person wants to remain on the same straight path along with the others, the person must also move sideways. If you can’t picture it, it looks like this:

Now, let’s set the geometry up so that it shows the position of the line of people on the sand right before they go in the water and the line of people on the water right after they left land. Again, the math is the same, but I thought I could make it more intuitive. Notice that the length of the shore is the same for the triangle above and below it. That will be the key to creating the ratio for Snell’s law:

Before we go on, I have to mention that angle 1 in triangle BFC and angle 2 in triangle ADE are the angles given in Snell’s law. Then how do we know that angle 1 in ABC and angle 2 in AEF are the same respectively? ABF and BFC are similar triangles. So is AEF and ADE. The proof of that is not the subject for this post, so I might as well leave it as your homework assignment 😉 .

Now, since we want to find the ratio, we want to find which line is the same for the triangle ABF and AEF. Line l is the same. So we want to find something that looks like this:

Well, we can use trigonometry for that. For triangle ABF:

1a)

1b)

Doing the same for triangle AEF gets us:

2)

As you can see equation 1=equation 2:

3)

Since distance=velocity*time:

4) and

Plugging it back in equation 3 gets us:

5)

Multiplying both sides by the speed of light c gets us the generic Snell’s law equation with n standing for index of refraction, which is equal c/v:

6)

Cool, the light bending its path due to the wave slowing down results in a path that is naturally the fastest possible route!

Interestingly, if you use the math of Snell’s law and imagine stacking infinitely thin layers of mediums, the ones next to each other being only infinitesimally different in their travel velocity, you can solve another interesting problem that involves the fastest route from top to bottom by gravity as we shall see in the next part of this series! And no, just letting go off the object so it falls straight down is not the answer (although it is A answer).