Snell's law and Fermat's principle

Question

I have read this sentence on a book about the Snell's law of refraction, referring on a ray that passes from air ($n_1=1$) to glass ($n_2=1.55$):

"Snell's equation can be derived from Fermat's principle of least time. The path length in air has been increased relative to the path length in glass since the speed in air ($c/n_1$) is greater than the speed in glass ($c/n_2$). This is analogous to the situation of a lifeguard who must rescue a swimmer who is down the beach and out to sea. Since the guard can run faster then he can swim, he should not head directly for the swimmer but should run at an angle so that the distance covered on the beach is greater than the distance in the water and the total time can be minimized"

I do not understand it. Why should we increase the path in air? From this sentence it seems that in order to reduce the total time (of travelling in air and in glass), we have to increase the path in air (and what does it mean?).

Answer 1

Let's do some math, shall we? Fermat's principle is actually a least-time principle. What it says is, that out of all possible trajectories between two points, light is going to follow the one that minimizes the time spent traversing that trajectory.

We know that light travels in straight lines if it's speed is constant throughout the medium, but at the interface of two media where light changes it's propagation speed we assume that we don't know how the light ray is gonna snap (in other words, we have unknown angles of incidence and refraction. We are going to derive Snell's law.

Suppose the light ray is emitted at point $P_1$ and wants to reach point $P_2$ in the least time possible. These points are fixed and their coordinates are $P_1(0,h_1), P_2(-h_2, D)$. Now we would like to minimize the time traversed on the set of possible trajectories. Assume that the light ray is incident distance $d_1$ away from $P_1$ and $d_2$ away from $P_2$. We know that $d_1+d_2=D$. Also the total time of travel between the two points is:

$$T(d_1,d_2)=t_1+t_2=\frac{L_1}{v_1}+\frac{L_2}{v_2}=\frac{1}{c}\Big[n_1\sqrt{d_1^2+h_1^2}+n_2\sqrt{d_2^2+h_2^2}\Big]~~, ~~ d_1+d_2=D$$

Now we want to minimize $T$ under the constraint written above. Using standard Lagrange multipliers procedures we minimize

$$\tilde{T}=T(d_1,d_2)-\lambda(d_1+d_2-D)$$

and we compute

$$\frac{\partial\tilde{T}}{\partial d_1}=\frac{n_1d_1}{c\sqrt{d_1^2+h_1^2}}-\lambda=0\\ \frac{\partial\tilde{T}}{\partial d_2}=\frac{n_2d_2}{c\sqrt{d_2^2+h_2^2}}-\lambda=0$$

Notice that $\sin\theta_1=\frac{d_1}{\sqrt{d_1^2+h_1^2}}$ and similarly for the angle of incidence (just swap index 1 for 2) and thus we get immediately that:

$$n_1\sin\theta_1=n_2\sin\theta_2=\lambda$$

I guess the take away here is what Fermat really says is that along a trajectory the time traversed is minimized. So when some authors say air path length, this actually means time. To understand why they say that, suppose the index of refraction is varying continuously in space. Then we need to minimize

$$T=\int_{trajectory}dt=\int{\frac{ds}{v(s)}}=\frac{1}{c}\int_{P_1\rightarrow P_2}n(x(s), y(s))ds$$

and this can be minimized using standard calculus of variations techniques. The last quantity (maybe omitting the speed of light) is the "path length" that they're talking about, because in the integral the actual path length $ds$ is considered, but weighted by a certain function of spacetime, the index of refraction.

Hope this is helpful!

Answer 2

You are asking about the speed of light in different media (air, glass).

The speed of light is c in vacuum, when measured locally.

Now in air, the speed of light reduces (when measured locally) relative to c.

In glass, the speed of light reduces (when measured locally) relative to c and relative to the speed of light in air.

There are two ways to think of it:

1. classically, air is denser then vacuum (and glass is denser then air) in a sense, that light needs to interact with the atoms and molecules of the material, this takes time, and this makes light seem to travel slower in thicker media

2. in QM, photons, the quanta of EM waves, as they travel through air (or glass), they interact with the atoms and molecules of air, though the wavefront of light travels slower then c, because the individual photons that make up the EM wave, travel actually a longer path through the media, thus it takes for them more time to travel the straight path you calculate for the wavefront (though individual photons travel at speed c in vacuum between atoms).

Now the denser the medium, usually the slower light propagates in it.

In your case, if you have a path that leads through air and glass, you have to select this path so, that light travels the shortest in glass and the longest part of the total path in air.

This way, the longer part of the total path will be traveled by light in air (faster then glass), and only a shorter path will be traveled in glass (slower then air).

So your total travel time for the EM wave will be the minimal.