Snell's Law of Refraction

Fermat's Principle of Least Time

In geometric optics one often makes the assumption that light travels from one point to another along the path requiring the least time. That is, light will always seek the path of minimal time from point A to point B. This fact is often referred to as Fermat's Principle of Least Time. Note that the path of minimal time need not be the path of minimal distance between points A and B.

It can be shown experimentally that light travels at different speeds in different mediums. In Figure 1, suppose that air is the medium above the line JG and let the speed of light in the air be represented by the constant v1. Let the region below the line JG be some other medium, say glass, and let the speed of light in glass be represented by the constant v2.

Imagine a ray of light starting at the point A, traveling through the air at a constant speed v1, then striking the boundary of the glass (the line JG) at the point P. The incoming ray of light makes an angle of a1 with the normal HI to the line JG at the point P. The light bends in the glass, making an angle of a2 with the normal HI, then travels through the denser medium to the point B at the constant speed v2.

Snell's Law

Snell's Law predicts the following relationship:

The following homework questions are designed to guide you through a proof of Snell's Law based on Fermat's Principle of Least Time.

Homework

1. If the speed of a particle is constant, then the distance it travels is given by the familiar formula d = st (distance equals speed times time). Equivalently, the time is found by dividing the distance traveled by the speed.
• Show that the time it takes a light particle to travel from point A to point B along the path APB is given by the formula T = (a²+x²)^1/2/v1 + (b² + (c - x)²)^1/2/v2.
2. The task before you is to minimize the time it takes a particle of light to travel the path from A to P to B. A standard result from differential calculus requires that any miminum value of T will occur at a critical value.
• Find dT/dx.
• Set dT/dx equal to zero and use the result to show that sin(a1)/v1 = sin(a2)/v2.