Refraction and Snell's Law

Snell's law relates to the refraction of light waves when they move from one optical medium to another.  An example of the refraction of light rays is shown in the following figure. 

Have you ever noticed that the bottom of a swimming pool, when filled with water, appears to be closer to the surface than is really the case?  The answer to this lies in the refraction of light from an object in the pool when the light reaches the water-air interface.  The result, as shown in the figure above, is that an object at the bottom of the pool has an apparent position that is part way up the side wall of the pool.

Through a series of experiments, Snell found (in 1622) that the angles of incidence and refraction are related by: 

The constant in Snell’s Law is related to the refractive indices of water and air, which in turn are related to the velocities of light in water and air.  Snell’s law can be written in terms of the velocities of light as follows: 

where v₁ is the velocity in medium 1 (water in the above figure) and v₂ is the velocity in medium 2 (air in the above figure).

Fermat's Principle 

The objective of this article is to see how this version of Snell’s law arises from a principle known as Fermat’s principle of least action.  This states that light follows a path from one point to another along the path that requires the least travel time. 

To investigate this further, consider the situation shown in the diagram below: 

This diagram shows a light ray moving from point A to point B, through two optical media with differing light velocities.  Our problem is to find the path from A to B that requires the least travel time.  To this end, assume that the ray crosses the boundary between media at a horizontal distance x from the point A.  The horizontal distance between A and B is a.

The total travel time is the sum of the travel times through the two media, which in turn is obtained from the distance travelled divided by the velocity.  Thus the total travel time for the ray in the above diagram is 

The path that minimises the travel time can be found by varying x until the travel time is a minimum.  The appropriate value of x is therefore found from: 

This provides us with the result that we seek.  It can be seen from this condition and the geometry of the diagram above that minimising the time required for the light ray to travel from A to B is equivalent to 

Generalisation of Snell's law

This result can be generalised.  Consider a light ray travelling through three or more optical media, as in the following diagram: 

For this light ray, it can easily be shown (in a similar manner to the case we considered above for two optical media) that the following condition holds if the light ray travel time is to be minimised: 

This result may be generalised even further.  Consider if a given optical medium is split up into a large number of very thin layers, then the requirement that light rays follow the path with the shortest travel time leads to the following constraint on the path: 

This condition applies to every point on the path, and is illustrated in the following diagram.  Note that this constraint does not apply only to light rays, it applies to any object that is following a path that minimises the travel time along that path. 

From the form of the constraint, if the velocity is constant then the angle α is a constant, and the path is a straight line.  If the velocity varies along the path, then α is not constant and the result is a curved path.