Snell's law

The numerical rule governing refraction was discovered by Snell, who must have collected experimental data something like what is shown on this graph and then attempted by trial and error to find the right equation. The equation he came up with was

The value of the constant would depend on the combination of media used. For instance, any one of the data points in the graph would have sufficed to show that the constant was 1.3 for an air-water interface (taking air to be substance 1 and water to be substance 2).

Snell further found that if media A and B gave a constant K_AB and media B and C gave a constant K_BC, then refraction at an interface between A and C would be described by a constant equal to the product, K_AC=K_ABK_BC. This is exactly what one would expect if the constant depended on the ratio of some number characterizing one medium to the number characteristic of the second medium. This number is called the index of refraction of the medium, written as "n" in equations. Since measuring the angles would only allow him to determine the ratio of the indices of refraction of two media, Snell had to pick some medium and define it as having n=1. He chose to define vacuum as having n=1. (The index of refraction of air at normal atmospheric pressure is 1.0003, so for most purposes it is a good approximation to assume that air has n=1.) He also had to decide which way to define the ratio, and he chose to define it so that media with their rays closer to the normal would have larger indices of refraction. This had the advantage that denser media would typically have higher indices of refraction, and for this reason the index of refraction is also referred to as the optical density. Written in terms of indices of refraction, Snell's equation becomes

but rewriting it in the form

n₁ sin θ₁ = n₂ sin θ₂ [relationship between angles of rays at the interface between media with indices of refraction n₁ and n₂; angles are defined with respect to the normal]

makes us less likely to get the 1's and 2's mixed up, so this the way most people remember Snell's law. A few indices of refraction are given in the back of the book.

Self-Check (1)	What would the graph look like for two substances with the same index of refraction?
Answer	If n 1 and n 2 are equal, Snell's law becomes sin θ1=sin θ2, which implies θ1=θ2. The graph would be a straight line along the diagonal of the graph.

Self-Check (2)	Based on the graph, when does refraction at an air-water interface change the direction of a ray most strongly?
Answer	The graph is farthest from the diagonal when the angles are large, i.e. when the ray strikes the interface at an oblique or grazing angle.

Snell's Law