Consider three incident rays of light encountering an interface between two media. In this example, the second medium is the slower medium and the rays are refracted towards the normal - note that angle A is greater than angle B in the diagram.

 

Since all rays are perpendicular to their respective wavefronts,

 

 

 

Since all normals are perpendicular to their respective interfaces,

 

 

 

Therefore, and so we can now examine the following new relationships:

where L is the distance along the interface between points P₁ and P₂ as shown in the diagram below.

 

Solving each equation for L yields:

Therefore

 

If d₁ and d₂ represent the distances traveled in the respective mediums during the same amount of time, then we can replace them with the expressions

But v₁ and v₂ represent the speed of the waves in each medium and can be replaced with the expressions

 

 

where n₁ and n₂ are the respective indices of refraction and c is the speed of light.

 

At this junction, we can now write

 

 

Canceling the common terms (c and t) yields

 

 

Or, as Snell's Law is more commonly expressed:

 

 

Notice that Snell's Law shows that the index of refraction and the sine of the angle of refraction are inversely proportional - that is, as the refractive index gets larger [n₂ > n₁] the sine of the refracted angle gets smaller [sinθ₂ < sinθ₁], since the product of the two terms must remain a constant.