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, m∠C = m∠A and m∠D = m∠B and 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:

 

d₁/sin(A) = d₂/sin(B)

 

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

 

d₁/sin(A) = d₂/sin(B)
v₁t/sin(A) = v₂t/sin(B)
(c/n₁)[t/sin(A)] = (c/n₂)[t/sin(B)]

 

Canceling the common terms (c and t) yields

 

sin(A)/n₁ = sin(B)/n₂
n₁ sin(A) = n₂ sin(B)

 

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

 

n₁ sin(θ₁) = n₂ sin(θ₂)

 

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 [sin(θ₂) < sin(θ₁)]. Since sin(θ) is an increasing function in the first quadrant, as sin(θ) decreases, so does θ.