Total internal reflection

When light propagates from air into glass or from glass in to air it may change its direction of travel.  Snell's law reveals the relationship between the directions of travel in the two media.
n₁sinθ₁ = n₂sinθ₂.
Consider light propagating in glass with index of refraction n₁ = 1.5 towards a glass-air boundary.  If the angle the light makes with the normal to the boundary in the glass is θ₁, then the angle it makes in the air is given by

sinθ₂ = (n₁/n₂)sinθ₁ = 1.5 sinθ₁.

If sinθ₁ > (1/1.5) = 2/3, or θ₁ > 41.8^o, then sinθ₂ is greater than 1 and there is no solution for θ₂.
The angle θ₁ for which sinθ₂ = 1 is called the critical angle θ_c.  We have sinθ_c = n₂/n₁.
For angles θ₁ greater than the critical angle there exists no solution for θ₂, and there is no refracted ray.
The incident light is totally reflected, obeying the law of reflection.  If n₂ = 1.5 and n₁ = 1 then the critical angle is θ_c = 41.8^o

Total internal reflection occurs only if light travels from a medium of high index of refraction to a medium of low index of refraction. 

For angles greater than the critical angle the incident light is totally reflected, obeying the law of reflection.

Problem:

A light source is at the bottom of a pool of water (n = 1.33).  At what minimum angle of incidence will a ray be totally reflected at the surface?

Solution:

• Reasoning:
  The angle θ_c for which sinθ_c = n₂/n₁ is the critical angle.  For angles greater than the critical angle there is no refracted ray.
• Details of the calculation:
  For total internal reflection sinθ ≥ sinθ_c = 1/1.33, θ ≥ 48.7^o.

Problem:

Assume you are using total internal reflection to make a corner reflector. If there is air outside and the incident angle is 45.0^o, what must be the minimum index of refraction of the material from which the reflector is made?

Solution:

• Reasoning:
  For total internal reflection we need sinθ ≥ sinθ_c = n₂/n₁.  Here n₂ = 1.  Therefore sinθ_c = 1/n.
• Details of the calculation:
  We need  sin(45^o) > 1/n, or  n > 1/sin(45^o) or n > √2 = 1.414.