A light ray is incident normally to the long face the hypote

A light ray is incident normally to the long face (the hypotenuse) of a 45degree -45degree -90degree prism surrounded by air, as shown in the figure below. Calculate the minimum index of refraction of the prism for which the ray will totally internally reflect at each of the two sides making the right angle.

Solution

This double internal reflection requires that the angle of incidence (45º) at the 1st prism side is at least = critical angle. This reflected ray then meets the other prism side at the same angle of incidence (45º). so we only need to consider the first internal reflection.

Applying [ n sin i = constant ] to the prism-air boundary.

(n_p= ref.index for prism material, n = ref.index for air =1.0 )
At a critical angle of incidence 45º, external angle = 90º

n_p sin 45 = n sin90
n_p = n sin90 / sin45=1 x 1 / 0.7071

n_p(min) = 1.414