Blog: What is the critical angle for a light ray traveling from diamond into glass?

The critical angle is defined as the highest angle of incidence that does not result in total internal reflection; it is the angle at which refraction will occur at pi/2 radians (90 degrees), the light traveling along the boundary between the two media.By Snell's Law, we know that angle of refraction and refractive index are related as follows: $\displaystyle{n}_{{1}}{\sin{{\left(\theta_{{1}}\right)}}}={n}_{{2}}{\sin{{\left(\theta_{{2}}\right)}}}$ At the critical angle, by definition $\displaystyle\theta_{{2}}=\frac{\pi}{{2}}$ ,...

By Snell's Law, we know that angle of refraction and refractive index are related as follows:

$\displaystyle{n}_{{1}}{\sin{{\left(\theta_{{1}}\right)}}}={n}_{{2}}{\sin{{\left(\theta_{{2}}\right)}}}$

At the critical angle, by definition $\displaystyle\theta_{{2}}=\frac{\pi}{{2}}$ , so $\displaystyle{\sin{{\left(\theta_{{2}}\right)}}}={1}$ .

$\displaystyle{n}_{{1}}{\sin{{\left(\theta_{{1}}\right)}}}={n}_{{2}}$

Solving for theta_1 gives us the critical angle in terms of the refractive indices:

$theta_1 = sin^{-1} (n_2 / n_1)$

Notice how this will be undefined if $\displaystyle{n}_{{2}}\gt{n}_{{1}}$ ; total internal reflection (and thus, a critical angle) only occurs when light travels to a medium with a lower refractive index. All we need now is to know the refractive indices of glass and diamond. Diamond we know quite precisely as 2.417; glass varies a bit depending on its precise composition, but is usually about 1.5.