Total internal reflection — conditions, critical angle, fiber optics, diamond sparkle

Total internal reflection happens when both conditions are met:

1. Light must travel from a denser medium to a rarer medium ($n_1 > n_2$)
2. The angle of incidence must be greater than the critical angle ($i > i_c$)

If either condition fails, TIR does not occur — you get partial reflection and partial refraction instead.

At the critical angle, the refracted ray grazes the surface (angle of refraction = 90 degrees). Using Snell's law:

$n_1 \sin i_c = n_2 \sin 90°$

$\sin i_c = \frac{n_2}{n_1}$

For glass ($n = 1.5$) to air ($n = 1$): $\sin i_c = \frac{1}{1.5} = \frac{2}{3} \implies i_c \approx 42°$

For diamond ($n = 2.42$) to air: $\sin i_c = \frac{1}{2.42} \implies i_c \approx 24.4°$

The higher the refractive index, the smaller the critical angle.

Diamond has an exceptionally high refractive index ($n = 2.42$), giving a very small critical angle (~24.4 degrees). This means most light rays entering the diamond hit internal surfaces at angles greater than 24.4 degrees and undergo TIR.

The diamond is cut with precise angles so that light bounces multiple times inside before exiting through the top face. Combined with dispersion (splitting of white light into colours), this produces the characteristic "fire" and brilliance.

An optical fiber has a core (higher $n$) surrounded by cladding (lower $n$). Light enters the fiber and hits the core-cladding boundary at angles greater than the critical angle, undergoing repeated TIR.

The light is "trapped" inside the core and can travel kilometres with very little loss. This is the basis of modern telecommunications and endoscopy in medicine.

Acceptance angle ($\theta_a$): the maximum angle at which light can enter the fiber and still undergo TIR inside:

$\sin\theta_a = \sqrt{n_1^2 - n_2^2}$

where $n_1$ = core refractive index, $n_2$ = cladding refractive index.