Total Internal Reflection

When light tries to leave a denser medium for a rarer one at a large enough angle, it cannot refract — it is reflected back entirely. This is total internal reflection (TIR).

Concept

Conditions for TIR:

Light is travelling from a denser medium ( $n_1$ ) towards a rarer medium ( $n_2$ , $n_2 < n_1$ ).
Angle of incidence (in the denser medium) exceeds the critical angle $i_c$ .

At $i = i_c$ , the refracted ray grazes the surface ( $r = 90^\circ$ ). From Snell's law:

$n_1 \sin i_c = n_2 \sin 90^\circ = n_2$

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

For glass-air ( $n_1 = 1.5$ ): $\sin i_c = 1/1.5 = 0.667$ , so $i_c \approx 41.8^\circ$ .

Derivation

Starting from Snell's law:

$\sin r = \frac{n_1}{n_2}\sin i$

For $n_1 > n_2$ , the ratio $n_1/n_2 > 1$ , so $\sin r$ can exceed 1 when $\sin i > n_2/n_1$ . Mathematically there is no real angle $r$ ; physically the wave is totally reflected. The critical angle is precisely where $\sin r = 1$ :

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

Above this incidence angle, all the energy is reflected (no losses, unlike ordinary mirror reflection where some light is absorbed).

Worked Example

The critical angle for diamond-air interface ( $n = 2.42$ ):

$\sin i_c = \frac{1}{2.42} = 0.413,\quad i_c \approx 24.4^\circ$

This small critical angle is why diamonds sparkle: a wide range of incidence angles undergoes TIR multiple times inside the stone before light exits, producing intense flashes.

Applications

Optical fibres. A glass core of high $n$ is surrounded by a cladding of slightly lower $n$ . Light entering one end undergoes thousands of TIR events inside the core and propagates with low loss.
Mirage. Hot air near a road has lower $n$ than cooler air above. A grazing ray from the sky bends gradually, then undergoes TIR in the hot layer near the surface, producing the illusion of a reflective pool.
Prism periscope and binoculars. Right-angled prisms use TIR to fold the light path, replacing mirrors.

Common Confusions

TIR happens only when light is in the denser medium going towards the rarer one. Reverse the direction, and the ray simply refracts.
"Total" means 100% reflection — no transmitted ray (in the geometrical-optics limit). Real systems have small evanescent fields but no energy transfer to the rarer medium.
The critical angle depends on the refractive indices of both media; it is not a fixed property of one substance.

Key Takeaways

TIR occurs for $i > i_c$ when light goes from denser to rarer medium.
$\sin i_c = n_2/n_1$ .
100% reflection — applications include optical fibres, prisms, mirages, diamond brilliance.