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:

1. Light is travelling from a denser medium ($n_1$) towards a rarer medium ($n_2$, $n_2 < n_1$).
2. 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.