Total Internal Reflection

When Light Gets Trapped Inside

You know from refraction that when light crosses from a denser medium into a rarer one, it bends away from the normal. But what if you keep increasing the angle at which the light hits the boundary? At some point, the refracted ray bends so far that it cannot escape into the second medium at all. The light bounces back entirely into the denser medium. This is total internal reflection, and it is the principle behind optical fibres, sparkling diamonds, and several other striking optical effects.

To build toward this idea, start with something simpler: ordinary internal reflection.

Internal Reflection: The Starting Point

When a ray of light travelling through a denser medium (say water) reaches the boundary with a rarer medium (say air), two things happen at the same time:

• Part of the light reflects back into the water. This is called internal reflection, because the reflection happens at an interface viewed from inside the denser medium.
• Part of the light refracts into the air, bending away from the normal since it is moving into a rarer medium.

This partial reflection, partial refraction behaviour is what you get at most angles of incidence. The reflected portion goes back into the denser medium, while the refracted portion escapes into the rarer medium.

Fig 9.11: Refraction and internal reflection of rays from a point A in the denser medium (water) at different angles of incidence

Increasing the Angle: From Partial to Total

Now picture several rays, all starting from point A inside the water, each hitting the water-air surface at a different angle:

• At a small angle of incidence (like ray $AO_1$), the refracted ray bends away from the normal and enters the air. A partially reflected ray also goes back into the water. Both are present.

• At a larger angle (like ray $AO_2$), the same thing happens, but the refracted ray bends even further from the normal. The angle of refraction is now bigger.

• At a special angle (ray $AO_3$), the refracted ray bends so much that it runs exactly along the surface, making an angle of refraction equal to $90°$. The refracted ray just barely grazes the interface. This angle of incidence is the critical angle, written as $i_c$.

• Beyond the critical angle (ray $AO_4$), something dramatic changes. The refracted ray would need to bend past $90°$, which is physically impossible. Snell’s law cannot produce a valid refraction angle. The result: no light passes into the rarer medium at all. Every bit of light bounces back into the denser medium. This is total internal reflection.

What Makes Total Internal Reflection Special

In everyday reflection from a mirror or any polished surface, the reflected ray is always somewhat weaker than the incoming ray. No matter how smooth the surface, some light gets transmitted through or absorbed. The reflected image is never quite as bright as the original.

Total internal reflection is different in a fundamental way: no light is transmitted at all. Every photon that hits the boundary bounces back. The reflected ray carries the full intensity of the incident ray, with zero loss to refraction. That is what puts the word “total” in total internal reflection.

The Critical Angle and Its Formula

The critical angle ($i_c$) is the angle of incidence in the denser medium at which the angle of refraction in the rarer medium becomes exactly $90°$. It marks the boundary between partial refraction and total internal reflection.

To find its value, apply Snell’s law at the critical angle. When the angle of incidence equals $i_c$, the angle of refraction $r$ equals $90°$:

$\frac{\sin i_c}{\sin 90°} = n_{21}$

Since $\sin 90° = 1$:

$\sin i_c = n_{21} \qquad \text{(9.12)}$

Here, $n_{21}$ is the refractive index of the rarer medium (medium 2) with respect to the denser medium (medium 1). Because the rarer medium has a lower refractive index, $n_{21} < 1$, which guarantees that $\sin i_c < 1$ and a valid critical angle exists.

You can also express this using the refractive index of the denser medium with respect to the rarer medium. Since $n_{12} = 1/n_{21}$ (the reciprocal rule from the previous topic):

$n_{12} = \frac{1}{\sin i_c}$

This form is often more convenient because $n_{12}$ is the familiar refractive index (greater than 1) that you look up in tables.

Why No Critical Angle in the Other Direction?

Notice that total internal reflection only works when light goes from denser to rarer. If light travels from a rarer medium into a denser one, the refracted ray bends towards the normal, making the angle of refraction smaller than the angle of incidence. The refraction angle can never reach $90°$, so there is no critical angle and no total internal reflection. The direction of travel matters.

Why Snell’s Law Breaks Down Beyond $i_c$

For angles of incidence greater than $i_c$, Snell’s law would require $\sin r > 1$. Since the sine of any real angle can never exceed 1, there is simply no valid angle of refraction. The law has no solution, which physically means refraction cannot happen. All the light must go back into the denser medium.

Critical Angles of Common Materials

The critical angle depends on the refractive index: a higher refractive index means a smaller critical angle, so total internal reflection kicks in at a smaller angle of incidence. Here are values for some common materials (with respect to air):

Substance	Refractive Index	Critical Angle
Water	1.33	48.75°
Crown glass	1.52	41.14°
Dense flint glass	1.62	37.31°
Diamond	2.42	24.41°

Notice how diamond has a remarkably small critical angle of just 24.4°. This means that light inside a diamond gets totally internally reflected over a wide range of angles. A skilled diamond cutter exploits this by shaping the facets so that light entering the top bounces around inside through multiple total internal reflections before finally exiting, producing the intense sparkle that diamonds are famous for.

Seeing It for Yourself: A Laser Demonstration

You can observe total internal reflection directly with a simple setup using a laser pointer and water.

Setup: Take a glass beaker and fill it with clear water. Add a few drops of milk or another suspension and stir gently. This makes the water slightly cloudy, which scatters enough laser light to make the beam path visible inside the water.

Step 1: Partial refraction. Shine the laser beam upward from below the beaker so it hits the top water-air surface. At a moderate angle, you will see two spots: one on the table below (from the partially reflected beam going back down) and one on the ceiling or roof (from the partially refracted beam that escapes into the air). Both reflection and refraction are happening together.

Fig 9.12: Observing total internal reflection in water with a laser beam

Step 2: Reaching the critical angle. Now direct the laser beam from the side of the beaker so that it strikes the upper water surface at a more oblique angle (a larger angle of incidence). Gradually adjust the direction. At some point, the spot on the ceiling vanishes completely and the beam bounces entirely back into the water. You have crossed the critical angle, and total internal reflection is happening right in front of you.

Step 3: Mimicking an optical fibre. Pour the turbid water into a long, narrow test tube. Shine the laser in from the top at an angle. Adjust the beam direction until it bounces off the inner walls of the tube repeatedly, zigzagging its way down the tube through a series of total internal reflections. This is exactly what happens inside an optical fibre: light enters one end and bounces along the walls, trapped by total internal reflection, until it emerges at the other end.

Safety note: Do not look directly into the laser beam, and do not point it at anyone’s face.