KAHIBARO
Table of Contents
When Refraction Stops and Reflection Becomes Complete
Total internal reflection is a special behavior of light that occurs when light tries to pass from one medium into another medium in which it would normally travel faster. Instead of partly refracting into the second medium, the light is reflected completely back into the first medium.
This effect happens only under specific conditions. It is one of the most important ideas in geometrical optics because it explains how optical fibers work, why diamonds sparkle strongly, and why underwater objects can see the outside world through a limited bright region.
The Two Conditions
For total internal reflection to happen, two conditions must both be true.
First, light must travel from a medium with higher refractive index to a medium with lower refractive index. In simple terms, it must go from an optically denser medium to an optically less dense medium. For example, from water to air, or from glass to air.
Second, the angle of incidence must be greater than a certain special angle called the critical angle.
If either condition is not satisfied, total internal reflection does not occur.
Total internal reflection occurs only when
$$n_1 > n_2$$
and
$$\theta_i > \theta_c$$
where $n_1$ is the refractive index of the first medium, $n_2$ is the refractive index of the second medium, $\theta_i$ is the angle of incidence, and $\theta_c$ is the critical angle.
The Critical Angle
The critical angle is the angle of incidence for which the refracted ray travels exactly along the boundary between the two media. That means the angle of refraction is $90^\circ$.
Starting from Snell's law,
$$n_1 \sin \theta_1 = n_2 \sin \theta_2$$
at the critical angle, we set
$$\theta_1 = \theta_c, \qquad \theta_2 = 90^\circ$$
Since $\sin 90^\circ = 1$, we get
$$n_1 \sin \theta_c = n_2$$
so
$$\sin \theta_c = \frac{n_2}{n_1}$$
and therefore
$$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$
This formula makes sense only when $n_1 > n_2$.
The critical angle is given by
$$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right), \qquad n_1 > n_2$$
If $\theta_i > \theta_c$, the light is totally internally reflected.
What Happens as the Angle Increases
If light goes from glass to air, the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction also increases. Eventually the refracted ray reaches $90^\circ$, which is the critical case. If the angle of incidence increases even more, no refracted ray can exist in the second medium, and all the light is reflected back.
This sequence is shown conceptually in the table below.
| Angle of incidence | Result |
|---|---|
| $\theta_i < \theta_c$ | Part of the light refracts, part reflects |
| $\theta_i = \theta_c$ | Refracted ray travels along the boundary |
| $\theta_i > \theta_c$ | Total internal reflection |
Physical Meaning
The second medium cannot accept a refracted ray at an angle larger than $90^\circ$. Since such a refracted ray is impossible, the light remains in the first medium and reflects from the boundary.
Even though we say the light is completely reflected, this is not the same as ordinary reflection from a mirror surface. In total internal reflection, the reflection happens because of the change in medium and the wave behavior at the boundary, not because of a metallic coating.
An important result is that the reflected light obeys the usual law of reflection,
$$\theta_r = \theta_i$$
where $\theta_r$ is the angle of reflection.
In total internal reflection, the reflected ray still satisfies
$$\theta_r = \theta_i$$
Common Examples
A familiar example is light traveling from water into air. If you are underwater and look upward, light from above can enter the water only within a certain angular region. Outside that region, light inside the water reflects from the surface instead of escaping.
Another important example is glass or plastic optical fibers. Light sent into the fiber strikes the inner wall at angles larger than the critical angle, so it keeps reflecting inside and can travel long distances with small loss.
Diamonds also show strong total internal reflection because their refractive index is high. This makes the critical angle small, so light inside the diamond is easily trapped and reflected many times before leaving.
| Boundary | Typical refractive indices | Critical angle |
|---|---|---|
| Water to air | $n_1 \approx 1.33$, $n_2 \approx 1.00$ | $\theta_c \approx 48.8^\circ$ |
| Glass to air | $n_1 \approx 1.50$, $n_2 \approx 1.00$ | $\theta_c \approx 41.8^\circ$ |
| Diamond to air | $n_1 \approx 2.42$, $n_2 \approx 1.00$ | $\theta_c \approx 24.4^\circ$ |
Example Calculation
Suppose light travels from glass into air. Let
$$n_1 = 1.50, \qquad n_2 = 1.00$$
Then
$$\sin \theta_c = \frac{1.00}{1.50} = 0.667$$
so
$$\theta_c = \sin^{-1}(0.667) \approx 41.8^\circ$$
This means that if the angle of incidence inside the glass is greater than $41.8^\circ$, total internal reflection occurs.
If the angle of incidence is $35^\circ$, there will still be refraction into the air. If it is $50^\circ$, the light will be totally internally reflected.
Optical Fibers
Optical fibers are one of the most important applications of total internal reflection. A fiber usually has a central core with refractive index slightly larger than the surrounding cladding. Light stays inside the core because it repeatedly undergoes total internal reflection at the core-cladding boundary.
This guiding of light is why fibers are used in communication systems, medical imaging tools, and sensors.
Comparison with Ordinary Reflection
Ordinary reflection can occur at any boundary, and usually some light is reflected while some is transmitted. Total internal reflection is different because under the right conditions nearly all the light remains in the first medium.
| Feature | Ordinary reflection | Total internal reflection |
|---|---|---|
| Can happen for any direction of travel | Yes | No |
| Requires $n_1 > n_2$ | No | Yes |
| Requires angle larger than critical angle | No | Yes |
| Transmitted refracted ray exists | Usually yes | No |
| Reflection can be nearly complete | Not usually | Yes |
Important Reminders
Students often make two common mistakes. The first is thinking that total internal reflection can happen when light goes from air into glass. It cannot, because this is from lower refractive index to higher refractive index. The second is confusing the critical angle with the angle of refraction. The critical angle is an angle of incidence in the denser medium.
Key ideas to remember:
$$\theta_c$$ is the angle of incidence in the higher-index medium for which the refracted angle is $90^\circ$.
Total internal reflection is impossible when light goes from lower refractive index to higher refractive index.
If $\theta_i > \theta_c$, no refracted ray emerges into the second medium.
Final Picture
Total internal reflection is the complete reflection of light at a boundary when light travels from a higher-index medium to a lower-index medium and the angle of incidence exceeds the critical angle. The essential formula is
$$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$
and the effect is central to many optical technologies, especially optical fibers.