25.4 Total internal reflection

Learning objectives

By the end of this section, you will be able to:

• Explain the phenomenon of total internal reflection.
• Describe the workings and uses of fiber optics.
• Analyze the reason for the sparkle of diamonds.

The information presented in this section supports the following AP® learning objectives and science practices:

• 6.E.1.1 The student is able to make claims using connections across concepts about the behavior of light as the wave travels from one medium into another, as some is transmitted, some is reflected, and some is absorbed. (S.P. 6.4, 7.2)

A good-quality mirror may reflect more than 90% of the light that falls on it, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on it. Interestingly, we can produce total reflection using an aspect of refraction .

Consider what happens when a ray of light strikes the surface between two materials, such as is shown in [link] (a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. (Since $n_1>n_2$ , the angle of refraction is greater than the angle of incidence—that is, ${\theta }_2>{\theta }_1$ .) Now imagine what happens as the incident angle is increased. This causes ${\theta }_2$ to increase also. The largest the angle of refraction ${\theta }_2$ can be is $\text{90º}$ , as shown in [link] (b).The critical angle     ${\theta }_c$ for a combination of materials is defined to be the incident angle ${\theta }_1$ that produces an angle of refraction of $\text{90º}$ . That is, ${\theta }_c$ is the incident angle for which ${\theta }_2=\text{90º}$ . If the incident angle ${\theta }_1$ is greater than the critical angle, as shown in [link] (c), then all of the light is reflected back into medium 1, a condition called total internal reflection .

Snell’s law states the relationship between angles and indices of refraction. It is given by

When the incident angle equals the critical angle ( ${\theta }_1={\theta }_c$ ), the angle of refraction is $\text{90º}$ ( ${\theta }_2=\text{90º}$ ). Noting that $\text{sin 90º}\text{=1}$ , Snell’s law in this case becomes

The critical angle ${\theta }_c$ for a given combination of materials is thus

Total internal reflection occurs for any incident angle greater than the critical angle ${\theta }_c$ , and it can only occur when the second medium has an index of refraction less than the first. Note the above equation is written for a light ray that travels in medium 1 and reflects from medium 2, as shown in the figure.