Chapter 25 Geometric Optics
25.4 Total Internal Reflection
Summary
- Explain the phenomenon of total internal reflection.
- Describe the workings and uses of fiber optics.
- Analyze the reason for the sparkle of diamonds.
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 Figure 1(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 [latex]{n_1 > n_2}[/latex], the angle of refraction is greater than the angle of incidence—that is, [latex]{\theta _1 > \theta _2}[/latex].) Now imagine what happens as the incident angle is increased. This causes [latex]{\theta _2}[/latex] to increase also. The largest the angle of refraction [latex]{\theta _2}[/latex] can be is [latex]{90 ^{\circ}}[/latex], as shown in Figure 1(b).The critical angle [latex]{\theta _c}[/latex] for a combination of materials is defined to be the incident angle [latex]{\theta _1}[/latex] that produces an angle of refraction of [latex]{90 ^{\circ}}[/latex]. That is, [latex]{\theta _c}[/latex] is the incident angle for which [latex]{\theta _2 = 90^{\circ}}[/latex]. If the incident angle [latex]{\theta _1}[/latex] is greater than the critical angle, as shown in Figure 1(c), then all of the light is reflected back into medium 1, a condition called total internal reflection.
Critical Angle
The incident angle [latex]{\theta _1}[/latex] that produces an angle of refraction of [latex]{90 ^{\circ}}[/latex] is called the critical angle, [latex]{\theta _c}[/latex].
Snell’s law states the relationship between angles and indices of refraction. It is given by
When the incident angle equals the critical angle ([latex]{\theta _1 = \theta _c}[/latex]), the angle of refraction is [latex]{90^{\circ}}[/latex] ([latex]{\theta _2 = 90^{\circ}}[/latex]). Noting that [latex]{\text{sin} \; 90^{\circ} = 1}[/latex], Snell’s law in this case becomes
The critical angle [latex]{\theta _c}[/latex] for a given combination of materials is thus
Total internal reflection occurs for any incident angle greater than the critical angle [latex]{\theta _c}[/latex], 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.
Example 1: How Big is the Critical Angle Here?
What is the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air?
Strategy
The index of refraction for polystyrene is found to be 1.49 in Figure 2, and the index of r