Lesson Explainer: Total Internal Reflection | Nagwa Lesson Explainer: Total Internal Reflection | Nagwa

# Lesson Explainer: Total Internal Reflection Physics

In this explainer, we will learn how to define total internal reflection (TIR) by describing rays produced by light rays incident at or past the critical angle for TIR.

When a ray of light arrives at an interface of two different materials, typically, part of the ray will be refracted and part will be reflected.

Refraction occurs when the light ray is redirected across the interface from one material into the other.

Reflection happens when the ray bounces off the interface and never leaves the first material.

The way an incoming ray of light reflects or refracts at an interface depends on two factors:

• The angle of incidence of the ray
• The ratio of the refractive index of the first material to the refractive index of the second material

Given two different materials and a ray of light traveling from the material with relatively higher refractive index to the one with relatively lower refractive index, increasing the ray’s angle of incidence will increase its angles of refraction and reflection.

Note that because material 2 has a lower index of refraction than material 1, the refracted portion of the ray bends away from the dashed line perpendicular to the boundary between the materials. Therefore, the angle of refraction is greater than the angle of incidence.

This means that if the ray’s angle of incidence is increased further, the angle of refraction will reach before the angle of incidence does.

At this specific angle of incidence, known as the critical angle, part of the incoming ray is refracted at —no light crosses the interface between the two materials. For any angle of incidence greater than the critical angle, all of the incident light is reflected, a phenomenon known as total internal reflection.

### Definition: Total Internal Reflection (TIR)

Suppose a ray of light travels from a medium with higher refractive index to an interface with a material of lower refractive index. If the ray reaches the interface at an angle of incidence greater than the critical angle for the materials involved, none of the ray will be refracted and all of it will be reflected. This complete reflection is called total internal reflection (TIR).

### Example 1: Determining Which Ray Leads to Total Internal Reflection

The rays A, B, and C are shown incident on a boundary between water and air, traveling in water. Which incident ray produces total internal reflection?

We see rays A, B, and C are all initially in the material (water) with higher index of refraction (compared to air). This means total internal reflection at this interface is indeed a possibility, requiring only a certain minimum angle of incidence.

Ray A is partly reflected and partly refracted at the water–air boundary. Since not all of the ray is reflected, ray A does not produce TIR.

Ray B is reflected at the boundary and also refracted, though the refracted ray follows the interface between the materials. We can say then that ray B reaches the boundary at the critical angle for water and air. For TIR to occur, it is necessary for a given ray’s angle of incidence to be greater than the critical angle. Therefore, ray B does not produce TIR.

The diagram shows that ray C has no refracted ray; it is entirely reflected. This indicates TIR, and our answer is therefore that ray C produces TIR.

### Example 2: Finding the Correct Ray Path from Air into Water

The diagram shows light rays in water being totally internally reflected at a boundary with air. The diagram also shows a light ray in air incident on the same boundary and at the same angle of incidence. Which of the paths A, B, and C would light incident on water from air follow?

Here we compare light incident from one side of an air–water interface to light incident from the other side. We see that the light moving through the water to reach the boundary is totally internally reflected. Total internal reflection (TIR) is only possible for light traveling from a higher-refractive-index material to a lower-refractive-index one.

Therefore, the light traveling through the air to reach the water at the same angle of incidence cannot be totally internally reflected. We know then that option C, by itself, cannot be a correct answer choice.

However, path C does show how light would reflect from the air–water interface. In general, when light is incident on a boundary, some of the light reflects. Thus, we expect that a portion of the light reaching the water surface in this instance will reflect as indicated by path C.

If the ray followed path B, that would mean the light was not refracted at the boundary between water and air. This is only possible if the two materials have the same index of refraction. Water, however, has a refractive index of about 1.3, while air’s index of refraction is nearly 1. Since the index of refraction across the boundary does change, the incident ray cannot continue in the same direction after crossing it. No portion of the ray will follow path B.

Path A shows the rays being refracted toward the dashed line perpendicular to the interface. Since water’s refractive index is higher than air’s, we would expect this type of redirection of the ray. Path A is one path that the incident ray would tend to follow.

Since we expect part of the incident ray to reflect from the interface and follow path C and part to refract into the water and follow path A, our final answer is that the incident ray follows paths A and C.

A common application of total internal reflection occurs in optical fibers.

Optical fibers are flexible cables that transmit light from one end of the fiber to the other with little loss.

A typical fiber consists of several layers of material, shown below.

The fiber core is made of glass or plastic; this is the medium in which light totally internally reflects as it travels along the fiber.

The fiber cladding is made of a material with a lower index of refraction than the core. As we have seen, this is a necessary condition for total internal reflection to occur.

The coating is designed to give the fiber strength and serve as a buffer material.

Optical fibers are used in long-distance telecommunications, sensing systems, and medical devices, with many other applications.

In the following two examples we consider light as it travels along an optical fiber core.

### Example 3: Comparing Angles of Incidence to the Critical Angle

The diagram shows two different light rays propagating through an optical fiber cable. Two angles are shown, one in green and one in blue. Which of the following statements about these angles is correct?

1. The blue angle is larger than the critical angle, but the green angle is smaller than the critical angle.
2. The blue angle is smaller than the critical angle, but the green angle is larger than the critical angle.
3. The blue angle and the green angle are both smaller than the critical angle.
4. The blue angle and the green angle are both larger than the critical angle.

Considering the green and blue rays of light in the optical fiber, we see that the green ray remains within the fiber while the blue ray escapes it.

We can say, therefore, that the green ray is totally internally reflected. This means that, at every encounter with the walls of the fiber, the green ray’s angle of incidence is greater than the critical angle.

Reviewing the diagram, we see the blue ray’s angle of incidence and the green ray’s angle of incidence for one encounter with the fiber’s wall.

The angle of incidence of the blue ray is visibly smaller than that of the green ray, confirming that a given ray’s angle of incidence must be greater than the critical angle for the ray to be totally internally reflected.

Because the blue ray is refracted out of the fiber, its angle of incidence must be less than the critical angle. The green ray’s angle of incidence, on the other hand, must be greater than that angle since it is totally internally reflected.

Answer option B is a correct statement about the green and blue angles indicated on the diagram.

### Example 4: Determining How a Ray Reflects and Refracts at an Interface

A light ray travels along an optical fiber by total internal reflection, as shown in the diagram. The light reflects from the internal surface of the fiber at an angle very slightly greater than the critical angle. What is the most correct description for what happens at point P if light is sent into the fiber when the fiber is bent?

1. Some of the light is transmitted out of the fiber at P because the angle of incidence there is greater than the critical angle.
2. Some of the light is refracted along the fiber surface at P because the angle of incidence there is equal to the critical angle.
3. Some of the light is transmitted out of the fiber at P because the angle of incidence there is less than the critical angle.
4. The light totally internally reflects at P because the angle of incidence there is greater than the critical angle.
5. The light totally internally reflects at P because the angle of incidence there is less than the critical angle.

The light totally internally reflected in the fiber is shown to be incident on the walls of the fiber at an angle just slightly greater than the critical angle. By inspection, this angle looks larger than .

The ray incident on the fiber wall at point P appears to have a much smaller angle of incidence than this.

Since the light traveling through the straight fiber was incident at an angle just marginally greater than the critical angle, we can reasonably assume the ray’s angle of incidence when encountering point P is less than the critical angle.

Therefore, the light hitting point P will not all be reflected; some will refract and escape to the outside of the fiber.

Considering our five answer options, the fact that the angle of incidence at point P is less than the critical angle eliminates three choices: A, B, and D.

We also recognize that incidence at less than this angle means some of the incident light will be refracted out of the fiber; not all of it will reflect back. This eliminates option E and indicates that option C is the most correct description of what happens to the light at point P.

### Example 5: Defining Total Internal Reflection

Which of the following definitions best describes the critical angle for total internal reflection?

1. The critical angle is the angle at which the refracted ray travels along the boundary from which the incident ray is reflected.
2. The critical angle is the angle of incidence minus the angle of refraction.
3. The critical angle is the angle at which all incident light at a boundary is reflected.
4. The critical angle is the angle of incidence plus the angle of refraction.
5. The critical angle is the angle of refraction minus the angle of incidence.

Let’s first recall what the critical angle for total internal reflection is.

When a ray of light is incident on a boundary between two materials of different refractive indices, part of that ray refracts and travels along the boundary when the ray is incident at the critical angle.

If the angle of refraction was less than , or there was no refracted ray at all, then the angle of incidence would not be equal to the critical angle. The hallmark of the critical angle is a ray refracted parallel to the interface between materials.

Looking at our answer options, note that options B, D, and E are incorrect because the critical angle is simply the angle of incidence of a ray when that angle leads to a angle of refraction.

Option C claims that the critical angle is the angle at which all light incident at a boundary is reflected, but the diagram above shows that at this angle a refracted ray still exists.

Therefore answer option A is all that remains. This option describes the situation shown above, where a ray incident on an interface at the critical angle is partly reflected and partly refracted at to a line normal to the interface.

### Example 6: Choosing the Best Description of Total Internal Reflection

Which of the following sets of conditions best describes total internal reflection?

1. Total internal reflection occurs when a light ray is completely reflected at the boundary between two media of different densities and the light ray is traveling in the lower-density medium.
2. Total internal reflection occurs when the reflected light ray moves along the same path as the incident light ray but in the opposite direction.
3. Total internal reflection occurs when a light ray is completely absorbed in an object.
4. Total internal reflection occurs when a light ray is completely reflected at the boundary between two media of different densities and the light ray is traveling in the higher-density medium.
5. Total internal reflection occurs when the angle of reflection of the reflected light ray is equal to the angle of refraction of the refracted ray.

For TIR to occur, a ray of light traveling in a more optically dense material must encounter a less optically dense material at an angle of incidence greater than the critical angle.

Under these conditions, the incident ray is completely reflected from the interface, with no light being refracted across the interface.

Reviewing our five answer choices, option E claims that even for TIR a refracted ray exists. This is not true, so option E is not our final choice.

Option A says that TIR involves a ray that is completely reflected from a boundary, which is true, but that this ray is traveling in the less-dense material of the two different materials making up the interface. Because of the way light refracts when moving from a medium with a lower to one with a higher index of refraction, TIR under this condition is not possible.

Answer B claims TIR involves a reflected ray of light following the same path as an incident ray. Because of the law of reflection, which states that the angle of incidence of a ray of light equals the angle of reflection of that ray, this option implies that TIR only happens when a ray is normally (i.e., perpendicularly) incident on a boundary.

Under this requirement, however, a refracted ray of light would exist, meaning the incident ray was not totally reflected.

Choice C says TIR involves a light ray being completely absorbed, but this would mean none of the ray was reflected, a necessary condition for TIR.

Lastly, option D claims that TIR involves complete reflection of an incident beam off an interface where the incident ray travels through the more optically dense medium at the boundary of the two. This best describes total internal reflection.

### Key Points

• For a light ray traveling through a medium of greater density toward a less-dense medium, the critical angle is the angle of incidence of that ray, which refracts along the boundary between the different media.
• A ray incident on such a boundary at an angle greater than the critical angle will be completely reflected from the interface. This phenomenon is known as total internal reflection (TIR).