What is the Critical Angle?

What is the Critical Angle and Why does it Occur?

The Quick Answer:

When a wave passes from one medium into another and changes speed, it must also change direction for the wave to be continuous across the interface. This is called refraction.

When passing from a medium where the wave is slower, such as glass, into a medium where the wave is faster, such as air, there are situations where it is impossible to have a wave on the fast side that is continuous with the wave on the slow side. In these situations, there can be no transmission and the wave is completely reflected. This is called total internal reflection (TIR). 

The critical angle is the angle of incidence where TIR begins to happen. All angles of incidence greater than the critical angle will be totally reflected. This happens with all kinds of waves including sound, electromagnetics, and light.

The longer and more complete answer:

To understand what a critical angle is and why it happens, it is most intuitive to visualize the wave itself. The critical angle can be explained through the context of light (an electromagnetic wave), but refraction and critical angles occur for all types of waves.

The speed of a wave inside of a medium is characterized by the medium’s refractive index. The refractive index \(n\) is the factor by which a wave slows down inside of a medium relative to the speed of light in a vacuum \(c_0\). If the refractive index is 2, then the wave propagates at half the speed of light. Air and the vacuum of space are good examples of mediums where waves travel fast. Water and glass are good examples of where waves travel slower than in air.

The speed \(v\), frequency \(f\) and wavelength \(\lambda\) of a wave are related through \(v = f \lambda\). Except in some exotic mediums, the frequency of a wave is constant no matter what medium it is in. So, when a wave changes its speed, the wavelength must change. The speed and wavelength changing as the wave passes through an interface between two mediums is animated in Figure 1 below. The red spheres track the oscillation of the wave as a function of time. Observe the frequency is constant on both mediums.

Figure 1 – A wave incident from material 1 onto material 2 that has a higher refractive index. The wave slows down in the second medium and the wavelength is reduced as a result. Frequency is tracked by the red spheres and is equal in both mediums.

Now, let the incident wave encounter the same interface at an angle, called the angle of incidence. This is illustrated in Figure 2. Observe the ripples of the wave are discontinuous at the interface when the same angle is used in both mediums. This is an impossible situation because it takes an infinite amount of energy to physically realize a discontinuity like this. The fields in both mediums must look exactly the same where they meet at the interface. Since the field in Figure 2 is discontinuous, it is recognized as being impossible and incorrect.

Critical angle fig 2 interface angle

Figure 2 – Same as Figure 1 but the incident wave encounters the interface at an angle. The same angle was used in both mediums causing the wave to be discontinuous at the interface. This is not physically possible.

So how can the discontinuity of the wave in Figure 2 be fixed? The transmitted wave must change its angle to make the wave continuous across the interface. A wave at the correct angle is illustrated in Figure 3. It turns out there is only one possible angle that satisfies continuity. The angle can be calculated using Snell’s law, but a mathematical discussion is not needed for the subject being discussed here.

Critical Angle fig 3 refraction

Figure 3 – Same as Figures 1 and 2 but the angle of the transmitted wave is changed so that the ripples of the wave are continuous across the interface. A change in the direction of a wave is called refraction and a change in speed is why refraction happens.

Figure 4 shows an animation of a wave passing from a lower refractive index medium into a higher refractive index medium over all angles of incidence. Observe the change in wavelength and observe there is always an angle on the transmitted side that makes the wave continuous across the interface. Refraction (bending of a wave at an interface) happens when a wave changes speed across an interface to keep the wave continuous.

Figure 4 – Animation of a wave incident from a low refractive index medium into a high refractive index medium. All angles of incidence produce a transmitted wave, but the angle changes due to refraction.

A more interesting situation happens when a wave passes from a higher refractive index medium into a lower refractive index medium. This is animated in Figure 5. Observe there are large enough angles of incidence where it is impossible to pick any angle in medium 2 that makes the field continuous across the interface. When no angle is possible, the wave in medium 2 is said to be cutoff and there is no transmitted wave. The only thing that can happen is total reflection because there is zero transmission. This is called total internal reflection (TIR).

Figure 5 – Animation of a wave incident from a high refractive index medium into a low refractive index medium. Only a certain range of angles produce a transmitted wave. Angles of incidence larger than the critical angle cannot produce a transmitted wave because there is no angle that makes the ripples of the wave continuous across the interface.

The critical angle is the smallest angle where TIR occurs. It is essentially the cutoff angle for TIR. It is interesting and useful to see that absolutely no math is needed to understand the critical angle and why it happens. It only requires the ripples of the wave to be visualized. 

I will leave you with something to think about. When an incident wave is totally reflected and does not pass into medium 2, how does it know to totally reflect? Maybe I will answer this in a future blog post.

What is the Critical Angle used for?

Perhaps the biggest and most important application of the critical angle is guiding of light inside of fiber optic cables. By 2020, over 5 billion kilometers of optical fiber had been installed around the world –  enough to completely encircle the earth 125,000 times. An optical fiber is like a strand of hair that is made of glass. The middle of the fiber, called the core, has a slightly higher refractive index than the rest of the fiber, called the cladding. Since the core has a higher refractive index than the cladding, light will stay trapped inside the core due to total internal reflection. Optical fibers can carry very high data rate signals over huge distances, like crossing oceans.

The critical angle is easy to measure and can be used to assess the optical properties of materials. For example, an ellipsometer can be used to measure the critical angle. From this, the refractive index of a material can be calculated.

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