In this explainer, we will learn how to determine whether two or more electromagnetic waves will interfere to form coherent or incoherent light.

A light wave, or electromagnetic wave, is a transverse wave; that is, a wave that oscillates in a direction perpendicular to the direction it is traveling in. We can describe these light waves as βsinusoidal,β which means that they look like a sine or cosine function.

Letβs imagine that we have a light wave traveling from left to right in empty space, as shown in the diagram below.

We can recall that this wave will have a certain amplitude, wavelength, and frequency. The amplitude and wavelength are shown in the above diagram.

The amplitude of a wave is the maximum displacement of the wave from the centerline (the dashed line in the diagram). This is the height to the top of a crest or the bottom of a trough.

The wavelength of a wave is the distance between any two adjacent equivalent points on the wave. This is also equal to the spatial distance covered by one complete cycle of the wave.

The frequency of a wave is the number of complete cycles of the wave passing a given point in space per second.

We may further recall that the frequency, , and wavelength, , of a light wave are related through the formula where is the speed of light.

Since the speed of light in a given medium is the same for all light waves, then we can see from this equation that two light waves that have the same wavelength as each other must also have the same frequency.

If we want to be able to compare two or more different waves to each other, then we also need to consider another property. This property is the phase of the wave.

The phase of a wave is a measure of how far through a cycle the wave is at a particular point in space. It is common to take a cycle to begin at the point marked in the diagram below, where the displacement is zero.

Then, at this position, we would say that the wave has a phase of zero.

Now, letβs consider the position shown by the pink line in the diagram below.

At the position shown by the pink line, the wave is one-quarter of the way through a cycle. In terms of phase, we could describe this as the wave having a phase of one-quarter of a cycle or, equivalently, one-quarter of a wavelength.

In a similar way, the phase of the wave may be defined at any position along it.

It is worth pointing out that we measure phase relative to the start of a cycle. Since the two points on the wave marked in the diagram below are identical to each other, the left-hand point defines the start of one cycle, while the right-hand point defines the start of the next cycle. These two points both have a difference of one wavelength between them. Points separated by one wavelength have the same phase as each otherβin this case, a phase of zero.

We said that an understanding of phase was necessary in order to be able to compare different waves to each other. So letβs now see how this works.

Let us imagine that we have two light waves traveling from left to right through empty space, as shown in the diagram below.

To compare these two waves, we could measure the distance between adjacent peaks on each wave to find the wavelength, as shown below.

In the diagram, we have labeled the wavelength of the upper wave as and the wavelength of the lower wave as . We can see from the diagram that the distance between two adjacent peaks on the upper wave is the same as the distance between two adjacent peaks on the lower wave. That is, these two waves have the same wavelength as each other. We could express this mathematically by saying that .

Recall that two waves with the same wavelength as each other also have the same frequency. This means we can also say that the two waves in the diagram must have the same frequency as each other.

Another way to compare these waves is by looking at their phases.

Specifically, we could pick a point along the direction of travel of the waves and see how far through a cycle each of the waves is at that point. This is illustrated for two different points in the diagram below.

At the position marked by the orange dashed line, we can see that each of the waves is at the start of a cycle. At the position marked by the pink dashed line, each wave is three-quarters of the way through a cycle. So, at both these positions, the waves have the same phase as each other.

In fact, it is easy to verify that we could draw a vertical line at any position along the waves, and we would find that both waves have the same phase as each other at that position.

When two waves have the same phase as each other at a particular point in space, we say that they have a phase difference of zero at that point.

More generally, the phase difference between two waves is the difference between the phase of one wave and the phase of the other.

Suppose that our two waves looked as shown below.

At the position marked by the orange dashed line, the upper wave has a phase of zero since it is at the start of a cycle. Meanwhile, the lower wave is a quarter of the way through a cycle. Therefore, the phase difference between these two waves at this position is a quarter of a cycle.

If we now look at the pink dashed line, we see that the upper wave is three-quarters of the way through a cycle. The lower wave is all the way through the cycleβor, equivalently, at the start of the next cycle. Therefore, the phase difference between the waves is also a quarter of a cycle at this position.

As with our two waves that had a phase difference of zero, we can verify that the phase difference between these two waves will be a quarter of a cycle no matter where we draw our vertical line.

When the phase difference between two waves is the same for all positions along the waves, we can say that those two waves have a constant phase difference between them. Here, βconstantβ means that the phase difference is not changing as a function of position along the waves.

Whenever we have two waves that have the same frequency as each other, we know that they also have the same wavelength. This means that both waves take the same distance to complete one cycle. In this case, whatever the phase difference between the two waves at a particular point is, this phase difference will be the same if we look at a second point at another position along the waves. This holds true no matter at what second point we choose to observe the phase difference. The phase difference between two waves with the same frequency will be the same at all positions along the waves.

This fact leads us to the definition of coherence.

### Definition: Coherence

Two or more waves are said to be coherent if they have the same frequency and, therefore, a constant phase difference from each other.

It is important to note that though the phase difference between two waves must be constant in order for them to be coherent, that constant phase difference does not have to be zero. Therefore, both pairs of waves we have considered so far would be coherent with each other.

Visually, the easiest way to identify if two waves are coherent is by looking at the distance each wave takes to complete one cycle of oscillation. If we can see that two waves complete one cycle over the same distance as each other, then we know that the waves have the same wavelength. This tells us that the waves must also have the same frequency as each other and a constant phase difference between them; that is, the two waves fit our definition of coherent waves.

Letβs look again at the two waves that had a quarter of a cycle phase difference between them.

On the diagram, we have indicated the distance over which each of the waves completes one cycle of oscillation. We have labeled the distance in each case as it is equal to the wavelength of the waves. We see that it is the same distance for both waves. This therefore provides a visual confirmation that the two waves are coherent.

It is worth pointing out that we can measure the distance for one complete oscillation between any two equivalent points on the wave. Here, we chose the position corresponding to the start of a cycle. However, we could equally have chosen to measure the distance between two peaks, or the distance between two troughs.

Two waves that are not coherent are said to be incoherent. In this case, the waves have a different frequency to each other, which means they also have a nonconstant phase difference.

So, what might two incoherent waves look like? Since literally any two waves that do not fit the definition of coherent waves are automatically incoherent, it turns out that there are lots of possibilities. The two waves in question do not have to be sinusoidal or to even have fixed frequencies.

However, here we are focused on electromagnetic radiation. In this case, we will always have sinusoidal waves, and those waves will have fixed frequencies. An example of two incoherent sinusoidal waves is shown in the diagram below.

In this case, we can see that the two waves have different wavelengths. Therefore, these waves have different frequencies and a nonconstant phase difference, so we know that the two waves cannot be coherent.

We can also confirm that the phase difference between the waves is not constant by considering two different horizontal positions on the waves as follows.

If we look at the position marked by the orange line, we see that both waves are at the start of a cycle. Therefore, both waves have the same phase at this position, and so the phase difference between them is zero. At the position marked by the pink line, the upper wave is halfway through the cycle, while the lower wave is three-quarters of the way through the cycle. So, at this position the phase difference between the waves is a quarter of a cycle.

Since we have a different phase difference at two positions along the waves, we see that they cannot have a constant phase difference between them.

Letβs have a look at a couple of examples.

### Example 1: Determining Which of a Group of Waves Is Not Coherent

The figure shows five light waves. Which light wave is **not** coherent with the
other four?

### Answer

This question shows us five different light waves and asks us to work out which of them is not coherent with the others.

We can recall that in order for waves to be coherent, they must have the same frequency and a constant phase difference from each other.

We can further recall that two waves with the same frequency must also have the same wavelength, and so we must look for the wave that does not have the same wavelength as the other four.

We can see this most easily by drawing two vertical lines onto the diagram, as shown in the diagram below.

At the position of the orange line, labeled 1, all five of the waves are at the start of a cycle.

At the position of the pink line, labeled 2, waves i, ii, iii, and v have all completed one cycle and are at the start of the next cycle. However, wave iv has only completed half a cycle between the two lines; at the position of the pink line it is halfway through a cycle.

Wave iv has a different wavelength to the other four, so wave iv is not coherent with the other four.

It is also worth pointing out that we can see that the five different waves have different amplitudes to each other; that is the maximum and minimum heights to which they oscillate differ. While this does mean that all the waves look different to each other, this property is not relevant to the coherence of the waves.

### Example 2: Determining Which Group of Light Waves is Incoherent

In each of the following diagrams, five light waves are shown. Which of the diagrams shows incoherent light?

### Answer

This question presents us with five different diagrams, each showing five different light waves, and asks us which of these five diagrams shows incoherent light.

We can recall that for two or more light waves to be coherent, they must have the same frequency and a constant phase difference from each other. When we have light waves that do not meet these criteria, those light waves are incoherent.

We can also recall that a way of visually identifying coherent waves is that waves that are coherent will all have the same wavelength.

Letβs look at each of the diagrams in turn in order to establish whether or not the waves are coherent.

We will begin with diagram A.

We can see that the distance taken for one complete cycle is the same for all waves. Each wave goes from the peak of one cycle to the peak of the next cycle over the distance marked by the two orange dashed lines. This means that the waves in diagram A are coherent.

Now letβs look at diagram B.

In this case, the top two waves and the bottom two waves all have the same wavelength. However, the middle wave has only completed approximately three-quarters of an oscillation over the same distance in which the others have completed a full cycle of oscillation. Therefore, the middle wave in this diagram is not coherent with the other four, and so this diagram shows incoherent light.

We still need to check whether the remaining diagrams show coherent light.

Letβs look at diagram C.

The distance taken for one complete cycle is the same for all waves. This means that the waves in diagram C are coherent.

Now letβs look at diagram D.

Again, all the waves take the same distance to complete one cycle. Therefore, the waves in diagram D are coherent.

Finally, we will look at diagram E.

Once again, the waves all take the same distance to complete one cycle. Therefore, the waves in diagram E are coherent. In fact, in this case, the waves also appear to have the same amplitude as each other. So, not only are these waves coherent, but they are also actually identical to each other.

In summary, we have found that the waves in diagrams A, C, D, and E are coherent. Meanwhile, the middle wave in diagram B is incoherent with the other four, so diagram B shows incoherent light.

We have seen that we may examine the shape of the wave to determine the properties of a particular light wave and how we may compare two such pictures in order to determine whether two waves are coherent or incoherent.

We can also do this in another way.

We said at the beginning of this explainer that light waves oscillate sinusoidally. This means that we can describe them mathematically using the sine function.

Consider the following two functions:

Plotting these two functions, we get the following graphs.

We see that the graph (the red line) oscillates between a displacement of 1 and a displacement of . Meanwhile, the graph of (the blue line) oscillates with twice the amplitude, between a displacement of 2 and a displacement of .

More generally, the function describes a sine wave that oscillates between and .

When using a sine function to describe a light wave, we can see that the value of in this function would describe the amplitude of the wave.

Now, instead, consider the following two functions:

If we plot these two functions, we get the following graphs.

The graph (the red line) takes to complete one cycle of oscillation. Meanwhile, the graph of (the purple line) takes half this distance, , to complete one cycle.

More generally, for the function the larger the value of , the smaller the distance over which the sine wave completes each cycle.

When using a sine function to describe a light wave, we can see that the value of is related to the wavelength of the wave. The bigger the value of , the shorter the wavelength.

Putting these two bits of information together, we may describe a light wave mathematically as follows.

### Definition: The Mathematical Description of a Light Wave

We can describe a light wave mathematically using the following equation:

In this equation, is the amplitude of the wave and is related to the wavelengthβthe larger the value of , the shorter the wavelength of the wave.

By looking at the mathematical functions describing two or more waves, we are able to determine whether or not those waves are coherent. Letβs see how this works by considering two light waves described by the following functions:

We can recall that in order for two waves to be coherent, they must have the same frequency and a constant phase difference between them. We have seen already that two light waves that have the same wavelength will have the same frequency, while two light waves with different wave lengths will have different frequencies.

We know that the s in these functions relate to the wavelengths of the waves. Therefore, for the two waves to be coherent, we must have that

By extension, if is not equal to , then the two waves must be incoherent.

Since the definition of coherence is independent of the amplitudes of the waves, the values of and have no effect on whether or not the two waves are coherent.

Letβs finish up by having a look at an example involving mathematical descriptions of light waves.

### Example 3: Determining Which Wave Is Not Coherent Using the Functions Describing the Waves

The following five functions can be used to model five light waves:

Which of the five waves is **not** coherent with the other
four?

### Answer

We are asked to work out which of the functions given to us describes a wave that is not coherent with the other four.

Letβs recall that in order to be coherent, waves need to have the same frequency and a constant phase difference from each other.

We can also recall that the value that is multiplied by inside the sine function is related to the wavelength of the wave and that waves with the same wavelengths as each other also have the same frequencies.

In the case of waves i, ii, iv, and v, the value that gets multiplied by is 1. Since this is the same for all four of these waves, they must all have the same wavelength as each other. Therefore, they must also have the same frequency as each other, which means that they are coherent.

However, for the wave iii, gets multiplied by a factor of 2. This means that the wavelength of wave iii is not the same as the other four, and so we know that this wave also has a different frequency to the others. Therefore, it cannot be coherent with them.

So, our answer to the question is that it is wave iii that is not coherent with the other four waves.

### Key Points

- The phase of a wave is a measure of how far through the cycle that wave is at a particular point in space and time.
- Two or more waves are coherent if they have the same frequency and the phase difference between them is constant.
- Any two waves that are not coherent are said to be incoherent.
- We may determine whether or not two waves are coherent by looking at the waveforms. In this case, we may draw lines through the waves perpendicular to their direction of travel to help confirm whether they have the same phase at the position of each line.
- We may also determine whether or not two waves are coherent by looking at the functions describing those waves. If we have two waves described by the functions and , then these two waves are coherent if . If this condition is not met, the waves are incoherent.