### Video Transcript

In this video, we will be learning about coherent light or, more generally, coherent waves. To understand what this means, let’s first start by recalling that light is an electromagnetic wave. We can therefore model light — which, in this case, is traveling in empty space, let’s say, from left to right — as a sinusoidal wave, a wave shaped like this. Let’s also recall that this light wave will have a certain frequency, amplitude, and wavelength.

Now, let’s imagine that we as observers stand at a certain point in space. Let’s say that we are standing here. And as our light wave moves from left to right, we start counting the number of complete cycles of our wave passing our observation point per unit time. This quantity that we’ve just measured is known as the frequency of our electromagnetic wave. And in this particular case, we’ve measured our frequency, which we will call 𝑓, to be two cycles passing our observation point per second. And we can recall that frequency has its own unit, the hertz. And so the frequency of our light wave here is two hertz.

Okay, now let’s think about our light wave in a slightly different way. Let’s recall that this light wave being an electromagnetic wave travels from left to right, as we’ve already said. And the reason that it can do this is because part of the electromagnetic field is oscillating in a direction perpendicular to this. Specifically, in this case, it’s oscillating up and down whilst the wave travels to the right. In other words, part of the electromagnetic field is displaced either upward or downward relative to this dotted line that we’ve drawn here. And the distance between the dotted line and the maximum displacement of our wave — so that’s this distance here — is known as the amplitude. In other words then, the amplitude is the maximum displacement of an oscillating point along a wave.

And finally, there’s a third property that we can measure which is the spatial distance covered by one complete cycle of the wave. In this case, this distance here for example, or this distance here, which is the distance between two adjacent equivalent points along the wave or, more importantly, the distance covered by one complete cycle of the wave. This is the wave’s wavelength, which means that we’ve now recalled three different properties of a wave. And all of these properties can be used to describe a wave in great detail. However, if we want to compare one wave to another, then we need to introduce another property to our list. We will see shortly why we would want to compare waves with each other. But first, let’s look at a new property that we can measure for a particular wave.

This new property, known as the phase of a wave, is a measure of how far through a cycle the wave is at a particular point in space. Now, to understand this a little bit more, let’s recall how we plot a sine curve on a graph. Specifically, if we start by building ourselves a set of axes, where we call the horizontal axis 𝑥 and the vertical axis 𝑦, and then we plot the curve 𝑦 is equal to the sin of 𝑥. Then, what we end up plotting looks very similar to our light wave from earlier. In fact, this is why the wave that we’ve drawn here is said to be sinusoidal, because it looks like a sine curve. And by the way, it’s worth noting that we’ve only plotted one complete cycle of our sine curve on our graph. And the reason for this is the following.

Let’s start by recalling the values of 𝑥 for which sin 𝑥 is equal to zero, sin 𝑥 is maximum, sin 𝑥 is equal to zero once again, sin 𝑥 has its minimum value, and sin 𝑥 is equal to zero once again. We can recall that these values are 𝑥 is equal to zero degrees, 𝑥 is equal to 90 degrees, 𝑥 is equal to 180 degrees, 𝑥 is equal to 270 degrees, and finally 𝑥 is equal to 360 degrees. And it’s these values of 𝑥 that are going to come in handy for us when we define the phase of our electromagnetic wave here. Specifically, what we’re going to do in this particular case is to say that the phase of our electromagnetic wave at, say, this point in space is zero degrees. Because, at this point in space, our electromagnetic wave looks exactly like our sine curve does when 𝑥 is equal to zero degrees.

And then at this point here, we will say that the phase of our wave is 90 degrees because, once again, our light wave at this point looks like the sine curve at 𝑥 is equal to 90 degrees. Specifically, we see maximum displacement for both waves in the upward direction. And then at this point, our electromagnetic wave has a phase of 180 degrees. At this point, it’s 270 degrees. And at this point, it’s 360 degrees, at which point we’ve defined the phase of one complete cycle of our electromagnetic wave.

Now, since electromagnetic waves are periodic, which means that they have a repeating pattern, we can choose to define this point also as zero degrees once again. And then, this point would have a phase of 90 degrees. This point would have a phase of 180 degrees, and so on and so forth. Or we could choose to continue with the convention we’ve decided earlier. Instead of zero degrees, this point would still be 360 degrees. And then this point would have a phase of 450 degrees. And this point would have a phase of 540 degrees and so on and so forth. But the point is that we can plot a curve of 𝑦 is equal to sin 𝑥 and use different values of 𝑥 to define the phase of our electromagnetic wave, which is also sinusoidal.

It’s worth mentioning by the way that we’ve mentioned only values of 𝑥 is equal to zero degrees, 90 degrees, 180 degrees, and so on. But the phase can actually be defined for all values even between zero degrees and 90 degrees and 180 degrees and so on. It’s also worth mentioning that sometimes we define phase in terms of the wavelength of our wave. A 90-degree phase is equivalent to a quarter of a wavelength, 180 degrees to half a wavelength, and so on. So that is what we mean by the phase of a wave. But why do we need this parameter in the first place? Well, like we mentioned earlier, this comes in handy when comparing multiple waves with each other.

Let’s now imagine that instead of just one light wave traveling from left to right, we’ve got two. In this particular case, the two waves that we’ve drawn look pretty much identical to each other in terms of their shape. Since we’re assuming that they’re both light waves traveling through empty space, in other words through a vacuum, they will both move at the same speed. They will move at the speed of light, which is labeled 𝑐. And essentially, what this means is that if we as an observer were to stand at a certain point in space and measure the frequency of these two waves, we would find that these two waves have the same frequency as each other. Now, the reason that we’ve drawn two waves with the same frequency is because this is an important condition for these waves to be coherent with each other. And we will discuss coherence later on.

But before we talk about coherence, let’s now imagine that we, as observers, are standing at this position here. And remember, the waves are moving from left to right. But for this particular instant in time, the instant for which we’ve drawn are waves, we can find the phase of each of the waves. To do this, we can once again draw our sine curve. And we can see that at this point here, for example, the displacement of the wave from the dotted line is zero. And therefore, that point of the first electromagnetic wave looks like this point on our sine graph. And that is the point that corresponds to 𝑥 is equal to zero degrees. So the phase of the first light wave here is zero degrees. And we can do the same thing at our observation point for the second wave. We draw in our dotted line and see that the phase is also zero degrees.

And at this point, we can define a quantity known as the phase difference between these two waves. And as we may imagine, it’s literally the difference between the phase of the second wave and the phase of the first wave. In fact, if we represent the phase of our first wave with the Greek letter 𝜙 subscript one and the phase of our second wave with 𝜙 subscript two. Then we can say that the phase difference between these two waves is equal to 𝜙 two minus 𝜙 one. At this particular observation point, we find that the phase difference 𝜙 two minus 𝜙 one is equal to zero degrees minus zero degrees, which is equal to zero degrees. And so we found the phase difference between these two waves at this particular observation point.

If instead we now move our observation point to here, we can find the phases of the two waves. We can see that the phase of the first wave corresponds to the sine curve when 𝑥 is equal to 270 degrees. So at this point, 𝜙 one is equal to 270 degrees. And actually, the same is true for the second wave. 𝜙 two is equal to 270 degrees, which means for this observation point, we can find the phase difference which is 𝜙 two minus 𝜙 one once again, which is 270 degrees minus 270 degrees, which is zero degrees. So now we can notice that the phase difference at this observation point is zero degrees and the phase difference at this observation point is also zero degrees and actually the phase difference at any observation point is going to be zero degrees if we look carefully.

Therefore, what we’ve determined is that for these two light waves, the phase difference is constant for all observation points. In this particular case, the value of that constant is zero degrees. But the important thing is that the phase difference at any point is zero degrees. Try calculating this for yourself. Try finding the phase difference at this observation point where the first wave reaches a maximum displacement and the second wave also reaches a maximum displacement. Now, this discovery that we’ve made, the fact that the phase difference for these two waves is constant for all observation points, leads us very nicely onto the concept of coherence. In fact, two or more waves are said to be coherent if their phase difference is constant.

So we saw earlier for these two waves, that the phase difference at any observation point was zero. And this is just one example of a pair of coherent waves. However, the phase difference does not necessarily need to be zero at different points along the wave. It just needs to be constant. It just needs to be the same as at other points along the waves. What about, for example, if we now consider these two waves? Once again, they’re waves of light traveling through a vacuum from left to right. But at this particular point in time, we can choose, for example, this to be our observation point. And then we can work out the phase of the first wave and the phase of the second wave.

To do this, we once again draw our 𝑦 is equal to sin 𝑥 curve, and then we can find the phase of our first wave. We see that this point corresponds to the maximum displacement in the upward direction on our 𝑦 is equal to sin 𝑥 graph. In other words, this corresponds to 𝑥 is equal to 90 degrees. So 𝜙 one is equal to 90 degrees. However, this time for the second wave, the phase at this particular point of observation is different. We see that this point is the maximum displacement in the negative direction, specifically corresponds to this point here, 𝑥 is equal to 270 degrees. So 𝜙 two is equal to 270 degrees. So what’s the phase difference at this observation point? That’s 𝜙 two minus 𝜙 one, which becomes 270 degrees minus 90 degrees. And so we find along our orange dotted line the phase difference is 180 degrees.

So now let’s shift our observation point to this second orange dotted line and find the phase of the first wave at this point here and the phase of the second wave at this point here. For the first wave, we can see that this point corresponds to a zero displacement. If we were to draw in our zero-displacement dotted line, we see that our point falls directly on this line. Also, as move from left to right, we see that the wave is going from an upward displacement to a downward displacement. And so we see that on our sin 𝑥 curve, the point that looks identical is this point here, zero displacement at this particular point. And also as we move from left to right along the sine curve, the displacement is going from upward to downward. This corresponds to 𝑥 is equal to 180 degrees. And hence, we can say that at this point 𝜙 one is equal to 180 degrees.

Moving on to 𝜙 two then, we see that this point also lies along the dotted line of zero displacement. However, this time moving from left to right, we see that the displacement of the wave changes from downward to upward. And to see this point on our sine curve, we have to slightly extend it below 𝑥 is equal to zero and above 𝑥 is equal to 360 degrees because the point that we’re looking for is either this one here or this one here. In other words, we’re either looking for 𝑥 is equal to zero degrees or 𝑥 is equal to 360 degrees. These points are equivalent because, remember, a sine curve as well as our sinusoidal waves here are periodic, so their cycles repeat.

So let’s say that we decided to go with 𝜙 two is equal to zero degrees. Well, in this case, we would find that the phase difference at this observation point 𝜙 two minus 𝜙 one is equal to zero degrees minus 180 degrees, which actually ends up being negative 180 degrees. So what’s going on here? Well, actually, when it comes to phases, zero degrees of phase is equivalent to 360 degrees of phase. Because, remember, like we said earlier, these waves repeat; they’re periodic. And so if we were to use a phase value of 360 degrees rather than zero degrees for the value of 𝜙 two, then we’d find a phase difference of positive 180 degrees. Which exactly matches the phase difference that we found at the other observation point.

And this just goes to show that we’ve found different ways of writing the same phase-difference relationship. In other words, both phases and phase differences that are out by 360 degrees from each other are equivalent because, as we’ve seen, a phase of zero degrees is the same thing as a phase of 360 degrees. And similarly, a phase difference of negative 180 degrees is the same thing as a phase difference of positive 108 degrees. But the point here is that we found the phase difference at one observation point and the phase difference at another observation point to be exactly the same.

And we could really find the phase difference at any observation point and still find it to be 180 degrees or another equivalent value, which means that these two waves here are coherent by this definition. They have a constant phase difference. Whatever observation point we choose, the phase difference between them is always 180 degrees. So this also means that these two waves which have the same frequency, by the way, are coherent. If two waves do not have the same frequency, they cannot be coherent. Let’s see an example of this. Let’s take a look at an example of a pair of waves that are not coherent.

When two waves are not coherent, they are said to be incoherent. And here is an example of a pair of light waves that are incoherent. We can notice that they do not have the same frequency. If we start by choosing this as our observation point, then we can find the phase of the first wave firstly. And we can see that this corresponds to 𝑥 is equal to 90 degrees. And so we can say that the phase of the first wave at this point is 90 degrees as well. And then we can do the same thing for the second wave. We find that this is sinusoidal also and at this particular point also corresponds to a maximum value of the sine curve if we imagine that the sine curve of our second wave looks something like this. Which means that we find that the phase of the second wave at this point is also 90 degrees.

Therefore, the phase difference 𝜙 two minus 𝜙 one at this particular observation point is equal to zero degrees because it’s 90 degrees minus 90 degrees. However, if we now choose this to be our observation point, then we see that for the first wave, the observation point once again corresponds to a maximum-displacement point which, along a sine curve, is 𝑥 is equal to 90 degrees. So our phase 𝜙 one is equal to 90 degrees here. Whereas for the second wave, we see that 𝜙 two must be 270 degrees. And this means that the phase difference between the two waves at this particular observation point is equal to 270 degrees, which is 𝜙 two, minus 90 degrees, which is 𝜙 one, and this is equal to 180 degrees.

So our first phase difference was zero degrees, and our second phase difference was 180 degrees. And these two are not equivalent because they are not one full cycle or 360 degrees apart from each other. And in fact, we could choose any random observation point and find the phase difference at this point and see that the phase difference was different to zero degrees or 180 degrees. Which means that the phase difference between these two waves at different points is different. It is not a constant value. So waves with different frequencies cannot be coherent. These two waves are incoherent.

And when talking about incoherent waves, we don’t have to restrict ourselves to sinusoidal waves either. We could try and find the phase relationship between the first wave and this newly drawn-in wave. And this wave doesn’t even display a periodic pattern. It doesn’t repeat in cycles, which means we can’t even define the phase at any point along this wave because it doesn’t look like a periodic wave such as a sine wave, for example. Therefore, we cannot say that the phase difference between the two waves along different observation points is constant. And therefore, these two waves are also incoherent.

So now that we’ve looked at coherence and incoherence, let’s summarize what we’ve talked about in this lesson. In this lesson, we firstly saw that 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. And we also saw that two or more waves are coherent if their phase difference is constant. A quick way to judge this is that if the waves do not have the same frequency, they cannot be coherent.