Video Transcript
In each of the following diagrams, five light waves are shown. Which of the diagrams shows coherent light?
Coherent light in this question means coherent waves, since light can be expressed as a wave. So, when this question asks which of the diagrams shows coherent light, it’s really asking which of the diagrams shows five light waves that are all coherent.
In order to figure this out, let’s recall what makes two or more light waves coherent. The waves in question must have the same frequency and a constant phase difference. Waves, plural, are coherent only in relation to each other. A wave cannot be coherent or incoherent by itself. So, what do these two qualifiers of same frequency and constant phase difference mean? And, more importantly, how can we recognize them in a diagram like this?
Let’s start by looking at waves that share the same frequency. Recall that a wave’s frequency is the number of cycles of that wave passing a fixed point per unit time. The standard unit of frequency is a hertz, abbreviated as Hz, with one hertz being equal to one wave cycle passing a fixed point per second. What this looks like is if we have some wave that is traveling forwards, we can assign a fixed point ahead of that wave so that as it propagates forwards, it reaches the start of this fixed point, which is when we start measuring the time. The wave then continues until the unit time has been reached, in this case one second. This wave has had one full wave cycle pass the fixed point in one second, meaning that this wave is one hertz. In order for it to be two hertz, there would have to be two full wave cycles passing the fixed point in one second.
Higher-frequency waves have more complete wave cycles per unit of time. So, by looking at the number of complete wave cycles on these waves, we can determine which ones have higher or lower frequencies. And since all of these waves are light waves, it means that they have the same speed, the speed of light 𝐶, provided that they’re all traveling through the same medium, which we’re assuming to be the case here, which means all of the light waves in these diagrams are traveling the same distance in the same amount of time, since they all have the same speed.
We don’t know the exact time it takes for the light waves to go from one end of the diagram to the other. But we don’t need any exact numbers. We just have to determine if they have the same frequency by looking at the waves, which means, do the waves have the same number of complete wave cycles starting from the same fixed point? With some waves, it’s easy to tell, since all of the parts of these waves may line up. The peaks and valleys of these waves line up. And closer inspection reveals that they both consist of one full wave cycle, meaning they are the same frequency.
But not all waves have the same phase. Consider now this third wave, which on close inspection also consists of one full wave cycle, which means that all three of these waves have the same frequency. The third wave just has a different phase. But whether waves are coherent or not also depends on phase, specifically whether or not waves have a constant phase difference between them. Let’s recall what the general phase of a wave is by looking at the first and third waves here.
The phase of a wave is used to measure and compare specific points on a wave and is typically measured in radians, like 𝜋 over two, or in degrees, like 90 degrees. Each part of a typical sine wave, like this one up here, has a value of phase associated with it. To save on space, we’re only going to list these measurements in degrees.
The beginning of a sine wave, which is at mid height going upwards, is zero degrees. The peak of a wave is 90 degrees. The midpoint, which occurs at half height going downwards, is 180 degrees. The lowest part, or valley, of the wave is 270 degrees. And the end of a wave, which is at half height going upwards, is 360 degrees. Or since the end of a wave is the start of a new wave, this point can also be zero degrees, but not every wave starts at zero degrees. The first point of this wave is at the peak of a wave, meaning that it actually starts at 90 degrees. Similarly, its second point is at the halfway height going down, so it is 180 degrees. The third point, the valley, is 270. Half height going up is 360 or zero, and the last point is at a peak again of 90 degrees. These two waves have different values of phase at the same points in time, meaning that they have a phase difference.
Light waves as they travel forwards repeat themselves and so eventually go through all the phases of a wave. But even though the phase becomes different with time, this is not what a phase difference is. A phase difference is when two or more waves have a different phase at the same point in time. The top wave has a phase of zero degrees, and the bottom wave has a phase of 90 degrees, at the same point in time. The phase difference between a phase of 90 degrees and zero degrees is of course just 90 degrees.
When looking at just two waves, this phase difference should remain constant across both waves for all points in time, provided of course that the waves themselves are consistent. We cannot make this assumption for irregular waves, which change their frequency as they travel. But for these two waves, it should be fine. Measuring the two different phases at any point in time should always give a difference of 90 degrees, because remember these waves are meant to be coherent.
But be careful when measuring the starts or ends of waves, the parts that could be zero or 360 degrees. For instance, measuring the phase at this point in time for both of the waves gives us 270 and zero or 360 degrees. If we choose the phase on the bottom wave to be zero degrees, then our phase difference would be zero minus 270 degrees, which comes out to negative 270 degrees, which is not 90 degrees. If we instead used a value of 360 degrees, then we see that it does come out to 90 degrees. So we’ve shown that these two waves have a constant phase difference of 90 degrees at all points. And we had shown earlier that they have the same frequency as well, which means that these two waves fit the requirements for being coherent with each other.
Finding if two waves have a constant phase difference to find if they’re coherent is pretty easy, since if they have the same frequency, it’s very likely they also have a constant phase difference. But we’re not looking at just two waves. We’re looking at five waves. The definition of whether these waves are coherent doesn’t change. They all still need a constant phase difference and the same frequency. But meeting these conditions is more difficult with more waves.
These three waves here have the same frequency, since they all consist of one complete wave cycle over the same time period. But all three of them do not have a constant phase difference. The first two waves have the same parts of the wave present at the same point in time. They are in phase with each other. This means that their phase difference is always zero degrees. But if we look at the phase difference between two and three or one and three, like we had done earlier, we would find the phase difference to be 90 degrees. And 90 degrees is not equal to zero degrees, meaning that these three waves have a nonconstant phase difference and are, thus, not coherent with each other.
With all of this in mind, let’s start looking at the diagrams containing the five light waves, starting with diagram (A). The waves in diagram (A) all have different phase differences. This one starts just before a peak, this one just before a valley, and these at other various points. These waves have a nonconstant phase difference and so are not coherent, even though they look pretty similar and might have the same frequency. So (A) is not the correct answer.
So let’s look at (B) instead. It, however, has a similar problem. The waves do not match in phase at the same time and, thus, all have different phase differences. In other words, just like (A), these waves have a nonconstant phase difference and, thus, are not coherent with each other.
Let’s see if (C) is any different. (C) contains waves of different frequencies, which is most obvious when we compare the central wave and the bottom wave. Waves with different frequencies cannot be coherent, so (C) is not the correct answer. And a cursory glance at the waves in diagram (D) also shows that they have different frequencies, meaning that (D) isn’t it either.
Looking now at diagram (E), we see that the waves are all lining up with each other, even though they all have different amplitudes. Looking at, for example, the second peaks of the waves, we see that they all occur at the same point in time, which means that the phase difference between any of these waves is zero degrees. So, across all five waves, there is a constant phase difference. And if we look at the total number of wave cycles from the fixed point at the start of these waves, we find that all of these waves contain about three and a half complete wave cycles, meaning that they’re all of the same frequency.
So, since all five of these waves together meet the two criteria for coherent waves, the diagram that shows coherent light for all five of its light waves is diagram (E).
