We have seen that we have to measure very tiny changes of distance with the passing gravitational wave. Typically we saw that the type of distance that we have to measure is 10 to minus 12 to 10 to minus 20 metre. So that seems an incredibly small distance, especially when you compare it with a proton radius 10 to minus 15 metre. it seems even an impossible task, but we have to realise that we are looking at bulk of matter and so it’s billions of protons, which are displaced, or seem to be displaced at the same time. This is why we can achieve such precision.
Now in order to get to that level of precision we have to rely on light because over the years, light has become the necessary tool to do very ultra fine, ultra precise metrology. So let me illustrate this by reminding you of the history of the definition of a metre.
The metre was introduced by the French Revolution in 1791 and from then on, one used a prototype metre, which was kept in the French city of Sevres. And that was precision enough until the 1960s. It was then realised that this was not a precise enough instrument and so one had to define the metre from the wavelength of a certain element, Krypton 86, which was used from then on. And so you see that this is when one had to resort to light in order to have a definition of the metre. And in 1983, it was even realised that this was not precise enough. And so from then on, one used the velocity of light.
So by definition, one decided that velocity of light is 299,792.458 kilometre per second. So that means that basically in the vacuum, the light is travelling a certain distance in a time, which corresponds to one over this number, 299 and so on. So you see that from the days of the French Revolution to nowadays one had little by little to resort to light in order to have ultra precise measurement of distances.
In order to do this type of very precise measurement, we will use the phenomenon of interference. So let me introduce you to this phenomenon and first take a very simple example of throwing stones into a pond. Now we have seen that with a single stone, now let us is imagine that we are throwing simultaneously two stones into the surface of a pond. Then each stone is developing a system of ripples which combine. And so you imagine that you have a figure of ripples on the surface of a pond with peaks and hollows, which are characteristic of what we call a figure of interference– interference between the shock of two stones.
We have a similar phenomenon with light. In this case you start with a coherent source of light– I’ll come back to, in a second, what does coherent mean. This light is getting to a screen. And imagine the screen is pierced with two slits. On the other side, the light is re-emitted by the two slits and is going to interfere. And we get, if we put a further screen, we get what we call an interference pattern, which means that we’ll have, alternatively, bright lines and dark lines. The bright lines correspond to coherent interference between the light emitted by the two slits. On the other hand, In the case of dark lines, you have destructive interference.
Now, you have here examples of this type of interference patterns on the left hand side, this is just what I described. On the right hand side it is in the case of a cylindrical beam light. In this case, the interference pattern has the same kind of cylindrical form and so you see this alternatively, again, these cylindrical bright lines and dark lines, which we call interference fringes. In order to have a good interference pattern– so that means sharply defined interference fringes– one needs to have a source of light with a very definite frequency, a very precise frequency. In order to understand that, let me return to the example of a stone, or two stones, thrown into a pond.
Just imagine that you send out many stones. Then instead of having, on the surface of a pond, a very definite pattern, you’ll have a sort of– all kinds of– noise on the– but no definite figure, no definite interference pattern. Well it is the same with light– the different patterns will sort of superimpose into nothing really geometric if you have too many frequencies in your light, like if you had, for example, white light. White light is a superposition of many frequencies. So we need a source with very definite, very precise frequency. And that’s why, in modern days, one is using a laser source.
For very precise measurements, one uses a set up, which was conceived in the 19th century, it’s called an interferometer. It was conceived by Michelson. And it was used by Michelson and Morley in 1887 in order to show that there was no ether and that is a starting point for the theory of special relativity. Now here is the setup we have used.
Here a laser– of course it was not available in the time of Michelson, but otherwise this is really the Michelson set up. So there is a light beam from the laser, which falls upon an optical device, which is called a beam
splitter, and which, precisely, splits the beam into two: a reflected beam that goes in a direction and a transmitted beam that goes in the other direction. Both beams are reflected on mirrors so they come back to of the beam splitter and from then on they are joined, and this is where they interfere. And so if we put a screen here, we’ll observe the same kind of interference pattern that we have just seen. Now why is this used for making very precise measurements of distances? Well, the point is the following– the interference pattern down there depends on the travel time it takes, or the distance it takes to go from the beam splitter to a mirror and back in both directions.
If we slightly change this distance, either of them, we change the interference pattern, which means that some fringes will move. For example, it will replace a bright fringe with a dark fringe. Now just imagine that a gravitational wave is moving towards the set up in the lab. Now we have seen that it changes the distances– it changes this distance, it changes this one– in a regular fashion, in a periodic fashion, because this is a wave. And so you’ll see periodic changes of fringes on the screen down there. And this way you’ll detect very tiny distances– very tiny changes of distance.
Now if you want, the wavelength of the light used, which is very small, is actually the sort of prototype metre that was used before. So it’s a very tiny distance and it allows to measure very tiny changes of distance.
Let us summarise with two historical pictures. In the first one, you see the setup of the Michelson-Morley experiment in 1887. You see that the whole experiment is lying on a single table. This is what we call a tabletop experiment. In the second one, Michelson is at his experiment and he’s counting the number of interference fringes.