 Sometimes when we are standing on the side of the road and we observe a headlight from a vehicle which is coming from far away, we usually might think, oh this could be a two wheeler because I see only one source of light. But as that vehicle comes closer, then you start seeing two headlights or two sources of light instead of one. And then you realize, oh this is actually a car, it's a four wheeler. Why does this happen? Why can't we at the very beginning, when we first saw the source, why can't we then say that it's a four wheeler, it's a car? Because we did learn in ray optics, we learned in ray optics that we get a point image for a point source. So if there is a source S1, there will be an image of S1 over here and if there is a source S2, there will be an image of S2 over here just like the two headlights of a car. So these two images should be formed on a retina and we should be able to clearly identify that these are two separate sources. But we can't do that, ray optics cannot really explain this. But turns out that wave optics can, we will use wave optics, the wave nature of light to explain this. And in the process, we will also learn about the diffraction limit of angular resolution. Alright, let's begin. We know that when light passes through a small opening or an aperture, it spreads out. It diffracts each point in the opening acts as a secondary source as per the Huygens principle and all of the secondary waves interfere with each other resulting in an interference pattern which can be obtained on a screen. This is a pattern when there is a vertical slit, when there is a vertical narrow opening. There is a central maxima and then the intensity decreases as you go away further from the center. But our eyes, they do not have vertical slits. We have a circular opening, a circular aperture that is called a pupil. The pupil is the aperture of our eye through which the light enters. Even, even for cameras, the aperture is roughly circular. So whenever light enters these circular openings, diffraction must also happen here. And it does, but turns out we do not get a pattern like this. If we have a circular opening or a circular aperture, the diffraction pattern, let me move this over here. The diffraction pattern formed by a circular aperture consists, it consists of a central bright spot which is surrounded by a series of bright and dark rings. The central bright spot over here, this is called an airy disc. Now we can describe this pattern in terms of the angle theta. Just like how we describe this pattern from a vertical slit in terms of theta. Here the angular position of the first minima, that is this angle right here, this angle theta is the angle between the first minima and the center of the central maximum. This angle theta 1, this was given by lambda divided by a, where a was the width of the slit and lambda is the wavelength of the light. Here the angle between the center of the central maximum and the first minimum is slightly more just because of the change in the geometry of the slit. This is circular and this was vertical. This angle, the angle between the center of the central maximum and the first minimum, this angle is given by theta, this is 1.22 lambda divided by capital D. Here capital D is the diameter of this opening, the diameter of this aperture. We will not be talking about how we got 1.22, the derivation is a bit complicated, we will not be going into that, but right now we can just accept that this is, it is 1.22 lambda divided by capital D. Now we get this pattern whenever a light passes through a circular aperture and even optical instruments, optical instruments can also undergo, they can also undergo diffraction. For instance a telescope, a telescope has an objective lens with very sharp edges at the corners and diffraction can happen at these sharp edges. So we cannot really ignore the wave nature of light, we cannot ignore the effects of diffraction, we need to consider that when we are working with optical instruments like a telescope or a microscope. For one source we see this diffraction pattern, but if we have one more source like the two headlights of a car, let's see what do we get then. Here we have two sources and they could be two distant stars or even the two headlights of a car. As the light from these two sources as it passes through the slit, two distinct bright spots, two different images can be seen on the screen. If no diffraction occurred, then we would get two point images on the screen. But because diffraction occurs, so we get something, something like this. Each source, each source is imaged, it is imaged as a bright central region which is surrounded by weaker bright and dark fringes. This is what we will see on the screen. If we draw the intensity pattern, draw the intensity pattern, this is what it would look like. We have one maxima over here and the other at some distance over here. We see two separate peaks in the resultant which is shown by this dashed line. Now if the two sources are far enough, are far enough apart to keep the central maxima from overlapping, we see that the central maxima of both the images, both the diffraction patterns, they aren't overlapping. Then we can say that these are two separate sources of light. We can easily identify that. That is when we say that the images are resolved. So in this case, we can easily identify that these are two separate sources because the central maxima of the diffraction patterns, they are not at all overlapping each other, they are at a good distance apart. We can easily tell that these are two separate sources, not one. So the image here is resolved. But if the objects are very close to each other, something like, something like, let's say in this case, then these diffraction patterns, even they will start coming close to each other. So as a result, the diffraction pattern on the screen could look like this. And if you draw the intensity graph, if you try to draw the intensity graph for this diffraction pattern, that could look somewhat, somewhat like this. These diffraction patterns, they start overlapping. The two central maxima overlap. In this case, instead of two, we will see only one source. And you can see that in the form of the resultant, the dashed line, we will see only one peak. So we don't see two central bright spots, we see only one. So it appears to us that there aren't two objects, there's only one object. In this case, we say that the image is not resolved, it's unresolved. Because we cannot distinguish, we cannot separate out, we cannot identify two separate objects. For us, they are just one object. There is a condition when the images are said to be just resolved. So if you have two sources, and the distance between these two sources is slightly less compared to the first case, but slightly more compared to the last case. Here, the diffraction pattern will be closer to each other compared to the first case. And the pattern could look somewhat like this. By looking at the image, we can say that these are two separate sources of light. The image is just, just resolved the right amount. Now to tell whether the two images are just resolved, the condition is whenever the central maximum of one image, when it falls on the first minimum, when it falls on the first minimum of another image, then the images are said to be just resolved. This limiting condition of resolution is called Rayleigh's criterion. This is when the limit of resolution has been reached. The objects cannot be closer to each other than this, otherwise then we won't be able to resolve them and we would just see them as one source. So what is the minimum separation that we can have? And here I'm talking about the minimum angular separation. I'm talking about this, I'm talking about this angle right here, theta, let's call it theta min, theta minimum. We know that for all circular apertures, the angular separation between the center of the central maximum and the first minimum is 1.22 lambda divided by D. So this distance right here, this is 1.22 lambda divided by capital D. You can see that is the angular separation between the center of the central maximum and the first minimum. So this is 1.22 lambda by D. So if you draw, if you draw this pattern over here, this is how it could look like. The center of one diffraction pattern or one image, it coincides or falls on the minimum of the second diffraction pattern or the second image. And this distance is the angular distance, this is the angle, and this is 1.22 lambda by capital D. The angular separation of the images is the same as the angular separation of the objects. So this theta min, this is also equal to 1.22 lambda by capital D. This is the minimum angle that the two sources can make at the slit so that the two sources are just resolved. So the Rayleigh's criterion of resolution is that the separation, and we are talking about angular separation here, the separation between the two sources of light, the two stars or the two headlights, should be equal to or more than 1.22 lambda by D. If it is less than that, then we won't be able to resolve it into two separate objects. This is what we call the diffraction limit on angular resolution. This is the diffraction limit on angular resolution. Now let's see what is the angular resolution of our eye. Let's say that light of wavelength 500 nanometers, let's take some average wavelength, 500 nanometers. It enters the human eye and the diameter of the pupil, the diameter of the pupil, we can take that, we can take that as, on average we can take that as 2 millimeters. So the limit of angular resolution, this angle is 1.22 into lambda divided by capital D and when we work this out, this comes out to be 1.22 into 500 into 10 to the power minus 9 divided by 2 into 10 to the power minus 3. When we work this out, this is 3 into 10 to the power minus 4 radians. Now to get a better feel for this, let's change radians to degrees and we can do that by multiplying this number with 180 divided by 3.14. So when we do that, this is 3 into 10 to the power minus 4 into 180 divided by 3.14 and this is 0.018 degrees. This is a very small number. So coming back to our original question of the two headlights, the reason why we see one source when the headlights are far apart from our eye, we see them as one source, is that the two sources, they must be making an angle which is less than 0.018 degrees. This angle will be less than this angle right here. So we cannot separate the two sources, we identify them as only one, but when the light sources they come closer to the eye, this angle you can see the value of this angle is slightly increasing and maybe at this point this is how the headlights look like. We are just able to resolve that these are two separate sources, this is not just one source. Over here we can see the angle would be equal to theta min, this would be equal to 0.018 degrees. That is the angle that the two sources are making at the eye and when these sources are closer to the eye, now they are making a big enough angle, we can clearly, very clearly see that these are two separate sources. You can imagine that if these two sources, S1, S2, let's say if these two sources are distant stars, then they would be making an extremely ridiculously small angle at the aperture. So we use telescopes to resolve that. And how do telescopes help? For telescopes the aperture, this D, is extremely large, that really decreases this angle, theta min can be very small. So this means that telescope can very finely resolve two distant objects even when they are making a very small angle at the aperture, this theta min is very small for telescopes. So telescopes can resolve distant, distant objects.