Hey guys. In this video, we want to look into more detail as to why exactly diffraction happens and we're going to analyze why diffraction happens by exploiting Hagen's principle for light. Okay? Let's get to it. Remember guys, what Hagen's principle says is two things about how a new wavefront is going to appear after an old wavefront. Okay?

The first thing that Hagen's principle says is that the old wavefront is going to produce these little spherical wavelets at every point along the wavefront. Okay? So if I consider this an old wavefront, then I have a point here where I'm considering these spherical wavelets to be produced. I have a point here where I'm considering these spherical wavelets to be produced. And I have a point here where these spherical wavelets are being produced. Okay? Now, this actually happens along the entire length of the wavefront instead of at the specific points that I've indicated. But just for clarity's sake, I'm only showing you three points. If I were to show you every single point, which is an infinite number, it would look crazy. You wouldn't be able to tell anything that was going on. Okay? So for the sake of clarity, I only showed three points.

Now considering this old wavefront right here, I can choose a point here and look at the spherical wave going out. I can choose a point here at the apex and draw spherical wavelets going out or I can choose a point down here and draw spherical wavelets going out. The point is that no matter the shape of the old wavefront, every point on the old wavefront is producing these spherical wavelets that are emanating from it.

The second point of Hagen's principle is that the new wavefront is a tangent line that crosses the apexes of each of the wavelets. Okay? So each of these wavelets has an apex here, an apex here, and an apex here. Okay? And then the new wavefront is just going to be drawn tangent through those apexes. So I'm just going to draw a line straight through. You see that drawing a line straight through here means that the new wavefront is parallel to the old wavefront. Okay?

Now for this circular wave, I have an apex here, I have an apex here, and I have an apex here. And drawing a line tangent through all of those means another circular wavefront. This new wavefront is also circular. Okay? And if we want to look at what those light rays would look like, just remember that light rays are perpendicular to every point on the wavefronts. This light ray is going to look like this, collimated light. This light ray is going to look like this, isotropic. The same in all directions. Okay? So that's just a refresher on Hagen's principle, and it's important to understand it to understand exactly how diffraction occurs.

Now, the important key for diffraction to understand is that the smaller the slit, what do I mean smaller relative to? I mean smaller relative to a constant wavelength. So let me remind myself for this. I have three figures, each of them showing wavefronts with identical wavelengths, right? The distance between two wavefronts by definition is the wavelength.

So three waves with identical wavefronts but the slit width is changing in each picture. Initially, we have a width much larger than the wavelength. Then we have one near the size of the wavelength. Maybe the wavelength is 1 millimeter. Maybe the width is 3 millimeters. It doesn't have to be identical. It just has to be on the same order of magnitude. And then I have a width much smaller than the wavelength. Okay? And the important thing to see here is that essentially the way to think about this is that as that width gets smaller and smaller and smaller, as the slit gets smaller, the number of wavelets that can pass through the slit decreases.

So you see in the first picture we have a ton of wavelets. Then when the slit gets smaller we don't have as many wavelets that can pass through. And then when the slit's really really really small we say essentially only a single wavelet can pass through. So what are the new wavefronts going to look like on the opposite side of the slit? Well, remember we have to draw a line through the apexes of each of these new wavelets. So here I'm just going to draw a line straight down. So this line is parallel to all the wavefronts before so it's just going to continue being parallel. And what we had initially, which was this collimated light we also have passing through the slit. Okay?

We have collimated light entering. We have collimated light leaving. There is clearly no diffraction here. Alright? Now, what are the wavefronts going to look like here in the second image? The thing to consider here is that these wavelets are small. They're fewer wavelets. Okay? So the effect that each wavelet has on the other is less. Here in this image, we said there are essentially so many that the line is basically straight. But now because there are fewer, the line actually curves a little bit at the edges. And then it's basically parallel in the center. So what are these rays going to look like? Well, these rays are clearly collimated. But what are the rays going to look like coming out? Well, near the center, they're basically still going to be collimated. Right? Collimated at the center. But what about the edges where those wavefronts get distorted a little bit? Well, then they aren't collimated anymore. They actually spread out a little bit. So we have spread at edges. What this means is that for light passing through the center of this slit, it's essentially passing through undiffracted. It's entering collimated and it's leaving collimated. For light passing through near the edges of the slit, it's diffracting as it passes through.

Now, what do the wavefronts look like when we only consider a single wavelet as shown in the figure on the right? Well, since it's only a single wavelet it's going to have to continue being spherical. There are no other wavelets to distort the wave front. It's going to continue being spherical. And now if we look at what happens to the light rays as it passes through the slit, these new light rays have to be perpendicular to the wavefronts at all points. The only way for that to be true is if the light spreads out isotropically. So it enters collimated and it leaves isotropic. So there's definitely diffraction here. Okay?

So on the left, we have no diffraction. On the right, we have full diffraction or a lot of diffraction, and in the middle, we have some diffraction. We have diffraction at the edges but not really at the center. Okay?

This process of diffraction is explained entirely using Hagen's principle. Okay? And a few consequences of this are important. Alright? If you're looking at a single slit as I showed in the above figures, you can never really make the slit so small that only a single wavelet is produced. Okay? I approximated that but in reality, wavelets occur everywhere on old wavefronts. Since they occur everywhere, you can never make a slit so small that you're going to isolate a single wavelet. Okay? So in reality, there will always be multiple wavelets passing through a slit no matter how small the slit is. Light then comes out at different angels from two different parts of the slit. Okay? So if we look at near the top of the slit, we actually have light coming out at multiple angels. And if we look at the bottom of the slit, we actually have light coming out at multiple angels. Right? This is exactly what I said happens when you have a few wavelets but not one. You have diffraction at the edges. That means that there's no guarantee that a light ray coming out of the top part of the slit is going to be the same angel as light coming out of the bottom of the slit. Okay?

And what this means is when you have all this light at different angels to one another, these are actually waves. Don't forget that. So you're going to have, let's say, a maximum here and then over here maybe you have a minimum. And when you have a maximum and a minimum at the same place interacting with one another, you have destructive interference. So the light passing through the slit is going to interfere with itself. Okay?

Now, if you have two slits, you can then assume, you can approximate that basically the light comes through the slit fully diffracted. That you only have a single wavelet passing through and so And the reason we can assume that is because we want to consider the actual size of the slit to be very very very small compared to how far apart the slits are. Okay, so compared to how far apart this actual length right here, how far apart those two slits are, we want to consider that bigger than the width of the slit itself. Okay?

Now once again, because light is coming out of all these different angles, the light travels a different distance. So maybe this top light has no displacement when it reaches there, and then this bottom light has a maximum displacement when it reaches there. And then the light is going to constructively interfere. The point is that as the light passes through these two slits, which is called the double slit, each light ray will interfere with each other. Okay? And I I'm sorry. I put a condition here that I did not address. That distance d is this width, this separation distance between the two slits. And the length l that I indicated there is different than the length l that I indicated in the figures above. That's actually the distance from the screen to the slits. Okay? So as long as the slit width, the distance between those two slits is very very small, that slit separation is very small compared to how far apart the screen is. You will get light coming out of those slits isotropically and each light ray coming out of the slit will interfere with one another. Alright? And this produces a very very important pattern that we're going to look at here which is called a diffraction pattern. Okay? This is what the light looks like when you assume diffraction occurs. Alright? So maybe light coming out of this slit at this point and light coming out of this slit at this point if we look right here, this is dark. Right? The brightness is very, very low. This means that there has to be destructive interference. Okay? If it's destructive, you're going to reduce the brightness of the light. And then if I consider another point over here, where now we have a peak. This is bright. That means that the two light rays reaching that point have to have constructive interference. Alright? And we get a similar brightness pattern in a single slit. The only real difference between the double slit and the single slit is how wide the central bright spot is. Alright guys, so that wraps up our discussion on diffraction specifically using Hagen's principle. Thanks for watching.