13.8 Diffraction and Interference

Okay, today's lesson, we're going to look at diffraction and how diffraction occurs due to the interference of waves. Some of it will be referring back to physics 20 when we looked at wave interference, but mostly we're going to be dealing with new stuff, new formulas, and so on. A little bit of history and review of what we started with. So remember, in Newton's thoughts, he believed that light was a particle. But then there were a few experiments that proved that it was actually more of a wave. Huygens was the first one to discover this. Remember, he was the one who found the Poisson spot and he figured out that waves would diffract around an obstacle or an opening, and we get blurry edges in terms of the shadow. We get the dots in the circle when it goes around the side, and so on. His idea was pretty key in showing that light was a wave. He drew a little diagram saying that if light is a wave, we can think of it as being just like water waves. When it hits an obstacle, it's going to go around it and the waves would continue on the other side of the obstacle. The hard part was they weren't really sure what the wavelength of light was. They assumed it to be small. So if you did it with too big of an opening, then you really didn't see much. The hard part was trying to figure out how small the actual wavelengths are and to get gaps small enough to actually cause some interference that they could measure. Thomas Young was the guy who first figured this issue out. So what he did was he made razor-thin cuts. For example, if you took a piece of paper and you made two little thin razor cuts into it as small as possible and as close together as possible, what we would think would happen is if light was a particle, the light would go through those two cuts and we would just get spray paint-like lines on the screen. But what actually happened, and this proved that light was a wave, is that the waves went through the thin cuts and we got wavefronts just like we see in this diagram here. So just like with water waves, we would get the round waves with overlapping. And anytime you have two waves overlapping, you get constructive interference. So everywhere we see a red line is where we have constructive interference, and the dark spots in between the lines are where we have destructive interference. So when the waves constructively interfere, we get these series of bright lines, and when they destructively interfere, the wave cancels out, resulting in dark spots. Thomas Young's experiment basically proved that light had to be waves. He tried different variations with thinner cracks, more cracks, less cracks, and so on. But the basic idea was that he always ended up with this band pattern, where you get lines and the middle one always tended to be the brightest. So we think about round waves interfering, it's always the middle that gets the most overlap. So you get the most constructive interference in the middle. That's why the middle one is the brightest. We call that one the central fringe, and we can count it as position zero. On either side, we have fringe numbers, positive or negative. So let's just imagine we have two waves going, and we already know the middle one is going to be where they overlap the most. So right down the center, we get the most overlap, and that would be the middle one. But then on Either side of that, you're going to get the next or second-most overlaps. When you see that one, it is a fairly bright line but not quite as bright, so we call that fringe one. One space on either side, then the next line would have fewer number overlaps, so it'll be not quite as bright. The third one, not left-right, and so on. Eventually, you can't see them anymore because there's not enough interference to make it visible. What Thomas Young figured out is if you shine white light in a certain type of double-slit, you end up getting different colors. So we know that that happens because each frequency of light, being 700 by Olivia 350, they're gonna refract differently, they're gonna interfere differently, and once they combine, different frequencies will join up and make a new frequency, so that you'd have a different color. It's not always going to be this color pattern, it's gonna depend on what the details of the slits are, what the distance R between the slits, and how far away and all those kind of things. So, the key thing to remember is we call our central fringe 0 and then we go 1, 2, 3, 4, and so on outwards from the center one. So what Young figured out was a formula to be able to calculate some of this stuff. The formula is wavelength equals XD over an L. So X is our distance between the fringes. So you can see in this top diagram, X would be the distance between 0 & 1, or we'd say X2, the distance between 0 & 2, or 0 & 3, and so on. So we just measure that X distance from the center 1 to whichever one we want. And the D distance is the distance between the slits that you cut. So that's usually given to the question that says there are a hundred lines per millimeter or something like that. So you'll get that distance. X is the distance between the images, the lines of the images, and N stands for what number we're going to from the center. To 1 we use N of 1, from the center to 2 we use N of 2, and so on. And then L is the distance between the diffraction grating and the screen. Let's do an example. So wavelength equals XD over L. It says we've got a screen that's thirteen point seven meters apart, so that's thirteen point seven L. It's asking for the third-order fringe, so that means we're looking for an N of 3. And it says we see these two point five centimeters from the center, so that would be 2.5 centimeters. N is -2, so 4X distance. And it says the D, the slits, are cut 0.096 centimeters. That's 10 to the negative 2 as well. And it's asking what the wavelength would be and what the color would be. So when we do this calculation, we get 5.84 times 10 to the negative 7 meters, which would be 584 nanometers. And we know that red light is 700, green light is above 500, so this would likely be an orange color. Okay, so the formula looks pretty easy and we can solve it for anything. You can solve for wavelength, you can solve for X, you can solve for any of those. There's also a slight variation of this formula, and we found out that this formula works good when the angle is small. If we're only going close to the central fringe, the numbers are pretty accurate. But if there are angles that get a little bit bigger, what happens is the numbers change a little bit, and it's because it actually gets based on our distances. It would be better if the screen was curved. So that's why a movie theater and things like that usually have a curved screen to get a clearer or sharper image. So once we get further out, that curves. Gonna make more of a difference, so then it's better to have a different formula. So if they give you a question, you don't have to wonder which one to use. You just base it on whatever the question is. So if they give you a question where they give you that angle instead, so we still have n's in this case, we would have an N of 2. We're going from central 2 to our D, which is still the distance between the diffraction grating, so that doesn't change. But now, instead of using X and L as part of the triangle, we're just going to use the angle. So it's pretty simple to use this one as well. Just plug in the numbers. For this case, it says yellow light has 540 nanometers, so that'd be 540 times 10^-9. And it said we have double slits, 2.2 times 10^-3 millimeters, so that'll actually be 2 times 10^-6 if you change it to meters. And it's looking for the angle. So it'd be sine theta, we want it from the central fringe to the second frame, so divided by 2. So we do that calculation, we get sine theta equals 0.49. So if we do the inverse sine of that answer, we should get about thirty degrees. You just double check them in Radian mode. I'll switch it to degree mode. So we get about thirty eight point eight or thirty one degrees approximately. So those are the two formulas. They work the same. Just one gives you distances, one gives you the angle. But just pick whichever one is appropriate. So a diffraction grading then is basically what we find out is the more slits we have, the more interference is going to occur and the more intense the better it's going to create the pattern. So if we only have two slits, we're likely only gonna see one or two lines. If you have three slits, four slits, or a million, the more you have, the better it's gonna get. The problem though is if we were to make razor cuts into an object, eventually it's going to be too many cuts and it would sort of fall apart. So what they do now for a diffraction grating is they actually take a piece of glass and they use diamond to etch lines in it. So the lines actually scratch the glass, so light doesn't pass through. So it sort of works the opposite, where the line blocks the light, and then where the line isn't, the light will pass through the grating. And so they work just the same. And if we look at a diffraction grating with white light, we'll see it differently in both colors because the different frequencies are diffracting differently. And if you use monochromatic light, like a red laser, something like that, then we usually just see a band. And they would just be dots or dashes that are all in red color. If we use diffraction gratings, we can actually measure the wavelength and get very, very accurate measurements. A true diffraction grating is made with glass and with diamonds. What we would typically use is just a piece of plastic or acetate that has the scratches in it, but it works pretty good as well. And a key thing to remember is that the violet end of the spectrum will diffract the least, and the red end will diffract the most. So if we were to shine a red laser and a violet laser at the same time, we'd see the red one at the top. So let's suppose the red scatters like that. Then if we did the exact same experiment with a violet, our center would be in the same spot, but we would see the second and third ones being a lot closer together, because they diffract least. So they're gonna be a lot closer to red. With stretching the most, and obviously green would be somewhere in the middle. And we can actually use a CD and a DVD to create rainbows and get these kind of things, because of the grooves on a CD. So if you think of a CD as just being a whole bunch... Of microscopic lines etched in with a laser, they basically work like a diffraction grating. As well, because the CD back is silver, it'll reflect what you see. So you can actually use a CD to bounce the laser off of and then reflect it onto a screen, and you would see a diffraction pattern occur exactly the same. And if you compare a CD to a DVD, the DVD will get a bigger refraction pattern or a diffraction pattern because there are more grooves. A CD can hold way less, so they have fewer lines cut in. A DVD has way more. So, the only thing you have to be careful of for a lot of these questions is quite often they'll give you the D number as how many lines per millimeter or how many lines per meter. In this example, I've got that the diffraction grating has 600 lines per millimeter. So what we need to do is figure out what that distance is between each line. So you just have to do the reciprocals. 600 lines per millimeter, we just have to go 1 divided by 600, and that gives us 0.0016 repeating. And that'll give us millimeters between our poor line. Then, all we have to do is change that to meters. If you divide by a thousand or times ten to the negative three, we would actually get 1.67 times ten to the negative six. So that'll be how many meters between each line. Quite often, we have to do that calculation first, and then we can do the rest. In this last example, we've got, "What is the value of D? Using the value of D, determine the color, or that wavelength." So we have D sine-theta over m. So we just calculated the D as 1.67 times ten to the negative six. Our angle is 17.5 degrees. Sine 17.5, and we're doing the first fringe, we just divide by one. So if we go 1.67 times ten to the negative six times sine 17.5, we get an answer of 5.02 times ten to the negative seven meters, which would be roughly 500 nanometers. And we know that 500 is green. It's got green light generated in this situation. So based on what we have got with our numbers, we get a 500 answer. We know that that's green. And that's it for the fraction.