Refraction and Refractive Index

Introduction:

So in the morning when you see a sunrise or in the evening when you sunset do you see the sunrise and sunset the moment they happen or do you seem them later or before? In other words, does light travel really, really fast? Almost infinitely fast. If you look at the sky, you see stars. Are you sure they are still there? Because when you look at them, does light instantaneously reach you from them or does it take a long time? Because when the distance becomes larger and larger, you might be able to notice that light takes some time. Because in real life, when you see around us, what we usually see is light is so quick and the distances are so small that speed of light is hardly a measure that we have to make. It’s almost infinite.

Does light really have speed, that is measurable?

In other words finite. Well, the truth is, in vacuum light travels at a very, very, very high speed of 3 × 108 m/sec, which makes it so that it’s almost instantaneous. But even with sun lights takes about 8 minutes to reach us. So in other words, if somebody turned off the sun, we wouldn’t know for about 8 minutes. Only then we will know, which means that if you take light and you allow it to travel a particular distance – it has a finite speed – but does light travel at the same speed in all media?

And what do I mean by media? Light can travel through glass, light can travel through water. There are some media through which light cannot travel. We have seen that. I put it through a table, it just gets absorbed. Doesn’t travel through it. But if it can travel through it, does it always travel at the same speed? It’s an interesting question. You might already know that light varies its speed depending on what medium it is in. But that is only a partly, you know, it’s a partly true statement. Because people confuse you all the time. They tell you that light travels at different speeds in different media. Why?

What is light?

It’s a form energy that’s travelling at some particular speed. Why does it slow down in a medium? Or does it even slow in the medium? Well the truth is, to understand this, we have to go into the atomic picture of something, right? Let’s say light is travelling through vacuum. What does it have to stop it? Nothing. So this energy, or light, which is actually electromagnetic wave, what does it really do? It’s going around in vacuum, happily, with nothing to stop it.

But the moment it encounters a medium, what really begins to happen?

There are atoms in the medium, which might get in the way of light and which is what happens. So light gets in. What usually happens is that light gets absorbed by one of the atoms and then gets reemitted again, gets absorbed by another one, gets reemitted again. So it’s not really that light is slowing down but light is getting absorbed and reemitted, absorbed and reemitted. Therefore, there is some time for the absorption, some time for the reemission.

So what does it look like to us from outside?

You know we are looking at it from so far away that it looks to us as if the light is just slowing down. For all practical purposes you can’t imagine light to be slowing down, but it’s nice to know what’s really happening inside. So the true analogy is that light doesn’t slow down, but light has a lot of stops. It’s like not really. Like if light was a bus, and the bus is going. As long as there are no bus stops, it’s quite fast. It enters the medium. Has the bus slowed down? Not really. Whenever the bus is moving, it’s moving with a particular velocity, but there are many stops. Each atom is a stop for the bus, and in this case, for light. So when it as if it’s moving inside the medium, it looks as if it’s slowed down.

So with this in mind, the biggest takeaway for us is that light has different speeds in different medium. And how is it determined in each media?

Depending upon how quickly they absorb and how quickly they reemit. Now, based on this, we have a definition. We say that a medium in which light travels very fast is less optically dense, and a medium in which light travels very slow is more optically dense. Now I want you to know that this density, optical density has got nothing to do with the usual mass density – which is what? Which is heavier? You take like 1 volume of, say like, liter of something. Which is heavier? That’s a different density. That’s mass density.

Depending upon how slow light is, in that particular medium, we have another term called optical density. A good example for you would be, if you take oil that oil floats on water. What does that mean? It means oil is less dense than water. But, if you find out the speeds of light in oil and in water, it’s a lot slower in oil. What does that mean? Oil is more optically dense than water. So here we have just shown you an example of where oil is more mass dense, but less optically dense. One example to show you that these two need not go together all the time. So with this in mind, let’s see what happens when light now changes from one medium to another, because what’s going to happen to it when it goes from one medium to another? Its speed is going to change.

So let’s see what happens.

So when we told that light always travels in straight line, you already know that light bounces off some surfaces and in one sense it’s bending, right?

• In another sense also it bends, so we kind of lied to you when we told you light always travels in a straight. It does travel in a straight line until and unless something stops it. When something stops it, it either gets absorbed, or bouncing off, is called reflection, or goes through.
• But when it does go through, does it continue to go in a straight line or does it do something else called bending?
• So will light bend when it goes from one medium to another? That’s the question. Now we know one thing. Everything in this chapter comes out of one fundamental principle. What is that? Light is in a hurry! Light wants to go from one point to another, as quickly as possible, so it takes the quickest path.
• Now how do we know this? Why is this true? We don’t know. But we call this Fermat’s principle of least time.

So let’s see if we can just extend the same analogy and understand whether light will bend, if it will bend how it will bend, which direction will it bend in?

All those things. So to do this, let’s take a very, very interesting analogy. Let us say 3 friends, right? Three of you, guys in this case, are walking along a shore or a river or of the sea, or whatever it is. And all of a sudden you watch a gorgeous woman sinking, right, she is screaming for help, somewhere in the middle of the water.

And all 3 of you know how to swim. And you decide this is your chance, right, to get her attention and become the hero because all of us watch these movies. So what do you decide to do? So three of you, or one of you is like extremely enthusiastic. You are the kind of guy who thinks like, you know, oh my, I have to do this. So you start running right at her.

So what kind of path will you take?

You start running right towards her. So you reach the water at some point and then start swimming. But if you are a little smarter, what would you do? Because you know that most people can run a lot faster than they can swim, right? So you would probably run a little bit more and then arrive at an angle and then start swimming towards her. So who’s going to win really? Is the guy who’s really enthusiastic, who runs high hatter, reaches the water, starts swimming. He has to swim a lot more, right, as you can see? So he starts off, he goes straight at her and then he starts swimming that much, right? Do you think he is going to reach first? Or do you think this guy who is going to do that is going to reach first?

Right, one of those paths is going to be more efficient than the other path which means that the quickest path need not always be the shortest path. And you already know this. Right? If you were to go, if you stay in a metro in India, or any other part of the world, you know that a straight line path that connects two points in a city might be through a main road and you will never ever reach on time but you can take like a little shortcut. Of course it’s a lot longer but you will reach faster because there is no traffic.

A very similar analogy here. So what do you observe? The shortest path need not be the quickest path. But what path does light always take? It takes the quickest path. So let us now begin to notice what is the quickest path? It’s not a simple problem right? You might also ask, how does light know it has to take the quickest path? We don’t know. How does light find the quickest path? That also we don’t know. Or rather if you really want to know about this, we will have to go and understand what light really does. When it behaves as a wave. Right? But in one sense here we really don’t know how light figures out, but it does figure out doesn’t it? So light always picks up the quickest path and as you can notice if this is land and this is water.

The light starts here. It’s not going to go straight that way. It is going to go here and then take that path because it wants to spend as much time as possible on land where it’s faster, and as little time as possible in water, where it is slower. So if people had asked you, why does light bend, then a trivial answer would be – because the speeds are different. But a much better answer is to say – speeds are different and light wants to take the quickest path, therefore, it bends that way. Of course light would not do something stupid like go and come back like this. Right? That would be really stupid.

Because the worst case path is, and down there straight, right? Now the question really is where will the bending be more? When will light bend a lot more? What appears to us as bending is when light wants to spend more and more time in the faster medium which means that if the speed, the ratio of the speeds, or the difference in speeds is a lot then the bending will be more. That makes a lot of sense, doesn’t it? So what is this bending proportional to? It’s proportional to the difference in the speeds or the ratio of the speeds of these two media. The slower and slower the second media is the more and more the light wants to spend in the first medium. Another, with this in mind, you keep this in mind. We will give you another analogy for you to be able to understand how light really bends. For you to think of this, think of yourself riding on the highway. You are riding a car on the highway. So you are going really fast and you see somebody, say in clay. So you are going to turn and go into the clay, let’s say. Right? So you are approaching. Let’s say you are a car and you are approaching clay from road. So the moment you hit your interface, which wheel touched the clay first? So what’s the assumption here? You can travel faster on the road than on the clay. Fair assumption. So your right wheel touches the clay first. Right?

Now what do you observe after that? That wheel gets slowed down. But who hasn’t got slowed down yet? The left wheel. So what happens? When one wheel slows down and the other wheel does not slow down, two wheels are going at different speeds. What happens when that happened? Turning, right? The car is going to turn that way and then start continuing straight because once both the wheels have touched the clay both are going at a same reduced speed. Turning only happens when one wheel is going slower or faster than the other wheel, right? So if you imagine it this way, it becomes very obvious, doesn’t it? Right? A car coming here, it’s hit and starts continuing at another angle. Now what’s really interesting is that look at the car as it comes straight down at the road, from the road to the clay.

Does it turn?

It won’t turn, because both the wheels will hit the clay at the same time.

Similarly, if you will imagine a light ray, even with our old analogy, if the light ray is approaching perpendicularly to the girl, right, if you are on the shore and if you see the girl drowning and if she is right at you, right? In other words she is perpendicular to the shore then all you have to do is run straight at her. Doesn’t matter. There is no angle to be worried about. So the more and more oblique you are the more and more you will be bending. So you have done a little bit of intuitions about why light bends, how much it bends and where does it all come from? The only principle we know for the entire chapter, Fermat’s principle of least time, or light’s in a hurry. So from that you can pretty much derive everything else.
So let’s drive a little law for refraction which is what it’s called because we haven’t really spoken about that word till now. We have kept saying bending, bending, bending, but what do we mean by light is bending. When light travels from one medium to another, when it bends, it’s called refraction. Yeah? It’s not because of refraction. Light doesn’t bend because of refraction. It bends, and it’s called refraction. Let’s see a little bit about it in our next module.

So we kind of qualitatively saw what light does when it enters from one medium to another. Because there is a speed change, light bends. Why?

Because it wants to take the quickest path. Now let us make this more mathematical and formal so that we can start making calculations based on it. So let us start naming certain things, and let’s take a light ray. Let’s say the light ray begins and goes through air, for example, and enters glass. Right? So, it’s going to go like this and it’s going to bend that way. Great! Now let us draw a normal. Right? And let us measure this angle as the angle of incidence and that angle as the angle of refraction and not reflection. So refraction. So you have i and you have r. Now how are these two related? We know that the greater and greater the difference between the speeds, the more and more the bending will be. So let us try to understand. Let’s take the ratio of the sin of these angles. And we will see why we do that. You kind of can see that, you know, the sin is the opposite by the hypotenuse. So we are kind of trying to take a measure of how much bending there really is. Right? So the more and more the sin is, the larger and larger the angle is. So in one sense measuring sin is like measuring the angle itself, because the sin is proportional to the angles.

So we will take the sin of the angle and if we take sin i by sin r, we find that it is a constant for a given two pair of media. And it’s called the refractive index. Of that two pairs of a media or that two pair of media. So you have media 1 and you have media 2. Then, sin i by sin r is called the refractive index of 2 with respect to 1. Because, the light entered from 1 into 2. Therefore, you represent that, you write sin i by sin r, as say, r 2 1. r of 2 with respect to 1. I am using a pretty ad hoc convention here, there is not really a very, very standard convention but we can do this. Now, what turns out to be even more beautiful is that that refractive index was a random number right? Yeah, for thousands of years we thought that refractive index is a random number which is a property of two media. But after Fermat’s principle of least time we understood something very, very, very interesting about it. And that is, that refractive index actually corresponds to a very, very, very physical representation of the speeds of light in those particular media.

1. So if you were to write this sin i by sin r equals a refractive index of 2 with respect to 1. That is also equal to the velocity of 1 by the velocity of 2.
2. Velocity of light in 1 by velocity of light in 2. Clearly there is a difference in velocities, otherwise, light wouldn’t bend.
3. So if we notice right now, what is this refractive index? It is nothing but the ratio of the speeds in the two media.

And the greater this ratio, the more the refractive index and, therefore, more the bending. Makes a lot of sense doesn’t it? So we don’t really have to remember this as a random thing, but, what is this really a measure of? How much does light bend and what does it depend on? The ratios of their velocities. So to bridge these two, we have a quantity called the refractive index, which is nothing but the ratio of their velocities.

Now with this in mind, that particular quantity that we derived, or that particular equation we wrote, sin i by sin r equals refractive index, which is also equal to the speed of light in the first medium by the speed of light in the second medium is called Snell’s Law. There is a pretty interesting derivation for this which we can do. We won’t go into it right now, right. You might want to learn about it. If you really want to learn about it, good, go ahead and do it. But the way to derive this is understanding a very fundamental principle about what light really does. What is light trying to do? Take the quickest path. Right? You go little deeper. What you will understand is if you calculate the quickest path and calculate sin i by sin r, you will get the ratios of those velocities. You can go ahead and do that. So what we have derived at right now is Snell’s Law.
Now, let’s take the opposite case. Because we know the law of reciprocity and what does it say? Whatever light does in one direction, if I invert the whole thing, it will do exactly in the other direction. So if light started off from glass and entered air, what would it do? It would come this way and then start going the other way. In other words it would bend away from the normal. What did it do in the first case?

It bent towards the normal. So if you take this into account, and you start observing what happens, the biggest takeaway from this little part is that, when light enters from a rarer medium to a denser medium, it starts and it enters a denser medium, it bends towards the normal and vice versa, if it starts from a denser medium to a rarer medium it bends away from the normal. Now that is one of the biggest takeaways because most of what we do in refraction can be boiled down to this fundamental idea. Does it bend towards normal or does it bend away from the normal. Now the really keen ones amongst you might have noticed something, right?

Let’s take a case where light enters from a denser medium to a rarer medium and look what happens? I make this. I start a ray of light that goes that way, bends away from the normal. I make it even more wider, it bends even more away from the normal. I keep doing this, what’s going to happen eventually. Light is going to hit and start going parallel to the medium and at that point we say light is at a critical angle because what light goes to the other medium? Nothing, right? All the light is going parallel between right at the interface itself. So then what would happen if I increased the angle even more, like you see, it’s going to get completely reflected back into the old medium itself. Was that a mirror? No! Just by refraction or where light bends, light bends so much that it’s going to start entering its own medium. So there is critical angle at which you can send light because at that angle, any greater angle than that light will not go to the other medium. It will just get reflected back. Interesting.

So this principle is called Total Internal Reflection.

Because all the light is totally reflected inside and it’s pretty useful in a lot of places whether you see a mirage, or whether you see, you know, little blocks of water on the road right? You have seen a very hot road on a long day, driving on the highway, you see little patches of water. You can explain that using this. How do optical fiber cable works, cables work – cable works I said – you know, how do optical fiber cables work? They carry communication throughout the world. They carry light such that it gets totally internal reflected. So these are all little parts that you might be really interested in. But the fundamental crux of this idea is that when light enters from one medium to another, it bends following a very, very reliable pattern of sin i by sin r equals the refractive index of the speed of light.
Now let’s take this one step further. What if, for every pair of media in the world we had to find out the refractive index with respect to each other? How many pairs of media, let’s say there are ten media in the world. Of course there are lot more. How many should you do? A lot, right? And if you are into permutations then its 10 c 2, and how much is that? 10 into 9 by 2 or 45. So, 45 different experiments you have to do. Okay how much is sin i by sin r here, how much is sin i by sin r there, but you don’t really have to do that.

Because of Fermat’s principle of least time. What does it tell us? The refractive index is not a random number. It is a ratio of the velocities, which means all you have to know is the refractive index of one material with respect to say a standard material, say vacuum. So you find out the refractive index of each material, or each media with respect to vacuum, then you can keep calculating for all the other possible pairs. So now if we have ten media out of which one is vacuum, how many times do we need to make an experiment? Only nine times. So now you know all the refractive indexes with respect to vacuum and then you can calculate for each other.

Example:

Let’s take an example. Let us say you have glass and let the refractive index of glass with respect to vacuum be called refractive index of glass and let the refractive index of water with respect to vacuum be called the refractive index of water. Now how would you find out the refractive index of glass with respect to water?

In other words when light enters from water into glass or from glass into water what’s going to happen? In order to do this what do you have to do? All you have to do is just divide these two quantities because we know that refractive index of water is going to be velocity of light in vacuum by velocity of light in water. So if we divide these two what do we get? Exactly the quantity that you want. Velocity of light in water by velocity of light in glass. Pretty simple right? So this kind of Fermat’s principle of least time makes it so that all you have to have is the refractive index with respect to some standard medium and then you can calculate for any pair of media, just by dividing the corresponding refractive indices.
So with all of this you’ve covered a good amount of ground here. You have done Snell’s Law and you’ve understood how refractive indices really work. By division. So now that we’ve understood what refraction is and we have also understood what Snell’s Law is, let’s take our first object to play with. Let’s take a glass slab.

What’s a glass slab?

It’s something that’s like a rectangular cuboidal piece of glass and let’s see how light really begins to behave when it passes through this glass slab. So we are beginning to play with our first object here. Let’s see what happens. So let us take a little glass slab. Let’s place it out here so that it’s here and let light start coming towards that. We know that light is entering from rarer to denser because air is rarer than glass. So it’s going to bend towards the normal. Great. Good up till here. But the glass slab ends after a point of time, so the light, it has to go from glass into air.

Now, that is denser to rarer. So what’s going to happen? It’s going to bend, that’s right, away from the normal. Wonderful. So we have a pretty simple picture here. Light enters, bends towards the normal, then goes away from the normal. Right? So, but what does it look like to you? Does it look like the light’s direction has been altered? Light came in a particular direction at some angle. Has it changed finally? Or is it going in the same direction that it came from? Of course you can see that it is a little split right? From wherever it would have gone, now it’s been a little deviated. Yeah. But is it still going parallel to its original direction or has this direction changed itself. We don’t know. It might look parallel but that doesn’t mean that it is parallel. So, let’s try and understand whether it is really parallel or not. So to do that it’s something very simple. Let’s mark out some angles here and what we will begin to do is that we know that the first angle, let’s call it i1, the second angle let’s call it r1. So sin i1 by sin r1, we know what it is. That’s the refractive index of glass with respect to air, or, vair by vglass.

Great. Now let’s name the other two angles i2 and r2 and that is what? sin i2 by sin r2 equals what? The refractive index of air with respect to glass, or the velocity of glass by velocity of air. When I am saying velocity of glass I think you know what I mean – velocity of light in glass or velocity of light in air. So you have those two equations. Great. Now what do you observe here? Top – velocity of air by velocity of glass and bottom – velocity of glass by velocity of air. They are reciprocals right? So you can forget them and then just equate these two as reciprocals and now that you have done that let’s observe one more thing. We’ve all learnt our geometry really well so we know that if there are two parallel lines, there’s a transversal that cuts it then those two angles, called alternate angles are equal. And we observed a pattern here. Is there an alternate angle here? Yes there is. So r1 and i2 are alternate angles. Great. So what do you have to do now? They are equals so we might as well replace them one with the other.

So if you do that what do you get? That’s right. You get sin i1 equals sin r2. In other words, i1 equals r2. What have we just proved here? Yeah. That particular angle that we began with is equal to this particular angle with which it leaves. So very clearly, very easily, if I extend that line out here and if I draw something like the normal x and the normal up there, you can very clearly see. Right? if you were to take this and draw it out there, you can very clearly see that what we have just proved. What have we proved? i1 equals r2 also clearly proves that the two lines are parallel. Which means that a glass slab really doesn’t bend light. All it does is displaces light a little bit. Right? While the light is in the glass it moves in a different direction but once the moment it’s out it’s back to its original direction. So with this in mind, we’ve kind of understood what a glass slab does. Why is this useful? We’ll see. And we have also proved – we have used this as an excuse to use our Snell’s Law. Get more hands on experience with it and prove that the light that comes out of the glass slab is parallel to its original direction, but with a little bit of a deviation.
Now it’s time to play with something very, very interesting from what you have just learnt. Because it’s always more fun to stretch ourselves beyond what we usually know. So let’s ask a simple question. I take a ray of light, send it at a particular angle, right? So, and then from air let’s say it goes into water, or glass. Let’s say from air it goes into glass. Right?

So, simple case. Light going from air into glass. We all know what happens. What happens?

It bends towards the normal. Clean. So you see it out here. Wonderful. Now let’s take another case. Just at over here, instead of air going, or the light going directly from air into glass what we are going to do to it is that we will put a layer of water in between. So, air goes through water, then into glass. Now the question here is. Both cases light emerges in glass, so it’s going to be travelling through glass. Would it travel at the same angle or the water in the middle, would it make a difference? And that’s the question. Right? So does the water in the middle really make some kind of difference to the file emergent angle of light? So should it or should it not? Now we can probably do this using math.

Yeah. sin i by sin r equals the refractive index. Over here sin i by sin r equals refractive index between water and air. sin i by sin r for glass and water, then you can calculate. You might get the answer to be that it does not make a difference, but let’s try and understand how we can do this in a more, maybe slightly more beautiful manner. Let us kind of imagine that between glass and water here, so you have air, you have water and you have glass in the second case right so you have three media.

Let’s just imagine for a moment that between the water and glass there is a very, very, very, very thin gap. An infinitesimal gap let’s call it a very, very small gap. And what do we know about glass slabs? Right? In this case what form is the glass slab or water? Because it’s any rectangular piece as good as a glass slab. So, light is going to enter from air into water, it’s going to bend towards the normal and just as it leaves water what is it going to do? It’s going to emerge outside and just in our previous module on glass slab we proved that these two rays are going to be parallel. What does that mean? It means that as far as the light is going to exit water in a parallel manner we can forget that the water exists. So what does it become equal to now? The second case becomes exactly equal to the first case, right? Because as long as the air, or light that enters water exits the same way as it came does it really matter if the water is there? So remove water. What do you get now? Light entering glass. So we are confirming that the two rays are going to go in a parallel manner. So the number of media, in other words generalize, I can start with media A and I can add n media in between – B, C, D, E, F – as long as light doesn’t get totally internally reflected back, when it emerges out, if it does emerge out of the last medium it will emerge out in the same direction as if all of them had been removed and just the two media are put together. Right? So that’s one interesting kind of corollary to understanding Snell’s Law in a very deep and fundamental manner.