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WEEK 4
Three interpretations of light
Note that the things we covered in the last lectures are just the math in the sense of its just a geometry and the algebra, the mathematics required to move points around in space.
So, today I want to talk about light and optical systems. So, I am going to be switching more towards physics. Two reasons for that, one of them is if we want to go from math to computer graphics so that we can talk about rendering in artificial worlds or virtual worlds then we need to understand how light propagates or if we want to maintain any kind of alternate world using the alternate world generator we want to understand how light behaves and propagates in these spaces. So, that is one reason.
The second reason why is that these virtual reality devices that we build exist in the physical world and there are optical systems associated with them. So, we have to understand how that is working because the light ultimately goes from the engineered device into your eye hitting the retina as part of the presentation of the stimulus as I said right, it is an artificial stimulus that is being generated.
So, let us talk about this.
(Refer Slide Time: 01:28)

So, we will talk about light and optical systems. Well, first of all I will just starting off with the absolute basics. I know you have all had physics before us from somewhere. We have three interpretations and we will find all of these useful in this class. So, three interpretations, one is light as a particle, in other words as photons and you may remember two more representations one of them is light as rays. So, we think about rays of light.
This representation is very popular among computer scientists for doing visibility kinds of calculations and computations. So, there is whole family of algorithms in computational geometry for example, called visibility algorithms in computer graphics there are what are called ray casting and ray shooting methods. So, rays of light become very important and at the same time waves are also important. So, like to think about propagation of waves. So, rays and waves are very closely intertwined very often in physics you hear more about the duality between particles and waves. But the rays perhaps are just another way to look at the waves which I will say in just a bit.
And also remember this, the frequency is equal to the speed of light divided by the wavelength, so just very very simple formula that hopefully you remember from somewhere frequency in hertz, speed of light. Anybody remember what that is roughly speed of light in a vacuum.
Student: three into ten to the eighth.
Three times 10 to the 8th meters per second yeah that is right good. So, I guess I remember too, right or I cheated and I heard what you said ok ah wavelength right. So, once you know the speed of a wave in a medium this is just a general property of waves, once you know the speed of the wave and a medium you can relate the wavelength through the frequency. So, you should be able to easily convert back and forth between those sometimes I might show you figures or plots that are using wavelength or frequency you can convert to the other one it is very straightforward assuming we know the speed. So, I want to relate rays and waves a bit here and then I will start to talk about optical systems.
(Refer Slide Time: 04:24)

So, let us think about a point source of light. So, we started a particular point as the name suggests and then we can think about waves propagating outward from that. Try to draw a circular wave fronts for as long as I can no guess I am already starting to mess up that should be far enough and so, we have these waves coming out. So, generates circular wave fronts assuming that there is no other obstructions right. So, we just have a point light source and were propagating through maybe a vacuum or the air which is closed in terms of propagation and the speed of light is only slightly lower in air and then we may think about rays of light the rays are perpendicular to the wave fronts.
So, whenever we consider rays of light and we draw this just think about them also corresponding to the way that the wave fronts are propagating right and, so at any given point a long array orthogonal to that should be the wave front that is propagating along all right. So, these two representations 2 and 3 are very closely intertwined and. So, if I start drawing pictures of lenses with rays of light going through them, think also about how the waves are propagating.
(Refer Slide Time: 06:26)

Now, one interesting thing about the physical world is that without mirrors or lenses the rays always diverge, all right. So, we could get we could get very far away from the source of light and the wave fronts are propagating way up when we get over here we are imagining these rays of light propagating out. They are still diverging right, no matter how far away we go they are still diverging do you agree with that.
Yet at the same time when we talk about optical systems everyone likes to talk about parallel light rays. So, I just want to point that out that parallel right where light rays are in some sense a kind of fiction unless you start using mirrors and lenses right away to fix that. But very often we pretend that we have a light source and we have parallel rays coming in and what is true about it only is that it is an approximation all right. So, what we often mean when we make a drawing and we claim there are parallel rays is that the light source is so far away that we may as well assume that the rays are parallel and if that happens what are the wave fronts look like.
So, they should be perpendicular and parallel as well. So, the parallel wave front case, so we have when we are very far away from point source then we almost have and you know the almost is important here that said it is not exactly right, parallel rays and what you might consider to be then flat wave fronts right.
(Refer Slide Time: 08:22)

So, the wave fronts should be curved in a circular way, but the radius of curvature keeps going down right, on the radius of curvature keeps going ups now radius of curvature goes which way let us see up or down, up sorry goes up and when it tries to tend toward infinity right then these look like lines, right. So, the circles are getting bigger. So, the radius is getting larger alright.
There are several names for parallel rays and just want to point out what these are, several names that you will find around if you look at other literature. You can also call them collimated, people say collimated light or collimated rays the situation is also called zero vergence, and also called rays to or from infinity which seems to make sense, right. If you could get infinitely far away in some kind of limiting case which as far as we know does not happen in our universe how do we know and experience anything like that then then the mathematical model would be correct all right. Questions about that?

Refraction
So, now let us take these rays or wave fronts or particles and have them interact with another material alright. So, if they start off by propagating either in a vacuum or through air, which we might consider to be almost like a vacuum, and then let us think about what happens when we when the waves interact with some kind of medium, they hit a medium that is different. (Refer Slide Time: 00:41)

So, what is going to happen essentially is that we use materials to bend rays or alter the wave fronts if you like because they are; obviously, very closely intertwined. So, I take some kind of material let us see we just draw rectangular block of material. So, if some kind of material and then we start up here and we send our light rays on their way and let us talk about the different things that can happen.
So, I am going to mention four different cases, one of them is that light comes around here to the corner, and then starts to bend around the corner what is that called its one diffraction, right. So, here is another case and this one will be the most important for optical systems, the light comes along, then its its course is bent in some way say maybe I will draw like this and then back out. So, hopefully that looked like it was bending maybe I did not put it enough bend into it, it should not look like it is going straight through make it a little more let me do the whole thing over again I am going to go here put some obvious bend into it, and come out like that. So, we have some kind of bending going on this is the case of transmission of the light through the material through the medium.
So, this is transmission and the bending of the rays is called refraction, refraction of the rays. Another thing that can happen the light could come into the material and just kind of and let us say the light comes in the material and never comes out again. So, just absorption so, the light is lost forever probably the material would heat up then right heat is usually the ultimate place where energy gets a kind of lost or the final form for it let us say. So, this part is called absorption, it is something important to keep in mind the more the more materials the thicker they are the more materials you put into your optical system, the more you will be just losing some light right, what case remains what else can happen to the light reflects away, right.
So, if this surface happened to be a shiny mirror then we get reflection. So, comes down like this angle of incidence should be equal to the angle of reflection alright. So, this angle and this angle are should be the same and off the ray goes. So, this is reflection. So, let us talk about a couple of these in a little bit of detail mainly I am going to be interested in refraction today.
But for reflection I should say a little bit about it I have drawn one kind of reflection there, this kind of reflection where these two angles are the same is called a specular, there is another kind of reflection just kind of redraw the material here, where let us say rays come in, but they may go in what seems to be various random directions alright. So, they bounce in all different sorts of ways, and this is called a diffuse, alright. So, these are two kinds of reflection, and these become extremely important in computer graphics for rendering. So, when you want to model surfaces do they look shiny like a mirror you have the specular case, does it look like let us say the carpet on the floor here then this would be more of a diffuse case right you cannot look down on the carpet and see your face on it alright. So it is diffusing the light all over the place, it does not look completely dark. So, light is clearly bouncing from it.
But its diffuse; and now what is the most important law for refraction between two materials Snell’s law alright. So, Snell’s law is the basic expression.
(Refer Slide Time: 06:42).

I have Snell’s law which is n 1 sin theta 1 equals n 2 sin theta 2 I guess is meaningless until I make a picture. So, I have two different materials I have an upper material and a lower material, and which one is air and which one is say a piece of glass, one could be water one could be air all sorts of possibilities right. You have noticed that a swimming pool can act like a lens sometimes right and you get very interesting effects from that. So, any two materials coming together, and now we draw a perpendicular line. So, these are two materials let us say material 1, material 2 draw a perpendicular line, and now we have the light ray coming into the boundary and then the light ray leaving.
Now I mark my angles here, say theta 1 say theta 2 these are both meant to be a positive angle. So, whichever way you and I guess I have not fulfilled the formula here in terms of putting in all of the components and defining them. So, the n’s I will put n sub i is equal to c the speed of light in a vacuum, over the speed of light in the medium. So, I will put in medium i since I have numbered my materials or media. So, sorry I have change the words a bit material or medium be a little more consistent medium one medium 2. So, n sub i is called the refractive index, and note that by the time you are done here what really matters at this interface is not the absolute value of the refractive index, but what matters is the difference between the two refractive indices correct let me just give some kind of simple qualitative examples, that correspond to the algebra here if you were to put these things in practice.
(Refer Slide Time: 09:46)

So, if I have a fast material and a slow material and the light ray is coming down going to make it look like it is about 45 degrees, what should happen to it? The waves are going to be propagating more slowly in the slow material. So, that causes it to bend upward or downward is that right down or up.
Student: (chorus) downward… down… downward.
Let us see I may have to actually go and do the calculations, that seems interesting that is material low material wave fronts yeah sorry thank you it is down yeah good I have a mistake here, but I see. So, alright; so, it goes downward. So, we draw a straight line through here and downward we go yep. I sometimes like to think of a kind of army marching along right and then it slows down the waves get closer together, but if you actually look at the normals, right which is the wave fronts then it propagates downward that is right. So, it is an interesting when I sort of think about imagine like a marching band coming along, but then they get slowed down they get compacted together, but yes the direction of the ray then ends up as perpendicular to the wave front ends up pointing more downward. If we go the other direction, I am slow too fast again at 45 degrees. So, in this case should be the opposite I hope right what do you think. So, I draw the rays here and should come off like this, couple more qualitative things and then we will start talking about lenses. (Refer Slide Time: 12:04)

So, let us go back to the fast slow case, and if the ray of light is very close to perpendicular, then when it leaves it is going to remain very close to perpendicular right. So, the amount of bending is not very much for a close to perpendicular ray right that is close to the perpendicular to this boundary between the two media. So, that is just something to pay attention to. So, when a light when a ray of light goes straight in or the wave fronts come right in parallel to the surface to this boundary, then there is effectively no change due to the propagation speed difference. So, the different refractive indices; however, when you come in at a larger angle, when the ray comes in at a larger angle we see significant bending correct.
So, just something to pay attention to. So, so two things become important the difference in refractive indices, and the angle at which the ray comes into the boundary. So, those are two things to pay attention to, you could make that angle. So, extreme. So, that when you take Snell’s law. (Refer Slide Time: 13:26)

And you try to figure out what the outgoing angle is let us say I do sin inverse n 1 over n 2 sin theta 1 if it turns out that n 1 is greater than n 2 and its significantly greater and let us say this sin theta one ends up being significantly large isn't it possible inside of here to get a value greater than 1 does that mean Snell’s law is wrong the way I mean it has to cant violate the laws of trigonometry right so.
So, it turns out that eventually it tries to bend so much that the light will not even enter the medium anymore. So, if it turns out. So, if n 1 over n 2 sin theta 1 is greater than 1 then rays do not traverse the medium. So, just one sort of extra warning there, so when you are close to perpendicular it looks very much like as if you had as if both media were the same as you start offsetting the direction that the rays are coming in you get more bending eventually there is some limit to where the rays will not reverse the medium and it does not have to be perfectly parallel just somewhere along the way based on this algebra.ok. So, now, I want to start to talk about simple lenses any questions so far? Have you seen this somewhere probably maybe a while ago maybe preparing for some exams somewhere who knows.

Simple lenses
(Refer Slide Time: 00:17)

So, I can make a simple prism, I could have a ray of light coming in here try to make a look like as if it were the parallel case parallel rays case, if the row coming in horizontally, then I get some bending here and I get some more bending coming out again each one of these bends here is based on Snell’s law which I just covered, I am not doing the exact algebra for this example on this drawing a kind of abstract picture and so, this is you know just a simple triangular shape; you can imagine that it just extends the same way in what we have been calling the z directions.
So, it is cylindrical in the z direction, if you like and has a triangular cross section. So, some kind of prism and I could make the upside down version of that may look something like this right. So, that is the behaviour of a single ray going through a prism where there is medium 1, then you go into medium 2, let us say and then you come back to medium 1. So, this is essentially the idea principle that is going to be applied in a continuous fashion across an entire lens.
So, if I make a simple spherical lens I get something like the following soon I am going to stop drawing pictures and switch to slides because it gets complicated to trace all visualize of light, but this will be one of the last pictures let us say that I try drawing myself. So, so these should be look like circular arcs and there is really another dimension to where these are spherical now.
So, these are spherical caps just like the way lenses look that you are familiar with whereas, this was supposed to be just the same outwards. So, there is no curvature I have my let us say I have a central axis for this kind of axis of symmetry and I have rays of light coming in and I am going assume this thing that I told you was fictitious which is there are parallel rays coming in and they are only fictitious if you live in a world with no mirrors or lenses. So, I guess we could have produced it through some other means, but if it came from a point light source then this is impossible, but a reasonable approximation.
So, we have these parallel rays coming in and when you think about the behaviour of these it is like when it comes in and when it comes out it is like having a slice of a prism right. So, that is a
kind of thing to think about the slice of a prism and if these rays are coming in at a different angle then I guess it will be a different slice of a prism right. So, so it represents all of that somehow it is this prism bending effect, but manifesting itself in very localized ways depending on exactly what the curvature is here, right, this is not a curved surface.
So, you can predict what is going to happen all the way up and down its going to look the same, but because this is curved the bending is going to depend on where you are add along the curve of here.
So, as it goes through the lens we end up with let me draw another axis here which just should be the center of the lens I am not very good at straight lines today.
(Refer Slide Time: 04:02)

You ever have a bad straight line day you know also. So, so what is going to happen is in this lens is that we effect of refraction is going to cause these rays to bend inward to start to converge and if we make a simple spherical lens they will converge to a point somewhere.
Let me just invent the location of that here they will converge to some point and this point is called the focal point whenever you hear the term focal point remember that that corresponds to what the lens would do for parallel rays and I guess I have to start to draw what happens to these rays as they come through its going be very difficult for me, but let me do my best here, right. I am going to draw all of them.
Of course, at the central part of the axis it just goes straight through without any bending right something like that guess I have to match these pictures a bit here oh I am not really sure I am probably not following the same rays, but get the idea of course, these come from somewhere inside the lens. So, do not think I have my arrows to the left matching exactly the ones to the to the right, but I think you understand the idea I will start providing some machine drawn pictures more correctly drawn pictures in just a bit.
So, that is the focal point that they all converge to and this distance here is called the focal length.
So, that is how we think of a simple lens in this case as a kind of engineered device that takes parallel rays and makes them all converge to a point somewhere they do not stop at that point they keep going right. So, it could also continue onward they meet at that point, but there is nothing to stop them from continuing onward unless once you put something here what might you put here like a screen perhaps I do not know. So, if you put something here then the light will be focused on whatever you put here remember that there is an interpretation in terms of waves draw the waves in blue here. So, we had parallel waves coming in and now after they get through here they are getting curved in some way, right. So, that is another way to look at it is when the rays are being forced to converge through the lens you are introducing curvature now to the wave fronts.
So, if you were going to play everything in reverse this starts to look like a point light source and the waves are propagating out like this now is it not that interesting. So, you get this strange effect, all right. One more, I’m kind of afraid to draw the picture let me see if I can just superimpose it on this; let us suppose I have rays coming in at an angle, but they are still parallel rays. So, let me let me draw rays coming in at some angle I realized is getting pretty cluttered now.
So, I just want to point out that just because they are parallel rays it does not mean that they come in perpendicularly, right, the rays are parallel, but they could be coming in at an angle because it could have I could imagine that there is this light source that is off at infinity, but it could be up really high right. So, if I am sitting out here in the open air theatre watching a big watching a movie on the big screen I could be and imagine that screen is gigantic and its really far away maybe it is 10 kilometers away and its enormous and it is a clear day and I can watch the movie like that there is still going to be the top part the rays are going come in they are almost parallel, but they are going be coming in at an angle, right.
So, in this case for an ideal spherical lens they will still converge, but they will converge somewhere else along this line. So, there may be these rays coming in to correspond to some distant feature or object the wave front has spread out. So, much that the curvature has as the curvature has turned out the radius turned out to be essentially infinity. So, that is when we have these parallel wave fronts coming in we have straight lines which is we have parallel lines which is represented by the pink here.
So, that everything comes in and we end up with the focal point being I am just going to draw in some arbitrary place down here somewhere the important thing to know you know is that they converge and they converge to another place along what is called the focal plane, right you know I think when I was talking about curvature going up and down; let us see curvature 0 means what straight line, but the radius of curvature would be infinite there, right. (Refer Slide Time: 09:02)

So, I got to be very careful I think it is part of the confusion in my speech, just a little bit ago I think that. So, I wanted to say that the curvature of the wave front is approaching 0, but the radius of curvature would be approaching infinity. So, make sure I have said the right things at different times or system there is those 2 are inverses of each other even though the names are very similar, all right.
So, there are simple formulas for determining the focal length of a simple spherical lens.
(Refer Slide Time: 09:58)

And so, I want to go into some pictures of that explain some optical properties and things will go back and forth between the board and some pictures. So, this should correspond pretty closely to the picture that I have drawn and oops don’t do that notice that we have the focal length as I have drawn it there is also a thickness of the lens d here and note that we have 2 different radii.
R1 and R2 that correspond to the curvature of the; one of them corresponds to the place where the light first hits which it looks like R2 in this case and then there is R1, oh God, sorry, sorry, hold on; the R1 corresponds to a circular disc and that is the place where the light first hits and then R2 corresponds to the place where the light hits second, but note that the center of the circle is before the light hits for the second circle and its after for the first circle. So, to determine the focal point we have the following the lens maker the lens makers’s equation.
(Refer Slide Time: 11:07)

So, one over the focal length is equal to let us say n 2 minus n 1, I am writing in a little more general form that I have in my notes here, but it should be 1 over R 1 minus 1 over R 2 plus n minus 1 nope ha, this is the place where I am doing it a little more generally n 2 minus n 1 d divided by n 2 n 1 R 1 R 2 and then for see n 1 is the index of refraction; let us say outside of the lens make it very clear the ambient the air outside of the lens. So, outside the lens usually we mean air and usually n 1 is approximately equal to 1.
(Refer Slide Time: 12:30)

But you can put your lenses under water and it will do something different right behave in a different way.
(Refer Slide Time: 13:11)

So, this is the index of refraction for the lens material there is a simple approximation that is often used to this and I will write that here it is called the thin lens approximation this is much more common now after this approximation where we get rid of this d parameter. So, here we say d is approximately equal to 0 in which case the lens maker’s equation simplifies to 1 over f equals n 2 minus n 1.
So, it is the difference in refractive indices 1 over R 1 minus 1 over R 2 and there is a peculiar sign convention that is used for these r. So, normally when you are right R in this kind of geometric picture that we have on the on the display here in ordinary geometry the R is would always be positive; however, there is a convention here which is for a converging lens which we have just made we have a sign convention which is that R 1 is greater than 0 and R 2 is less than 0. (Refer Slide Time: 14:08)

So, this is for a converging lens I upper lens type here that is only kind I have shown you. So, far for a converging lens one way to remember that is that the R 1 case is positive because you hit the surface that that circle corresponds to before you hit the center of the circle and the R 2 cases backwards is you hit the rays of light that are coming into the lens have hit the center of the circle before they actually hit the surface. So, in other words, R 1 corresponds to the case of a front side which is an entrance and R 2 corresponds to the case of an exit. So, so these are these are for the converging lenses, it ends up giving you these signs.
(Refer Slide Time: 15:15)

So, there is also diverging lenses if you seen optics before it should not be surprising. So, there is a diverging case where we can make rays of light instead of converging spread apart and I will just mention the sign convention in this case. So, R 1 in this case is less than 0 and R 2 is greater than
0 and so, you can directly apply the lens maker’s equation in the original form and in the thin lens approximation to determine the focal point.
However notice that if its diverging it seems like it should not even make sense to talk about a focal point I have parallel rays coming in and then they diverge, but you can trace them backwards that is what these dash lines correspond to too to give a kind of let us say fictitious focal point right. So, because once these rays spray up spread apart they will still all point backwards if you reverse their direction to a common point. So, that is called the focal points of a diverging lens.
So, so the diverging lens does not cause focus to happen it is the opposite of that, but you can just reverse the direction we are on everything backwards and to the left of the lens you end up with the focal point right questions about that I want to let us see I want to now talk about a very useful representation called diopter which becomes very handy when combining lenses talking about optical power and then I will give an example of it in terms of the human eye human eye is going to be important because it is part of the virtual reality system.
Diopters
(Refer Slide Time: 00:17)

So, this is a convenient unit is called the diopter, its units are 1 over m. So, 1 over meters is the unit of diopters, and what it tells you is the converging, it can also express diverging power of a lens. So, how hard does this lens work to bend rays and in which direction. (Refer Slide Time: 01:05)

So, if I have parallel rays coming in, and I have let us say a really thin lens here that is. So, thin that I am not going to draw it and then these rays converge, to some point here that is supposed to meet, let us suppose this distance this is a simple example let me write 20 centimeters.