 Welcome to this last lecture in Physics 1308, Introduction to Electricity and Magnetism. In this lecture, I'm going to cover thin lenses and the human eye. To begin the discussion of thin lenses, we have to look back at the law of refraction, also known as Snell's law, and the way in which light moves from one material into another. So to refresh here, we could imagine that we have a situation where we have one material, like air, over on one part of our picture, and then there's a boundary between air and another material. So for instance, we could imagine that we have here some glass. Now what distinguishes air from glass for the purpose of optics is the index of refraction of these two materials. So we have N1, which is the index of refraction of air, and that is pretty much very close to 1.000, so we'll just write that as 1.0. And we have N2, which is the index of refraction for glass, and that varies depending on the type of glass, but we could imagine that we have an index of refraction of about 1.6 for the glass in this example here. Now what I'm going to do is I'm going to send in a ray of light, and that ray of light is going to originate in the air, and it's going to strike the surface of the glass. Some of it will reflect, but I'm interested in the part of the light that is going to refract that is passed through the interface between the air and the glass, and then travel inside of the glass. So for instance, I might imagine sending an array at an angle like this, and we know that we then have to consider the relationship between that light ray and the normal to the surface of the glass where it strikes. So for instance, I've drawn a nice red dot here to indicate where the ray strikes the glass, and I can just put a little dotted line through here that represents the normal to the surface. So this right here at this point where I've drawn the ray striking the glass makes an angle of 90 degrees with the surface of the glass. Now we can write the angle in the air part of the problem as theta 1. That's the angle between the ray and the normal to the surface where the refraction is going to occur. And the ray is going to exit, and it's going to exit at a different angle. The light ray will bend. We know from the discussion of Snell's law the light ray will bend because the speed of light inside of glass is different than the speed of light inside of air, and this causes the wavelength of the light to shift slightly. And the net effect of this is that the trajectory of the light ray changes when it goes from air to glass, or if it were to go from glass to air, that would also happen. So we have some different angle here, and we have the refracted ray. So here is our refracted ray, and it's going to make an angle with respect to the normal of theta 2 inside of the glass, and Snell's law allows us to relate the two. So Snell's law is simply n1 times the sine of theta 1 must be equal to n2 times the sine of theta 2. Now we can rearrange this equation to write the ratio of the sines in terms of the ratio of the indices of refraction. So let's go ahead and do that. What I'd like to do is I'd like to get the ratio of sine theta 1 to sine theta 2 in terms of the ratio of the index of refraction of the glass over that of the air. So with a simple rearrangement I can go ahead and do this. I can say sine theta 1 divided by sine theta 2 is just equal to n2 over n1, and because of the indices of refraction that I've picked for this example, this ratio is just 1.6 because it's 1.6 divided by 1. This then allows us to look at the ratio of the sines. We see that the ratio of the sines of the two angles is going to be 1.6. Now the sine of a small angle is a small number. The sine of a larger angle as you approach 90 degrees for instance is a larger number. So what this tells us is that the sine of theta 1 divided by the sine of theta 2 is a number greater than 1. And that implies that the angle theta 1 is greater than theta 2 because the sine of theta 1 is greater than the sine of theta 2 according to this equation, and that implies, and if you're not sure about this go and play around with the sine for a little bit, this implies that theta 1 is greater than theta 2, which is exactly what I've drawn in this picture. So in this refraction we have that this angle theta 1 is a larger angle than this angle theta 2 because we're going from a material with a low index of refraction into a material with a higher index of refraction. And in general this will always be true. If you go from a material with a low index of refraction and you pass into a material with a high index of refraction relative to the original medium in which the light was propagating, you will reduce the angle with respect to the normal that the light ray is making. Now let's take the principle that we've just looked at with refraction where you go from a medium with a low index of refraction to a high index of refraction and let's apply this to what's known as a lens. A lens is nothing more than a system of materials designed to change the path that light takes. So light will enter the material at one angle and it will exit the material on the other side at a different angle. And by adjusting the properties of the material one can cause the light to follow very well defined predefined paths that allow you to focus light to a point or defocus light away from a point. And by doing all of this we can build up optical transportation systems. So let me sketch a lens. I'm going to make the lens slightly rounded on one side just a little bit. And then I'm going to have just a really thick bit of material in here and then I'm going to have that same slightly rounded face on the other side. So these slightly rounded faces for the purposes of this course I'm going to keep a nice simple shape. I'm going to make them semicircular. That is these are part of some very large circle with radius for instance for the right hand side R2. And then similarly over here you can imagine that this face is part of some other large circle with radius R1. So the radii of curvature of the two sides of this glass system let's just call this glass some index of refraction N2 these are semicircular faces of glass that you can imagine having been cut from a very large sphere of glass with very large radii R1 and R2. Now just like in the mirror case I'm going to define the optical axis to be a line that passes dead center through the lens. So let's label this the optical axis. The optical axis is a convenient reference point for images and objects and focal points and focal lengths and all that stuff that we used for mirrors and we're going to repeat that now for not for reflection but for refraction. Now the whole purpose of this is to show you what the transportation of light looks like through this system. So let me send an array that comes in parallel to the optical axis. So here's my incident ray and it's parallel to the optical axis which is this black dotted line here. Now it's going to refract when it strikes the air-glass interface. So let's write this N1 and we can call this air. We've already handled that situation before that's an index of refraction of about one. The glass is an index of refraction of something like 1.6. It's a bigger index of refraction. So we know that once we draw the normal to the interface between the air and the glass whatever the angle is between the incident ray in air and the normal it will be a smaller angle inside the glass. So the normal to the surface of a circular interface is going to lie along a radial line of the circle from which the glass is cut. So this is a very large radius circle. So we might imagine that the normal looks something like this. It makes an angle with respect to the optical axis that is not zero but it's very slight as we can see here. We know that the refracted ray is going to come out at a very slight angle to the normal inside of the glass. Let me exaggerate that by having it lie almost on top of that normal. See it has a slight divergence there. There's my ray and there's my refracted ray. So we have an incident ray making an angle theta 1 with respect to the normal and we have the outgoing ray inside the glass making an angle theta 2 that's a lot smaller than the angle between the incident ray and the normal on the outside. That ray is going to continue to travel inside of the glass and it's going to strike the glass on the other side where it meets the air and we have to just repeat the refractive process again. We have to draw a normal to the surface of the air glass interface and our refracted ray will make a larger angle outside of the glass than it did inside of the glass because now we're going from a large index of a fraction material to a lower index of a fraction material and there we go. So we'll call this angle theta 4 and this angle theta 3. So the relationships that we expect between all these angles is that theta 2 is less than theta 1 and theta 3 is less than theta 4. So this is the basic principle by which any kind of lens works. Parallel rays coming in from the left will refract at the first boundary they will bend inside the material as a result of that refraction they will experience a second refraction over on the other face of the material and then they will bend even further and in this particular configuration this results in a parallel ray being bent toward the optical axis on the other side of the material and this is known as a converging or focusing lens. It takes light that comes in parallel and it bends it down toward the optical axis on the other side. You could imagine that another ray that enters down here at the bottom following a similar set of arguments, so here's our normal we get a refraction that results in a smaller angle on this side and then we have a second refraction here and we get some refracted ray on the other side. We see that a ray that enters the bottom of the lens on the left side will be bent up toward the optical axis on the right and a ray that enters at the top on the left side will be bent down toward the optical axis on the right and you can see why this gets its name as being a focusing lens. Now I didn't do a great job of drawing this if I drawn this as a perfectly smooth circle and got my normals exactly right then of course these rays should have converged maybe right on the optical axis but you get the idea. This is an example of a converging lens and here you have convex glass faces on both sides and the effect of those convex glass faces is to cause a light rays to bend toward the optical axis if they came in parallel on the left side of the setup. That was an example of a converging lens system but we can also imagine a diverging lens system. Here instead of having a convex glass face we just have to make a slight concave glass face on one side lots of material and then a similarly slight convex face on the other. I will draw my optical axis and then I will send in a light ray up here parallel to the optical axis. Now in this case the circular face bends away from the light ray so the radial line that points back toward the center of the circle points back this way unlike in the case of the converging face where it sort of tilted down here it tilts up. We get a slightly smaller angle with respect to the face we get a second refraction over here and this ray will bend away. Sending a light ray down here again we draw our normal to the surface we have a slight angle here and we have the normal that way and we get an even more pronounced angle out of the other side of the interface. This is what is known as a diverging lens and you can see why. It causes parallel rays from the left to pass through the glass diverging away from the optical axis on the right and here whereas the focal point was on the right hand side of the lens before here if we extend our rays back to where we think they came from on the other side we see that they converge actually on the side of the lens where the light originated. This is a very handy point to define real and virtual images for lens systems. So let's go back to the concept of images again. A real image just like in a mirror a real image for a lens system is located in a place where it physically could in fact be observed with a screen. You could put a screen in that location and you could actually see the image of the light from the object on the other side of the lens. Real images form on the side of the lens opposite the object. That is if I have light from an object starting out over on the left side I expect for a real image for a lens system that the focused light, the focal point for the refracted light will lie on the other side of the lens. So on the right hand side of the lens. A virtual image on the other hand it lies on the same side as the object. You can see already that a converging lens has the potential to form a real image on the opposite side of the lens and a diverging lens forms a virtual image on the same side of the lens where the object is actually located. If you try to put a screen there you would of course block the incident light from the object and you wouldn't be able to form the image on the screen. Nonetheless, diverging lenses are extremely important in optical systems and you need both converging and diverging lenses in order to transport light for all kinds of applications like medical imaging and non-invasive, well invasive but non-surgical imaging techniques like endoscopy. Now before moving on to a demonstration of lensing systems what I would like to do is I would like to discuss thin lenses which are the things we're actually going to use for the rest of the class. So I've been showing you essentially thick lenses to motivate the refraction process but we can instead switch to a related class of lenses known as thin lenses. And these are lenses whose thickness is much smaller than the distance of the object from the lens. That's it. So if you imagine having an object like a person standing on the optical axis of a lens and we were to put a thick lens far enough away from them so that the thickness didn't really show we could imagine that a converging lens could be represented simply as two semi-circular faces glued together with very little material in between them or similarly we could imagine a diverging lens again here's the person a diverging lens simply being represented as two semi-circular faces but concave rather than convex. So here we have converging and then we have diverging and it's actually much easier to represent these lenses using a simple symbol and a typical convention for converging thin lenses is to simply draw them like this and for diverging thin lenses is to simply draw them like this. So this just represents a thin lens a lens whose thickness is far smaller involved for instance of the object to the lens system and so you can use a line with two arrow heads for converging and a line with two reversed arrow heads for diverging and the idea here is that any light ray that enters will be bent converging lens will bend the light toward the optical axis diverging lenses will bend the light away from the optical axis so you can very quickly sketch these pictures and draw some light rays and you'll be good to go. Now just as in mirror systems it's very convenient to analyze lens systems in the way that light moves through them using this concept of principle rays so for instance if I imagine that I have an object over here some distance P away from the lens system so this is my object I'm just going to use a simple upright arrow to represent that object in this case you can use principle rays to analyze the transport of light through the system so for example if you have a lens that has a focal point given by F that point might be located over here and because lenses for instance are symmetric it's the same distance on either side so light entering from the left can pass through the focal point on the right and light entering from the right can pass through the focal point on the left lenses are typically some kind of symmetric or semi-symmetric system and so the focal point of a lens applies equally on either side of the lens this is distinct from a mirror because light can't penetrate a mirror the focal point lies on either one side of the mirror but for a lens it can be on either side now the principle rays that we will use to look at the transport of light through a thin lens system one will just use parallel rays that's one of the rays we looked at with mirrors and we'll look at it again with thin lenses a parallel ray comes in parallel to the optical axis so if I imagine a ray coming from this point here and I draw it coming in parallel to the optical axis these light rays will be bent through the focal point of the lens so here is my parallel ray now another kind of principle ray is a central ray this is a ray that passes dead center through the optical axis of the lens so in this case that would be a ray that starts out at this point passes through the dead center of the lens and actually passes through undiverted that's a central ray so any ray that passes through the dead center of the lens right through its optical center will essentially look like it's undiverted and it will continue on its way on the other side of the thin lens system now of course you can also have rays that go through the focus, the focal point so we will have focal rays we can look at that as a principle ray that's another ray that you are free to draw that's a ray that passes through the focal point of the lens on the object side and then comes out parallel to the optical axis on the other side so this one is a focal ray now I didn't do a great job of trying to draw the focal point as symmetric on either side but basically where these rays converge over here represents the location on the image that is equal to the location that I started with on the object and so I would here draw my image over here and we see that it's inverted so this is a converging lens and we see that when we have the object to the left of the focal point of the converging lens we wind up with an image that is real and inverted now we can continue to analyze lens systems using these principle rays and to do that what I'd like to do is not keep drawing these by hand I'd rather have a computer do all this stuff for me so that my pictures are prettier and more accurate so what I'm going to do is I'm going to switch now from this hand drawing nonsense to using a simulation of optical effects that is provided by the University of Colorado's FET demonstration system so this is the optics lab demonstration system provided by the FET simulation suite you can see here that we have a very nice system we have an object over here that we can move and I've elected to have the simulation show principle rays so here you see the focal ray passing through the focal point marked with an X of this converging lens it comes out parallel to the optical axis on the other side we see the central ray passing through the optical center of the lens undiverted and then coming through the other side and we see a parallel ray coming in and then being bent through the focal point on the right hand side of the converging lens and they all meet up over here you see when a computer does this it looks a whole lot prettier than what I do I've actually set the refractive index of this lens to be 1.6 and you'll notice it has so that just changes basically this height of the lens so that just changes the diameter of the lens itself and you can affect the radius of curvature of the glass and you can alter the curvature in such a way that you also alter the optical properties of the lens itself so I'm going to leave that at 0.8 meters for the curvature radius that basically means that this glass has a rounded surface that is as if it were cut from a big sphere of glass whose radius is about 0.8 meters now you'll see that I have it set up to show virtual images when those occur we haven't gotten to that quite yet but we will get to that in a moment and what I'm going to do now is I'm going to change our little object here so we have something a little more akin to what we've been using, an arrow and I'm going to put that arrow pretty firmly right on the optical axis so I've now laid that arrow right out on the optical axis and you see here that my little sketch that I did a moment ago for a converging lens when the object is located to the left of the focal point that's not too bad, I actually got that pretty good considering how crummy my drawing was you see that the resulting image is in fact inverted on the other side and it's real, it's on the opposite side of the lens from the object which is on the left, image is on the right now let's see what happens as I move my arrow, my object closer and closer to the focal point we see that what's happening is that as I move my arrow closer to the focal point the rays are having to diverge away from one another, never mind this this is just an effect here to show you how the ray would bend if it passed through the lens on that end it's getting bigger the image is getting larger on the right hand side it's still inverted, but it's a much larger image so you can kind of see that if you want to enlarge an image of something if you have an object like writing on a piece of paper over here and your eye is over here on the right and you want the writing on the paper to be bigger you want to move the paper closer and closer and closer to the focal point of the lens system now let's see if I can rain this in a little bit so I'm going to make this a big refractive index here let's take it up to like 1.8 just so we can see everything still on the screen here alright I'm going to continue to move my object closer and closer to the focal point and right at the focal point we get into an interesting situation at the focal point all the rays coming out the other side are parallel that is they don't focus anywhere on the right hand side when you stick an object right at the focal point of a converging lens you can't form an image on the other side we'll see this reflected in a bit when we look at the lens equation the thin lens equation which is very similar to the mirror equation now as I move my object inside the focal point we see that now the rays do converge but they don't converge on the right hand side of the lens rather they point back to a focal point on the left hand side of the lens now we're making a virtual image there's no way I could put a screen over here and get this image to form alright so what's kind of neat about this is that this is essentially the operating principle of a magnifying glass so a magnifying glass surfaces of glass you can hold it in your hand and the surface of the glass is curved such that the focal point is very far away from the central point in the lens if you lay the magnifying glass over a piece of paper you will see on the other side of the glass a large and enlarged upright image of the magnifying for instance that you're attempting to magnify so this is a magnifying glass that I've just created here so if my eye is over here on the right hand side I will appear to see the rays from the object converging at a point behind the object but enlarged and upright and that's exactly what we want from a magnifying glass we don't want a magnifying glass to invert text and enlarge it then we can't read it we want it instead to have the text be upright we can put our eyeballs over here and we can see the light rays that are refracted they appear to come from an object that's behind where the actual object is located much bigger and upright so this is the operating principle of a magnifying glass you put your object inside the focal length of the magnifying glass lens and you put your eye on the other side and the rays that come out all appear to come from a point this is a virtual image but this is how you observe that virtual image you put your eye over here and you'll see the rays as if they appear to converge back behind the lens where the paper is located so that's how you get a virtual image so this very nicely demonstrates how you can use a lens and an object to create different kinds of images you can create a real inverted reduced image if you move your object far from the focal point if you move your object closer to the focal point you can get a real enlarged and inverted image and if you move your object within the focal point you can get a virtual enlarged upright image that winds up being on the same side as the object itself and that's the basics of how a converging lens will work a diverging lens will only ever form a virtual image that lies on the same side as the lens as the object so that's one of the neat things about a diverging lens the converging lens has a much more rich set of structures that much more rich set of outcomes that can occur and that's why I demonstrated here but a diverging lens is also useful but it always forms a virtual image on the same side as the object itself now the workhorse of thin lenses is an equation and it won't look too surprising to you it's simply that one over the object distance plus one over the image distance is equal to one over the focal length of the lens now again, sign conventions play a very important role in this business P which is the object distance is always positive now whereas for mirrors where P defined the side where positive image distances were to be defined rather for lenses we have i greater than zero when it's on the opposite side from the object so this is the key distinction between mirrors and lenses for mirrors are real that is i is a positive number when the image forms on the same side as the object but for lenses the image is real that is i is positive when that image forms on the opposite side of the lens from where the object is located so just to complete this i less than zero you're on the same side as the object and this defines real i positive just like before and i negative defines virtual now for focal lengths f is greater than zero for a converging lens and that's because parallel rays focus on the side of the lens opposite the object so you can kind of see a pattern here where the image forms is positive when it's on the side opposite the object the focal length of a lens is positive when it focuses parallel rays on the side opposite the object f less than zero is a diverging lens and that's because parallel rays focus on the same side as the object so that's the thin lens equation and this allows you to locate images given focal lengths or figure out the focal length of the lens given the image distance and the object distance so this is the kind of thing you'll have to play around with now of course in addition we have the concept of magnification again and just like with mirrors the absolute value of the magnification is equal to the height of the image divided by the height of the object so this is the image height this is the object height and m is a signed quantity and again we have a situation where the sign you know recalling again that the real images for a converging lens formed on the opposite side of the lens and were inverted so those eyes were positive but to get an inverted image you have to have a negative number so this is just the negative of the ratio of the image distance over the object distance so this gives you both the magnification factor and a sign that means upright or inverted so if m is greater than zero we have an upright image and if m is less than zero we have an inverted image that's all that sign means otherwise take the absolute value of m and you get the magnification factor magnification factor so it's the same game as with mirrors m equals negative i over p and the absolute value of m gives you the ratio of the heights of the image and the object and putting all of this together you can go back and think about that thin lens situation so let's think about a thin converging lens here you have a focal length that's greater than zero so f is a positive number of course we have that p is a positive number and let's look at the lens equation for different situations so the lens equation is 1 over p plus 1 over i equals 1 over f now let's focus on images so if p is greater than f that is if the object is located very far from the focal point of the converging lens then we can figure out what the image is going to do so we have 1 over i is equal to 1 over f minus 1 over p now let's put in some numbers let's say that the focal length of this lens is a convenient 1 meter and let's put the object distance at a whopping 4 meters away from the lens so that is 4 times the focal length now we can plug into the lens equation we have 1 over i equals 1 over 1 meter minus 1 over 4 meters so we have 1 minus 1 fourth which is 3 fourths so we have 1 over i equals 3 fourths inverse meters or 1 over meters so we can solve i is equal to 4 thirds of a meter so we see that we have a real image that forms as we saw in the computer simulation we have a real image that forms and it forms at a distance of 4 thirds meters which is just slightly longer than 1 meter so it forms a little bit closer to the lens than the object but still further than the focal length on the other side and we can also see that this is going to be an inverted image because m equals negative i over p and this is going to be negative 4 thirds over 4 which is negative one third so we wind up with a inverted reduced image compared to the object it's one third the height of the object that's the one third and it's a negative sign in front of that so it's inverted so we can play around with the equation and very quickly see what we saw in the computer simulation if you put an object at a larger distance than the focal length then you get an inverted reduced image that's real you can continue to play this game so again let's imagine that f is one meter and p is now two meters so we have one over i equals one over one meter minus one over two meters and that leaves us with one over two meters or one half meters to the minus one so that i is equal to two meters so we see here that again we get a real image and it's magnification factor negative i over p is negative two meters over two meters is negative one so when you put an object at twice the focal length equals two f something interesting occurs you still get a real image it is inverted but it is un-magnified it's the same height as the object so that's a special case when p is equal to two f you wind up with a real inverted image whose height is exactly the same as the object now let's go inside the focal length we put f equal to one meter and p equal to 0.5 meters or one half of a meter so now we have one over i equals one over one meter minus one over a half is two meters to the minus one so we wind up with one meter to the minus one one meter to the minus one minus two meters to the minus one is negative one meters to the minus one so i equals negative one meters so here we have a virtual image because i is a negative number we can look at the magnification factor this is negative of negative one meter divided by one half meter which is equal to negative two which is equal to positive two so here you get an enlarged upright image of the object that's virtual so a virtual enlarged upright image its height is greater than that of the object by a factor of two and it's upright, we have a positive sign on the magnification and it's virtual because it forms on the same side of the lens as the object at a negative distance so i hope that this gives you the groundwork for playing around with thin lenses and the last thing that i'd like to do before closing out the formal material in this course is to talk about the human eye one application of lenses including even the simplified concept of thin lenses is to look at the behavior of the human eye now the human eye is a very complex optical system but nonetheless with the basic information that we've learned so far using the thin lens concept we can look at very common situations involving the human eye such as near-sightedness and far-sightedness and even the blind spot effect of the human eye and we can begin to understand those from both an optical perspective and from the perspective of the limitations of measurement in the natural world and how one compensates for problems in a measurement system so if we go back to those drawings i was doing with thick lenses these are better drawings with lots of decomposition of angles and rays and so forth we can think for a minute about what it means to have rays emanating from a common point O they diverge outward from that point so this shows lots of rays of light that might be scattered off the point O out toward this surface they then strike another medium they strike the boundary between the initial medium where the light travels and a second medium so this could be air and this could be water or glass and they refract and we know from a curved surface that bows outward a convex surface of lens material that these rays will be bent inward inside of the material and if they never reach the other interface they will simply converge at some point I within the material so you can already see using a very simple set of media air and glass for instance that it's possible to have rays diverging out from a point source outside of the the material but then because of the shape of the material and the fact that it has a different index of refraction than the medium outside that you can then get bending of rays inward now if you were going to make an optical imaging system like the one that's implanted in your skull this is a pretty sensible place to start and in fact it's no surprise that the eye has evolved separately many times in nature because it takes very little effort to actually construct an optical imaging system the earliest forms of life that were multicellular some of those had single cell optical imaging systems that actually tell the difference between a light spot and a dark spot and those systems eventually developed into more complicated systems where you could actually see the finer get the finer detail of the light and dark spots and then eventually what we see in more complex organisms is the development of a lens that's a simple bulging of biological material that's transparent but has an index of refraction that's different from the air of life evolving in water than the water around it and so we are able to see in many forms of life that exist on the planet now the various stages of the development of the eye as an optical imaging system across millions and millions of years so this is a cartoon of a human eye as I'll point out a little bit later the human eye is different from the eyes of other animals it actually has some flaws that other animals like the octopus do not have but basically this structure is repeated in many places in nature and as I said it has evolved separately at least a few times that we're aware of so it's pretty clear that this on our planet is a convergence of biological evolution and development that seems to have served a very useful purpose providing capabilities that are much higher over animals that don't have a developed optical system like this the idea is simple outside of the eye you have some medium like air or water if you've ever dived underwater and opened your eyes you know that your eyes still function underwater for focusing light although not in the same way that they do in air after all we are primarily we are primarily organisms that function in an atmosphere that's mostly nitrogen partly oxygen which we require to live of course but we can dive into the water we can hold our breath we can take scuba tanks down we can open our eyes underwater we can still see so our eyes still functions even in media where it's not at its peak performance it still functions in other places like underwater now the eye has a few parts which are fairly common there's the sclera there's the tissue that basically encapsulates the fluids and other materials of the eye on the front of the eye you have this first lens system called the cornea and the cornea is effectively a fixed lens system its shape is not supposed to change it's not biologically hooked up to anything that can significantly affect the shape of the cornea but as we know as I'll point out a little bit later we do know that the cornea can become the shape and this can lead to sight problems in human beings as well as other animals as well we have just inside the cornea we have the first liquid that we're going to encounter known as the aqueous humor and then we reach the pupil the pupil is essentially a hole with a lens behind it and it is that hole into which light can pass and then eventually reach our optic nerve at the back of the eye we have what's called the lens and this lens is actually designed to change shape it's capable it's attached to musculature above and below and as those muscles relax or contract they can stretch or squeeze the lens changing its shape that's why we're able to focus at different distances as you'll see in a bit so this is why you're able to look at text up close and then turn your head and look at an object tens or even hundreds of meters away and your eye is able to adjust for the fact that it's now looking for light coming from a completely different distance with completely different angles and yet it can still focus it okay so those are the major components the iris which is capable of opening and closing over the pupil in order to allow in less or more light this is how we handle light bright and dark situations if we're in a very bright situation the iris tightens it's a muscle it's a musculature if we're in bright situations the iris tightens it's a musculature system and so it's capable of closing down the opening of the eye so that less light has to come in we don't get blinded by bright light as a result but for in a dark room the iris will relax and open up and let in the very little available light that might be present in the environment and so this is why even under extremely low light conditions the human eye is still able to basically function as long as there's some visible light in the environment even if it's a very low intensity source the human eye can adjust over 15 or 20 minutes and is capable of imaging very low light levels but this is it this is the primary system so again you have the cornea the aqueous humor light then passes through the iris into the lens whose properties geometric properties can be adjusted to change the focal point of that lens then you have the vitreous humor through which light continues to propagate it's then capable of striking nerve cells on the back of the eye and those nerve cells are wired into the optic nerve cable which then connects to your brain at the back and this here is just a blow up of the different kinds of light sensing cells that are present at the back of the eye this entire system is fed by a blood supply the blood vessels come in through the optic nerve and fan out and all the connections to these light sensing cells they actually come in out of this bundle here and then fan out and connect into the cells at the back of the eye and I'll have a picture of what that looks like in a moment the back of the eye where you have the light sensing capability is known as the retina now from a physicist's perspective the eye is just a whole collection of different indices of refraction so outside of the the eyeball you have air for the most part unless you dive under water and open your eyes but let's say you have air with an index of refraction of basically one the cornea has an index of refraction that's just slightly greater than water so clean water has an index of refraction of 1.333 nice convenient number to remember the cornea has an index of refraction that is slightly higher than that at 1.376 and so this is part of why your eye is still able to function even under water that's because there is still a difference in the indices of refraction the water outside of your eye and the materials that make up the lensing systems in your eye so the fixed lens the cornea has an index of refraction that's just a hair higher than water inside of the system between the cornea and the lens you have these humors that have indices of refraction that are actually very close to water 1.336 and 1.337 inside of the eye and then you have the lens and the lens's index of refraction varies it can go between about 1.386 and 1.406 and you see that these are higher than the cornea and both higher than air or water and it's this very non-linear this non-uniform lens system that is really the key ingredient in being able to focus at short and long distances with an eye like ours down here you have information about the radius of curvature of the different optics systems so the cornea and the lens and you can see that they have different radii of curvature just like that I implied in my lecture earlier that you could make the front and back of the lens have different radiuses of curvature here the eye actually realizes that so the front of the cornea has a radius of curvature of 7.259mm while the back is about 2mm smaller than that in its radius of curvature that these surfaces curve at different radii you can see that they diverge out here and they're closer up here and you can very much see the different radiuses of curvature of the lens system and again it's these features which nature has tested out over time and found that some work better than others because they confer a greater survival advantage those changes that happened the lens changing its radius of curvature slightly that's genetic information passed down from parent to child over many generations and after generations and generations of populations coming into being and passing out of being those that are more fit for survival those that are able to more capable handle the challenges of the environment they're the ones that get past their genes along to the next generation whereas other people die off at a higher rate and so over long periods of time in the case of the evolution of the human eye hundreds of thousands of years today and it presumably could continue to change over time both technologically as humans continue to develop medical technology for altering their biology but also biologically there are chances of course that a better eye could happen in a population and that could be passed along to the next generation and so forth and so on I should note over here by the way there's this unit of power which I didn't mention in my earlier lecture but which I'll define here power is merely one over the focal length units of meters to the minus one or one over meters so power is one over meters gets its own name diopters and so if you ever go to a store to buy reading glasses and you see that they're listed in terms of powers whose units are diopters those are merely one over the focal length of the lenses in the reading glasses so you can very easily convert power or diopters into focal length you just invert the number it's very simple and you can see the other distance scales involved in here the cornea thickness is about half a millimeter the lens thickness is about five millimeters the cornea to lens distance is about 2.3 millimeters so this distance right here and then finally from the front surface of the cornea back to the retina the imaging system of the eye you have about a 24 millimeter distance so this gives you the sort of the engineering and physics of the eye this is what a physicist would look at and go I can analyze this with ray tracing you see here the musculature that attaches to the top and bottom of the lens and allows it to be pulled up and squeezed thinner or pushed down and fattened to cause more or less refraction to light and thus alter the focal length of this dynamic lens system so here's in a cartoon example of what I mean by what I just said so if you imagine you have parallel rays that are coming in from a great distance so maybe there's some object 500 meters away that you're trying to look at at the time the rays from that object reach you they're mostly entering parallel to your cornea parallel to the optical axis of your cornea you need a lens that's capable of focusing that light down but gently you don't have to bend it too far to get it to focus on the retina and so here what you do is your eye when you look at a distant object the muscles will pull the ends of the lenses away from each other to thin the amount of material that's in the path of the light so the cornea will focus these parallel light rays down and then you'll get a little bit of extra focusing from the lens and the goal of course is to put a nice focused spot right on your retina for imaging to your brain now on the other hand if you're looking at a very close by object that object could be right in front of your face maybe 30 centimeters in front of your face and so the light rays from that are diverging very strongly away from the optic axis of your eyeball is going to be quite large and so your cornea may not be capable of bending that light enough to focus it on the back of the eye it may only be capable for instance of making those rays now enter parallel inside the humors of the eye so what will happen now is that the musculature on the top and bottom of the lens will squeeze it to make it fatter and this will put more material in the path length of the light give you a stronger bending angle and focus on the back of the eye so when the eye is operating under optimal conditions basically the closest that you can see something and still focus on it so you could try this right now whether you have perfect vision, 2020 vision or whether you have contact lenses or glasses that correct your vision to 2020 take your hand, put it about a meter extend it to arms length in front of your face and then slowly pull your hand in and keep trying to focus on the palm of your hand and what you'll find is that about 22 centimeters from your eye your hand will get blurry and you just won't be able to focus enough to actually form an unblurred image of your hand on the back of your retina on the back of your eye that 22 centimeters is the common point for almost all human beings for the distance at which you can still focus close by information so that's something to keep in mind when you're doing vision, you're basically correcting it so that that near point in your vision is 22 centimeters and that's exactly what corrective optics do so let's imagine near-sighted people so near-sighted people have the following problem they can see things that are close up but they're unable to focus parallel rays from distant objects enough to form an image on the back of the eye on the retina so this cartoon sort of illustrates the issue you might have some kind of temporal buckle in the eye that misshapes the eye in a way, deforms it and what happens is parallel rays coming in parallel to the optical axis they will be over focused by the eye so that the focused image appears somewhere in the middle of the humor inside the eye and then the rays diverge before they hit the retina that's bad because now at best you're going to see a blurred version of the world in front of you at a great distance so for instance, I am near-sighted I can't focus distant objects I have to wear corrective lenses that compensate for this problem and so I wear contact lenses and glasses depending on whether my eyes are irritated or not and what these corrective optics do for instance, contact lenses do is you lay them on the surface of the eye right over the cornea and they are crafted so that you have a convex surface on the front and a concave surface on the back and what this does is it creates a virtual image in front of the lens on the same side as the parallel rays coming in light appears to come from that virtual image and then it passes through your cornea and then it goes through the lens and then it can be focused on the retina so for an eye that over focuses what you have to do is get those rays from the distant objects to spread out more away from the optical axis and then the focusing eye will focus them perfectly on the back of the retina and this is exactly what an eye doctor tries to do when they're engineering optics to correct your sight problems so again, I'm nearsighted I can read text up close I can read text that's 22, 30 cm away from my eye but if you put a sign on the road 50 meters away from me it's a blur to me and that's because my eye over corrects the parallel rays of my eye ahead of the retina so all I see is a blurred image of distant objects so I have corrective optics and all nearsighted people can get corrective optics that puts a virtual image just in front of your eye where you can focus light very easily so that that then focuses on your retina and you can see very clearly alright so nearsighted individuals need corrective optics that make objects look closer than they actually are so the eye can focus on them this doesn't distort your vision in any way to normal now far sighted people are people that can't see close objects so this is the opposite of the problem I have a far sighted individual has no problem focusing distant parallel rays but rays that come from close by that diverge strongly before they reach the cornea they have the problem they can't focus them enough and so the rays focus behind the eye some place so when they hit the retina of course they're blurry it's not a focused point image from a single point in space in front of the eye and you get a blurry image so you cannot see close objects as a result of this and so here for corrective optics what you do is you put a double focusing lens in front of the eye and so what this does is you again design the lens so that it puts a virtual image in front of the eye that's further away than the object you're trying to see so if you're trying to hold the book up to your face and read it but you can't read it because the text is blurry you can do one of two things you can pull the book away from your face and put it at arm's length and try to read that way so if you ever see a person who's holding a book really far from their face that means they're far sighted and they need corrective optics to compensate for that so what you do is you put a lens in front of the eye that takes the close image, the close object images a virtual image further away and that's where their eye can actually focus rays and then that allows them to put a nice crisp focused real image on the back of the eye from the virtual image that's created in front of the eye so far sighted individuals need corrective optics that make objects look further away than they actually are so their eye can focus them because their eye is a weak focusing system it can't focus strongly diverging rays from close by objects now there are all kinds of neat procedures that human beings have developed to correct the eye not just putting lenses, contact or otherwise in front of the cornea but you can even go in and you can do reshaping of the cornea directly to correct certain kinds of vision problems so Lasik surgery is a very common surgery these days it's not available to everybody and it's still expensive but if you can afford it and if your eyesight issues are amenable to being corrected by Lasik surgery you can go in and have the procedure done and so it's done while you're awake which for me is a little scary because I do not like to have my eyeball touched the eye is anesthetized with some topical eye drops so that you don't feel any pain from the eyes the eye does have nerve receptors in it so you do have to numb it before you do surgery this is then followed by the use of a basically a razor a very precise slicing device called a micro keratome and you slice gently through part of the cornea to create a flap on the outer layer of the eye that can be lifted away that exposes the underlying corneal tissue and then a laser is used to reshape the cornea under the flap and the flap is simply replaced and the eye simply heals itself no stitches are required you just have to take care and avoid infection things like that but if you follow the care procedures after the surgery if the surgery was done well you should have your eye healed and your vision corrected very quickly on the time scale of having the surgery done now one last thing I want to point out is that many people make the claim that the human eye is a perfectly engineered system and it couldn't possibly have evolved on its own in fact the biological evidence that you see in the world around us is that the eye as I said earlier has evolved many times in many different organisms along many different tracks and it often converges to the same basic principles but not always the same way in different animals in human beings we have a flaw in the eye and that is the way that it's wired up so this is actually a photograph of the back of my eye it was taken through my iris with a very bright camera flash very high density pixel camera to image the light that scatters off the back of my eye so what you're seeing here is an image essentially of my retina this bright spot here is the opening from the optic nerve if you were to follow this tunnel back into the slide you'd eventually reach my brain you can see the bundle of blood vessels that comes out and feeds the eye so this is how my eye gets oxygen and then it's very hard to see on here but there are these very fine almost glass like filaments that come out of these dense bundles all around the blood vessels and they fan out and they connect into all of the cells that form the back of my eye the retina these are essentially the optical connectors that go from the retina cells to the brain to the fiber optics of your eye now if you were an engineer and you were going to build a camera system with a whole bunch of cables coming in that connect to the pixels at the back of the camera how would you connect them up would you connect them fanning out and then into the front of the pixels or would you connect them into the back of the pixel so that they don't obscure the camera image think about that for a second now I'll bet you thought about even if you're not an engineer I'll bet you thought about good principles of engineering if you were Apple and you were going to make the best possible camera for a camera phone would you obscure the image by having your connections to the camera pixels going through the front of the pixels no your customers would complain endlessly about that they would get crummy photos they would hate every minute of using that camera and that's of course not what companies like Apple want to do you'd connect your fiber optics into the back of the pixels so that they stay out of the way of the light and they don't blur the image they don't obscure any of the fine detail of the image the camera is trying to take unfortunately the human eye is wired in a back-asswards way the human eye is wired from the front so the connective fiber optic cables that come out from our eye the optic nerve bundle into the retina they connect into the front and so the consequence of this is that the human eye has a blind spot it has a flaw let's look at the next slide here's what we're going to do we're going to locate the location of the blind spot in your right eye all you have to do is stare at this slide covering your left eye so cover your left eye with your hand close it and cover it and I want you to sit far back from the slide I want you to sit back at least a meter from the slide so this is going to be on YouTube so put yourself a meter away from your screen very clearly an A and O and an X I want you to focus your right eye on the O stare at the O using your peripheral vision but staring at the O you should still see the A and the X even if they're a little bit blurry now what I want you to do is very slowly move your head in closer and closer keep your left eye covered keep your right eye focused on the O and at some point probably about 35 or 40 centimeters from the screen you're going to suddenly not be able to see the X in your peripheral vision it's going to vanish your eye is going to perceive that the slide is blank except for the A and the O so back out again and the X will pop into existence slide in slowly toward the slide in about 35 or 40 centimeters let's say you're going to get to a situation where the X vanishes from your peripheral vision slide in closer keep going and eventually the X will reappear and for me the X reappears at about 20 to 15 centimeters away from the screen so again back out about a full meter meter and a half then lean in slowly focusing on the O at about 40 centimeters the X vanishes and as you close in on about 20 to 15 centimeters the X reappears in your peripheral vision you've just located the angle of the blind spot in your right eye you can repeat this with the left eye cover your right eye go way back from the screen slide in slowly toward the O and at about 40 centimeters or so the X will vanish from your peripheral vision keep going keep going keep going and the X reappears at about 20 centimeters or so you've just located the blind spot on your left eye the X reappears in both eyes the blind spot for the right eyes off to the right and for the left eyes off to the left if we were single-eyed creatures we would have a significant survival disadvantage because anything attacking us from a specific set of angles would basically have the complete element of surprise on us the fact that we have stereoscopic vision two eyes allows for the left eye to compensate for the blind spot of the right eye and the right eye to compensate for the blind spot and that is how when you have both eyes now go ahead and do this test one more time stare at the O put yourself a meter or so away from the screen close in slowly with both eyes open focusing on the O focus on the O and the X will never vanish from your peripheral vision and that's because your right eye and left eye together can compensate for their mutual blind spots but if one of them is damaged you will have a blind spot in your vision and this is very dangerous for walking driving any common activities that require you to have full control of your peripheral vision the blind spot creates a whole set of problems for you now interestingly the eye separately evolved in the octopus and the octopus has its fiber optic cables its optical nerve fibers connected into the back of its retina so it doesn't suffer this blind spot problem that human beings have so anybody who argues that the eye how biology has to be designed by some alien or some supernatural being is nonsense the eye is as flawed as the appendix or the way that our organs and our abdomen are attached to our spine rather than to our rib cages which causes all kinds of back problems it's as flawed as wisdom teeth and other things that human beings have that cause all kinds of problems the differences that our stereoscopic vision has a mechanism, an optical mechanism for compensating for the flaws of either one of the two cameras but if you were going to build a single camera like on an iPhone and aim that thing at a picture you don't want it to have a blind spot and so you would wire it up correctly because you're designing it from scratch but unfortunately in nature which conducts experiment after experiment after experiment when it develops biological systems these flaws can often remain in the system for hundreds of thousands if not millions of years and it doesn't matter they're not fixed because they don't really cause a significant disadvantage but if there was ever a predator that could take advantage of our blind spot by subverting our stereoscopic correcting mechanisms we'd be screwed or we would find a way to technologically or biologically evolve so that that blind spot goes away well I want to thank you for what has been for me an extremely stimulating semester of introductory physics I know introductory physics can often seem very bland and dry but what I find simple principles in hand conservation of energy, conservation of charge you know Coulomb's law the Biot-Savart law, these laws of electromagnetism we started to explore this semester with some basic information about geometry and optics you can really begin to look at the world in a whole new way and see why things are the way that they are not just that they are but why they are and for me that's what I love about physics it gives you answers to why's yes it raises questions, everything every active exploration that yields answers raises more questions but that's no reason to shy away from it and what I like about physics is that it is the science that is capable of explaining why at the most fundamental levels of the cosmos so I've enjoyed teaching I hope you've enjoyed learning and I look forward to our last final exam together and I wish you many great semesters ahead or if you're graduating good for you and enjoy the rest of your life thank you very much