 Light is an electromagnetic phenomenon that can travel at extremely high speed, even through empty space. But what happens when light is suddenly forced to encounter a new material, a material it had not already been traveling in? What will that light do? We will begin to explore the transport of light through materials, and the results of what happens when light encounters boundaries between materials in this portion of the course. In this lecture, I'm going to cover the basics of optics, that is the transport of light from one place to another, and specifically we're going to concentrate on the very fundamental behaviors of light in the presence of a medium. The behaviors are reflection and refraction of the light. If you recall from the special lecture on the nature of light, light is an example of an electromagnetic wave. Electromagnetic waves are merely waves that consist of oscillating electric and magnetic fields. They propagate, that is they travel at a velocity in a direction that is perpendicular to the oscillations of the electric and magnetic fields, which themselves oscillate in directions perpendicular to one another. The distance between the peaks of the wave, that is the places where for instance the electric field is at its strongest maximum value for a given wave. That distance is known as the wavelength, and that's represented here on this graphic. And the number of points per second that are common in the wave, for instance the number of peaks per second that pass you at a specific point in space, is known as the frequency. The frequency can be written as f, the wavelength as the Greek letter lambda, and the product of these two things is a speed. It has units of meters per second, and that speed is the speed of the wave, and in this case for an electromagnetic wave that is the speed of light. The speed of light is a number. In vacuum it is 2.998 times 10 to the 8th meters per second. So all the behaviors of light will arise from the fact that it is fundamentally a wave phenomenon that travels from one point to another point in space. And it does so along straight lines, unless the path of the wave is interrupted by the presence of a change in medium or a medium other than the one that it started in. This photograph illustrates very nicely the kinds of phenomena that we're going to look at that are common to all electromagnetic waves and are most easily seen by our eyes with visible light, which is the fraction of the electromagnetic spectrum that our bodies are sensitive to. We have here a laser beam which has begun its journey in air at the top of the picture. It then strikes an air vegetable oil boundary. This is merely the top of the surface of the vegetable oil layer that has been poured into this beaker. The light appears to travel through that boundary, and you see it goes from being green to being orange. This is a special phenomenon that occurs for green light traveling through olive oil, specifically. You get a wavelength shift, but that's not something we're going to explore here. It merely illustrates the presence of the vegetable oil layer in the picture. The light then approaches and passes through a vegetable oil water boundary. So water is at the bottom of the beaker. Vegetable oil is less dense than water, so it floats on top of it. And the air is on top of the vegetable oil, so we have a three layer system so far. Light begins in air, penetrates the boundary of the air vegetable oil system, travels through the vegetable oil, where it is seen as being orange. It then penetrates the water vegetable oil boundary and travels through the water where it's restored to being green light again. And then it strikes the bottom of the beaker, which is made of glass. And we see what occurs there appears to be a reflection. That is, rather than passing through the glass, we primarily see the laser light as scattering at some angle with respect to the floor of the beaker. And then it travels back up through the water, strikes the water vegetable oil interface, travels through the vegetable oil, strikes the vegetable air, vegetable oil air interface, and continues on its way. So we see here combined in one photograph the fundamental behaviors of electromagnetic waves, specifically light, that we are going to study. We have reflection at the bottom of the beaker where it strikes the water glass interface, and we have refraction where it passes through a boundary between two media. So for instance, between air and vegetable oil or between vegetable oil and water. Let's begin by looking at the phenomenon of reflection or specifically specular reflection. The specular reflection is where an incident beam of light, which is here represented by a very straight, narrow laser beam of red light. Where that laser light can strike the surface and reflect at a well-defined angle with respect to a normal to the surface. So what we see on the left here is our incident laser beam, which I have indicated with a white arrow pointing in the direction that the laser beam was aimed. There is a reflective surface represented by the horizontal white line with no arrows on it. That surface can be characterized, as all surfaces can, by a line that is perpendicular to that surface at all points. This is a flat surface, so there is simply one of these. It's called a normal. We've encountered these already in the discussion of magnetic fields and magnetic flux. The normal to the surface is indicated by the vertical dotted line. And because I can draw this with a precise drawing program, such as that which comes with Libre office, a free and open source office suite. I can make the angle between the dotted vertical line and that solid horizontal line precisely 90 degrees. Now what we observe is that the majority of the laser light that we see is reflected up into the right. And that's indicated with the second arrow on the right hand side. That is referred to as the reflected laser beam. And we notice something very interesting. The reflection, the bouncing of the laser light off the interface between the first medium and the second medium, the angle that the light comes off, not only places it entirely within the first medium, but on the other side of the normal from where it started. So if we denote the angle of the incident laser beam with respect to the normal to the surface as theta sub one, which I've indicated here on the drawing. And we then indicate the so-called reflection angle, that is the angle that the reflected laser light makes with respect to the normal as theta one prime, which I've also drawn here on the picture. Then we can ask if there is a relationship between these two angles, theta one and theta one prime in the case of specular reflection. Such that we can solve problems, for instance, in the propagation of light through a series of mirrors. Each mirror is a reflecting surface. And if we knew the rule that related theta one and theta one prime, we could then, for instance, very easily solve problems. We could predict the path that light beams will take through a system of our own designs, for instance, medical optical instrumentation. Well, what's very interesting about using a precision drawing program like that that comes with LibreOffice, really any office suite, is that you can draw the first arrow, as I've done here, and then you can draw the second arrow and see if there's any numerical relationship between those two. And in fact, the relationship that you see here is a very simple one. In order to get the right hand arrow, all I had to do was take the left hand arrow, copy it, and flip it about the horizontal axis, basically making a mirror image of the first arrow in the second. And you'll see that without drawing the second arrow, I was able to perfectly trace the path of the reflected laser beam. My white arrow on the right hand side perfectly follows the red line of laser light. I didn't even have to hand trace that and then figure out the relationship. Why? Because the law of reflection states that theta 1 is equal to theta 1 prime. And in fact, we see that here borne out beautifully merely by drawing one arrow, flipping it, and drawing a second arrow. So the law of reflection can be stated very simply. If a beam of light enters a surface and strikes the surface, if we draw normal to the surface at the point where the laser light strikes the surface, and we denote the angle with respect to that normal as theta 1 for the incident laser light or the incident ray of light, then theta 1 prime, the angle of reflection of that ray of light, will be equal to theta 1. Now, the reason I emphasize specular reflection is because there's another kind of reflection that you're actually probably more familiar with, but don't even realize it's occurring, and that is diffuse reflection. The human eye only sees things because light is scattered off of surfaces and then eventually strikes our eye. So for instance, light from the lamps in the ceiling of the room I'm recording in right now is flooding the room and it's striking the table and it's bouncing off the surface of the table and some of that light is entering the cavity in the front of my eyeball which itself is a lens system. And that light is eventually turned into electrical information that's interpreted by my brain. But in order to see things, they have to either give off light or reflect light. Materials that neither give off light nor reflect light cannot be seen by the human eye. Specular reflection is when incident light entering at a specific angle will absolutely come out at the same well-defined angle given by the law of reflection every time. Diffuse reflection on the other hand says that is a phenomenon that occurs when incident light coming in on a well-defined path may be scattered in one of many possible directions and that's indicated in this cartoon here by the red arrows. What fundamentally causes diffuse reflection is unevenness in the surface. That is, there's no guarantee that if light comes in on the same path that when it strikes a point on the surface that that point on the surface will be angled exactly the same as it was either a moment ago or as any neighboring points happen to be oriented. And we can see that a bit more carefully on the next picture. So here you have a very jagged and uneven surface. Light rays can come in parallel to the horizontal but it depends where they strike the material, regardless of the fact that they all come in parallel, influences where the outgoing scattered ray will point. And we see that here the outgoing rays appear to go in a jumble of directions. So a surface that's uneven where maybe each point on the surface is part of a little plane but those planes are not all oriented in the same directions. That is their normals don't all point in the same directions. A non-smooth jagged surface is an excellent diffuse reflector and you probably have a diffuse reflector sitting by you right now. So for instance, white paper. Blank paper is a diffuse reflector and the reason white paper appears white is because if you send in light that contains all possible visible frequencies. Roy G. Biff, right? Red, orange, yellow, green, blue, indigo and violet. If you send in light with all of those frequencies like light from the sun or light from a fluorescent bulb in the ceiling above you that's white light that appears white to our eyes and when it strikes the paper it scatters in all directions and you get a diffuse emission of white light off the paper. Go ahead and move a piece of white paper around in a well lit room and that paper will look basically the same from all angles and that's because your eye is simply picking up the diffusely scattered light that's scattered at all kinds of different angles regardless of whether or not it comes in parallel from the source. Diffuse reflectors don't form images and we'll get to images in the next lecture when we talk about mirrors but you don't see your reflection in a piece of white paper because light coming in off your skin is not reflected back at your eye always at the same well-defined angle which is a minimum requirement for making a very well-defined image or sharp image. So diffuse reflection produces a wide scattering of light. You can't necessarily tell where the light came from once it's scattered because of the unevenness of the surface. Specular reflection allows you to make very specific predictions based on the outgoing ray where the incoming ray came from and that allows you to do something called ray tracing which is the basis of identifying the formation of images and optical systems and we'll get to that more when we talk about mirrors and thin lenses. We've been looking at reflection but there's another equally important phenomenon that occurs with waves of light and that is refraction. Now I've been very loosey-goosey in talking about light. Light is fundamentally a wave but you can see here from this picture that we can represent the information in the wave merely by drawing an arrow that follows the direction that the wave fronts are traveling. So if you think of a wave as a series of maximum crests followed by minimum troughs followed by maximum crests and so forth a repeating structure of maxima, minima, maxima, minima then we can represent, for instance, just key points on the wave. We could pick the locations of maximum electric strength electric field strength along the wave and those here are indicated by these black parallel lines. Those are simply called wave fronts and they are locations of maximum strength of the electric field on the wave. Now when an electromagnetic wave strikes the boundary between two materials, two media, different media here indicated one by white and one by blue coloring. So for instance, this could be air and water or air and glass or air and ice or vacuum and water, something like that. When an electromagnetic wave strikes the medium something happens to it. Some of it will reflect. You can get reflections, for instance, off the surface of water. You've probably seen these at a pool or at a pond. You can see the sky in a pool or a pond if you're standing at the right angle but you know that if you dive into the pool or pond and you open your eyes, you can see light coming from outside the surface of the water and penetrating inside. The process of a light wave or any electromagnetic wave passing through the boundary between two media is known as refraction and it's the opposite of reflection. Reflection keeps the light in the original medium from which it came. Refraction is when light enters a second medium passing through the boundary between the first medium and the second one. Refraction carries with it an interesting optical phenomenon which is that the path of the light ray will change. Now the ray here is merely the arrow that indicates the direction that wave fronts are traveling and we see in this picture that the wave fronts inside the blue medium are more closely spaced than they were in the white medium and their direction is tilted at a steeper angle with respect to the interface between the two materials. So the red line here is a perpendicular line that makes a 90 degree angle with respect to the boundary between the two media. And the rays, that is the arrows that indicate where the wave fronts are traveling, those rays are closer to the normal inside the blue medium than they were outside the blue medium in the white medium where the rays start its journey. This is called refraction. When the difference between the properties of two media cause light rays to bend when they pass through the surface between the two media. This only occurs at the boundary between two media. If you have a light ray that starts in air and it continues in air and then it continues further in air, refraction will not occur. But if you have a light ray that begins in air, strikes an air glass interface, say at a window, travels through the glass and then strikes the glass air interface, you can get refraction at both of those interfaces. You can get the light ray bending. You've probably seen this up close before. This is an optical effect that's caused by refraction of light through different media. So for instance, in order to see the straw outside the glass, all we have to do is receive light scattered off the straw, traveling through air and striking our eyes. In order to see the straws inside the glass, but still in air, the light rays have to travel through the glass and then back into air and then strike our eyes. If we want to see the straws when they're inside the water in the drinking glass, light has to reflect off of the straws and then travel through the water and through the glass to reach our eyes. And the more boundaries that you add for that journey, so for instance, if the light starts outside the glass, travels through the glass, through the water strikes, the straw reflects off the straw, travels through the water, travels through the glass and then through the air to our eyes. The more boundaries you put in the path of light, the more extreme refractions you can get. And you can see here what the effect of the refracting of the light is. The straw not only appears to be magnified inside the water, but its body appears to be physically disconnected from where it actually enters the water. It's as if physical reality has been altered. But all we're getting here is an optical effect. It's an illusion created by the refraction of light on its journey from the source to where it scatters to our eye. And the more surfaces you put in between the source, the scattering surface and your eye, the more refractions can occur and the more distorted images can become, the more deceived we can be about the location, for instance, of the body of the straw. And you've probably experienced this before. If you've ever accidentally dropped something into a pool like your keys and you try to reach down into the pool to get the keys, you're looking at the keys, you know, based on light that comes from the sun, strikes the surface of the pool, refracts, reflects off the keys, refracts again at the water-air interface and then reaches your eyes. And you know that when you reach your hand and your hand appears to bend at a weird angle. And so you have a very hard time for a moment trying to figure out exactly where the keys are because you're disoriented by the fact that light is being bent at funny angles before it reaches your eye. It takes a moment for your eye to sort of correct and catch up to the fact that basically light is coming from someplace other than it appears to be originating from when you're reaching for your keys underwater in a pool while your head is still out of water. So you have to kind of correct for these weird optical phenomena, but they're all based on reflection and refraction. And that's the origin of where all of these observations come from. Now the law of refraction is a little bit more complicated than the law of reflection. The law of reflection, specular reflection specifically, says that light striking a reflective surface will bounce back into the medium from where it started at an angle of reflection that is equal to the angle of incidence with respect to a normal to the surface. We've looked at that already. The law of refraction, also known as Snell's law, is a little bit more complicated because it takes into account the difference of the properties of the two media as well as those angles, the incident angle of light coming into the boundary and the outgoing angle, the refracted angle of light leaving the boundary in the second medium. Now you can look at the history of this law and you can, you know, with modern techniques and all the information we have available to us for many historical records, you'll find that this observation of the geometric rules that relate the incident angle to the refracted light angle, they predate Willebrod Snell Van Royen, who is the person after which Snell's law is actually named. He did his work in 1621 and basically codified what we now think of as the modern law of refraction. And he lived from 1580 to 1626. But in fact, you can go all the way back to, for instance, manuscripts, for instance, from Ibn Sahl in a roughly 984 AD, showing that multiple cultures, for instance, his was a culture from what we now consider the Middle East, showing that they had discovered the law of refraction mathematically. This was a period of high mathematical and technological progress in the region of the world now covered by Iran and Iraq and other countries in the Middle East. They had come up with all kinds of mathematical things that people in Europe would only discover a long time afterward. But nonetheless, you can see that multiple cultures using the same investigations of the natural world came to understand the law that unites the incident angle of light and the refracted angle of light, even going back to maybe 984, and possibly earlier than that, although it's not clear that there was a good enough mathematical understanding prior to that to really nail all this down. So we have here collected on one page the laws of reflection and refraction, and now you'll get to see Snell's law, the law of refraction written out. Reflection, just to remind you, again, as long as the light reflects back into the original medium material from which it started. So if light comes in from air and when it strikes the interface with another material like glass, part of it will bounce in such a way that it stays in air. That's called reflection. And for that, we know that theta incident is equal to theta reflected. That is, the incident angle with respect to the normal to the surface is equal to the reflected angle with respect to the normal to the surface, always with respect to the normal vector to a point on the surface where the light strikes. The law of refraction is a little bit more complicated and boiled down to its final essence. It tells us that the property of the incident material, that is, for instance, air, which is known as the index of refraction, multiplied by the sign of the angle with respect to the normal to the interface between medium one and medium two, will be equal to the index of refraction of the second material times the sign of the angle of refraction. Now, N is what I refer to as the index of refraction, and all it tells you is it tells you the ratio of the speed of light in vacuum to the speed of light inside the material in which light is traveling. So one of the things you can see if you go back a little bit earlier to the picture of the wave fronts coming in from the white medium and then refracting in the blue medium, one of the things you can see there is that the speed with which the wave travels is not the same in both medium. What is the same in both media is the frequency of the waves. Their wavelength changes, their speed changes, but their frequency remains the same. That is the number of wave fronts passing a given point in a given amount of time is the same between the two, but the wavelength shortens in the picture I showed earlier and the speed of light correspondingly changes. So if the wavelength gets shorter, the speed of light in that medium is smaller. The fastest that light can ever travel is when it's traveling in vacuum. That is empty space, and by definition, the index of refraction of the vacuum is one, because since the speed of light in vacuum is the fastest light can ever travel and its speed in vacuum is that, the ratio of those two is, by definition, one. In any other medium, the speed of light will slow down compared to its speed in vacuum. So for instance, in water or in air or in glass, light is traveling a little bit, or in many cases, a lot of bit, slower compared to its speed in empty space, vacuum, where there in principle are no particles off of which it can scatter or be absorbed and re-emitted which would slow down its journey. So in any other material, C over V is going to be a number bigger than one because V will be smaller than C, the speed of light in vacuum, so C over V will be a ratio that is always greater than one in any other medium other than vacuum. So N1 is the index of refraction, that is the ratio of the speed of light in vacuum to the speed of light in that material in medium one. Theta one is the angle of incidence from medium one with respect to the normal to the boundary between the two surfaces. And two is the index of refraction, that is the ratio of the speed of light in vacuum to the speed of light in that material for material two, and sine of theta two is the sine of the angle of refraction in the second medium. Again, with respect to the normal to the boundary between the two media at different angles. And this is the entire basis of something called prism or the phenomenon which we observe called a rainbow. We'll go into that more in a moment, but before I do that I wanted to talk a little bit about indices of refraction and common materials. So what I've shown here are selected refractive indices at a wavelength of 589 nanometers. For, these are pulled off of Wikipedia so you can go ahead and look at the Wikipedia article on index of refraction and there's an extended list of refracted indices which appears as a hyperlink in the original caption of the original table. And as I said before, vacuum, empty space has an index refraction of one because the fastest light can ever travel is when it's in vacuum and that is when it's traveling at C 2.998 times 10 to the eighth meters per second. So there you have C over C which is N and N is thus equal to one in vacuum. Now for instance we could look at gases. Now gases are very sparse in the amount of material they have but of course there are more atoms in gas than there are in vacuum because vacuum is empty space where there are no atoms. So air, helium, hydrogen gas clouds, carbon dioxide clouds, those all have indices of refraction that are almost one but not quite. So you see that air has an index of refraction of 1.000293. You can see that's a very small change in the speed of light in air compared to the speed of light in vacuum but nonetheless it's not zero and it's measurable. Helium has a slightly smaller index of refraction so the speed of light in helium is faster than it is in air. In hydrogen it's also a little bit faster than it is in air. Carbon dioxide however slows down light much more. So the speed of light in carbon dioxide is slightly slower than it is in just air which is a mix of nitrogen, oxygen, argon, carbon dioxide and other trace gases. What about liquids? Well liquids at 20 degrees Celsius are roughly room temperature. Water has an index of refraction of 1.333. We see that basically light is 33.3% slower in water than it is in vacuum. Ethanol is 1.36, olive oil is 1.47 and you can go back to that picture I showed you earlier and you can see indeed water and air and olive oil and glass which I have down here in the bottom of the table. They all have different indices of refraction and you would expect light rays to bend at the interfaces between air and oil, oil and water, water and glass and so forth. It's a little hard to see in the photo I showed you earlier but if you got out very precise measuring instruments you could see that there's a slight bending of the rays of light as it travels through those different surfaces. Solids, well common solids like ice have indices of refraction of 1.309, plexus lower in diamond than it does in vacuum, empty space with an index of refraction of 2.42. And you can see here refraction beautifully illustrated in this photograph which I also got from Wikimedia Commons. Light comes in through air from the left. It strikes the air plexiglass interface. It bends and you see that the light makes a smaller angle with respect to the normal to the surface of the air plexiglass interface than it did outside. Then it strikes the plexiglass air interface on the other side and it bends away from the surface of the, from the normal to the surface of the plexiglass. And you see actually that the path of the laser beam in terms of the vector that represents its direction of travel is the same when it comes out the other side. So this is why images don't appear to be distorted when they travel through glass windows, for instance, unless the glass is uneven. If the glass is very even and uniform, then images traveling through the glass are preserved, they're shifted slightly maybe to the left or to the right of where they were outside, but all the relative positions of objects that you see through a window are preserved when light travels through a window. On the other hand, you can see that in fact, the light ray doesn't come out at exactly the same point that it came in. And so as a result of that, you do. So now one comment I wanna make here is that color and refraction, they do go hand in hand. So for instance, the angle of refraction when you go from medium one to medium two is a color dependent phenomenon. Red light will not necessarily scatter when it passes through the boundary at the same angle as blue light. And so the reason we see red light as red is because it has a particular wavelength of about 750 nanometers. And the reason we perceive blue light as blue is because it has a shorter wavelength and that is much closer to 400 nanometers or thereabouts. Wave length enters Snell's law. If you go back and take a look at Snell's law n one sine theta one equals n two sine theta two. And we substitute in for the ends, the ratio of the speed of light in vacuum to the speed of light in material. Then what we see is that wavelength enters very directly into that. And because of that, we'll find what will be related between the two media by Snell's law when relating n one and n two is really fundamentally the wavelength of the light, lambda one and lambda two. So what we'll find is that different wavelengths of light will be refracted at different angles. And this is the entire basis of something called a prism or the phenomenon which we observe called a rainbow. Expect images could be potentially distorted, especially if color is a big factor in the angle of refraction. Here's a great example of that. So here is white light coming into this device called a prism. It's just a triangle of glass. Newton played with these and was one of the first people to understand that white light is composed of other fundamental colors that taken together appear as white to our eye. And he did so using a prism. We see here that the different indices of refraction are laid bare. The blue light comes out on the other side of the glass at a different angle than red light. And this is exactly the phenomenon that we perceive during a rainstorm as a rainbow in the sky. What's going on in a rainbow? Well, if you're standing on the ground at the observer point O and there are water droplets that are suspended in the air, maybe from fog or just after a heavy rain, something like that. And suddenly the sunlight peeks through the clouds behind you, strikes the water drops and that light can be refracted down to your eye because different wavelengths of light are refracted at different angles inside of the water droplets at the water air interface. You get the effect that the white light is spread out into its constituent wavelengths. And when it strikes your eye, the blue light is coming in at a slightly different angle to your eye than the red light. And so you perceive that as the white light having been split into its constituent colors. Normally all of these wavelengths would come into your retina at roughly the same point. But if you put something like water drops up in the air and observe those water drops at a 42 degree angle with respect to the surface of the earth, you will perceive the phenomenon known as the rainbow. And so this is why the angle of the rainbow, for instance, the angle between the ground and the top of the rainbow, you will see it as 42 degrees because that's the angle, the critical angle at which total internal reflection will occur inside the water droplets and spread the light out and bring it down to your eye. That's what's going on in the back of the water droplet and we're gonna explore that phenomenon in detail more in a moment. But it's actually possible to have water come in, have light come in through the water, strike the water air interface and completely reflect back into the water. And this is known as total internal reflection. It's a phenomenon based on refraction, but it results in a perfect reflection of light from a boundary. So here's a beautiful picture of a rainbow. If you squint very carefully at, there is a second angle at which you should be able to see a second rainbow. So here you have the phenomenon known as the double rainbow. The primary rainbow is seen at the bottom. And again, at the angle you're gonna make with respect to the center of that curve there is 42 degrees. And then you see the secondary rainbow, the double rainbow, much fainter and up above the first primary rainbow. Under ideal viewing conditions, you can even see the very hard to perceive third refraction, which gives you the triple rainbow effect. But you can't see that in this picture. It's very hard to get that to occur in such a way that it can be seen with the eye or even photographed. So let's look at total internal reflection. We'll close out the lecture with this topic. This is a very special case of refraction. It occurs when the first material shown here in blue has a higher index of refraction than the second. So for instance, when light travels into water droplets, it refracts at the air-water boundary. And now the light is traveling inside the water. When it gets to the back of the water droplet, it strikes the water-air boundary. So it's coming in a material, it's traveling through material that has an index of refraction of about 1.33. And it strikes the boundary between water and air. And air has an index of refraction of 1.0002 or so. So a very different index of refraction in much lower than that of water. And under that special case, light travels through a medium, strikes a boundary with another medium that has a lower index of refraction. It is possible for the light not to pass through the boundary, but to completely reflect inside the original medium. When does this occur? Well, it's illustrated in the picture here at the right. Light comes in from the blue medium, strikes the interface between the blue and the white medium. Here, N1 is greater than N2. And if you tilt the incident ray until you achieve a special angle known as the critical angle, that is theta one equals theta C, then it's possible for the refracted ray to have an angle of refraction that's 90 degrees. That is, it travels entirely along the boundary layer between the two media and it never really enters the second medium. And that's illustrated in this picture. And you can solve for the critical angle. The critical angle theta C will be that angle that achieves a theta two of 90 degrees. That is pi over two as illustrated in this picture. N1 is greater than N2. If N1 is greater than N2, then the ratio N1 over N2 is a number bigger than one. When you move that number to the other side of the equation to solve for theta critical, theta one, whatever that critical angle is, then you'll have a number less than one multiplying sine of pi over two, which is just one. So you'll have an equation that says sine of theta critical equals a number less than one. You know that the sine of an angle can never be a number less than negative one or greater than one. If N2, on the other hand, is greater than N1, when you move N1 over N2 to the right-hand side of this equation, you will have N2 divided by N1, which is a number greater than one, and you'll try to take the arc sine of a number greater than one and you will get basically a big frowny face from your calculator. You'll get not a number or something like that, some warning from your calculator that you've tried to do something nonsensical. You can never have total internal reflection if medium one has an index of refraction that is less than medium two. There is no critical angle at which this phenomenon occurs, but if medium one has an index of refraction greater than medium two, there is a critical angle at which total internal reflection can occur and you can solve for it. It may be very big, it may be very small, it depends on the ratio of N1 and N2, but it is a solvable thing. Total internal reflection is absolutely instrumental in modern medical imaging technology. Not only is total internal reflection the basis of modern telecommunications, so for instance, fiber optic cable, which is used to create high speed networks upon which we rely not only for voice transmission but video transmission, for instance through the internet backbone. Total internal reflection allows signals to travel over vast distances with very little loss simply by sending light in one end of a tube and having it totally internally reflect inside the tube and come out the other side with very little loss. This instrument pictured here is a scope which can be for instance, inserted down the throat of a patient and can go into their stomach. This can be used to do visual imaging by human eye of for instance, a person who has extreme stomach pain. You can look for the presence of ulcers in the stomach or other dangerous phenomena that might be present in the gastrointestinal system of a human being. Medical endoscopy relies completely on total internal reflection to bring reflected light back from inside the patient up into the camera in the scope and then out to a monitor or a computer where it can be recorded. So for instance here, you have a portable light source that's on the tip of the scope that lights up whatever's in front of the optical fiber that's present in the scope. So you get light emitted by the end of the scope, bounces off the stomach lining, reflects off the stomach lining through diffuse reflection back into the glass tube, the fiber optic tube that's present inside this long actual scope end. That light can be totally internally reflected even at crazy angles all the way back up to the camera which is sitting outside the human being and then you can image what you see coming through the tube. This is much more economical than shoving a camera down a human being's throat. The throat is a constricted vessel. It has a limited size. There are small cameras now, but it carries much less risk to stick an optical instrument down the throat that is merely a vessel for light to travel up to the camera which sits outside the human being. All the computers and imaging equipment sits on the outside of the person. The scope goes inside and merely transmits reflected light from the light back up the scope's length to the camera on the outside. It's not exactly non-invasive imaging but it's certainly minimally invasive imaging and that no one has to be cut open to do this procedure.