 Now, we have some image from physical reality. We have an interface that is air and glass. What I see here is that there is reflection and there is reflection. So in case not everyone understands the term, reflection is least reflection on. And reflection is reflection, right? With my horribly broken German. Thank God this course is not in German, everyone would cry. But you guys and girls have it easy because in Hungarian you would say, think this is a very dish and think the dish. So I think reflection and reflection is much better than that. At least much more convenient. Question, which is stronger? Which effect is stronger? Raise your hand if you think that reflection that this is this effect is stronger. You are asking this question about this example in the example. Yes. So I can see that the reflection has a more pronounced effect than reflection here. Because we will deal with the cilistackrum. And what we can do is we can actually write up the vectors that we have been talking about for a case like that. This is towards the direction of the light source. There is a surface normal. It looks like this so the normal looks upwards. This is where it is reflected. This is where it is transmitted. We don't have a vector for that. And we have the different angles for this. Wonderful. So let's take a look at the simplified version of the analysis equation. At the simplified version of the analysis equation, this one at least is called Schlich's approximation. We have no idea what this is about. This is not so complicated thing. What this gives me is the probability of reflection. So R of theta is the probability of reflection. So as I have seen the image, I am interested in what is the probability of reflection and reflection. Because I imagine that the probability of reflection is higher in this case. And I would like to compute it in some way in my computer program. Well, let's take a quick look at this. R of theta is the probability of reflection. And this is important to remember because during our calculations, I will forget this approximately 15 times. So may I ask your name? Lisa. Lisa, OK. Well, if I ask you what is R theta, you will tell me that this is the probability of reflection. The probability of reflection. Yes, and exactly. Because seriously, I will be forgetting it all the time. R0 is the probability of reflection on normal incidence. This means that light is coming from, if it would be coming from above, what are the chances that it gets reflected? And R0 can be given by this expression. We will be talking about this. N1 and N2 are basically indices of reflection. We will have examples with that too. But let's quickly go through this and see if physics makes any sense. Well, for an R vacuum medium, we have, let's say, the index of reflection of R is 1. And this N1 is the medium that we go into. So for instance, here glass. And let's see what happens there. But before we do that, T is the probability of transmission. Obviously, if the light is not, we forget absorption for now. If the light is not reflected, then there is reflection. It's simple as that. So if I add up these two probabilities, I give 1. So let's play with it. R at 0 degrees is R0. Why? Because cosine of theta, so the cosine of 0 degrees is 1. So on the right side, I have 1 minus 1. So the second term is killed by the 0. Therefore, I will have R0. And R at 0 degrees, this is the theta is the degrees, the angle of the incoming light, 0 degrees. So it means that it comes from upwards. What is it? This is the probability of reflection of normal incidence. So this is basically the very same thing. So if it comes like this, what is the probability of it coming back, bouncing off of the glass? What's up with 90 degrees? Well, 90 degrees, the cosine of theta is 0. Therefore, I will have both of these terms. Very simple. And this is going to be 1. So the probability of reflection at 90 degrees is 1. Because imagine that I'm coming from above. Then this means that there is a very likely chance that I'm going to get through. So imagine a super crowded bus in the morning. And you just cannot fit in there. How would you like to go in if you don't care about the health and the effectiveness of the other people? Well, you will just run in there and hopefully they will make some space. I have the best probability if I run towards them. If I would be running from the side, it would be very likely that they would just push me back. There is a high chance that I would be reflected. So I want reflection. I want to get on the bus. So as I raise this angle from normal incidence, the more probability there is for the rate of bounce back. So this, for now, seems to make some sense. But is it still reflection at 90 degrees and not just the much itself? You have to think in terms of limits. So what is the probability at 89 degrees? It's going to get reflected. If you just raise that, you are going to approach the probability of 1. So this is a bit boggling the mind that you can see that there is a continuous transition. From here, there is a high probability for reflection. As I go towards 90 degrees, there is more probability for reflection. And we define that at 90 degrees, we say that it is reflection because I am moving along the boundary and entering the glass. By the way, that's a great question. I was thinking about this too. So let's say that index of reflection of glass is 1.5. Let's compute this thing quickly. It's 0.5 over 2.5 squared. And I do the very same substitution for the rest of the equation. But before I get this, what do I expect from the plot? That's another important mathematical principle. Do this all the time. Before you compute the result, state what you would expect from the result. Because this gives you a much higher level of understanding. Well, I'm interesting in R of theta. What does R of theta mean? How about building the reflection? Excellent. Please note it again. So the probability of reflection at 0 degrees, I would say, is something very low. So I have written this here. That R of 0 is less than 0.1. Because it's the probability of... I mean, R 0 is the probability of... Oh, sorry. So the probability of reflection is low. I would say less than 10%. So if I come from upwards, reflection is likely. I'm very likely to get on the bus if I ran into the people from the front. What's up with, for instance, 60 degrees? Well, we know exactly what happens at 60 degrees. Because how many degrees do we have here? What is the incidence now? It's 60. And we can see that at 60 degrees, there is a chance for reflection and reflection. And the reflection chance is... Higher. Higher, exactly. So what I would imagine is that at 60 degrees, there is a higher chance of reflection. There is... Then the previous one, I mean, but reflection is still stronger. As you can see on the image, we are going to compute this and we are going to be... We're going to let nature be our judge whether the calculation is correct or not. So 60 degrees, I could be converted to radiance. That's more or less 1. And so R of 1 is 0.2. This means 20% chance of reflection, 80% of reflection. It seems to be in line of what I see here. But this is just my expectation. And what we have been talking about, 5 or 2, this means 90 degree angle. I would expect it to be 1. So if I convert it to radiance, then this is 1.7. So R of 1.7, I expect it to be 1. Let's put all of these together and let's do what engineers do all the time. We'll come from Alpha and try to plot this. So I imagine that R of 0 is less than 0.1. So getting on the bus easily, well, R at 0 is less than 0.1. So far so good. R at 1 is less than 0.2. This is the 60 degrees, both later, reflection. Well, R at 1 is less than 0.2. So checkmark. And R at around 1.7 is indeed around 1. So apparently physicists are small people. And physics makes sense. So, particularly. But there is something fishy about this. So this is correct. I mean, what we see here is in line with our expectations. But there is something fishy about this plot. Raise your hand if you know what it is. Okay, I'll give you a hint. This plot R of theta, which is. This is the probability of reflection. Okay. What happens after, I don't know, if I would just extrapolate. What would happen if not 1.5 or 2? What would I measure? I think it would be something like the very least minus. It's going upwards. But it would be at least 3. I don't know about you, but I don't know about probabilities that can be larger than 1. So this would give me some fishy results if I would substitute something like that. So let's try to share some more light on it. What if I have a vacuum, vacuum interaction? So below, I don't have glass anymore. I have vacuum. Well, the index of reflection of vacuum is 1. So let's just substitute 1 here. So this is going to be 0. And I'm going to have the second term. 1 minus cosine of theta to the fifth. Why? Because this is 0. And this is 1 minus 0. So I will keep this term. Okay. Engineering mode. What do we expect from this plot? I have vacuum or air, if you wish, if you will. And vacuum again. I start a way of light. What will happen with probability of what? Reflection or refraction? Raise your hand if you know. There should be no reflection. There should be no reflection. Exactly. Why? Because there is no interaction of doing medium. Exactly. So the definition of vacuum is again, nothing. There's nothing in there. There's nothing that could reflect this light, this ray of light back. If there is vacuum, we expect rays of light to travel in vacuum indefinitely. There is no way that it could be reflected. So since r0 is the probability of......thin this r theta. What do I think this should be? If there is only refraction, then this r of theta will be fantastic. Well, let's plot this. This looks like 1 minus cosine theta to the fifth. This is already fishy. But let's take a look. Blah blah blah. This is what I've been talking about. So I expected to be 0. Let's plot it. And this looks like this, which is not constant 0 by any stretch. So the question is, you know, what went terribly wrong here? And it's very... See, the sleek approximation is an approximation. It's good for interfaces which are vacuum or air and something. And not something, something. So A and B. It works well, but it's a quick approximation because this is the original Fresnel equation. And this is much more expensive to compute. And this other one was much quicker, but it's a bit limited in use. So let's give it a crack. So what would this say about a vacuum interaction? Well, I would substitute n1 and n2 equals 1. This is the index of refraction of vacuum. Again, the very same thing back just the n1s and n2s are gone. I'm going to use a trigonometric identity, which says that the square root of 1 minus the sine of square something is at the side. So let's substitute these for the cosines. So what I see here is the cosine of theta minus cosine theta. So what is this expression exactly? How much is it? 0. Exactly. And this is what I was expecting. So apparently physicists, again, are smart people.