One of the last things that we need to remember, again you should have seen this in a physics course before, is that when these plane waves intersect planar boundaries, they reflect and refract. And of course the direction of reflection and refraction is described by Senkrecht's Law. The amplitude, the efficiency of the light transmitting or reflecting is given by the Fresnel Equations. These are found from solving the electromagnetic boundary conditions at this point or boundary for a plane wave. They formally only apply to a plain wave. I've copied them here so you have them for reference, and I'm going to remind you of the sometimes confusing terminology, is that we have two possible conditions, because we have two possible polarization states. One is that the electric field is out of the plane of incidence. Here I've drawn the light coming in and bouncing and I've drawn a plane through that, the magnetic fields in the plane of incidence and the magnetic electric field is out of the plane. Or the opposite, the magnetic field is out of the plane and the electric field is in. There's three different names, at least, for this. The most common is that this is s polarization. And that actually comes from the German for, well, as written there. Another way, and probably the easier way to remember it, is in this polarization, from the perspective of the electric field, which is what we care about because we use the electric field on our wave equation. The electric field never changes direction, it's always out of the plane. So it could be described by a scalar which is just the magnitude of the electric field. Possibly positive or negative if it changes sign. But unlike the magnetic field it's not actually changing direction. So you can think about s as the scalar case for the electric field. I don't actually care about the electric field direction. P polarization is the one where the electric field does change its direction. Polarization for the electric field is important in this case, and you actually have to keep a little more track of that. So s, it's not formal, but s can also be thought of the scalar case for electric field. And there's some interesting terms in these things. One of these is that this particular term right here can go to infinity. And that tells you that the reflection coefficient in this P case can actually go to 0, that's probably important and we'll pop the next. So, you can go plot those coefficients, most of the time in this course we're mainly going to be concerned with where did the light go? Where did it focus? And we're not going to be too concerned with how much light got there. In real engineering practice, you are very concerned with how much light got to the end of your system, and these Fresnel coefficients are a big deal. That just sort of, that's typically something that will be handled almost entirely numerically for you by something like Optic Studio, so we don't deal with that too much in design. But you should know that they're there, so when we run across these things, you understand what the physics is. So this shows you those reflection and transmission coefficients we just plotted here for, notice this is also called perpendicular because e is perpendicular to the plane. So this is the reflection coefficient for s, transmission coefficient for S and the two coefficients for B. And reflection coefficients and transmission coefficients do something interesting. Particularly right here for that P polarization, you can have zero reflection. That's called the Brewster's angle and that's that tan theta that we pointed out before. Another thing that's probably important is that the two coefficients, reflection and transmission should be the same when the angle of incidence is zero. Because you can't tell what your two different polarizations are. When your angle of incidence is normal to the surface, it's a degenerate case. And that's actually what you do see. The transmission coefficients are the same. The two reflection coefficients are the same within a sign. And that sign is just from the definitions of the coordinate systems. And so indeed they make sense that they go to the same values at normal incidence. And notice, at 90 degrees, that's grazing incidence. Weird things happen. First the transmission event essentially goes to zero. When you go to grazing incidence, you get higher and higher reflection. One efficiency reflection and lower and lower transmission. And you know this from everyday life. Grazing reflection off of the surface of water or something like that can be just as efficient and just as strong as the light coming in directly to your eye. The Brewster's angle can be understood just through some simple geometry, which I've given there. Again, this is things you should know from previous classes. So we've just summarized them here so you understand that these coefficients depend on angle. Which might be 10 or 20 degrees here, you notice that they don't change very much, so that we can ignore it. But as you start designing lenses at high numerical aperture that get ray angles hitting surfaces at large degrees, now you may care a lot. And the polarization dependence can be quite different, and you may need to deal with that. Finally, just so you have a complete plot of these. This shows you what the phase of those functions do. And the most important thing is that the reflection coefficient does have a different phase, depending on polarization. What that means is that if you have light bouncing off of a glass surface, or a mirror, it can pick up a phase shift that's different between the two polarizations. And if phase and polarization are important to you, like you're using polarization to encode some bit of information, the fact that the polarization phases change is really important. So you need to understand in those cases that you can change the polarization state of a beam when it undergoes reflection. And then finally, the last bit of physics, we've seen this before in the lab, is that if the medium you're coming out of has a higher refractive index than the material you're going into. Then something happens, is that the second interesting angle after the Brewster angle is that you undergo total internal reflection. There is a critical angle beyond which everything blows up and you simply don't transmit into the material at all, that these are larger angles. That of course can be found just by solving Snell's law, for example. But everything has to asymptote to infinity, basically, right at the critical angle. And that's important to you again if you're using a prism or something near the critical angle. Then your transmission and reflection coefficients can depend, really strongly, on polarization. And that may or may not be something you want.
