It would be very typical for a course like this to start with Maxwell's equations. They're the foundation that describes everything we're doing. And if we can't relate how our modelling, and how our design relates to Maxwell's equations, then you probably shouldn't be doing it. We chose not to do that in this course, because we wanted to dive right in to useful design tools, and really get into the meat of optical engineering. But we have to do it someplace, so we're going to do it now, and we have two purposes. One is this basically is a quick summary of the expected prerequisites for this course, the stuff you need to know. We're going to go through that relatively fast, under the assumption that if you don't know it, now you know exactly what to go dig into some background material and find. The second point and goal is to derive the plane wave. Because the plane wave solution to Maxwell's equations is so foundational, that we can use it to understand an enormous amount of what we're doing, and we're going to need it In the following bits of this course. So, I'm going to go through this relatively quickly, I've given you equations, and units. If this is all quite familiar with you, then you're good. If some part of this doesn't feel comfortable to you, then that's a flag to stop the video, and go find your old textbooks, or an online course or something and get a little more background. We have to start with Maxwell's equations, because that's what describes how light propagates. There are four of them, two Curl laws that describe how time rate of change of flux through some area gives rise to electric or magnetic field. And then two that deal with the divergence of those fields due to free electric charge, or the nonexistent free magnetic charge. And that is all the units, and hopefully things like Curls and Divergences are familiar to you. Those are incomplete without the Constitutive relations, that tells you how materials interact with these fields. In general, we care about the electrical properties of materials at optical frequencies. Rarely do we care about the magnetic properties of materials. And the full Constitutive relationship, which you may or may not have ever seen written down before this way, it really has a set of pieces. First it's a convolution integral, and that it expresses that a dielectric response of the material. The polarizability of molecules let's say, is an impulse response that must be therefore convolved with the electric field to come up with the displacement field. And real materials have dielectric responses that aren't impulse responses. That is there is a shape, and we'll talk about that in just a moment, and that's going to give rise to things we've already seen, dispersion. So, it's important to remember how it relates back to Maxwell's equations. Real materials may also be Anisotropic, not the same in all directions, and that would be codified by the dielectric response of the material. Being a second rank or three by three tensor, such that when you push on a material in an electric field, the displacement current, and the displacement field may not be in exactly the same direction. So quartz crystals, and things like that we do use in optics need this form of Anisotropic dielectric response. Most of the time in this course, we'll make the Isotropic assumption. We'll turn this tensor into a scalar, and it becomes a simple multiplication. Magnetic fields are described by the same relationship, but almost always the relative permeability of the material can be taken as one. Materials generally don't respond to magnetic fields at these frequencies, and so we can ignore that term. And though we do have materials that have absorption or loss, in many cases we'll assume there's absolutely no loss, and therefore the conductivity is zero, and the current is zero. We don't normally deal with terms in the time domain here, of course we Fourier transform. And so, there's the Fourier transform integral. It says that we can represent these functions of time as a function of frequency, and of course this transform goes back both ways. That should be a two-pi right there. The key thing is we can transform between domains. If you're not intimately familiar with the Fourier transform, that would be a quick thing to go review as well. And that's essentially equivalent to saying that we have only monochromatic fields, and so we're only dealing with one frequency component at a time anyway. So in many cases in optics, we do have lasers, or something that are pretty good approximations of monochromatic. And so the physical version of this is you're only dealing with one temporal frequency, or one wavelength at a time. Mathematically, when we take this Fourier Transform, if we take a time derivative of a variable, it now just takes the time derivative right there. So, all time derivatives turn into a multiplication. Since when you take the derivative of that, you get j omega. They turn into a multiplication by j omega, and the key is that it reduces all of these equations that have derivatives in them in time, to be just multiplications. And that's the kind of reason we like Fourier Transforms. So, there's the same Maxwell's equations again, now in the temporal monochromatic domain, and the temporal frequency domain. And importantly, the constitutive relations also can be Fourier transformed. So if we take that Convolution integral, the Convolution theorem of the Fourier transform, says that it reduces to a multiplication. There's still the, in this case, for dealing with antisubtropic materials, there's still the matrix multiply here. In the Isotropic case, it's just a multiply, but the Convolution interval goes away, and that is such a dramatic simplification, that it almost motivates, or it transforms all by themselves. So the key thing that's going to be important, and has been important for understanding things like prisms that we saw earlier. Is that the material response, impulse response, how it goes boing to light when it's hit with an impulse, now turns into a dielectric constant that is a function of temporal frequency. And that's why for example, index varies with wavelength, or with frequency because this function, the dielectric, relates directly to the refractive index. And fits to the time derivative, or the time impulse response of the dielectric. If the impulse response was an impulse, with a delta function, then the Fourier transform of it would be a constant, and there wouldn't be dispersion. So, the fact that there's a time response to the polarizability of materials, leads to the refractive index, or the polarizability, being frequency dependent. Which leads to lenses not operating the same for every color, so it's relatively important.
