Apparently, these rays, these things that go in straight lines, are important. So we better figure out exactly what they are, because they are going to be the foundation for this entire specialization. What Euclid and various other people would have told you if you asked, is that somehow a ray is a pencil beam of light. It's a little tiny aperture, let's say, a hole that you shine some sunlight on. Or in the modern day, you might describe it as a laser pointer. And this isn't wrong, but we're going to need better in this class. The goal of this video is to derive precisely what a ray is, mathematically. We begin with the scalar wave equation. You can derive this from Maxwell's equations, and you do that in most undergraduate electromagnetics courses. If you haven't seen that in awhile, or if this equation looks completely unusual to you, hit Wikipedia or your old E and M textbook. And just remind yourself how you take the two first-order Maxwell's equations, combine them to eliminate the magnetic field, and end up with this scalar version of a wave equation. Note the k here is called the wave number. That's simply two pi over the local wavelength, or that's equivalent to the temporal radian frequency omega over the speed of light. We're going to, in this class, describe solutions to this electromagnetic equation. And conveniently, not ever come back to this equation again, cause we're going to solve it with geometry. We're going to start out describing solutions as a local amplitude and a complex phase, which seems like a reasonable thing to do. And we're particularly interested in the phase part. And notice I've divided the phase up in two terms. I have the wave number of free space here, which is two pi over the vacuum wavelength. And that's got units of meters, right, or sorry one over meters. And then I've got a quantity s, which we're going to call the optical path length, which must therefore have the inverse units, or meters. So, for example if this was a plane wave, something you should have seen in a basic electromagnetics class, again, if not, look up plane wave on Wikipedia. Then, the total phase of the wave of k-nought times this optical path length would be k dot r, where the vector k is the direction in which the plane wave is propagating. The other solution we'll use a lot is a spherical wave, and now k times this optical path length is, k times a scalar r, which is the radial coordinate. And I'll show you some pictures of these in just a minute. So the main point is, we have an amplitude and a phase. And we've divided out this vacuum wave number, so that we have a phase quantity here that's in units of meters. So, it turns out that if you write the electric field that way, you plug it into the wave equation, and you make one approximation. This is really important, because it's the approximation the entire class is based on. If you make the approximation that that amplitude, capital E, changes slowly, and that means derivatives of it can be neglected in the right approximation, then you get an equation only for the phase. Only for this optical path length quantity. That's critical to what we're doing for the class. The idea is, is that actually, the phase of the light encodes almost all of the interesting information, and the amplitude, not so much. Or you can rederive where the amplitude is high and low from the phase. So, we're actually going to be primarily during the whole class, calculating just on what's the phase of the optical wave. This is the first equation you get. We don't directly use it in this class. It's actually called the ray equation, which is a bit of a misnomer, because there's no rays in it. And it's worth thinking about it a little bit if you like to think about math, of how this describes how the phase evolves, given a distribution of refractive index in space. And this little picture here might help you think about that. We don't actually need it for the class. What we do need is to note that from this, we can learn what this quantity S is, this optical path length. It turns out it's simply the path integral, ds, where s is some little coordinate along a ray, along this distance, the trajectory over which the ray goes, of the refractive index itself. So we're just going to integrate up along normally, a line, the refractive index. So, if this was a fixed distance, and n was a constant, this would be n times L, some distance between points A and B. And we're going to use this quantity the optical path length a lot. We don't actually need this particular equation or its solutions, we need the next one. And the next one is derived from the ray equation. The derivation is not at all straightforward, but it's just calculus. And it comes up with what's called the eikonal equation, yet another equation. Eikonal was German for image, and that's beginning to tell us, this is the equation which is going to tell us how images evolve. And the important thing about this equation, is this r coordinate here, it's actually the path of a ray, how the light travels, if it was, for example, a little pencil beam. And the key thing about this equation is, if we assume homogenous space, let's just say we have an index n sub 0. And so we don't have any spacial gradients, any spacial variation to n. Then this term on the right, the gradient of n is 0. And this equation is easily integrated to the trajectory of the ray along its path length s is simply a constant times s, that's the equation for a straight line. So that was a long way around for a simple conclusion, but it's very important that you understand where this came from. We have learned what you could already know in 260 BC, that rays in homogeneous media, media that don't have any gradients of refractive index, travel in straight lines. But we know something he didn't know, two things, several things. First, the rays are solutions to Maxwell's equations. We are solving the partial differential equation. If the amplitude in some region of space the we're interested in, the amplitude of the electric field varies slowly, and we're going to find in most of our space that we're interested in it does, and there's always some places that it doesn't. And so our solution is going to be valid in the first case and invalid in the second case. So you definitely need to know where those points are, and it's based on this slowly varying envelope approximation. So that's one thing, the [INAUDIBLE], these get equations, these rays are solutions to Maxwell's equations in most places. Next, the rays themselves have a meaning, and what that meaning is is that they are the normals to the phase fronts. The phase is evolving, and the rays are simply a set of lines that are normal to the phase. And I'll show you next why that's a much nicer description of the optical field and its phase, than the phase fronts itself. We get this definition that the optical path length is the integral along that ray of the refractive index. And finally what Euclid already knew, is if the refractive index is constant in space, rays travel in straight lines. That's the foundation of everything we're doing, so it's important you understand each of those pieces. [BLANK AUDIO] So let's look at some examples now of those rays. Let's take our plane wave again, which is the simplest solution there is to Maxwell's equations or to the wave equation. So if we draw a typical picture of a plane wave, this might be the real amplitude of the electric filed, then of course the distance between any two points of constant phase, say the peaks of the wave, is by definition the wavelength. The k vector, kx, ky, kz, travels orthogonally, or points orthogonally to those phase fronts. So in this case the k vector, the wave vector and the rays are essentially the same thing. And the rays are just any set of lines you'd like to draw as long as they're orthogonal to these phase fronts, their direction generally the direction the wave is propagating. A circle wave is something we'll also use a lot in this class, little point radiators radiates their cowaves, for example. The solution to a point radiator in a scale of rate equation is e to the j, k, r over r. In that case the phase fronts are circles or spheres, and the rays are simply the radial set of lines that travel out from the origin. So you notice that one set of information gives you the other, and that's key. We're going to work with rays, but you always want to be able to reconstruct these phase fronts, because the phase fronts are the electric field and the solution to Maxwell's equations. So why would we use rays? It seems like we've done a lot of derivations there, and defined a lot of terms, and they're the same thing. The rays and phase fronts really contain the same information. And they do, but this picture illustrates a very typical case, and shows you why rays are much more friendly to use. If we had a plane wave that traveled into a lens such that it came to a focus, on the left side of the lens we have a plane wave. The lens converts that into ideally a converging spherical wave which then goes into through the focus to a diverging spherical wave. You notice the phase fronts are kind of complicated shapes. Sometimes they're planes, sometimes they're convex spheres, sometimes they're concave spheres. And in perial lenses, they won't even be spheres. But the rays are always straight lines. And that's the key. We're going to be able to draw straight lines and trace straight lines with things like Snell's Law, that describe the phase fronts, which, under that one approximation of slowly varying envelope, are solutions to Maxwell's equations. And as a preview here, in this region, the amplitude is changing slowly so the solution is good. Out here there's no amplitude at all, so it's changing slowly, so the solution is good. But right on this boundary here is where the solution actually would not be good. And that's where defraction would happen. And we'd have some more complicated optical physics going on here. We can in most cases, either ignore that, or patch our solutions back up later. But anywhere along these boundaries between field and no field is where the solution is suspect. Everywhere else the solution is good.