In our last class we traced rays through optical systems, and learned rules using first-order design principles for how those rays would move, and eventually, say, converged at the focus of a lens. There's clearly something we did wrong there, and the focus of this second class is to fix this mistake we made. In particular, we had ray traces that looked like this, where all of the light came to a focus of infinitely small size. And that must have some problem, because if you did that, you'd create your own Big Bang, and though exciting, is generally a poor design principle. So we must have done something wrong, and the thing that we're going to learn in this class is that we need to think about more than single rays. We actually have to look at the ensemble of rays, and think about the fact that light has a shape. And the core of this course is really understanding the shape of light, as it moves through optical systems. And this will limit the ability to focus light to an infinitely small point, so it's going to give us resolution in things like imaging or lithography systems. And it's also very important in understanding how you get light off of, let's say, a lightbulb, in through a system, and how much light you can get, how bright it can be. It turns out, we weren't completely wrong, there was just a region of space where these solutions that we derived in the last class are incorrect. And we can understand that by remembering that we derived rays from an equation called the Eikonal, from Maxwell's equations, but we made one critical assumption as we went through that derivation. And that was that the amplitude of the light was slowly varying, the derivative of the amplitude of e with space was slow. And that's generally true in the middle of our optical systems, or outside of our optical systems. But at the edges where we're going from this kind of designs, from no intensity to lots, to a step function at this edges, clearly that assumption is violated. So the first thing to remember is that in the kind of derivations and ray traces that we did last time, the edges of all of them are a bit suspect. And of course all those edges all come together at that focus, where we noted momentarily ago that we had infinite energy density. That explains that our solution using rays is invalid at that spot, and so we need to fix that. And another example of that is, instead of having a little point radiator, an incoherent source like a little screaming electron on a hot body, what if instead we had something like a laser beam? This is a finite sized coherent object, and that's going to launch a set of rays, or a set of fields, that are not coming from an infinitesimally small point. And it turns out in that case, pretty much the entire solution that we have going through this rays can be, if not wrong, at least not perfectly correct and we're going to fix that. So this is a very common sort of thing when you are imaging laser beams, or optical fibers, or maybe we have small coherent sources, which is a very common and important optical problem. So we need to understand how to deal with systems now, where we think about the shape of the laser beam, or the shape of the optics, and deal with this problem that our edges in our previous solutions were incorrect.