In our previous lectures, we introduced first the simple thin lens equations. Which relate object and image distances to a single lens. We then graduated to the y-u per axial ray tracing, in which, in a tabular format, we pushed a ray consisting of a height and angle descriptor through a system. And most recently, we introduced an a, b, c, and d matrices. Which takes that tabular format, the system of linear equations, and captures them as two-by-two matrices. The real advantage of that last approach is we can now cascade an entire system description up. Multiply out those matrices either numerically or symbolically. And end up with descriptions of entire optical systems. That's a very powerful approach for first order design. And so here we're going to explore how we can use system descriptions given by these matrices to put constraints on a system. And it turns out the conjugate matrix N, that we defined earlier, is the way to do that. And each of the four terms in that matrix have very important properties. So we're going to go through those one at a time. So I've written out here the matrix equation for N. And remember, what it does is it relates the object given by its object height and the angle coming off of the object to the image given by the image height and the angle coming in to the image. So what we're going to do is look at each of the four terms of N. And set them = 0 one at a time. And we'll discover that enforces a constraint or a condition on the overall conjugate condition on the system that we're looking at. So let's look at the first one. If the term N12 = 0, then just looking at the top equation here, we see that the image height y depends only on the object height, y0. And it has no dependence on the object angle. So pictorially, that's the situation that's shown here. We launch a bunch of rays from the object with arbitrary angles. And they all come back to the same image height, independent of the angle. And that, of course, is our imaging condition. So if we calculate or someone gives us a conjugate matrix N, all we have to do to enforce the system to be in focus is set this term = 0. And that's an example of how we'll use this for design. Next let's look at setting N22 = 0. And again, by the same logic, just looking at the bottom equation, we find that the output angle, the angle at the K+1, or image plane, now depends only on object height and not on the object angle. So pictorially, we launch a set of rays from a fixed height, y0, and a set of ray angles, u0-prime. And they all seem to come out the back of the system at the same angle. Well, there's only one place that can occur, or one way that can occur. If the object, or surface 0, is at the front focal point of the system. So again, if you are given an N matrix, or if you calculate this conjugate matrix, and you want to find the front focal plane, all you have to do is set N22= 0. Similarly, N11, and you might guess by symmetry what's going to happen here. If we set that term = 0 and look at the top equation, we find that the image height depends only on object angle. But has no dependence on object height. So we can come over and draw that picture. We see here a bunch of ray heights drawn in blue. But the same ray angle coming off the object, u0-prime. And they all come to the same point in the K+1, or the image plane. And therefore, that plane is the back focal plane of the system. So that's three important things. We can find the focal plane, we can find the back focal plane. And we can enforce an imaging addition, or bring the system into focus. Finally, what is the last term, N21, if that one = 0. Well, now what we see is the ray angles coming out of the system depend only on the ray angles coming into the system. And don't depend on the object height at all. So pictorially, and I've drawn it in two different colors here for two different ray angles, we have a bunch of object height, y0. We don't depend on that. But if we have a single object angle, u0-prime, then we come out at a fixed angle in the image plane uK+1. That system doesn't focus at all. Rays come in parallel, and they come out parallel. That's what that bottom equation says. And we have a name for that, that's the afocal condition. So that's actually a system that sometimes you'll want for transforming objects in certain ways. So we've learned four things about our system now. And when we're going to use this in design, we can do a symbolic trace through an optical system. And then by setting these various terms = 0, we can constrain that design. And enforce, for example, that the system be in focus.