Before we get into using first order retracing techniques to design optical systems, we need to do a little housekeeping. First, we need a little bit of language. We'll be building the language of optical design continually. We need the first steps. And most importantly, we need a sign convention. And this is one of the things that distinguishes engineers from basic sciences like physics is we need to agree on detailed ways of keeping track of where everything is because of the complexity of the optical designs we'll be using. So first some language, conjugates is our word for the day. In optics, two conjugate planes mean that if you have an object at one of the planes, you will find an image at the next plane. In things like microscopes, there can be three, or four, or five conjugate planes, places you could find the image of the object throughout the system. So, if this is an object, this is an image. A finite conjugate system is one where both of the distances of the object lens and the image lens are finite which contrasts with infinite conjugate systems. In the case of your satellite dish, for example, which is essentially a lens, the satellite is effectively infinitely far away. If you're looking at the sun, it's effectively, infinitely far away. The image can be at finite distance. And of course this could also be an object and we could be sending radio waves or an optical signal out to the satellite. So an object here could produce an image infinitely far away. So a system is called infinite conjugate if one of the object or image distances is infinite. And then the special case where they're both infinite is called an afocal system, and you can kind of see why here. If we have light coming in from infinite distance, so the rays are parallel to the axis, and it comes to a focus, we can tell that because of the way it bends down towards the axis, then that's a focal system. An afocal system, on the other hand, is one where the array comes in parallel to the axis, that is from an object at infinity. It can go to all sorts of interesting things inside the system. But when it leaves the system again, it's once again going parallel to the axis so the image would be at infinity. Afocal systems have a lot of important properties, and so we give them their special name. So, that's the first bit of jargon, and now sign convention. Everybody groans about this and it, in some of its applications, can be slightly headache inducing. But it's super important. The basic reason for this is if you have a sign convention that tells you, you calculate, for example, where is the image. And if you get a positive number, perhaps the image is on one side of the lens. But if you get a negative number, it simply tells you that the image is on the other side of the lens. So it's really important to have a consistent, self-consistent set of sign conventions that we really slavishly apply. And sometimes it's going to be annoying. And if you've studied basic optics in a physics course or something, they didn't use a sign convention because you didn't get confused when you only had one lens of where anything was. But we're going to have a lot of lenses and so we can't afford to do that. So, the convention is, light always travels left to right. If you've ever worked with an optical designer, you start drawing things on the board. You start left and you get to the right. It's just the way we do things always. Unless you hit a mirror but it starts out from left to right. Perhaps most important, and from this you can kind of derive everything else, is that optic axis is basically a number line. So if you go to the right, if you're measuring something that's to the right of you, positive distances down the number line, then those are positive distances. If you go backwards, if you have to find something that's to the left of you, that's a negative distance. Again, just like a number line. In the coordinates up and down, you will have positive heights if they're above the axis and negative heights if the're below the axis. That one's pretty obvious. Now, we're going to specify the radii of curvature. We already saw this in the optics studio applications. And just like this distances convention, if the center of curvature of a surface is to the right of you, that would be a positive distance, then we call that a positive radius of curvature. If, on the other hand, the center of curvature is to the left of you, backwards on the number line, we'll call it a negative radius of curvature. Focal lengths are going to be positive if the lens is converging, and negative if the lens is diverging. And then, after a mirror, and this is the one that causes lots of pain, but it makes everything work very nicely. After a mirror, we're going to flip all these conventions. Because the mirror sends light backwards, now it's going right to left. And the way we'll deal with that is distances will flip their direction and the index of refraction will flip it's sign. It will actually have a negative index of refraction. We'll do some examples of this one. So that's the hard one, but the rest are basically just a number line. So let's look at this in pictures, it's a much easier way to understand it. So, here is that number line. If you have a coordinate system origin, it could just be the origin of your axis but it could be the surface of the lens, and you have to move to the right, you call it a positive distance. If what you're looking at, if the distance you're measuring is to the left, we'll give that a negative number. Well, let's say this is negative 12 centimeters, and this is positive 12 centimeters, just like a number line. Angles, we'll measure the same way. An angle that goes up, we'll call a positive angle. An angle that goes down, a negative angle. If we have an object type of field, the point as we saw before, then if it's above the axis, it's positive. And if it's below the axis, it's negative. Here's those radii of curvature that are just an application of the number line approach. If the center of the radius of curvature, this point here which is important, is to your right, then this is a positive radius of curvature. And if your center of curvature is to the left, this is a negative radius of curvature. And I will generally note the quantities, as I've done here, to remind you the sign convention. I will label this, -t on the drawings, just to remind you it's in that negative direction.