In previous lectures, we learned how to relate curved pieces of glass and their reflective indices or curved mirrors to an equivalent thin paraxial element that of course just has an optical focal length or power. We learned how to cascade up multiple such elements through the lens maker's equation. Those are your first connection between thin optics and real optics that of course have thickness. Now we are going to make the sort of formal connection between thin and thick optics. This is very very important because it is literally how you will take a design that's done in the first order approximation and the first order means paraxial because cyme data its first order approximation is data and that is the paraxial approximation. So will take and do designs in thin paraxial systems. But then we need to translate them to real optics. You might do with thin lens design some night in the lab and come up with, you need a 25 millimeter Focalin glance. You then look in your lens catalog put a real lens and you have to thicken your design. You have to put the real lens in and that will often involve for the cases most positive ones is the pushing some space open on the optic axis and inserting this now thick lens replacing a thin lens design. But just as importantly the same concepts can be used backwards. And this is a concept it often takes students a lot of to get. So try to think about. We could take a thick lens perhaps when we looked up in a catalog or one we have in our drawer and ask how can I retrace this with my thin paraxial lens techniques? Perhaps graphical retracing. So it turns out we are about to learn here which is called Gaussian optics can be used to take either a thin lens and thicken your design, make the lenses real. Or take real thick lenses and apply thin lens retracing to them. You can go either way and both concepts are very very important. So let us dive into the simple concept. It is relatively simple, and it's all in this slide. So here is some real landside drawing by convex symmetric cones here but this would apply to any of our elements and we are going to trace two rays to understand the equivalent thin lens properties of this real thick lens. We are still paraxial here. So we are not worrying about aberrations, we are not worrying about when we hit the surface. It is not a quite the same Z distance as the axis because we are still thin. So we bring a ray in, an axial ray, one that is parallel to the axis. We'll imagine that it refracts at the front surface and bends down. It refracts again at the second surface and bends down on even steeper angle and comes to a focus by hitting the axis. So the first thing we are going to do is, we're going to put a little circle around that spot and say, that is the rear or back focal distance. And in our lab demos we talk about discovering the focal length of a lens by essentially doing exactly this experiment and critically you don't actually know where to measure the focal length capital F from, you know that this is the focal point but where is the lens? If the lens is thick, it is not really quite too clear where you measure back to. So when you do this lab experiment, when you shine light into a lens with a collimated source and figure out where the back focal point is, all you really know is what is called the back focal distance, which is the distance from the apex of the glass to the focal length. That generally does not equal the focal length because we don't know where to measure back to, where it is the lens precisely? The way we do that graphically or the geometric optics is we take this ray that we traced and we project it backwards until it intersects the path of the initial incoming ray. The plane and it will be a plane and the paraxial approximation, where these two rays intersect we call the rear or back principle plane. And the reason is, is it looks like the ray refracted right here. We could ignore the glass. It seems like, and just refract the ray once as if this was a single thin lens of focal length the capital F that refracted at this principle plane P plane. And that is exactly what we are going to use the concept for. Now you might be wondering does it work the same backwards? And the answer is sort of. So to discover that, we are going to trace the same ray backwards or the same idea we are going to come in with an axial ray that is parallel to the axis and we are going to trace backwards here. Once again it refracted the first surface and then down, it refracted the second surface bend down even more, in the case of this biconvex lens and we label the point where we hit the axis as the front focal point, let us put a little circle on that in our diagram. Again we don't know exactly that that tells us the focal length of the lens. It just tells us the front focal distance, the distance from the apex of the lens to the focal point. And that in general does not equal the effective focal length or the power of the lens. And because we don't know where to measure back to, from this focal point because the lens is thick. But if we project the rays backwards to their intersection then we find the front principle plane of the lens. So you can see here that when we are going forward or backward, we have to treat the lens slightly differently. What we do find is that the effective focal length does not care about which direction you go. That is important but it is not completely obvious in the paraxial limit if you go forward or backwards through a lens, the effective focal length positive or negative, whatever it is will be the same. But in general P and P prime will be in different places.