Now, that we have an expression for how a single curved surface reflects light, we can relate the radii of curvature of that surface, and the two effective indices to a one's power, or focal length. Of course, we'd like to have lenses with two sides. So we'd like two surfaces. So now let's work out what you do, when you have two optical elements in the path. The first thing to do is to back up two or thin per axial approximation, and see what that tells us. So here we have our traditional imaging problem. But now let's imagine, that instead of a single ends, we've contacted two lenses of power V-1 and V-2, and they're thin, so they can be in intimate contact. Presumably, they should operate like some equivalent single lens of some equivalent focal length. The question is what is that focal length? So, once again, optical path length is the way to solve this problem. And remember, optical path length is just an integration along the ray path of the index refraction times the distance along the path. And the paraxial version of that expression is simply integration in Z. So in other words, if we know the optical path length of these two lenses, then the sum is what the effective ones would look like. We reasonably, add the optical path length of the two lenses. So that's a super simple calculation. We just add the two optical path lengths. If we see one over F-1, and one over F-2 here, of course, we can factor that out. This expression looks just like the original expression. So quite simply, one over the focal length of the combined lenses is one over F-1 plus one over F-2. And if we turn those all into powers by inverting them, and this is probably the primary motivation for using optical power, we see that powers add. Pause the video here for a moment, and think about the following question. If you're in the lab, and you have two 50 millimeter lenses and you put them back to back, what's the focal length then of the pair of lenses? Put them in mid contact. The answer to that, if you've thought about it, is 25 millimeters. Because one over 50, plus one over 50, must equal one over 25. So that's again, why you have to think in powers, powers are the things that had linearly, and that trick by the way is something you often do in the lab. You don't have all lenses you need, you can roughly paraxially, synthesize what you need, by stacking up other lenses of longer total focal length that always will shorten for positive lenses course. Okay. Now we're ready to do something they would say is real. If we have a lens, let's put it in vacuum to start out, to make life simple, so indexing one. It has a glass of a particular refractive index, and it's bound by two radii of curvature, R-1 and R-2. We will first simplify that to our equivalent thin lens system. We're not going to worry about these surfaces. We actually hit them at odd positions of the axis. So we'll make that an equivalent thin lens system. And so that's V-1 and V-2 whatever those are, index or reflection. And in middle separated by some distance D. So here's the expression we have from before. The power of that equivalent two lens system, if the lenses were in intimate contact D equals zero, is V-1 plus V-2. It turns out that the correction to that, and there's a finite distance is this term here, we don't quite have the tools yet to derive that in a convenient way. So please just accept it right now. We'll come back and fix that fairly quickly. So this is the expression for what's the power of a two lens system. This two lens system, when there's a finite distance D, between the two lenses. So if you'll believe that for a moment, then we know the power of, equivalent power thin lens surfaces for single pieces of glass, so we can substitute V-1 and V-2 throughout, and we get this expression here, which is called the Lens maker's equation, and it's actually the place you start when you were designing real lenses. If you want a particular power V, you had a particular index N, you see that you have two degrees of freedom here. Well, sorry three degrees of freedom, C-1, C-2, and D. That gives you the idea of where optics studio might come in, or more sophisticated design techniques to choose what those three variables would be. Because here you have one constraint on those three variants. This is how you'd start to build real lenses. If the lens is reasonably thin, D is smaller than both the radii of curvature. We can throw away this term, and we get back to a simpler expression. And now you see it's just the difference in the radii of curvature, and the difference in their refractive indices. If we didn't have vacuum here, all of the one's here would turn into the refractive index of the medium in which were a must. So that equation is really important, because it tells you how to build real lenses. And it's also got this nugget in it that you were left with degrees of freedom, radii of curvature, and distances. These are your degrees of freedom, as a designer, and this is now one constraint, that gives you a paraxial power on this singlet lens. Singlet lenses are so common that it's worth giving all the different possible shapes names. If both surfaces are convex, It's a Biconvex. If one of the surfaces has an infinite radius of curvature, it's flat. We call it a Plano-convex, or Plano-concave. If both surfaces are curved with the same sign, you see these are both positive radii of curvature, then the lens is called a meniscus. That's kind of like a physical object, like a meniscus water or film. And finally, correct the other end, if both surfaces are concave, and so Biconcave lens. Those are just words that's useful to have.