Now that we have an equation which relates the radii of curvature and the index of refraction of a lens to the paraxial power or focal length, we'd like the same thing for our other primary way of focusing light which are mirrors. So let's derive the equivalent of the lens makers' equation for mirrors. It turns out, there's a fairly easy way to do this. If we imagine that we put an object at the radius of curvature, such that rays launched from this object hit the mirror at normal incidence, or equivalently since rays represent the normal to the face fronts, we're launching a spherical face front, a spherical wave, and that spherical wave would perfectly match the curvature of the mirror once it got out of the mirror. And of course that that turns around the ray or the wave front, and heads right back to the original position, the origin of the coordinate system. Thus that particular condition is an imaging condition, and we just write our thin lens equation for that imaging condition. And then we'll know how to figure out what focal length is. The only tricky bit here of course is the sun convention for mirrors. So let's review that again. This is not something you're going to use too often, because it's often the case you simply unfold the system. But you will have to use this in optics studio, at least the negative distances, and so it's worth one more review. So first we have our mirror here, the radius of curvature is negative because the center of the radius of curvature is to the left of the vertex of the mirror. The infinite light is coming from an object here at the origin of the radius of curvature, the origin of the sphere. And because that point is to the left of the vertex, the object distance T is negative. And I've written therefore a minus T because I write positive quantities on my diagrams. Of course the refractive index of air, or vacuum, or whatever we're in, is index N. When we hit the mirror, the image shows up back here over to the left of the vertex, and therefore we have a minus T prime. That's just the standard number line coordinate system. But if we didn't do anything else, that minus T prime wouldn't work in our thin lens equation because our thin lens equation would normally want try to work for a lens for example and, in that case, T prime would be positive for a real image. And so the way we correct for that, to make a thin lens equations work, is the index of refraction after we hit the mirror is taken to be negative. In optics studio, this is done for you automatically, you don't have to worry about the index, you only have to put in the negative distance. And that will be obvious if you done it wrong because you'll have rays on the wrong side of the mirror. So let's take that condition, and now put this into our thin lens equation. N prime over T prime, minus N over T, equals 1 over F. But N prime is minus N, T prime is R, both negative quantities. And this N and T here is again, or again both negative quantities. And by the way you'll notice that this is a positive quantity, this is a positive quantity, N over minus R. And this is a positive point. So all the actual ratios here, if you include this minus sign, are all positive. And it is worth going through and convincing yourself that you've got all those signs correct. So given the sign convention now, we can put these two quantities together because they're equal. And we find that one over the focal length, which is the power of the mirror, is simply two N over minus R. Or the focal length is R over two. And that's pretty simple, it's pretty obvious, because what we have here is the one to one imaging condition, we have two F to the optic. Then another two F to the image. And that's our simple one to one imaging condition. So when you have an object at the center of the radius of curvature, it re-images upside down, negative magnification back to that same image. Remember that the radius of curvature in this coordinate system is negative. So the minus sign here tells us that overall, we have a positive focal length mirror, which is consistent with our sign convention that a ray coming in parallel to the axis is bent down towards the axis upon reflection. So you've seen this in the demo videos, but let's just quickly remind ourselves, if we sent rays into a lens with a positive radius of curvature, its center of curvature is to the right to the element. The rays would be deviated off the axis and we'd see what is clearly a negative element with a virtual image. Conversely, if we have a negative radius of curvature, because the radius of curvature is to the left of the element, then the rays that come in parallel are sent towards the axis. They must of course converge to the focal point. We know though that would be halfway between the mirror and its center of curvature because the focal length is the radius over two. But there's that important minus sign there, that this negative radius of curvature corresponds to the positive focal length. Let's put our elements together just as a practice. There's a particular optical element called the Mangin mirror. It's nothing more than a meniscus singlet, where the back side, the outer surface of the singlet has been mirrored. I've shown that with a big mirror here but, in reality, you just deposit some metal on the outside of the lens. It turns out that this is a convenient real optic, it's put into telescopes and it cancels a particular aberration colts or collaboration. We'll learn about that later. So let's just work through figuring out what the paraxial power of this system is. Now we could unfold this, remember that it's always a good place to start. And if we did, we'd have three lenses or three optical elements in intimate contact. We'd have the first meniscus, then a mirror that has power, and then of course we go backwards as the ray chunk here, through the same meniscus again. But if there are intimate contact, that's easy. That just means the power of this total element is the sum of the powers of the three elements. We can drop in what those powers are, here we have the curvature of that first surface, times index minus one. Then we don't accept the glass right? So we don't really even see the second radius of curvature from the mirror, because we stay inside the glass. Then we hit the mirror. So there is it's power, two C, if the index was one. And then we come back and hit the first surface again. So there's the second exit or exiting the surface. And work through this and make sure you understand all the sign conventions. Of course the first and last term look a lot the same. So what this appears to be is actually a surface here twice, there's the two, plus the effect of the mirror. So this is an example of how you apply what we've just learned to construct the equivalent paraxial power of more complex elements.