Up to now we've discussed all imaging systems, as containing lenses. But, it turns out you can also use curved mirrors as equivalent imaging systems or imaging components. And there's good reason to do this, as a matter of fact. Newton, and if you remember Newton's Optica course, there's the iconic image of the prism spreading the light out into the rainbow due to dispersion, the change of index of the glass with wavelength. Newton, when he explored this, then realized that lenses look very much like prisms, up to the top, if you will. And therefore, the inevitable change of refractive index with wavelength, with color was going to cause lenses to have different focal properties as a function of a wave length. And his conclusion was that lenses were therefore never going to be particularly good imaging systems, if color was important to you, and he thus turned to mirrors to solve this problem. It turns out, by the way, he was wrong, one of the very few times he was, and we'll come back to that later in the course. So, it turns out, if you remember, there are two versions of Snell's Law. One, has to do with refraction across a boundary between two refractive indices that are different. But, the other version of Snell's Law has to do with reflection off of the mirror. And in that case the two angles incident and reflected are equal and they're equal independent of wavelength, there's no refractive index to change the color here, change trees, change the direction of the outgoing rate. So, Newton concluded that one should be able to do imaging with mirrors that didn't depend on color, and he invented, coincidentally, the Newtonian telescope, an all mirror based telescope, for exactly that purpose. So, we will get later in the course into understanding exactly how the radius of curvature of a mirror relates to its focal length. But, at the moment, we're just doing this simplified design with graphical symbols on paper. We can simply draw a new simple little arc. Everything works from planes. So, I actually put a little line at the base of the arc and all of the power of the mirror, all of the ray bending, will happen at this point, because we're in the simplification now, where all optics are infinitely thin you'll remember. But, we can take this imaging system that we've been looking at, with quite a lot of detail now, either a one-to-one imaging system with magnification minus one, and T prime equals T, or a nought unity magnification system, perhaps with a larger object than image distance, so, we have a minification, a reduction of the image size, and we can turn that into a mirror system, and the only difference is the light turns around at the mirror. Lenses transmit the light, you keep on going to the right. Mirrors reflect the light, you turn around and go left. So, at a simple level, mirror systems are no different than anything we've been doing before, and the first way to design and analyze any system, that you might want to have a mirror in it, is to unfold the system to take the mirror system, and flip the whole system around the mirror, replace the mirror with an equivalent focal length lens, and in our praxial design environment, the optical function of that system is identical, and that's a good way to do design. Now, when you come back to actually build the system, you're going to worry about running into yourself, maybe, there's components here that only come on when you come back, when you were flat, you hit or you run into. But, we're not worried about that in the initial design stages. We're worried initially about just getting the optical function. So, in the case of real images and objects this is a fairly straightforward operation, and it turns out it is with virtual objects or virtual image as well. So, if we have a object that is inside the focal length of our positive focal length mirror system, then it will form a virtual image and you might be a little confused of where the heck that image would appear to be. You could do that one way by tracing rays, following our normal rules, it's just that the rays instead of coming out the back of the system turn around and go backwards. They flip their direction to mirror or you could unfold the system into the equivalent lens system, this is one we already know how to deal with, and we can do everything in this system and if you'd like, go back to the mirror to check your intuition, and that works for positive mirrors with a object placed inside the focal length, or negative focal length mirrors with an object placed outside the focal length, that works just the way we did before. So, this is the way to start. Equivalently, if you have any flat mirrors in your system, we often do this to fold the system up to make it fit in the box, basically. Then, those have absolutely no optical function at all, they're there simply to turn the whole system around and send it off in a new direction. This is exactly the same technique you'd use there. You'd first design the optical system, lay it out on a straight line and then like in that camera we saw at the beginning of this course series, you'd fold the system up to get it to fit in the camera or the box. That's the first way you approach the system. But then, you might want to actually be modeling a system, be thinking about a system, in which the light does go backwards and our sign convention can help us do that. So, let's remind ourselves of that last rule in the sign convention, and what it said, which seems a bit confusing, is if you're going to go left, that is you hit a mirror, because you are going left to right, that's our convention, we started out from left to right. But, at some point, if you're going in the negative direction, because we've hit a mirror, we take all the distance conventions, and all of our indices of refraction, and we flip their sign. Now, we have the techniques to see how that works and why. So, here's our lens system that we've been looking at over and over, and here's, Gauss's version of the thin lens equation. What we want to do now is replace the thin lens with a thin mirror and have the same equation apply. We want to use all the same sign conventions, all the same equations that describe where objects and images land held. What's the hype of the object? Etc. But, now we're going backwards. Well, it turns out if you think about this, first, notice that here, has drawn T prime as a positive quantity, and by our convention, T prime has to be a negative quantity in this case because the image is showing up in the coordinate system of the mirror to the left, and so T prime has to be negative. That's what our first, or second, or third, I can't remember which one it is, but that's what our sign convention tells us. So, how can we make Gauss's law still work? Well, notice T prime always appears, as I mentioned earlier, with an index and prime. So, if we flipped the sign of the index, let's say, this is vacuum, then this would be minus one, this was air and this could be minus 1.0003. Again, we write number of zeros there. If this happened to be glass, this might be about minus 1.5. If we just go ahead and flip the sign of the index, we notice that the equation is consistent and we have to do that, because by our sign convention, T prime itself is now a negative quantity, and we still want this equation to work. So, in optics studio or any other ray trace program, if you put a mirror in the system and you leave T prime positive, you will find that the ray tracer puts light out here to the right of the mirror. It doesn't know what to do when you told it, keep on going, keep tracing forward, and that's sort of a nonsense system now, because you told it, I have a mirror here. To get the system to operate correctly, to program to operate correctly, you have to give it a negative distance, because that's our convention. If you want to go backwards, you use minus signs. You won't have to worry about negative refractive indices, it will know about that. When it hits a mirror, it will go ahead and do all that for you, but you have to tell it what direction to go. And in this case, that means you have to give it a minus sign. So, the point is we now can trace systems that consist of positive and negative mirrored, negative focal length lenses, positive and negative focal length mirrors with real and virtual objects and images, and we have a set of equations, which we just describe the throw, where the objects are, how big they are, and angular magnification as well. All of those things work, and they're all consistent, given the sign convention and the way we've described how to use it.