All right. Now, let's talk what you can do to the optics themselves. In this course, a number of times, I have mentioned that a particular equation is a constraint. Now, I'll show what I really mean by that. Let's say we're designing a singlet lens. We have the first radius of curvature, the thickness, the second radius of curvature and the refractive index. That's all we have for variables assuming object and image distances are fixed, somehow I've got a defined imaging problem. I don't have a lot of range on refractive index, we'll look at that in a minute, and often that I can't get a lot out of it. I can get something but there's not enormous range of refractive index. It turns out thickness is also not a very strong variable. I have to keep the lens kind of a reasonable shape and size, and I'll find that the performance the lens doesn't depend too dramatically on thickness. It's important but not dramatically. So, really the two dominant degrees of freedom I have as a designer are the first and second curvatures. So, if I have a constraint on the focal length, then I have given that because I've particularly imaging problem to solve, then the lens makers equation which has the curvatures, the refractive index, and the thickness in it takes my four variables and now I have three. I have to work in a three-variable subspace because the relationship between those variables has to be fixed to give me the focal length I want. It turns out that since thickness and index refraction aren't too powerful, I really have two significant design variables I can use, and the focal length constraint through the lens maker's equation means I'm left with one free variable. Well, I want to choose, I want to spend that variable wisely to minimize my aberrations. That's called bending the lens. As I show here, I could make a set of singlets where I'm changing the front curvature, keeping the thickness and the refractive index the same, solving for the back curvature to fix the focal length. If I do that and plot the spherical aberration and coma, I find that those aberrations depend strongly on the overall shape of that lens. So, I want to choose the overall shape of the lens, I want to bend the lens keeping its focal length the same to minimize my aberrations. When I do that, it's called a best form lens. It turns out for the case of infinite conjugate ratio, I'm coming in from infinity at focusing, the best form lens is somewhere around here. Notice that coma and spherical conveniently are minimum at about the same point, but that front surface curvature isn't too far from flight. Now, if aberrations really matter to you, you go look up best form lenses and you can buy that. If you're not at infinite conjugate ratio, this point shifts and now you really might want to invest form lens, but you're actually not too far off to be using a lens that's plano-convex, which way it goes matters. So, it turns out the way this is designed, I'll llustrate over here, that your aberrations are minimum when the curved surface is facing the straight raise so that you first bend the light at the current surface, and then bend it again because now the razor going at an angle at that second service. If you take that simple same lens and flip it around, the aberrations will be terrible. So, lenses have fronts and backs and this little diagram right here should be something you remember, that if you have a plano-convex lens, you want to put the curved surface towards the not curved wavefront, and that's pretty close to best form. Over here, I've just gone to explore that in OpticStudio to make sure it works. I did exactly the same calculation here. I fixed 100 millimeter focal length BK7 lens, working at infinite conjugate. I picked a particular diameter, then I just changed the front surface curvature keeping the back surface such that I got the focal length that I want. I worked only on axis. So therefore, I only could have transverse spherical aberration and simply measured the largest ray angle or radiation, the how far off the axis this most extreme ray landed, the particular way I measure it had the opposite sign convention is the structure here. But notice, I get exactly the same shape and it predicts a little less than 0.02 front surface curvature is the place to be, which is exactly the same thing I got here. So, bending the lens, this is how you optimize in something like OpticStudio. But here, we see that it's not just turning the optimizer loose on all possible degrees of freedom that we realize that once we fix the focal length, we've spent one degree of freedom, and we only have a certain number of curvatures, thicknesses, indices. This is why you end up adding elements to a lens is if I define that I want the focal length fixed and some other condition; I want a particular field curvature, I want to minimize aberrations in some way, I may spend my second radius of curvature, and then I'm done, I have no more strong knobs to turn the performance of this lens, I may have to add another piece of glass.