Now we're going to sort of teach a set of tricks, sort of an approach to design that you might sort of, when you're first building a system, you might think, well, how might I build it in order to just immediately move towards higher performance? And a symmetric system, if you can do it, and you can't always, depending upon your geometry, is a really good thing. A little side note here, the way you more formally approach understanding how individual end surfaces contribute to aberrations, I mentioned this before, called the Seidel aberration coefficient, see these capitals S's on these slides. They are a systematic approach, a bit tedious in algebra, that tell you that each surface contributes a certain amount to each aberration. I never calculate them by hand because the algebra is too tedious. They do exist, there's a table that it can be degenerated by codes like OpticStudio. And you should go look at that once you start doing system design because it will tell you, man, I'm dominated by astigmatism in this system, and this particular lens is the one that's dominating, it's contributing most of the astigmatism. It's very useful in trying to understand why a particular system has particular aberrations, and it guides discussions like the one we're having now on what kind of symmetry conditions or what kind of surfaces would help. So I'm going to give you, on these slides, the Seidel coefficients, just so that you know what they are, and if you dig a little deeper, you got that more deep information out here. I'm not actually going to actually talk about them because we're not educing them in this course, but there are on the slides for your reference. So let's say you want two image from A to B, a very common object image plane, you've tried a single lens, a singlet, let's say, and it just won't do the job. So now it's time to add some glass and you're going to be use two pieces of glass. So you could design a contacted doublet, like the achromatic doublet we discussed, or more like the Huygens eye piece. You could think, well, maybe I have two separate piece of glass, what should they be, where should they go? Turns out symmetry is a good thing, if you can. And we can prove through those Seidel coefficients that a system that's prolate symmetric between object and image plane has no COMA or distortion, and that's pretty darn handy. So to show you that, I've walked through a system here, where I dropped two symmetric singlets into the system, but I didn't drop them in symmetrically. Notice, here is my aperture stop, you can tell it's the aperture stop because all the ray bundles go right through there. And I dropped the two lenses in just sort of randomly relative to the stop. I see, On axis, you can see this is the object that is at zero millimeters, I see I've got little bit of spherical that looks like, and as I get off axis, I get all sort of stuff. So I've got field curvature and astigmatism and COMA, and I recognize those, for example, the effect that I've got, this looks positively curved and this is not, that's COMA. The fact that this slope is different than that, that's astigmatism. So the fact that they're both, I've got what looks like defocus on both x and y that I don't see over here at zero field point, that's field curvature. So I can just look at those curves and immediately say, I can see I got those four aberrations. You want to be able to do that too. So in other words, lots and lots of aberrations. And if you look at the scale here, this is the number here, the scale bars here are 2 millimeters. So these are really poorly focused spots, that's bad optics. So let's make it symmetrical, all I did is I shifted the two lenses and the stop such that I guess I just shifted the stop. So I left the two lenses at the same place, then I just pushed the stop over so that it's halfway between the lenses. And now I get pretty significant improvement, the scale is still the same here, but my aberrations have halved. And the reason is the COMA, which was one of the dominating terms here, went away. How can I tell? See this curvature here, that's coma because it's in the ey, the error ray aberration for y, but not in x. And now these two look the same, so I've lost the COMA. So I dropped all my COMA just by shifting that stop into that symmetrical position. And then, and we'll discuss this just in a minute, I went ahead and put field curve flatteners in here, that's going to get rid of this significant field curvature, this big slope in the onathic axis point. And I optimized these lenses to be best formed, which is almost that plane or convex case. Notice that I've got the curve side towards the collimated beam. And now my scale went to 200 microns, and all I've got left is spherical. Notice that the aberration looks the same on axis as off axis, which is what the only aberration that would do that is spherical, little bit of astigmatism over here that these curves aren't identical. So I've made an ordered magnitude improvement in my aberrations. Now I pulled several tricks in a row here, we're about to discuss field curvatures, but this is the kind of sequence you'd like to go through. So symmetry is your friend, gets rid of COMA and distortion.