Here's a quick little calculation that tells us what field curvature we have for a single lens. And this is an important concept, so rather than just state the result, I'm going to do a derivation here because this is one of the things you need to know about field curvature. We'd like to find some way of estimating how this surface here, how it's curved. And it's clearly a non-paraxial effect. That's, of course, how we got here. Maybe we could notice the following. If I have an object point here that's off the axis, sufficently far that I really am no longer praxial, maybe I could think about this central array here from the object to the image as a new object, a new optical axis. I just happen to go through a lens which is tilted. But maybe what I can point out is look, the object distance along this line is bigger than the actual object distance by 1 over cosine theta. So maybe the way to think about, this is true. What field curvature is, is I know as I make my object distance larger in this typical positive lens imaging condition, my image distance must get smaller. That's something we learned in course 1. So a, that's the way to think about this, that's good logic. And b, maybe we could actually solve for, let's say, what's the radius of curvature of this surface. So let's do that because it works. So I'm going to use quantities with a tilda over them to be these sort of, these tilted quantities. The object distance along this dark grey here, and the image distance along the dark grey. And then the actual object in image distances T0 and T1 will be all along the optic axis as they should be. So i'm going to write down my Gaussian thin lens equation for paraxial optic even though i'm using it non-paraxially but it will give us the right answer. And I am going to assume that I can do this trick, that I can assume that the object distance along the dark ray corresponds to an image distance along the dark ray even though they are tilted. Well okay then, let's just put it in. The object distance along this hypotenuse here is going to be the actual on axis object distance over cosine theta. Then I'd like to get rid of object distance, so I'm going to substitute in the regular Gaussian thin lens equation here, that's going to give me another 1 over f and t1. And so now I've got an expression for this distance along this hypotenuse, T1 tilde in terms of focal length, the angle, and T1, the distance just along the optic access. because I used the regular thing lines equation to get rid of T0. Let's first solve for this axial error. This longitudinal aberration where the focus is relative to the praxial focus point. That would be by definition, the difference between T1 lens to praxial image point along the axis and T1 tilda O times cosine theta. That would be this distance, this dark line right here. That would define delta, this error. So I can take this expression right here, which is kind of icky, and I can shove it in right there. And I get something with a whole lot of cosine thetas in it. And I don't want to deal with that, so I'll then just do a power series expansion of that and keep the lowest ordered term, which turns out to be quadratic. Okay, now I have an expression for what this deviation is between the praxial image plane and the best focus plane in terms of theta. And what I would really like is, I would like to get an expression for what's this radius of curvature because I think this is got a quadratic shape and that I could near the axis color cluster. So do that, let me do a little more trig. Lock me right down the equation for circle in terms of this coordinate and this coordinate, the x and y coordinates of that circle. And if I do that, that's this expression right here to lowest order. And now I'll set the two equal. What I've just derived for this ray error and what the longitudinal ray error would be in terms of the image heighth and this Petzval surface radius of curvature. And notice that t1 theta in the paraxial sense is h prime. That's the kind of math we're used to doing. So the neat thing that pops out is near the axis, which is the only place that this quadratic surface looks like a sphere, we find that the radius of curvature of the Petzval focus is equal to the focal length of the lens. And that term is correct generally. And it means something important. It means that if you have a positive lens imaging system, you have to have a Petzval curvature. And how much curvature you have is about equal to the focal length. So if you're doing very compact imaging, very short throw for your system, then you're also unavoidably going to have quick take curvature on that surface. The other thing this means is, and this comes from side elaboration coefficients. Is if you'd like to flatten out your field, you're going to have to add elements into the system that have negative focal length somewhere because this is going to be true for every element in the system roughly. So only positive elements means you have this curvature. But some negative elements in there can be used to cancel it out and we'll actually look at some of those later.
