In the previous modules we have worked through how the finite cone of light that gets off of the object relates to, for example, the point spread function, how big is the focused spot, the modulation transfer function, how do spatial frequencies get through the system? And those concepts were all done a little bit vaguely in terms of how they related to the optical system itself. This pupil concept, for example, it's the right formal wording but where was this pupil? So, in this module, we're now going to go through with our first order design principles, and figure out where these pupils are and more precisely what they mean. And that's going to take the material we've just developed, and now really put it into the optical system. So formally we're now talking about finite aperture optics. This is the new thing we're putting in. Our paraxial lenses up to now really have been infinite in extent. And we haven't worried too much about the fact that they had edges. Or maybe there were intentional apertures and stops put within a system to define how big the cone of rays or how large the object physically could be that could get into an optical system. So now we're going to put all that in. So we're still in paraxial first order world. But now we're going to put in the fact that optics are finite. And we've seen that that's important because that's what determines the point spread function and the modulation transfer function. And it turns out this is pretty simple. We have all the tools, and we now just have to apply them. So when you meet an optical system that you've never been introduced to before, it turns out there's a very mechanical set of things you want to do to understand these finite apertures. And, principally, it involves tracing two particular rays. And, these rays are very equivalent to the two rays we use to understand Gaussian beams, which was one of the other reasons I like that two ray mechanism for Gaussian beams. So let's figure out what those are. The first one is the marginal ray. We're going to start at the object. And you can tell this is my object right here because I have in light gray, my little single headed arrow. And I'm going to start a ray off of the axis. So this would be like my divergence ray from the Gaussian beam language. So it's going to start at the intersection of the object. It depends on where the object is then. And I think of this as just tilting this ray up. I always hear a little mechanical sound in my head [SOUND] as I tilt this ray up. And I would tilt this ray up. And as I'm doing that, I'm tracing all the way through the optical system. And we've done a simple one here. But if I had 38 lenses and all sorts of stops distributed through this system, I would trace this ray initially right down the axis. So it obviously would get through the system. And that I would keep increasing it until I hit the first stop anywhere in the system that limited that ray. And that is therefor the largest possible angle. I'll call it alpha here. That I can get off of the center of the object, and to my system. That ray is really important. It defines, just like the divergence ray for the a Gaussian beam, the numerical aperture of the system. And we've seen from the previous module, a numerical aperture is a hugely important concept, and quantity about an optical system. The aperture or the stop that I hit, is called the aperture stop. And the numerical aperture is just a measure of the aperture stop, via the sign of the angle that I get through it. In this particular case, the aperture stop is here in object space. That is there is nothing between the object and the aperture stop. But that doesn't always have to be the case. Notice there can be another stop, or diaphragm, or hole somewhere in this system. And this one isn't the aperture stop. And so one thing that's important here is that in our sort of formal design process there can be only one. There is an aperture stop. And if I thought about I could make this hole smaller until I sort of just touched this ray I have to invoke engineering here and say they can't both be perfectly the size necessary to just intersect that ray. One of them's gotta be just slightly smaller. And so formally we say, there is only one aperture stop. because we want to know where that aperture stop is. It turns out it's very important. So tilt this ray up. That's called the marginal ray because it goes through the margin of the system. It's kind of the edge. And the angle that it makes with the optical axis defines the numerical aperture of the system. And that would tell us what's the point spread function that we can see here in object space. We might have a different numerical aperture over here in image space. Of course the ratio is the angular magnification of the system. So as always numerical aperture is a property of cones of light. And so when we say numerical aperture, we have to say are we talking about, say, the object or the image. So that's one of the concepts that's related to aperture stop. The other are pupils. And pupils are simply the view that you'd get through binoculars that we talked about before. In other words, if I'm here in object space and I look at my system I see this particular aperture stop, because this aperture stop is in the opposite object space. So this would be my entrance pupil, just like we were talking about mathematically in the previous module, that would determine the range of angles, and therefore spatial frequencies that can get from my object. But if I'm over here at the image space, and I look back at the aperture, I don't see the aperture directly because there's a lens in the way. So if I turned my binoculars around, and I put my eyeball over here and I look backwards, what I would see would be an image of the aperture stop. And it would be an image caused by whatever set of lenses happened to be between the aperture stop and my image point where I'm looking from. So to find the exit pupil what I would see if I looked backwards into my binoculars, I have to take the marginal ray and project backwards to find where's the image of the aperture stop in image space. And I know how to do that, for example. This is now just another graphical raytracing problem or raytracing problem in general. I need to shoot at least two rays from, let's say, the tip of my aperture stop right here, out into image space and see where they intersect. Well, I already have one of them, because I've already traced the marginal array, so that's easy. And the really obvious other one to use would be our favorite, the ray that goes from this thing where I want to find the image of, the edge of the aperture stop, through the center of a lens. I find in this case that those two rays are diverging in image space. And that tells me I need to project backwards. And the exit pupil happens to be over here in object space. That's okay. So to reiterate, if I put my eyeball here and looked into this system and asked where do I see the edge of a piece of metal, that limits my ability to get light into the system. I, in this case, would see the aperture stop directly. And so I would also call that the entrance pupil. If I turned it around and I look from this end with my eyeball, and I tried to look back and ask, what limits the cone of rays and therefore the numerical aperture in image space, I find here that I don't directly see the aperture stop. Instead, I see an image of the aperture stop that because this particular optical layout seems to be back here a little bit further behind the aperture stop and a bit larger. But that's what I would see, and so I call it the exit pupil. The entrance and exit pupils are the ways that light gets into the system. And you can see here it appears that this determines, this exit pupil determines how light gets out of the system and to the image. And the distance to those pupils, and the size of those pupils is what determines the numerical aperture in object space and image space.