Let's use these concepts to show how simply you can constrain the design of a very typical imaging system. I called this a projector and that could be a slide projector, it could be the conference room projector that you might have seen hanging on a ceiling in a conference room, it could be a lithographic projector sending an image off of the mask down onto a chip. The general idea if we want to start with a lamp, we would gather some light off of the lamp within an imaging system or an optical system, typically called a condenser, so that we illuminate an object that could be the chip in your conference room projector that then is imaged somewhere else, we'll call that the screen. The question is, how do we begin to design that problem and how do we use these radiometric concepts to immediately constrain and put some bounds on the problem? I've drawn our traditional two rays here and I've used the red and blue colors that we typically use. But there's an important little distinction here is that they actually flip their character going through the system and this is super common, microscopes use this same idea. So here's my lamp. Let's say and I will tilt up a paraxial marginal ray up off of that lamp, and I'll be limited by my numerical aperture that I captured from the lamp by the condenser. Of course whatever aperture that I stopped, that i having this condensing system, will appear as an entrance pupil, perhaps just the edge of the condenser lens itself. So my chief ray for the condensing system would be centered in the entrance pupil of the condenser. So I've drawn that, and that chief ray is going to determine the edge of my field at the lamp. The thing that's odd is I typically will put my, I've called it slide here, but there was a slide projector or this would be the chip of your conference room projector. This is my object. Note here, I have switched the two rays. I have my chief ray for the condenser at the middle of my object. The reason for that is, is if I did it the other way around, if I imaged the lamp to the object then I would see all of the humps and bumps and filaments or whatever shape I had in my lamp in my object and eventually I'd see that projected out of the screen and that's not what I would like. So instead I flipped the nature of these two rays. Now, every point on the object is illuminated by every point on the lamp. So really what's sprayed out here across the object is the angular spectrum of the lamp. You can see the blue ray here. That's the angular content of the lamp and that is what is spread out in space across the object. So that's a common way to get a bumpy lamp like a wire filament to look very uniform here at my object. Then, my object now the red ray is my marginal ray. I tilt it up until I hit some aperture stop in my system, perhaps the aperture of the projection lens that can be viewed as an entrance pupil here. Notice my little dotted lines as I'd find the image of my lamp in the entrance pupil of my projection lens and the blue ray now for the projection lens is my chief ray and so it determines the edge of field for my object. So, they're the same two rays it's just they flipped their character going through. All right. So how do we design this system? It turns out that radiometry is fundamentally terribly simple. You probably have a specification that you need a certain amount of power on the screen that could be because it's a camera and it needs to get a certain amount of photons, it's a photosensitive material it needs a certain amount of power or you're sitting in a conference room in the eyeballs that are sitting out in that conference room need a certain amount of power. It's got a certain area A, and so you divide those two and that gives you the irradiance that you need on that screen and it makes sense that we're talking about irradiance on the screen because that's watts per unit area. Well, now we can jump all the way back to the fact that the best we can do, and of course this is probably not actually the case, you probably need to put efficiency factors in, but the best we can do is, the total power we got on the screen was exactly the total power we managed to accept from the lamp. Once we know that, we can say well, the lamp has a certain radiance spec. I can look that up or know that from the lamp and that's going to give me a Lagrange invariant H that I have to have for my condenser and the rest of my system in order to get the total power from the lamp to the screen. As soon as I have Lagrange invariant, I can pick anywhere in that system I'd like maybe the object and say well, if it's got a certain size H and that's going to give me a numerical aperture and I'm done. That then defines the Lagrange invariant through the whole system. Now, the height or numerical aperture at every point on the system that product had better be at least that if I'm going to have Lagrange invariant big enough to carry this power all the way to the screen. There's some discussion here of which those would you like to make slightly larger. In general, let's say the conference room projector. You're going to find that, that chip whether it's a liquid crystal or Deformable Mirror Device, DMD, that's going to be your limiting aperture. It's going to have a certain size and you can't make that bigger because that's the chip and it's going to have a certain acceptance angle. Again, you can't make that bigger, that's going to be on the physics of whatever limits, whatever how about chip works. So that's going to determine a Lagrange invariant right there. That tells you that your condenser had better have that Lagrange invariant, maybe a little bit more, but certainly not less because if your condenser had a lower Lagrange invariant, then this limiting aperture, you're imaging chip or your display chip, then you'd not be optimally gathering light up off the lamp. On the other hand, if you made your condenser have a much bigger Lagrange invariant than that of your limiting object here, your display chip, you would just either be putting light outside the display chip in area or bigger in angle than the chip could work on and you'd be throwing it away. You'd be wasting it. There's no reason to make the condenser Lagrange invariant significantly bigger than your display chip. The projection lens probably needs to have a Lagrange invariant a little bit bigger than the limiting aperture, let's say you're imaging chip. If it's smaller, you're going to vignette or you're going to throw away power and that would be silly. So that's a very typical thing, your theaters that use digital chips now, that's the calculation all the way through. It's you have this enormous bulb back in that projection chamber, you have a very expensive projection lens but the thing that determines how the whole system works. The total optical power you can get onto the screen is really determined by the Lagrange invariant it which is set by the smallest Lagrange invariant in your system, the display chip and then everything else in the system you simply have to design to have at least that. The point is, radiometry isn't super hard if you use these concepts of course the details are more difficult, but fundamentally, the question we asked at the beginning of this module, how do I understand, how much light I can get off of that light bulb and out of an optical system? This is the math for it. It's really just understanding these units that we've defined carefully and the super important concept of conservation of radiance and how it relates to the Lagrange invariant.