Most of what we've been talking about up to now in terms of the impact of finite apertures in optics, use resolution and diffraction limit. But we went through this, even clearer back when we were talking about Gaussian beams has been the other important concept that these finite apertures bring in. And now we're going to deal with that formally. We referred to this in the postdocs sad tale as well and it turns out to be very important not in terms of the resolution of a system though it talks about that too, but it often comes in when you're worried about how much light can a system transmit. Can I illuminate an object with enough power? And if you're going to call yourself an optical engineer, you'll have to know this quantity. It's really important and it's easy to miss and to not pay attention to. So this one slide is a pretty darn important part of this whole course. It's got a lot of names, optical invariant, or lagrange invariant, let's go through and derive it and see what it means. So I've drawn a simple double and telecentric imaging system here, with the marginal and chief rays. I want to think about just one of these lenses. This rule holds in general, but if we just look at one lens we can derive it in one step and that's nice. And I want to write down our paraxial refraction equation, how we get across that lens. And remember, what this is, is over here, we see Snell's law. If the lens had no power v was equal to zero. We'd have n prime, u prime, equals nu, that's the paraxial version of Snell's law. A lens simply adds some extra re-bending. It's proportional to the power of the lens and the height that we hit it and the fact that the ray bending is linearly proportional to ray height is wide paraxial one does work perfectly. So you can form all those triangles and all the rays intersect exactly at the same image point. So let's break this down for the marginal ray shown here in blue and the chief ray. And what I'll do here a very common notation is also note to the chief ray, all quantities associated with it, with a bar over it. So for example, here's u, the angle of the marginal ray, but out here, here's u bar prime, the angle of the chief ray after the lens. Then there's u bar prime right there. So let's write those two things down. Remember the course that you're ray heights, your marginal ray, or the chief ray don't change at the lens, so I don't need primes otherwise that the same before and after lens. And now let's gather up all the terms that are after the lens and put them on one side of an equation and all the terms that are before the lens and put them on the other side of the equation. And what we find with those two things is then, when we do that, the equations we get are identical. In other words, this finite quantity, the refractive index, times something which involves the angle of the marginal ray times the height of the chief minus the angle of the chief times the height of the marginal, doesn't change as we go across that lens. And as a matter of fact, it doesn't change on transfer either and that means paraxially, it's a conserved quantity, and conserved quantities are important. So we're going to call that the lagrange invariant and we can calculate it anywhere because it's a conserved quantity. When we talk about like the Gaussian beam and we have those two rays and once we understood with those rays went, we knew the Gaussian beam properties everywhere. This is the similar quantity for the optical system. The chief ray out here in image base, sets the height of the field while the marginal ray determines the aperture, the angular content that we get off the field. Notice when we go to the aperture stop. Those two ray properties have flipped, here in the aperture stop, the marginal ray determines how big the bundle of rays is and the chief ray is what setting the maximum angle of all the rays. However, if you calculate this quantity, you'll find it's conserved. The point is, and this goes back to my postdoc said tail, once you have the chief and the marginal ray you know this quantity all the way through the system and you can't make it go up. You can make it go down by losing light but you can't get more angle or more field than that quantity tells you. It's convenient to think about this. It's old, it's simpler in object and image planes and that's the version I used when I was talking to my sad post-doc. So let's figure that out. At this object plane right here, notice that for this simple system, the angle of the chief ray is zero, u bar equals zero, and the height of the marginal ray is always zero, so that's right there. So in object planes, I'm always going to have the height of the marginal ray is equal to zero because that's what the marginal ray, that's how it's defined, it comes off of the object. So that means that this thing simplifies and we have one terminate and I now have just the index times the angle of the marginal array and the height of the chief ray. Well, let's use some of our other knowledge here and see if we can learn something about that quantity. That's a great expression to remember anyway, but let's go a little farther. nu is paraxially the numerical aperture, n time data. Y bar is the height of the field, I call it L over two. Numerical aperture could be related to the diffraction limit and spot size, point six Land over an A. So I'll replaced numerical aperture with spot size. So now I have the total field size over the size of the spots that I can see in that field, that's the number of spots this lens will carry. The maximum information content linearly. You square this if you have a square field or use pot pie R-squared to get the total number spots in 2D. This is a super important quantity. If you want to have a megapixel camera, so a thousand pixels on a side, you'd better have your optical system carry at least a thousand resolvable spots in it's field or the lens will be limiting you. There was a quite a popular field of optical computing several years ago or at this point, several decades ago, and you could use optical systems like this to vector matrix multiplies and all sorts of really cool things. And the reason they were cool is it was all done in parallel at the speed of light and that was cool. The information you could carry though, the size of that vector or matrix was of course limited but the number of spots, the number of independent degrees of freedom information capacity of your system. So that is a pretty important quantity and you can see here, H has quantities of units of distance because u is universe, endless universe. So if you take H and you divide by wavelength over two, that tells you the number of spots. You can make H bigger and therefore number spots bigger by letting more angles through the system which decreases your diffraction limited spot size or by increasing the field which would increase the field size here.
