Now let's imagine, we take our optical system, and we hold it up to our eye, and we look through it, binoculars, for example. We presumably see a clear region, through which the binoculars transmit, and then outside that it will be black. And somewhere in the optical system this represent a metal aperture with a round hole in it holding up one of our lenses. We're going to call this image, this view of some aperture that could be in front of the lens, it could be buried on the optical system, we're going to call that the pupil. The same as this pupil in your eye that's back behind your cornea. It limits the light that gets into the system. And it's that same limiting aperture we just used to find the cutoff spatial frequency for the system. In the case of this round aperture, we can now use that same function we just did, but now we'll go ahead and do a 2D system. So we got a radius r, and the function is 0 outside some diameter D, and 1 inside the diameter D, and that's what that function says. We could now map that function to the spatial frequencies that get through the system using the logic we just used. So we know that low spatial frequencies, that is rays at low angles, get through the aperture. Rays at high angles, therefore, high spatial frequencies, hit the metal. They don't get through. And right in this boundary is where the cutoff happens. These are just the rays that get through the system, and they correspond to NA over lambda, the spatial frequency cutoff of the system. That means we could take this pupil function that's written in spatial coordinate, and we could transform it to spatial frequency, again, using the logic and the math we just wrote down. And that would tell us, overall, all of the spatial frequencies through the system, either launched toward the system, what gets through just by using this scaling. So we can scale our pupil function with this relationship of spatial frequencies on the object to position somewhere in the optical system. And that is then, the transfer function of the system. I just plug this number into here, pupil function in x and y, but I now use this scaling relationship. And that gives me capital H. The transfer function of the system, in terms of spatial frequencies, off the object here, f sub r, is a radial spatial frequency and a square root of fx squared plus fy squared, because this is a round pupil. What that means is if forward your transform, my object, to understand the spatial frequencies being launched from it, I can then multiply by this transfer function, which is just a simple circular function here. And there's the spatial frequencies of my object. And that tells me what light gets through my system. I find that, as in my previous slide, high spatial frequencies are cut off. And so this is going to enforce a finite resolution in my system because I'm low pass filtering my object. This is known, therefore, as the coherent transfer functional system. We're talking single frequencies here, so laser beams, if you will. And we'll turn next to what incoherent objects do. But this is a super important concept. We need just enough of this to use it, and relate it to retracing. This is really a Fourier optics concept and taught in math class, but we'll use some of these concepts here. And the neat fundamental relationship is that these rays tell us about plane waves, which in turn tell us about spatial frequencies. So there's this nice one to one mapping between rays, and their angles, and spatial frequencies. And once we have spatial frequencies, it's a pretty nice and useful thing to think about transfer functions, because that brings the power of linear systems into thinking about optics.