That was a lot of material in a short amount of lecture time. So let's stop, gather it all up and review. This one slide is sort of your cheat sheet that summarizes everything we've just talked about in the last few lectures. And is enough to really understand this for us to go forward, and to use these tools. So, I'm going to walk through the whole process again, but now we can see it all in one slide. These are all plotted for one dimension because it's easier to make the plots. So it is easier slits instead of circles, for example. So we started with the pupil function. This characterizes the fact that, let's say we have some sort of imaging system on the front lens of that system like a telescope or something, is limiting the amount of light that gets into the system. There's an object, some finite distance T away from that lens, then the object can be thought of as launching spatial frequencies f sub x. And those get into the pupil at a coordinate x that relates to the distance to the object, and the wavelength of the light. So, if we go ahead and make this substitution, we can turn the pupil function, that might be some sort of slit, into a function which describes the spatial frequencies that get off of the object and into the system. And if the pupil function was let's say a rectangle function, then so is the transfer function of this coherent optical system. And there is an, where the edge of the pupil, that would correspond to a particular cut-off spatial frequency, which is simply the numerical aperture of the lens as seen from the object over the wavelength. So this is a pretty important concept. It tells you the highest spatial frequency that gets off of a coherent object and into the pupil and it can go on through the system. Now we know from linear systems theory, that if you have the transfer function of a system, you can relate that to the impulse response, little h, through Fourier transform. So if we take this transfer function, and inverse Fourier transform at from fx to x, we find the impulse response. This is the area disc, this is the one dimensional version of an area disc where it's sin X over x. The two dimensional is Bessel function of the first kind over x. So this has the traditional central hump and nulls and loops, it looks like a target in two dimensions. The first null of that function is in one dimension point five lambda over in a, in two dimensions that simply points six. Or, the f number times lambda. So, f number which is one for a really powerful lens, and 10 to 20 for weaker lenses, is simply the size of the diffraction limited spot in units of the wavelength, very handy thing to know. This is for coherent light. If we then think about how would incoherent light things off of lamps, let's say instead of lasers, what would the impulse response look like? Well, it would be the same impulse response, but instead of the electric field h, we would look at the intensity, and so we take the absolute magnitude of this area disc squared. So it looks much the same, it has the first null at the same spot, but all of the humps and bumps are positive, because now we're talking about how intensity moves through a system. Up here we were talking about how coherent electric field would move through a system. So, that relationship seems pretty obvious and we see much the same area disc, it's just all positive now. Then conveniently, we can inverse the same relationship we had before. We can take that forward Fourier transform of this function right here, to find the incoherent transfer function. And there is a Fourier transform rule that tells us if the impulse functions are related by absolute magnitude squared, the transfer functions in the four E-Space are related, by the auto correlation, and that's what this little circle X means here. So we take this rect, and we auto correlate it. We correlate it with itself. We make a copy, we slide one copy by another, and we calculate the overlap. And so when the two are separated by twice f nought, they begin to overlap and they're backs along at the center, and we slide by another twice f nought to get off to the edge. So the incoherent transfer function, which is referred to as the optical transfer function, you'll also see the MTF, or magnitude modulation transfer function. That's just the absolute magnitude of this guy because it can be complex. So the MTF for the OTF, the incoherent function here generally has this triangle shape, and it's got twice the overall bandwidth of the coherent transfer function because of this auto correlation operation. So this is a brief tour through Fourier optics. The key thing it tells us is, when we have optics with finite pupils that results in finite sized impulse responses, we can very easily calculate the cutoff frequency, and that's a nice thing if we want to think about how test patterns, and spatial frequencies get through a system, or the inverse of that within a factor of two, is the size of the diffraction limited point spread function impulse response, and this is now how big our points are. In summary, this solves the problem we pointed out at the beginning of the course. That the universe is going to blow up because we had infinite energy density to point. Now we see by Fourier concepts that because apertures are always finite, impulse responses are always a finite size.