In the last several lectures, we understood, from Gaussian beams and other coherent laser beam-like systems, how to think about spatial frequency and impulse response. And how the transfer function of a system related to the impulse response. And that was a beautiful and really valuable set of concepts. But most of them applied narrowly to systems with single temporal frequency light, as you would find inside a laser beam. If you're not working with lasers, maybe you're working with photography, and so you're taking a picture of something. Those concepts need to be modified a little bit. And the way to think about that is to look at the object. Let's say we have that Gaussian laser beam at the front focal point of our lens. And to imagine taking two points on that object and asking, how are the electric fields correlated in time? And we'd find they're very correlated. They're the same temporal frequency, so they might be in phase, they might be out of phase. One could be bigger, but they're perfectly correlated. And we call that coherent light. Now instead, imagine that we had a filament in an incandescent lamp. So it's a black body at very high temperature. And there's little atoms screaming with their thermal radiation. And they're doing so completely independently. So if I walk a millimeter up the filament on an incandescent lamp, and I look at two points, I don't see any correlation in the phase. The intensity coming off those could be nice and constant with time. But the phase of the electric field has nothing to do with the phase somewhere else. So that's the formal definition of an incoherent light source. A light bulb, sunlight on a scene, etc., could be well described as incoherent light. So we need to think about now how the concepts we just derived work now. And it turns out that there's just this little modification, it's pretty simple. And we can take what we just learned and transform it to an incoherent case. So the way to do that is to think about the impulse response. So I've plotted here, in the darker color, (h)x. That's in the one-dimensional case, sine x over x. In the two-dimensional case, the area disk, the Bessel function over r. And it has negative values. And that's fine when we're looking at an electric field. A negative electric field is simply one pointing the other way. So negative e, no problem. But if we'd like to describe how a picture of it, a system of incoherent light, maybe we have a picture of a car or something. And now we'd like to have the impulse response of that system. In intensity, negative values don't make any sense, because I don't know what negative intensity means. It turns out, and this shouldn't be really surprising, that the impulse response of an incoherent system describing how intensity moves through an optical system is nothing more than the absolute magnitude squared of the impulse response of the electric field, describing how coherent light moves through a system. And that shouldn't seem real shocking. If I think about the relationship between field and intensity, that's exactly what I have. And there's scaling factors and intensity of the units, right? But that's not the important thing here. We always normalize impulse responses. So we're going to have four different quantities by the time we're done here. The impulse response and transfer function for coherent and incoherent light. We've already done the first two, the impulse response for coherent light, the area disk. The transfer function, that's just the pupil, the 010. Now we've learned the third function, the impulse response for incoherent light. And conveniently, it's just the absolute magnitude squared of the area disk, or whatever our impulse response would be. For a Gaussian beam, we'd have the Gaussian beam of the electric field would be the impulse response for the coherent light. And then if for some reason we had an incoherent distribution of light that looked like an intensity, that was a Gaussian shape, its impulse response would be simply the absolute magnitude squared of the impulse response for the system incoherent light. So that's three of the four functions we want.