Let's look again at the Gaussian beam transform that we found with one f, one f lens system. And we noticed there that the divergence ray on one side of the lens became the waist stray on the second side and vice versa. The waist ray becomes the divergence ray, and the product of the quantities the waist size times its divergence appeared to be a constant, because notice that the f's cancel out. And that has the flavor that we found before in the first order design class, that transverse an angular magnification or inverses. So, the hint here is that there is a conserved quantity. And that conserved quantity is between two rays, and is a very important part of this class. And the idea that it's bundles of rays that have this conserved property that we're looking for in this class. So, in this slide we're going to motivate that and then we'll return to it more formally a little bit later. One of the ways I like to motivate it is quantum mechanics. The transverse profile of our beam here has a few will an uncertainty in its position, and that's something like w naught, or two w naught, or something like that. And of course this could represent a photon. So, we might be curious if there is an associated uncertainty in momentum associated with this wave. And if you remember your quantum, momentum p is related to the wave vector for the probability wave k by simply the constant h bar. So, this k is exactly rk that we've had before, or looked at before. It's two pie over the transverse wavelength. Kx equals two pie over lambda x. And that's given by two pie over sin of well, the angle. And this represents the sort of the bound on, or the uncertainty on the range of k vectors, that make up the fourier transform of this beam. And those three h bar can be related to a range of transverse momentum uncertainty. Well, we have some ability to take sin theta and replace it with something else here because in the paraxial elements sin theta is theta, and theta naught is lambda over pie w naught. So the quantum mechanical momentum uncertainty of a Gaussian beam with a particular divergence angle is h bar over the waist size. And so the uncertainty in position times the uncertainty in momentum is h bar, to within some numerical constants the order of unity that depends on exactly where you measure. So, in other words there's a conservation here between the divergence angle of a Gaussian beam and its waist size. And that's actually a reflection of the Heisenberg uncertainty principle. Again indicating there's something fairly fundamental. We will find later that the actual conserved quantity is given by this combination of the hypth locally with whatever Z and the angle again locally at whatever Z of the two rays that we're looking at here. And this particular quantity is the Lagrange invariant, and it's a very important part of the class because it describes an invariant to the system, something you cannot change, which we all find not only describes how Gaussian beams move through the system, but in general how the entire optical flux moves through a system. This is going to limit the efficiency of optical systems or how much power that you can get through them. So, it's very very fundamental. And that's just a hint of what's coming. But the important part is that we can see this emerging even in this two ray description of a Gaussian beam. And we're going to have analogies of these two rays to describe all of the light that can get through a system, not just one Gaussian beam.