So we just used that little map lab, graphical tool to understand the plane wave in some greater detail. The only other solution we're going to use is a circle wave, and we've seen this before. That's what foci look like around lenses And it is also an eigensolution to the scalar wave equation. It's the eigensolution in spherical coordinates. And there's other terms out here if you include angular variation but we'll ignore those spherical harmonics. If you've ever heard of that math, you run into it in quantum and things like that. Mainly, we care about this fundamental simplest version of the spherical wave right here, which looks kind of like a plane wave, except notices that k.r, we just have kr. That means we have spheres because the r is just the magnitude of the position vector. And so now, we have sort of rays proceeding out everywhere with a wave number 2 pi over lambda k. But to conserve energy, that field better fall off like 1 over r so the power falls somehow like 1 over r squared. We won't need to sort of dig into the spherical wave with as much intuition as we need to understand plane waves but it is worth remembering that what lenses do is transform the curvature of waves. For example, it will turn a plane wave, or chunk of a plane wave, into first a converging and then a diverging spherical wave. Note that this is all sort of in-depth no diffraction solution that we started out with in first order design. So once again, we have infinite energy density right at the origin. We have now learned that we should replace this in the paraxial limit with a Gaussian beam, but far away from the focus, more than two, or three, or four, well, it ranges. It would look like a converging circle wave. Goes to some mysterious diffraction right in the middle, and then turns back into a diverging spherical wave. So this is consistent with our Gaussian beams that we had before.