Let's use the two ray system to trace some typical Gaussian beam transforms and see how it works. To help, I wrote myself a little MatLab program, which traced the two primary rays, the divergence and waist ray. Locally calculated the beam parameter q. And plotted the intensity of the Gaussian beam. And so I've put the two rays on top of the intensity of the Gaussian to help us understand what it looks like. If you're going to only remember one geometry for transforming Gaussian beams, it would probably be this one. We have one focal length in front of the lens, one focal length behind the lens, and we've put the front waist at that front focal point. Let's trace our two rays. The divergence ray, here shown in blue, comes out parallel behind the lens. And the waist ray, shown here in red, will go through the back focal plane. Remembering that the local waist radius w of z is calculated as the square root of the sum of the squares of the two ray heights, it's obvious that the final waist is at this back focal plane. Because the blue ray is parallel to the axis and the red ray goes through zero at that back focal plane. So really cool. Gaussian beams, if you put the the waist at the front focal plane, you recreate a waist at the back focal plane. Notice that's dramatically different than the imaging condition. Because, of course, any object placed at the front focal plane would form an image at infinity. So clearly, a completely different system here. Images and waists are not the same thing. The reason I like the two ray system is how easily, then, I can calculate the properties of this second or new Gaussian beam. Because notice that the divergence ray coming in becomes the waist ray on the output side. And the waist ray for the [INAUDIBLE] Gaussian becomes the new divergence ray. So now I can simply calculate that the new waist, by simple triangles as we often do in retracing, is the focal length times the incident divergence angle. And the new divergence angle is the old waist divided by the focal link, this triangle here. That right there is enough to motivate to me to two ray systems. So I can very quickly calculate the properties of the Gaussian beam on the output side. Something perhaps more important or as important is notice that if we multiply the waist times the divergence, we can conserve quantity. This is very fundamental, and is one of the two primary concepts of the course. That there's this quantity, height times angle, which is conserved, and the Gaussian beam exhibits that. And that's going to be one of our very major important concepts in the class. Maybe having looked at the 1F:1F case, let's do the classic 1 to 1 imaging case where we have 2F in front of the lens and 2F behind the lens. So the image is now back here in 2F, let's see what happens to the Gaussian beam. So we'll launch again, divergence ray in blue, that will come back to the axis at that image plane because it should. The waist ray comes through the back focal plane and continues on. And if you take the square root of the sum of the squares of the two ray heights here, it's not minimum at 2F at the image plane. And as a matter of fact, it's minimum where the two rays have the same magnitude of the ray height, which is in front of the image plane. So this is a pretty common thing, that you find the waist inside the image plane. And again, as an example of that, waists and images are different things. And also why you need to know about Gaussian beams. Because if you'd used the concepts of course one, simple first order design, you might have thought the waist was going to carry here at the image plane. And that could lead you to a poor design. For example, coupling back into a fiber wouldn't be optimal. That would be better at the waist. Well, what if we don't go 1:1? So here I've pulled the waist back to 3 f. Which brings the image plane to three-halves f. And what you find very quickly is the waist is still in front of the image, but it moves rapidly towards the image plane as the waist moves farther away from the lens. And there are several ways of thinking about this. But one is simply that the spherical wave fronts that are hitting the lens, once I get multiple z naughts away from the waist, begin to look like spheres emanating from a point. I can't really tell there was a waist there. And so they tend to launch spherical waves that are pointed back at what would've been the image plane. So it's moving towards our more classical 1 f, first order geometrical optics. Finally, just so that you know this works, here's that same example, the 3 f, three-halves f case, but I've moved the incident Gaussian off the axis. I tend to think of this as a two ray system, but formally it's three. Because we have the axis here that I've drawn in gray. And formally, the heights of the divergence and waist ray are measured relative to this axial ray. Which in all previous cases was on the optic axis. Simple description of what happens is, of course the Gaussian beam, before it hits the lens, simply propagates shifted off the axis, as you'd expect. And that you can see because the gray ray here is indeed shifted off the axis. And then behind the lens, it's all tilted. And it's tilted by just exactly how much you'd expect, based on this gray ray coming along the central axis of the Gaussian beam. So it doesn't move in this paraxial limit, the location of the waist, so I tend to simply always trace Gaussian beams on axis. Now if I need to know or understand what happens off axis, I simply kind of patch it up by noticing or thinking about that principle axial ray. And just mapping the Gaussian beam to that ray. But without actually formally calculating all of the ray heights, because they are normal to the ray back here and the math gets slightly uncomfortable. But the point is, these rays are very easy to calculate. They allow you, through simple first order geometry, to calculate the new properties out here of this Gaussian beam to find the location of the waist and the waist itself, the z naughts, so the discreptors of the Gaussian beam. So I find this a very handy back-of-the-envelope way to design with Gaussian beams.