This lecture is if you will optional, if you feel like you understand the Do Gaussian beams and are comfortable with the fact that they behave differently than our first order practical ray tracing, you don't really need it. But the whole point of this first module is to try to use Gaussian beams to build intuition for this shape of light. And I get curious myself, of if I used that two ray trace that we just learned. To trace Gaussian beams through an optical system, how did that actually relate to imaging? And so, this is the answer to that question, I find it actually fascinating, and you might find it useful or not. It's not really critical, it's not design technique, but it might be useful. So, what I've done here is I've launched a waist ray, and here in blue, divergence ray, and I've calculated from them the local Gaussian beam radius w(z), and I have gone through a lens and done it again. So I just wrote myself a little Matlab program that used ABCDs, to move the two primary rays, and I took the square of the sum of the squares of the ray height to come up with w(z). And so that gives me a beautiful little Gaussian beam ray tracing. Then I printed those out, and I started doing graphical ray tracing on them. For example, what if I had an object at the waist here. Well, I will launch the three rays that we'd like to use for graphical ray tracing parallel to the axis through the lens center, and since we're at the front focal point here, off a bit front focal plane. And they all converge to the new image plane over here, and shockingly, that seems to live right on w(z). So the point is, is if I have a Gaussian intensity profile here with a particular radius, I find another Gaussian profile in intensity here that has -1, it's symmetrical, so I can't tell, so has exactly the same radius, therefore obeying the paraxial imaging condition. It just turns out for this particular Gaussian beam, that there's another place in front of this image, where the waist is actually smaller. So, this point isn't the waist, but it does image, it is the point where the Gaussian beam here with a radius w0, I find another Gaussian beam here with a local radius given by this w0, it just doesn't happen to be the waist. Well, that's intriguing. Let's just take that same diagram and pick another random point. So, I just drew a little random object here again, drew my three rays for example here, it's the one right there that's going through that front focal plane, comes up parallel, and they converge to another point but that's also on the Gaussian beam. In this particular case, I have a negative magnification. That's less than one, and this new waist height here w(z), is smaller than this ray waist height here w(z), by the magnification. So, in that sense, Gaussian beams do obey our paraxial ray equations. It's just that waists and images are different things. And that's maybe another way of expressing why Gaussian beams don't seem to obey the paraxial imaging questions. And it's because again, waists and images are not the same thing. The waists go from this point to that point, but images, for example here and here, are not the same. Well, that was a real object to a real image. Let's check those other cases, just because so far we're doing well, maybe this works for everything. And you may have to go back and review virtual objects and images, but that's a good thing. So, if you do stop, go back and do that. So, I've got exactly the same condition here, but now I'm going to imagine that with the dotted colored lines, that there was no lens, and so off here somewhere, I could define a virtual object. It's what the object, the Gaussian beam over here, would have looked like if the lens wasn't present. And so therefore is somewhere to the right hand side of the lens. And I'll pick a random point here on this curvy profile w(z), and ask where does it get image two. Well, to do that, again I have to do a graphical retrace, I'll just shoot two rays here, I'll shoot one towards the virtual object which would be here, dotted black line, but of course in reality, it hits the lens goes through the back focal point on and out. And then I'll shoot one everyone loves, because it doesn't do anything, the one through the center of the lens that would hit the virtual object. I find these two rays in image space intersect on the Gaussian beam radius again. So, this portion of the Gaussian beam in image space or on the right hand side of the lens, actually is the image of virtual parts of the Gaussian beam object. But it still works, and the magnification is positive, and the ratio is right, and that will make sense. And finally, how about virtual images? We're on a roll. So, again, to remind you, I'm now going to take the Gaussian beam over here in image space, I'm going to take its two rays, which I got from my little mobile program. I'm going to extend them back into object space as if the lens wasn't there. So, I get for example this ray right there, and this my blue ray was my divergence ray, and then I'll trace out the square to the sum of squares, which is the Gaussian beam. So, now I can say that a point on w(z) for this image space Gaussian beam, I can pick that as a virtual image, and I can trace graphical retrace off that. So, let's take the easy one first from the tip of my virtual image, I can trace a ray through the center of the lens. I could also take the ray that appears to go through the center of the image, and hit the back focal plane that of course, I actually have a lens there, you can just barely see the dotted line, would have been parallel to the axis in object space parallel in back focal point out, and so I intersect those two rays, and I find a real object which of course lands on w(z) over here on object space. So, this portion of the Gaussian beam in object space can be thought of as coming from virtual images of the Gaussian beam, the image space, and all that's perfectly analogous with where those various regions image to any of the objects are for graphical ray tracing, for paraxial ray tracing. So the point is, every single point on this Gaussian beam here including those that are behind the lens, the virtual objects, images to the Gaussian beam and image space, sometimes parts of the Gaussian beam image space that are in object space that is their virtual, with exactly the right magnification. So, you can think of the profile of this Gaussian beam, as being little source objects that always image over to the exact same part of the same w(z), the same profile distance whenever e, e in like a field, whenever e_squared in intensity of the image, Gaussian beam. So, that's actually kind of cool, and it's just another way of thinking about Gaussian beams and how they work in comparison to paraxial ray tracing. And one last time, the key idea is that imaging, points, and waist points, are simply different things. And if you separate them in your head, then it's not so confusing as we showed that waist don't show always up at image planes, because they're simply not the same thing.