Up to now, we have been using the Gaussian beam to build some intuition on resolution. How the waist size depends on other Gaussian beam parameters. On how we can use first order design principles to move these Gaussian beams through optical systems. And lastly we discovered the Lagrangian variant, that the size of the beam and its divergence have this constraint. And we can understand that through those two rays that we used before. The waist and divergence ray. But this is fundamentally a design course. And so we have to end up teaching you how to design something with Gaussian beams because it turns out they're really useful for that. Again, any coherent single mode system, fibers, lasers, things like that, Gaussian beams are a pretty good approximation to the light that comes out of those. And so they're really common in coherent systems, laser beam systems. So, we're going to end up here with doing a design example. And I can tell you from experience that it's pretty easy to get lost in a pile of equations, designing systems of Gaussian beams. And we've tried to here present a very simple set of equations which will keep you from getting lost. And we're going to use this as an example. So the problem is, we have a single mode fiber here, that this was the telecom band of 1.5 microns. The core of the fiber is about nine or ten microns across. And so we have a source here which isn't an infinitesimal point source, it's maybe nine microns in diameter. We're going to want to let the light out of the fiber, probably go through some sort of lens which will at the moment in first order design world, think of thin paraxial lens. And it would expand that beam to something bigger where we are going to put some sort of optic, a wave plate or a modulator, or something that's going to interact with the light. Then we want to refocus the light, couple it with high efficiency back into the same fiber, thus we want a waist here, same as the incident waist that's maybe 9 or 10 microns in diameter. The question is, given that we have certain constraints on the overall package length, given we have an optic in the middle that has to have a certain diameter, what lens power should we choose? And what distances should we choose, there's our design problem. So the first thing to do is to realize that we can simplify this problem by noticing that through symmetry, if we get to the middle and we've put the waist in this space of the Gaussian beam right half way through the system, then the remainder of the optics are the same thing, only backwards. So we don't actually have to go from fiber to fiber, we can go from fiber to the symmetry plane in the middle of this system. And as long as we make sure that is a symmetry plane, then we've simplified the analysis by a factor of two, and that's a pretty handy thing. So let's do that. Let's write down the system matrix M, ABCD matrix that goes transfer for distance d, right there. Then, refraction through a lens of some arbitrary power of Fe. And then transfer some unknown distance L, and that's easy. We can write that down, and that's what that ABCD system matrix looks like. Now, a couple of slides ago, we showed that if you have an ABCD matrix and you have an input q parameter, describing your Gaussian beam. Well, that's really easy in this case, because remember that one way of writing q is z, the distance from the waist plus j z naught the Rowley range of that particular beam, or at the waist here, the Gaussian beam emerges from the fiber face. And the face fronts are parallel right at the fiber face. So the incident q is just j z 0. So we can take that q, which is just an imaginary number, so it's simpler than normal, and put it into that simple expression for how we calculate a new q, at some distance L. And it's at a distance L, because that's what the system matrix described. There's that transfer, last transfer of the distance L. And now this is the q at some plane here, which fully describes a new Gaussian beam at the dotted line, and there it is. I just plug in that q into the ABCD matrix expression for how you would advance cues. What we wanted this to be a symmetry plane. And that means that this position here, the dotted line, the distance L, must be the waist as I've shown in the picture. Well it's really easy to take this new expression for q and constrain it to be at the waist because by the same logic we used here the real part of this new q must be zero at the waist. So I'm simply going to take the real part of this expression, and set it equal to 0, and solve for distance L prime. The precise distance which is at the waist. So that's trivial, I just take the real part of this expression, set it equal to 0 and solve for L. And I get this expression here. And now if I'm at this distance L prime, no matter what my distance d was, no matter what my initial Gaussian beam relay range z0 was, and no matter what my lens was, I've put a constraint on this system, that I'm at the waist, as long as I use this new L prime. If I take that L prime, shove it into my expression for q, I should get a purely imaginary number, because that's what this bit of math did, and I do. And that imaginary number is j times the new Rayleigh range, the Rayleigh range in this space. And, of course, I have expressions which relate the rally range to the waist size. So I can shove this Rayleigh range Z0 new into that expression, and now I have the waist size in this new space. And I'm done. I have all the constraints I need now to choose distances. The total throw of this package, assuming infinite length in the lenses, is 2d plus 2L prime, the size of the beam, in this, the waist beam, I got here, I have an expression for. And now I have some certain knobs that I can use to meet packaging constraints or whatever optic I want to put in here. So this is an example of how to use these expressions. That we've shown you to do a true design problem. And this is a pretty typical design problem. This idea of using symmetry is a clever thing, remember that. I could use these same techniques to figure out in a laser resonator with two reflective mirrors, what would be the range of Gaussian beams that might fit in that resonator or where might that resonator be stable? So, overlaps with laser design and is an example of how you now design, which of course is the goal of the course.