So q allows us to write this equation for the electric field. Or you could take a square of it make two squares of it, the intensity of a Gaussian beam in a simpler way. That's okay. If that was all it did, it wouldn't be worth writing down really. But, it turns out that q also can now be used to advance a Gaussian beam through an optical system. So an example, we'll come back to towards the end of these lectures, is a light coming out of a fiber. It goes through a lens and maybe through some optics, then maybe you want a couple back through a lens and back into another fiber. What should those distances and focal lengths be? That's an example of a calculation if you did it without knowing about Gaussian beams. If you just did the first-order ray-tracing, where the light comes from an infinitely small focus of that first fiber, you'd get it wrong. And you'd find that you don't get very much light back into that second fiber. Which is a very common problem in telecommunications. So, what we'd like to do now is use this q parameter and somehow get it back into the world of design that we understand, some way to manipulate it as it goes through an optical system, so that we could for example, design that fiber colorimeter. And this is the first of two rather remarkable things to me, is that just a few minutes ago in these lectures, I said that these Gaussian beams violate the assumptions we made in deriving the ray equations, and so we can't use rays. Well, it turns out now that we know something like the q parameter, we can figure out how to use the same concepts and the same math, that we derive for rays and we can advance the q parameter. In other words, we can still use those first order design tools. We just have to use them with a bit more sophistication. And now, we can advance not just single rays as we did in the first course, but we can advance entire Gaussian beams through a system. And that's sort of remarkable, it's not obvious to me that should work, but it does and it's really important. So we're going to learn two ways: the first is, we're going to find that the q parameter evolves with our transfer and refraction equations that we learned for the Yu ray tracing and for the ABCD matrices. And that means we can use those tools for Gaussian beams. So, how do we do that? Well, let's just ask how q changes under transfer? That is free space propagation. And we'll use our expression for q, because we've got a couple, that's the complex distance. Some distance from the focus plus an imaginary part that's given by the brawly range of the particular Gaussian beam. Well, it's some new distance q2 and some new distance z2 of course, we simply add the distances. That's pretty obvious from that expression. So, we get a super simple transfer equation that q changes on transfer, that is propagation between a plain one and a plane two, just like the distance between those two points. So, that's pretty easy. Now, let's ask how the q parameter changes upon refraction, as it goes across the paraxial thin lens. And we're going to stay in the paraxial, then first order design world here. And that's self-consistent with the Gaussian beam itself, because it's a solution to the paraxial. We have equation, so we're saying self-consistent. So it's more convenient now to use the expression for one over q, because what changes are going to cross lenses is the radius of curvature. And so, this is a different expression. This is why it's nice to have all these different expressions for q. And of course the waste itself does not change when you go across the lens. You don't move rays around, you just change the radius of curvature as those rays bend. So it's really this first time we just have to figure out. So, I've gone back into course one and I've pulled an expression here that tells you how the radius of curvature of light changes, when it grows across the lens of a particular focal length or one of the focal length to the power. And we found that, the power was just simply a way to express the change in the radius of curvature of the beam. So that's clearly what we need here. The one complication, is that the sign convention for the Gaussian beam this R(z) here, the Gaussian radius is actually backwards from our sign convention that we use in geometrical optics. And there's just no good way to fix this. So, if you look at this, the signs in this expression are different than the one I used before, because this quantity R(z) is got a different sign convention. If you go back and look at the expression for R(z) from our previous lecture just a few minutes ago, you'll find that when the origin is to the left, that is your positive Z, you have a positive radius of curvature for the Gaussian beam and we would have find this negative. So, I wouldn't worry about that too much. If this expression bothers you, you can go back and figure it out. If not, just accept that the power of the lens changes the radius of curvature. And I've patched the signs up here to make it work for Gaussian beams. If you accept that, it's pretty easy then to see that one over the radius of curvature changes like one over the focal length then q prime, the expression that describes a Gaussian beam after the lens, differs from q through this expression here. We just change the inverse q by the amount one over f. So that we can solve for q prime, because it's nice to have an actual expression and that gives us the refraction equation, how light changes in a Gaussian beam as it moves from one side of the lens to the other. Well that's exciting. Now, we have both transfer and refraction equations. Could we put those in the ABCD format? Because if we could, then we can take all of those matrices that we used before, with all of their nice properties we understood and we can now instead of shoving single rays through an optical system, we could ask how a Gaussian beam of all through that system described with the same matrices. And the answer, somewhat strangely to me is yes. So, let's take our transfer matrix and this would normally again, this would be how rays move a certain distance. And ask, could we force this to describe the Gaussian beam transfer equation that we just derived? And the answer is yes. Because remember, for the Gaussian beam equation, all we did is we added a distance. That's what Tk prime here means. It's the distance that you traveled and that just in the Gaussian beam transfer equation we just had to add that distance. So working backwards, it looks to me like I could write a new version of this evolution equation for q, using these four terms by just well, it just can take up the matrix down here. And I need one to be multiplied by q and I need to add two Tk prime. So it looks like if I just wrote a little matrix here that the vector that had q in it in the right way, I could indeed get q here and Tk prime here, because that was the easy one. What about refraction? Well it turns out, I can find the same thing. So here's my refraction equation before. Again this described how a ray coming in with a particular height and angle Yu changed its height and angle Y prime, U prime when it went across the lens. I would like to get this equation out, because that's the refraction equation we just derived for the Gaussian beam. And I can just kind of see how those terms would pop in, I'd need a q there and a q there. So, it turns out that if I walk all the way through a system and I derive these ABCD matrices, these two equations tell me how to use ABC and D. ABC and D go right there and ABC and D go right there. So if I do that, I can now take any q coming in. That's our Gaussian beam illuminating a system, described by ABCD and use this single equation. I've written right up here at the top, q comes in. This tells me q prime coming out. And I just pop AB and CD into this funny little equation, and this gives me the new q prime. So the point is, I can now evolve a Gaussian beam all the way through a system. I can go find the distances anywhere and evolve the beam in distance and space, if I want through T prime. I can go through as many lenses as I want. And this expression gives me a new q prime. I plug this q prime back into my expression for the Gaussian beam and boom I've got my Gaussian beam. Or, I can use the q prime to describe or derive the local waste, the local depth of focus, the local beam divergence. So I can understand the Gaussian beam parameters, that come out of a system.
