The expression for the Gaussian beam, and I've written it again here, has a lot comes out and it's convenient to express those terms in a different way which reduces the number of things you need to keep track of down to one. And this is, I call the Q parameter. It's a common way to rephrase and recast that equation for the Gaussian beam. And it turns out, it's pretty darn convenient when we start doing design with Gaussian beams. For example, how do you couple from a fiber through an optic and back into another fiber, or how do you design a laser cavity. So right now, what we're going to do is translate the Gaussian beam expression we have to this new parameter. So basically, what this new parameter is, is it expresses as a complex number. Remember the waste, the radius here and this extra Gouy phase were all real. So we're going to write a new thing called the complex radius of curvature and you can see it's related to R, but we add an imaginary part that turns out to be just right to capture two of these terms. So the real part, as a matter of fact, of the expression is going to express the radius of curvature here and then this imaginary part is going to capture the amplitude of the beam transversely. And that's a good start. And that's just substitution right here. But it turns out, the Q parameter can be expressed in all sorts of different ways. And once you start doing that, you can kind of see why it's a good idea. So let me just substitute in for the expression off of the previous slides for this Gaussian beam radius of curvature and the local waste, and so I do that here. And I go through a tiny bit of Algebra, and I've discovered that Q also is simply a complex distance, its distance from the focus Z plus J times the Rylie range Z_knot. So that's the second way of thinking of what Q is, it's simply complex distance. If we ask, well, what's the phase of this term right here, of Q itself? We find that it's just the arc-tangent of the oringinal alternator or the real part, and that is a lot like this Gouy phase here. So it also captures that. And finally, if you're taking divide Q by Z_knot, you find that that reconstructs or gives you back the waste expression. So another way of thinking about what Q is, is just a normalized view or measure of the local waste. I mention all of these things, because there handy little bits to have when you start using Q, which we're going to show how to do, to design Gaussian beam systems. It's nice to know you can express it rationally and reasonably as one over Q or as Q, what the phase of Q means, or the magnitude of Q is given by the beam radius. So, Q, whenever Q, the phase, and the amplitude all tell you useful things, so that kind of suggests it's a useful parameter. And then here, what we finally do, is we see how we can now write the Gaussian beam. So for complete generality here, I've written the Gaussian beam in either one or two dimensions. A one-dimensional beam would be like a sheet of light out of a cylindrical lens and that's not something you use a huge amount of time, but I wanted to put this expression in your notes because it does change the normalizations just a little bit. And sometimes, you can get yourself confused. So most of the time, this term amplitude, in front is important. And we see the whole Gaussian beam amplitude now has collapsed down to really being fairly simple. We have E to the minus jk rho squared over Q and nothing else. And out here for the amplitude term, we just have one over Q. Carrying along with this finally, is the plane wave phase. And lot of times, we drop this term when doing design or analysis because we know, of course, it's a propagating wave and it's oscillating with a period lambda and that fast oscillation is something. It's actually nice to kind of get rid of and so that's an extra term that we carry along here. But really, the most of the Gaussian beam now is just characterized by this much, much simpler expression.