What I'm more interested in and what I think is super exciting is,

turns out once you have Q, you can from that, go get the expressions from locally.

What is the divergence of my beam?

That, of course, doesn't change very often, it only changes when I go across

the lens and what's the waist size of my Gaussian beam?

That, of course, changes dramatically as I move through space.

And what you find is, is that this is the y and u,

the height and angle of these two rays, delta and omega.

You find that the local Gaussian beam divergence and

the local Gaussian beam radius are given this by the square root of a sum of

the squares of the heights and angles of these two rays.

And that is really cool.

It means that, and I've kind of shown this here, that if I'd like to know

the local waist of my Gaussian beam, w(z), I just

take the height of these two rays and take the square root of the sum of the squares.

So for example right here at the origin, the height of the Divergence ray is

zero and so my beam waist must be, my local beam radius must be at the waist.

Check, that's easy.

As I go towards this lens, notice that the divergence ray is increasing.

And if I take the square root of the sum of the squares,

that tells me how the local w(z), the local Gaussian beam radius increases.

If for example, now imagine this lens wasn't here, and so

I keep going with these two rays.

Eventually, the Waist ray, which stays parallel to the origin,

is much, much smaller than the Divergence ray, which keeps getting bigger.

And that means my w(z) asymptotically converges to the Divergence ray,

which is just what it should do.

That makes sense.

If I put a lens in place instead, so now that my Waist ray bends down through

the origin and my Divergence ray comes over here to the image plane, I just take

everywhere and take the absolute magnitude squared about these two quantities.

And that allows me to trace out this w(z), the local Gaussian beam radius.

Notice something super important.

Right here at the focal plane, where the Waist ray on the right

hand side of the lens has gone to zero, its height has gone to zero,

that means that the local Gaussian beam radius must touch the Divergence ray.

That's not where the Gaussian beams minimum, the new waist occurs.

It occurs a little further down, a little further to the right.

And is a matter of fact, you can figure out that the place that this quantity

is minimum, that would be where the waist of the Gaussian beam is over here.

Is actually where the two rays, the Divergence ray height and

the Waist ray height are equal.

And that occurs here-ish or so, equal in amplitude.

And this is a super important concept, it tells you that the waists of a Gaussian

beam, in this case, are occurring neither at the back focal plane,

or at the paraxial image plane, but instead somewhere in between.

And that's another good example of how these Gaussian

beams do not obey the simple paraxial ray equations.

And that we'd screw up if you thought I've got a Gaussian beam over here,

it must have its waist at the image plane, you're wrong.

And that's why this is important.