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.