We just argued that a lens with

one F in front and one F behind is a Fourier transform machine.

Let's see if that works for the Gaussian beam,

which we've already used in exactly that same geometry.

So, I've painted the intensity of a Gaussian beam here in front of and behind the lens.

I used a different color table in front of and behind the lens,

otherwise, we wouldn't be able to see any of this at all.

It will be very, very low intensity.

And we have a waist at the front focal plane, and of course,

we know that that transforms to a new waist at the back focal plane.

And I've drawn the divergence and waist rays here and reminded ourselves of that

the new waist W_0-prime is equal to

the focal length of the lens times the initial divergence, theta_knot right there.

So we know that already.

We know from our Gaussian beam relationships that we could relate

the divergence of the Gaussian beam

in front of the lens to the waist in front of a lens by lambda over pi W_knot.

So just substituting in that Gaussian beam expression.

So now we know how to waist size behind

the lens relates to the initial waist size in front of the lens.

Let's see if that works given

the proposal that this system can also be understood through a Fourier transform.

So let's take the electric field in front of

the lens that's just e_to_the_minus X_over_W_knot either.

That's the initial waist squared.

So that's my Gaussian field right along there.

Now let's take its Fourier transform.

And I've dropped all terms relating to the amplitude,

because they don't really matter here, just care about the shape.

Gaussian beams or Gaussian functions,

Fourier transform to Gaussian functions.

So, the spatial frequency related to this space that transform is here,

and notice the units are right.

We have one over distance here and W_knots and unit of distance.

And to check, that's this Fourier transform of the field along here.

To understand how it shows up in the X-prime coordinate here,

I have to make the scaling. I have to make the substitution.

And that we just saw was that the spatial frequency times F lambda equals X-prime,

or spatial frequency equals X-prime over F lambda.

Again, notice that it's a quantity if one over distance,

so that's appropriate to be spatial frequency.

And now, I've got a function that tells me how the electric field is

distributed along the X-prime variable, because there's X-prime.

So let me rearrange that a little bit and I can write this looking like a new Gaussian,

so I have X-prime and then I have all this junk,

which is always terms gathered up.

So this must be my W_0-prime.

This must be my new waist coordinate.

And what that tells me is the size of the new waist is F lambda over pi W_knot.

The old waist, which is just what we found here.

So, convincingly, or at least conveniently,

if we take the Fourier transform of the field on the front side of the lens,

we get a distribution of spatial frequencies that relate to this field.

We make the substitution,

so that we know how those spatial frequencies

paint themselves along the back of the lens here.

And we get, in the case of a Gaussian, a new Gaussian.

And, we accurately predict what the waist size should be.

The point is, we should be able to do that now for any spatial distribution of

electric field at the front of the lens to

predict how it focuses at the back of the lens.