Well, what if we don't go 1:1?

So here I've pulled the waist back to 3 f.

Which brings the image plane to three-halves f.

And what you find very quickly is the waist is still in front of the image, but

it moves rapidly towards the image plane as the waist moves farther away from

the lens.

And there are several ways of thinking about this.

But one is simply that the spherical wave fronts that are hitting the lens,

once I get multiple z naughts away from the waist,

begin to look like spheres emanating from a point.

I can't really tell there was a waist there.

And so they tend to launch spherical waves that are pointed back at

what would've been the image plane.

So it's moving towards our more classical 1 f,

first order geometrical optics.

Finally, just so that you know this works, here's that same example, the 3 f,

three-halves f case, but I've moved the incident Gaussian off the axis.

I tend to think of this as a two ray system, but formally it's three.

Because we have the axis here that I've drawn in gray.

And formally, the heights of the divergence and

waist ray are measured relative to this axial ray.

Which in all previous cases was on the optic axis.

Simple description of what happens is, of course the Gaussian beam,

before it hits the lens, simply propagates shifted off the axis, as you'd expect.

And that you can see because the gray ray here is indeed shifted off the axis.

And then behind the lens, it's all tilted.

And it's tilted by just exactly how much you'd expect,

based on this gray ray coming along the central axis of the Gaussian beam.

So it doesn't move in this paraxial limit, the location of the waist,

so I tend to simply always trace Gaussian beams on axis.

Now if I need to know or understand what happens off axis, I simply kind of patch

it up by noticing or thinking about that principle axial ray.

And just mapping the Gaussian beam to that ray.

But without actually formally calculating all of the ray heights, because they

are normal to the ray back here and the math gets slightly uncomfortable.

But the point is, these rays are very easy to calculate.

They allow you, through simple first order geometry, to calculate the new

properties out here of this Gaussian beam to find the location of the waist and

the waist itself, the z naughts, so the discreptors of the Gaussian beam.

So I find this a very handy back-of-the-envelope way to design with

Gaussian beams.