Focused Spot Diameter of a Gaussian Beam

This is the working equation of laser materials processing: $d_f = 4M^2\lambda f/(\pi D)$, the $1/e^2$ diameter that a collimated Gaussian beam of diameter $D$ is brought to by a lens of focal length $f$. It is really the divergence equation seen from the other end — the lens converts the input beam's width into an angle, and that angle sets the new waist — but writing it this way puts the four numbers an engineer actually controls on one line.

Three of them are levers and one is not. A shorter lens gives a smaller spot, in proportion. A wider input beam gives a smaller spot, in inverse proportion. A shorter wavelength gives a smaller spot, in proportion — and that ratio is most of the reason fibre lasers at 1064 nm displaced CO₂ at 10.6 µm for cutting thin metal, since the same optics give a ten-times-smaller spot and a hundred-times-higher irradiance. The fourth, $M^2$, is not a lever at all. Nothing in the optical train reduces it, and a beam expander that improves your spot by three does not improve your $M^2$ by anything.

Expanding the input beam is usually the cheapest of the three. A 3× expander thirds the focused spot without touching the focusing lens or the working distance. What it costs is aperture: every mirror mount, scanner aperture and protective window downstream now has to clear the wider beam with room to spare, and a Gaussian needs about 1.5 times its $1/e^2$ diameter of clear aperture before the clipping stops mattering. Below that, the wings you cut off come back as rings around the focus.

Now the disagreement, because this page and the Airy-disc page will hand you different numbers for the same lens and both are right. Feed both $f = 100$ mm, $D = 10$ mm and $\lambda = 1064$ nm: this equation gives 13.55 µm and $d = 2.44\lambda f/D$ gives 25.96 µm. The ratio is $2.44 \div (4/\pi) = 1.92$. They are not competing estimates of one quantity. The Airy formula describes a pupil illuminated UNIFORMLY, edge to edge, and reports the first-null diameter of the resulting ring pattern. This one describes a Gaussian that is bright on axis and down to 13.5% at the rim, and reports a $1/e^2$ diameter. Different illumination, different width criterion, two exact answers to two different questions.

You may have heard the two quoted as "about 30% apart", and that comparison is a real one — it just assumes something this page does not. If a Gaussian is deliberately over-filled so that its $1/e^2$ diameter is $D/1.5$ and its wings actually reach the aperture edge at $D$, the Gaussian spot widens by half, to $6\lambda f/(\pi D) = 1.91\lambda f/D$, and the two answers land within 28% of each other. That truncation is what a real focusing head does. So the honest summary is: at equal $D$ the two differ by 1.92, and at equal APERTURE with a realistically truncated Gaussian they differ by about a quarter. If you computed both and got different numbers, you have learned something about your illumination rather than found an error.

Finally, the number the spot equation hides: depth of focus. The Rayleigh range of the new waist is $\pi (d_f/2)^2/(M^2\lambda)$, and it falls with the SQUARE of the spot. On a cutting head it is usually the depth of focus, not the spot, that decides which lens goes in.