Rayleigh Range (z_R = πw₀²/λ)

Ask how far a laser stays collimated and you are asking a question with no exact answer, because the beam is widening everywhere and never stops. The Rayleigh range is the honest way to give the question a number anyway: it is the distance from the waist at which the beam RADIUS has grown by $\sqrt2$. Since the radius has grown by $\sqrt2$, the area has exactly doubled, and the irradiance on axis has exactly halved. One number, three readings, all the same fact.

The whole character of the equation is in the square. $z_R = \pi w_0^2/\lambda$ — double the waist radius and the collimated run goes up FOURFOLD. That is why a beam is expanded before it is sent anywhere, and why the expansion is worth the aperture it costs downstream. It is also the bill that a tight focus pays: halve the spot to double the irradiance and you keep only a quarter of the depth of focus. On a laser cutting head that trade is usually decided by the material thickness rather than by the spot, because a beam that is only in focus over a fraction of the plate cuts a tapered kerf.

Some vocabulary that trips people up. The confocal parameter $b = 2z_R$ counts both sides of the waist, and is what most people mean when they say "depth of focus" out loud. Some older texts define the Rayleigh range at the half-power point of the intensity rather than the $1/e^2$ radius, which is the same thing said differently, but a few define a "collimation length" at some other criterion entirely — if a supplier quotes a number that does not match this one, ask what criterion it used before assuming either of you is wrong.

The inverse dependence on wavelength deserves a sentence of its own. For a given waist, a CO₂ laser at 10.6 µm holds its width about one sixteenth as far as a HeNe at 633 nm. Long wavelengths diffract harder, always, and optical quality has nothing to do with it. That is the arithmetic behind why infrared systems are built around larger optics for the same job, and why the resolution of a radio telescope is what it is despite the dish being enormous.

One conceptual point worth holding onto. The wavefront is flat at the waist and flat again at infinity, and it is at its most strongly curved right at $z = z_R$. A beam is not "collimated then diverging" — it is doing both continuously, and $z_R$ is simply where the character changes over. Inside it the width is set by the waist; outside it the width is set by diffraction, and the beam has forgotten how wide it started.