How this instrument works
Beam radius at distance z answers a narrow, practical question: past the tightest point of a focused beam, how wide has it actually become by the time it reaches a given plane? The formula w(z) = w₀√(1 + (z ⁄ z_R)²) is a hyperbola in disguise — squared spot radius grows in quadrature with distance from the waist, the same quadrature-sum pattern that shows up whenever two independent contributions to a spread combine. Near z = 0 the second term is negligible and w(z) sits close to w₀; far from the waist the term dominates and w(z) grows almost in a straight line, the geometry stretching out into a cone.
The pivot between those two regimes is the Rayleigh range, z_R = π w₀² ⁄ λ, and it is the second number this formula quietly needs. It is the distance at which the beam has grown to √2 times its waist radius — about 41 percent wider — and it depends on the waist squared, so halving the waist quarters the Rayleigh range. A beam focused down to a tight 20-micron spot for cutting or engraving keeps that tightness for barely a couple of millimetres either side of focus; a beam focused to a loose few millimetres, as in a beam-expander output, might stay collimated for tens of metres. That squared relationship is the whole reason focus tolerance gets brutal as spots get smaller.
The result is exact only for an ideal single-transverse-mode (TEM00) Gaussian beam with beam-quality factor M² equal to 1, propagating under the paraxial approximation. Real diode bars, multimode fiber lasers, and beams that have picked up aberrations from bad optics spread faster than this predicts; their true radius is what this formula gives once w₀ is replaced with an M²-scaled effective waist. Push the waist down toward the wavelength itself, as with some tightly confined nanophotonic sources, and the paraxial assumption this equation rests on stops holding.
- Enter the Beam waist radius — the 1/e² intensity radius at the beam's narrowest point, in µm for tight focus spots or mm for wider ones.
- Enter the Wavelength of the laser, in nm for visible and near-IR sources or µm for mid-infrared ones like a CO₂ laser.
- Set Distance from the waist to the plane you care about — the workpiece, sensor, or screen — in mm, cm, or m.
- Read Beam radius at distance z; at z = 0 it equals the waist exactly, then grows following the hyperbola as distance increases.
- Switch the output unit to mm once the spot has grown past a few hundred micrometres, to keep the reading easy to place on a ruler.
Worked example — a 20 µm waist, 500 mm downstream
A diode laser is focused to a beam waist radius of 20 µm at a wavelength of 650 nm — a tight spot typical of the focusing optics on a small engraving or marking laser. Converting units, w₀ = 2 × 10⁻⁵ m and λ = 6.5 × 10⁻⁷ m, which puts the Rayleigh range at z_R = π(2 × 10⁻⁵)² ⁄ (6.5 × 10⁻⁷) ≈ 1.933 × 10⁻³ m, or about 1.93 mm. That is the entire zone in which this beam stays reasonably tight.
Now ask what the spot looks like 500 mm downstream — z = 0.5 m, roughly 259 times the Rayleigh range, deep into the far field. The formula gives w(0.5) = (2 × 10⁻⁵)√(1 + (0.5 ⁄ 1.933 × 10⁻³)²) = 0.0051725743161 m, which the field reports as about 5.173 mm. A spot that started at 20 µm has swollen to over 5 mm wide, roughly 259 times its waist radius, over just half a metre of drift. That is precisely why a laser cutter or engraver's workpiece has to sit within a millimetre or two of the focal plane: stray outside the Rayleigh range and the tight, energy-dense spot needed to cut or mark cleanly has already spread into a faint smear.
Questions
Why is there a square root and a squared term in the formula?
Because the two contributions to the spot's size — the waist itself and the spreading caused by distance — combine in quadrature, not by simple addition, the same pattern used whenever two independent spreads sum. Squaring both, adding them, and taking the square root is what produces the hyperbola w(z) traces as distance from the waist increases in either direction.
What does the Rayleigh range actually represent?
It is the distance from the waist at which the beam radius has grown to √2 times w₀, about 41 percent wider, and it marks the rough boundary between the near field, where the beam stays close to its narrowest size, and the far field, where it spreads almost linearly. Twice the Rayleigh range is often called the depth of focus.
What is the spot size exactly at the beam waist, where z equals zero?
Exactly w₀. When z = 0 the term (z ⁄ z_R)² vanishes, leaving w(0) = w₀√1 = w₀ — the definition of the waist as the single narrowest point the beam ever reaches along its path, before or after which it can only widen.
If I double the distance, does the spot size simply double?
Only once you are well past the Rayleigh range, where the linear term dominates and growth becomes nearly proportional to distance; a beam at 1,000 mm from the waist described here reaches about 10.35 mm, close to double the 5.173 mm at 500 mm. Close to the waist the relationship is far from linear, since w(0) stays flat at w₀ no matter how you nudge z near zero.
How is this different from a beam-divergence angle calculator?
A divergence figure gives only the far-field cone angle and is a poor estimate close to focus, where the beam radius barely changes with distance. This formula returns the true radius at any distance, near or far, by combining the flat behaviour close to the waist with the linear spreading of the far field into one continuous curve.
Does this work for a real, non-ideal laser beam?
Only exactly for a diffraction-limited TEM00 Gaussian beam with M² = 1. Multimode diode bars, poorly collimated fiber outputs, and beams degraded by bad optics spread faster; for those, substitute an M²-scaled effective waist or expect the true spot to be larger than this instrument reports, never smaller.