Laser Beam Spot Size Calculator

Background.

Point a laser at a wall a few metres away and the dot is bigger than the opening the light came out of. How much bigger depends on three things: how tight the beam was at its narrowest point, the waist; how far the wall is from that waist; and the wavelength. For an ideal Gaussian beam the relationship is exact, and it has a distinctive shape: almost flat at first, then bending into a steady straight-line spread.

The hinge between those two regimes is the Rayleigh range, z_R = πw₀²/λ, the distance over which the beam's cross-sectional area doubles and its radius grows by √2. Inside it the beam is close to parallel, which is why a focused beam holds its small spot only over a short working distance. Several Rayleigh ranges out, the radius grows almost in proportion to distance. For the default 1,064 nm beam with a 0.5 mm waist radius, z_R is about 0.738 m.

The surprising consequence is that a smaller waist does not give a smaller spot far away. The Rayleigh range shrinks with the square of the waist, so a tightly focused beam balloons quickly: at 2 m, a 0.1 mm waist grows to about 6.77 mm, nearly five times the 1.444 mm reached from a 0.5 mm waist. For any target distance there is a best waist, √(λz/π), which puts the target exactly one Rayleigh range away.

The radius reported is the 1/e² radius, where intensity has dropped to about 13.5% of the centre value and inside which about 86.5% of the power travels; double it for the diameter. The model is the ideal TEM₀₀ beam with M² = 1. Real beam quality, astigmatism, lenses, clipping and air turbulence all lie outside it, as the scope note beside the result says.

What is laser beam spot size calculator?

Laser spot size means the radius or diameter of the beam at a given plane. For a Gaussian beam the usual convention is the 1/e² radius, w: the distance from the axis at which intensity falls to e⁻², about 13.5% of the peak. Here w(z) is that radius at a distance z along the axis from the beam waist, the plane where the beam is narrowest, for light of the given vacuum wavelength travelling through free space.

How to use this calculator.

1. 01Enter the waist radius w₀ in metres: 0.5 mm is 0.0005. Datasheets often give a beam diameter, so halve it, and check that it is a 1/e² figure rather than a FWHM one.
2. 02Enter the distance from the waist, not from the laser housing. The datasheet usually says where the waist lies; for many collimated lasers it is close to the output, and after a lens it is near the focal point.
3. 03Enter the vacuum wavelength in metres: 1,064 nm is 0.000001064, 532 nm is 0.000000532 and 632.8 nm is 0.0000006328.
4. 04Read the ideal beam radius and double it for the spot diameter. For a real laser with a known M², multiply the wavelength by M² before entering it to estimate the real beam's size.

The formula.

w(z) = w₀ × √(1 + (zλ / (πw₀²))²). w₀ is the waist radius, z the distance from the waist and λ the vacuum wavelength, all in metres. The group πw₀²/λ is the Rayleigh range z_R, so the formula can also be read as w(z) = w₀√(1 + (z/z_R)²).

Two limits make it easy to sanity-check. Close to the waist, where z is much smaller than z_R, the square-root term is near 1 and the radius barely changes. Far away, where z is many times z_R, the 1 becomes negligible and w ≈ λz/(πw₀), a straight line whose slope is the far-field divergence half-angle. At exactly z = z_R the radius is √2 × w₀.

Wavelength and distance enter only as their product, so doubling either has the same effect. The waist appears twice, once as the starting size and once, squared, inside z_R, which is why small changes to w₀ swing the far-field result so strongly.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 11146-1:2021, laser beam widths, divergence angles and propagation ratios (BIBLIOGRAPHIC/PAYWALL).
2. [2]RP Photonics Encyclopedia, Gaussian Beams.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.