A beam leaving an aperture of diameter D₀ with a full-angle divergence θ grows as it propagates. For a Gaussian beam the diameter at distance z follows from the beam waist relation, and the two terms add in quadrature rather than linearly.

θ is the full-angle divergence in radians and z the distance, so θ in mrad multiplied by z in metres gives millimetres directly. Both forms converge in the far field: at 100 m the linear form runs about 0.5 % high for the default inputs, at 10 m about 5 %, and at 1 m about 34 %. The linear form is never below the true value, which is why it is the one to use where a conservative figure is wanted, such as an eye safety assessment.

Full angle or half angle

Divergence is quoted both ways and the two differ by a factor of two, which is the most common error with this calculation. A half-angle figure is measured from the optical axis; a full angle spans the whole cone. Check the datasheet before entering a number, and note whether the figure is defined at the 1/e² intensity points, at full width half maximum, or at some other criterion, because those definitions differ by another 20 % or so for a Gaussian profile.

Beam quality

Diffraction sets a floor on how slowly a beam of a given diameter can diverge. M² expresses how far a real beam sits above that floor, and a perfect Gaussian has M² = 1. A value below 1 is not physically possible, so it means the entered diameter, divergence and wavelength do not describe the same beam. The Rayleigh range is the distance over which the beam area doubles, and it marks the boundary between the near field, where the beam is still collimated, and the far field, where it grows linearly.

What this does not cover

The calculation is for free propagation of a circular beam in a uniform medium. It says nothing about atmospheric turbulence, thermal blooming, scattering, or clipping on an aperture along the path. An elliptical or astigmatic beam needs the two axes treated separately, since divergence generally differs between them.