Beam divergence

Every beam of finite width spreads as it travels, because a beam confined to a width $w$ must contain a range of propagation directions of order $\lambda/w$. Close to the beam waist the width barely changes; beyond a few Rayleigh ranges it grows linearly with distance, and the divergence is the angle of that linear growth. It is the number that decides how large a spot a beam makes on a distant target and how much of it a lens or fiber downstream can collect.

For an ideal Gaussian beam the far-field half angle, measured to the $1/e^2$ intensity radius, is

where $w_0$ is the waist radius. A real beam with beam quality factor $M^2$ diverges $M^2$ times faster for the same waist, $\theta = M^2\lambda/(\pi w_0)$, and the product of waist and divergence is the beam parameter product.

Typical values

A helium-neon laser with a 0.4 mm waist radius at 632.8 nm has a half angle of 0.50 mrad, or a full angle of 1.01 mrad; its Rayleigh range is 0.79 m, so the linear far-field regime begins a few metres from the laser. The beam from a fiber collimator with an 11 mm focal length, launched from single-mode fiber with a 5.2 µm mode-field radius at 1550 nm, diverges with a half angle of 0.47 mrad. Edge-emitting laser diodes are at the other extreme: their fast axis spreads over tens of degrees, and the far-field divergence entry covers that case, where the angle is usually quoted as a full width at half maximum.

Conventions

Datasheets differ in two ways that together can change a quoted number by a factor of more than two. Some quote the half angle and some the full angle; laser manufacturers most often quote the full angle, while beam-optics formulas use the half angle. Some use the $1/e^2$ intensity points and some the full width at half maximum; for a Gaussian profile the half-maximum width is $\sqrt{2\ln 2} \approx 1.177$ times the $1/e^2$ radius, so a full-width-at-half-maximum full angle equals 1.177 times the $1/e^2$ half angle. ISO 11146 defines divergence from second-moment widths, which coincide with the $1/e^2$ widths for a Gaussian but not for beams with side lobes or pedestals.

Measurement

Two methods are common. In the far-field method, the beam width is measured at two or more distances well beyond the Rayleigh range, and the slope of width against distance gives the angle. In the focal-plane method, a lens of focal length $f$ is placed in the beam and the width $w_f$ is measured in its back focal plane; the half angle is $w_f/f$, independent of where the lens sits, which makes a far field of a few centimetres available on a bench. Widths come from a camera profiler or a knife-edge measurement. The full ISO procedure, which fits the whole caustic and gives $M^2$ as well, is described in the M² measurement article.

References: A. E. Siegman, Lasers (University Science Books, 1986), Ch. 17; ISO 11146-1:2021, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios.