Glossary entry

Gaussian beam

The beam of a single-mode laser or fiber, whose intensity profile is Gaussian at every distance. It is set by its waist radius w₀ and wavelength: nearly collimated within the Rayleigh range πw₀²/λ of the waist, and diverging at the half-angle λ/(πw₀) beyond it.

Optics & beams Optics fundamentals Lab practiceUpdated October 2026

A Gaussian beam is a light beam whose intensity across its width follows a Gaussian curve at every distance along it. It is the beam emitted by single-mode lasers and single-mode fibers, the fundamental TEM₀₀ mode, and it keeps its Gaussian shape through free space and ideal lenses, changing only in width and wavefront curvature. At its narrowest point, the beam waist, its $1 / e^{2}$ radius is $w_{0}$ , and away from the waist the radius grows as

w (z) = w_{0} \sqrt{1 + {(\frac{z}{z_{R}})}^{2}}, z_{R} = \frac{π w_{0}^{2}}{λ} .

Within the Rayleigh range $z_{R}$ of the waist the beam is nearly collimated; far beyond it the beam spreads at the constant half-angle $θ = λ / (π w_{0})$ , its divergence. A helium-neon laser beam with a 0.4 mm waist stays collimated over 0.79 m and diverges at 0.50 mrad, while the 5.2 µm mode leaving a single-mode fiber at 1550 nm diverges at 95 mrad, or 5.4°, and has doubled its area 55 µm from the fiber end.

Intensity profile and beam width

A beam of power $P$ and radius $w$ has the intensity profile

I (r) = \frac{2 P}{π w^{2}} \exp ⁣ (- \frac{2 r^{2}}{w^{2}}),

so its peak intensity $2 P / (π w^{2})$ is twice the power divided by the area $π w^{2}$ . The fraction of the power inside a radius $r$ is $1 - \exp (- 2 r^{2} / w^{2})$ : 86.5% within $w$ , 98.9% within $1.5 w$ , and 99.3% within an aperture of diameter $π w$ , the usual rule for sizing optics that must pass the beam without clipping.

Beam widths are quoted in several conventions, all fixed multiples of $w$ for a Gaussian:

Width	In terms of the $1 / e^{2}$ radius $w$
$1 / e^{2}$ intensity radius	$w$
$1 / e$ intensity radius ( $1 / e$ field radius)	$0.707 w$
Half width at half maximum	$0.589 w$
Full width at half maximum	$1.177 w$
Second-moment (D4σ) diameter, used by ISO 11146	$2 w$
10–90% distance of a knife-edge scan	$1.28 w$

The radius is measured with a camera beam profiler or by a knife-edge scan. For the mode of a single-mode fiber, the mode field diameter is $2 w$ .

Propagation

Besides the radius $w (z)$ , a Gaussian beam has a wavefront radius of curvature and a phase that change along the beam:

R (z) = z [1 + {(\frac{z_{R}}{z})}^{2}], ζ (z) = \arctan \frac{z}{z_{R}} .

The wavefront is flat at the waist, most strongly curved at $z = \pm z_{R}$ , where $R = 2 z_{R}$ , and approaches a spherical wave centered on the waist far away. The Gouy phase $ζ$ adds an extra phase shift of $π$ in total as the beam passes through a focus; higher-order transverse modes accumulate more of it, which shifts their resonance frequencies in a cavity. The distance $2 z_{R}$ over which the beam stays within $\sqrt{2}$ of its waist radius is the confocal parameter, the usual measure of the depth of focus of a laser beam.

The product of waist radius and divergence angle is fixed: $w_{0} θ = λ / π$ . A beam can be made small or well collimated but not both, and this product, the beam parameter product, is the smallest any beam of that wavelength can have. Through lenses and mirrors, the beam is propagated with the complex beam parameter $q$ , defined by $1 / q = 1 / R - i λ / (π w^{2})$ , which transforms as $q_{2} = (A q_{1} + B) / (C q_{1} + D)$ with the ray-transfer matrix of each element; the ABCD matrix entry describes the method.

Beam	Wavelength	$w_{0}$	$z_{R}$	$θ$
Helium-neon laser output	633 nm	0.4 mm	0.79 m	0.50 mrad
Single-mode fiber output (MFD 10.4 µm)	1550 nm	5.2 µm	55 µm	95 mrad (5.4°)
Focus of a lens with NA 0.1	1550 nm	4.9 µm	49 µm	0.1 rad
Focus of an objective with NA 0.5	633 nm	0.40 µm	0.81 µm	0.5 rad

The last row is outside the range where these paraxial formulas are accurate: once the waist approaches the wavelength, they give only an estimate of the focal spot.

Focusing and collimating with a lens

A collimated beam of radius $W$ focused by a lens of focal length $f$ forms a waist of radius

w_{0}^{'} = \frac{λ f}{π W},

provided the input's Rayleigh range is long compared with $f$ . A 1064 nm beam 5 mm in diameter focused by a 100 mm lens has a 13.5 µm waist radius and a confocal parameter of 1.08 mm. The same relation run backward collimates a fiber output: the 5.2 µm mode of a single-mode fiber at 1550 nm, collimated by an 11 mm lens, becomes a beam of 1.04 mm radius whose Rayleigh range is 2.2 m. When the input waist is near the lens, or the Rayleigh range is comparable with the focal length, the output waist size and position differ from what geometric imaging predicts; the Gaussian Beam Calculator applies the exact thin-lens transform for any waist, distance and $M^{2}$ .

Real beams and M²

A real laser beam that is not a pure TEM₀₀ mode spreads faster than a Gaussian of the same waist. Its beam quality factor $M^{2} = π w_{0} θ / λ$ is 1 for a Gaussian and larger for every other beam, and the beam's radius follows the Gaussian formula with the Rayleigh range reduced to $π w_{0}^{2} / (M^{2} λ)$ . $M^{2}$ is measured by fitting the beam radius at several distances through a focus, following ISO 11146, as described in the M² measurement article.

Coupling between Gaussian beams

The fundamental mode of a single-mode fiber or waveguide is close to Gaussian, so the efficiency of coupling one into another follows from the overlap of two Gaussians. For aligned beams with their waists in the same plane,

η = {(\frac{2 w_{1} w_{2}}{w_{1}^{2} + w_{2}^{2}})}^{2},

which gives 28% (5.5 dB of loss) between the 10.4 µm mode of standard fiber and a 3 µm waveguide mode, the reason chip couplers use spot-size converters and lensed fibers. Offsets, tilts and waist separations reduce it further; the mode mismatch loss entry gives the full expressions, and coupling a free-space beam into single-mode fiber works through the practical case.

Common questions

What is the Rayleigh range of a Gaussian beam?

The distance from the waist at which the beam radius has grown by $\sqrt{2}$ and its area has doubled, $z_{R} = π w_{0}^{2} / λ$ . A 1 mm waist radius at 1550 nm gives 2.0 m; a 5 µm waist gives 51 µm.

What is the difference between the beam waist and the beam diameter?

The waist $w_{0}$ is the smallest radius of the beam, at its focus. The beam diameter $2 w (z)$ is the width at any given position, equal to $2 w_{0}$ only at the waist.

How do you calculate the focused spot size of a Gaussian beam?

For a collimated input of $1 / e^{2}$ radius $W$ and a lens of focal length $f$ , the focused waist radius is $λ f / (π W)$ ; in diameters, $4 λ f / (π D)$ . A beam of real quality $M^{2}$ focuses $M^{2}$ times larger.

Why does a Gaussian beam stay Gaussian?

The Fourier transform of a Gaussian is a Gaussian, and paraxial diffraction and ideal lenses act on the beam through such transforms, so the profile keeps its shape and only its width and wavefront curvature change. The higher-order Hermite-Gaussian and Laguerre-Gaussian modes keep their shape in the same way. Any other profile is a mixture of these modes, and it changes shape as it propagates because each mode picks up a different Gouy phase.

References: H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966). A. E. Siegman, Lasers (University Science Books, 1986). S. A. Self, "Focusing of spherical Gaussian beams," Applied Optics 22, 658 (1983). ISO 11146-1, Lasers and laser-related equipment: test methods for laser beam widths, divergence angles and beam propagation ratios.