Glossary entry

Beam waist (w₀)

The narrowest point of a Gaussian laser beam, and the beam's radius there, w₀, measured to the 1/e² intensity points. It sets the Rayleigh range, πw₀²/λ, and the far-field divergence, λ/(πw₀); a lens of focal length f focuses a collimated beam of radius w to a waist of about λf/(πw).

Optics & beams Optics fundamentals Lasers & gainUpdated October 2026

The beam waist of a Gaussian beam is the plane along its axis where the beam is narrowest, and the waist radius $w_{0}$ is the beam's radius there, measured to the points where the intensity has fallen to $1 / e^{2}$ , 13.5% of its peak, which is where the field has fallen to $1 / e$ . The wavefronts are flat at the waist. Away from it the radius grows as

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

where $z_{R}$ is the Rayleigh range and $λ$ the wavelength in the medium, $λ_{0} / n$ . The radius is $\sqrt{2} w_{0}$ at $z = z_{R}$ and $2 w_{0}$ at $\sqrt{3} z_{R}$ , and far from the waist it grows linearly, at the half-angle $θ = λ / (π w_{0})$ . A small waist therefore goes with a short Rayleigh range and a fast-spreading beam: the mode leaving standard single-mode fiber at 1550 nm, with a waist radius of 5.2 µm, has a Rayleigh range of 54.8 µm and spreads at 94.9 mrad (5.4°), while a beam with a 1 mm waist radius stays within $\sqrt{2}$ of it for 2.0 m on either side.

Beam-size conventions

Measure	Value for a Gaussian beam
$1 / e^{2}$ intensity radius	$w_{0}$
$1 / e^{2}$ intensity diameter	$2 w_{0}$
Second-moment diameter, D4σ (ISO 11146)	$2 w_{0}$
Full width at half maximum of the intensity	$\sqrt{2 \ln 2} w_{0} = 1.177 w_{0}$
Knife-edge 10 to 90% distance	$1.282 w_{0}$
Radius containing 86.5% of the power	$w_{0}$

ISO 11146 defines beam widths by the second moment of the intensity distribution, D4σ, because it applies to a beam of any shape and because second-moment widths follow the same growth law as a Gaussian beam's, with $M^{2}$ included. For a Gaussian beam it equals the $1 / e^{2}$ diameter. For other beams the conventions give different widths, so a beam size can only be compared with another measured the same way.

Focusing a beam

A lens of focal length $f$ focuses a collimated beam of radius $w$ to a waist of radius

w_{0} \approx \frac{λ f}{π w} = \frac{2 λ N}{π},

where $N = f / D$ is the working f-number, with $D = 2 w$ the $1 / e^{2}$ diameter of the beam at the lens. The focused beam converges at the half-angle $w / f$ , and a Gaussian beam with that far-field angle has the waist above, so a smaller spot takes a larger beam at the lens or a shorter focal length, and the spot diameter $2 w_{0} = 4 λ N / π$ depends only on the wavelength and the f-number. For a beam with $M^{2} > 1$ the spot is $M^{2}$ times larger. Focused spot diameters $2 w_{0}$ for $M^{2} = 1$ :

f-number $N$	Half-angle $w / f$	532 nm	1064 nm	1550 nm
5	0.10 rad	3.39 µm	6.77 µm	9.87 µm
10	0.050 rad	6.77 µm	13.5 µm	19.7 µm
20	0.025 rad	13.5 µm	27.1 µm	39.5 µm
50	0.010 rad	33.9 µm	67.7 µm	98.7 µm

The formula is exact when the input waist lies in the front focal plane of the lens, and the new waist then lies in the back focal plane. For an input waist at the lens, Self's thin-lens formula gives

w_{0}^{'} = \frac{w}{\sqrt{1 + (z_{R, in} / f)^{2}}},

where $z_{R, in} = π w^{2} / λ$ is the Rayleigh range of the input beam, and the new waist forms at $f / [1 + (f / z_{R, in})^{2}]$ , inside the focal length. The simple formula holds when $z_{R, in}$ is much longer than $f$ . A beam with a 0.5 mm waist radius at 1064 nm has $z_{R, in}$ = 738 mm; a 100 mm lens focuses it to 67.1 µm at 98.2 mm, within 1% of the simple value of 67.7 µm, while a 500 mm lens focuses it to 280 µm at 343 mm, where the simple formula gives 339 µm at 500 mm. The Gaussian Beam Calculator applies Self's formula for an input waist at any distance from the lens.

The Gaussian beam formulas are paraxial and lose accuracy beyond a half-angle of about 0.1 to 0.2 rad, a waist radius of a few wavelengths. At high numerical aperture the focus is no longer Gaussian: $λ_{0} / (π NA)$ gives 121 nm for a 1.4 NA oil-immersion objective at 532 nm, but the real focal spot is set by vector diffraction and is described in the Airy disk and diffraction limit entries.

Choosing and orienting the lens, and the irradiance and fluence at the focus, are covered in Focusing a Laser Beam.

Coupling and mode matching

Two beams couple well only if their waists match in size and position. Two Gaussian modes with waist radii $w_{1}$ and $w_{2}$ in the same plane couple with efficiency $[2 w_{1} w_{2} / (w_{1}^{2} + w_{2}^{2})]^{2}$ , so a 20% difference in size costs 0.14 dB; the mode mismatch loss entry gives the general case, including offsets and tilts. The mode field diameter of a single-mode fiber is $2 w_{0}$ of its nearly Gaussian mode, 10.4 µm for standard fiber at 1550 nm. A lensed fiber focuses that mode to a waist 2 to 8 µm across at a working distance of 5 to 30 µm, depending on the tip, to match the mode of a laser facet or of a silicon inverse taper, which is about 2 to 3 µm across for a tip 170 to 190 nm wide.

Beam quality

A beam that is not an ideal Gaussian has a product of waist radius and far-field half-angle $M^{2}$ times larger,

w_{0} θ = \frac{M^{2} λ}{π},

the beam parameter product, which no lens can reduce. For a given waist such a beam spreads $M^{2}$ times faster, and a lens focuses it to a spot $M^{2}$ times larger (beam quality). The output of a single-mode fiber and of a TEM₀₀ laser has $M^{2}$ close to 1. A multimode fiber with its core and numerical aperture fully filled has a beam parameter product of about the core radius $a$ times the NA, so $M^{2} \approx π a NA / λ$ : about 37 for a 105 µm core with NA 0.22 at 976 nm and 130 for a 400 µm core, also with NA 0.22, at 1064 nm.

Measuring it

A waist is measured by recording the beam width at a series of positions through it, with a knife edge, a scanning slit or a camera, and fitting the hyperbola $w (z)$ , which gives the waist radius, its position and $M^{2}$ together. ISO 11146 asks for at least ten positions, about half within one Rayleigh range of the waist and half beyond two Rayleigh ranges; M² beam quality measurement describes the procedure.

Common questions

What is the beam waist of a laser?

The narrowest point of the beam, and the beam's radius there, $w_{0}$ , measured to the $1 / e^{2}$ intensity points. The Rayleigh range and the far-field divergence both follow from it and the wavelength.

How is the beam waist of a focused beam calculated?

As $w_{0} \approx λ f / (π w)$ for a collimated beam of radius $w$ focused by a lens of focal length $f$ , or $2 λ N / π$ in terms of the working f-number $N = f / 2 w$ , multiplied by $M^{2}$ for a beam that is not an ideal Gaussian. At 1064 nm and f/10 the spot is 13.5 µm across.

What is the difference between beam waist and beam diameter?

The beam diameter is twice the $1 / e^{2}$ radius at any plane along the beam, $2 w (z)$ ; the waist is the plane where it is smallest. Datasheets quote either the waist radius $w_{0}$ or the waist diameter $2 w_{0}$ , so the convention is checked before numbers are compared.

Where is the beam waist of a laser?

Inside or close to the laser. In a cavity with one flat mirror the waist lies on that mirror, in a symmetric two-mirror cavity at its center, and for most diode lasers at the output facet. A lens that focuses the beam forms a new waist near its focal plane.

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 11146-1:2021, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios.