General · EdgepediaSeptember 17, 20268 min read

Gaussian beam

A Gaussian beam is a beam of light whose transverse electric field and intensity profiles are Gaussian functions, with wavefronts of spherical curvature that are flat at the beam waist, the narrowest point of the beam.¹ It is the fundamental transverse mode (TEM $_{00}$ ) of a laser resonator and the standard reference for beam quality: an ideal Gaussian beam has the diffraction-limited beam quality, corresponding to an $M^2$ factor of 1.¹^•²

Key fact	Detail
Defining profile	Transverse field and intensity are Gaussian; intensity falls to $1/e^2$ of its peak at the beam radius $w$ from the axis.³
Free parameters	The whole beam profile is fixed by two parameters, the wavelength $\lambda$ and the beam waist radius $w_0$ .³
Rayleigh length	$z_R = \pi w_0^2 / \lambda$ , the distance from the waist at which the beam radius grows by $\sqrt{2}$ and the cross-sectional area doubles.⁴
Far-field divergence	$\theta = \lambda / (\pi w_0)$ ; the beam parameter product $w_0 \theta = \lambda/\pi$ depends only on wavelength.¹
Gouy phase	$\zeta(z) = \arctan(\lambda z / \pi w_0^2)$ , an extra phase shift accumulated passing through the waist; it sets the resonance frequencies of optical cavities.³^•¹
Beam quality	$M^2 = 1$ for an ideal Gaussian beam; helium-neon gas lasers typically fall between 1 and 1.1.²
Validity	The standard formulas rest on the paraxial approximation, which requires a waist radius large compared with the wavelength.¹

Field description

Within the paraxial approximation, a monochromatic Gaussian beam propagating in the $z$ direction has a complex electric field amplitude

$E(r,z) = E_0 \frac{w_0}{w(z)} \exp\left(-\frac{r^2}{w(z)^2}\right) \exp\left(i\left[kz - \arctan\frac{z}{z_R} + \frac{k r^2}{2 R(z)}\right]\right)$

where $E_0$ is the peak amplitude at the waist, $k = 2\pi/\lambda$ is the wavenumber, $w(z)$ is the beam radius, $R(z)$ the wavefront radius of curvature, and the arctangent term is the Gouy phase.¹ The intensity profile is Gaussian, $I \propto e^{-2r^2/w^2}$ , so the radius $w$ marks where the intensity has fallen to $1/e^2$ of its on-axis value.³

The beam radius evolves as

$w(z) = w_0 \sqrt{1 + \left( z/z_R \right)^2}$

with the Rayleigh length

$z_R = \frac{\pi w_0^2}{\lambda}$

which sets the length over which the beam propagates without diverging significantly. Older literature often uses the confocal parameter $b$ , which is twice the Rayleigh length.¹^•⁴ As the beam expands, total power is conserved, so the peak intensity decreases in proportion.³

Wavefront curvature and divergence

The wavefronts are flat at the waist and curved elsewhere. The radius of curvature follows

$R(z) = z \left[ 1 + \left( z_R / z \right)^2 \right]$

so $R \to \infty$ as $z \to 0$ ; the curvature is weak both very close to the focus and far from it, and is greatest around $z = z_R$ .¹^•³

In the far field, where $z$ is much larger than $z_R$ , the divergence half-angle is

$\theta = \frac{\lambda}{\pi w_0}$

so a smaller waist radius or a longer wavelength gives stronger divergence. The product of waist radius and divergence angle, the beam parameter product, equals $\lambda/\pi$ for an ideal Gaussian beam and is unchanged by any aberration-free optical system that transforms one Gaussian beam into another.¹^•⁵ For a 1064 nm beam, this minimum beam parameter product is about 0.339 mm-mrad.²

The Gouy phase

The term $-\arctan(z/z_R)$ in the field expression is the Gouy phase shift, an additional phase retardation relative to an on-axis plane wave that accumulates as the beam passes through its waist. Compared with a plane wave, Hermite-Gaussian modes have a slightly slower phase velocity, especially close to the waist.¹^•⁶ For the fundamental beam the Gouy phase is $\zeta(z) = \arctan(\lambda z / \pi w_0^2)$ .³

The Gouy phase has practical consequences. It determines the resonance frequencies of optical resonators, since each transverse mode accumulates a mode-dependent Gouy phase per round trip.¹ For a Hermite-Gaussian mode of order $n$ and $m$ in the two transverse directions, the Gouy phase is $(n + m + 1) \arctan(z/z_R)$ , so higher-order modes accumulate more phase and their resonance frequencies are separated from the fundamental.⁷

Several interpretations of the phase anomaly exist. One analysis relates the finite Gouy phase reduction of a Gaussian mode to the averaged phase retardation of the tilted plane-wave components in its Fourier spectrum compared with an on-axis plane wave, providing a unified view of resonance conditions in cavities with plane and curved mirrors; other interpretations are based on propagation path lengths in real space, Berry's phase, or a geometrical quantum effect.⁸

Higher-order modes

Solutions of the paraxial wave equation form two main families. Hermite-Gaussian modes multiply the fundamental Gaussian by Hermite polynomials in $x$ and $y$ , giving intensity patterns with rectangular symmetry; Laguerre-Gaussian modes are the cylindrical-coordinate solutions, with circularly symmetric doughnut-shaped profiles for nonzero azimuthal order.³^•⁹^•¹⁰ Each family forms a complete basis, so any laser beam can be described as a superposition of either.⁹

Laguerre-Gaussian beams with nonzero azimuthal index $l$ carry orbital angular momentum and have a dark spot at the center due to a phase discontinuity; their field amplitude varies as $\exp(i l \phi) \exp(-i(2p + l + 1)\arctan(z/z_R))$ , showing the mode-dependent Gouy phase.¹⁰ Hermite-Gaussian modes can be converted into Laguerre-Gaussian modes with cylindrical-lens mode converters that exploit the different Gouy phase accumulated in the two transverse directions of an astigmatic beam.⁹ Higher-order modes share the same Rayleigh length as the fundamental Gaussian mode with the same waist parameter $w_0$ , because their profiles scale with $w(z)$ .⁴ Pure higher-order Laguerre-Gaussian modes have also been investigated for gravitational-wave detectors, where they can reduce the impact of mirror thermal noise on sensitivity.⁶

Beam quality and real beams

Real laser beams deviate from the ideal Gaussian model. The $M^2$ factor, defined in the ISO 11146 standard from second-moment beam widths, compares a beam's propagation to that of a pure TEM $_{00}$ Gaussian beam: $M^2 = 1$ is diffraction-limited, values above 1 indicate faster divergence and poorer focusability, and a value below 1 is impossible.² For a non-ideal beam the effective Rayleigh length is $z_{R,\text{eff}} = \pi w_0^2 / (M^2 \lambda)$ , shorter than that of an ideal beam with the same waist radius.⁴

Measuring $M^2$ requires more than a single profile: ISO 11146 specifies five separate beam-radius measurements at various positions in the near field and far field, because a multimode combination can look Gaussian in one plane yet propagate quite differently.² For this reason a simple Gaussian fit to a beam profile can be deceptive, suggesting propagation properties the beam does not have.¹¹

Transformation by optics

The state of a Gaussian beam at a position $z$ can be summarized in a complex $q$ parameter,

$q(z) = z - i\, z_R$

Free-space propagation simply adds to $q$ , and passage through any paraxial optical element is described by the ABCD matrix transformation

$q' = \frac{Aq + B}{Cq + D}$

so Gaussian beam optics can be computed with the same matrices used for ray tracing.¹ The results sometimes contradict ray-optics intuition: a Gaussian beam waist cannot be imaged to infinity, and when the object waist sits one focal length from a lens, the image waist appears one focal length on the other side rather than at infinity as the thin-lens equation predicts.¹²

Practical use and limits

The Rayleigh length is the working quantity in focusing problems: it essentially determines the depth of focus, and when pumping a laser crystal or a nonlinear frequency-doubling crystal it is often advisable to make the Rayleigh length comparable to the crystal length, since tighter focusing raises intensity only over a shorter length before divergence takes over.⁴ For collimation, the required waist follows directly from the formulas: a 1064 nm beam collimated over 1 m needs a Rayleigh length of the order of 1 m, implying a Gaussian beam diameter of 1.2 mm or larger.¹³

The paraxial model has limits. It assumes the waist radius is large compared with the wavelength, so that divergence stays moderate and the Rayleigh length substantially exceeds the beam radius. For very tightly focused beams the approximation fails, and a more complex treatment is needed that may also account for the vector character of the field, including a longitudinal polarization component.¹ Within the model, the polarization of a Gaussian beam may be linear, circular, or elliptical, but it must be uniform across the profile; radially or azimuthally polarized beams are excluded because they require zero intensity at the beam center.¹

References

Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Quantum optics and photonics › Laser physics

Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026

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