Optics / 02 / Gaussian beam

From the waist
to the far field.

Enter a wavelength, a waist and the beam quality. See how the beam grows with distance, where it stays collimated, and how fast it diverges.

Free space after a waist. One round, stigmatic beam in a uniform medium. Lenses, astigmatism and elliptical beams are not modelled. Runs in your browser; nothing is sent or stored.

Beam

Vacuum wavelength λ₀

Waist given as

Radius W₀ Diameter 2W₀

Waist radius W₀ (1/e² intensity)

For M² > 1, the second-moment radius (ISO 11146: d = 2W).

Beam quality M² 1 for an ideal TEM₀₀ beam; measured value otherwise.

Refractive index n of the medium 1 for vacuum; about 1.000 27 for air near 1 µm.

Evaluate at distance z from the waist

Negative before the waist. The plot spans at least ±3 z_R.

Result

Rayleigh range z_R: —
Divergence, half-angle θ: —
Full angle 2θ: —
Beam parameter product W₀θ, in the medium: —
Reduced BPP n W₀θ = M²λ₀/π, as in air: —
Radius W(z) at the entered z: —
Diameter 2W(z): —
Wavefront radius R(z): —
Wavelength in the medium λ₀/n: —

Solid lines: ±W(z), the 1/e² (second-moment) envelope. Dashed vertical lines: ±z_R, where the radius is √2 W₀. Marker: the entered distance. Values are in the table below.

Values in a table

Beam radius and wavefront radius at selected distances
z	z / z_R	W(z)	R(z)

Model and limits

z_{R} = \frac{π n W_{0}^{2}}{M^{2} λ_{0}}

Rayleigh range

W (z) = W_{0} \sqrt{1 + {(\frac{z}{z_{R}})}^{2}}

beam radius at distance z from the waist

θ = \frac{M^{2} λ_{0}}{π n W_{0}}

far-field half-angle divergence

BPP = W_{0} θ = \frac{M^{2} λ_{0}}{π n}

beam parameter product inside the medium; n W₀θ = M²λ₀/π as in air

R (z) = z (1 + {(\frac{z_{R}}{z})}^{2})

wavefront radius of the embedded Gaussian beam

W is a radius: where a TEM₀₀ intensity falls to 1/e² (13.5 %) of its peak. For M² > 1 it is the second-moment radius of ISO 11146, which equals the 1/e² radius only for a pure Gaussian.
M² > 1 uses the embedded-Gaussian model (Siegman 1990): the real beam has the same waist position and Rayleigh range as a TEM₀₀ beam of waist W₀/√M², with radii √M² larger. R(z) is the curvature of that embedded beam.
Paraxial: the results assume small angles. A note appears above θ = 0.1 rad, where tan θ and θ differ by 0.3 %; for M² = 1 that is a waist of about 3 wavelengths, for larger M² a proportionally larger one.
W₀θ is smaller inside a medium by the factor n. The reduced product n W₀θ = M²λ₀/π does not change at an interface and equals the BPP of the same beam in air; data sheets quote that value.
One plane, round beam, uniform medium, no lenses or apertures. An elliptical or astigmatic beam needs a calculation per axis; a beam with M² entered per axis is not the same as this round model.
The waist position is the origin. The tool does not find a waist from a measured radius at another plane.

Sources

A. E. Siegman, Lasers (University Science Books, 1986), ch. 17.
A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2 (1990).
ISO 11146-1:2021, beam widths, divergence angles and beam propagation ratios.
Same formulas as the LaserShark module physics/m2.py.

From the waistto the far field.

Beam

Result

Model and limits

Sources

From the waist
to the far field.