Tool · Beams and focusing

Gaussian Beam Calculator

How large is a laser beam at a given distance from its waist, how far does it stay collimated, and how small a spot will a lens focus it to? The calculator takes the wavelength, the beam quality M² and either the waist radius or the far-field divergence, and reports the Rayleigh range, divergence, beam radius, wavefront curvature, Gouy phase, peak intensity and the power through an aperture at any distance, then the size and position of the new waist behind a thin lens. Background: Gaussian beam, Rayleigh range, and beam quality (M²).

Beam

Wavelength λ (nm)

Beam quality M² 1 for an ideal Gaussian

Set the beam by

Waist radius w₀ (1/e²)

Far-field half-angle θ (mrad)

At a distance from the waist

Distance z

Power P (mW)

Aperture radius a

Thin lens

Focal length f (mm)

Waist to lens distance d (mm)

d = 0 puts the input waist at the lens, the usual case for a collimated beam.

Presets

Paraxial theory for a beam of circular cross-section. A real beam of quality M² is treated as an embedded Gaussian: its Rayleigh range is divided by M² and its divergence multiplied by it. The lens is thin and aberration-free, and its aperture does not clip the beam.

Readouts

Beam radius versus distance from the waist

beam radius w(z), 1/e²far-field asymptote θzRayleigh range and distance z

Through the thin lens

input beamoutput beamlens

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The closed forms are compared with values worked out by hand, the lens transform with an independent propagation of the complex beam parameter through the ABCD matrix of the lens, and the aperture and intensity formulas with a numerical integration of the Gaussian profile.

Check	Expected	Computed	Tolerance

The expected values are the Gaussian beam relations of Siegman, Lasers, chapter 17, the M² scaling of ISO 11146-1 and the thin-lens transform of Self (1983), evaluated by hand for the stated beams. The tolerance is the largest relative difference from Expected that still passes.

The model

A Gaussian beam of vacuum wavelength $\lambda$ and waist radius $w_0$ , measured where the intensity falls to $1/e^2$ of its peak, keeps its Gaussian profile as it propagates while its radius, wavefront curvature and phase change with the distance $z$ from the waist:

w(z) = w_0\sqrt{1 + \left(\frac{z}{z_R}\right)^2}, \qquad R(z) = z\left[1 + \left(\frac{z_R}{z}\right)^2\right], \qquad \psi(z) = \arctan\frac{z}{z_R}

with the Rayleigh range and far-field half-angle

z_R = \frac{\pi w_0^2}{M^2\lambda}, \qquad \theta = \frac{M^2\lambda}{\pi w_0}

For an ideal beam $M^2 = 1$ . A real beam is treated as an embedded Gaussian: its second-moment radius follows the same law with the Rayleigh range shortened and the divergence raised by $M^2$ , so the beam parameter product $w_0\theta = M^2\lambda/\pi$ is fixed by the source. The peak intensity and the power inside a centred aperture of radius $a$ are

I_0 = \frac{2P}{\pi w^2}, \qquad \frac{P(r<a)}{P} = 1 - e^{-2a^2/w^2}

and hold only for a Gaussian profile. A thin lens of focal length $f$ placed a distance $s$ after the waist forms a new waist a distance $s'$ behind it (Self, 1983):

s' = f + \frac{(s-f)\,f^2}{(s-f)^2 + z_R^2}, \qquad w_0' = \frac{f\,w_0}{\sqrt{(s-f)^2 + z_R^2}}, \qquad z_R' = \left(\frac{w_0'}{w_0}\right)^2 z_R

When $z_R$ is short compared with $s - f$ this reduces to the ray-optics lens law; when the waist sits at the front focus the output waist sits at the back focus with $w_0' = f\theta$ . The theory is paraxial and loses accuracy once the half-angle exceeds about 0.1 rad. It assumes a beam of circular cross-section and a thin, aberration-free lens large enough not to clip the beam; an astigmatic beam such as the raw output of an edge-emitting laser diode needs the calculation done separately in each axis.

Worked example

The default is a collimated 1550 nm beam 2 mm across, $w_0 = 1\ \mathrm{mm}$ . Its Rayleigh range is $\pi w_0^2/\lambda = 2.027\ \mathrm{m}$ and it spreads at 0.4934 mrad. One metre from the waist the radius has grown only to 1.115 mm, the wavefront radius is 5.108 m, and a 1 mm-radius aperture passes 79.98 % of the power. A 50 mm lens at the waist focuses it to a waist radius of 24.66 µm, 49.970 mm behind the lens, just inside the focal length, with a depth of focus $2z_R'$ of 2.465 mm. The estimate $f\theta = 24.67\ \mu\mathrm{m}$ is close because the input Rayleigh range is 40 times the focal length.

References: A. E. Siegman, Lasers, University Science Books (1986), chapter 17. S. A. Self, “Focusing of spherical Gaussian beams,” Applied Optics 22, 658–661 (1983). ISO 11146-1, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley (2007).