Gaussian Beam Optics for Engineers

The Gaussian intensity profile

A laser oscillating in its lowest-order transverse mode (TEM₀₀) has a transverse intensity profile that is very nearly Gaussian:

\[ I(r) = I_0 \exp\!\left(-\frac{2r^2}{w^2}\right), \qquad I_0 = \frac{2P}{\pi w^2} \]

Here \(w\) is the 1/e² radius, where intensity falls to 13.5 % of its peak, and \(P\) is the total power. The 1/e² circle contains 86.5 % of the power. Three consequences come up repeatedly:

The peak is twice the average. On-axis intensity \(I_0 = 2P/(\pi w^2)\) is twice the power divided by the 1/e² area. The same factor of 2 applies to fluence. Whether a datasheet or paper quotes peak or average fluence is one of the most common sources of disagreement about thresholds.
Width conventions differ. The full width at half maximum is \(1.177\,w\), not \(2w\). The second-moment diameter D4σ used by ISO 11146 equals \(2w\) for a perfect Gaussian, but it weights the wings heavily and becomes larger than the visual 1/e² width for beams with side lobes or a noisy background.
A knife edge measures w directly. The distance between the 16 % and 84 % transmission points equals \(w\). The 10 %–90 % distance is \(1.28\,w\). This is the quickest check on a focused spot when you don't have a camera with small enough pixels.

One watt at 1064 nm focused to an 18.6 µm diameter gives a peak irradiance of about 7 × 10⁵ W/cm². The same average power in 10 ns pulses at 100 kHz gives a peak of about 7 × 10⁸ W/cm². The fluence & irradiance calculator works through these numbers in both peak and average form.

Waist, Rayleigh range, and divergence

A Gaussian beam keeps its Gaussian shape as it propagates. Its radius follows a hyperbola about the narrowest point, the waist \(w_0\):

\[ w(z) = w_0\sqrt{1 + \left(\frac{z}{z_R}\right)^2}, \qquad z_R = \frac{\pi w_0^2}{M^2\lambda}, \qquad \theta = \frac{M^2\lambda}{\pi w_0} \]

The beam radius \(w(z)\) is a hyperbola. It equals \(w_0\) at the waist, grows to \(\sqrt{2}\,w_0\) at \(\pm z_R\) (twice the area, half the peak fluence), and approaches straight asymptotes at the far-field half-angle \(\theta\). At \(3z_R\) the radius is already \(3.16\,w_0\).

The Rayleigh range \(z_R\) is the distance from the waist at which the beam area has doubled. The wavefront radius of curvature \(R(z) = z\,[1 + (z_R/z)^2]\) is infinite at the waist and smallest (equal to \(2z_R\)) at \(z = \pm z_R\). In the far field the beam diverges at the half-angle \(\theta\). The phase also picks up an extra Gouy shift of \(\arctan(z/z_R)\), which matters in resonators and interferometers but rarely in materials processing.

A concrete case is a 1064 nm beam with M² = 1.1 focused to \(w_0 = 10\) µm. Then \(z_R = 0.27\) mm and \(\theta = 37\) mrad. The beam stays tight for only about a quarter of a millimeter on either side of focus, and that is typical of precision ablation. The Gaussian beam propagation calculator plots \(w(z)\) for your own numbers.

M² and the beam parameter product

Real beams are not perfect TEM₀₀ modes. The beam quality factor M² (the "beam propagation ratio" in ISO 11146) measures how much faster a beam diverges than an ideal Gaussian with the same waist. The product of waist radius and far-field half-angle, the beam parameter product, is

\[ \mathrm{BPP} = w_0\,\theta = \frac{M^2\lambda}{\pi} . \]

BPP is conserved by ideal lenses, mirrors, and telescopes. You can trade waist for divergence, but you cannot make the product smaller without throwing away power with a spatial filter. For 1064 nm the diffraction limit is 0.34 mm·mrad. Single-mode fiber lasers typically have M² below about 1.1. A fully filled 50 µm core, 0.22 NA delivery fiber has a BPP of about 25 µm × 220 mrad = 5.5 mm·mrad, roughly 16 times the diffraction limit. That is fine for welding and cutting thick sections, but it rules out 20 µm spots at practical working distances.

M² is measured, not inferred from a single profile. ISO 11146 calls for second-moment (D4σ) widths at ten or more positions along the caustic, about half within one Rayleigh range of the waist and half beyond two Rayleigh ranges, followed by a hyperbolic fit. A single camera image near focus says almost nothing about M².

Focusing a collimated beam

When a collimated beam of 1/e² diameter \(D\) passes through a lens of focal length \(f\) and its Rayleigh range is much longer than \(f\), the focused waist sits at the back focal plane with diameter

\[ d_0 = 2w_0 = \frac{4 M^2 \lambda f}{\pi D}. \]

An 8 mm beam at 1064 nm (M² = 1.1) through a 100 mm lens gives an 18.6 µm spot. The formula tells you which knobs exist: a shorter wavelength (the same optics at 355 nm give 6.2 µm), a shorter focal length, a larger input beam, or better beam quality. Run it in reverse to size the optics. A 20 µm spot from a 160 mm f-theta lens needs an input beam of \(D = 4M^2\lambda f/(\pi d_0) \approx 11.9\) mm, which in turn sets the beam expander ratio and the minimum scanner aperture.

The price is depth of focus. If the process tolerates a spot-size growth \(\varepsilon\), the usable focal depth is

\[ \mathrm{DOF} = 2z_R\sqrt{(1+\varepsilon)^2 - 1} \;\approx\; 0.50\,\frac{d_0^2}{M^2\lambda}\quad(\varepsilon = 5\,\%). \]

For a 20 µm spot at 1064 nm that is ±86 µm, or 0.17 mm in total. Because DOF scales with \(d_0^2\), halving the spot cuts the depth of focus by four. In production, height variation of the part, fixture flatness, and thermal drift of the optics all draw on that budget. This is why tightly focused processes often need autofocus or height tracking, or a deliberately larger spot with more pulse energy. The focused spot size calculator reports spot, \(z_R\), and DOF together.

Shorter lens or bigger beam?

Both routes shrink the spot in the same proportion, and for a given final spot size the depth of focus is the same either way. It depends only on \(d_0\), \(M^2\), and \(\lambda\). The choice therefore comes down to everything else:

Working distance and scan field. A longer focal length keeps the lens farther from spatter and fume, leaves room for gas nozzles and fixtures, and for an f-theta lens gives a larger scan field, which grows in proportion to \(f\).
Aperture size and speed. A larger beam needs larger optics, and in a galvo scanner larger mirrors with more inertia, which cost acceleration and marking speed.
Aberrations. Fast lenses (low f-number) introduce spherical aberration sooner. A larger beam on a longer lens can reach the same f-number, so check the lens's specified spot size at your beam diameter.
Optic damage. Expanding the beam lowers fluence on every optic downstream of the expander, which adds margin against laser-induced damage and contamination burn-in.

The beam expander and f-theta lens calculators let you compare the two approaches with your own numbers.

Apertures and truncation

The Gaussian tails extend indefinitely, so every aperture clips some power. A circular aperture of radius \(a\) centered on a beam of radius \(w\) transmits

\[ T_P = 1 - \exp\!\left(-\frac{2a^2}{w^2}\right). \]

An aperture equal to the 1/e² diameter passes 86.5 %. One 1.5 times larger passes 98.9 %, and one twice as large passes 99.97 %. Clipping costs more than the lost power, though. The hard edge diffracts light into rings, the focal spot gets larger than the Gaussian formula predicts, and the clipped light heats mounts and apertures. For a beam filling its aperture at the 1/e² diameter, the spot is about 1.83 \(\lambda f M^2/D\) instead of 1.27 \(\lambda f M^2/D\), roughly 44 % larger. A ratio of beam to aperture diameter of 0.6–0.8 is a reasonable compromise for galvo scanners. The beam clipping tool computes the transmitted power, and the focused-spot tool includes the exact truncation factor.

When the focus is not at f

The simple focusing formula assumes the incoming beam's Rayleigh range is much longer than the focal length. For a 4 mm radius beam at 1064 nm, \(z_R\) is about 43 m, so the assumption holds easily. It fails for small beams and long focal lengths. S. A. Self's thin-lens equation for Gaussian beams makes this explicit. With the input waist a distance \(s\) in front of the lens, the output waist lies at \(s''\) given by

\[ \frac{1}{s + z_R^2/(s - f)} + \frac{1}{s''} = \frac{1}{f}, \qquad w_0'' = \frac{w_0}{\sqrt{(1 - s/f)^2 + (z_R/f)^2}} . \]

Take a 1 mm diameter beam (\(w_0 = 0.5\) mm, \(z_R = 0.74\) m at 1064 nm) with its waist at a 1000 mm lens. The output waist forms 353 mm behind the lens, not 1000 mm, and its radius is 0.40 mm instead of the 0.68 mm the simple formula predicts. The furthest the output waist can ever lie from the lens is \(f + f^2/(2z_R)\). When a long-focal-length system "focuses short," this is usually the reason, not a mislabeled lens.

Practical rules of thumb

State the width convention. Write 1/e² diameter, FWHM, or D4σ, and peak or average fluence, in every specification and report.
Size optics from the BPP. Spot size and working distance are linked through \(M^2\lambda/\pi\). If the specified spot, working distance, and laser BPP don't fit together, no lens will make them fit.
Budget depth of focus explicitly. Sum part flatness, fixture tolerance, stage runout, and thermal focus shift, and compare the total with the ±ε DOF before choosing the spot size.
Keep beam-to-aperture ratios near or below 0.7. Beyond that, a larger input beam gives little improvement in spot size, and pointing errors start to cost power.
Measure the caustic, not one plane. A beam profiler stepped through focus is the fastest way to find a degraded M², a thermal lens, or a focus that has moved after a lens change.

References

A. E. Siegman, Lasers, University Science Books (1986), chapters 16–17.
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), chapter 3.
S. A. Self, “Focusing of spherical Gaussian beams,” Applied Optics 22, 658–661 (1983).
ISO 11146-1:2021, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios.