Gaussian beam diameter converter

Data sheets, beam profilers and papers state a laser beam’s size in different ways: at 1/e² of the peak intensity, at 1/e, at half maximum, by a knife-edge scan, or by the power it contains. For an ideal Gaussian beam each is a fixed multiple of the others. Type the size you have in its box, and the others fill in.

Sizes are

1/e² intensitythe same as 1/e of the field (amplitude); for an ideal Gaussian also the ISO 11146 second-moment (D4σ) width, the 86.5 % encircled-power size, and a single-mode fiber’s mode-field diameter 2.0000 w

1/e intensity 1.4142 w

Full width at half maximum (FWHM)also the 50 % encircled-power size; as a radius, the half width at half maximum (HWHM) 1.1774 w

10–90 % knife-edge widththe distance a knife edge moves to go from 10 % to 90 % of the power; a width along one axis, not a diameter 1.2816 w

90 % encircled power 2.1460 w

95 % encircled power 2.4477 w

99 % encircled powera common size for clear apertures that must pass nearly all the beam 3.0349 w

Any one unit for all the boxes: µm, mm or m for a beam’s size, or µrad for its far-field divergence (an angle converts the same way). The last column is each size as a multiple of the 1/e² radius w.

How it is calculated

A round Gaussian beam (the TEM₀₀ mode of a laser, or the output of a single-mode fiber) has intensity

I(r) = I₀ exp(−2r²/w²),

where w is the radius at which the intensity falls to 1/e² (13.5 %) of its peak. Every other measure is a fixed multiple of w:

At a fraction f of the peak intensity the radius is w √(ln(1/f)/2). The diameter is √2 w ≈ 1.4142 w at 1/e, and √(2 ln 2) w ≈ 1.1774 w at half maximum.
Encircled power: the fraction of the power inside radius r is 1 − exp(−2r²/w²), so a fraction p lies within r = w √(ln(1/(1−p))/2). The 1/e² circle holds 86.5 % of the power, and the FWHM circle exactly half.
Knife edge: along one axis the beam’s profile is a normal distribution with standard deviation σ = w/2. A knife edge passes from 10 % to 90 % of the power over 2 × 1.2816 σ = 1.2816 w.
Second moment (ISO 11146): the standard beam width, D4σ = 4σ, equals the 1/e² diameter 2w for an ideal Gaussian.

When not to use it

The ratios hold only for an ideal Gaussian beam. A multimode laser (beam quality M² well above 1), a beam clipped by an aperture, or one with aberrations has a different shape, and its 1/e², FWHM and knife-edge widths no longer convert by these factors. For such beams the second-moment width of ISO 11146 is the one that stays well defined, and it should be measured rather than converted.

References

A. E. Siegman, Lasers (University Science Books, 1986), on the properties of Gaussian beams. ISO 11146-1, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios.