Beam diameter (1/e², FWHM and D4σ)

A laser beam has no sharp edge, so its diameter is defined by a criterion applied to the intensity profile, and the number depends on the criterion. The three in common use are the $1/e^2$ diameter, the distance across the beam between the points where the intensity has fallen to $1/e^2$ (13.5%) of its peak; the full width at half maximum (FWHM), taken at 50% of the peak; and the second-moment or D4σ diameter, four times the standard deviation of the intensity distribution. For a Gaussian beam with $1/e^2$ radius $w$, the $1/e^2$ and D4σ diameters are both $2w$ and the FWHM diameter is $1.177\,w$. A beam with $w$ = 1 mm is therefore 2.00 mm across by the first two definitions and 1.18 mm by the third.

Definitions and conversions for a Gaussian beam

The intensity of a Gaussian beam carrying power $P$ is

$$I(r) = \frac{2P}{\pi w^2}\,\exp\!\left(-\frac{2r^2}{w^2}\right).$$

Setting $I = I_0/2$ gives the half-maximum radius $w\sqrt{\ln 2/2}$, so

$$\text{FWHM} = \sqrt{2\ln 2}\;w \approx 1.177\,w,$$

and the FWHM diameter is 0.589 times the $1/e^2$ diameter; in the other direction, $2w$ is 1.699 times the FWHM. The $1/e^2$ convention matches the $w$ that appears in the formulas for beam waist, Rayleigh range and beam divergence, which is why it is the default in laser datasheets.

The D4σ diameter along $x$ is $4\sigma_x$, with $\sigma_x^2$ the intensity-weighted variance of $x$ about the centroid. For the Gaussian above, $\sigma_x = w/2$, so D4σ $= 2w$. This second-moment width is the one defined in ISO 11146 and used to define $M^2$ (beam quality), because second-moment widths of any beam, Gaussian or not, grow with distance by the same hyperbolic law.

Power contained within a radius

The fraction of the power inside a circle of radius $r$ is

$$\frac{P(r)}{P} = 1 - \exp\!\left(-\frac{2r^2}{w^2}\right).$$

Computed values:

The 86.5% figure gives rise to a fourth definition, the "86% power-content diameter", the diameter of the circle that passes 86.5% of the power. It equals $2w$ for a Gaussian and is used for beams that are far from Gaussian, such as the output of high-power multimode lasers. The last row is the usual rule for sizing an aperture: an optic of clear diameter $\pi w$ passes all but 0.7% of the beam.

Measurement

A camera beam profiler records the full two-dimensional profile and can report any of these widths. A knife-edge scan records the transmitted power as a blade crosses the beam; for a Gaussian, the distance between the 10% and 90% points is $1.282\,w$, so the $1/e^2$ diameter is 1.561 times the 10–90% width, and the distance between the 16% and 84% points is $w$, half the $1/e^2$ diameter. A beam with $w$ = 1 mm gives a 10–90% width of 1.28 mm. The mode field diameter of a single-mode fiber is the same $1/e^2$ quantity applied to the guided mode.

Pitfalls

Background in the D4σ width. Because the second moment weights intensity by $x^2$, a small offset far from the center has a large effect. Computed for a Gaussian with $w$ = 1 mm on a square integration window 6 mm on a side, a uniform residual background of 0.1% of the peak inflates the D4σ diameter from 2.00 mm to 2.23 mm, and a background of 1% inflates it to 3.50 mm. Profiler software therefore subtracts a measured dark frame and restricts the integration area, typically to about three times the beam width.

Clipping. An aperture that cuts the wings reduces the second moment. A Gaussian with $w$ = 1 mm passed through a 2 mm diameter aperture keeps 86.5% of its power, but its D4σ diameter computed just after the aperture falls to 1.66 mm, 17% below the true value; with a 3 mm aperture the error is 2.6%.

Non-Gaussian beams. For a uniform circular flat-top beam the $1/e^2$, FWHM and D4σ diameters all equal the geometric diameter. For beams with side lobes, a pedestal or several transverse modes, the definitions diverge, and a diameter can only be compared with another measured the same way.

Common questions

What is the 1/e² beam diameter?

The distance across the beam between the points where the intensity is $1/e^2$, 13.5%, of the peak, equal to $2w$ for a Gaussian beam. It contains 86.5% of the beam power.

How do I convert FWHM to 1/e² diameter?

For a Gaussian beam, multiply the FWHM by 1.699. A beam with a 1.0 mm FWHM has a 1.70 mm $1/e^2$ diameter. The factor does not hold for other profiles.

Why does a profiler report D4σ larger than the 1/e² diameter?

For a true Gaussian the two are equal. A larger D4σ usually means uncorrected background, a pedestal or weak higher-order modes in the wings, which the second moment weights heavily.

References: A. E. Siegman, Lasers (University Science Books, 1986); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966).