Full width at half maximum (FWHM)

The full width at half maximum (FWHM) is the distance between the two points on either side of a peak where the signal has fallen to half of its maximum value. It is the default width of a spectral line, pulse, beam profile or resonance because it can be read off a trace without fitting a model. A telecom DFB laser has a linewidth FWHM of roughly 1–10 MHz, a mode-locked Ti:sapphire pulse an intensity FWHM of 10–100 fs, and the mode leaving standard single-mode fiber an intensity FWHM of about 6.1 µm at 1550 nm, for a 1/e² radius of 5.2 µm. The half-width at half maximum (HWHM) is half of the FWHM and is the natural parameter of a Lorentzian.

Formulas for common line shapes

The FWHM is tied to each shape's own width parameter by a fixed factor. For a Gaussian $\exp[-x^2/(2\sigma^2)]$ with standard deviation $\sigma$,

$$\text{FWHM} = 2\sqrt{2\ln 2}\;\sigma \approx 2.355\,\sigma .$$

A Gaussian beam is usually described by its 1/e² intensity radius $w$, for which the intensity is $\exp(-2r^2/w^2)$, and then

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

so the 1/e² diameter $2w$ is 1.70 times the FWHM. A Lorentzian $1/[1 + (x/\gamma)^2]$ has FWHM $= 2\gamma$, where $\gamma$ is the HWHM; it describes homogeneously broadened lines and the transmission peaks of a high-finesse cavity. A sech² pulse $\mathrm{sech}^2(t/T_0)$ has FWHM $= 1.763\,T_0$, and a Gaussian pulse written as $\exp(-t^2/T_0^2)$ has FWHM $= 1.665\,T_0$, the convention used in the pulse duration entry. Mixing these conventions produces errors of a factor of 1.18, 1.41 or 2.

Converting spectral FWHM between wavelength and frequency

For a narrow line, a width in wavelength converts to one in frequency as

$$\Delta\nu = \frac{c\,\Delta\lambda}{\lambda^2} .$$

At 1550 nm, 0.1 nm corresponds to 12.48 GHz, and 0.8 nm to 99.8 GHz, which is why the 100 GHz DWDM grid is often described as 0.8 nm spacing. In the other direction, a 1 MHz laser linewidth at 1550 nm is 8.0 fm, far below the resolution of a grating optical spectrum analyzer, so narrow linewidths are measured with heterodyne or self-heterodyne methods instead (see How to Measure Laser Linewidth). The Photonics Unit Converter converts linewidths between GHz and nm at any center wavelength.

Measuring FWHM from data

From a sampled trace:

1. Subtract the baseline. The half-maximum level is halfway between the baseline and the peak, so a trace that peaks at 1.00 V on a 0.10 V offset has its half level at 0.55 V; using 0.50 V widens the result.
2. Find the two crossings by linear interpolation between neighboring samples. On a spectrum sampled every 0.05 nm, with 0.48 V at 1549.80 nm and 0.62 V at 1549.85 nm, the rising crossing is at 1549.825 nm. With 0.63 V at 1550.15 nm and 0.49 V at 1550.20 nm, the falling crossing is at 1550.179 nm.
3. Take the difference: 0.354 nm, or 44.1 GHz at 1550 nm.

Instrument resolution adds to the measured width. For Gaussian shapes the widths add in quadrature, so a 50 ps pulse recorded with a 25 ps system response reads about 55.9 ps; Lorentzian widths add linearly. On a beam profiler, pixels larger than about a tenth of the FWHM bias the result upward. A width measured on a logarithmic (dB) display should be taken 3 dB below the peak, which is the half-power point.

Where FWHM is the wrong summary

The FWHM ignores the wings. A Lorentzian is still at 20% of its peak two half-widths from center, where a Gaussian of the same FWHM has fallen to 6.25%, and at 10% at three half-widths against 0.2% for the Gaussian. The area under a peak is 1.064 times peak × FWHM for a Gaussian and 1.571 times for a Lorentzian, which matters when integrating a spectral line to get its power. For beams with side lobes or pedestals, the ISO 11146 second-moment (D4σ) width is the standard measure of beam size (see M² Beam Quality Measurement), and for pulses with satellites an autocorrelation FWHM can look short while much of the energy lies outside it.

Common questions

How do you calculate FWHM from standard deviation?

Multiply by $2\sqrt{2\ln 2} = 2.3548$. A Gaussian with $\sigma$ = 1 nm has a FWHM of 2.355 nm. The relation holds only for a Gaussian.

Is FWHM the same as the 3 dB bandwidth?

For a power or intensity spectrum, yes: half the peak power is 3.01 dB down. For an amplitude or field quantity, half maximum corresponds to 6 dB, so the 3 dB width of a field profile, taken at 70.7% of the peak amplitude, is narrower than its half-amplitude width.

What is the FWHM of a Gaussian laser beam?

1.177 times the 1/e² radius $w$, or 0.589 times the 1/e² diameter. A beam with $w$ = 1 mm has an intensity FWHM of 1.18 mm.

How does FWHM relate to the time-bandwidth product?

For a transform-limited pulse, the product of the spectral FWHM in hertz and the temporal FWHM is 0.441 for a Gaussian and 0.315 for a sech² shape; the time-bandwidth product entry covers chirped pulses.

References: Saleh & Teich, Fundamentals of Photonics 3rd ed. 2019; Siegman, Lasers 1986; Hecht, Optics 5th ed. 2017; ISO 11146-1:2021, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios, Part 1: Stigmatic and simple astigmatic beams.