Tool · Units and conversions

FWHM to Sigma Converter

How does a full width at half maximum compare with a standard deviation, a 1/e² radius or a pulse’s shape parameter? Enter any one width of a Gaussian, Lorentzian or sech² profile and the converter gives all the others, the share of the area inside the FWHM and, for a pulse duration, the transform-limited bandwidth. Background: FWHM, time-bandwidth product and pulse duration.

Width

Profile

The width you have

Value

Unit

Half widths are measured from the center. For a Gaussian the shape parameter is σ; for a Lorentzian it is the HWHM γ; for sech²(t/τ) it is τ. A time unit adds the transform-limited bandwidth of a pulse of that duration.

Presets

Readouts

The profile and its widths

Horizontal lines at half maximum, 1/e and 1/e² (13.5 %) of the peak, each spanning the full width at that level.

Three shapes with the same FWHM

Blue Gaussian, orange Lorentzian, dark sech², all with the current FWHM. The Lorentzian keeps far more of its area in the wings.

Learn with it

Two 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 width ratios and area fractions are compared with their closed forms.

Check	Expected	Computed	Tolerance

The expected values are the closed-form ratios for each profile: 2√(2 ln 2) for the Gaussian, 2 arccosh √2 for sech², and the standard time-bandwidth products 2 ln 2/π and (2 arccosh √2/π)². The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

For a Gaussian $f (x) = e^{- x^{2} / 2 σ^{2}}$ the half-maximum points are at $x = \pm σ \sqrt{2 \ln 2}$ , so

F W H M = 2 \sqrt{2 \ln 2} σ = 2.354820 σ

and the 1/e and 1/e² half widths are $\sqrt{2} σ$ and $2 σ$ . For a laser beam the 1/e² half width is the beam radius $w$ , so $σ = w / 2$ and $F W H M = 1.1774 w$ . A Lorentzian $1 / (1 + x^{2} / γ^{2})$ has FWHM $2 γ$ and 1/e² half width $γ \sqrt{e^{2} - 1}$ , but no finite standard deviation, because its tails fall only as $1 / x^{2}$ . A sech² pulse ${s e c h}^{2} (t / τ)$ , the shape of a soliton and of most passively mode-locked lasers, has

F W H M = 2 a r c c o s h \sqrt{2} τ = 1.762747 τ, σ_{r m s} = \frac{π τ}{2 \sqrt{3}}

A transform-limited pulse has the intensity time-bandwidth product $Δ ν Δ t = 2 \ln 2 / π = 0.4413$ for a Gaussian and 0.3148 for sech², with both widths taken as FWHM. All the ratios are for the profile itself; a measured width also includes the instrument’s own response, which for two Gaussians adds in quadrature.

Worked example

A Gaussian peak with FWHM 2.355 has $σ = 2.355 / 2.354820 = 1.0001$ ; 76.10 % of its area lies between the half-maximum points, against 68.27 % within $\pm σ$ . A 100 fs Gaussian pulse at its transform limit has a spectral width of $0.441271 / 100 f s = 4.4127 T H z$ , 9.420 nm at 800 nm; a 100 fs sech² pulse needs only 3.148 THz, 6.721 nm.

Related: the knife-edge beam width calculator measures a beam radius, the coherence length calculator uses line shapes, and the measuring ultrashort pulses article covers autocorrelation. See also autocorrelation.

References: J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed., Academic Press (2006), chapter 1.