Definitions

# Full width at half maximum

A full width at half maximum (FWHM) is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value.

FWHM is applied to such phenomena as the duration of pulse waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers.

The term full duration at half maximum (FDHM) is preferred when the independent variable is time.

When the considered function is the normal distribution of the form

$f\left(x\right) = frac\left\{1\right\}\left\{sigma sqrt\left\{2 pi\right\} \right\} exp left\left[-frac\left\{\left(x-x_0\right)^2\right\}\left\{2 sigma^2\right\} right\right]$

where $sigma$ is the standard deviation and $x_0$ can be any value (the width of the function does not depend on translation). The relationship between FWHM and the standard deviation is

$mathrm\left\{FWHM\right\} = 2 sqrt\left\{2 ln 2 \right\} ; sigma approx 2.35482 ; sigma.$

Another important function, related to solitons in optics, is the hyperbolic secant:

$f\left(x\right)=operatorname\left\{sech\right\} left\left(frac\left\{x\right\}\left\{X\right\} right\right).$

Any translating element was omitted, since it does not affect the FWHM. For this impulse we have:

$mathrm\left\{FWHM\right\} = 2 ; operatorname\left\{arsech\right\} left\left(frac\left\{1\right\}\left\{2\right\} right\right) X = 2 ln \left(2 + sqrt\left\{3\right\}\right) ; X approx 2.633 ; X$

where arsech is the inverse hyperbolic secant.

## References

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