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In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the y -axis which are half the maximum amplitude.
Gaussian functions are the Green's function for the (homogeneous and isotropic) diffusion equation (and to the heat equation, which is the same thing), a partial differential equation that describes the time evolution of a mass-density under diffusion.
The Gaussian function has a 1/e 2 diameter (2w as used in the text) about 1.7 times the FWHM.. At a position z along the beam (measured from the focus), the spot size parameter w is given by a hyperbolic relation: [1] = + (), where [1] = is called the Rayleigh range as further discussed below, and is the refractive index of the medium.
Comparison of Gaussian (red) and Lorentzian (blue) standardized line shapes. The HWHM (w/2) is 1. Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
The FWHM of the Gaussian profile is = (). The FWHM of the Lorentzian profile is =. An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is: [10]
The 1/e 2 width is important in the mathematics of Gaussian beams, in which the intensity profile is described by () = (). The American National Standard Z136.1-2007 for Safe Use of Lasers (p. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0 ...
All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists. The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal ...
In fiber-optic communication applications, the usual method of specifying spectral width is the full width at half maximum (FWHM). This is the same convention used in bandwidth, defined as the frequency range where power drops by less than half (at most −3 dB). The FWHM method may be difficult to apply when the spectrum has a complex shape.