two points closest to the maximum are chosen. The 1/e2 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 (0.368) times that of the peak power per unit area. This is the beam diameter definition that is used for computing the maximum permissible exposure to a laser beam. In addition, the Federal Aviation Administration also uses the 1/e definition for laser safety calculations in FAA Order JO 7400.2, Para. 29-1-5d. Measurements of the 1/e2 width only depend on three points on the marginal distribution, unlike D4σ and knife-edge widths that depend on the integral of the marginal distribution. 1/e2 width measurements are noisier than D4σ width measurements. For multimodal marginal distributions (a beam profile with multiple peaks), the 1/e2 width usually does not yield a meaningful value and can grossly underestimate the inherent width of the beam. For multimodal distributions, the D4σ width is a better choice. For an ideal single-mode Gaussian beam, the D4σ, D86 and 1/e2 width measurements would give the same value. For a Gaussian beam, the relationship between the 1/e2 width and the full width at half maximum is , where is the full width of the beam at 1/e2. D4σ or second-moment width The D4σ width of a beam in the horizontal or vertical direction is 4 times σ, where σ is the standard deviation of the horizontal or vertical marginal distribution respectively. Mathematically, the D4σ beam width in the x dimension for the beam profile is expressed as where is the centroid of the beam profile in the x direction. When a beam is measured with a laser beam profiler, the wings of the beam profile influence the D4σ value more than the center of the profile, since the wings are weighted by the square of its distance, x2, from the center of the beam. If the beam does not fill more than a third of the beam profiler's sensor area, then there will be a significant number of pixels at the edges of the sensor that register a small baseline value (the background value). If the baseline value is large or if it is not subtracted out