Editing Beam diameter (section)

=== 1/e<sup>2</sup> width ===
The 1/e<sup>2</sup> width is the distance between the two points on the where the intensity falls to 1/e<sup>2</sup> = 0.135 times the maximum value. If there are more than two points that are 1/e<sup>2</sup> times the maximum value, then the two points closest to the maximum are chosen. The 1/e<sup>2</sup> width is important in the mathematics of [[Gaussian beam]]s, in which the intensity profile is described by <math display=block>I(r) = I_{0} \left( \frac{w_0}{w} \right)^2 \exp \! \left(  \! -2 \frac{r^2}{w^2}\right ).</math>

The American National Standard Z136.1-2007 for Safe Use of Lasers (p.&nbsp;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. The Federal Aviation Administration also uses the 1/e definition for laser safety calculations in FAA Order JO 7400.2, Para. 29-1-5d.<ref>[https://www.faa.gov/documentLibrary/media/Order/7400.2L_Bsc_w_Chg1_dtd_10-12-17.pdf FAA Order JO 7400.2L, Procedures for Handling Airspace Matters], effective 2017-10-12 (with changes), accessed 2017-12-04</ref>

Measurements of the 1/e<sup>2</sup> 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/e<sup>2</sup> width measurements are noisier than D4σ width measurements.  For [[transverse mode|multimodal]] marginal distributions (a beam profile with multiple peaks), the 1/e<sup>2</sup> 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/e<sup>2</sup> width measurements would give the same value.

For a Gaussian beam, the relationship between the 1/e<sup>2</sup> width and the full width at half maximum is <math display=block>2w = \frac{\sqrt 2\ \mathrm{FWHM}}{\sqrt{\ln 2}} = 1.699 \times \mathrm{FWHM},</math> where <math>2w</math> is the full width of the beam at 1/e<sup>2</sup>.<ref name=zemax>{{cite web |url=https://support.zemax.com/hc/en-us/articles/1500005488161-How-to-convert-FWHM-measurements-to-1-e-2-halfwidths |title=How to Convert FWHM Measurements to 1/e-Squared Halfwidths |first=Dan |last=Hill |date=March 31, 2021 |work=Radiant Zemax Knowledge Base |access-date=February 28, 2023}}</ref>