Editing Beam diameter (section)

=== 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 <math> I(x,y) </math> is expressed as<ref name=Siegman>{{cite web |first=A. E. |last=Siegman |url=http://www.stanford.edu/~siegman/beams_and_resonators/beam_quality_tutorial_osa.pdf |title=How to (Maybe) Measure Laser Beam Quality |date=October 1997 |access-date=July 2, 2014 |archive-url=https://web.archive.org/web/20110604095354/http://www.stanford.edu/~siegman/beams_and_resonators/beam_quality_tutorial_osa.pdf |archive-date=June 4, 2011}} Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California.</ref>

:<math> D4\sigma = 4 \sigma = 4 \sqrt{\frac{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) (x - \bar{x})^2 \,dx \,dy} {\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y)\, dx \,dy}}, </math>

where

:<math> \bar{x} = \frac{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) x \,dx \,dy}{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) \,dx \,dy} </math>

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, ''x''<sup>2</sup>, 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 of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by ''x''<sup>2</sup>.  Therefore, baseline subtraction is necessary for accurate D4σ measurements.  The baseline is easily measured by recording the average value for each pixel when the sensor is not illuminated.  The D4σ width, unlike the FWHM and 1/e<sup>2</sup> widths, is meaningful for multimodal marginal distributions — that is, beam profiles with multiple peaks — but requires careful subtraction of the baseline for accurate results.  The D4σ is the ISO international standard definition for beam width.