Editing Beam diameter (section)

=== D86 width ===
The D86 width is defined as the diameter of the circle that is centered at the centroid of the beam profile and contains 86% of the beam power. The solution for D86 is found by computing the area of increasingly larger circles around the centroid until the area contains 0.86 of the total power.  Unlike the previous beam width definitions, the D86 width is not derived from marginal distributions.  The percentage of 86, rather than 50, 80, or 90, is chosen because a circular Gaussian beam profile integrated down to 1/e<sup>2</sup> of its peak value contains 86% of its total power.  The D86 width is often used in applications that are concerned with knowing exactly how much power is in a given area.  For example, applications of high-energy [[laser weapon]]s and [[lidar]]s require precise knowledge of how much transmitted power actually illuminates the target.

==== ISO11146 beam width for elliptic beams ====

The definition given before holds for stigmatic (circular symmetric) beams only. For astigmatic beams, however, a more rigorous definition of the beam width has to be used:<ref name="ISO11146-3">ISO 11146-3:2004(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods".</ref>
:<math> d_{\sigma x} = 2 \sqrt{2} \left( \langle x^2 \rangle + \langle y^2 \rangle + \gamma \left( \left( \langle x^2 \rangle - \langle y^2 \rangle \right)^2 + 4 \langle xy \rangle^2 \right)^{1/2} \right)^{1/2} </math>
and
:<math> d_{\sigma y} = 2 \sqrt{2} \left( \langle x^2 \rangle + \langle y^2 \rangle - \gamma \left( \left( \langle x^2 \rangle - \langle y^2 \rangle \right)^2 + 4 \langle xy \rangle^2 \right)^{1/2} \right)^{1/2}. </math>
This definition also incorporates information about ''x''–''y'' correlation <math> \langle xy \rangle </math>, but for circular symmetric beams, both definitions are the same.

Some new symbols appeared within the formulas, which are the first- and second-order moments:
:<math> \langle x \rangle = \frac{1}{P} \int I(x,y) x \,dx \,dy, </math>
:<math> \langle y \rangle = \frac{1}{P} \int I(x,y) y \,dx \,dy, </math>
:<math> \langle x^2 \rangle = \frac{1}{P} \int I(x,y) (x - \langle x \rangle )^2 \,dx \,dy, </math>
:<math> \langle xy \rangle = \frac{1}{P} \int I(x,y) (x - \langle x \rangle ) (y - \langle y \rangle ) \,dx \,dy, </math>
:<math> \langle y^2 \rangle = \frac{1}{P} \int I(x,y) (y - \langle y \rangle )^2 \,dx \,dy, </math>
the beam power 
:<math> P = \int I(x,y) \,dx \,dy, </math>
and 
:<math> \gamma = \sgn \left( \langle x^2 \rangle - \langle y^2 \rangle \right) = \frac{\langle x^2 \rangle - \langle y^2 \rangle}{|\langle x^2 \rangle - \langle y^2 \rangle|}. </math>

Using this general definition, also the beam azimuthal angle <math> \phi </math> can be expressed. It is the angle between the beam directions of minimal and maximal elongations, known as principal axes, and the laboratory system, being the <math>x</math> and <math>y</math> axes of the detector and given by 
:<math> \phi = \frac{1}{2} \arctan \frac{2 \langle xy \rangle}{\langle x^2 \rangle - \langle y^2 \rangle }.</math>