Editing Beam diameter (section)

==== 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>