Editing Gaussian beam (section)

==Power and intensity==

=== Power through an aperture ===
With a beam centered on an [[aperture]], the [[power (physics)|power]] {{mvar|P}} passing through a circle of radius {{mvar|r}} in the transverse plane at position {{mvar|z}} is<ref name="melles griot">{{Cite web |url=http://www.pa.msu.edu/courses/2010fall/phy431/PostNotes/PHY431-Notes-GaussianBeamOptics.pdf |title=Melles Griot. Gaussian Beam Optics |access-date=2015-04-07 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304031525/http://www.pa.msu.edu/courses/2010fall/phy431/PostNotes/PHY431-Notes-GaussianBeamOptics.pdf |url-status=dead }}</ref>
<math display="block">P(r,z) =  P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right],</math>
where
<math display="block">P_0 = \frac{ 1 }{ 2 } \pi I_0 w_0^2</math>
is the total power transmitted by the beam.

For a circle of radius {{math|1=''r'' = ''w''(''z'')}}, the fraction of power transmitted through the circle is
<math display="block">\frac{P(z)}{P_0} = 1 - e^{-2} \approx 0.865.</math>

Similarly, about 90% of the beam's power will flow through a circle of radius {{math|1=''r'' = 1.07 × ''w''(''z'')}}, 95% through a circle of radius {{math|1=''r'' = 1.224 × ''w''(''z'')}}, and 99% through a circle of radius {{math|1=''r'' = 1.52 × ''w''(''z'')}}.<ref name="melles griot"/>

=== Peak intensity ===
The peak intensity at an axial distance {{mvar|z}} from the beam waist can be calculated as the limit of the enclosed power within a circle of radius {{mvar|r}}, divided by the area of the circle {{math|''πr''<sup>2</sup>}} as the circle shrinks:
<math display="block">I(0,z) = \lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} .</math>

The limit can be evaluated using [[L'Hôpital's rule]]:
<math display="block">I(0,z)
         = \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)} 
         = {2P_0 \over \pi w^2(z)} .</math>