Peak Fluence of a Guassian Laser Beam

[GalaxyM31]

I am working on theoretical laser amplifier model for a pulsed input beam. However, my model is based on a top hat pump beam and laser beam profile, I want to update to a super gaussian and gaussian distribution respectively. The parameters I know are:

beam radius w
Pulse Energy E
and therefore the Fluence of the seed beam if it is a top hot profile E/Area
Pulse Width tp

I know a gaussian beam takes the form f(r) = (-2*e^((r/w)^2))).

My ultimate question is, for a guassian distribution, how do I calculate fluence E0 for the equation:

E = E0*f(r)

where E is the fluence (energy / area)?

I feel this should be obvious, and I'm missing something very easy.

 

[gtoguy97]

The answer is that the fluence E0 for a Gaussian beam is simply the pulse energy E divided by the area of the beam A, just as with the top hat profile, i.e., E0 = E/A. However, for a Gaussian beam profile, the area A is not constant, but is instead given by A = πw^2, where w is the beam radius. Therefore, the fluence E0 for a Gaussian beam is given by: E0 = E/(πw^2).