Energy density expression of a Gaussian pulse

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SUMMARY

The energy density of a Gaussian pulse is expressed through the formula F(r_i, r_i+1) = 2*totalEnergy/[(r_i+1^2 - r_i^2)*w^2]*integral(r*exp(-r^2/w^2), from r_i to r_i+1). This equation calculates the energy density for the ith annular section of the pulse, where r spans from 0 to theoretically infinity in n steps, and w represents the beam waist size. The average energy density is given by F_average = 2*totalEnergy/[pi*w^2]. The discussion seeks to derive F(r) from the provided annular energy density expression.

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Hi,

Energy density of the ith annular section of a Gaussian pulse is written as

F(r_i, r_i+1)=2*totalEnergy/[(r_i+1^2-r_i^2)*w^2]*integral(r*exp(-r^2/w^2),from r_i to r_i+1)

where r spans from r=0 to r=r_n (theoretically infinity) in n steps, w is the waist size of the beam.

This equation is for i. annular part. How can I write the whole energy density of a Gaussian pulse?
I know F_average = 2*totalEnergy/[pi*w^2]

What is the equivalance of F_average in its exact form considering Gaussian spatial profile?

Any answer will be highly appreciated!

Fulya
 
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The question can be misunderstood.
I know I = I0*(w/w0)^2*int(r*exp(-2r^2/w^2)*int(-2t^2/tho_laser) and
F = I*tho_laser.

My question is how or.. can I find F(r) from F(r_i, r_i+1) [the equation above] ?
 

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