Energy density expression of a Gaussian pulse

[serenade]

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

 

[serenade]

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] ?

 

[charng]

Hello Fulya,

Thank you for sharing your expression for the energy density of a Gaussian pulse. It is important to consider the energy density of a pulse when studying its properties and interactions with other systems.

To calculate the total energy density of a Gaussian pulse, you can integrate the expression you provided over all annular sections. This will give you the total energy density of the pulse in terms of its waist size and the total energy. So the expression for the total energy density would be:

F_total = integral(F(r_i, r_i+1), from i=1 to n)

This integral will give you the energy density at each annular section and summing them up will give you the total energy density of the pulse.

As for the equivalence of F_average in its exact form for a Gaussian spatial profile, it is equal to the average energy density over the entire beam profile. This can be calculated by dividing the total energy density by the area of the beam profile, which is given by pi*w^2. So the expression for F_average would be:

F_average = F_total/(pi*w^2)

I hope this helps to answer your questions. Let me know if you have any further inquiries. Good luck with your research!