Hello,
I have a fiber and a lens as shown below:
 

After the lens I propagate using POP for 1km (10 surfaces with 100m each time). I am interested in the Irradiance along the beam.  I use Gaussian Waist, total power and waist X and Y to define my beam.
What I would like to do is to optimize the irradiance at a particular distance after the lens (focusing), by changing the distance between the fiber end and the lens distance. I would like to maximize the below quantity, where I(x,y) is the irradiance and Po is the  total power defined in the beam definition.

I have used POPD operator in the merit function to optimize the peak power or beam width. I also tried GBPS and ray based focus optimization. But when I trace the above equation from lens to the image surface (1km), I don’t see the irradiance being maximum.

Can someone help me to understand this or provide a method to better optimize the irradiance on a a given surface place?

Thank you very much in advance for your help.