I have a 2D normal distribution representing a laser profile. The width of the beam is expressed as the diameter that encloses 1 - 1/e^2 ~ 82%. This is a standard way of describing beam width (just 1/e^2). The laser goes through a hols which cuts of the outer most part of the beam (the "tail" of the distribution)
I would like to calculate how much is lost to this cut off. So I guess it boils down to integrating normal distribution out to some fixed bound. It seems obvious that I should use polar coordinates. So I have two questions:
how do I solve this integral?
How do I deal with the 1/e^2 width? Do I convert it to a standard deviation? This doesn't seems right.