Beam shaping optics transform input intensity distributions into some desired output distribution. A common example is to take the Gaussian irradiance distribution produced by a laser and transform it to a Top Hap (flat top) output.

An efficient way of optimizing the sag of such a lens is to use geometrical ray targets in the merit function. In this approach, we compute where in the output plane a given input ray should land, and express this through ray targets entered into thte Merit Function Editor.

The diagram below represents such a system with output top hat beam of K radius and input Gaussian beam of W waist.

First, we need to determine analytically the output radial distance S for a given input coordinate X such that the encircled energy B at the output is the same as the encircled energy A at the input.