Design of an axisymmetric optical element generating a prescribed illuminance distribution and wavefront

Design of an axisymmetric refracting optical element to transform an incident beam into an output beam with the prescribed illuminance distribution and prescribed wavefront is considered. The wavefront of the output beam is represented by the eikonal function defined in some plane behind the optical element. The computation of the optical element surface is reduced to the solution of two explicit ordinary differential equations of the first order. As examples, we consider the design of two optical elements to transform a spherical beam from a point Lambert light source into a beam with the uniform illuminance and plane wavefront and a beam with the complex wavefront to focus into an on-axis line.