Design of an axisymmetrical refractive optical element generating required illuminance distribution and wavefront

The design of an axisymmetrical refractive optical element transforming a given incident beam into an output beam with prescribed illuminance distribution and wavefront is considered. The wavefront of the output beam is represented by the eikonal function defined in a certain plane behind the optical element. The design of the optical element is reduced to the solution of two explicit ordinary differential equations of the first order. These equations can be easily integrated using conventional numerical methods. As examples, we consider the design of two optical elements transforming a spherical beam from a point Lambertian light source into the uniform-illuminance beams with a plane wavefront and with a complex wavefront providing the subsequent focusing into a line segment on the optical axis.