Ray mapping approach in double freeform surface design for collimated beam shaping

Laser beam shaping enables the simultaneous redistribution of the irradiance and phase of a laser beam. The desired shape of the laser beam is thereby determined by the respective application. One possible way to achieve the desired irradiance and phase at the same time is from double freeform surfaces. We investigate the numerical design of double freeform surfaces for collimated beam shaping with arbitrary irradiances by using ray-mapping techniques, where at first a proper ray mapping between the source and target irradiance is calculated, and in a subsequent step the freeform surfaces are constructed. The difficulty thereby is to find an integrable ray mapping which leads to two continuous surfaces. Combining the law of refraction and an integrability condition, we derive a condition for a ray mapping and show that it can be fulfilled in a small-angle approximation by a mapping derived with optimal mass transport. As a consequence the design process decouples into the separate calculation of the ray mapping as well as both freeform surfaces. A quantitative estimate for the approximate integrability of the optimal mass transport mapping is derived. The decoupling of the design process offers an efficient way of constructing both freeform surfaces by solving linear advection equations. The efficiency of the design algorithm is demonstrated by applying it to a challenging design example, furthermore the limitations of the numerical approach are investigated.