Ray mapping approach in double freeform surface design for collimated beam shaping beyond the paraxial approximation

摘要

Numerous applications require the simultaneous redistribution of theirradiance and phase of a laser beam. The beam shape is thereby determined bythe respective application. An elegant way to control the irradiance and phaseat the same time is from double freeform surfaces. In this work the numericaldesign of continuous double freeform surfaces from ray mapping methods forcollimated beam shaping with arbitrary irradiances is considered. These methodsconsist of the calculation of a proper ray mapping between the source and thetarget irradiance and the subsequent construction of the freeform surfaces. Bycombining the law of refraction, the constant optical path length and thesurface continuity condition, a partial differential equation (PDE) for the raymapping is derived. It is shown that the PDE can be fulfilled in a small-angleapproximation by a mapping derived from optimal mass transport with a quadraticcost function. To overcome the restriction to the paraxial regime we use thismapping as an initial iterate for the simultaneous solution of the Jacobianequation and the ray mapping PDE by an optimization. The presented approachenables the efficient calculation of compact double freeform surfaces forcomplex target irradiances. This is demonstrated by applying it to the designof a single-lens and a two-lens system.