  Numerous applications require the simultaneous redistribution of the
irradiance and phase of a laser beam. The beam shape is thereby determined by
the respective application. An elegant way to control the irradiance and phase
at the same time is from double freeform surfaces. In this work the numerical
design of continuous double freeform surfaces from ray mapping methods for
collimated beam shaping with arbitrary irradiances is considered. These methods
consist of the calculation of a proper ray mapping between the source and the
target irradiance and the subsequent construction of the freeform surfaces. By
combining the law of refraction, the constant optical path length and the
surface continuity condition, a partial differential equation (PDE) for the ray
mapping is derived. It is shown that the PDE can be fulfilled in a small-angle
approximation by a mapping derived from optimal mass transport with a quadratic
cost function. To overcome the restriction to the paraxial regime we use this
mapping as an initial iterate for the simultaneous solution of the Jacobian
equation and the ray mapping PDE by an optimization. The presented approach
enables the efficient calculation of compact double freeform surfaces for
complex target irradiances. This is demonstrated by applying it to the design
of a single-lens and a two-lens system.
