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

## TL;DR

This paper presents a numerical method for designing double freeform surfaces for collimated beam shaping that surpasses the paraxial approximation, enabling complex irradiance and phase control in laser beam applications.

## Contribution

It introduces a combined approach using optimal mass transport and optimization to solve the PDE for non-paraxial beam shaping with double freeform surfaces.

## Key findings

- Effective design of complex beam shapes demonstrated
- Method surpasses paraxial approximation limitations
- Applications include single-lens and two-lens systems

## Abstract

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.