Bösel Christoph, Worku Norman G, Gross Herbert
Appl Opt. 2017 May 1;56(13):3679-3688. doi: 10.1364/AO.56.003679.
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 a root-finding algorithm. The presented approach enables the efficient calculation of double freeform lenses with small distances between the freeform surfaces for complex target irradiances. This is demonstrated by applying it to the design of a single-lens and a two-lens system.
许多应用需要同时重新分配激光束的辐照度和相位。光束形状由此由各自的应用决定。一种同时控制辐照度和相位的巧妙方法是利用双自由曲面。在这项工作中，考虑了用于具有任意辐照度的准直光束整形的基于光线映射方法的连续双自由曲面的数值设计。这些方法包括计算源辐照度和目标辐照度之间的适当光线映射以及随后构建自由曲面。通过结合折射定律、恒定光程长度和表面连续性条件，推导出用于光线映射的偏微分方程（PDE）。结果表明，该PDE可以通过从具有二次代价函数的最优质量传输导出的映射在小角度近似下得到满足。为了克服对傍轴区域的限制，我们使用此映射作为通过根查找算法同时求解雅可比方程和光线映射PDE的初始迭代。所提出的方法能够有效地计算自由曲面之间距离较小的双自由曲面透镜，以实现复杂的目标辐照度。将其应用于单透镜和双透镜系统的设计中证明了这一点。
