Beam of light shaping process can be considered ultimate, if both irradiance and wavefront spatial distributions are under control and both can be shaped arbitrarily. In order to keep these two quantities determined simultaneously, it is required to apply at least two powered refractive or reflective surfaces. In this paper, a fully geometric design method of double-freeform beam shapers is discussed briefly. The presented algorithm is based on two stages. First, integrable input-output ray mapping is calculated by the application of the novel GATMA (Geometric Approach to Monge–Ampere equation) method. It allows us to determine the shape of the first freeform surface. Then, according to the condition of constant optical path length between input and output plane, corrected by wavefront phases at those planes, the second surface is determined. GATMA algorithm combines advantages of Monge–Ampere (MA) equation and ray-tracing efficient apparatus. Compared to the state-of-the-art freeform design methods, GATMA does not need to solve MA equation directly but uses this equation as an error function. Such approach makes the computation algorithm simpler and more robust and convergent. The application of the proposed method in a challenging design example of a beam shaper, transforming uniform collimated beam into a beam having a triangular cross section and flat wavefront, is presented as a case study.

中文翻译：

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自由束成形器设计的简化几何方法

如果辐照度和波前空间分布都处于受控状态，并且两者都可以任意成形，则光束成形过程可以认为是最终的。为了同时确定这两个量，需要施加至少两个带电的折射或反射表面。本文简要讨论了双自由光束成形器的全几何设计方法。所提出的算法基于两个阶段。首先，通过使用新颖的GATMA（Monge–Ampere方程的几何方法）方法来计算可积分的输入-输出射线映射。它使我们能够确定第一自由曲面的形状。然后，根据输入平面和输出平面之间的恒定光程长度的条件，通过在这些平面上的波前相位进行校正，确定第二表面。GATMA算法结合了蒙格安培（MA）方程和光线追踪高效设备的优点。与最新的自由格式设计方法相比，GATMA不需要直接求解MA方程，而可以使用该方程作为误差函数。这种方法使计算算法更简单，更健壮和收敛。作为案例研究，提出了该方法在具有挑战性的光束整形器设计示例中的应用，该示例将均匀的准直光束转换为具有三角形横截面和平坦波前的光束。这种方法使计算算法更简单，更健壮和收敛。作为案例研究，提出了该方法在具有挑战性的光束整形器设计示例中的应用，该示例将均匀的准直光束转换为具有三角形横截面和平坦波前的光束。这种方法使计算算法更简单，更健壮和收敛。作为案例研究，提出了该方法在具有挑战性的光束整形器设计示例中的应用，该示例将均匀的准直光束转换为具有三角形横截面和平坦波前的光束。