In far-field light shaping, one of the design methods is based on a one-to-one map between the irradiance of the source and target. However, an integrability issue may occur in this kind of algorithms, either in the ray mapping method for designing a freeform surface or in those geometric-optics-based methods for achieving a required output phase. We introduce another mapping-type algorithm to tackle the integrability problem, which instead of establishing a mapping between both the source and target irradiance in the space domain, the mapping is assumed on electric fields of a Fourier pair between the space domain and the spatial-frequency domain. By solving the mapping from the Fourier pair, the gradient of the output phase is achieved, that the gradient is equivalent to the obtained mapping function. Moreover, the existence and the characterization of the mapping guarantees the integrability of the gradient so that a smooth output phase can be directly integrated. Based on the obtained smooth output phase, a freeform surface can then be designed for the light-shaping task. Numerical examples are demonstrated for the comparison of the approaches with different mapping assumptions.

中文翻译：

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基于傅里叶对映射的光整形可积分解决方案

在远场光整形中，其中一种设计方法是基于源和目标辐照度之间的一对一映射。然而，在这种算法中可能会出现可积性问题，无论是在用于设计自由曲面的射线映射方法中，还是在那些用于实现所需输出相位的基于几何光学的方法中。我们引入了另一种映射类型的算法来解决可积性问题，它不是在空间域中建立源和目标辐照度之间的映射，而是假设映射是在空间域和空间域之间的傅立叶对的电场上进行的。频域。通过从傅立叶对求解映射，得到输出相位的梯度，即梯度等价于得到的映射函数。而且，映射的存在和表征保证了梯度的可积性，从而可以直接对平滑的输出相位进行积分。基于获得的平滑输出相位，然后可以为光整形任务设计自由曲面。数值示例用于比较具有不同映射假设的方法。