二阶支持二次曲面法用于设计产生预定辐照度分布的自由曲面折射面
Second-order supporting quadric method for designing freeform refracting surfaces generating prescribed irradiance distributions
AI总结：
本文提出了一种二阶支持二次曲面法，用于设计产生预定辐照度分布的自由曲面折射面，并展示了其在非成像光学问题中的应用。
摘要：
我们考虑了计算产生预定远场辐照度分布的折射面的逆问题，对于入射的平行光束。该问题可以表述为一个具有二次成本函数的质量运输问题（MTP）。为了解决这个问题，我们提出了一种支持二次曲面法（SQM）的变种，其中二次参数的计算被减少到最小化一个凸函数的问题。我们获得了该函数二阶导数的简单解析表达式，使得可以利用二阶优化方法计算二次参数。这使我们能够将所提出的方法称为二阶SQM。我们通过设计几个产生复杂辐照度分布的光学表面来展示该方法的高效性。我们还考虑了将二阶SQM应用于由具有非二次成本函数的MTP描述的非成像光学问题。
英文摘要：
We consider the inverse problem of calculating a refracting surface that generates a prescribed irradiance distribution in the far field for a collimated incident beam. This problem can be formulated as a mass transportation problem (MTP) with a quadratic cost function. To solve this problem, we propose a version of the supporting quadric method (SQM), in which the calculation of the quadric parameters is reduced to the problem of minimizing a convex function. We obtain simple analytical expressions for the second derivatives of this function, making it possible to calculate the quadric parameters using second-order optimization methods. This allows us to refer to the proposed method as the second-order SQM. We demonstrate high efficiency of this approach by designing several optical surfaces that generate complex irradiance distributions. We also consider the application of the second-order SQM to nonimaging optics problems described by MTPs with a non-quadratic cost function.
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