Koç Aykut, Ozaktas Haldun M, Hesselink Lambertus
Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA.
J Opt Soc Am A Opt Image Sci Vis. 2010 Jun 1;27(6):1288-302. doi: 10.1364/JOSAA.27.001288.
We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non-orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes approximately N log N time, where N is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.
我们报告了一种用于二维不可分离线性规范变换（2D-NS-LCT）数值计算的快速且准确的算法。这类积分变换也被称为二次相位积分，它代表了一大类光学系统，包括自由空间中的菲涅耳传播、梯度折射率介质中的传播、通过薄透镜以及这些情况的任意数量的任意串联，包括变形/像散/非正交情况。一般的二维不可分离情况带来了一些在一维情况和可分离二维情况中不存在的挑战。该算法耗时约为N log N，其中N是信号的二维空间带宽积。我们的方法在二维中正确地跟踪和控制空间带宽积，以便在算法的任何阶段实现理论上足够的信息但不过度冗余的采样，从而重建基础连续函数。此外，我们给出了一般二维不可分离线性规范变换的另一种定义，该定义根据其十个参数明确显示了其核，并将这些参数与传统的ABCD矩阵参数双向关联。
