结构化光场的关联几何与拓扑
Correlation geometry and topology of structured optical beams
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- University of Miami（迈阿密大学）
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中文总结 AI 辅助
本文揭示随机标量光束中轨道角动量模式的关联几何与拓扑分别由轨道化矩阵的实部和虚部决定，并定义了轨道各向异性与轨道化程度，展示了多模OAM光中独有的新关联结构。
中文摘要 AI 辅助
本文证明，在携带轨道角动量（OAM）且包含 $L$ 个模式的随机标量光束中，其关联几何与拓扑分别与轨道化矩阵（OM）的实部和虚部相关联。该矩阵通过在极坐标傅里叶基下，对每个OAM指标对在给定截面和半径处对交叉谱密度进行滤波而获得。OM的对称实部具有对角规范形式，确定了光束的椭球状关联几何。OM的反对称虚部具有达布（Darboux）规范形式，可分解为 $N\leq L/2$ 个相互正交的循环态和 $L-2N$ 个平凡的（无循环）态，从而定义了与光束关联拓扑相关的 $N$ 维轨道化环面。随后，利用这些规范形式定义了线性和圆轨道各向异性的程度及稳定性，以及线性和圆轨道化的程度，这与二维和三维偏振理论中的对应量类似。这些结果证明了在随机多模OAM光中出现了根本性的新关联结构，这些结构在低维系统中无法获得，且在确定性极限下不存在。
英文摘要
Correlation geometry and topology in a random, scalar beam carrying Orbital Angular Momentum (OAM) in $L$ modes are shown to be linked to the real and imaginary parts, respectively, of its orbitalization matrix (OM). The OM is obtained by filtering the cross-spectral density in the polar Fourier basis at a given cross-section and radius for each pair of OAM indices. The symmetric real part of the OM has the diagonal canonical form, specifying the ellipsoid-like correlation geometry of the beam. The skew-symmetric imaginary part of the OM has the Darboux canonical form, enabling decomposition into $N\leq L/2$ mutually orthogonal circulation states and $L-2N$ trivial circulation-free states, mutually orthogonal and $L-2N$ trivial circulation states thereby defining the $N$-dimensional orbitalization torus associated with the correlation topology of the beam. The canonical forms are then used to define the degrees of linear and circular orbital anisotropy and stability, and the degrees of linear and circular orbitalization, in analogy with corresponding quantities in 2D and 3D polarization theory. These results demonstrate the emergence of fundamentally new correlation structures in random multimode OAM-carrying light, not accessible in lower-dimensional systems and absent in the deterministic limit.
