We show theoretically and numerically that an n-th order vector light field containing the central V-point singularity of indefinite linear polarization with polarization singularity index n, with a 'flower' 2(n – 1)-petal polarization pattern centered on it, produces an intensity pattern with 2(n – 1) local maxima at the tight focus. Meanwhile, a vector light field with polarization singularity index –n, leading to a 'web' of polarization singularities composed of 2(n + 1) cells, is tightly focused into an intensity pattern with 2(n + 1) intensity maxima. At the intensity nulls at the focus, either 2(n – 1) or 2(n + 1) V-points with alternating + 1 or –1 indices are produced. In addition, we study more general vector fields of the order (n, m) and analytically derive their Poincare-Hopf indices for many values of n and m. Application areas of such light fields with polarization singularities are laser information technologies, laser material processing, microscopy and optical trapping.

我们在理论上和数值上证明了一个n阶矢量光场，它包含不定线性偏振的中心 V 点奇点，其偏振奇点指数为n，其中心为“花”2（n – 1）花瓣偏振图案，在紧密焦点处产生具有 2( n – 1) 个局部最大值的强度模式。同时，具有偏振奇异性指数 - n的矢量光场导致由 2( n  + 1) 个单元组成的偏振奇异性“网络” ，紧密聚焦成具有 2( n  + 1) 个强度最大值的强度模式。在焦点处的强度零点处，2( n – 1) 或 2( n + 1) 产生具有交替 + 1 或 –1 索引的 V 点。此外，我们研究了更一般的 ( n , m )阶向量场，并针对n和m 的许多值分析导出了它们的 Poincare-Hopf 指数。这种具有偏振奇点的光场的应用领域是激光信息技术、激光材料加工、显微镜和光阱。