The fundamental polarization singularities of monochromatic light are normally associated with invariance under coordinated rotations: symmetry operations that rotate the spatial dependence of an electromagnetic field by an angle θ and its polarization by a multiple γθ of that angle. These symmetries are generated by mixed angular momenta of the form J_γ = L + γS, and they generally induce Möbius-strip topologies, with the coordination parameter γ restricted to integer and half-integer values. In this work we construct beams of light that are invariant under coordinated rotations for arbitrary rational γ, by exploiting the higher internal symmetry of ‘bicircular’ superpositions of counter-rotating circularly polarized beams at different frequencies. We show that these beams have the topology of a torus knot, which reflects the subgroup generated by the torus-knot angular momentum J_γ, and we characterize the resulting optical polarization singularity using third- and higher-order field moment tensors, which we experimentally observe using nonlinear polarization tomography.

单色光的基本偏振奇异性通常与协调旋转下的不变性相关：对称操作将电磁场的空间相关性旋转一个角度θ，并将其极化旋转一个角度的多个γθ。这些对称性是由形式的混合角动量产生Ĵ _γ  = 大号 +  γS，并且它们通常诱导莫比乌斯条拓扑，与协调参数γ限于整数和半整数的值。在这项工作中，我们为任意有理γ构造了在协调旋转下不变的光束通过利用不同频率的反向旋转的圆偏振光束的“双圆”叠加的更高的内部对称性。我们表明，这些光束具有一个环面结，这反映了环面结角动量产生的子组的拓扑Ĵ _γ，我们使用三阶和更高阶的场矩张量，表征所得到的光学偏振奇异我们实验使用非线性极化层析成像进行观察。