Knotted polarizations and spin in three-dimensional polychromatic waves

We consider complex three-dimensional polarizations in the interference of several vector wave fields with different commensurable frequencies and polarizations. We show that the resulting polarizations can form knots, and interfering three waves is sufficient to generate a variety of Lissajous, torus, and other knot types. We describe the spin angular momentum, generalized Stokes parameters, and degree of polarization for such knotted polarizations, which can be regarded as partially polarized. Our results are generic for any vector wave fields, including, e.g., optical and acoustic waves. As a directly observable example, we consider knotted trajectories of water particles in the interference of surface water (gravity) waves with three different frequencies.

Figure 1

Examples of knotted polarizations $\mathcal{F}\left(t\right)$ in the interference of three waves with different frequencies and polarizations (shown to the right from the knots). (a) The Lissajous knot (1) with $A_1=A_2=A_3$, ${\omega }_2/{\omega }_1=5/2$, ${\omega }_3/{\omega }_1=3/2$, ${\varphi }_1-{\varphi }_2=0.04$, and ${\varphi }_3-{\varphi }_2=1.8$ (the $5_2$ or three-twist knot [35]). (b) The torus knot (2) with $p=2$, $q=3$ (trefoil knot). (c) Figure-eight knot (3). The spins of the interfering waves and resulted knotted polarizations are shown in magenta, whereas the 3D degree of polarization for the knotted states (a)–(c) are $P=0$, $P=1/2$, and $P\simeq 0.67$, respectively [36].

Figure 2

Interference of three surface water waves, one propagating in $x$ and two standing along $y$ [Eq. (6)] with different frequencies and phases corresponding to Fig. 1. The instantaneous surface shapes (grayscale) and 3D water-particle trajectories (colored) are shown for the interference field (a) (see also its animated version [36]) and each of the interfering waves (b). Some of these trajectories correspond to trefoil torus knots [Fig. 1], while others are “unknots.” The $(x,y)$ regions with knotted and unknotted trajectories are shown in (c).