Polychromatic electric field knots

The polarization of a monochromatic optical beam lies in a plane and, in general, is described by an ellipse, known as the polarization ellipse. The polarization ellipse in the tight-focusing (nonparaxial) regime forms nontrivial three-dimensional topologies, such as Möbius and ribbon strips as well as knots. The latter are formed when the dynamics of specifically structured polarization states are studied upon propagation. However, optical knots can also exist within another form: The electric field's tip can be made to locally oscillate along a knotted trajectory. We propose an intuitive technique to generate and engineer the path traced by the electric field vector of polychromatic beams to form different knots. In particular, we show examples of how tightly focused beams with at least three frequency components and different spatial modes can cause the tip of the electric field vector to follow, locally, a knotted trajectory. Our study may provide insight in designing current densities for structured polychromatic electromagnetic fields that interact with matter.