Tying Knots in Light Fields

We construct analytically, a new family of null solutions to Maxwell’s equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is shear free, preserves the topology of the knots and links. Our approach combines the construction of null fields with complex polynomials on ${\mathbb{S}}^3$. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.

Figure 1

Hopfion solution: field line structure (a)–(c) and time evolution (d)–(e). Field lines fill nested tori, forming closed loops linked with every other loop. (a) Hopf link formed by the circle at the core (orange) of the nested tori, and one of the field lines (blue). (b) The torus (purple) that the field line forming the Hopf link is tangent to. (c) Nested tori (purple) enclosing the core, on which the field lines lie. (d) Time evolution of the Poynting field lines (gray), an energy isosurface (red), and the energy density (shown via projections). (e) Time evolution of the electric (yellow), magnetic (blue), and Poynting field lines (gray), with the top view shown in the inset.

Figure 2

Structure of magnetic field lines, (a)–(c) Trefoil knots ($p=2$, $q=3$), (d)–(f) Cinquefoil knots ($p=2$, $q=5$), (g)–(i) 4 linked rings ($p=2$, $q=2$). (a),(d),(g) Core (orange) field line(s) forming (a) a trefoil knot (d) a cinquefoil knot (g) 4 linked rings. (b),(e),(h) Field line(s) (blue) wrapping around the core (orange) confined to a knotted torus (purple) enclosing the core. (c),(f),(i) Knotted nested tori (purple) enclosing the core, on which the field lines lie.

Figure 3

Time evolution of magnetic field lines and energy density for (a) the trefoil knot ($p=2$, $q=3$), (b) the cinquefoil knot ($p=2$, $q=5$), and (c) the 4 Hopf-linked rings ($p=2$; $q=2$). Shown is the topology preserving, fluidlike evolution of the core field line(s) (orange), and field line(s) (blue) lying on tori (purple) enclosing the core.