Smooth Evolutions of Knots of Electromagnetic Fields in Light Photons

Maxwell's equations for electrodynamics are a set of four partial differential equations, which relates the space and time derivatives of electric and magnetic fields. Together with Lorentz force law, the Maxwell's equations form the essence of classical electrodynamics. The decouple solutions of these equations were used by Maxwell to predict that light is an electromagnetic wave which were confirmed by subsequent experiments by Hertz in 1886. Further experimental and theoretical work with electromagnetic fields, over a century, had done little to illuminate the fundamental nature of light. But, in 1989, Rañada discovered some remarkable solutions for Maxwell's equations for electromagnetic fields for light which have linked and knotted electromagnetic field lines [1]. These topological solutions were based on Hopf fibration and evolve with time. The time evolution of these electric and magnetic filed lines was found to preserve the topological invariants [2]. The preservation of the topological structure of the field lines in these solutions has previously been ascribed to the fact that the electric and magnetic helicities, a measure of the degree of linking and knotting between field lines, are conserved [1]. Another work [3] suggested that the smooth evolution of the field lines is due to the stricter condition that the electric and magnetic fields be everywhere orthogonal . Here, a simple argument is given for the closed knotted field line structure of light and it is shown that light always contain the continuous electric and magnetic field lines in form of knot structures which evolve smoothly with time.