Perturbing the polarisation of Riemann-Silberstein electromagnetic vortices

In optics the electric field E usually receives more attention than the magnetic field B because it interacts more strongly with matter. But for the propagation of waves in free space E and B should be given equal weight. From the two real vector fields E and B one first constructs the vector field {F}={E}+i{B} F = E + i B and then the complex scalar field F. F. It is this complex scalar field that contains the Riemann-Silberstein (RS) vortex lines. The optical vortices in monochromatic waves that are usually discussed are stationary in space, but these RS vortices are normally in constant motion. However, exceptionally, if the plane waves whose interference gives the field are circularly polarised and are all of the same hand, it is known that the RS vortices are stationary. We study here what happens when this polarisation condition is gradually relaxed, using a 4-wave model. RS vortices are Lorentz invariant and this makes it possible to define a coordinate frame in which there is spatial repetition, so that results can be exhibited in a repeating 3D unit cell. As the perturbation grows, a given point on an RS vortex at first traces out an elliptical orbit; then a critical value is reached where reconnections and disconnections occur at two definite times during the cycle. The changes thereafter are like those seen in a general RS vortex field.