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Phase saddles in light beams

In a section through a monochromatic light beam the contour map of phase contains saddle points. It has already been shown that a two-dimensional model of two superposed coaxial Gaussian beams, in antiphase and having different waist sizes, contains two saddles that perform an elaborate dance as the ratio of the amplitudes of the beams is altered. The present paper explains why this choreography is qualitatively identical to that found in a symmetrical version of an earlier and simpler two-dimensional model, where a plane wave is modulated by a quadratic polynomial. If wavefronts are defined as lines of equal phase, successive wavefronts pinch together in these models, and change their connectedness as they pass through two fixed saddle points on the axis. Although the idea of a phase saddle is not generally applicable in three dimensions, it can be extended to three dimensions in axially symmetric models, for example, two superposed coaxial Gaussian beams. The saddles are features of the set of azimuthal planes, and can either form rings around the axis or be on the axis itself. The action here as a parameter changes takes a more dramatic form, because it involves both a vortex ring and two saddle points on the axis, which collide and explode into a concentric saddle ring. The physical significance of saddles is that they change the topology of the wavefronts. In two dimensions a moving wavefront line passing through a fixed saddle point on the axis undergoes reconnection. As it meets an off-axis saddle the same process occurs but in a different orientation. In three dimensions as a wavefront passes through a saddle point on the axis, its local form changes from a hyperboloid of two sheets to a hyperboloid of one sheet, or vice versa, via a cone of angle . As it passes through a saddle ring a similar transition occurs simultaneously at all points of the ring. The changes in the topology of a wavefront as it encounters a monkey saddle are also interesting. The behaviour of the saddle points and rings in two superposed three-dimensional Gaussian beams is also similar to that found in a simpler quadratic polynomial model.