Phase saddles in light beams

Abstract

In a section through a monochromatic light beam the contour map of phase contains saddlepoints. It has already been shown that a two-dimensional model of two superposed coaxialGaussian beams, in antiphase and having different waist sizes, contains two saddles thatperform an elaborate dance as the ratio of the amplitudes of the beams is altered. Thepresent paper explains why this choreography is qualitatively identical to that found in asymmetrical version of an earlier and simpler two-dimensional model, where a plane wave ismodulated by a quadratic polynomial. If wavefronts are defined as lines of equal phase,successive wavefronts pinch together in these models, and change their connectedness asthey pass through two fixed saddle points on the axis. Although the idea of aphase saddle is not generally applicable in three dimensions, it can be extendedto three dimensions in axially symmetric models, for example, two superposedcoaxial 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 actionhere as a parameter changes takes a more dramatic form, because it involvesboth a vortex ring and two saddle points on the axis, which collide and explodeinto a concentric saddle ring. The physical significance of saddles is that theychange the topology of the wavefronts. In two dimensions a moving wavefront linepassing through a fixed saddle point on the axis undergoes reconnection. As itmeets 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 onesheet, or vice versa, via a cone of angle . As it passes through a saddle ring a similar transition occurs simultaneously at all pointsof the ring. The changes in the topology of a wavefront as it encounters a monkey saddleare also interesting. The behaviour of the saddle points and rings in two superposedthree-dimensional Gaussian beams is also similar to that found in a simpler quadraticpolynomial model.