Three-dimensional representation of two-dimensional vortex dynamics in lasers

We observe that the dynamics of vortices in paraxial optics, including vortex pair-creation and pair-annihilation, can be represented in a simplified manner by the motion of a locus obtained from the complex optical field, for which the real and imaginary parts of the field are equal. When the locus lies within the transverse plane a dark line is present in the optical field with a phase change of π across. When the locus intersects the transverse plane, a vortex exists at the intersection point. Thus the locus line appears like a three-dimensional vortex line. Examples are given ranging from emission in low order laser mode families to patterns calculated for plane mirror resonators