Topology of optical vortex lines formed by the interference of three, four, and five plane waves

When three or more plane waves overlap in space, complete destructive interference occurs on nodal lines, also called phase singularities or optical vortices. For super positions of three plane waves, the vortices are straight, parallel lines. For four plane waves the vortices form an array of closed or open loops. For five or more plane waves the loops are irregular. We illustrate these patterns numerically and experimentally and explain the three-, four- and five-wave topologies with a phasor argument