The coherent superposition of a Gaussian beam with an optical vortex can be mathematically described to occupy the complex plane. We provide a simple analogy between the mathematics, in the form of the complex plane, and the visual representation of these two superimposed optical fields. We provide detailed instructions as to how one can experimentally produce, measure, and control these fields with the use of digital holograms encoded on a spatial light modulator.

In this work we construct coherent superpositions of Gaussian and vortex modes which can be described to occupy the complex-plane. We demonstrate how these fields can be experimentally constructed in a digital, controllable manner with a spatial light modulator. Once these fields have been generated we illustrate, with three separate techniques, how the constituent components of these fields can be extracted, namely by measuring the intensity of the field at two adjacent points; performing a modal decomposition and a new digital Stokes measurement.