Stereographical visualization of a polarization state using weak measurements with an optical-vortex beam

We propose a stereographical-visualization scheme for a polarization state by two-dimensional imaging of a weak value with a single setup. The key idea is to employ Laguerre-Gaussian modes or an optical vortex beam for a probe state in weak measurement. Our scheme has the advantage that we can extract information on the polarization state from the single image in which the zero-intensity point of the optical vortex beam corresponds to a stereographic projection point of the Poincare sphere. We experimentally perform single-setup weak measurement to validate the stereographical relationship between the polarization state on the Poincare sphere and the location of the zero-intensity point.