Mapping Relations Between the Polarization, the Higher-Order and the Orbital Angular Momentum Poincaré Spheres for Inner Points

A polarization state can be regarded as the coherent superposition of a right- and left-handed polarization state, and it is represented on the polarization Poincaré sphere (PS). Similarly, the coherent superposition of two spatial modes with opposite orbital angular momentum (OAM) lies on the OAM PS (OAMPS) [1]. In addition, coherent superpositions of vortex vector beams (VBs) can be described as points on the higher-order PS (HOPS) [2]. The mapping between spheres has been studied for points on the surface of the spheres. Here instead, we provide an intuitive description of the mapping process for points inside the spheres, which we regard as incoherent superpositions of points on the surface of their respective sphere [3] (polarization states for the PS, VBs for the HOPS and spatial modes for the OAMPS). Figs. 1(a)-(c) depict the mapping process. They show that a state inside the PS can be converted to a state in the HOPS with a q-plate (a half-wave plate whose eigenaxes follow $q$ times the azimuthal coordinate) and that a point in the HOPS transforms into a point in the OAMPS using a linear polarizer. We implemented this process experimentally using the setup in Fig. 1(d) and designed a new polarization state generator (PSG) to create partially polarized states. First, an unpolarized laser is split into two orthogonal polarization components using a geometric phase (GP) blazed grating. Their relative intensity is then modified with an intensity filter, and they are lastly recombined using another GP grating. As a result, we generate an incoherent superposition of two polarization states with the desired relative intensity that is sent to a waveplate, therefore achieving every point inside the PS.