Mapping partially polarized light to incoherent superpositions of vector beams and vortex beams with orbital angular momentum

. Fully polarized light, cylindrical vector beams, and beams with opposite orbital angular momentum (OAM) and their superpositions are respectively represented as points on the Poincaré sphere (PS), the higher-order Poincaré sphere (HOPS) and the OAM Poincaré sphere (OAMPS). Here, we study the mapping of inner points between these spheres, which we regard as incoherent superpositions of points on the surface of their respective sphere. We obtain points inside the HOPS and OAMPS by mapping incoherent superpositions of points on the PS, i.e., partially polarized states. To map points from the PS to the HOPS, we use a q-plate, while for mapping points from the HOPS to the OAMPS, we use a linear polarizer. Furthermore, we demonstrate a new polarization state generator (PSG) that generates efficiently partially polarized light. It uses a geometric phase (GP) blazed grating to split an unpolarized laser into two orthogonal polarization components. An intensity filter adjusts the relative intensity of the components, which are then recombined with another GP grating and directed to a waveplate, thus achieving every point inside the PS. The proposed PSG offers advantages over other methods in terms of energy efficiency, ease of alignment, and not requiring spatial or long-time integrations.


Introduction
Coherent superpositions of polarization states are represented as points on the surface of the polarization Poincaré sphere (PS).Points inside the sphere represent partially polarized light, which can be regarded as incoherent superpositions of fully polarized states.Similarly, coherent superpositions of vortex vector beams are represented as points on the surface of the higherorder PS (HOPS) [1], while coherent summations of vortex beams carrying orbital angular momentum (OAM) are described as points on the OAM PS (OAMPS) [2].Mapping relations between points on the surface of the three spheres have been studied, and several setups have been proposed to perform such mapping.A simple setup involves using a q-plate (a half-wave retarder whose fast axis distribution follows q times the azimuthal coordinate) to map from the PS to the HOPS and a linear polarizer to map from the HOPS to the OAMPS [3].
In this work, we extend this method to map the inner points between the three spheres [4] and extend the degree of polarization (DoP) concept to vector beams and OAM modes.We achieve an incoherent superposition of vector beams and scalar beams by performing an incoherent superposition of polarization states (thus obtaining a partially polarized state), where the control of each beam in the superposition is determined by the DoP of the input polarization mixed state.
Additionally, we present a new design of a polarization state generator (PSG) that makes use of a pair of geometric phase (GP) linear blazed gratings to produce partially polarized light.Our PSG design offers several advantages compared to other systems [5,6], as we will demonstrate in the experimental section.

Theory of the spheres mapping
Figure 1 illustrates the mapping of inner points between the PS, HOPS, and OAMPS using a q-plate and a linear polarizer [4].By applying these two optical elements to a point inside the PS, we obtain its position in the HOPS and OAMPS, represented by the Stokes vector   , where  corresponds to the PS, HOPS, or OAMPS.Each point inside its respective sphere can be expressed as the incoherent superposition of two points on the surface,  � and  � ⊥ , which represent orthogonal states.This superposition is weighted by the degree of polarization (DoP)  of the initial polarization state: The parameter  corresponds to the radius of each sphere and can be viewed as the length of the Bloch vector that represents a mixed state in a Bloch sphere.In the HOPS,  can be interpreted as the DoP of the entire vector beam, while in the OAMPS, it represents the relative weight of an incoherent mixture of two scalar beams.We describe the transformations for Laguerre-Gaussian and Hermite-Gaussian beams, but the OAMPS can also be used for other scalar beams carrying OAM and their superpositions.

Experimental setup and results
The experimental setup is described in Fig. 2. We use a randomly polarized He-Ne laser in which the orientation of the polarization state changes rapidly on a nanosecond timescale.The laser beam is split into two orthogonal circular components using a GP linear polarization grating.A variable intensity filter, placed after the grating, controls the relative amount of each component.The beams are then recombined using another grating, resulting in an incoherent superposition of two orthogonal polarization states.The position inside the Poincaré sphere is controlled by the weight of each polarization component (Eq.( 1)).A linear retarder, placed after the recombined beam, allows us to achieve any polarization state, unpolarized, fully or partially polarized.This new PSG offers several advantages over other systems.For instance, it avoids the use of beamsplitters, which makes it more energy-efficient.Additionally, the resulting setup is an in-line system, relatively easy to align.Moreover, since the system exploits very fast changes in the polarization state of the laser, it does not rely on long temporal [5] or spatial integrations [6].However, let us note that the utilization of Wollaston prisms instead of GP gratings could lead to higher efficiency, since they do not suffer from diffraction losses.Behind the PSG, a q-plate transforms the state from the PS to the HOPS, and the resulting vector beam is analyzed using a polarization state analyzer (PSA) built with a polarization camera and quarter-wave plate.The polarization camera allows us to capture four linear polarization components in a single shot since it has pixels made up of four micropixels with a grid polarizer oriented at 0º, 90º, 45º, and -45º.The position of the spatial mode in the OAMPS is determined by the equivalent Stokes parameters in such a sphere [2].These were obtained by measuring the correlation of the generated beam with the beams on the  1 ,  2 and  3 (Stokes parameters) axes of the OAMPS (Laguerre and Hermite-Gaussian beams in our case) using a spatial light modulator and a regular intensity camera instead of the PSA in Fig. 2. We generated fully polarized, partially polarized, and unpolarized light using the PSG.For each case, Fig. 3 illustrates the Stokes vector of the input polarization state, the polarization and DoP maps, as well as the average DoP ( parameter) of the vector beam after the q-plate.The intensity profile and position in the OAMPS of the scalar beam after the linear polarizer are also shown.The results demonstrate that the mapping between the spheres maintains the value of the parameter  in Eq. (1).

Fig. 1 .
Fig. 1.Mapping between the inner points of the (a)PS, the (b) HOPS, and the (c) OAMPS using a q-plate and a linear polarizer.

Fig. 3 .
Fig. 3. Results of the mapping of a point inside the PS to the HOPS and OAMPS.DoP (degree of polarization).The ellipses have been reescaled with the DoP at each point.