Fig. 1. Coherence Poincaré sphere of partially coherent optical vector beams^[105]. (a) Coherence and polarization properties of an arbitrary partially coherent vector beam can be represented by two coherence Poincaré vectors q12 and q21. The radius of the coherence Poincaré sphere is PΩμ2; (b) when the beam reduces to a fully polarized partially coherent beam, q12=q21=q and the radius reduces to μ2; (c) when the beam reduces to a fully coherent beam, the coherence Poincaré sphere reduces to a polarization Poincaré sphere. The polarization properties of the beam are described by the polarization Poincaré vector s. The radius of the polarization Poincaré sphere is equal to the degree of polarization P of the beam

Fig. 2. Synthesis of partially coherent vector beams via generalized van Cittert-Zernike theorem^[92]. (a) Experimental setup. A linearly polarized beam passes through a neutral density filter (NDF) and a beam expander (BE), and then goes into a 4f common path interferometric system composed of two thin lenses L1 and L2. A spatial light modulator (SLM) is placed in the input plane of the 4f system with a fork-shaped grating loaded in it. In the frequency plane (rear focal plane of L1), a V-shaped filter is placed to filter out +1 and -1 diffraction orders of the light from the SLM. After the two diffraction orders are filtered out, the beams then go into the quarter-wave plates (QWPs). After the QWPs, two optical beams become the left-hand and right-hand circularly polarized beams and then are superposed by a Ronchi grating (RG) into a vector beam with higher-order polarization state. After passing through a rotating ground glass disk (RGGD), the beam becomes incoherent. After passing through the thin lens L4 and the Gaussian amplitude filter (GAF), the incoherent beam becomes partially coherent. The lens L5 focuses the beam, and the QWP, linear polarizer (LP), and the CCD are used to measure the polarization state of the beam during transmission; (b) the measured Stokes parameters and polarization state for the synthesis of fully coherent vector beam; (c) the measured spatial coherence structure for the partially coherent vector beam

Fig. 3. Synthesis of partially coherent vector beams via coherent-mode superposition. (a) Schematic for synthesis of the beams. The screen of the DMD is split into two parts, which load computer generated holograms to create the modes Txn(r) and Tyn(r). By dynamically controlling the holograms, the partially coherent vector beams can be synthesized; The light intensity, polarization, and coherence structure can be controlled by adjusting the weight and spatial distribution of vector modes^[126]; (b) experimental setup of synthesis of partially coherent vector beams based on the principle of mode superposition^[126]; (c) the intensity distributions of the partially coherent vector beams with nonuniform coherence structures synthesized by pseudo-mode superposition^[125]; (d) controlling the amplitude and phase of Bxyr1,r2 by random-mode superposition^[126]

Fig. 4. Experimental measurement for the spatial coherence of the partially coherent vector beams. (a) Measuring the electromagnetic degree of coherence of partially coherent vector beams based on subwavelength double scatterer interference^[129]; (b) measuring the real and imaginary parts of the complex spatial coherence of partially coherent vector beams based on the generalized Hanbury Brown-Twiss effect^[133]; (c) the experimental results for the measured real and imaginary parts of the partially coherent vector beams with special spatial coherent structure, as well as the modulus of the degree of coherence^[133]

Fig. 5. Propagation properties of partially coherent vector beams in free space. (a) Distribution of light intensity and the energy ratio of polarized and unpolarized parts of radially polarized beams with Gaussian Shelter-mode coherent structure during paraxial transmission^[136]; (b) the evolution characteristics of light intensity, polarization state, polarization degree distribution, and global polarization degree in paraxial transmission of partially coherent vector beam with characteristic spatial coherent structure varying with transmission distance^[92]

Fig. 6. Robust of propagation of partially coherent vector beams^[92]. (a) Intensity and polarization properties of a partially coherent vector beam passing through a sector-shaped obstruction; (b) the Stokes parameters and polarization state for an obstructed fully coherent vector beam at focal plane; (c) the distributions of light intensity and polarization state of a partially coherent vector beam passing through a turbulent atmosphere and the experimental results of linear polarization intensity; (d) experimental results of linear polarization intensity component of completely coherent vector beam passing through turbulent atmosphere

Fig. 7. Distributions for the eigenvalues of the real part of the polarization matrix for the 3D optical fields^[141]. (a) a1≫a2,a1≫a3, the polarization dimension D(r)≈1; (b) a1≈a2≫a3, the polarization dimension D(r)≈2; (c) a1≈a2≈a3, the polarization dimension D(r)≈3

Fig. 8. Characteristic decomposition of 3D polarization state. The polarization matrix Φ can be decomposed into fully polarized state Φ^p, middle-component state Φ^m, and 3D unpolarized state Φ^u

Fig. 9. Beam shaping in the partially coherent tightly focused field^[148]. (a) Schematic of tight focusing of the partially coherent beam; (b) the total intensity, transverse intensity, and longitudinal intensity for the tightly focused Gaussian Schell-model (GSM) beam, multi-Gaussian Schell-model (MGSM) beam, Laguerre-Gaussian Schell-model (LGSM) beam, and Hermite-Gaussian Schell-model (HGSM) beam with radial polarization; (c) the corresponding intensity distributions of the tightly focused field along longitudinal direction

Fig. 10. 3D polarization properties for the partially coherent tightly focused fields^[150-151]. (a) Polarization dimension D for the tightly focused radially polarized Gaussian Schell-model beams with different spatial coherence lengths; (b) the spatial distributions for the 3D degree of polarization P3D and polarization dimension D along x axis; (c) the powers for the fully polarized state, middle-component state, and 3D unpolarized state, as well as the degree of nonregularity along x axis in 3D characteristic decomposition

Fig. 11. Generation of optical 3D unpolarized lattices and channels in a tightly focused field based on coherent structure control of light field^[152]. (a) Average intensity distributions at various propagation distances near the focal region for the tightly focused beam; (b) the spatial distributions of the polarimetric dimension and 3D degree of polarization near the focal field; (c)(d) the spatial distributions of three-dimensional unpolarized array of focal field are controlled based on coherent structure reverse design

Fig. 12. Spatial distributions of fully polarized spin vectors and intermediate spin vectors in a partially coherent tightly focused light field. (a) Incident beams are radially polarized Gaussian Schell-model beams with different initial spatial coherence lengths^[150]; (b) the incident beam is composed by two uncorrelated plane waves with x and y linear polarization states. The degrees of 2D polarization are 0.8, 0.2, and 0, respectively^[153]