Coherence and polarization are two intrinsic properties of optical fields. The investigation of optical coherence has boosted the development of partially coherent optics, while the study of polarization properties has led to the discovery and application of optical structured vector fields. For a long time, the coherence and polarization properties of optical fields were generally treated as independent degrees of freedom and often studied separately. Since the 1990s, researchers have gradually realized the inherent correlation between the coherence and polarization properties of optical fields. It has been recognized that coherence and polarization properties can interact during light beam propagation or in the interaction of light with complex media. The joint control of coherence and polarization has driven the study of partially coherent vector optical fields. However, previous research mainly focuses on electromagnetic Gaussian Schell-model beams, whose coherence structure follows a Gaussian distribution. Recently, with the emerging research on light field manipulation and structured light, and the development of theories and technologies for controlling the coherence structure of optical fields, the research focus on partially coherent vector beams has gradually shifted toward those with special spatial coherence structures. Due to the control of vectorial coherence structures, these beams exhibit characteristics during propagation that are completely different from traditional electromagnetic Gaussian Schell-model beams. They have potential applications in far-field polarization shaping and optical super-resolution imaging. Furthermore, with the rapid development of nano-optics, research on three-dimensional optical fields has emerged. Studies have indicated that due to the modulation of coherence, partially coherent vector fields exhibit rich three-dimensional polarization characteristics. We review the research progress on the joint control of coherence and polarization in optical fields, with a focus on the characterization and synthesis of two-dimensional partially coherent vector optical beams with special spatial coherence structures, and their robust transmission properties in complex environments. By combining developments in nanophotonics, we present the extension of two-dimensional partially coherent vector beams to three-dimensional partially coherent vector fields.

We start by reviewing the characterization, synthesis, measurement, and propagation of two-dimensional partially coherent optical beams. In the characterization of two-dimensional partially coherent vector beams, the utilization of two-dimensional coherence and polarization matrices is common. Various polarization characteristics of the beams and construction of polarization Stokes parameters and Poincaré sphere are obtained by the two-dimensional polarization matrix. Although the coherence Stokes parameters similar to the polarization Stokes parameters can be constructed using the two-dimensional coherence matrix, the lack of Hermitian symmetry in the coherence matrix prevents the direct construction of a coherence Poincaré sphere. To this end, Set?l? et al. from the University of Eastern Finland proposed a method using the Gram matrix to construct a coherence Poincaré sphere as shown in Fig. 1. This sphere can fully describe the coherence and polarization characteristics of a partially coherent vector optical beam between points r₁ and r₂ using the coherence Poincaré sphere vectors q₁₂ and q₂₁. Concerning the construction and synthesis of partially coherent vector optical beams, we primarily review methods for synthesizing partially coherent vector optical beams with novel coherence structures. This includes the scheme based on the generalized van Cittert-Zernike theorem (Fig. 2) and the method based on vector-mode superposition (Fig. 3). The former method based on the generalized van Cittert-Zernike theorem is suitable only for synthesizing vector optical beams with spatially uniform coherence structures and has low optical efficiency due to the utilization of rotating ground glass to synthesize spatially incoherent light. The latter method based on vector-mode superposition solves the low efficiency and the inability to synthesize spatially non-uniform coherence structures, providing a significant advantage in synthesizing high-power spatially non-uniform coherence structures. In terms of measuring partially coherent vector optical beams, traditional methods based on Young's double-slit interference have low spatial resolution and measurement speeds. While the Hanbury Brown-Twiss (HBT) experiment based on intensity correlation resolves the limitations of Young's double-slit interference, it only allows for the absolute value measurement of coherence structures. To this end, Chen et al. proposed a generalized Hanbury Brown-Twiss experimental scheme (Fig. 4), which introduces a vector fully coherent reference light to achieve simultaneous and rapid measurement of the real and imaginary parts of the coherence structures of partially coherent vector optical beams. Regarding the propagation of partially coherent vector optical beams, studies indicate that due to the modulation of vectorial coherence structures, these beams exhibit completely different propagation characteristics compared to traditional electromagnetic Gaussian Schell-model beams. The former shows a gradual increase in polarization degree during propagation, while the latter exhibits a gradual decrease in polarization degree during propagation (Fig. 5). Meanwhile, it is demonstrated that vector optical beams with special coherence structures exhibit robust propagation characteristics in complex media (Fig. 6) to present potential applications in far-field polarization shaping. Additionally, we review the research on three-dimensional partially coherent vector optical fields. In the characterization of three-dimensional partially coherent vector fields, three-dimensional coherence and polarization matrices are employed. Unlike fully coherent vector optical fields, partially coherent vector fields exhibit rich three-dimensional polarization characteristics due to the coherence modulation, with polarization dimensions exceeding 2 (Fig. 7). In contrast, fully coherent vector optical fields localized in a plane at a determined spatial position only exhibit two-dimensional polarization characteristics. Furthermore, for clearer presentation of three-dimensional polarization structures in partially coherent vector optical fields, characteristic decomposition is utilized to decompose the three-dimensional polarization matrix into fully polarized state, middle-component polarization state, and three-dimensional unpolarized state (Fig. 8). The middle-component state is generally considered as the two-dimensional unpolarized state, but under complex polarization matrix of the middle-component state, it exhibits three-dimensional polarization properties, which can be characterized by the concept of the degree of nonregularity.

It is shown that rich three-dimensional polarization structures are presented in partially coherent tightly focused fields. In studying the three-dimensional polarization characteristics of partially coherent tightly focused fields, the first challenge is the rapid calculation of the tightly focused fields. Traditional methods using the Richard-Wolf vector diffraction integral formula for direct integration typically take hundreds of hours. To enhance computational efficiency, Tong et al. proposed a method based on random-mode expansion in 2020 to achieve rapid computation of partially coherent tightly focused fields. Subsequently, researchers from Spain (Carnicer et al.) and China (Chen et al.) separately put forward convolution algorithms to fast calculate the tight focusing properties of partially coherent vector optical beams with a Schell-model correlation function. Compared to random mode expansion algorithms, the advantage of convolution algorithms is that the computation time is independent of the coherence length of the incident partially coherent vector beams, providing a significant advantage in computing the tightly focused characteristics of low-coherence optical fields. However, the four-dimensional convolution algorithm can only compute the tightly focused characteristics of partially coherent optical fields with Schell-model correlations. The random mode expansion algorithm is still required to improve computational efficiency and thus compute the tightly focused characteristics of partially coherent fields with spatially non-uniform correlations. Additionally, the four-dimensional convolution algorithm can only rapidly compute the polarization characteristics of tightly focused fields. The mode superposition algorithm is still required to compute the coherence characteristics between two or more points in the tightly focused field. By adopting fast algorithms, it is discovered that coherence structures play a critical role in shaping tightly focused fields. Research indicates that the transverse and longitudinal intensities of the tightly focused field can be controlled by the coherence structure of the incident light (Fig. 9). Furthermore, fast algorithms help discovered that in the tightly focused field of a radially polarized Gaussian Schell-model beam, three-dimensional polarization states with polarization dimension greater than 2 and three-dimensional degree of polarization less than 0.5 can be observed. By controlling the coherence length of the incident beam, the polarization dimension and three-dimensional degree of polarization of the focused field can be controlled (Fig. 10). Additionally, by introducing coherence structure control, three-dimensional unpolarized lattice and channels with specific spatial distributions can be designed near the focus (Fig. 11). Due to the rich three-dimensional polarization structures in partially coherent tightly focused fields, the spin angular momentum vector of the field can be decomposed into contributions from the fully polarized state and middle-component state. Under the nonregular middle-component state, the spin angular momentum will be carried. Research indicates that for the classical Gaussian Schell-model beams, the focused field exhibits three-dimensional nonregular polarization characteristics under moderate coherence length. Therefore, the spin angular momentum is contributed by both the fully polarized state and the nonregular middle-component state. Since the coherence structures and radial polarization of the optical field exhibit rotational symmetry, the generated spin angular momentum in the focused field has a vortex distribution with rotational symmetry as well. When the coherence structure or polarization state exhibits spatial asymmetry, it is found that the directions for spin vectors of the fully polarized state and middle-component state can be completely different (Fig. 12).

We review partially coherent vector optical fields, including two-dimensional partially coherent vector beams and three-dimensional partially coherent vector fields. Meanwhile, we emphasize the basic principles and experimental techniques for controlling and measuring the two-dimensional coherence structure of partially coherent vector beams and analyze the propagation characteristics of beams with novel vectorial coherence structures. Results show that partially coherent vector optical beams controlled by coherence structures can maintain robust propagation characteristics in complex environments and have potential applications in far-field optical polarization shaping. Additionally, in conjunction with the development of nanophotonics, we discuss the extension of two-dimensional partially coherent beams to three-dimensional partially coherent fields. Specifically, we introduce the three-dimensional polarization structure, three-dimensional nonregular polarization state, and spin angular momentum structure caused by optical coherence in vector optical fields. Genuine three-dimensional polarization structures are discovered in partially coherent tightly focused fields, with the influence of coherence on polarization dimensions, three-dimensional degree of polarization, degree of nonregularity, and spin angular momentum structure analyzed. Optical coherence as a novel degree of freedom plays a crucial role in the control and application expansion of vector optical fields. With the development of temporal and spatio-temporal joint control techniques in the optical field, temporal or spatio-temporal structured optical fields play a significant role in fields such as ultra-fast optics, quantum optics, and nonlinear optics. Currently, the spatial coherence structure control of vector optical fields has been widely studied, but research on their temporal or even spatio-temporal joint control is limited. Optical coherence as an intrinsic property of the optical field is expected to provide a novel degree of freedom for spatio-temporal structured optical fields and thus expand the application range of such fields. Additionally, we specifically review the joint control of coherence and polarization parameters. Optical coherence plays a crucial role in the joint control of more parameters. Research suggests that coherence plays an important role in the spin (polarization)-orbital angular momentum (phase) coupling of light. In the case of three-dimensional partially coherent vector optical fields, coherence not only induces three-dimensional polarization structures in tightly focused fields but also plays a significant role in controlling the evanescent waves and surface plasmon polaritons. Finally, this has led to the research on physical properties and potential applications of partially coherent surface waves.