Significance   Based on the spatial manipulation technique of polarization, the unique spatially structured properties of polarization make vector optical beam show great research values and application potentials in optics and its interdisciplinary fields. Previous studies mainly focused on the polarization manipulation in the transverse plane, but the longitudinal (propagation) direction is also an important dimension for manipulating optical field. The unique beam, with customized polarization distribution in the longitudinal direction, has attracted increasing attentions in recent years. Beyond enriching the diversity of vector beam, the variation of polarization along the direction of propagation provides increased scope for the light-matter interaction, especially in the optical nonlinear effect and spin-orbit coupling. Moreover, it also offers applied advantages in remote polarimetry, material deep processing and three-dimensional micromanipulation.

  Progress   First, the generation principles of vector beam with longitudinally varying polarization are introduced. In order to realize the variation of polarization in the longitudinal direction in free space, the direct method to modulate the propagation environment of the polarized beam are unsuitable and the possibility of modulating the beam at the initial plane to indirectly control the longitudinal distribution of polarization should be taken into consideration. Two main methods reported to achieve longitudinally varying polarization are the construction of varying phase difference and amplitude difference in the propagation direction. The longitudinally varying phase difference is achieved by discrepant initial radial phase modulations on the orthogonally polarized components, while the construction of varying amplitude difference in the propagation direction is achieved by different spatial spectrum filters for the customized amplitude relations between orthogonally polarized components. Relevant experimental methods are summarized which can be divided into the modulation of phase and the filtering of the spatial spectrum. The phase modulation method includes the single-path generation method based on phase mask (Fig.3-6) and double-path generation method based on holographic gratings (Fig.7-9). As the most common method to generate vector beam with longitudinally varying polarization, the phase modulation method has the problem of controllability on the polarization variation. On the one hand, this controllability is reflected in the accuracy of the longitudinal manipulation of polarization. Due to the different initial phase modulation, corresponding variation occurs in the amplitude of orthogonally polarized components during propagation, which will inevitably affect accuracy of polarization manipulation especially for the high-frequency longitudinal variation of polarization. On the other hand, the controllability is reflected in the flexibility of the longitudinal spatial modulation of polarization. Except for the variation of polarization along the equator and meridian of the Poincaré sphere, the continuous longitudinal variation of polarization can also track other trajectories on the Poincaré sphere. The recent works to improve the controllability of longitudinal manipulation of polarization are discussed in detail.

  Conclusions and Prospects  Methods to generate vector beam with longitudinally varying polarization along propagation direction have been rapidly developed in recent years. Although many approaches to achieve the longitudinal manipulation of polarization have been demonstrated, there are still some problems to be solved. On the one hand, the generation method which balances efficiency and flexibility will contribute to research and practical applications of vector beams with longitudinally varying polarization. On the other hand, based on the novel spatial manipulation dimension of polarization, the interaction between the longitudinally varying polarization and matter still need to be further studied to give full play to its longitudinal polarization "ruler" role, and the application of unique vector beam in laser depth machining, laser measurement and optical micromanipulation needs to be expanded. The research of this paper aims to provide some reference for the design and generation of vector beam with longitudinally varying polarization. The generation theories and experimental methods of vector beam with longitudinally varying polarization are summarized, and the development prospects are also forecasted, which may be helpful for the manipulation techniques of optical field and its applications in laser fabrication, laser measurement and optical micromanipulation.