###### Abstract  
The vector optical beam with longitudinally varying polarization during propagation in free space has attracted significant attention in recent years. Compared with traditional vector optical beams with inhomogeneous distribution of polarization in the transverse plane, manipulating the longitudinal distribution of polarization provides a new dimension for the expansion of the applications of vector optical beams in volume laser machining, longitudinal detection, and in vivo micromanipulation. Two theoretical strategies for achieving this unique optical beam are presented in the way of constructing the longitudinally varying phase difference and amplitude difference. Relevant generation methods are reviewed which can be divided into the modulation of complex amplitude in real space and the filtering of the spatial spectrum. In addition, current problems and prospects for vector optical beams with longitudinally varying polarization are discussed. vector optical beam; structured light; Bessel beam; Poincare sphere +
Footnote †: journal:  
_Applied Sciences_  
**Victor Optical Beam with Controllable Variation of Polarization during Propagation in Free Space: A Review. **  
**[PERSON], T.-Y. **  
**[PERSON]-[PERSON], [PERSON], [PERSON]Q. Vector Optical Beam with Controllable Variation of Polarization during Propagation in Free Space: A Review. **  
**[PERSON], J.-Q. Vector Optical Beam with Controllable Variation of Polarization during Propagation in Free Space: A Review. **  
parabolic- [27] and elliptic- [28] symmetry, double-mode [18], and fractal VOFs, [29] have been generated. However, with a degree of spatial freedom, the propagation direction is also a significant dimension to construct inhomogeneous distributions of polarization. The unique beam, with customized polarization distribution in the longitudinal direction, was first generated in 2015 [30] and has attracted increasing attention in recent years. Beyond enriching the forms of vector beam, the variation of polarization along the direction of propagation provides increased scope for the interaction between light and matter, especially in nonlinear effect and spin-orbit coupling. Moreover, it also offers advantages in the applications of material deep processing, remote polarimetry, and three-dimensional micromanipulation. In order to realize the variation of polarization in the longitudinal direction, a direct method is to modulate the propagation environment of the polarized beam, including by artificially constructing the anisotropic media with a spatially modulated optic axis [31; 32; 33; 34], or by utilizing the self-induced nonlinear effects of optical beams in a nonlinear medium [35; 36; 37; 38]. However, for situations in which the propagation environment cannot be manipulated, such as in free space, the direct methods 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. In recent years, many indirect methods have been proposed to generate vector optical beams with longitudinally varying polarization in free space based on different theoretical and experimental strategies. The schematic diagram of vector optical beams with longitudinally varying polarization in free space is shown in Figure 1. Here, we focus on the most recent research into the unique vector beam with controllable polarization distribution along the longitudinal direction in free space generated by indirect methods. Different theories and experimental methods for generating this vector optical beam are reviewed in detail. Based on experimental results, the variation of polarization and its rate during propagation in free space are discussed with respect to existing problems and possible development directions of the vector optical beam with longitudinally varying polarization.