## 4 Discussion  
Methods to generate vector optical beams with polarization longitudinally varying along propagation direction have been rapidly developed in recent years. Although an increasing number of approaches to achieve the longitudinal manipulation of polarization in free space have been demonstrated, there are still some problems to be solved. For practical purposes, a highly effective experimental method with accurate and flexible control of the longitudinal distribution of polarization in free space has always been sought. Although the single-path generation method has relatively high efficiency, the flexibility of this method is limited by the modulation depth of the phase mask. For the applications of microfabrication or micromanipulation, the variation of polarization with high oscillation frequency in a small region is desired, which requires a large value for the radial wave number and much greater modulation depth. This problem may be solved by the double-path generation method based on digital holographic grating, because the phase modulation of diffraction beams is only determined by the structure of the grating. However, the existing double-path method is confronted with the problem of low efficiency, which is mainly caused by the high loss of digital holographic grating and may be solved by optimizing the design of phase grating. A generation method which balances efficiency and flexibility will contribute to practical applications of vector optical beams with polarization longitudinally varying along the propagation direction. The accurate and flexible customization of the longitudinal distribution of polarization is a promising direction for the development of this unique vector optical beam. Most of the reported vector optical beams with longitudinal polarization distribution exhibit uniform variations of polarization along trajectories corresponding to the equator or meridian of the Poincare sphere, as shown in Figure 10a and b, respectively. For customizing the longitudinal distribution of polarization, the variation of polarization is predicted to trace arbitrary trajectories on the Poincare sphere, as shown in Figure 10c, and the rate of the variation of polarization in longitudinal direction is predicted to be flexibly adjusted. The arbitrary variation of polarization in ellipticities and the long axis orientation will bring additional degrees of freedom to engineer the optical beam which may provide further possibilities for the interaction between light and matter, as well as broadening the application region of vector beams. In fact, the variation of polarization along arbitrary circular trajectories of the Poincare sphere can be realized in experiments by controlling the parameters of superposed beams, including the polarized bases and the amplitude radio [63]. Figure 9: **(a)** The diffraction intensity distributions and phase profiles of anomalous Bessel beams under different propagation distances. **(b)** The intensity and polarization distribution of the generated anomalous VBBs at different propagation distances. Adapted with permission from [65], copyright Elsevier, 2021. Benefiting from in-depth research into the characteristics of VOFs, the characteristics of inhomogeneous polarization distributions in the transverse plane have been widely explored in many applications including nonlinear optics [4; 5; 69; 70], microfabrication [12; 13], and quantum information [8].To date, the research into the vector optical beam with longitudinal distribution of polarization in free space has focused on the generation method of this special beam. However, effective methods to achieve the longitudinal manipulation of polarization in free space provide the technical basis for constructing diverse and desired longitudinal distributions of polarization, which may induce special responses in various mediums.