### Modulation on the Complex Amplitude in Real Space
#### 3.1.1 Single-Path Generation Method Based on Phase Mask
The polarization distribution of the generated 1 st-order VBB where \(l_{1}=-l_{2}=1\) are analyzed by the intensity profiles after passing through polarizers in which two lobes are displayed. The orientation of the lobes rotates with propagation distance, indicating the rotations of localized linear polarization during propagation. After a propagation distance of \(Z/2\), the radially polarized VOF turns to an azimuthally polarized VOF with the same charge. The rotating rate of linear polarization in the VBB is approximately uniform at 0.18 deg/mm. [PERSON] et al. also attempted to change the topological charge of the vector beam along the propagation axis by using multi-mode phase modulation. Each phase mask with fixed radial phase distribution is divided into different radial sectors where different topological charges of the spiral phase are attached. Experiments were carried out by dividing the phase mask into five radial sectors, with different charges \(l_{1}=-l_{2}=1\), 2, 3, 2, 1 corresponding to different propagation distances. Two, four, and six lobes can be observed after an analyzer, corresponding to the charges of VOF of 1, 2, 3, which means the charge of localized linear-polarized VOF changes along the propagation axis. Apart from dividing the phase mask and modulating the optical beam multiple times based on the polarization selectivity of SLM, vector optical beams with longitudinally varying polarization can also be generated by cascading SLMs in a single-path system. [PERSON] et al. demonstrated a Bessel-type vector beam, named oscillating polarized (OP) vector beam, with a spatially oscillating polarization along the optical axis based on the cascade of SLMs [50]. To obtain the variable phase shift along the propagation axis, they used the combination of SLM1 and SLM2 displaying the different phase patterns shown in Figure 4a. When a plane wavefront illuminates an axicon-like phase mask uploaded on SLM1, it will be converted to a conical wave. Then, the beam passes through the SLM2, therefore achieving different additional phase modulation. The schematic diagram is shown at the top of Figure 4a. The oscillating phase shift \(\delta(z)\) along the optical path will be obtained in the overlap region. Based on the oscillating phase shift in the longitudinal direction, [PERSON] et al. constructed an experimental setup to achieve the variation of polarization along the propagation direction in free space, which is shown in Figure 4b. The horizontal linearly polarized fundamental Gaussian beam illuminates the phase mask on the SLM1 which performs as the combination of an axicon and a \(l\)th-order spiral phase modulation. An approximate Bessel beam with an azimuthal topologic charge of \(-l\) will be generated because of the reflective SLM1. The half wave plate (HWP) is placed at an angle of 22.5\({}^{\circ}\) to the horizontal direction in order to rotate the polarization orientation of the generated Bessel beam at 45\({}^{\circ}\). After reflection by SLM2, only the horizontal polarization component will be modulated. Except for the difference in radial phase modulating with SLM1, the SLM2 will provide another spiral phase modulation with a topologic charge of \(2l\) by the uploaded phase mask. A 45\({}^{\circ}\) QWP is used to transform the two linearly polarized Bessel beams into left and right circularly polarized beams. The experimental results of generated 1 st- and 2 nd-order OP vector beams are shown in Figure 4c. It can be observed that the orientation of polarization  
Figure 3: The generation system of a VBB whose polarization varies with propagation [49]. in the localized linearly polarized VOF rotates with the propagation distance. The rotation angle has a relation with the oscillating phase shift and azimuthal topologic charge as  
\[\theta=\delta(z)/2l \tag{11}\]  
which is coincident with the measured rotation angle at different propagation distances, as shown in Figure 4d. The highest rotating rate achieved by this method is \(\sim\)0.15 deg/mm.