### Modulation on the Complex Amplitude in Real Space
#### 3.1.1 Single-Path Generation Method Based on Phase Mask  
In 2015, [PERSON] et al. proposed a type of VBB whose polarization varies with propagation distance, by adding an extra radial phase retardation profile between two orthogonal linear-polarized components [30]. The schematic diagram is shown as Figure 2a. In theory, a collimated light enters a diffractive axicon which acts as a radial diffraction grating to produce an approximate Bessel beam. The modulation of radial diffraction grating can be expressed as  
\[g(r)=\exp\left(-i\frac{2\pi r}{d}\right) \tag{9}\]  
where \(d\) denotes the grating period. If a birefringent axicon is placed in front of the diffraction grating, a spatially varying phase shift will be introduced between two orthogonal linear polarizations as a function of radius, which further influences the polarizations at different longitudinal positions. This configuration is realized by an experiment arrangement, as indicated in Figure 2b. By using a 4f system combined with a spatial light modulator (SLM) based on a divided screen, two orthogonal polarization components are modulated by different phase patterns attached on the SLM in the single path. The initial 45\({}^{\circ}\) linear-polarized beam illuminates a half of the SLM and only the polarization component parallel to the liquid crystal director can be modulated. The beam is then reflected to the other half of the SLM by means of a lens and a mirror. The initial polarization undergoes reversal due to the function of the quarter wave plate (QWP). The unmodulated polarization component will pass through the other half of the SLM and experiences a different phase modulation compared to the first passage. Both the phase patterns encoded on different sides of the SLM act as axicons. However, the tilt angles of these two 'axicons' are different which leads to a difference between the radial components of wave vector on these two polarizations. Finally, by synthesizing two orthogonal polarization components, a Bessel beam whose polarization changes with propagation distance in free space is generated. The polarization of the generated zeroth-order Bessel beam at different propagation distances are measured. It changes from linearly polarized at 45\({}^{\circ}\) to LCP to linearly polarized at 135\({}^{\circ}\) to RCP within the propagation region of 75 cm. It is demonstrated that higher order Bessel beams with varying polarization during propagation in free space can also be achieved by adding a spiral phase to the radial diffractive grating [30] which can be expressed as  
\[g(r,\phi)=\exp\left[-i\left(\frac{2\pi r}{d}+l\phi\right)\right] \tag{10}\]  
where \(l\) denotes the topological charge of the spiral phase. Unlike for the zeroth-order Bessel beam, the resulting nondiffracting beam shows a dark center and a surrounding ring corresponding to the \(l\)th-order Bessel function, but the variation of polarization at the central ring is the same with those in zeroth-order Bessel beam. [PERSON] et al. further generated higher order VBSs with the distribution of polarization varying during propagation in free space [49]. By means of adding different spiral and radial phases onto two polarization components, they achieved VBSs with variable charge and polarization in the longitudinal direction. In addition, by tailoring the additional phase shift between the two orthogonal linear polarizations as a function of the distance, the rotation rate of localized linear polarization of the VBSs can be controlled. The experimental setup is shown in Figure 3, which is similar to Figure 2b. The phase masks encoded on SLM have additional spiral phases with opposite topological charges, and extra wave plates are inserted in the optical path. Figure 2: **(a)** Schematic diagram of line focus with continuous variation of the polarization along the propagation distance. **(b)** The generation system of a Bessel beam whose polarization varies with propagation [30].