### Filtering of Spatial Spectrum  
Phase modulation in real space is the commonly used method to manipulate the polarization distribution both in the transverse and longitudinal directions. However, for the vector optical beam with varying polarization along propagation direction in free space, the initial phase modulation will inevitably influence the intensity distribution of the propagating field. In order to improve the axial intensity distribution of the generated beams with longitudinally varying polarization, [PERSON] et al. utilized the Sagnac interferometer [19; 56; 61; 62] to construct the superposition of two beams with complementary axial intensities and orthogonal SoPs based on the spatial spectrum optimization approach proposed by [PERSON] and [PERSON] [47], thereby generating a quasi-Bessel beam with uniform axial intensity but varying polarization upon propagation [46]. The schematic of the theoretical configuration is shown in Figure 8.  
Figure 8: Schematic of the theoretical configuration of reshaping the axial intensity distributions of quasi-Bessel beams [46]. A quasi-Bessel beam composed of two orthogonally polarized beams with linearly varying axial intensity, for which the axial fields can be expressed as  
\[E_{H,V}(z)\exp(i\delta_{H,V})=\sqrt{I_{0}(1\mp az)/2}\exp(\pm i\pi/4) \tag{23}\]  
where \(a\) is a constant determining the varying period. With the axial intensities of two polarization components constantly varying, the polarization also changes correspondingly. The results at \(z=-15\), \(-10\), \(0\), \(10\), \(15\) cm are measured respectively, and the ellipticity of the polarization experiences a variation from increase to decrease, thereby the transformation of the polarization of this quasi-Bessel beam continuously moving along the meridian of the Poincare sphere. More importantly, because of the complementary axial intensity distributions of these two polarization components, the total axial intensity has an approximately uniform profile in the non-diffractive region. Nevertheless, the variation of the ellipticity angle varies with different angles when propagating over the same distance. In order to achieve uniform variation of polarization, another axial envelope is designed as  
\[E_{H}(z) =\sqrt{I_{0}}\sin(2\pi bz) \tag{24}\] \[E_{V}(z) =\sqrt{I_{0}}\cos(2\pi bz) \tag{25}\]  
where \(b\) denotes the varying period of envelopes. As the phase difference is zero, the quasi-Bessel beam is always linearly polarized. the polarization direction has a relationship with the longitudinal distance as  
\[\tan(\pi/2-\psi)=E_{V}/E_{H}=\tan(2\pi bz) \tag{26}\]  
The distributions of Stokes parameters at \(z=0\), \(1.9\), \(3.8\), \(5.6\), and \(7.5\) cm are measured. The results demonstrate the quasi-Bessel beam retains a linear polarization upon propagation that shows a periodic variation of polarization along the equator of the Poincare sphere. Recently, a theoretical model to obtain anomalous VBBs with varying polarization order during propagation was demonstrated by [PERSON] et al. [65]. Compared with changing the charge of VOF by dividing the axicon into various radial sectors [49], the method introduced a continuous phase delay by designed spiral slits [66; 67] is more flexible such that arbitrary polarization orders, including integers and fractions, can be generated along the propagation axis. This approach was inspired by the idea that a zeroth-order Bessel beam can be thought of as the Fourier transform of an annular slit [68]. The diffraction intensity distributions and phase profiles of anomalous Bessel beams at different propagation distances are shown in Figure 9a. A right-/left-handed circularly polarized plane wave illuminates the spiral slit corresponding to \(l=3\) and \(-3\), respectively. It can be observed that the topological charge of the anomalous Bessel vortex beam decreases with the propagation distance. When these two beams are collinearly superposed, the generated anomalous VBB is shown in Figure 9b.