### Modulation on the Complex Amplitude in Real Space
#### 3.1.2 Double-Path Generation Method Based on Holographic Gratings
Considering the superposition of two collinear beams with different radial components of wave vector and orthogonally circularly polarized bases, the generated beam in cylindrical coordinates at initial plane can be expressed as  
\[\mathbf{E}(r,\theta,z=0)=E_{01}e^{-ik_{r}r}\mathbf{e}_{l}+E_{02}e^{-ik_{r}2r} \mathbf{e}_{r} \tag{16}\]  
the propagating equation can be expressed under the paraxial approximation as  
\[\mathbf{E}(r,z)=e^{i\frac{\mathbf{k}_{r}^{2}}{2z}}\left[\sqrt{I_{1}}e^{i(k- \frac{k_{r}^{2}}{2k})z}\mathbf{e}_{l}+\sqrt{I_{2}}e^{i(k-\frac{k_{r}^{2}}{2k}) z}\mathbf{e}_{r}\right] \tag{17}\]  
where [64]  
\[I_{q}(r,z)=\frac{2\pi z}{k}E_{0q}^{2}k_{rq}^{2}I_{0}^{2}(k_{rq}r),q=1,2 \tag{18}\]  
Since the Bessel function is influenced by the radial wave vector \(k_{r}\), a square relation between \(k_{r}\) and the intensity \(I\) can also be found. It is evident that an equal-amplitude superposition at the initial plane cannot be maintained during propagation, which may further influence the accuracy of the longitudinal distribution of polarization. Figure 6: (**a**) Schematic representation of the experimental setup based on a Sagnac interferometer combined with the 4f system. (**b**) Schematic of the variation of polarization along the equator of the Poincaré sphere. (**c**) Schematic representation of the polarization distribution of higher-order VBB with varying polarization during propagation [56]. To solve this problem, [PERSON] et al. proposed an amplitude-phase joint modulation method based on a two-dimensional (2-D) binary phase holographic grating (BP-HG) and 4f system. The experimental setup is shown in Figure 7a. The horizontal linearly polarized beam illuminates a phase-only SLM after expanding and collimating. The SLM is located at the input plane of the 4f system which is composed of a pair of lenses (L1 and L2). The designed 2-D BP-HG imprinted on the SLM diffracts the incident beam into different orders, and only the two +1 st-orders are allowed to pass through the Fourier plane of L1 and are converted by two QWP into the left- and right-hand circularly polarized beams. Then the two +1 st-order diffraction beams are recombined by a Ronchi grating (RG), which is placed at the rear focal plane of L2. The complex amplitude transmission function of the BP-HG can be expressed as  
\[t(x,y)=\left\{ \begin{array}{c}e^{i\varphi_{1}},\text{ when rect}\Big{(}\frac{x}{d-a_{1}}\Big{)}\ rect\Big{(}\frac{y}{d-a_{2}}\Big{)}\otimes\text{comb} \Big{(}\frac{x}{d}+\frac{\psi_{1}}{2\pi},\frac{y}{d}+\frac{\psi_{2}}{2\pi} \Big{)}=0,\\ e^{i\varphi_{2}},\text{ otherwise}.\end{array} \right. \tag{19}\]  
where rect(\(x\)) and comb(\(x\)) are rectangular and comb functions, \(\otimes\) is a convolution operator, \(\varphi_{1}\) and \(\varphi_{2}\) are the binary phases of the grating, \(\psi_{1}\) and \(\psi_{2}\) are the phase modulations adding on the two +1 st-orders, \(d\) is grating period, while \(a_{1}\) and \(a_{2}\) are the stripe widths in \(x\) and \(y\) direction, respectively. The two +1 st-order left- and right-handed circularly polarized diffraction beams carrying the respective space-varying phases can be rewritten as  
\[\mathbf{E}_{+1,0}\propto\frac{a_{1}}{d}\Big{(}1-\frac{a_{2}}{d}\Big{)}\text{ sinc}\Big{(}\frac{a_{1}}{d}\Big{)}\exp(i\psi_{1})\mathbf{e}_{l} \tag{20}\]  
\[\mathbf{E}_{0,+1}\propto\frac{a_{2}}{d}\Big{(}1-\frac{a_{1}}{d}\Big{)}\text{ sinc}\Big{(}\frac{a_{2}}{d}\Big{)}\exp(i\psi_{2})\mathbf{e}_{r} \tag{21}\]  
Figure 7: Generation of localized linearly polarized vector beams with longitudinally varying polarization based on the amplitude–phase joint modulation method. (**a**) Generation system based on two-dimensional binary phase holographic grating and 4f system. (**b**) Intensity profiles of generated vector beams at different propagation distance. (**c**) Dependence of the orientation angle of local linear polarization on propagation distance [63].