### Modulation on the Complex Amplitude in Real Space
#### 3.1.1 Single-Path Generation Method Based on Phase Mask
The above approaches to generate vector optical beams with longitudinally varying polarization in free space all rely on digital phase mask [51], enabled by SLM, and require a particular SoP at the input plane. Recently, a form-birefringent was demonstrated by [PERSON] et al. to impart the arbitrary polarization along the propagation direction without a priori knowledge of the incident polarization [52]. The metasurface is composed of dielectric nanopillars with structured birefringence in the way of subwavelength-spaced arrays. By controlling the constituent and arrangement, the metasurface can manipulate the incident waveform converting to any desired output [53] based on the Jones calculus [54] and dual phase holography [55]. Optical metasurfaces have become a current subject of intense research due to their unprecedented capability in manipulating the polarization at subwavelength resolution. The desired polarization distribution along the propagation direction can be achieved by the superposition of two (or more) monochromatic waveforms of the same wavelength but with different SoP and spatial frequencies, which can be expressed in the Jones vector as  
\[\mathbf{E}(z)=\mathbf{E}_{1}(z)+\mathbf{E}_{2}(z)=e^{ikz}\left[ \begin{array}[ ]{c}E_{1,x}e^{-i\frac{ikz}{2}z}+E_{2,x}e^{i\frac{ikz}{2}z}\\ E_{1,y}e^{-i\frac{ikz}{2}z}+E_{2,y}e^{i\frac{ikz}{2}z}\end{array} \right] \tag{12}\]  
where \(\Delta k_{z}=k_{z2}\)-\(k_{z1}\) and \(k_{z}\) is the average spatial frequency (\(k_{z2}+k_{z1}\))/2. Metasurfaces are a particularly convenient platform for obtaining the desired polarization response. The metasurfaces proposed by [PERSON] et al. can act as \(z\)-dependent polarizing devices, as shown in Figure 5a, which can also be treated as a \(z\)-dependent 2 \(\times\) 2 Jones matrix that  
Figure 4: Generation of an OP beam with spatial oscillating polarization. (**a**) Schematic diagram of generating oscillating phase shift along the propagation direction by using two SLMs. (**b**) Experimental setup for generating OP vector beams. (**c**) Experimental results of generated OP vector beams. (**d**) The angle of rotation as a function of propagation distance. Adapted with permission from [50]. implements a transformation from an incident scalar light into a vector beam with varying polarization along the optical axis. The transformation function can be expressed as  
\[\widetilde{U}(r,z)=e^{-i\omega t}\sum_{m=-N}^{N}A^{(m)}J_{0}\left(k_{r}^{(m)}r \right)e^{ik_{z}^{(m)}z} \tag{13}\]  
where \(\tilde{A}^{(m)}\) are 2 \(\times\) 2 matrix-valued coefficients and can be obtained from the equation exhibiting similarity with a Fourier integral that  
\[\left[ \begin{array}{cc}A_{11}^{(m)}&A_{12}^{(m)}\\ A_{21}^{(m)}&A_{22}^{(m)}\end{array} \right]=\frac{1}{L}J_{0}^{L}\left[ \begin{array} []{cc}F_{11}(z)&F_{12}(z)\\ F_{21}(z)&F_{22}(z)\end{array} \right]e^{-i\tilde{\omega}_{z}^{m}z}dz \tag{14}\]  
where \(\widetilde{F}\) may take the form of an arbitrary \(z\)-dependent Jones matrix with a periodicity of \(L\) along the propagation direction. Based on the structure of dielectric nanopillars with structured birefringence, the desired Jones matrix function can be implemented and the corresponding metasurfaces can be locally represented by a spatial arrangement of linearly birefringent wave-plate-like elements as  
\[\widetilde{J}(x,y)=R(-\phi(x,y))\left[ \begin{array}{cc}e^{ik_{x}(x,y)}&0\\ 0&e^{ik_{y}(x,y)}\end{array} \right]R(\phi(x,y)) \tag{15}\]  
where \(R(\phi)\) is the 2 \(\times\) 2 rotation matrix. Due to anisotropy of dielectric nanofins, unit cells of this topology provide two propagating modes that experience different phase delays, \(\delta_{x}\) and \(\delta_{y}\). By changing the dimensions of the nanopillars and the pillar's angular orientation in the transverse plane, the phase retardation and rotational angle can be adjusted. Furthermore, to break the constraints that limit the possible polarization behaviors, a multidimensional matrix reconstruction is implemented based on dual matrix holography.