### Modulation on the Complex Amplitude in Real Space
#### 3.1.2 Double-Path Generation Method Based on Holographic Gratings  
The single-path generation method is simple and highly effective. However, when using SLM as the phase modulation device, the phase mask is limited by the modulation depth of the SLM which constrains the frequency of the longitudinal polarization oscillation. Although the metasurface may solve the problem of phase modulation depth, this manufactured element lacks flexibility compared to the digital modulation method based on SLM. The generation method based on digital holographic gratings is a valid way to promote the modulation depth without losing the flexibility of the longitudinal manipulation of polarization. [PERSON] and [PERSON] et al. demonstrated an experimental setup using a Sagnac interferometer combining with the 4f system [56]. In theory, the Pancharatnam-Berry (PB) phase [57; 58; 59; 60] is introduced to account for the variation of polarization in the longitudinal direction. To achieve the longitudinally varying PB phases, the modified Sagnac interferometer is used [19; 61; 62], as depicted in Figure 6a. The system comprises two main parts. One is the interferometer section which is made up of two mirrors (M1 and M2) and a reflective phase SLM used to prepare transverse PB phases, while the other part is the transverse-to-longitudinal structuring based on an axicon. An input expanded beam is divided into two orthogonally polarized components by a polarization beam splitter (PBS), and then enters the Sagnac interferometer. The designed holographic grating placed at the front focal plane of L3 is imprinted on the SLM. The \(\pm\)1 st-order diffraction beams are filtered through the same open aperture A placed on the Fourier plane of L3. After the collimating lens L4, two collinear beams with orthogonal polarizations and different phase modulations pass through an axicon to achieve the transverse-to-longitudinal transformation of the PB phase, and the VBB with spatially varying polarization in the propagation direction is generated. When the orthogonal circular polarizations are chosen as bases and a QWP is inserted after the PBS, the variation of polarization along the equator of the Poincare sphere can be achieved, as shown in Figure 6b. It has been demonstrated that the generated central linear polarization distribution changes from horizontal linearly polarized to linearly polarized at 45\({}^{\circ}\) to vertical linearly polarized to linearly polarized at 135\({}^{\circ}\) and back to horizontal linearly polarized within the propagation region of 11.2 cm. The rotation rate of the orientation of linear polarization can be estimated as \(\sim\)1.6 \({}^{\circ}\) / mm. When choosing the horizontal and vertical polarization as bases, the variation of polarization along a meridian of the Poincare sphere can be achieved. In this case, an HWP is inserted after the PBS. The generated central polarization experiences a variation from 45\({}^{\circ}\) linearly polarized to right circularly polarized, to 135\({}^{\circ}\) linearly polarized, to left circularly polarized, and back to 45\({}^{\circ}\) linearly polarized within a region of 112 mm. Based on this experimental setup, a higher-order VBB with varying polarization during propagation can also be generated. As shown in Figure 6c, the orientation of polarization in the localized linear-polarized VOF rotates with propagation distance at a uniform rate. The axicon-like radial phase modulation in the initial plane has been widely used to achieve the oscillating phase delay and to further drive the variation of polarization along the propagation direction. However, [PERSON] et al. demonstrated that the radial wave number of axicon-like phase modulation will perform an additional amplitude modulation to the Bessel profile, which determines the overall intensity [63].