# Fast 3D Content Update for Wide-Angle Holographic Near-Eye Display
## 4 Update of the Overlap View Region  
The second part of hologram updating for the next viewpoint is to fill in the overlapping area. In general, overlapping parts of original and updated holograms lie in different planes. Therefore, the data of the original hologram cannot be directly copied. However, the original data can be reproporated to the plane of the new hologram to fill the overlap region. This is faster than employing the PAS to generate a new CGH. For the investigated case of wide viewing angles, the hologram is small. Thus, the calculation for updating the overlapping area should consider slight rotation of the initial hologram plane by the angle \(\alpha\) and axial shift by a short distance, \(z_{p}\). Both geometrical operations can be realized by propagation with the AS, where calculations are performed in frequency space. The corresponding calculations of data transfer from the initial hologram to the updated hologram plane (\(x_{2},y_{2}\)) can be described as  
\[O_{p_{2}}(x_{2},y_{2})=\int\widetilde{O}_{p_{1}}\big{(}f_{x_{2}},f_{y_{2}}\big{)} \exp\big{\{}2\pi iz_{p}f_{z}\big{\}}\exp\big{\{}2\pi i\big{(}xf_{x_{2}}+yf_{y_ {2}}\big{)}\big{\}}df_{x_{2}}df_{y_{2}}, \tag{5}\]  
where \(O_{p1}\) and \(O_{p2}\) are complex optical fields of the overlap areas at the initial and propagated planes, respectively. The tilde denotes FT operation, and \(f_{z}\) is the longitudinal frequency component, which is  
\[f_{z}=\sqrt{\lambda^{-2}-f_{x_{2}}^{2}-f_{y_{2}}^{2}}. \tag{6}\]  
To account for the tilt operation by the angle \(\alpha\), the components of the spectra \(\tilde{O}_{p1}(f_{x1}f_{y1})\) are projected onto \(\tilde{O}_{p1}(f_{x2}f_{y2})\) using a rotation matrix describing the new frequency coordinates  
\[f_{x_{2}}=f_{x_{1}}\cos\alpha-\sqrt{\lambda^{-2}-f_{x_{1}}^{2}-f_{y_{1}}^{2}} \sin\alpha, \tag{7}\]  
and \(f_{y2}=f_{y1}\) as the rotation is around the \(y\)-axis. To compute the overlapping data for the second hologram, the rotation operation is determined first, followed by propagation in the axial direction. Because of the non-linearity of the new spatial frequency coordinates obtained after the tilting operation, interpolation of the processed signal is necessary, which is time-consuming [39; 40]. However, for the case of a wide-angle hologram, where angle \(\alpha\) is small, the corresponding calculations of \(\tilde{O}_{p1}(f_{x2}f_{y2})\) can be simplified. The spectrum casting can be represented as a frequency shift operation, \(f_{x2}=f_{x1}+f_{diff}(f_{x1}f_{y1})\), where \(f_{diff}\) can be found by analyzing Equation (7). Notably, \(f_{diff}\) is a function of the transverse frequency coordinates. However, for the case considered, we can find a constant value \(f_{s}\) approximating\(f_{diff}\) with a small error within the entire range of spatial frequencies. To find the optimal value of \(f_{s}\) requires a closer look at the distribution of \(f_{diff}\) shown in Figure 3a. The presented distribution is normalized to the frequency sample size of \(\Delta f_{xu}=1/N_{xu}\Delta\). First, as can be noticed, the required frequency shifts are low; accordingly, the corresponding frequencies of the hologram can be placed in the correct frequency location by selecting proper \(f_{s}\). Notably, the tilt operation should not give an error larger than \(\frac{1}{2}\Delta f_{xu}\). Second, it should be noted that for wide-angle holograms, the point object is encoded in a small frequency area. For example, the hologram's corner frequency corresponds to the object's corner location in the FoV. Thus, we propose an approach where a major part of the hologram has relatively small errors, and the largest errors are obtained at the corner points. This does not mean that the center point will be reproduced with the smallest error.