# A Fast Computer-Generated Holographic Method for VR and AR Near-Eye 3D Display
## 2 Methods  
The concept of hologram calculation is shown in Figure 1 with two projection planes for ease of explanation since the projection of 3D object is multidirectional. In Figure 1a, A and B are two points in the space, and layer1 and layer2 are two projection planes. On each projection plane, the projected rays from the spatial 3D object points are in the same direction. p\({}_{\text{1A}}\) and p\({}_{\text{1B}}\) are the projected points of the spatial points A and B on layer\({}_{\text{1}}\), respectively. p\({}_{\text{2A}}\) and p\({}_{\text{2B}}\) are the projection points of space points A and B on layer2, respectively. The projection of 3D object points on one projection plane forms a projected image corresponding to this projecting angle. In the hologram calculation process, the illumination direction of each point on different projection planes is set as the propagation direction, as shown in Figure 1b. The complex amplitude distribution on the holographic plane H is a superposition of light field from all projection planes. When the hologram is reconstructed, the beamlets in these directions are diffracted from the hologram. Similar to integral imaging 3D displays, the overlapping regions of the different beamlets form a 3D object for 3D display [22; 23], where the propagation direction of the beamlets determines the outcome of the 3D display. To simplify calculation, the projection layers are all assumed to coincide in the same plane. On this projection plane, projected images from the 3D object are obtained with the designed projection angles. The light in different directions of the 3D object is calculated onto holographic plane H from the projected images. The direction of propagation of light on a projected image is \(\theta_{ix}\) and angular interval is \(\Delta\theta_{ix}\) as shown in Figure 2. The distance from the projection plane to the holographic plane H is z. The boundary of the propagation of the light from point \(p\) on the image on the holographic plane falls between \(x_{i01}\) and \(x_{i02}\), respectively. Assume the point \(p\) is an on-axis point and the PSF of light propagating from the projection plane to the holographic plane can be expressed as:  
\[psf_{i,j}=\exp[i\frac{2\pi}{\lambda}(\sqrt{{x_{h}}^{2}+{y_{h}}^{2}+{z}^{2}})] \tag{1}\]  
where \(x_{i01}\leq x_{h}\leq x_{i02},y_{j01}\leq y_{h}\leq y_{j02}.x_{i01}\) and \(x_{i02}\) are the boundary of the light field on the holographic plane limited by the projection angle and angle interval in the \(x\) direction, respectively. \(i\) and \(j\) are the index of projected images in \(x\) and \(y\) directions, respectively. \(y_{j01}\) and \(y_{j02}\) are the boundary of the light field on the holographic plane limited by the projection angle and angle interval in the \(y\) direction, respectively. The boundary of the light field distribution area on the holographic plane limited by the projection angle \((\theta_{ix},\theta_{jy})\) and angle interval \((\Delta\theta_{ix},\Delta\theta_{iy})\) can be expressed as:  
\[ \begin{array}{l}x_{i01}=z\tan(\theta_{ix}+\frac{\Delta\theta_{ix}}{2});x_{i 02}=z\tan(\theta_{ix}-\frac{\Delta\theta_{ix}}{2})\\ y_{j01}=z\tan(\theta_{iy}+\frac{\Delta\theta_{ix}}{2});y_{j02}=z\tan(\theta_{ jy}-\frac{\Delta\theta_{ix}}{2})\end{array}  \tag{2}\]  
Figure 1: The concept of hologram calculation. **(a)** Projection of 3D object to different layers; **(b)** the calculation of complex amplitude on holographic plane H.  
Figure 2: The hologram projection angle and boundary. where \(\theta_{iy}\) and \(\Delta\theta_{jy}\) are the projection angle and angular interval in the \(y\) direction, respectively. The complex amplitude on the holographic plane H is the superposition of the convolution of each projected image with its corresponding PSF, which can be expressed as:  
\[U(x,y)=\sum_{i=1:I}\sum_{j=1:I}im_{i,j}\otimes psf_{i,j} \tag{3}\]  
where \(I\) and \(J\) are the number of projected images in x and y directions, respectively.