## 2 Layer-Based Shifted Fresnel Diffraction
 \tag{8}\]The sampled 2D Fresnel diffraction can be expressed as:  
\[U(m,n) =\frac{\exp(j\kappa z)}{j\lambda z}\exp\left[j\frac{k}{2z}\big{(}{u_{ m}}^{2}+{v_{n}}^{2}\big{)}\right]\sum\limits_{p}\sum\limits_{q}U(p,q)\exp\left[j \frac{k}{2z}\big{(}{x_{p}}^{2}+{y_{q}}^{2}\big{)}\right]\exp\left[-j\frac{2\pi}{ \lambda z}\big{(}{x_{p}}{u_{m}}+{y_{q}}{v_{n}}\big{)}\right]\] \[=\frac{\exp(j\kappa z)}{j\lambda z}\exp\left[j\frac{k}{2z}\big{(}{u _{m}}^{2}+{v_{n}}^{2}\big{)}\right]\exp\left[-j\frac{2\pi}{\lambda z}\big{(}{x_ {0}}m\Delta u+{y_{0}}n\Delta v\big{)}\right] \tag{9}\] \[\sum\limits_{p}\sum\limits_{q}U(p,q)\exp\left[j\frac{k}{2z}\big{(}{ x_{p}}^{2}+{y_{q}}^{2}\big{)}\right]\exp\left[-j\frac{2\pi}{\lambda z}\big{(}{x_ {p}}{u_{0}}+{y_{q}}{v_{0}}\big{)}\right]\exp\left[-j\frac{2\pi}{\lambda z} \big{(}{pm\Delta x}\Delta u+q\pi\Delta y\Delta v\big{)}\right].\]  
Let:  
\[2 pm=p^{2}+m^{2}-\left(m-p\right)^{2}, \tag{10}\] \[2 qn=q^{2}+n^{2}-\left(n-q\right)^{2}.\]  
According to the convolution theorem, the optical wave field \(U(m,n)\) can be calculated as:  
\[U(m,n)=C\cdot F^{-1}[F[a(p,q)]F[b(p,q)]], \tag{11}\]  
where:  
\[C=\frac{\exp(j\kappa z)}{j\lambda z}\exp\left[j\frac{k}{2z}\big{(}{u_{m}}^{2}+ {v_{n}}^{2}\big{)}\right]\exp\left[-j\frac{2\pi}{\lambda z}\big{(}{x_{0}}m \Delta u+{y_{0}}n\Delta v\big{)}\right]\exp\left[-j\pi\left(\frac{\Delta x \Delta u}{\lambda z}m^{2}+\frac{\Delta y\Delta v}{\lambda z}n^{2}\right)\right], \tag{12}\]  
\[a(p,q)=U(p,q)\exp\left[j\frac{k}{2z}\big{(}{x_{p}}^{2}+{y_{q}}^{2}\big{)} \right]\exp\left[-j\frac{2\pi}{\lambda z}\big{(}{x_{p}}{u_{0}}+{y_{q}}{v_{0}} \big{)}\right]\exp\left[-j\pi\left(\frac{\Delta x\Delta u}{\lambda z}p^{2}+ \frac{\Delta y\Delta v}{\lambda z}q^{2}\right)\right], \tag{13}\]  
\[b(p,q)=\exp\left[j\pi\left(\frac{\Delta x\Delta u}{\lambda z}p^{2}+\frac{ \Delta y\Delta v}{\lambda z}q^{2}\right)\right]. \tag{14}\]  
By imposing the required sampling rules in the source plane and the observation plane, diffraction patterns with different scale factors can be reconstructed with the help of shifted Fresnel diffraction. Figure 2 numerically demonstrates the shifted Fresnel diffraction of a square aperture, with three different scale factors. The size of the aperture is \(2.4\) mm \(\times\) 2.4 mm. The sampling number in the source plane is \(1024\times 1024\), and the sampling interval is 8 \(\mu\)m. The wave propagation distance is 300 mm, and the wavelength used in the simulation is 633 nm. The diffraction patterns in the observation plane are computed with different scale factors. Figure 2a-c shows the diffraction patterns, with \(\Delta u=6\)\(\mu\)m, \(\Delta u=8\)\(\mu\)m, and \(\Delta u=10\)\(\mu\)m, respectively. Figure 1: Geometry of shifted Fresnel diffraction. In layer-based 3D CGH synthesis, the 3D scene is firstly partitioned into multiple layers, which are parallel to the CGH plane. Then, the wave fields diffracted from the partitioned layers to the CGH plane are calculated. By using shifted Fresnel diffraction in the diffraction simulations from each layer to the CGH plane, the sampling parameters addressed on the 3D scene can be adjusted accordingly. The diagram of the layer-based shifted Fresnel diffraction is shown in \[=\frac{\exp(j\kappa z)}{j\lambda z}\exp\left[j\frac{k}{2z}\big{(}{u _{m}}^{2}+{v_{n}}^{2}\big{)}\right]\exp\left[-j\frac{2\pi}{\lambda z}\big{(}{x_ {0}}m\Delta u+{y_{0}}n\Delta v\big{)}\right] \tag{9}\]0 The 3D model is partitioned into different parallel layers, based on its depth information. For each layer, amplitude distribution is extracted according to the rendered image. In order to simulate the diffusive effect of the object surface, random phase distribution is added to each layer. The complex distribution of each layer can be calculated by using shifted Fresnel diffractions with preset sampling rules. The complex distribution of the CGH plane can be acquired after adding the contributions from all the layers [27].