# Computer-Generated Hologram Based on Reference Light Multiplexing for Holographic Display
## 2 Principles and Methods  
The calculation for the proposed method contains eight steps, and the detailed description is as follows:  
1. Acquire target complex amplitude distribution The complex amplitude distribution of each image is initialized as follows: \[E_{i}(\xi,\eta)=A_{i}(\xi,\eta)exp[j\varphi(\xi,\eta)]\] (1) where \(A_{i}(\xi,\eta)\) is the target amplitude of the \(i\)th image, and \(\varphi(\xi,\eta)\) is random phase, which is distributed between 0 and \(2\pi\). 2. Propagate to CGH plane The object light wave propagates toward the CGH plane using angular spectrum diffraction, which divides the optical wavefront into several plane waves with different spatial frequencies and superimposes them after propagating distance \(z\), as shown in Equation (2): \[O_{l}(x,y)=\text{IFFT}\big{\{}\text{FFT}[E_{i}(\xi,\eta)]\cdot T(f_{\xi},f_{ \eta})\big{\}}\] (2) where \(T(f_{\xi},f_{\eta})\) is the angular spectrum transfer function and can be described as: \[T(f_{\xi},f_{\eta})=\exp(jkz\sqrt{1-\lambda^{2}f_{\xi}{}^{2}-\lambda^{2}f_{ \eta}{}^{2}})\] (3) where \(f_{\xi}\) and \(f_{\eta}\) are spatial frequencies. \(\lambda\) is wavelength and \(k=2\pi/\lambda\) is wave number. 3. Define reference light wave Usually, the reference light of the CGH is a parallel light wave. However, in the proposed method, cylindrical and spherical light waves are used as reference light waves to reduce crosstalk among different images and increase information capacity. The reference light wave can be provided by:  
\[R_{c}(x,y)=\exp(jk\sqrt{x^{2}+r^{2}}) \tag{4}\] \[R_{s}(x,y)=\exp(jk\sqrt{x^{2}+y^{2}+r^{2}})\]  
where \(R_{c}(x,y)\) and \(R_{s}(x,y)\) are cylindrical and spherical reference light waves, respectively. \(x\) and \(y\) are horizontal and vertical horizontal coordinates in the CGH plane, respectively. \(r\) is the distance between the light source and CGH plane. In numerical simulations, the light sources of different reference light waves locate at different positions along \(z\)-direction, which creates a more compact system in practical application. Record CGH  
The process of recording is shown in Figure 1. \(E(x,y)\) indicates the complex amplitude distribution of the image. \(H(x,y)\) is the complex amplitude distribution of CGH. \(R(x,y)\) represents the reference light waves. In the CGH plane, different object light waves interfere with respective reference light waves; the complex amplitude distribution of each sub-CGH can be described as:  
\[H_{i}(x,y)=O_{i}(x,y)\cdot R_{i}(x,y) \tag{5}\]  
5. Generate MCGH  
The complex amplitude distribution of MCGH is generated by adding the complex amplitude distribution of all sub-CGHs, and is provided by  
\[H(x,y)=\sum_{i=1}^{N}H_{i}(x,y) \tag{6}\]  
Then the amplitude of MCGH is normalized and the phase is retained, which is provided by:  
\[H^{\prime}(x,y)=\exp\{jangle[H(x,y)]\} \tag{7}\]  
6. Reconstruct image  
The process of reconstruction is shown in Figure 2. \(H^{\prime}(x,y)\) is the complex amplitude distribution of MCGH. \(R^{*}(x,y)\) represents the conjugate light waves of reference light waves. \[O_{l}(x,y)=\text{IFFT}\big{\{}\text{FFT}[E_{i}(\xi,\eta)]\cdot T(f_{\xi},f_{ \eta})\big{\}}\]0 Schematic of the recording process. In the reconstruction process, the conjugate light waves of reference light waves are used as reconstructed light waves. MCGH under the illumination of reconstructed light waves propagates backward to the object plane using inverse angular spectrum diffraction to obtain the corresponding reconstructed images as shown in Equation (8):  
\[O_{l}(x,y)=\text{IFFT}\big{\{}\text{FFT}[E_{i}(\xi,\eta)]\cdot T(f_{\xi},f_{ \eta})\big{\}}\]1  
7. Decide to stop the iteration process  
The iteration algorithm uses forward and backward propagation of the light field between the object plane and CGH plane. The iteration operation aims to produce an appropriate MCGH and obtain the target images. Therefore, it is essential to evaluate the quality of reconstructed images and determine whether to stop the iteration process.