# Accurate Image Locating by Hologram Multiplexing in Off-Axis Digital Holography Display
## 2 Principle
The former three terms on the right side of Equation (6) are all centered at the origin of frequency plane in Fourier domain, but the centers of the last two terms are shifted to the coordinates of \((\pm u_{p},\pm v_{p})\), especially the term \(A_{O1}Ar\ \delta(u\pm u_{p},\ v\pm v_{p})\) are two delta functions with obvious extreme values at points \((\pm u_{p},\pm v_{p})\), which makes them easy to detect by searching for the maximum spectrum values. After the points on the frequency plane are fixed, we can calculate the off-axis angle by their coordinates [16]:  
\[\sin\theta_{x}=\lambda\frac{u_{p}}{M_{x}},\ \sin\theta_{y}=\lambda\frac{v_{p}}{M _{y}}, \tag{7}\]  
where \(M_{x}\), \(d_{x}\) and \(N\), \(d_{y}\) are the pixel numbers and pixel size along horizontal and vertical directions correspondently. Since \(A_{r}\) is a real constant, after we fixed the TARW, the plane wave of the reference can be expressed as  
\[R(x,y)=A_{r}\exp\left[-i\frac{2\pi}{\lambda}(x\sin\theta_{x}+y\sin\theta_{y}) \right]. \tag{8}\]  
When the crucial TARW is achieved, the original object wave can also be reconstructed exactly by the inverse diffraction automatically by [18]  
\[O(x,y)=FT^{-1}\{FT\ [R(x,y)\cdot\{x,y\}]\cdot w(u,v)\cdot H(u,v)\}, \tag{9}\]  
where \(FT^{-1}\{\}\) represents the operator of inverse Fourier transform, \(w(u,v)\) is a window functions to choose only one spectrum area on Fourier spectrum domain in the last term of Equation (6), and \(H(u,v)\) is the transfer function of the inverse diffraction system. The original object image can be reconstructed though an inverse Fresnel diffraction operation on \(O(x,y)\). The merits of the proposed method can be concluded as the following: First of all, the recording configuration is comparatively simple than the setup used before [16] and only one CCD is employed during the recording process. Secondly, the off-axis parameter and the interference information are carried by only one single hologram which decreases the computer storage load and of course is easy to transfer. Thirdly, the TARW information  
Figure 1: Recording configuration for the OADHM principle. is concealed in the hologram so that the data are much safer than keeping it separately since this storage manner can avoid data loss and theft. Lastly, the multiplexing method proposed here can make the whole process of hologram recording and display automated and intelligent. The whole process can complete without human intervention. To illustrate the method of OADHM more clearly, the whole process for the holography display can be summarized as the steps below:  
Step 1: Perform FT operations on the hologram \(I\) in Equation (5) and detect the spatial frequency points corresponding to the fourth term on the right side of Equation (6) by searching for the brightest points in the Fourier frequency plane (exclude the direct current point at the coordinate origin). Step 2: Locate and store the coordinates of the points (\(-u_{p}\), \(-v_{p}\)) or (\(u_{p}\), \(v_{p}\)). Step 3: Calculate the value of the TARWs by Equation (7) and the coordinates (\(u_{p}\), \(v_{p}\)) stored in step 2. Step 4: Construct the reference wave \(R\) by Equation (8) and the TARWs achieved in step 3 and then do FT operation on \(RI\). Step 5: Set up the window function in Equation (9) to select the objective spectra around the origin and filter out the spectrum at the origin point. Step 6: Recover the object wave by Equation (9) and the frequency window function in step 5. Step 7: Recover the original object image by performing the inverse Fresnel transform at displace z.