# Accurate Image Locating by Hologram Multiplexing in Off-Axis Digital Holography Display
## 3 Computer Simulations
The spectra from the object wave do not locate at the center area, but a place deviates symmetrically to the spectra area of its conjugate wave about the original point of the frequency plane because of the off-axis recording. Figure 3: Object wave and holograms for simulation. (**a**) intensity; (**b**) phase of the object in \(\mathrm{P_{O}}\) and holograms in \(\mathrm{P_{H}}\) with (**c**) \(A_{O1}\) = 0.1 \(A_{r}\); (**d**) \(A_{O1}\) = 0.5 \(A_{r}\). Figure 2: Schematic diagrams for simulation. The extreme values of the two point \(\delta(u\pm u_{p}\). \(v\pm v_{p})\) make it possible for us to find the exact coordinate of the spectra center by searching for the maximum absolute value of the spectrum in Fourier domain plane. Searching for the maximum value of the spectrum intensity in Figure 4a, we gained the coordinates \(u_{p}\) = 362.5 and \(v_{p}\) = 362.5. Substituting the coordinates \(u_{p}\) = 362.5, \(v_{p}\) = 362.5, \(\lambda\) = 0.532 \(\upmu\)m, the pixel size \(d_{x}\) = \(d_{y}\) = 5 \(\upmu\)m, and the total pixel number \(M\) = \(N\) = 1690 into Equation (7), the TARWs of the reference wave in the \(x\) and \(y\) directions can be calculated, and the results are both approximately 1.3 degrees. The reference wave can be recovered by Equation (8) and the calculation results of sin\(\theta_{x}\) and sin\(\theta_{y}\). Another FT operation is carried out on the production \(RI\) and the intensity distribution of the spectra is shown in Figure 5a. The whole spectra group is shifted accordingly to the TARW because of the linear phase distribution of the recovered reference wave \(R\). One lobe of the frequency spectra (corresponding to the object wave) is shifted to the center of the spatial frequency plane, which makes it possible to select this lobe and filter out other spectra. The intensity distribution of the spectra of the selected lobe is shown in Figure 5b. Wave front \(O\) is retrieved by using Equation (9) and the reference \(R\) with the tilt angles of about \(\theta_{x}\) = 0.023 rad and \(\theta_{y}\) = 0.023 rad. Then, the object image can be displayed after the inverse Fresnel diffraction is performed. It should be noted that, in the spectrum lobe kept at the center of the frequency plane, there still exists one delta function of \(\delta(u,v)\) in Figure 5b. These spectra come from the fourth term of Equation (6) and always add unwanted effects  
Figure 4: The intensity distributions of the spectra from hologram \(I\) for simulation. (**a**) two points \(\delta(u\pm u_{p},v\pm v_{p})\); (**b**) spectra areas. Figure 5: The intensity distribution of the spectra from the Fourier transform of \(RI\) for simulation. (**a**) before and (**b**) after spectra filtering. on the hologram display. The larger the \(A_{O1}\), there is the more serious bad effect on the image display. To investigate the bad effect of them, three cases with \(A_{O1}=0.1\)\(A_{r}\), \(A_{O1}=0.3\)\(A_{r}\), and \(A_{O1}=0.5\)\(A_{r}\) are introduced in our investigation. The display results are described in Figure 6a-c, respectively. Naturally, when \(A_{O1}=0.1\)\(A_{r}\) is used in Figure 6a, the impact of the \(\delta(u,v)\) spectra is little and the transparent area in resolution target image is acceptable. However, for the cases of \(A_{O1}=0.3\)\(A_{r}\), \(A_{O1}=0.5\)\(A_{r}\), and other cases with even larger \(A_{O1}\), the bad impacts are serious and the displayed image in Figure 6b,c are unacceptable. To eliminate the negative impact, these spectra of \(\delta(u,v)\) in Figure 5b are filtered out by improving the window function. Another problem should be mentioned too. With the enlarging of \(A_{O1}\), the on-axis hologram takes a large proportion of energy, which incurs the disturbance on the hologram display accordingly. The test results from four different cases with \(A_{O1}=0.5\)\(A_{r}\), \(A_{O1}=1\)\(A_{r}\), \(A_{O1}=2\)\(A_{r}\), and \(A_{O1}=3\)\(A_{r}\) are listed in the figures from Figure 7a-d, respectively.