# Accurate Image Locating by Hologram Multiplexing in Off-Axis Digital Holography Display
## 3 Computer Simulations  
Several sets of numerical simulations are carried out by the software Matlab on a personal computer to investigate the applicability of the proposed method. The recording setup is illustrated in Figure 2. The three illuminating beams are all plane wave filtered and expanded from a laser source with 532 nm wavelength. In the original object plane Po, the picture of a resolution target in Figure 3a is used as the intensity of the object wave. To simulate a transparent sample, the dark area of the resolution target figure has a transmittance of 20%. In these numerical simulations, different elliptical equations controlled by two parameters \(R_{1}\) and \(R_{2}\) were employed as the phase of the transmitting objects (Figure 3b). We set the phase distribution on the original object plane Po, as  
\[\varphi(x,y)=\frac{2\pi}{\lambda}\bigg{(}\frac{x^{2}}{R_{1}}+\frac{y^{2}}{R_{ 2}}\bigg{)}. \tag{10}\]  
The object wave \(O(x,y)\) in the recording plane \(\mathrm{P_{H}}\) is generated by the two-dimensional Fresnel diffraction on the complex amplitude in plane Po. The plane wave \(O_{1}(x,y)\) is an on-axis plane wave with the amplitude of \(A_{O1}\) and the reference wave \(R\) is a tilt plane wave along both the \(x\)-axis and the \(y\)-axis with reference tilt angle of \(\theta_{x}\) and \(\theta_{y}\) correspondingly. Mostly, the intensity of \(A_{O1}\) is set smaller than that of the reference wave of \(A_{r}\). The hologram is generated by three wave interference in computer. In these simulations, we set two TARW values as \(\theta_{x}\) = 0.023 rad and \(\theta_{y}\) = 0.023 rad, respectively, and we give only a few results with \(R_{1}\) = 1.5 m and \(R_{2}\) = 2.5 m as an example. It should be noted that the object in plane \(\mathrm{P_{O}}\) can be a resolution target with a PCM membrane attached or a spatial light modulator with the intensity and phase signals modulated by computers so that the object wave in plane \(\mathrm{P_{O}}\) has a complex value. In the simulation setup in Figure 2, there are 1690 \(\times\) 1690 pixels with 5 \(\upmu\)m \(\times\) 5 \(\upmu\)m size on the CCD working chip and the recording distance is set as z = 79.4173 mm (all the parameters used satisfy the requirement of the sampling theory to avoid aliasing problem [4]). The holograms are computer generated by interference of three waves as Equation (4). Holograms from the three waves interference with \(A_{O1}\) = 0.1 \(A_{r}\) and \(A_{O1}\) = 0.5 \(A_{r}\) are shown in Figure 3c,d, respectively, where we can see that there are not only the hologram but also the fringes of the two plane wave interference. The straight fine stripes in Figure 3d are more obvious than that in Figure 3c because of the strong interference between the two plane waves. In fact, there are two sets of holograms in either Figure 3c or Figure 3d--on-axis hologram and off-axis hologram. The off-axis hologram contributes the information to the object wave; however, the on-axis part contributes only to zero order term and can be removed by spatial frequency filtering. Fast Fourier Transform (FFT) operations on the interferogram \(I\) in Figure 3d are performed in computer and the intensity distributions of the Fourier domain spectra from this hologram are shown in Figure 4a. The values in the areas of zero frequency have been decreased for display purpose and the two points corresponding to the fourth term of Equation (6) are zoomed into the two circles, respectively, for the convenience of observation. In Figure 4a, because the two points collect all the energy of the periodic distribution in Figure 3d, they are so bright that other spectra cannot appear correctly. To display other spectra, the two points are depressed in Figure 4b. The spectra from the on-axis part of the hologram are weaker than that from the off-axis hologram because \(A_{O1}\) is relatively small compared with \(A_{r}\).