# Accurate Image Locating by Hologram Multiplexing in Off-Axis Digital Holography Display
## 2 Principle  
Figure 1 shows the geometry setup for the OADHM design, where BS, M, MO, PH, and L note beam splitter, mirror, microscopic object lens, pinhole, and collimator lens, respectively. To write the TARW information in the hologram, a plane wave is added in the traditional rectangular recording configuration. The new beam denoted as \(O_{1}(x,y)\) is reflected by BS\({}_{2}\), focused, and filtered by MO\({}_{2}\) and PH\({}_{2}\), collimated by L\({}_{2}\), reflected further by M\({}_{2}\) and BS\({}_{3}\) consequently, and interfered with the object wave \(O(x,y)\) and the reference wave \(R(x,y)\) on a CCD chip at last. Both beam \(O_{1}(x,y)\) and the illuminating beam are set as on axis, but the tilt reference beam has a tilt angle needed to separate the original image for display. Actually, \(O_{1}(x,y)\) can be seen as a real constant \(A_{O1}\) because it is an on-axis plane wave. Based on the description above, the complex amplitudes of object wave \(O(x,y)\) and the reference wave \(R(x,y)\) in the working plane of the recording CCD can be written as  
\[O(x,y)=A_{o}(x,y)\exp[-i\varphi_{o}(x,y)], \tag{1}\]  
\[R(x,y)=A_{r}\exp[-i\varphi_{r}(x,y)], \tag{2}\]  
where we also have assumed \(A_{r}\) a constant for brevity. However, in Equation (2), the phase of the reference wave \(\varphi_{r}(x,y)\) should not be a constant but a linear distribution  
\[\varphi_{r}(x,y)=\frac{2\pi}{\lambda}(x\sin\theta_{x}+y\sin\theta_{y}), \tag{3}\]  
where \(\lambda\) is the wavelength of the laser, and \(\theta_{x}\) and \(\theta_{y}\) are the TARWs along the \(x\) and \(y\) directions. In fact, the multiplexing plane wave \(O_{1}\) is a constant and can be represented by a small number \"\(A_{O1}\)\" because it is a normal incident on-axis plane wave. From Equations (1) and (2), the interferogram intensities for the three waves interference can be expressed as  
\[I=[A_{O1}+O(x,y)+R(x,y)]\cdot[A_{O1}+O(x,y)+R(x,y)]^{*}, \tag{4}\]  
where symbol \"\(\star\)\" is the operator of conjugation. From Equation (4), we have  
\[I=A_{O1}^{2}+|R|^{2}+|O|^{2}+A_{O1}O^{*}+A_{O1}O+A_{O1}R^{*}+A_{O1}R+RO^{*}+ R^{*}O \tag{5}\]  
In Equation (5), '\(\mid\)' represents the absolute value, and the nine terms on the right side of the equation are the interference results of the three waves in Figure 1. Terms \(A_{O1}^{2}\), \(|R|^{2}\), and \(|O|^{2}\) contribute to the zero order term during the hologram reconstruction. The 4 th and 5 th terms are interfering results between \(O_{1}\) and \(O\), which belongs to on-axis interference and can be seen as the zero order terms also when holography display is carried out. The sixth and seventh terms are the interference between the plane waves \(O_{1}\) and \(R\), and the intensity distribution for this interference is the straight stripes modulated by the cosine function, which can be used to record the TARW information. The 8 th and 9 th terms are the interference terms between reference wave \(R\) and object wave \(O\), which contribute the conjugate image and the original image. To find the TARW information of the hologram, Fourier Transform (FT) can be done on Equation (6), and then we have [8]  
\[ \begin{array}{c}FT[I]=\Big{(}A_{O1}^{2}+|R|^{2}\Big{)}\delta(u,v)+F_{o}(u,v) \oplus F_{o}(u,v)+A_{O1}[F_{o}(u,v)+F_{o}^{*}(u,v)]\\ \hskip 14.226378 pt+A_{O1}A_{r}\delta\big{(}u\pm u_{p},v\pm v_{p}\big{)}+A_{O1}A _{r}F_{o}\big{(}u\pm u_{p},v\pm v_{p}\big{)}\end{array}  \tag{6}\]  
where \(\delta(u,v)\) is a delta function on the original point of the spatial frequency plane, \(F_{o}(u,v)\) is the spectrum distribution of the object wave recorded, and \(F_{o}(u,v)\oplus F_{o}(u,v)\) is the auto-correlation of \(F_{o}(u,v)\).