## 2 Geometry of the System  
Figure 1 shows the configuration of our optical system. A programmable SLM is placed a distance \(p=100\) cm from an input object. The object (a slide transparency with the word \"PROFESSOR\") is illuminated by a collimated beam from a helium-neon laser, with the \(\lambda=633\) nm wavelength. A second SLM is placed a distance \(d=140\) cm from the first. Finally, the camera is located a distance also approximately \(q=100\) cm from the second SLM. The output patterns were recorded with a CCD camera, model XC-37 (Sony, USA), having 491 \(\times\) 384 pixels on a 8.86 \(\times\) 6.60 mm\({}^{2}\) sensor. Note that zoom lenses used in photography are designed to image distant objects, so \(p\) is a very large distance. Tunable zoom lenses have also been demonstrated for holographic projection systems [17; 18]. There, the \(p\) is a short distance, while \(q\) should be larger. Here we selected an intermediate approach where the object and image are located at finite close distances. Using a standard ray matrix approach [19], this system can be represented by the product of three translation matrices and the two lens matrices as in Equation (1). The product of these matrices is given by the final ABCD matrix:  
\[\left| \begin{array}{c}r_{2}\\ r_{2^{\prime}}\end{array} \right|=\left| \begin{array}{cc}1&q\\ 0&1\end{array} \right|\times\left| \begin{array}{cc}1&0\\ -1/f_{2}&1\end{array} \right|\times\left| \begin{array}{cc}1&d\\ 0&1\end{array} \right|\times\left| \begin{array}{cc}1&0\\ -1/f_{1}&1\end{array} \right|\times\left| \begin{array}{cc}1&p\\ 0&1\end{array} \right|\times\left| \begin{array}{cc}r_{1}\\ r_{1^{\prime}}\end{array} \right|=\left| \begin{array}{cc}A&B\\ C&D\end{array} \right|\times\left| \begin{array}{c}r_{1}\\ r_{1^{\prime}}\end{array} \right| \tag{1}\]  
Figure 1: Schematic of the zoom lens optical system. \(r_{i}\) and \(r_{i}^{\prime}\) (\(i=1,2\)) denote the ray height and angle coordinates. The calculation of the ABCD matrix in Equation (1) leads to the following result:  
\[\left| \begin{array}{c}r_{2}\\ r_{2}^{\prime}\end{array} \right|=\left| \begin{array}{cc}1-\frac{d}{f_{1}}- \frac{q}{F}&p\Big{(}1-\frac{d}{f_{1}}-\frac{q}{F}\Big{)}+d+q\Big{(}1-\frac{d}{ f_{2}}\Big{)}\\ -\frac{1}{F}&-\frac{p}{F}+1-\frac{d}{f_{2}}\end{array} \right|\times\left| \begin{array}{c}r_{1}\\ r_{1}^{\prime}\end{array} \right| \tag{2}\]  
where \(F\) denotes the focal length of the composed system, given by:  
\[\frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}}-\frac{d}{f_{1}f_{2}} \tag{3}\]  
The imaging requirement is met by forcing the matrix element \(B=0\). This condition leads to the following relation  
\[f_{2}=\frac{f_{1}q(d+p)-pdq}{f_{1}(p+d+q)-p(d+q)}=\frac{f_{1}q(D-q)-pdq}{f_{1} D-p(D-p)} \tag{4}\]  
where \(D=p+d+q\) is the total distance of the system, which we want to keep constant, so the system has no moving elements. The magnification is given by the matrix element \(A\) in Equation (2), i.e.,  
\[m=1-\frac{d+q}{f_{1}}-\frac{q}{f_{2}}-\frac{qd}{f_{1}f_{2}} \tag{5}\]  
The system is limited by the geometry of the SLMs. In this work, the first lens is encoded onto a CRL model XGA-3 transmissive twisted nematic SLM operating in a phase-only mode by means of the rotated eigen-polarization state technique [20]. This device has \(1024\times 768\) pixels with a pixel spacing of 18 microns. The second lens is formed on a Hamamatsu model X10468-01 reflective parallel-aligned SLM, also operating in a phase-only mode. This device has 800 \(\times\) 600 pixels with a pixel spacing of 20 microns. Because this second SLM is a reflective liquid-crystal on silicon (LCoS) display, we include a beam splitter in the system to allow a reflective geometry in the second part of the system in Figure 1. Both displays use eight bits for codification of phase levels, i.e., they have \(N=256\) quantization levels.