## 2 Geometry of the System
The first order diffraction efficiency caused by the quantization of phase levels is given by \(\eta_{Q}=[\sin(\pi/N)/(\pi/N)]^{2}\) which for \(N=256\) levels results in a value of exceeding 99% [21]. Thus, quantization is not an issue for this system. Both SLMs have been tested and produce a phase-only response versus addressed gray level, with no cross-modulation effects. In addition, the phase modulation versus addressed gray level is linearized. On the contrary, the limited spatial resolution imposes drastic losses and limits. Although both devices have similar pixel pitch, the LCoS technology presents a much higher fill factor (FF). The fill factor is defined as FF = (\(w/\Delta\))2, where \(w\) stands for the pixel width and \(\Delta\) stands for the pixel spacing [22]. The CRL display has a fill factor of approximately only FF\({}_{\text{CRL}}\approx 35\%\). In contrast, the Hamamatsu LCoS display presents a much higher value of approximately FF\({}_{\text{HAM}}\approx 98\%\). This difference makes a major impact between the two devices. First, the fraction of the incident intensity that is transmitted is given directly by the FF. Secondly, the SLM rasterized structure created a 2D grid of diffraction orders. The central principal zero order has the highest intensity, which is roughly proportional to FF\({}^{2}\)[22]. Since we are using the zero diffracted order in the zoom lens system, the total intensity transmission is reduced by this effect to a factor of \(\eta_{\text{FF}}=(\text{FF}_{\text{CRL}}\times\text{FF}_{\text{HAM}})^{2} \approx 12\%\). But, even more relevant for the application to the zoom lens system is the limitation imposed by the limited spatial resolution on the minimum focal length that the SLMs can encode. We have previously discussed [13] the limits imposed by the spatial resolution of the SLM when we implement a lens function having a quadratic phase dependence as:  
\[Z^{*}(r,f)=\exp\left(-i\frac{\pi r^{2}}{\lambda f}\right) \tag{6}\]where \(f\) denotes the focal length of the lens, \(\lambda\) is the wavelength and \(r\) denotes the radial coordinate at the SLM plane. Next, we summarize the main effect, for its implication in the zoom lens system. We assume that the SLM has an array of \(N\times N\) pixels where each pixel has a finite size given by \(\Delta\). As the focal length \(f\) of the encoded lens decreases, the spacing between the phase at the edges of the device can become smaller than the Nyquist limit of 2 pixels causing aliasing effects. These considerations lead to the Nyquist focal length (\(f_{N}\)) for a lens encoded onto the SLM as given by [12]:  
\[f_{N}=\frac{N\Delta^{2}}{\lambda} \tag{7}\]  
Lens functions as in Equation (6) with focal lengths shorter (in absolute value) than the Nyquist focal length \(|f|\leq f_{N}\) are affected by aliasing and will not be well reproduced. Figure 2 shows three diffractive lenses where the focal lengths are greater and smaller than the Nyquist limit. If the focal length is clearly greater than \(f_{N}\), the lens function is encoded perfectly. However, for shorter focal lengths the effective central area of the lens decreases and the lens begins to act as a low pass spatial filter. In addition, the strengths of the replica focal lengths caused by the aliasing increase. For our SLMs, the effective Nyquist focal lengths are about 38 cm, considering the smaller dimensions. These considerations have important consequences for our optical system because the SLMs have difficulty encoding focal lengths smaller than this value. This limit requires the large values for the distances in our experimental system in Figure 1. These large dimensions of the zoom lens system make it impractical for real applications.