## 3 Liquid Crystal on Silicon (LCoS) Calibration Procedure and Phase Modulation Responses
From this measurement, a new LUT was established (\"modified gamma\", line with circles in Figure 3b) in order to achieve a linear phase modulation (\"linear ph. mod.\", line with circles in Figure 3c). Figure 3 depicts the modified LUT and the corresponding linear performance for up to 8\(\pi\) phase modulation range for an illuminating wavelength of \(\lambda_{\text{G}}=514\) nm. In this condition, we measured the response of the display for another wavelength in the green region of the spectrum, \(\lambda_{\text{G}}=520\) nm (Figure 4). Note that the SLM phase modulation response with the new calibration LUT (or modified gamma curve) does not vary remarkably when a different, but close, illuminating wavelength is used. Thus, we assume that the phase modulation response is resistant to slight variations in the illuminating wavelength with respect to the one used to determine the new gamma curve. Figure 3: (**a**) Interferogram obtained when two different gray levels are sent to the top and bottom halves of the SLM display (Figure 2b). The phase difference is measured from the detected fringe shift. (**b**) Look-up tables (LUTs), also named gamma curves, to control the response of the SLM display and (**c**) measured phase modulation depth. Line curves with cross markers from (**b,c**) correspond to the initial measurement when the gamma curve provided by the manufacturer is used (default gamma). The corresponding phase modulation slightly exceeds 8\(\pi\) radians but with a nonlinear response. Line curves with circles from (**b,c**) correspond to the modified gamma curve that achieves a linear phase modulation of the display. The LCoS display was illuminated with \(\lambda_{\text{G}}=514\) nm. Next, we measured the light reflected by the SLM display when a uniform image of a varying grey level was depicted as a full screen on the display. Light coming from the flat mirror was blocked at this time. The CCD camera was replaced by a photodetector so that the detector cell received the light reflected from the SLM display and integrated it. Figure 5a sketches the setup, and Figure 5b shows the modulation detected on the reflected light for the three illuminating wavelengths used sequentially in this experiment. This effect is due to the Fabry-Perot interference on the round-trip propagation inside the liquid crystal layer as already described in Refs. [6; 14]. The number of oscillations in each plot accounts for the corresponding phase modulation range that approximates \(6\pi\), \(8\pi\), and \(9\pi\) for \(\lambda_{\text{R}}=647\) nm, \(\lambda_{\text{G}}=514\) nm, and \(\lambda_{\text{B}}=476\) nm, respectively. As shown in Figure 4b, the experimental results obtained with our LCoS-SLM are in very good agreement with those reported by [PERSON] et al. for another LCoS-SLM device (Hamamatsu X10468-08) [6]. We were able to obtain a variety of linear phase modulation ranges, from \(\pi\) up to \(8\pi\), by properly setting the configuration parameters of the SLM control driver. In particular, optimized calibration curves were obtained for \(\pi\), \(2\pi\), \(3\pi\), \(4\pi\), \(6\pi\), and \(8\pi\) phase modulation ranges. An analogous calibration procedure was followed in all cases. For those calibration tasks, the illumina wavelength was set  
Figure 4: Phase modulation of the LCoS for two green wavelengths \(\lambda_{0}=514\) nm (used in the characterization) and \(\lambda=520\) nm when the SLM display operates with the LUT that linearizes the \(8\pi\) phase modulation range for \(\lambda_{0}=514\) nm (Figure 3b,c). Figure 5: (**a**) Scheme of the experimental setup. (**b**) Reflectance modulation versus varying grey level addressed to the SLM. to 530 nm, which coincided with the maximum intensity of an available green LED source that was going to be used in subsequent experiments (display of Fresnel diffraction lenses). Once the calibration LUT was established, measurements for 647 and 476 nm, as representative of the available red and blue LED sources, were also taken.