## 4 Overcoming the Nyquist Limit for a Lens with Low F-Number
Theoretically, all three lenses would have a common focalization plane at \(f_{0}=80\) mm with 100% efficiency on its corresponding order of diffraction (first-order for the conventional modulo \(2\pi\) lens, second-order for the modulo \(4\pi\) lens, and third-order for the modulo \(6\pi\) lens). The setup sketched in Figure 7 was used in this experiment. It is similar to the Michelson interferometer used for characterization (Figure 2) but with some variations. In this experiment, a green LED source (\(\lambda_{\mathrm{G}}=530\) nm) and a 100 micron pinhole were used to illuminate the setup, and a polarizer served to select the correct polarization plane of the light beam incident onto the SLM display. Only the SLM arm was operative since light coming from the mirror was blocked with an opaque mask. And, finally, a microscope objective was attached to the CCD camera in order to acquire the focalization spots at the focal planes with appropriate magnification. This configuration, already used for device characterization, assures good control of the phase modulation for the on-axis operating lenses of our experiment (in the case of off-axis illumination with incident angle deviations within \(\pm 10\) degrees, no substantial changes in the modulation properties would be expected either [12]). The three aforementioned diffractive lenses were sequentially displayed on the SLM screen, and the obtained experimental results are summarized in Figure 8. Figure 8a depicts the profile of the focalization spots located at \(f_{0}\) = 80 mm for the three analyzed lenses. Profiles are cropped so that only the central area of the output zoomed plane is plotted. Intensity profiles plotted in Figure 8 were normalized to the maximum intensity peak obtained for the modulo \(4\pi\) lens. As expected, the second-order lens produced a more intense focalization spot than the conventional modulo \(2\pi\) lens, since the latter was below the Nyquist limit to reproduce the smallest details at the edges of the lens, whereas the modulo \(4\pi\) lens avoided the aliasing effect. According to the graph, the focal spot of the modulo \(2\pi\) lens was affected by an intensity reduction of about 10%  
Figure 8: (**a**) Coincident focalization planes at \(f_{0}\) = 80 mm for first-, second-, and third-order diffraction lenses. Calibrations curves for the linear \(2\pi\) (Figure 6b), \(4\pi\) (Figure 6d), and \(6\pi\) (Figure 6e) responses were used, respectively. Different diffraction orders, at different locations, for (**b**) the second-order diffraction lens with \(f\) = 160 mm and (**c**) the third-order diffraction lens with \(f\) = 240 mm. Figure 7: Setup for diffractive lens evaluation. Illumination is done with an LED source and a 100 micron pinhole. A polarizer selects the appropriate incidence on the SLM display. A microscope objective is attached to the charged coupled device (CCD) camera to capture the focal planes. Light coming from the mirror was blocked with an opaque mask. with respect to the modulo \(4\pi\) lens. In the same Figure 8a, the intensity profile for the modulo \(6\pi\) lens showed that this diffractive lens obtained the lowest focalization peak. It experienced a reduction around 20% of the maximum intensity. A possible reason for the latter result can be the fact that the LCoS display has a maximum of 256 grey levels to address the diffractive optics. The same number of grey levels is used to reproduce a phase range of \(4\pi\) and \(6\pi\), so that the second-order lens can be implemented more precisely, in particular, at the central part of the lens aperture compared to the coarser reproduction of the third-order lens. Thus, the third-order lens would have larger quantization error than the second-order lens. Figure 8b,c show the profiles of diffraction orders other than the one with a maximum efficiency for both the second-order lens and the third-order lens, respectively. In both cases, there was a small amount of light going to these orders, and both multiorder lenses had similar behaviors.