## 4 Overcoming the Nyquist Limit for a Lens with Low F-Number  
Conventional modulo \(2\pi\) diffractive lenses, which entail phase jumps of \(2\pi\) for the design wavelength on either side of each zone boundary (Figure 1b), are designed to work efficiently in the first diffractive order [3; 4]. One of the problems with the phase profile of a diffractive lens is that the phase function changes very rapidly at the outer part of the lens aperture. Thus, implementation is prone to undersampling and aliasing effects when the lens is displayed on a pixelated device (see, for instance, the bottom left corner of Figure 1b). As a rule of thumb, the range of focal lengths must meet the condition \(f_{N}<f<50f_{N}\). The inferior limit is the Nyquist focal length [7; 11], which is given by  
\[f_{N}=\frac{N\Delta^{2}}{\lambda}, \tag{4}\]  
where \(\Delta\) is the pixel pitch, and \(N\times N\) is the size of the pixel array. Multiorder diffractive lenses with phase echelettes higher than modulo \(2\pi\) (see Figure 1b) may overcome the limitations to implement first-order lenses. In that sense, new LCoS displays with large dynamic ranges in phase modulation offer advantages in comparison to older SLM technology. If a first-order lens is displayed on a square area of \(N=1000\) pixels of the Holoeye LCoS SLM (\(\Delta=8\) microns pixel pitch) and is illuminated with \(\lambda=530\) nm, the corresponding Nyquist focal length--as derived from Equation (4)--will result in \(f_{N}=120\) mm (or equivalently, an optical power of 8.3 dipeters). Thus, undersampling problems will arise at the periphery of the aperture if one implements a lens of shorter focal length (or higher optical power) on the display operating at a phase modulation range modulo \(2\pi\). In consequence, aliasing effects and poor performance would result. Let us overcome this situation by designing multiorder diffractive lenses. Let us suppose we want to implement a diffractive lens with \(f_{0}=80\) mm focal length (or, equivalently, 12.5 dipeters). Note that \(f_{0}<f_{N}\) for the Holoeye LCoS-SLM. A possible solution would be to design a second-order lens of \(f=160\) mm that would obtain maximum efficiency on its second diffraction order located at \(f/2=f_{0}=80\) mm. Similarly, a third-order lens of \(f=240\) mm would concentrate the vast majority of the energy on its third diffraction order at \(f/3=f_{0}=80\) mm. Both multiorder diffraction lenses would have larger phase jumps at their edges in comparison to the conventional modulo \(2\pi\) lens, and, in addition, both focal lengths (160 and 240 mm) would be above the Nyquist limit. Under these circumstances, the performances of second- and third-order lenses can be superior in comparison to conventional diffractive lenses. We experimentally compared the performance of both multiorder lenses with their equivalent modulo \(2\pi\) lens. We first considered a conventional lens with \(f_{0}=80\) mm focal length for a design wavelength \(\lambda_{0}=530\) nm. We displayed it on the LCoS display with the linear \(2\pi\) calibration (Figure 6b) obtained for such wavelength. Note that the focal length of this lens is shorter than the focal length corresponding to the Nyquist limit (\(f_{N}=120\) mm), so that undersampling of the lens function will appear at the periphery of the lens. To avoid aliasing, we proposed a multiorder lens with larger focal length (\(f>f_{N}\)). We compared the performance of such a conventional lens with two different multiorder diffraction lenses: a second-order lens with \(f=160\) mm and a third-order lens with \(f=240\) mm computed for the design wavelength \(\lambda_{0}=\lambda_{\mathrm{G}}=530\) nm. The second-order diffraction lens was displayed on the LCoS by using the calibration LUT for a linear \(4\pi\) phase variation (Figure 6d), whereas we implemented the third-order diffraction lens by using the linear \(6\pi\) phase variation curve (Figure 6e) for the same design wavelength \(\lambda_{0}=\lambda_{\mathrm{G}}=530\) nm.