## 2 Multiorder Diffractive Lenses: Theoretical Background  
Multiorder diffractive lenses [1; 2; 3; 4] are commonly built so that the optical path difference (OPD) at the boundaries of adjacent zones is a multiple of \(\lambda_{0}\), (or equivalently, a phase jump multiple of \(2\pi\) rad).  
Thus, \(OPD=(f_{0}+jq\lambda_{0})-[f_{0}+(j-1)q\lambda_{0}]=q\lambda_{0}\) (Figure 1), where \(\lambda_{0}\) is the design wavelength, \(f_{0}\) is the focal length when the illumination wavelength is \(\lambda=\lambda_{0}\), and \(q\) is an integer that multiplies \(2\pi\) at the phase jump. A conventional diffractive lens with facets reaching \(2\pi\) phase variation (also called a modulo \(2\pi\) lens) has \(q=1\).  
The condition for obtaining a constructive interference in the focal plane \(F_{0}\) of the lens determines the radial position of the \(j\)th diffractive zone, which is computed through the Pythagorean theorem  
\[r_{j}^{2}=(f_{0}+jq\lambda_{0})^{2}-f_{0}^{2}\approx 2 jq\lambda_{0}f_{0}. \tag{1}\]  
The maximum variation of the phase profile in each zone is given by \(q2\pi\). The lens has an infinite number of focal lengths according to  
\[f(\lambda)=\frac{q\lambda_{0}f_{0}}{m\lambda}. \tag{2}\]  
The interested reader is referred to [2] for a detailed explanation of how Equation (2) is obtained. Note that Equation (2) justifies the highly dispersive nature of diffractive lenses.  
For a conventional modulo \(2\pi\) diffractive lens (\(q=1\)) illuminated with the design wavelength (\(\lambda=\lambda_{0}\)), an infinite number of focal planes \(f=f_{0}/m\) exist for the different diffraction orders [8]. If one considers a multiorder diffractive lens (\(q>1\)), the optical path length \(q\lambda_{0}\) for the design wavelength may coincide with other wavelengths for which the phase change between adjacent steps of the lens is a multiple of \(2\pi\). These wavelengths, which are called the resonant wavelengths [1], fulfill \(\lambda=q\lambda_{0}/m\), and have a common focal plane \(f(\lambda)=f_{0}\).  
Not only does the location of the focal plane coincide for the resonant wavelengths but also the diffraction efficiency of the respective diffraction orders, which is given by [4]  
\[\eta_{m,q}=\text{sinc}^{2}(\alpha q-m). \tag{3}\]  
Coefficient \(\alpha\) is a wavelength detuning factor that can be approximated by \(\alpha=\lambda_{0}/\lambda\) when low-dispersive materials, or even no dispersion, are considered. The maximum diffraction efficiency \(\eta_{m,q}=1\) is achieved for resonant wavelengths at their common focal plane of a multiorder diffractive lens. For first-order diffractive lenses with \(q=1\), all the energy concentrates on the first diffraction order (\(m=1\)) with a quick but smooth decrease of the efficiency for wavelengths that differ from the design wavelength. As the phase profile increases its modulation range, that is, when \(q>1\)  
Figure 1: (**a**) Zone construction of a multiorder (\(q\)th-order) diffractive lens whose optical path difference (OPD) between consecutive zones is \(q\) integer times the design wavelength \(\lambda_{0}\). (**b**) Profiles for a diffractive lens with \(2\pi\) rad (\(q=1\)) and \(6\pi\) rad (\(q=3\)) phase modulation depths.  
resonant wavelengths get closer in the spectrum, and diffractive lenses of higher orders can be obtained; however, their diffracted intensity drops even faster with wavelength [4].