# Orbital Angular Momentum Generation and Detection by Geometric-Phase Based Metasurfaces
## 4 Holographic Metasurfaces for OAM Detection  
We adopt the holographic concept for the detection of multiple OAM-beam using a single metasurface [89]. The detection process is summarized in Figure 12. The metasurface converts the incident wave to multiple waves, only one of which is Gaussian. The radiation direction of the Gaussian wave is distinguishable according to the order of incident OAM. Consequently, by locating the Gaussian wave, the incident OAM can be conveniently determined. The transmission function of the desired metasurface is given by (5). Then, the far-field response of the metasurface illuminated by an incident wave carrying OAM of order \(l_{0}\) is calculated by doing the Fourier transform  
\[E=F\{E_{in}\cdot t\}=\sum_{m}A_{m}F\{E_{\text{OAM}(l_{m}+l_{0})}(k_{xm},k_{ym})\}. \tag{10}\]  
Figure 10: (**a**) The metasurface is constructed by drilling a silver film with multiple periods of annular rings, whose radius is defined as \(R_{n}=R_{1}+(n-1)P\), where \(n\) and \(P\) denote the number and the period of the apertures. The annular apertures can be taken as two-dimensional extensions of a set of nanoslits with spatially varying orientation. Reproduced with permission from [87], Copyright American Chemical Society, 2016; (**b**) OAM generators based on catenary arrays. The topological charges from up to bottom are \(-3\), \(-6\), and \(12\) (\(s=1\)), respectively. The first column represents the scanning electron microscopy (SEM) images of the fabricated samples. The second column shows the spiral phase profiles. Reproduced with permission from [88], Copyright The American Association for the Advancement of Science, 2015; (**c**) Top, geometry of the subwavelength gratings for four topological charges. Bottom, image of a typical grating profile taken with a scanning-electron microscope. Reproduced with permission from [96], Copyright Optical Society of America, 2002. Multiple waves with the OAM of order \(l_{m}+l_{0}\) at the k-space position \((k_{xm},k_{ym})\) are observed. When \(l_{M}+l_{0}=0\), the beam is Gaussian and its beam axis is at \((k_{xM},k_{yM})\). It is known that OAM wave has a singularity at its beam axis. Therefore, by examining the field intensity at the positions of \((k_{xm},k_{ym})\), we can identify the gaussian beam, i.e., identify \(M\). Then the incident OAM order \(l_{0}\) can be determined. As a proof of concept, a five-beam case with \(l_{m}=2,1,0,-1,-2\) at the directions of \(\theta=40^{\circ}\) and \(\phi=90^{\circ},18^{\circ},306^{\circ},234^{\circ},162^{\circ}\) is demonstrated. The calculated transmittance \(t(r,\phi)\) is implemented using the unit cell as shown in Figure 4a. Full-wave simulated radiation patterns are shown in Figure 13. It can be seen that the maximum radiation direction (axis of the gaussian beam) when \(l_{0}=-2,-1,0,1,2\) is at \(\theta=40^{\circ}\) and \(\phi=90^{\circ},18^{\circ},306^{\circ},234^{\circ},162^{\circ}\), respectively, which is as expected. Figure 11: (**a**) Schematic of the nanofins azimuthal distribution in the inner part of metasurface device with interleaved patterns that generate collinear beams having topological charges \(|l|=5\) and \(|l|=10\). The device has a 500 \(\upmu\)m diameter and contains more than 700 interleaved radial rows of nanofins. Reproduced with permission from [90], Copyright Optical Society of America, 2017; (**b**) Schematic of shared-aperture concepts using interleaved 1D phased arrays and the schematic far-field intensity distribution of wavefronts with positive (red) and negative (blue) helicities. Reproduced with permission from [97], Copyright The American Association for the Advancement of Science, 2016. Figure 12: Schematic representation of multiple OAM-beam detection by a single metasurface. Reproduced with permission from [89], Copyright IEEE, 2017. A modified transmission function is proposed to lower the side-lobe level:  
\[t_{mod}(r,\phi)=\sum_{m}e^{i(l_{m}\phi+k_{zm}x+k_{ym}y+\alpha_{m})}. \tag{11}\]  
We see an additional phase term \(e^{i\alpha_{m}}\) in (11).