# Orbital Angular Momentum Generation and Detection by Geometric-Phase Based Metasurfaces
## 3 Metasurfaces for OAM Generation
### Metasurface for OAM-Carrying Vector Beams Generation  
In the previous discussion, the metasurfaces generate circularly polarized waves with OAM. In the following, we will review several pieces of research work dealing with vector fields carrying OAM [93; 94]. The polarization of vector fields is represent by \(\alpha(\phi)=m\phi+\alpha_{0}\), where \(m\) is the polarization order and \(\alpha_{0}\) is the initial polarization. To generate a vector field, a linear polarizer shown in Figure 8 is used. It composes of rectangular apertures and the transmitted wave is linearly polarized along the direction that is vertical to the long axis of the aperture. When the linear polarizers are oriented to different directions according to their locations, the polarizations are varied locally. The aperture with \(\alpha=0\) is modelled by Jones matrix:  
\[\mathbf{J}_{lin\_pol}^{l}=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}. \tag{6}\]  
Then, the Jones matrix for a rotated aperture is written as  
\[\mathbf{J}_{lin\_pol}^{l}(\alpha)=\begin{pmatrix}\cos^{2}(\alpha)&\sin(\alpha )\cos(\alpha)\\ \sin(\alpha)\cos(\alpha)&\sin^{2}(\alpha)\end{pmatrix}. \tag{7}\]  
If the incident wave is circularly polarized, i.e., \(E_{in}^{\pm}=\frac{1}{\sqrt{2}}[1\ \pm i]^{T}\) (plus sign for LHCP and minus sign for RHCP), the transmitted field is expressed by  
\[E_{out}^{+}=\mathbf{J}_{lin\_pol}^{l}E_{in}^{+}=\frac{1}{\sqrt{2}}e^{i\alpha} \begin{pmatrix}\cos(\alpha)\\ \sin(\alpha)\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1\\ i\end{pmatrix}+\frac{1}{2}e^{2i\alpha}\begin{pmatrix}1\\ -i\end{pmatrix}. \tag{8}\]  
Figure 8: Geometry of a one-dimensional inhomogeneous anisotropic metamaterial composed of 10 rectangular holes with the orientation changed stepwisely from 0 to \(\pi/2\). The inset is the geometry of the unit cell that is a square metal slab punched into a rectangular hole. Reproduced with permission from [93], Copyright The Optical Society, 2012.  
\[E_{out}^{-}=\mathbf{J}_{lin\_pol}^{l}E_{in}^{-}=\frac{1}{\sqrt{2}}e^{-ia}\left( \frac{\cos(\alpha)}{\sin(\alpha)}\right)=\frac{1}{2}e^{-2 ia}\left(\frac{1}{i} \right)+\frac{1}{2}\left(\frac{1}{-i}\right). \tag{9}\]  
From (8) and (9), we can notice that the output wave is linearly polarized with a geometric phase of \(e^{\pm ia}\). The linearly polarized wave can be decomposed into two circularly polarized waves. The co-circular polarization does not have the phase term and the cross-circular polarization has the term of \(e^{\pm 2 ia}\). It should be emphasized that unlike the previous derivation that the conversion efficiency can reach 1, the transmitted power after the linear polarizer is only half of the power in the incident wave and each transmitted circularly polarized component takes half of the total transmitted power.  
In Figure 9, two rings of slots are put on a gold film with thickness of 200 nm. The rotation angle of the slots \(\alpha\) satisfies \(\alpha(\phi)=l\phi+\alpha_{0}\). Thus, under the excitation of RCP light, according to Equation (9), the spatially variant factor in \(E_{out}\) is \(e^{-i(l\phi+\alpha_{0})}\). The generated OAM order is \(-l\). The structure can be fabricated using electron-beam lithography [95].