# Orbital Angular Momentum Generation and Detection by Geometric-Phase Based Metasurfaces
## 2 Geometric Phase  
When a light changes its initial polarization state to a final polarization state along different paths on a Poincare sphere, the final polarization states will have difference phases, known as geometric phase [77]. In the following, we derive the geometric phase when the SAM shifts from \(\pm\hbar\) to \(\mp\hbar\) through an anisotropic scatterer with different orientations.  
The behavior of a scatterer can be modelled by Jones matrix \(\mathbf{J}\). It connects the polarization state of the scattered wave with that of an incident wave:  
\[\begin{pmatrix}E_{x}^{s}\\ E_{y}^{s}\end{pmatrix}=\begin{pmatrix}J_{xx}&J_{xy}\\ J_{yx}&J_{yy}\end{pmatrix}\begin{pmatrix}E_{x}^{i}\\ E_{y}^{i}\end{pmatrix}=\mathbf{J}^{i}\begin{pmatrix}E_{x}^{i}\\ E_{y}^{i}\end{pmatrix}\,, \tag{1}\]  
where \(E_{x}^{i}\) and \(E_{y}^{i}\) are the \(x\) and \(y\) components of the incident electric field. \(E_{x}^{s}\) and \(E_{y}^{s}\) are the corresponding components of the scattered electric field.  
Jones matrix under the circular basis can be obtained by coordinate transformation,  
\[\mathbf{J}^{c}=\begin{pmatrix}J_{++}&J_{+-}\\ J_{-+}&J_{--}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}(J_{xx}+J_{yy})+i(J_{xy}- J_{yx})&(J_{xx}-J_{yy})-i(J_{xy}+J_{yx})\\ (J_{xx}-J_{yy})+i(J_{xy}+J_{yx})&(J_{xx}+J_{yy})-i(J_{xy}-J_{yx})\end{pmatrix}\,, \tag{2}\]  
where \(+\) and \(--\) represent the right circularly polarized (RCP) and left circularly polarized (LCP) components.  
If the scatterer achieves a perfect conversion between RCP wave and LCP wave, it must satisfy \(J_{yy}=-J_{xx}\) and \(J_{yx}=J_{xy}=0\), so that the Jones matrix is written as,  
\[\mathbf{J}_{pol}^{l}=\begin{pmatrix}J_{xx}&0\\ 0&-J_{xx}\end{pmatrix},\quad\mathbf{J}_{pol}^{c}=\begin{pmatrix}0&J_{xx}\\ J_{xx}&0\end{pmatrix} \tag{3}\]Clearly, there are only off-diagonal entities in \(\mathbf{J}_{pol}^{c}\), indicating the flip between RHCP and LHCP. Axially rotating the scatterer by an angle of \(\alpha\) will result in a new Jones matrix,  
\[\mathbf{J}_{pol}^{l}(\alpha)=J_{xx}\begin{pmatrix}\cos(2\alpha)&\sin(2\alpha)\\ \sin(2\alpha)&-\cos(2\alpha)\end{pmatrix},\quad\mathbf{J}_{pol}^{c}(\alpha)=J _{xx}\begin{pmatrix}0&e^{-2i\alpha}\\ e^{2i\alpha}&0\end{pmatrix} \tag{4}\]  
\(\mathbf{J}_{pol}^{c}(\alpha)\) also has only the non-zero off-diagonal items. Besides, an additional phase factor \(e^{\pm 2i\alpha}\) is introduced. This phase is the so-called geometric phase. The scatterers on a geometric-phase metasurface cover a \(2\pi\) phase shift by changing the rotation angle \(\alpha\).  
To generate OAM, geometric phase is utilized to construct the required phase profile, \(e^{il\phi}\) of an OAM wave. Apparently, to generate an OAM of order \(l\), scatterers on a metasurface with azimuthal location \(\phi\) should be designed to provide a phase change of \(l\phi\), i.e., \(\alpha=\pm l\phi/2\). The sign depends on the incident circular polarization state. The value of \(\alpha/\phi\) is known as the topology charge \(q\) of the metasurface. Apparently, the OAM order \(l\) is determined by \(q\) and should be double of its value.  
The geometric phase itself does not depend on the frequency so that broadband metasurfaces could be potentially designed. For a perfect (100%) conversion, the condition \(J_{yy}=-J_{xx}=\pm 1\) and \(J_{yx}=J_{xy}=0\) should be strictly satisfied, i.e., \(J_{yy}\) and \(J_{xx}\) should have the same unit amplitude and \(\pi\) phase difference. Achieving these conditions by scatterers on resonance will inevitably limit the bandwidth. When the requirement is not strictly satisfied, the co-circularly polarized component needs to be filtered out. Only the cross-circularly polarized component carries OAM.