# Holographic Silicon Metasurfaces for Total Angular Momentum Demultiplexing Applications in Telecom
## 2 Materials and Methods  
### Phase Pattern Calculation  
A diffractive optical element designed to analyze the OAM spectrum in the set of OAM values \(\{\ell_{j}\}\) presents a phase pattern \(\Omega(u,v)\) which is given by the linear superposition of \(n\) orthogonal OAM modes \(\{\psi_{j}\}\) as follows [35]:  
\[\Omega(u,v)=\arg\{\sum_{j=1}^{n}c_{j}\psi_{j}^{*}\exp\{i\alpha_{j}\mu+i\beta_{j }v\}\}, \tag{1}\]  
being \(\{\psi_{j}=R_{j}(\rho,\delta)\exp(i\ell_{j}\vartheta)\}\), where \(\vartheta=\arctan(v/u)\), \(\rho=\sqrt{u^{2}+v^{2}}\), \(\{R_{j}(\rho,\delta)\}\) describe the field spatial distributions and depend on the family of modes. \(\{(\alpha_{j},\beta_{j})\}\) are the \(n\) vectors of carriers spatial frequencies in Cartesian coordinates, and \(\{c_{j}\}\) are complex coefficients whose modulus is usually unitary, and the phases are fitted so that Equation (1) is an exact equality [23]. The set of parameters \(\{c_{j}\}\) is calculated with the following integral:  
\[c_{j}=\int\limits_{-\infty}^{+\infty}du\int\limits_{-\infty}^{+\infty}\psi_{j} \exp(i\Omega)\exp\Bigl{(}-i\alpha_{j}u-i\beta_{j}v\Bigr{)}dv, \tag{2}\]  
The diffractive element is basically a computer-generated hologram originated from the linear combination of \(n\) fork-holograms. Each term in Equation (1) is given by the interference pattern of the \(j\)th OAM-mode with azimuthal phase term \(\exp(i\ell_{j}\vartheta)\) with the tilted plane-wave \(\exp(i\alpha_{j}\mu+i\beta_{j}v)\) defined by the corresponding carrier frequency. In the Fourier plane, the carrier frequencies manifest as separate spatial coordinates \(\{(x_{j},y_{j})\}\) given by:  
\[ \begin{array}{l}x_{j}=\alpha_{j}\frac{f}{k}\\ y_{j}=\beta_{j}\frac{f}{k}\end{array} , \tag{3}\]  
being \(f\) the focal length of the lens which is used for far-field reconstruction in \(f\)-\(f\) configuration, and \(k=2\pi/\lambda\), where \(\lambda\) is the working wavelength. When the optical element is illuminated with an integer OAM beam, the projection of the beam is optically performed over the selected OAM set, and a bright spot appears at the position corresponding to the input OAM value in far field (Figure 1) [23]. In a metasurface realization of the optical element, the phase pattern \(\Omega(u\),\(v)\) is obtained by fabricating an inhomogeneous and anisotropic effective medium, whose extraordinary-axis orientation \(\theta(u\),\(v)\) changes point-by-point and is equal to half the local phase value \(\Omega(u\),\(v)\). The two orthogonal circular polarizations exhibit a different behavior, as it follows [32]:  
\[T(u,v)\left( \begin{array}{c}1\\ \pm i\end{array} \right)=\cos\!\left(\frac{\delta}{2}\right)\!\left( \begin{array} []{c}1\\ \pm i\end{array} \right)-i\sin\!\left(\frac{\delta}{2}\right)\!\exp\!\left[\pm i \Omega(u,v)\right]\!\left( \begin{array}{c}1\\ \mp i\end{array} \right) \tag{4}\]  
being \(T\) the transmission matrix of the optical element, \(\delta\) the phase delay between the ordinary and extraordinary axes of the metasurface effective medium, [1, +\(i\)] and [1, -\(i\)] the vectors of right-handed and left-handed circular polarizations in Jones matrix formalism, respectively (the normalization factor \(1/\sqrt{2}\) has been omitted). In particular, when the metasurface is engineered in order to achieve the condition \(\delta=\pi\) (\(\pi\)-delay between the two optical axes), the zero-order term is cancelled out and a total polarization conversion is obtained:  
\[T\!\left( \begin{array}{c}1\\ \pm i\end{array} \right)=-i\exp(\pm i\Omega)\!\left( \begin{array}{c}1\\ \mp i\end{array} \right) \tag{5}\]  
Figure 1: Scheme of the working principle of Pancharatnam–Berry optical elements (PBOEs)for orbital angular momentum (OAM) demultiplexing with the method of OAM-beam projection. If a circularly-polarized OAM-beam illuminates the optical element, a bright spot appears in the far field, at a position which depends on the polarization handedness and on the carried OAM.