## 2 Theoretical Principle  
As it is well known [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17], guiding metasurfaces (MTSs) support the propagation of surface waves (SWs) characterized by a phase velocity (\(\
u=\omega/\beta\)) lower than the free space velocity (\(c=\omega/k_{0}\)). Thus, the phase constant of the SWs (\(\beta\)) is larger than the free space wavenumber (\(k_{0}\)): \(\
u<c\rightarrow\beta>k_{0}\). When a cylindrically symmetric MTS is in the \(xy\) plane and these modes are excited from the center of the structure, they propagate in a radial direction, bounded to the surface, following \(e^{-j\beta\rho}\) (with \(\hat{\rho}\) being the radial unit vector). Away from the structure, the field is evanescent, and it is defined by an exponential decay \(e^{-jk_{2}\omega z}\), fulfilling: \(k_{0}^{2}=\beta^{2}+{k_{2}}^{2}\), where \(\Im(k_{2\omega})=-\alpha_{z0},\ \alpha_{z0}>0\).  
The goal of this work is to implement a single-layer metasurface that supports the propagation of a transverse magnetic mode (TM) and a transverse electric mode (TE). As has been widely explained in [8; 9; 10; 11; 12; 13], a single-layer MTS can be analyzed in terms of its equivalent circuit model. According to the transverse resonance equation, the surface impedance relating the fields of each SW in the surface (\(\mathbf{Z}_{surf,i}\)) can be related to the free space impedance of each mode at the air-surface interface (\(\mathbf{Z}_{0,i}\)) as: \(\mathbf{Z}_{0,i\ =\mathbf{TM,TE}}=-\ \mathbf{Z}_{surf,i\ i=\mathbf{TM,TE}}\). Thus, the surface impedance of each mode can be described by the following expressions:  
\[\mathbf{Z}_{surf,TM}\ =j\,\mathbf{X}_{surf,TM}\ =-\xi\frac{k_{2\omega}}{\beta} \tag{1}\]  
\[\mathbf{Z}_{surf,TE}\ =j\,\mathbf{X}_{surf,TE}\ =-\xi\frac{\beta}{k_{2\omega}} \tag{2}\]  
where \(\xi=120\pi\) is the free space impedance. From Equations (1) and (2), it can be concluded that the surface impedance of the desired MTS must have an inductive reactive term in order to allow the propagation of a TM SW ( \(X_{surf,TM}\ >0\)), while the surface impedance must have an inductive reactive term ( \(X_{surf,TE}\ <0\)), to support the TE mode propagation.  
Both modes must propagate in the structure decoupled and synchronized in phase. The condition for TM and TE modes phase synchronism can also be related to the equivalent surface reactance values as:  
\[\mathbf{\beta}_{\mathbf{TM}}=\mathbf{\beta}_{\mathbf{TE}}\ \rightarrow\ \mathbf{X}_{surf,TM}\,\mathbf{X}_{surf,TE}\ =\ -\xi^{2} \tag{3}\]  
Exciting the rotationally symmetric MTS from the center by a magnetic dipole (slot in the ground plane in a practical implementation) the TM and TE modes propagate radially from the center. In the MTS configuration shown in Figure 1, the magnetic dipole (depicted as a black arrow) is oriented towards the \(y\)-axis. In this case, the TM SW is azimuthally excited following a \(\cos(\varphi)\) function, while the field related to the TE mode meets a \(\sin(\varphi)\) excitation. If the phase synchronism of the modes is ensured all over the surface and the mode amplitudes are equal, the total field on the surface, obtained as the combination of both modes, provides a linearly polarized \(E_{x}\) field:  
\[\mathbf{E}^{(y)}=\ \mathbf{E}_{\mathbf{TM}}\ ^{(y)}+\mathbf{E}_{\mathbf{TE}}\ ^{(y)}\ =-\mathbf{E}_{\mathbf{TM}}\ ^{(y)}\cos( \varphi)\hat{\mathbf{\rho}}+\mathbf{E}_{\mathbf{TE}}\ ^{(y)}\sin(\varphi)\hat{\mathbf{\varphi}}=-\mathbf{E}_{0}\hat{ \mathbf{x}} \tag{4}\]  
where \(E_{TM}\ =E_{TE}\ =E_{0}\) for circular polarization.