# Experimental Demonstration of Beam Scanning of Dual-Metasurface Antenna
## 2 Materials and Methods
### Design Method  
The design was carried out in terms of standard guided-mode analysis [25] via the Transverse-Resonance Equation (TRE). As depicted in Figure 1b, the total admittance from the antenna-free space boundary was controlled by the variable impedance layer, resulting in a controllable wavenumber of the guided wave. The use of the usual \"adiabatic\" approximation for slow variations of the top impedance allowed us to approximate the antenna with a transverse equivalent network (TEN) (see Figure 1b) at any given sample point of the modulated structure. From this transmission-line representation of the structure, the _local_ value of the surface impedance \(Z_{\mathrm{s}}(x)\) can be retrieved by solving the corresponding TRE:  
\[Y_{\mathrm{UP}}(x)+Y_{\mathrm{DOWN}}(x)=0, \tag{3}\]  
where \(Y_{\mathrm{UP}}\) and \(Y_{\mathrm{DOWN}}\) are the admittances looking up and down from a reference plane located at the interface between the antenna and free space. Indicating with \(d_{1}\) the thickness of the dielectric layer between the two metasurfaces and with \(d_{0}\) the air gap between reconfigurable metasurface and ground plane, standard transmission-line theory yields [26]:  
\[Y_{\mathrm{UP}}=Y_{0}^{\mathrm{TM}} \tag{4}\] \[Y_{\mathrm{DOWN}}=Y_{\mathrm{MMIS}}+Y_{1}^{\mathrm{TM}}\frac{Y_{ \mathrm{EGP}}+\mathrm{j}Y_{1}^{\mathrm{TM}}\tan\left(k_{21}d_{1}\right)}{Y_{ 1}^{\mathrm{TM}}+\mathrm{j}Y_{\mathrm{EGP}}\tan\left(k_{21}d_{1}\right)}\] (5) \[Y_{\mathrm{EGP}}=Y_{\mathrm{RMIS}}-\mathrm{j}Y_{0}^{\mathrm{TM}} \cot\left(k_{20}d_{0}\right), \tag{6}\]  
where \(Y_{\mathrm{MMIS}}\) is the admittance of the upper metasurface, \(Y_{\mathrm{RMIS}}\) is the admittance of the lower (reconfigurable) metasurface, and \(Y_{\mathrm{EGP}}\) is the admittance of the equivalent ground plane looking down from just above the tunable impedance plane (Figure 1b). In (4)-(6), \(Y_{0}^{\text{TM}}\) and \(Y_{1}^{\text{TM}}\) are the TM wave admittances in free space and dielectric substrate:  
\[Y_{0}^{\text{TM}} =\frac{\omega e_{0}}{k_{z0}} \tag{7}\] \[Y_{1}^{\text{TM}} =\frac{\omega e_{1}}{k_{z1}}, \tag{8}\]  
where \(k_{z0}\) and \(k_{z1}\) are the wavenumbers in the transverse direction:  
\[k_{z0} =\sqrt{k_{0}^{2}-k_{x}^{2}} \tag{9}\] \[k_{z1} =\sqrt{k_{1}^{2}-k_{x}^{2}}, \tag{10}\]  
and \(k_{x}\) is the longitudinal wavenumber. From (5) and (6), it can be seen that the solution of the TRE depends on the sheet admittances of both metasurfaces (\(Y_{\text{MMTS}}\) and \(Y_{\text{RMTS}}\)). Since all quantities in (4)-(6) can be expressed as a function of \(k_{x}\), it is possible to solve (3) for \(k_{x}\) and then obtain the (local) surface impedance as:  
\[Z_{\text{s}}=\text{j}X_{\text{s}}=\frac{1}{Y_{\text{DOWN}}}. \tag{11}\]  
It is worth noting that the longitudinal wavenumber \(k_{x}\) considered above is used only to compute the local surface impedance based on the adiabatic approximation, and does not correspond to the actual wavenumber of the leaky wave that arises from the modulation of such impedance. The average value of \(X_{\text{s}}\) is linked to the radiation angle through (2). This means that tuning \(Y_{\text{RMTS}}\) directly affects the beam direction, thus validating the concept of this architecture. Once a desired radiation angle for the \(n=-1\) harmonic is chosen, the modulation period \(p\) is retrieved from (2) after selecting a proper value of \(X_{\text{s,ave}}\) among those physically attainable with the considered structure. The design process now requires the choice of the geometry of both metasurfaces and the computation of their sheet admittances \(Y_{\text{MMTS}}\) and \(Y_{\text{RMTS}}\), in order to solve (3). For the upper layer, a unit cell was chosen such that its admittance can be easily modulated by varying only one geometric feature, e.g., the gap width between two adjacent metal strips, as in [23]. For the implementation of the tunable metasurface, the unit cell shown in Figure 2 was used. This shape was inspired by the geometry described in [27].