# Characterization of Tiled Architecture for C-Band 1-Bit Beam-Steering Transmitarray
## 2 Transmitarray Model  
The model of a transmitarray, first presented by the authors in [10], is given below for consistency. In a spatially phase-modulated transmitarray, the normalized wave amplitude received by a unit cell from the focal source reads:  
\[a_{mn}=\frac{\lambda e^{ikR_{mn}}}{4\pi R_{mn}}\mathbf{F}_{mn}^{fs}\cdot \mathbf{F}_{mn}^{ucr} \tag{1}\]  
where \(m=1\), \(2\),..., \(M\) and \(n=1\), \(2\),..., \(N\) are the row and column indexes of the array which define the position of the unit cell with respect to the reference one, \(k\) and \(\lambda\) are free-space wavenumber and wavelength, \(\mathbf{F}_{mn}^{fs}\) is the complex vector field pattern of the focal-plane source transmitting in the direction of the unit cell defined by the corresponding polar and azimuthal angles of the local coordinate system (CS) with the origin at the focal point, \(\mathbf{F}_{mn}^{ucr}\) is that of the unit cell on receive in the direction of the focal plane source defined by the respective angles of the local CS with the origin at the center of the unit cell (note that the antenna pattern on receive is conjugate of that on transmit due to the reciprocity), the dot symbol denotes the Hermitian inner product of the two complex vector patterns, and \(R_{mn}\) is the distance between the focal-source and unit-cell CSs. The unit cells are assumed to be matched to the incoming wave at all angles of incidence determined by the angular aperture of the transmitarray. The effects of the element coupling and finite aperture of the transmitarray can, in principle, be accounted for in the unit-cell antenna patterns by infinite array analysis, [12], or embedded element technique, [13]. Figure 1: One-bit dual-polarized tiled transmitarray architecture: (**a**) design of the array tile (vertically exploded view), comprising two identical proximity-coupled square-ring radiators on the opposite sides of the tile connected via two U-shaped feed loops; (**b**,**c**) schematic view of the surface currents on the proximity coupled feed loops and square-ring patches in two phase states, respectively (the ground plane is not shown); (**d**) a section of the tiled transmitarray partially assembled; (**e**) a section of the integral single-panel transmitarray; (**f**) proposed device architecture, [10], including individual unit cells (different colors indicate one of the two phase states) to be mounted in the plastic grid frame and spatially fed by a focal-source patch antenna. The focal-source and unit-cell antenna patterns, in the case of linear polarization, reduce to scalar-valued functions. After sampling and retardation of the incident spherical wavefront, the complex amplitude antenna pattern of the transmitarray, \(F(\theta,\phi)\), can be calculated by the pattern multiplication principle, as follows:  
\[F(\theta,\phi)=\sum_{m=1}^{M}\sum_{n=1}^{N}b_{mn}F_{nm}^{uc}(\theta,\phi)e^{ \psi_{mn}^{uc}(\theta,\phi)}e^{ikd(msin\&cos\phi+nsin\&sin\phi)} \tag{2}\]  
where \(F_{nm}^{uc}(\theta,\phi)\) and \(\psi_{mn}^{uc}(\theta,\phi)\) are, respectively, the unit-cell amplitude and phase patterns on transmit, \(\theta\) and \(\phi\) are azimuthal and polar angles in the spherical CS with the origin at the center of the transmitarray aperture and the polar direction aligned with the transmitarray optical axis, \(b_{mn}=T_{mn}a_{mn}\)-complex amplitudes of the waves radiated by each unit cell, and \(T_{mn}\)-the corresponding complex transmission coefficients. Equation (2) enables accounting for the effect of the finite array on the standalone pattern of the element, [14]. Also, the unit-cell radiation pattern is assumed to be independent of the transmission coefficient, i.e., of the specific phase shift for the phase-modulated transmitarray. The above model can be adopted for the design of the proposed tiled transmitarray by suitably adjusting the unit-cell transmission coefficients for given focal-source and unit-cell antenna patterns.