## 3 Near Field Models of Reflectarray and Transmitarray Antennas  
### Field at the Aperture  
Before presenting the two near field models, the field at the aperture needs to be calculated, since it is used as the source from which the radiated field is obtained. First, a feed whose phase center is placed at \(\vec{r}_{f}=(x_{f},y_{f},z_{f})\) in the ACS generates a tangential incident field on the surface of the array, which can be written for the \(i\)-th element as:  
\[\vec{E}_{\rm inc}(\vec{r}_{i}^{\,\prime})=E_{\rm inc,x}(\vec{r}_{i}^{\,\prime} )\hat{s}_{a}+E_{\rm inc,y}(\vec{r}_{i}^{\,\prime})\hat{g}_{a}, \tag{1}\]  
with \(i=1,\ldots,N\). In order to alleviate the notation, the dependence on \(\vec{r}_{i}^{\,\prime}\) of the field components will be dropped from here on. From the incident field, the field at the aperture may be calculated as:  
\[\vec{E}_{\rm ap}(\vec{r}_{i}^{\,\prime})=M_{i}\vec{E}_{\rm inc}(\vec{r}_{i}^{ \,\prime}) \tag{2}\]  
where \(M_{i}\) is a \(2\times 2\) matrix of complex numbers which depends on the considered array. For reflectarrays, \(M_{i}\) is the matrix of reflection coefficients:  
\[M_{i}=\left( \begin{array}{cc}\rho_{xx,i}&\rho_{xy,i}\\ \rho_{yx,i}&\rho_{yy,i}\end{array} \right), \tag{3}\]  
and the field at the aperture \(\vec{E}_{\rm ap}\) would correspond to the reflected field. In the case of transmitarrays, \(M_{i}\) is the matrix of transmission coefficients:  
\[M_{i}=\left( \begin{array}{cc}\tau_{xx,i}&\tau_{xy,i}\\ \tau_{yx,i}&\tau_{yy,i}\end{array} \right), \tag{4}\]  
and the field at the aperture would correspond to the transmitted field. For both cases, \(M_{i}\) is computed using a full-wave analysis tool assuming local periodicity. In addition, \(M_{i}\) depends on the working frequency, unit cell topology, periodicity, substrate characteristics, and angles of incidence of the plane wave employed in the analysis [27]. Once the tangential electric field at the aperture has been obtained with (2), the tangential magnetic field may be calculated with:  
\[\vec{H}_{\rm ap}(\vec{r}_{i}^{\,\prime})=\frac{\vec{k}\times\vec{E}_{\rm ap}( \vec{r}_{i}^{\,\prime})}{\omega\mu_{0}}, \tag{5}\]  
where  
\[\vec{k}=k_{x}\hat{s}_{a}+k_{y}\hat{g}_{a}+k_{z}\hat{z}_{a}=-k_{0}(\sin\theta_{ i}\cos\varphi_{i}\hat{x}_{a}+\sin\theta_{i}\sin\varphi_{i}\hat{g}_{a}-\cos \theta_{i}\hat{z}_{a}), \tag{6}\]  
\(\omega=2\pi f\), \(f\) is the frequency, \(\mu_{0}\) is the free-space permeability, \(k_{0}\) is the free-space wavenumber, and \((\theta_{i},\varphi_{i})\) is the angle of incidence of the plane wave for the \(i\)-th array element (see Figure 1). The \(\hat{z}\) component of \(\vec{E}_{\rm ap}\), necessary to solve (5), is obtained by solving the plane wave equation \(\vec{k}\cdot\vec{E}_{\rm ap}=0\). Then, the components of the reflected tangential magnetic field are:  
\[H_{\rm ap,r} = -K_{1}E_{\rm ap,r}-K_{2}E_{\rm ap,y}, \tag{7a}\] \[H_{\rm ap,y} = \phantom{-}K_{3}E_{\rm ap,x}+K_{1}E_{\rm ap,y}, \tag{7b}\]with:  
\[K_{1} =\frac{k_{x}k_{y}}{\omega\eta_{0}k_{z}}, \tag{8a}\] \[K_{2} =\frac{k_{y}^{2}+k_{z}^{2}}{\omega\eta_{0}k_{z}},\] (8b) \[K_{3} =\frac{k_{x}^{2}+k_{z}^{2}}{\omega\eta_{0}k_{z}}. \tag{8c}\]  
In this way, the tangential field at the aperture is fully characterized by:  
\[\vec{E}_{\text{ap}}(\vec{r}_{i}^{\,\prime}) =E_{\text{ap},x}\hat{x}_{a}+E_{\text{ap},y}\hat{y}_{a}, \tag{9a}\] \[\vec{H}_{\text{ap}}(\vec{r}_{i}^{\,\prime}) =H_{\text{ap},x}\hat{x}_{a}+H_{\text{ap},y}\hat{y}_{a}, \tag{9b}\]  
where \(\vec{E}_{\text{ap}}(\vec{r}_{i}^{\,\prime})\) is obtained from (2) and \(\vec{H}_{\text{ap}}(\vec{r}_{i}^{\,\prime})\) from (5) and may be readily employed as a source for the calculation of the radiated field. Sometimes, the electric (\(\vec{J}\)) and magnetic (\(\vec{M}\)) current densities are required. Since planar arrays are considered, the notation \(\vec{J}_{s}\) and \(\vec{M}_{s}\) is employed.