## 3 Near Field Models of Reflectarray and Transmitarray Antennas
### Near Field from the Radiation Equations  
The first near field model is based on the radiation equations obtained as superposition of the contributions of the vector potentials \(\vec{A}\) and \(\vec{F}\)[25]:  
\[\vec{E}_{\text{NF}}(\vec{r}) =\vec{E}_{A}(\vec{r})+\vec{E}_{F}(\vec{r}), \tag{11a}\] \[\vec{H}_{\text{NF}}(\vec{r}) =\vec{H}_{A}(\vec{r})+\vec{H}_{F}(\vec{r}), \tag{11b}\]  
where  
\[\vec{E}_{A}(\vec{r}) =E_{A_{x}}\hat{x}_{a}+E_{A_{y}}\hat{y}_{a}+E_{A_{z}}\hat{z}_{a} \tag{12a}\] \[\vec{E}_{F}(\vec{r}) =E_{F_{x}}\hat{x}_{a}+E_{F_{y}}\hat{y}_{a}+E_{F_{z}}\hat{z}_{a}\] (12b) \[\vec{H}_{A}(\vec{r}) =H_{A_{x}}\hat{x}_{a}+H_{A_{y}}\hat{y}_{a}+H_{A_{z}}\hat{z}_{a}\] (12c) \[\vec{H}_{F}(\vec{r}) =H_{F_{x}}\hat{x}_{a}+H_{F_{y}}\hat{y}_{a}+H_{F_{z}}\hat{z}_{a} \tag{12d}\]  
In (12) and subsequent equations, the dependence of the field components on the observation point \(\vec{r}\) has been dropped to alleviate the notation. There are a total of 12 field components in (11), for which an integral of field sources must be evaluated. An example of one such component is:  
\[E_{F_{z}}=-\frac{1}{4\pi}\iiint_{V}\left[(y-y^{\prime})M_{x}-(x-x^{\prime})M_ {y}\right]\frac{1+j\beta R}{R^{3}}e^{-j\beta R}dx^{\prime}dy^{\prime}dz^{\prime}, \tag{13}\]  
where \(R\) is the distance from a source point in the volume \(V\) to the observation point, and \(\beta\) is the phase constant that in this case is equal to the free-space wavenumber \(k_{0}\). The rest of the integrals may be consulted in [25] (pp. 285-286). Since the arrays are planar, (13) can be reduced to:  
\[E_{F_{z}}=-\frac{1}{4\pi}\iint_{S}\Big{[}(y-y^{\prime})M_{x}-(x-x^{\prime})M_{y }\Big{]}\frac{1+j\beta R_{i}}{R^{3}}e^{-j\beta R_{i}}dx^{\prime}dy^{\prime}, \tag{14}\]  
where \(S\) is the surface of the array. The array is divided into \(N\) unit cells, where the elements of matrix \(M_{i}\), and thus the field at the aperture and surface currents, are computed. Thus, we express the integral as a sum of the integral over each array element:  
\[E_{F_{z}}=-\frac{1}{4\pi}\sum_{i=1}^{N}\iint_{S_{i}}\Big{[}(y-y^{\prime})M_{x_{ i}}-(x-x^{\prime})M_{y_{i}}\Big{]}\frac{1+j\beta R_{i}}{R^{3}_{i}}e^{-j\beta R_{i} }dx^{\prime}dy^{\prime}, \tag{15}\]  
where \(S_{i}\) is the surface of the \(i\)-th unit cell, and:  
\[R_{i}=|\vec{r}-\vec{r}_{i}^{\prime}|=\sqrt{(x-x_{i}^{\prime})^{2}+(y-y_{i}^{ \prime})^{2}+z^{2}}. \tag{16}\]  
To finally obtain \(E_{F_{z}}\) in (15), the double integral may be evaluated by any quadrature method, for instance, the mid point quadrature. With this technique, the integral is approximated by a double sum:  
\[\vec{H}_{\text{NF}}(\vec{r}) =\vec{H}_{A}(\vec{r})+\vec{H}_{F}(\vec{r}), \tag{11b}\]0  
where surface \(S_{i}\) is divided into \(N_{x}N_{y}\) rectangles of length \(\Delta_{x}=p_{x}/N_{x}\) and width \(\Delta_{y}=p_{y}/N_{y}\), and:  
\[\vec{H}_{\text{NF}}(\vec{r}) =\vec{H}_{A}(\vec{r})+\vec{H}_{F}(\vec{r}), \tag{11b}\]1  
with:  
\[\vec{H}_{\text{NF}}(\vec{r}) =\vec{H}_{A}(\vec{r})+\vec{H}_{F}(\vec{r}), \tag{11b}\]2 \[\vec{H}_{\text{NF}}(\vec{r}) =\vec{H}_{A}(\vec{r})+\vec{H}_{F}(\vec{r}), \tag{11b}\]3  
Thus, the function is evaluated at the center of each rectangle. By substituting (17) into (15), we finally obtain:  
\[\vec{H}_{\text{NF}}(\vec{r}) =\vec{H}_{A}(\vec{r})+\vec{H}_{F}(\vec{r}), \tag{11b}\]4  
The same process can be applied to the other \(11\) integrals in [25] (pp. 285-286), taking into account that the planar arrays are placed at the plane \(z^{\prime}=0\) and that the longitudinal components of the surface currents are zero (\(J_{z}=M_{z}=0\)).