## 3 Near Field Models of Reflectarray and Transmitarray Antennas
### Comparison of the Near Field Models  
To compare the near field predicted by both models, let us consider a center-fed square planar transmitarray comprised of \(60\times 60\) elements in a regular grid of periodicity \(3.84\,\mathrm{mm}^{2}\). The feed is modelled as a \(\cos^{q}\theta\) function [29] with \(q=22\) and placed at \(\vec{r}_{f}=(0,0,-180)\,\mathrm{mm}\) in the ACS (see Figure 1). The working frequency is \(39\,\mathrm{GHz}\). For the analysis, the phases of the direct transmission coefficients in (4) are set to:  
\[\angle\tau_{xx}=\angle\tau_{yy}=k_{0}(d_{i}-(x^{\prime}\cos\varphi_{0}+y^{ \prime}\sin\varphi_{0})\sin\theta_{0}), \tag{33}\]  
where \(d_{i}\) is the distance from the feed phase center to the \(i\)-th array element, and \((\theta_{0},\varphi_{0})\) is the pointing direction of the far field pencil beam defined by the phases of \(\tau_{xx}\) and \(\tau_{yy}\). For the present example, \((\theta_{0},\varphi_{0})\) is set to \((0^{\circ},0^{\circ})\). The radiated near field will be calculated with the two models in planes parallel to the array with \(z\) constant and values \(z=0.1\,\mathrm{m}\), \(1\,\mathrm{m}\). Thus, the matrix of change of coordinates from the ACS to the NFCS is the identity. In addition, in the case of the model based on the radiation equations, it will be evaluated for several values of \(N_{x}\) and \(N_{y}\), specifically for \(N_{x}=N_{y}=1,3,5\) to assess the convergence of the integrals. Finally, each plane is divided into a grid of \(150\times 150\) points (22,500 in total) where the near field is computed. Figure 3 shows the main cuts in \(x\) for \(y=0\) for the near field component \(E_{x}\) in magnitude and phase for both \(z\) planes. The agreement between both models is in general very good. In the case of the magnitude, the model based on the radiation equations with \(N_{x}=N_{y}=1\) presents some discrepancies in the plane \(z=0.1\,\mathrm{m}\) with the rest of the models, albeit for very low values, \(60\,\mathrm{dB}\) lower than the peak magnitude. In the case of the phase, the model based on the superposition of the far field radiated by each array element presents a small discrepancy in the plane \(z=0.1\,\mathrm{m}\). In this case, the shape of the phase is the same as in the other models, but it has a shift of approximately \(1.3\,\mathrm{deg}\). Both discrepancies disappear when the field is calculated further away from the array. Figure 4 shows the magnitude and phase of the field components \(E_{x}\) and \(E_{y}\) as calculated by the models based on far field superposition (top row) and the radiation equations with \(N_{x}=N_{y}=5\) (bottom row) for the plane \(z=0.1\,\mathrm{m}\). As can be seen, visually the results are practically the same. A better way of assessing the similarities between both models is to compute the relative error in the calculated near fields. To that end, the following expression is employed:  
\[\mathrm{RE}=100\frac{\left\|E_{x|y|z}-\tilde{E}_{x|y|z}\right\|}{\left\|E_{x| y|z}\right\|}, \tag{34}\]  
where \(\|\cdot\|\) is the Euclidean norm; \(E\) is one component of the near field computed with the model based on the radiation equations with \(N_{x}=N_{y}=5\), which is considered as reference to compute the relative error in percentage; and \(\tilde{E}\) is the same component of the near field computed with any other model. Table 1 shows the relative error for both models. It shows the relative errors for the three components of the near field for both planes. In the case of the model based on the radiation equations, even a value of \(N_{x}=N_{y}=1\) shows a sufficiently low error. As the near field is computed further away from the array, the error decreases. The use of \(N_{x}=N_{y}=1\) means that the double integrals in (15) are just evaluated at the center of the array element, i.e., considering only one sample, effectively reducing the triple summation in (21) to a single sum, considerably accelerating computations without barely affecting accuracy in the prediction of the near field.