# HERAS: A Modular Matlab Tool Using Physical Optics for the Analysis of Reflector Antennas
## 2 Numerical Techniques for the Analysis of Reflector Antennas  
As discussed previously, there are several options commercially available for the analysis of reflector antennas that employ different numerical methods. Finite integration technique (FIT), finite element method (FEM), and method of moments (MoM) are widely used by antenna engineers to solve electromagnetic problems for electrically small structures [26; 27; 28]. They are generally referred to as \"full-wave\" or \"exact\" numerical techniques as they solve Maxwell's equations, thus providing highly accurate results when convergence is achieved. However, these methods lead to matrices that are of increasing sizes for larger radiating structures (such as reflector antennas), and the computational times and/or the required memory eventually become unmanageable. Other methods have been proposed over the years to reduce the computational requirements when solving this kind of structure, which are based on geometrical optics (GO) and generally referred to as \"asymptotic\" numerical techniques because of the associated approximations. GO describes the propagation of waves by rays, which is a valid approximation in the case of electrically large structures. However, this ray tracing approximation is not well suited to evaluate interference and diffraction phenomena, as is the case when computing the far-field of an antenna. To this end, hybrid numerical techniques, which combine GO and full-wave techniques, have been developed and are referred to as physical optics (PO). In particular, two methods have received some attention for the analysis of reflector antennas: the aperture and the current distribution methods. The aperture distribution method obtains the radiated far-field of the reflector antenna from an aperture field distribution derived from ray tracing [29; 30]:  
\[\vec{E}=jk\frac{e^{-jkr}}{4\pi r}\hat{r}\times\iint_{S_{a}}-2\hat{z}\times \vec{E}_{a}e^{jk\vec{r}_{a}\cdot\hat{a}_{r}}\mathrm{d}S_{a} \tag{1}\]  
where \(\vec{E}\) is the far-field vector, \(k\) is the wave number, \(r\) is the radial component, \(\hat{a}_{r}\) the unit vector for the far-field direction, and the double integral is evaluated over the aperture plane \(S_{a}\) on the aperture field \(\vec{E}_{a}\) with the integration vector \(\vec{r}_{a}\). The aperture plane \(S_{a}\) is commonly defined as the projected aperture of the reflector surface \(S\) onto the focal plane, i.e. the plane orthogonal to \(2\), defining the focal axis, and containing the focal point of the parabola. Essentially, this method applies ray tracing from the feed to the reflector and up to an assumed flat surface, which is then considered to be the radiating aperture. The main advantage of this method is that the integral can be computed with discrete Fourier transform (DFT) which allows the use of algorithms, such as fast Fourier transform (FFT), that significantly reduce the computational cost [31]. In addition, the evaluation can be performed independently of the feed position and illumination pattern. Nevertheless, this method can apply a false symmetry in some cases, which leads to beam squint and side lobe errors [30; 32]. The current distribution method is well-established as the preferred approach to model the radiation characteristics from reflector antennas [15]. Similar to the aperture distribution method, ray tracing is applied in order to obtain the currents on the reflector. For a perfectly conducting surface, the currents can be estimated following the next equation:  
\[\vec{J}_{s}=2\hat{n}\times\vec{H}^{i} \tag{2}\]where \(\vec{f}_{s}\) is the linear current density, \(\hat{n}\) the unit vector, normal to the reflector surface \(S\), and \(\vec{H}^{i}\) the incident magnetic field.