Wave propagation
Far-Field Pattern
The far field radiated or scattered by an antenna can be calculated numerically from the near field on an arbitrary surface completely enclosing the antenna structure, denoted here as $S_{NTF}$. This procedure is referred to as the near-to-far-field (NTF) transformation. Given the near-zone electric and magnetic fields on $S_{NTF}$, it can be shown that the electric field in the far zone can be expressed as
\begin{align} E_\theta(\mathbf{r})&=-\frac{\jmath k_0 e^{-\jmath k_0 r}}{4\pi r} (L_\phi + Z_0 N_\theta) \label{E_elevation}\\ E_\phi(\mathbf{r}) &= \frac{\jmath k_0 e^{-\jmath k_0 r}}{4\pi r} (L_\theta - Z_0 N_\phi) \label{E_azimuth} \end{align}
where $Z_0$ is the free-space impedance and \begin{align} \mathbf{N}(\mathbf{\hat{r}}) &= \int\int_{S_{NTF}} \mathbf{J}(\mathbf{r'}) e^{\jmath k_0 \mathbf{r'}\cdot \mathbf{\hat r}} \, dS' \\ \mathbf{L}(\mathbf{\hat{r}}) &= \int\int_{S_{NTF}} \mathbf{M}(\mathbf{r'}) e^{\jmath k_0 \mathbf{r'}\cdot \mathbf{\hat r}} \, dS' \end{align}
where $\mathbf{J} = \hat{n} \times \mathbf{H}$ and $\mathbf{M} = -\hat{n} \times \mathbf{E}$ are equivalent surface electric and magnetic currents, respectively. The near-to-far-field transformation surface $S_{NTF}$ must be a closed surface containing all the sources inside. If an infinite ground plane is present (e.g., in the case of monopole antennas), the near-to-far-field transformation surface $S_{NTF}$ and its image $S'_{NTF}$ together comprise a closed surface. The equivalent surface currents on $S'_{NTF}$ can easily be obtained by invoking the image theory.
