Near and far field

Definition

Let time-harmonic sources of angular frequency $\omega$ be contained in a ball of diameter $D$ centred at the origin. Let the medium be homogeneous and lossless, with wavenumber $k = 2\pi/\lambda$ and wave impedance $\eta$. Let $(\mathbf{E}, \mathbf{H})$ be the radiating solution of Maxwell's equations outside the ball. Then there is a vector field $\mathbf{F}$ on the unit sphere, the far-field pattern, with $\hat{\mathbf{r}} \cdot \mathbf{F}(\hat{\mathbf{r}}) = 0$, such that $$\mathbf{E}(r\hat{\mathbf{r}}) = \frac{e^{ikr}}{r}\left( \mathbf{F}(\hat{\mathbf{r}}) + O(1/r) \right), \qquad \mathbf{H} = \frac{1}{\eta}\, \hat{\mathbf{r}} \times \mathbf{E} + O(1/r^2), \qquad r \to \infty.$$ The far field, or Fraunhofer region, is the region where the leading term is an accurate approximation. By the usual engineering convention it is $r > 2D^2/\lambda$ together with $r \gg D$ and $r \gg \lambda$. The near field is the region close to the source where the $O(1/r)$ correction is not negligible. Its inner part $r \lesssim \lambda/(2\pi)$ is the reactive near field, dominated by the terms of order $1/r^2$ and $1/r^3$.