Green's Functions

The Green's function $G(\mathbf{r}, \mathbf{r}')$ satisfies

$$\nabla^2 G + \kappa_{0}^{2} G = -\delta(\mathbf{r} - \mathbf{r}'),$$

with the Sommerfeld radiation condition at infinity. In 2D and 3D:

$$G_{\text{2D}}(\mathbf{r}, \mathbf{r}') = \frac{j}{4}H_0^{(2)}(\kappa_{0}|\mathbf{r} - \mathbf{r}'|),$$

$$G_{\text{3D}}(\mathbf{r}, \mathbf{r}') = \frac{e^{-j\kappa_{0}|\mathbf{r} - \mathbf{r}'|}}{4\pi|\mathbf{r} - \mathbf{r}'|},$$

where $H_0^{(2)}$ is the Hankel function of the second kind (consistent with the $e^{-j\omega t}$ convention). The Green's function propagates a point source at $\mathbf{r}'$ to the observation point $\mathbf{r}$.

For the full vector Maxwell equations, the Green's function becomes a dyadic (tensor) $\overline{\mathbf{G}}(\mathbf{r}, \mathbf{r}')$ satisfying:

$$\nabla \times \nabla \times \overline{\mathbf{G}} - \kappa_{0}^{2} \overline{\mathbf{G}} = \mathbf{I}\,\delta(\mathbf{r} - \mathbf{r}'),$$

where $\mathbf{I}$ is the $3 \times 3$ identity tensor. The electric field due to a current source $\mathbf{J}$ is:

$$\mathbf{E}(\mathbf{r}) = j\omega\mu_0 \int \overline{\mathbf{G}}(\mathbf{r}, \mathbf{r}') \cdot \mathbf{J}(\mathbf{r}')\,d\mathbf{r}'.$$

In the far field, $\overline{\mathbf{G}}$ reduces to the scalar Green's function times a projection operator onto the transverse plane.