Figure 1

Illustrating the notation for scattering in free space. The incident field, here taken to be a single plane wave, is represented by a solid line in the direction of the wave vector of the plane wave. A plane wave component of the scattered field is represented by a dashed line and is labeled with the appropriate amplitude.Reuse & Permissions

Figure 2

Illustrating the notation for the half-space problem. In (a), an incident mode ${\varphi }_i^{+}(\mathbf{r},{\mathbf{k}}_1,{\mathbf{k}}_1^{\prime })$ associated with sources in the lower half space is represented by a solid line indicating the wave vector of the three planewave components of the mode. A mode of the scattered field is represented by a dashed line. The two plane wave components of the scattered field in the upper half space combine to produce an outgoing plane wave with wave vector ${\mathbf{k}}_2$ and amplitude proportional to $A_{+}^{+}({\mathbf{k}}_1,{\mathbf{k}}_1^{\prime },{\mathbf{k}}_2,{\mathbf{k}}_2^{\prime })$. The plane wave component of the scattered mode in the lower half space is proportional to $A_{-}^{+}({\mathbf{k}}_1,{\mathbf{k}}_1^{\prime },{\mathbf{k}}_2,{\mathbf{k}}_2^{\prime })$. In (b), the notation is similarly illustrated, here with a different mode of the incident field, ${\varphi }_i^{-}(\mathbf{r},{\mathbf{k}}_1,{\mathbf{k}}_1^{\prime })$, generated by sources in the upper half space.Reuse & Permissions