Electromagnetic scattering

Remarks

The scattered field has the asymptotic form $\mathbf{E}^{\mathrm{sc}}(r\hat{\mathbf{x}}) = \dfrac{e^{ikr}}{r}\left( \mathbf{E}_\infty(\hat{\mathbf{x}}) + O(1/r) \right)$, and $\mathbf{E}_\infty$ is the far-field pattern. For an incident plane wave of unit amplitude, the scattering cross-section is $C_{\mathrm{sca}} = \int_{S^2} \abs{\mathbf{E}_\infty}^2 \, d\Omega$, the scattered power divided by the incident intensity. The radiation condition singles out outgoing waves and makes the problem uniquely solvable for a perfect conductor. It does the same for a medium whose permittivity has non-negative imaginary part, under mild regularity assumptions. Example: a small dielectric sphere in a plane wave gives Rayleigh scattering, whose far-field pattern is that of an oscillating dipole. Non-example: the field $\mathbf{E}^{\mathrm{inc}}$ itself is not a scattered field, since a plane wave does not satisfy the radiation condition. The optical theorem relates the total power removed from the incident wave to the forward value of $\mathbf{E}_\infty$.