Point scatterer

A point scatterer idealizes an object as a localized coupling at one position when its physical size is small compared with the relevant wavelength and observation distances. For a three-dimensional scalar time-harmonic wave, an outgoing contribution has the formwhere the source strength $q$ depends on the scatterer's response and the exciting field at its position. This convention gives $({\nabla }^2+k^2)G=-\delta$.

The idealization keeps a finite scattering response while discarding the resolved surface geometry. It is useful for dilute assemblies of small objects and models of multiple scattering. The Green function is singular at the source, so using the total field at that same point requires a prescribed treatment of the self-field, such as short-distance regularization of a resonant point scatterer. Replacing a finite object by a point without specifying its response does not define a complete physical model.

An outgoing three-dimensional Green function behaves as $e^{ikr}/r=1/r+ik+O(r)$. Evaluating a point-coupled discrete-level equation at cutoff distance $\delta$ produces a real shift proportional to $1/\delta$ and an imaginary outgoing-wave term proportional to $ik$. The bare level must depend on the cutoff to keep the measured detuning finite.