Convergence criterion for the time-domain iterative Born approximation to scattering by an inhomogeneous, dispersive object

A convergence criterion is derived for the iterative Rayleigh–Gans–Born approximation (or Neumann expansion) applied to the time-domain integral equation for the scattering of transient scalar waves by an inhomogeneous, dispersive object of bounded extent, embedded in a homogeneous, nondispersive medium. The criterion is independent of the size of the object and contains only a bound on the maximum absolute value of the contrast susceptibility of the object with respect to its embedding. Three types of contrast susceptibility relaxation function are considered in more detail: one for an instantaneously reacting (i.e., nondispersive) material, one for a dielectric with a Lorentzian absorption line, and one for a dispersive metal. For the last two cases the convergence proves to be unconditional if the object is embedded in vacuum. The proof makes use of the time Laplace transformation with a real, positive transform parameter and Lerch’s theorem on the uniqueness of the one-sided Laplace transformation, which implies that causality of the wave motion plays an essential role.