Electromagnetic scattering beyond the weak regime

Abstract

In many optical measurement methods one aims to retrieve the shape or physical properties of an unknown object by measuring how it scatters an incident optical field. Such an inverse problem is often approached by solving the corresponding direct scattering problem iteratively. Despite the existence of numerical methods, a perturbation approach could be a powerful way to solve those direct problems. In such an approach the field is expressed as a series, known as the Born series, in which the higher-order corrections represent multiplescattering effects. This approach has the advantage that each term of the series has a clear physical meaning and that it can unveil much more about the scattering process than a purely numerical approach. Under (very) weak-scattering conditions the Born series converges and then often the first or first few terms of the series are sufficient to model the scattering. However, except under very special circumstances such as for x-rays where optical contrasts are extremely low, multiple scattering effects are very important and the Born series in general does not converge. Although multiple scattering could, in principle, allow resolving subwavelength details of the unknown object, this is in practice impeded by the divergent nature of the Born series.

In this thesis, we show how Padé approximation can be employed in electromagnetic problems to retrieve an accurate evaluation of the scattered field even under strong-scattering conditions. Padé approximants are rational functions that can offer improvements in two ways, namely, accelerating the rate of convergence of (already) converging series and analytic continuation of series outside its region of convergence. We apply the method to three scalar scattering problems. In one dimension we consider an infinitely thin slab and a slab of finite thickness, and in two dimensions an infinitely long cylinder. In particular, we study cases in the strong-scattering regime for which the Born series diverges. It will be shown that for all cases studied, Padé approximation retrieves an accurate result.

The presented method integrates multiple-scattering effects one by one and can therefore represent an important building block to the application of the Born series to direct and inverse problems, with potential applications in superresolution, optical metrology, and phase retrieval.