Investigation of the effect of super-resolution in nonlinear inverse scattering

The idea that a solution to a nonlinear inverse scattering problem (ISP) can contain information about the target on a subwavelength scale and thus allow one to achieve super-resolution (spatial resolution beyond the diffraction limit) has been around since the 1990s. However, a solid mathematical theory of super-resolution in nonlinear image reconstruction is still lacking. In this paper, we investigate the effect of super-resolution in nonlinear ISPs (both analytically and numerically) by analyzing several inverse problems in which the limit of spatial resolution can be defined precisely. The conclusions we obtain are not optimistic. Although it is possible to create examples of exactly solvable models in which account of nonlinearity in the ISP results in additional mathematically independent equations (one such example is shown herein), our results indicate that super-resolution is not achievable in any practical sense. Rather, we find that the linear subspace of possible solutions to a band-limited linearized ISP is transformed into a more general curved manifold due to the effects of nonlinearity. In the one-dimensional problem with realistic interaction that we have considered, the manifold can have a slightly smaller dimensionality that the subspace of solutions to the linearized problem but it does not contract to a point and the effect is practically insignificant.