Near-Field Imaging of Infinite Rough Surfaces in Dielectric Media

This paper is concerned with an inverse surface scattering problem in near-field optical imaging, which is to reconstruct the scattering surface of a dielectric medium with a resolution beyond the diffraction limit. It is a nontrivial extension of the authors' work on near-field imaging of infinite rough surfaces from impenetrable to penetrable media [G. Bao and P. Li, SIAM J. Appl. Math., 73 (2013), pp. 2162-2187], where a more sophisticated transmission problem needs to be considered. The scattering surface is modeled as a small and smooth deformation of a plan surface. Based on a transformed field expansion, an analytic solution, which is given as a power series, is derived for the direct scattering problem. By neglecting high order terms in the power series, the original nonlinear inverse problem is linearized; explicit and unified reconstruction formulas are deduced for both reflection and transmission configurations. A spectral cut-off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components, which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near-field regime. The method requires only a single illumination at a fixed frequency and is realized efficiently by the fast Fourier transform. Numerical results show that the method is simple, stable, and effective in reconstructing scattering surfaces of dielectric media with subwavelength resolution.