Computational Imaging of Small-Amplitude Biperiodic Surfaces with Negative Index Material (CSIAM Transactions on Applied Mathematics, 2025)
This paper presents an innovative approach to computational acoustic imaging of biperiodic surfaces, exploiting the capabilities of an acoustic superlens to overcome the diffraction limit. We address the challenge of imaging physical entities incomplex environments by considering the partial differential equations that governthe physics and solving the corresponding inverse problem. We focus on imaging infinite rough surfaces, specifically 2D diffraction gratings, and propose a method thatleverages the transformed field expansion. We derive a reconstruction formula connecting the Fourier coefficients of the surface and the measured field, demonstratingthe potential for unlimited resolution under ideal conditions. We also introduce an approximate discrepancy principle to determine the cut-off frequency for the truncatedFourier series expansion in surface profile reconstruction. Furthermore, we elucidatethe resolution enhancement effect of the superlens by deriving the discrete Fouriertransform of white Gaussian noise. Our numerical experiments confirm the effectiveness of the proposed method, demonstrating high subwavelength resolution even under slightly non-ideal conditions. This study extends the current understanding ofsuperlens-based imaging and provides a robust framework for future research.
Keywords: Inverse scattering problems; superlens; metamaterial; superresolution; diffraction gratings.