Energetic vs Inference-Based Invisibility: Fisher-Information Analysis of Two-Layer Acoustic Near-Cloaks

Abstract: Near-cloaks based on passive coatings can strongly suppress scattered-field energy in a narrow frequency band, yet an observer's ability to infer object parameters from noisy measurements need not decrease proportionally. We develop a fully theoretical two-dimensional (2D) framework for a coated acoustic cylinder in an air background. Using an exact cylindrical-harmonic solution of the Helmholtz equation, we compute the modal scattering coefficients a_m(omega) for a core of radius a surrounded by two concentric effective-fluid layers, and we design the coating to cancel the dominant low-order multipoles (monopole m=0 and dipole m=+/-1) at a target frequency, yielding a narrowband near-cloak. Beyond the conventional energetic metric (total scattering width), we quantify information-based detectability through the Fisher information matrix (FIM) and the associated Cramer-Rao lower bounds (CRLBs) for joint estimation of the size-material parameter vector x=[a, rho1, c1]^T from noisy far-field data. A representative air-background study exhibits an approximately 25 dB reduction in total scattering width near the design frequency, while tr(FIM) decreases by only a few dB, demonstrating that energy-based and inference-based notions of invisibility are distinct objectives. We further provide a low-order analytic argument clarifying the mechanism behind this energetic-informational decoupling and report design-space and local-robustness diagnostics that highlight persistent trade-offs between scattering suppression and parameter identifiability.