Localization Landscapes based on the Green's Matrix of Non-Hermitian Optical Scattering Systems

Abstract

The Filoche--Mayboroda (FM) localization landscape predicts where the spatially localized modes of a self-adjoint elliptic operator reside without diagonalizing it, but the construction fails for the complex, non-normal Green's matrix G of an open scattering array of electric dipoles. We propose three real-valued, positive landscapes built from the Feinberg--Zee hermitization of G: a regularised Gramian inverse (Gram), the diagonal Schur block that carries the Metz--Neri--Rogers spectral density (SY), and the full hermitized inverse contracted against a polarization-isotropic source pair (Herm). The three form a progression from spectrum-blind to fully hermitisation-faithful, agree in the limit $z\to 0$, and diverge where it matters: at nonzero spectral shift SY resolves modes that Gram collapses, and Herm resolves modes that SY collapses. We demonstrate the progression on random and deterministic aperiodic scattering arrays with a wide range of structural correlations, extend to the vectorial case of transverse electric polarization (TE), and provide rigorous justifications along with mathematical proofs and a complete numerical recipe in the appendices and in the supplementary material.