Use of loss limit approach to zero in scattering-based parameter retrieval of elastic micro-structured media

Abstract

The parameter retrieval method based on scattering data is used to derive dynamic constitutive parameters of solids with periodic structure. There are inherent ambiguities in the real part of the retrieved wavenumber and the sign of impedance, though the latter has a one-to-one correspondence with the direction of energy flux. Moreover, for lossless structures there can be multiple solution branches that satisfy passivity and continuity of the wavenumber as a function of frequency, leading to potentially double positive or double negative (in terms of density and modulus) overall descriptions of the micro-structured medium. The continuity of wavenumber for lossy unit cells is used to unambiguously determine their constitutive parameters and by taking the limit when loss approaches zero, one can determine the stable solution branches of lossless micro-structures. The two loss models of nonzero damping and complex modulus are compared in terms of their energy loss characteristics and retrieved parameters. These models are employed in both time and frequency domain calculations. The lossy solutions demonstrate that from double-negative and double-positive solutions, only the latter is the stable solution branch in a pass-band of a lossless 1D unit cell. A 2D photonic crystal example is used to demonstrate that the wavenumber appears to jump, discontinuously, at the transition point between two consecutive stop-bands, hence rendering the existing methods based on continuity of wavenumber ineffective. In contrast, the proposed approach based on taking limit of lossy solutions can successfully be used to determine stable overall properties of such media. Finally, certain features of half- and full-cycle stop-bands are discussed.