On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static Approximation

The paper considers the transmission problems for Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero the fields may blow up. When the speed of the blow up is suitable, this is called the Anomalous Localized Reconance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak Anomalous Localized Reconance (w-ALR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-ALR or ALR may, or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary neither ALR nor w-ALR appears with non-zero frequencies (i.e. in the dynamic range) in dimensions $d\ge 3$. In the case when the boundary of the negative material contains a flat subset we show that the w-ALR always happens with some point sources in dimensions $d\ge 2$. These results, together with the earlier results of Milton et al. ( [22, 23]) and Ammari et al. ([2]) show that for strictly convex bodies ALR may appear only for bodies so small that the quasi-static approximation is realistic. This gives limits for size of the objects for which invisibility cloaking methods based on ALR may be used.Comment: 30 pages, 7 figure

On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static Approximation

This paper considers transmission problems for the Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero, the fields may blow up. When the speed of the blow up is suitable, this is called the anomalous localized resonance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak anomalous resonance (w-AR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-AR or ALR may or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary, neither ALR nor w-AR appears with nonzero frequencies (i.e., in the dynamic range) in dimensions d >= 3. In the case when the boundary of the negative material contains a flat subset, we show that w-AR always happens with some point sources in dimensions d >= 2.Peer reviewe

Rocking Subdiffusive Ratchets: Origin, Optimization and Efficiency

We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale