2014/06/24 by Henrik Kettunen, Kettunen, Henrik, Matti Lassas +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Microwave Imaging and Scattering Analysis #Numerical methods in engineering #Numerical methods in inverse problems #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1406.6224
30 pages, 7 figures
openalex publication_date 2014/06/24 · arxiv created 2015/04/14 · arxiv updated 2015/04/15 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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≥ 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≥ 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.