Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime

Abstract

Cloaking a source via anomalous localized resonance (ALR) was discovered by Milton and Nicorovici in [14]. A general setting in which cloaking a source via ALR takes place is the settting of doubly complementary media. This was introduced and studied in [19] for the quasistatic regime. In this paper, we study cloaking a source via ALR for doubly complementary media in the finite frequency regime as a natural continuation of [19]. We establish the following results: 1) Cloaking a source via ALR appears if and only if the power blows up; 2) The power blows up if the source is "placed" near the plasmonic structure; 3) The power remains bounded if the source is far away from the plasmonic structure. Concerning the analysis, we extend ideas from [19] and add new insights on the problem which allows us to overcome difficulties related to the finite frequency regime and to obtain new information on the problem. In particular, we are able to characterize the behaviour of the fields far enough from the plasmonic shell as the loss goes to 0 for an arbitrary source outside the core-shell structure in the doubly complementary media setting.

Negative index materials and their applications: Recent mathematics progress

“…In comparison with results in [19], Theorem 1.1 is stronger: no conditions on the blow up rate of the power is required. The proof of Theorem 1.1 is in the spirit of [19].…”

“…One of the key points in the proof is the definition ofû δ in (2.11) after introducing u 1,δ and u 2,δ as in [19]. In Ω 3 \ Ω 2 , we remove u 1,δ − u 2,δ from u δ .…”

“…The proof of the first statement of Proposition 1.3 is based on a kind of removing singularity technique and has roots from [19]. A key point is the construction of the auxiliary function W δ in (5.8).…”

“…The proof of Proposition 1.1 essentially uses the ideas in the one of [21, Lemma 10] which has root from [19]. It is clear that Proposition 1.1 implies Property P2).…”

“…The proof of Theorem 1.1 is in the spirit of [19]. Nevertheless, we incorporate two important ingredients.…”