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

Cloaking a source via anomalous localized resonance (ALR) was discovered by Milton and Nicorovici in \cite{MiltonNicorovici}. 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 \cite{Ng-CALR} 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 \cite{Ng-CALR}. 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 \cite{Ng-CALR} 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 {\bf arbitrary source} outside the core-shell structure in the doubly complementary media setting.