Department of Mathematics
We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions N ≥ 2 via the approach of transformation optics. There are four major ingredients in our proposed theory: (1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation that blows up a subset Kε in the virtual space, where ε ≪ is an asymptotic regularization parameter. Kε will degenerate to K0 as (Formula Presented), and in our theory K0 could be any convex compact set in RN, or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. (2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. (3) The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. (4) In order to achieve a cloaking device of compact size, particularly for the case when (Formula Presented) is not “uniformly small”, an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components. Within the proposed framework, we show that the scattered wave field (Formula Presented) corresponding to a cloaking problem will converge to u0 as (Formula Presented), with u0 being the scattered wave field corresponding to a sound-hard K0. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of K0. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are also conducted to illustrate our theoretical results.
wave scattering, Helmholtz equation, invisibility cloaking, transformation optics, partial and full cloaks, asymptotic estimates
Source Publication Title
Communications in Mathematical Physics
The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-015-2318-8
The work of Jingzhi Li is supported by the NSF of China under the grant No. 11201453 and No. 11371115. The work of Hongyu Liu is supported by FRG and start-up funds from HKBU. Luca Rondi is partly supported by Università degli Studi di Trieste through Finanziamento per Ricercatori di Ateneo 2009 and by GNAMPA, INdAM, through 2012 projects. The work of Gunther Uhlmann is partly supported by NSF and the Fondation des Sciences Mathématiques de Paris.
Link to Publisher's Edition
Li, J., Liu, H., Rondi, L., & Uhlmann, G. (2015). Regularized transformation-optics cloaking for the Helmholtz equation: From partial cloak to full cloak. Communications in Mathematical Physics, 335 (2), 671-712. https://doi.org/10.1007/s00220-015-2318-8