Singular partial differential operators and pseudo-differential boundary conditions in invisibility cloaking

Matti Lassas, T. Zhou

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

2 Citations (Scopus)

Abstract

Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum parameters waves around a region without penetrating it, so that this region is hidden from external observations. The material parameters used to describe these devices are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. These singular material parameters correspond to singular coefficient functions in the partial differential equations modeling these constructions and the presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we give a review on mathematical theory of cloaking. Moreover, we consider the two-dimensional cloaking, that is, cylindrical cloaking, for Maxwell's equations. For this case, we present results that generalizes earlier analogous results for the two-dimensional cloaking for the scalar equations. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. Using such truncation we analyze the limit when the approximate cloaking approaches the ideal cloaking. We show that a non-local pseudo-differential boundary condition appears on the inner cloak interface. This effect in the two-dimensional (or cylindrical) invisibility cloaks, which seems to be caused by the infinite phase velocity near the interface between the cloaked and uncloaked regions, is very different to the behavior of the solutions in the three-dimensional cloaks.

Original languageEnglish
Title of host publicationFourier Analysis - Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential Equations
PublisherSpringer International Publishing AG
Pages263-268
Number of pages6
Volume63
ISBN (Print)9783319025490
Publication statusPublished - 2014
MoE publication typeA4 Article in a conference publication
EventInternational Conference on Fourier Analysis and Pseudo-Differential Operators - Helsinki, Finland
Duration: 25 Jun 201229 Jun 2012

Publication series

NameTrends in Mathematics
Volume63
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Conference

ConferenceInternational Conference on Fourier Analysis and Pseudo-Differential Operators
Country/TerritoryFinland
CityHelsinki
Period25/06/201229/06/2012

Keywords

  • Invisibility cloaking
  • Pseudodifferential boundary conditions

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