Full-Wave Invisibility of Active Devices at All Frequencies
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There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] cloaks passive objects, i.e., those without internal currents, at all frequencies k. Due to the singularity of the metric, one needs to work with weak solutions. Analyzing the behavior of such solutions inside the cloaked region, we show that, depending on the chosen construction, there appear new “hidden” boundary conditions at the surface separating the cloaked and uncloaked regions. We also consider the effect on invisibility of active devices inside the cloaked region, interpreted as collections of sources and sinks or internal currents. When these conditions are overdetermined, as happens for Maxwell’s equations, generic internal currents prevent the existence of finite energy solutions and invisibility is compromised.
We give two basic constructions for cloaking a region D contained in a domain \(\Omega\subset\mathbb R^n, n\ge 3\) , from detection by measurements made at \(\partial\Omega\) of Cauchy data of waves on Ω. These constructions, the single and double coatings, correspond to surrounding either just the outer boundary \(\partial D^+\) of the cloaked region, or both \(\partial D^+\) and \(\partial D^-\) , with metamaterials whose EM material parameters (index of refraction or electric permittivity and magnetic permeability) are conformal to a singular Riemannian metric on Ω. For the single coating construction, invisibility holds for the Helmholtz equation, but fails for Maxwell’s equations with generic internal currents. However, invisibility can be restored by modifying the single coating construction, by either inserting a physical surface at \(\partial D^-\) or using the double coating. When cloaking an infinite cylinder, invisibility results for Maxwell’s equations are valid if the coating material is lined on \(\partial D^-\) with a surface satisfying the soft and hard surface (SHS) boundary condition, but in general not without such a lining, even for passive objects.
KeywordsWeak Solution Riemannian Manifold Neumann Boundary Condition Helmholtz Equation Energy Solution
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