Abstract
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.
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References
Adams R. and Fournier J. (2003). Sobolev Spaces. Pure and Applied Mathematics 140. Academic Press, New York-London
Alu A. and Engheta N. (2005). Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72: 016623
Astala K. and Päivärinta L. (2006). Calderón’s inverse conductivity problem in the plane. Annals of Math. 163: 265–299
Astala K., Lassas M. and Päiväirinta L. (2005). Calderón’s inverse problem for anisotropic conductivity in the plane. Comm. Partial Diff. Eqs. 30: 207–224
Belishev M. and Kurylev Y. (1992). To the reconstruction of a Riemannian manifold via its spectral data (B-method). Comm. Part. Diff. Eq. 17: 767–804
Calderón, A.P.: On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Rio de Janeiro: Soc. Brasil. Mat., 1980, pp. 65–73
Cummer S., Popa B.-I., Schurig D., Smith D. and Pendry J. (2006). Full-wave simulations of electromagnetic cloaking structures. Phys. Rev. E 74: 036621
Greenleaf A., Lassas M. and Uhlmann G. (2003). The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction. Comm. Pure Appl. Math. 56(3): 328–352
Greenleaf A., Lassas M. and Uhlmann G. (2003). Anisotropic conductivities that cannot detected in EIT, Physiological Measurement (special issue on Impedance Tomography) 24: 413–420
Greenleaf A., Lassas M. and Uhlmann G. (2003). On nonuniqueness for Calderón’s inverse problem. Math. Res. Let. 10(5-6): 685–693
Hänninen, I., Lindell, I., Sihvola, A.: Realization of Generalized Soft-and-Hard Boundary. Progress In Electromagnetics Research, PIER 64, 2006, pp. 317–333
Kachalov A. and Kurylev Y. (1998). Multidimensional inverse problem with incomplete boundary spectral data. Comm. Part. Diff. Eq. 23: 55–95
Kachalov, A., Kurylev, Y., Lassas, M.: Inverse Boundary Spectral Problems, Chapman and Hall/CRC Monogr. and Surv. in Pure and Appl. Math. 123. Boca Raton, FL: Chapman and Hall/CRC, 2001
Kildal P.-S. (1988). Definition of artificially soft and hard surfaces for electromagnetic waves. Electron. Let. 24: 168–170
Kildal, P.-S.: Artificially soft and hard surfaces in electromagnetics. IEEE Transactions on Antennas and Propagation 38, 10, 1537–1544 (1990)
Kohn, R., Shen, H., Vogelius, M., Weinstein, M.: In preparation
Kohn, R., Vogelius, M.: Identification of an unknown conductivity by means of measurements at the boundary. In: Inverse Problems, SIAM-AMS Proceedings., 14 (1984)
Kato T. (1995). Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin
Kilpeläinen T., Kinnunen J. and Martio O. (2000). Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3): 233–247
Kurylev Y. (1993). Multidimensional inverse boundary problems by the BC-method: groups of transformations and niqueness results. Math. Comput. Modelling 18: 33–46
Kurylev Y., Lassas M. and Somersalo E. (2006). Maxwell’s equations with a polarization independent wave velocity: Direct and inverse problems. J. Math. Pures et Appl. 86: 237–270
Lassas M. and Uhlmann G. (2001). Determining Riemannian manifold from boundary measurements. Ann. Sci. École Norm. Sup. 34(5): 771–787
Lassas M., Taylor M. and Uhlmann G. (2003). The dirichlet-to-neumann map for complete Riemannian manifolds with boundary. Comm. Geom. Anal. 11: 207–222
Lee J. and Uhlmann G. (1989). Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math. 42: 1097–1112
Leonhardt, U.: Optical Conformal Mapping. Science 312, 1777–1780, 23 June, 2006
Leonhardt U. and Philbin T. (2006). General relativity in electrical engineering. New J. Phys. 8: 247
Lindell I. (2002). Generalized soft-and-hard surface. IEEE Tran. Ant. and Propag. 50: 926–929
Maz‘ja V. (1985). Sobolev Spaces. Springer-Verlag, Berlin
Melrose R. (1995). Geometric scattering theory. Cambridge Univ. Press, Cambridge
Milton G., Briane M. and Willis J. (2006). On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8: 248
Milton G. and Nicorovici N.-A. (2006). On the cloaking effects associated with anomalous localized resonance. Proc. Royal Soc. A 462: 3027–3059
Nachman A. (1988). Reconstructions from boundary measurements. Ann. of Math. (2) 128: 531–576
Nachman A. (1996). Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. 143: 71–96
Pendry J.B., Schurig D. and Smith D.R. (2006). Controlling electromagnetic fields. Science 312: 1780–1782
Pendry J.B., Schurig D. and Smith D.R. (2006). Calculation of material properties and ray tracing in transformation media. Opt. Exp. 14: 9794
Schurig, D., Mock, J., Justice, B., Cummer, S., Pendry, J., Starr, A., Smith, D.: Metamaterial electromagnetic cloak at microwave frequencies. Science Online, 10.1126/science.1133628, Oct. 19, 2006
Serrin J. (1964). Local behavior of solutions of quasi-linear equations. Acta Math. 111: 247–302
Sun Z. and Uhlmann G. (2003). Anisotropic inverse problems in two dimensions. Inverse Problems 19: 1001–1010
Sylvester J. (1990). An anisotropic inverse boundary value problem. Comm. Pure Appl. Math. 43: 201–232
Sylvester J. and Uhlmann G. (1987). A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125: 153–169
Uhlmann, G.: Scattering by a metric. In: Encyclopedia on Scattering, R. Pike and P. Sabatier, eds. Chap. 6.1.5, London-New York: Academic Press, 2002, pp. 1668–1677
Vogelius, M.: Lecture, Workshop on Inverse Problems and Applications, BIRS, August, 2006
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Greenleaf, A., Kurylev, Y., Lassas, M. et al. Full-Wave Invisibility of Active Devices at All Frequencies. Commun. Math. Phys. 275, 749–789 (2007). https://doi.org/10.1007/s00220-007-0311-6
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DOI: https://doi.org/10.1007/s00220-007-0311-6