# American Institute of Mathematical Sciences

June  2011, 31(2): 525-543. doi: 10.3934/dcds.2011.31.525

## Two dimensional invisibility cloaking via transformation optics

 1 Department of Mathematics and Statistics, University of Reading, Whiteknights, Reading RG6 6AX, United Kingdom 2 Department of Mathematics, University of California, Irvine, Irvine, CA 92697, United States

Received  June 2010 Revised  April 2011 Published  June 2011

We investigate two-dimensional invisibility cloaking via transformation optics approach. The cloaking media possess much more singular parameters than those having been considered for three-dimensional cloaking in literature. Finite energy solutions for these cloaking devices are studied in appropriate weighted Sobolev spaces. We derive some crucial properties of the singularly weighted Sobolev spaces. The invisibility cloaking is then justified by decoupling the underlying singular PDEs into one problem in the cloaked region and the other one in the cloaking layer. We derive some completely novel characterizations of the finite energy solutions corresponding to the singular cloaking problems. Particularly, some `hidden' boundary conditions on the cloaking interface are shown for the first time. We present our study for a very general system of PDEs, where the Helmholtz equation underlying acoustic cloaking is included as a special case.
Citation: Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 525-543. doi: 10.3934/dcds.2011.31.525
##### References:
 [1] A. Alu and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings, Phys. Rev. E, 72 (2005), 016623. doi: 10.1103/PhysRevE.72.016623. [2] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Commu. Math. Phys., 275 (2007), 749-789. doi: 10.1007/s00220-007-0311-6. [3] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics, SIAM Review, 51 (2009), 3-33. doi: 10.1137/080716827. [4] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the SHS lining, Opt. Exp., 15 (2007), 12717. doi: 10.1364/OE.15.012717. [5] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693. [6] U. Hetmaniuk and H. Y. Liu, On acoustic cloaking devices by transformation media and their simulation, SIAM J. Appl. Math., 70 (2010), 2996-3021. doi: 10.1137/090771077. [7] V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Springer-Verlag, New York, 2006. [8] R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the Helmholtz equation, Commu. Pure Appl. Math., 63 (2010), 0973-1016. [9] R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electrical impedance tomography, Inverse Problems, 24 (2008), 015016. doi: 10.1088/0266-5611/24/1/015016. [10] U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493. [11] H. Y. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 045006. doi: 10.1088/0266-5611/25/4/045006. [12] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 2000. [13] G. W. Milton and N.-A. P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. Roy. Soc. A, 462 (2006), 3027-3095. doi: 10.1098/rspa.2006.1715. [14] H. M. Nguyen, Cloaking via change of variables for the Helmholtz equation in the whole space, Commu. Pure Appl. Math., 63 (2010), 1505-1524. doi: 10.1002/cpa.20333. [15] H. M. Nguyen, Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking, preprint, 2011. [16] A. N. Norris, Acoustic cloaking theory, Proc. R. Soc. A, 464 (2008), 2411-2434. doi: 10.1098/rspa.2008.0076. [17] J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. doi: 10.1126/science.1125907. [18] G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, Ch. 19 in "Harmonic Analysis and Partial Differential Equations" (M. Christ, C. Kenig and C. Sadosky eds.), [19] M. Yan, W. Yan and M. Qiu, Invisibility cloaking by coordinate transformation, Ch. 4 in "Progress in Optics," Elsevier, 2008, 261-304. [20] B. Zhang, H. Chen, B. I. Wu and J. A. Kong, Extraordinary surface voltage effect in the invisibility cloak with an active device inside, Phys. Rev. Lett., 100 (2008), 063904. doi: 10.1103/PhysRevLett.100.063904.

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##### References:
 [1] A. Alu and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings, Phys. Rev. E, 72 (2005), 016623. doi: 10.1103/PhysRevE.72.016623. [2] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Commu. Math. Phys., 275 (2007), 749-789. doi: 10.1007/s00220-007-0311-6. [3] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics, SIAM Review, 51 (2009), 3-33. doi: 10.1137/080716827. [4] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the SHS lining, Opt. Exp., 15 (2007), 12717. doi: 10.1364/OE.15.012717. [5] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693. [6] U. Hetmaniuk and H. Y. Liu, On acoustic cloaking devices by transformation media and their simulation, SIAM J. Appl. Math., 70 (2010), 2996-3021. doi: 10.1137/090771077. [7] V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Springer-Verlag, New York, 2006. [8] R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the Helmholtz equation, Commu. Pure Appl. Math., 63 (2010), 0973-1016. [9] R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electrical impedance tomography, Inverse Problems, 24 (2008), 015016. doi: 10.1088/0266-5611/24/1/015016. [10] U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493. [11] H. Y. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 045006. doi: 10.1088/0266-5611/25/4/045006. [12] W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 2000. [13] G. W. Milton and N.-A. P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. Roy. Soc. A, 462 (2006), 3027-3095. doi: 10.1098/rspa.2006.1715. [14] H. M. Nguyen, Cloaking via change of variables for the Helmholtz equation in the whole space, Commu. Pure Appl. Math., 63 (2010), 1505-1524. doi: 10.1002/cpa.20333. [15] H. M. Nguyen, Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking, preprint, 2011. [16] A. N. Norris, Acoustic cloaking theory, Proc. R. Soc. A, 464 (2008), 2411-2434. doi: 10.1098/rspa.2008.0076. [17] J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. doi: 10.1126/science.1125907. [18] G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, Ch. 19 in "Harmonic Analysis and Partial Differential Equations" (M. Christ, C. Kenig and C. Sadosky eds.), [19] M. Yan, W. Yan and M. Qiu, Invisibility cloaking by coordinate transformation, Ch. 4 in "Progress in Optics," Elsevier, 2008, 261-304. [20] B. Zhang, H. Chen, B. I. Wu and J. A. Kong, Extraordinary surface voltage effect in the invisibility cloak with an active device inside, Phys. Rev. Lett., 100 (2008), 063904. doi: 10.1103/PhysRevLett.100.063904.
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