American Institute of Mathematical Sciences

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

Two dimensional invisibility cloaking via transformation optics

Hongyu Liu 1, and  Ting Zhou 2,

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.
Keywords: finite energy solutions, Invisibility cloaking, singularly weighted Sobolev space., transformation optics.
Mathematics Subject Classification: Primary: 35J, 35Q; Secondary: 35R, 78.
Citation: Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete & Continuous Dynamical Systems - A, 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).  doi: 10.1103/PhysRevE.72.016623.  Google Scholar

[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.  doi: 10.1007/s00220-007-0311-6.  Google Scholar

[3]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics,, SIAM Review, 51 (2009), 3.  doi: 10.1137/080716827.  Google Scholar

[4]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the SHS lining,, Opt. Exp., 15 (2007).  doi: 10.1364/OE.15.012717.  Google Scholar

[5]

A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem,, Math. Res. Lett., 10 (2003), 685.   Google Scholar

[6]

U. Hetmaniuk and H. Y. Liu, On acoustic cloaking devices by transformation media and their simulation,, SIAM J. Appl. Math., 70 (2010), 2996.  doi: 10.1137/090771077.  Google Scholar

[7]

V. Isakov, "Inverse Problems for Partial Differential Equations,", 2nd edition, (2006).   Google Scholar

[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.   Google Scholar

[9]

R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electrical impedance tomography,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/1/015016.  Google Scholar

[10]

U. Leonhardt, Optical conformal mapping,, Science, 312 (2006), 1777.  doi: 10.1126/science.1126493.  Google Scholar

[11]

H. Y. Liu, Virtual reshaping and invisibility in obstacle scattering,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/4/045006.  Google Scholar

[12]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000).   Google Scholar

[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.  doi: 10.1098/rspa.2006.1715.  Google Scholar

[14]

H. M. Nguyen, Cloaking via change of variables for the Helmholtz equation in the whole space,, Commu. Pure Appl. Math., 63 (2010), 1505.  doi: 10.1002/cpa.20333.  Google Scholar

[15]

H. M. Nguyen, Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking,, preprint, (2011).   Google Scholar

[16]

A. N. Norris, Acoustic cloaking theory,, Proc. R. Soc. A, 464 (2008), 2411.  doi: 10.1098/rspa.2008.0076.  Google Scholar

[17]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields,, Science, 312 (2006), 1780.  doi: 10.1126/science.1125907.  Google Scholar

[18]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper,, Ch. 19 in, ().   Google Scholar

[19]

M. Yan, W. Yan and M. Qiu, Invisibility cloaking by coordinate transformation,, Ch. 4 in, (2008), 261.   Google Scholar

[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).  doi: 10.1103/PhysRevLett.100.063904.  Google Scholar

show all references

References:
[1]

A. Alu and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings,, Phys. Rev. E, 72 (2005).  doi: 10.1103/PhysRevE.72.016623.  Google Scholar

[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.  doi: 10.1007/s00220-007-0311-6.  Google Scholar

[3]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics,, SIAM Review, 51 (2009), 3.  doi: 10.1137/080716827.  Google Scholar

[4]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the SHS lining,, Opt. Exp., 15 (2007).  doi: 10.1364/OE.15.012717.  Google Scholar

[5]

A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem,, Math. Res. Lett., 10 (2003), 685.   Google Scholar

[6]

U. Hetmaniuk and H. Y. Liu, On acoustic cloaking devices by transformation media and their simulation,, SIAM J. Appl. Math., 70 (2010), 2996.  doi: 10.1137/090771077.  Google Scholar

[7]

V. Isakov, "Inverse Problems for Partial Differential Equations,", 2nd edition, (2006).   Google Scholar

[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.   Google Scholar

[9]

R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electrical impedance tomography,, Inverse Problems, 24 (2008).  doi: 10.1088/0266-5611/24/1/015016.  Google Scholar

[10]

U. Leonhardt, Optical conformal mapping,, Science, 312 (2006), 1777.  doi: 10.1126/science.1126493.  Google Scholar

[11]

H. Y. Liu, Virtual reshaping and invisibility in obstacle scattering,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/4/045006.  Google Scholar

[12]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000).   Google Scholar

[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.  doi: 10.1098/rspa.2006.1715.  Google Scholar

[14]

H. M. Nguyen, Cloaking via change of variables for the Helmholtz equation in the whole space,, Commu. Pure Appl. Math., 63 (2010), 1505.  doi: 10.1002/cpa.20333.  Google Scholar

[15]

H. M. Nguyen, Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking,, preprint, (2011).   Google Scholar

[16]

A. N. Norris, Acoustic cloaking theory,, Proc. R. Soc. A, 464 (2008), 2411.  doi: 10.1098/rspa.2008.0076.  Google Scholar

[17]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields,, Science, 312 (2006), 1780.  doi: 10.1126/science.1125907.  Google Scholar

[18]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper,, Ch. 19 in, ().   Google Scholar

[19]

M. Yan, W. Yan and M. Qiu, Invisibility cloaking by coordinate transformation,, Ch. 4 in, (2008), 261.   Google Scholar

[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).  doi: 10.1103/PhysRevLett.100.063904.  Google Scholar

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