American Institute of Mathematical Sciences

Advanced
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, 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.  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), 749-789. 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), 3-33. 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), 12717. 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), 685-693.  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), 2996-3021. doi: 10.1137/090771077.  Google Scholar

[7]

V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Springer-Verlag, New York, 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), 0973-1016.  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), 015016. doi: 10.1088/0266-5611/24/1/015016.  Google Scholar

[10]

U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493.  Google Scholar

[11]

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

[12]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 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), 3027-3095. 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), 1505-1524. 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), 2411-2434. doi: 10.1098/rspa.2008.0076.  Google Scholar

[17]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. 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 "Progress in Optics," Elsevier, 2008, 261-304. 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), 063904. 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), 016623. 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), 749-789. 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), 3-33. 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), 12717. 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), 685-693.  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), 2996-3021. doi: 10.1137/090771077.  Google Scholar

[7]

V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Springer-Verlag, New York, 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), 0973-1016.  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), 015016. doi: 10.1088/0266-5611/24/1/015016.  Google Scholar

[10]

U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493.  Google Scholar

[11]

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

[12]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations," Cambridge University Press, Cambridge, 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), 3027-3095. 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), 1505-1524. 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), 2411-2434. doi: 10.1098/rspa.2008.0076.  Google Scholar

[17]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. 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 "Progress in Optics," Elsevier, 2008, 261-304. 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), 063904. doi: 10.1103/PhysRevLett.100.063904.  Google Scholar

[1]

Guillermo Reyes, Juan-Luis Vázquez. The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1275-1294. doi: 10.3934/cpaa.2008.7.1275
[2]

Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731
[3]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365
[4]

Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[5]

Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[6]

Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
[7]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002
[8]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[9]

Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427
[10]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[11]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[12]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[13]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[14]

Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
[15]

Yutian Lei. On finite energy solutions of fractional order equations of the Choquard type. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1497-1515. doi: 10.3934/dcds.2019064
[16]

Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[17]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469
[18]

T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105
[19]

Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574
[20]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences