Two dimensional invisibility cloaking via transformation optics

Previous Article
Topological pressure and topological entropy of a semigroup of maps
DCDS Home
This Issue
Next Article
Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions

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

Abstract
Related Papers

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 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," 2^nd 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, ().
[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.

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.
[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," 2^nd 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, ().
[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.

[1]	Guillermo Reyes, Juan-Luis Vázquez. The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions. Communications on Pure and 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 and Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731
[3]	Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics and Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[4]	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
[5]	Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[6]	Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Jian Zhang. Sequences of high and low energy solutions for weighted (p, q)-equations. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022114
[7]	Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
[8]	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 and Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002
[9]	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 and Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[10]	Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427
[11]	Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[12]	Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[13]	Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[14]	Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[15]	Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
[16]	Yutian Lei. On finite energy solutions of fractional order equations of the Choquard type. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1497-1515. doi: 10.3934/dcds.2019064
[17]	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 and Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[18]	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 and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469
[19]	T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105
[20]	Doyoon Kim, Kyeong-Hun Kim, Kijung Lee. Parabolic Systems with measurable coefficients in weighted Sobolev spaces. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022062

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Hongyu Liu Ting Zhou
on Google Scholar
Hongyu Liu Ting Zhou

[Back to Top]