Abstract

We adapt tools of transformation optics, governed by a (elliptic) wave equation, to thermodynamics, governed by the (parabolic) heat equation. We apply this new concept to an invibility cloak in order to thermally protect a region (a dead core) and to a concentrator to focus heat flux in a small region. We finally propose a multilayered cloak consisting of 20 homogeneous concentric layers with a piecewise constant isotropic diffusivity working over a finite time interval (homogenization approach).

© 2012 OSA

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  4. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
    [CrossRef]
  5. R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
    [CrossRef]
  6. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [CrossRef] [PubMed]
  7. B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
    [CrossRef]
  8. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
    [CrossRef] [PubMed]
  9. A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
  10. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
    [CrossRef]
  11. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
    [CrossRef] [PubMed]
  12. A. Diatta and S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011).
    [CrossRef]
  13. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).
  14. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007).
    [CrossRef]
  15. H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
  16. A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008).
    [CrossRef]
  17. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006).
    [CrossRef]
  18. M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
    [CrossRef]
  19. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
    [CrossRef]
  20. G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London, Ser. A462,3027–3059 (2006).
    [CrossRef]
  21. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994).
    [CrossRef]
  22. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008).
    [CrossRef]
  23. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133–11141 (2007).
    [CrossRef] [PubMed]
  24. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
    [CrossRef]
  25. L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953).
    [CrossRef]

2011 (1)

A. Diatta and S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011).
[CrossRef]

2009 (2)

M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[CrossRef]

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

2008 (6)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
[CrossRef]

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
[CrossRef]

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008).
[CrossRef]

A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

2007 (5)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
[CrossRef]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007).
[CrossRef]

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).

Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133–11141 (2007).
[CrossRef] [PubMed]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[CrossRef] [PubMed]

2006 (4)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006).
[CrossRef]

2005 (1)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[CrossRef]

2003 (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

1953 (1)

L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[CrossRef]

Briane, M.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006).
[CrossRef]

Brun, M.

M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[CrossRef]

Chan, C. T.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).

Chen, H.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

de Lustrac, A.

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Diatta, A.

A. Diatta and S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011).
[CrossRef]

Engheta, N.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[CrossRef]

Feng, Y.

Genov, D. A.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Germain, D.

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).

Guenneau, S.

A. Diatta and S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011).
[CrossRef]

M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[CrossRef]

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008).
[CrossRef]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[CrossRef] [PubMed]

Huang, Y.

Jiang, T.

Jikov, V. V.

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Kanté, B.

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Kohn, R. V.

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
[CrossRef]

Kozlov, S. M.

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994).
[CrossRef]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

Milton, G. W.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006).
[CrossRef]

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London, Ser. A462,3027–3059 (2006).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Movchan, A.B.

M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[CrossRef]

Nicolet, A.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008).
[CrossRef]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[CrossRef] [PubMed]

Nicorovici, N. A.

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London, Ser. A462,3027–3059 (2006).
[CrossRef]

Nirenberg, L.

L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953).
[CrossRef]

Norris, A.

A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008).
[CrossRef]

Oleinik, O. A.

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994).
[CrossRef]

Pendry, J. B.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Shen, H.

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
[CrossRef]

Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Sun, C.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).

Vogelius, M. S.

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
[CrossRef]

Weinstein, M. I.

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
[CrossRef]

Willis, J. R.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006).
[CrossRef]

Zhang, S.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Zolla, F.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008).
[CrossRef]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009).
[CrossRef]

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).

Commun. Math. Phys. (1)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007).
[CrossRef]

Commun. Pure Appl. Math. (1)

L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953).
[CrossRef]

Inverse Probl. (1)

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008).
[CrossRef]

J. Opt. (1)

A. Diatta and S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011).
[CrossRef]

Math. Res. Lett. (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

New J. Phys. (3)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008).
[CrossRef]

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007).
[CrossRef]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Letters (1)

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008).
[CrossRef]

Photonics Nanostruct. Fundam. Appl. (1)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Phys. Rev. B (1)

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009).
[CrossRef]

Phys. Rev. E (1)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Proc. R. Soc. London (1)

A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008).
[CrossRef]

Science (3)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
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Other (4)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London, Ser. A462,3027–3059 (2006).
[CrossRef]

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994).
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A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).

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Figures (4)

Fig. 1
Fig. 1

Principle of an invisibility cloak and a concentrator for heat flux. (a) Sketch of the transformed metric (dotted lines) for an invisibility cloak of inner and outer boundaries described by radii R1(θ) and R2(θ) depending upon the angular variable θ. The metamaterial consists of a coating (in blue) with anisotropic heterogeneous conductivity as described in section 3.1. The invisibility region is shown in red; (b) Sketch of the transformed metric (dotted lines) for a concentrator of inner and outer boundaries described by radii R1(θ) and R3(θ) depending upon the angular variable θ. The boundary R2(θ) (in pink) is virtual and only serves the purpose of the geometric transform. The metamaterial consists of an inner region (in red) and a coating (in blue) filled with anisotropic heterogeneous conductivities as described in section 3.2.

Fig. 2
Fig. 2

Diffusion of heat from the left on a cloak with R1 = 2.10−4m and R2 = 3.10−4m. The temperature is normalized throughout time on the left side of the cell. Snapshots of temperature distribution at t = 0.001s (a), t = 0.005s (b), t = 0.02s (c), t = 0.05s (d). Streamlines of thermal flux are also represented with white color in panel (d). The mesh formed by streamlines and isothermal values illustrates the deformation of the transformed thermal space: the central disc (‘invisibility region’) is a hole in the metric, which is curved smoothly around it.

Fig. 3
Fig. 3

Diffusion of heat from the left on a concentrator with R1 = 1.10−4m, R2 = 2.10−4m and R3 = 3.10−4m. The temperature is normalized throughout time on the left side of the cell. Snapshots of heat distribution at t = 0.002s (a), t = 0.005s (b), t = 0.01s (c) and t = 0.02s (d). Streamlines of thermal flux are also represented with white color in panel (d). The mesh formed by streamlines and isothermal values illustrates the deformation of the transformed thermal space: it is squeezed into the central disc.

Fig. 4
Fig. 4

Diffusion of heat from the left on a multilayered cloak of inner radius R1 = 1.5 10−4m and outer radius R2 = 3 10−4m, consisting of 20 homogeneous layers of equal thickness and of respective diffusivities (in unit of 10−5.m2.s−1) 1680.70, 0.25, 80.75, 0.25, 29.39, 0.25, 16.37, 0.25, 10.99, 0.25, 8.18, 0.25, 6.50, 0.25, 5.40, 0.25, 4.63, 0.25, 4.06,0.25 in a bulk material of diffusivity 10−5m2.s−1 (inner disc and homogeneous region outside the concentric layers). Importantly, one layer in two has the same diffusivity. Snapshots of heat distribution at t = 0.01s (a) and t = 0.05s (b) show that normalized temperature is non vanishing inside the inner disc (invisibility region), but it never exceeds a half of the maximum value of temperature (even for t > 0.05s). We note the similarities with Fig. 2.

Equations (25)

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ρ ( x ) c ( x ) u t = ( κ ( x ) u ) + p ( x , t ) ,
ρ ( x ) c ( x ) det ( J ) u t = ( J T κ ( x ) J 1 det ( J ) u ) + det ( J ) p ( x , t ) .
κ _ _ = J T κ J 1 det ( J ) = κ J T J 1 det ( J ) = κ T 1 ,
Ω ρ c u t ϕ dxdy + Ω ( ϕ κ u ) dxdy Ω ( κ u n ϕ κ ϕ n u ) dl + < p , ϕ > = 0 ,
Ω { ( ρ c u t ϕ ) det ( J ) } d x d y + Ω { ( J 1 ϕ κ J 1 u ) det ( J ) } d x d y Ω { ( κ J 1 u n ϕ κ J 1 ϕ n u ) det ( J ) } d l + < det ( J ) p , ϕ > = 0
{ r = α ( θ ) r + β ( θ ) θ = θ
T 1 = R ( θ ) ( ( r β ) 2 + c 12 2 . α 2 ( r β ) r c 12 . α r β c 12 α r β r r β ) R ( θ ) T ,
c 12 = 1 ( R 2 ( θ ) R 1 ( θ ) ) 2 { d R 1 ( θ ) d θ ( R 2 ( θ ) r ) ( R 2 ( θ ) R 1 ( θ ) ) 2 R 2 ( θ ) + d R 2 ( θ ) d θ R 1 ( θ ) ( R 1 ( θ ) r ) }
J x x = J x r J r r J r x = R ( θ ) diag ( 1 , r ) diag ( α 1 , 1 ) diag(1 , 1 / r ) R ( θ ) = R ( θ ) diag ( α 1 , r / r ) R ( θ ) ,
T 1 = J x x T J x x 1 det ( J x x ) = R ( θ ) T diag ( α , r / r ) R ( θ ) T R ( θ ) 1 diag( α , r / r ) R ( θ ) 1 α 1 r / r = R ( θ ) diag ( α r / r , α 1 r / r ) R ( θ ) 1 ,
κ _ _ = κ T 1 = R ( θ ) diag( κ r ' , κ θ ' ) R ( θ ) T ,
κ r ' = r R 1 r , κ θ ' = r r R 1 ,
ρ c = det ( J r r ) ρ c = r r ρ c α = r R 1 r ( R 2 R 2 R 1 ) 2 ( ρ c ) ,
c 12 = r θ = r R 1 ( θ ) 2 ( R 2 ( θ ) R 1 ( θ ) θ R 1 ( θ ) R 2 ( θ ) θ ) ,
c 12 = 1 ( R 2 ( θ ) R 1 ( θ ) ) 2 { R 1 ( θ ) θ ( R 3 ( θ ) r ) ( R 3 ( θ ) R 2 ( θ ) ) + R 2 ( θ ) θ ( R 1 ( θ ) R 3 ( θ ) ) ( R 3 ( θ ) r ) R 3 ( θ ) θ ( R 2 ( θ ) R 1 ( θ ) ) ( R 1 ( θ ) r ) } .
κ r ' = 1 , κ θ ' = 1 , if  0 r R 2 κ r ' = r + R 3 R 2 R 1 R 3 R 2 r , κ θ ' = r r + R 3 R 2 R 1 R 3 R 2 , if  R 2 r R 3
ρ c = det ( J ) ρ c = { ( R 2 R 1 ) 2 ρ c , if 0 r R 1 , r + R 3 R 2 R 1 R 3 R 2 r ( R 3 R 2 R 3 R 1 ) 2 ρ c , if R 1 r R 3 ,
κ r = ( R 2 R 2 R 1 ) 2 ( r R 1 r ) 2 , κ θ " = ( R 2 R 2 R 1 ) 2 ,
( κ _ _ u ) = det ( J ) ( κ _ _ u ) + κ _ _ ( det ( J ) u ) .
ρ η c η u η t = ( κ η u η ) ,
u η ( x ) = i = 0 η i u ( i ) ( x , x η ) ,
< ρ c > u 0 t = ( κ hom _ _ u 0 ) ,
κ hom _ _ = Diag ( < κ 1 > 1 , < κ > ) .
1 κ r = 1 1 + η ( 1 κ A + η κ B ) , κ θ = κ A + η κ B 1 + η , < ρ c > = ρ A c A + η ρ B c B 1 + η ,
det ( J ) ( i h ¯ Ψ t ) = h ¯ 2 2 ( J T m 1 ( x ) J 1 det ( J ) Ψ ) + det ( J ) V Ψ ,

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