Abstract

Transformation optics is a powerful concept for designing novel optical components such as high transmission waveguides and cloaking devices. The selection of specific transformations is a non-unique problem. Here we reveal that transformations which allow for all dielectric and broadband optical realizations correspond to minimizers of elastic energy potentials for extreme values of the mechanical Poisson’s ratio ν. For TE (Hz) polarized light an incompressible transformation ν=12 is ideal and for TM (Ez) polarized light one should use a compressible transformation with negative Poissons’s ratio ν = −1. For the TM polarization the mechanical analogy corresponds to a modified Liao functional known from the transformation optics literature. Finally, the analogy between ideal transformations and solid mechanical material models automates and broadens the concept of transformation optics.

© 2010 OSA

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  5. 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]
  6. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 314, 1780–1782 (2006).
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  7. J. B. Pendry, “Taking the wraps off cloaking,” Physics 2, 95-1 – 95-62009.
    [Crossref]
  8. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
    [Crossref]
  9. D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (2008).
    [Crossref]
  10. M. Rahm, D. R. Smith, and D. A. Schurig, “Transformation-optical design of reconfigurable optical devices, Patent application publication no.: US 2009/0147342 A1, 14 pages, http://www.freepatentsonline.com/y2009/0147342.html (2009).
  11. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  15. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105-1 – 241105-3 (2007)
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  24. G. W. Minton, The Theory of Composites, (Cambridge University Press, Cambridge, 2002).
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    [Crossref]
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2009 (7)

J. B. Pendry, “Taking the wraps off cloaking,” Physics 2, 95-1 – 95-62009.
[Crossref]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
[Crossref] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[Crossref] [PubMed]

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17, 14872–14879 (2009).
[Crossref] [PubMed]

J. Hu, X. Zhoua, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Computational Materials Science 46, 708–712 (2009).
[Crossref]

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Physical Review 79, 033102-1 – 033102-4 (2009).

M. Brun, S. Guenneau, and A. B. Movchan2, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903-1 – 061903-3 (2009).
[Crossref]

2008 (5)

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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref] [PubMed]

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008).
[Crossref]

2007 (1)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105-1 – 241105-3 (2007)

2006 (6)

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New Journal of Physics 8, 247-1 – 247-18 (2006).
[Crossref]

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777 – 1780 (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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 314, 1780–1782 (2006).
[Crossref]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New Journal of Physics 8, 248-1 – 248-20 (2006).
[Crossref]

2000 (1)

O. Sigmund, “A new class of extremal composites,” Journal of the Mechanics and Physics of solids 48, 397–428 (2000).
[Crossref]

1998 (1)

B. Hassani and E. Hinton, “A review of homogenization and topology optimization I-homogenization theory for media with periodic structure,” Computers & Structures,  69, 707–717 (1998).
[Crossref]

1997 (1)

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio,” Journal of Micromechanical Systems 6, 99–106 (1997).
[Crossref]

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

1994 (1)

O. Sigmund, “Materials with prescribed constitutive parameters: an inverse homogenization problem,” International Journal of Solids and Structures,  31, 2313–2329 (1994).
[Crossref]

1987 (1)

R. Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science 235, 1038–1040 (1987).
[Crossref] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
[Crossref] [PubMed]

Belytschko, T.

T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

Bouwstra, S.

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio,” Journal of Micromechanical Systems 6, 99–106 (1997).
[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 Journal of Physics 8, 248-1 – 248-20 (2006).
[Crossref]

Brun, M.

M. Brun, S. Guenneau, and A. B. Movchan2, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903-1 – 061903-3 (2009).
[Crossref]

Chan, C. T.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105-1 – 241105-3 (2007)

Chen, H.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105-1 – 241105-3 (2007)

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[Crossref] [PubMed]

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[Crossref] [PubMed]

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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[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]

Enoch, S.

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Physical Review 79, 033102-1 – 033102-4 (2009).

Farhat, M.

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Physical Review 79, 033102-1 – 033102-4 (2009).

Guenneau, S.

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Physical Review 79, 033102-1 – 033102-4 (2009).

M. Brun, S. Guenneau, and A. B. Movchan2, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903-1 – 061903-3 (2009).
[Crossref]

Hassani, B.

B. Hassani and E. Hinton, “A review of homogenization and topology optimization I-homogenization theory for media with periodic structure,” Computers & Structures,  69, 707–717 (1998).
[Crossref]

Hinton, E.

B. Hassani and E. Hinton, “A review of homogenization and topology optimization I-homogenization theory for media with periodic structure,” Computers & Structures,  69, 707–717 (1998).
[Crossref]

Hu, G.

J. Hu, X. Zhoua, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Computational Materials Science 46, 708–712 (2009).
[Crossref]

Hu, J.

J. Hu, X. Zhoua, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Computational Materials Science 46, 708–712 (2009).
[Crossref]

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[Crossref] [PubMed]

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]

Knupp, P. M.

P. M. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC-Press, 1993, ISBN 978-0849-38987-0).

Lakes, R.

R. Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science 235, 1038–1040 (1987).
[Crossref] [PubMed]

Landy, N. I.

Larsen, U. D.

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio,” Journal of Micromechanical Systems 6, 99–106 (1997).
[Crossref]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New Journal of Physics 8, 247-1 – 247-18 (2006).
[Crossref]

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777 – 1780 (2006).
[Crossref] [PubMed]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
[Crossref] [PubMed]

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008).
[Crossref]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[Crossref] [PubMed]

Liu, W. K.

T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

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 Journal of Physics 8, 248-1 – 248-20 (2006).
[Crossref]

Minton, G. W.

G. W. Minton, The Theory of Composites, (Cambridge University Press, Cambridge, 2002).

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[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]

Moran, B.

T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

Movchan, A. B.

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Physical Review 79, 033102-1 – 033102-4 (2009).

Movchan2, A. B.

M. Brun, S. Guenneau, and A. B. Movchan2, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903-1 – 061903-3 (2009).
[Crossref]

Padilla, W. J.

Pendry, J. B.

J. B. Pendry, “Taking the wraps off cloaking,” Physics 2, 95-1 – 95-62009.
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[Crossref]

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 314, 1780–1782 (2006).
[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]

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New Journal of Physics 8, 247-1 – 247-18 (2006).
[Crossref]

Rahm, M.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref] [PubMed]

M. Rahm, D. R. Smith, and D. A. Schurig, “Transformation-optical design of reconfigurable optical devices, Patent application publication no.: US 2009/0147342 A1, 14 pages, http://www.freepatentsonline.com/y2009/0147342.html (2009).

Roberts, D. A.

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref] [PubMed]

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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 314, 1780–1782 (2006).
[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]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

Schurig, D. A.

M. Rahm, D. R. Smith, and D. A. Schurig, “Transformation-optical design of reconfigurable optical devices, Patent application publication no.: US 2009/0147342 A1, 14 pages, http://www.freepatentsonline.com/y2009/0147342.html (2009).

Sigmund, O.

O. Sigmund, “A new class of extremal composites,” Journal of the Mechanics and Physics of solids 48, 397–428 (2000).
[Crossref]

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio,” Journal of Micromechanical Systems 6, 99–106 (1997).
[Crossref]

O. Sigmund, “Materials with prescribed constitutive parameters: an inverse homogenization problem,” International Journal of Solids and Structures,  31, 2313–2329 (1994).
[Crossref]

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
[Crossref] [PubMed]

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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[Crossref]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008).
[Crossref] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 314, 1780–1782 (2006).
[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]

M. Rahm, D. R. Smith, and D. A. Schurig, “Transformation-optical design of reconfigurable optical devices, Patent application publication no.: US 2009/0147342 A1, 14 pages, http://www.freepatentsonline.com/y2009/0147342.html (2009).

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]

Steinberg, S.

P. M. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC-Press, 1993, ISBN 978-0849-38987-0).

Taylor, R. L.

O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, vol 2., 5th edition, (Butterworth Heinemann, Oxford, 2000).

Torquato, S.

S. Torquato, Random Heterogeneous Materials, (Springer, New York, 2001).

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
[Crossref] [PubMed]

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[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 Journal of Physics 8, 248-1 – 248-20 (2006).
[Crossref]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
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J. Hu, X. Zhoua, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Computational Materials Science 46, 708–712 (2009).
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O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, vol 2., 5th edition, (Butterworth Heinemann, Oxford, 2000).

Appl. Phys. Lett. (3)

D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (2008).
[Crossref]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105-1 – 241105-3 (2007)

M. Brun, S. Guenneau, and A. B. Movchan2, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903-1 – 061903-3 (2009).
[Crossref]

Computational Materials Science (1)

J. Hu, X. Zhoua, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Computational Materials Science 46, 708–712 (2009).
[Crossref]

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J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
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O. Sigmund, “A new class of extremal composites,” Journal of the Mechanics and Physics of solids 48, 397–428 (2000).
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Nature Materials (1)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009).
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New Journal of Physics (2)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New Journal of Physics 8, 247-1 – 247-18 (2006).
[Crossref]

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New Journal of Physics 8, 248-1 – 248-20 (2006).
[Crossref]

Opt. Express (3)

Photonics and Nanostructures Fundamentals and Applications (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 Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008).
[Crossref]

Phys. Rev. Lett. (2)

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008).
[Crossref]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008).
[Crossref]

Physical Review (1)

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Physical Review 79, 033102-1 – 033102-4 (2009).

Physics (1)

J. B. Pendry, “Taking the wraps off cloaking,” Physics 2, 95-1 – 95-62009.
[Crossref]

Science (5)

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777 – 1780 (2006).
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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 314, 1780–1782 (2006).
[Crossref]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009).
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Other (6)

P. M. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC-Press, 1993, ISBN 978-0849-38987-0).

G. W. Minton, The Theory of Composites, (Cambridge University Press, Cambridge, 2002).

T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, vol 2., 5th edition, (Butterworth Heinemann, Oxford, 2000).

S. Torquato, Random Heterogeneous Materials, (Springer, New York, 2001).

M. Rahm, D. R. Smith, and D. A. Schurig, “Transformation-optical design of reconfigurable optical devices, Patent application publication no.: US 2009/0147342 A1, 14 pages, http://www.freepatentsonline.com/y2009/0147342.html (2009).

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Figures (8)

Fig. 1.
Fig. 1.

Field plots of the magnetic field Hz for simple and complicated geometries with and without transformation optics. Perfect transmission in the simple geometry ω with frame of reference xi (a). Response in the complicated geometry Ω with frame of reference Xi = xi + ui with transformation optics and excellent transmission (b). Response in the complicated geometry Ω without transformation optics and a resulting poor transmission (c). The colorbar defines the amplitude of the wave.

Fig. 2.
Fig. 2.

Response for TE (Hz ) polarized light in the complicated geometry (a) where only the impermittiviy η is transformed and the transformation is the minimizer of the elastic energy potential for a nearly incompressible material with K/G = 1000. The error measure ∣Hz H ref z ∣/∣H ref z ∣ where H ref z is the response where both η and μ 33 are transformed (b). The colorbars define the amplitude of the wave and the error measure, respectively.

Fig. 3.
Fig. 3.

Anisotropy of ε = (η′)−1 illustrated with crosses oriented in the principal directions (a). The relative difference between the principal directions (b) defined as ( q 1 q 2 ) ( 1 2 ( q 1 + q 2 ) ) where qi are the eigenvalues. In the straight entrance and exit of the geometry the permittivity is isotropic with a value of ε inout = 3 such that the eigenvalues in Ω satisfy 1.31 ≤ q 2q 1 ≤ 6.89. The transformation is nearly incompressible and governed by Eq. (7).

Fig. 4.
Fig. 4.

Response for TM (Ez ) polarized light in the complicated geometry (a) where only ε 33 is transformed and the transformation is the minimizer of the elastic energy potential for a negative Poisson’s ratio material with G/K = 1000 cf. Eq. (12). The relative difference between the (ε33ε inout)/ε inout where ε inout = 2 is the background permittivity of the straight entrance and exit of the geometry (b). The transformed permittivity belongs to the interval 1.25 ≤ ε33 ≤ 10.2.

Fig. 5.
Fig. 5.

Response for TM (Ez ) polarized light in the complicated geometry (a) where only ε 33 is transformed and the transformation is the minimizer of the modified Liao functional cf. Eq. (13). The relative difference between the (ε33ε inout)/ε inout where ε inout = 2 is the background permittivity of the straight entrance and exit of the geometry (b). The transformed permittivity belongs to the interval 1.20 ≤ ε33 ≤ 8.92.

Fig. 6.
Fig. 6.

A carpet cloaking example [12] where the transformation is generated by the modified Liao functional (a) with the amplitude of the wave given in the colorbar. The spatial variation of the permittivity (b), illustrated by the measure (ε33ε inout)/ε inout where ε inout = 2.

Fig. 7.
Fig. 7.

A carpet cloaking example [12] where the transformation is generated by the elastic energy potential for a negative Poisson’s ratio material with G/K = 1000 (a) with the amplitude of the wave given in the colorbar. The spatial variation of the permittivity (b), illustrated by the measure (ε33ε inout)/ε inout where ε inout = 2.

Fig. 8.
Fig. 8.

Response for TE (Hz ) polarized light in the complicated geometry where only η is transformed (a) and without transformation optics and ε 11 = ε 22 = 4 (b). The principal directions of ε = (η′)−1 are shown (c) and the relative difference between the eigenvalues ( q 1 q 2 ) ( 1 2 ( q 1 + q 2 ) ) where qi are the eigenvalues (d). The transformation is the minimizer of the elastic energy potential for a nearly incompressible material with K/G=1000. The relative error measure of the amplitude is ∣max∣Hz ∣ − ∣H ref z ∣∣/∣H ref z ∣ = 1.2%. In the straight entrance and exit of the geometry the permittivity has been scaled to ε inout = 4 such that the eigenvalues of ε in Ω satisfy 3.05 ≤ q 2q 1 ≤ 5.25. The colorbars define the amplitude of the waves and the relative difference between the eigenvalues, respectively.

Equations (16)

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ω ( ( ν · η H z ) + k 0 2 μ 33 ν H z ) d x = 0 ν 𝒲
Ω ( ( x v · η x H z ) + k 0 2 μ 33 v H z ) d X = 0 v 𝒲
Ω ( ( Χ v · ( Λ η Λ T ) Χ H z ) + k 0 2 μ 33 v H z ) d X det ( Λ ) = 0 v 𝒲
η = Λ η Λ T det ( Λ ) , μ 33 = μ 33 det ( Λ )
Ω ( ( X v · ( Λ B Λ T ) X E z ) + k 0 2 ε 33 v E z ) d X det ( Λ ) = 0 v 𝒲
B = Λ B Λ T det ( Λ ) , ε 33 = ε 33 det ( Λ ) .
ψ = ω ( 1 2 G ( I 1 2 2 ln J ) + 1 2 K ( J 1 ) 2 ) d x
r q = q 1 q 2 1 2 ( q 1 + q 2 )
Λ Λ T = V 1 [ γ 1 0 0 γ 2 ] V
det ( Λ Λ T ) = det ( Λ T Λ ) = γ 1 γ 2 = J 2 , trace ( Λ Λ T ) = I 1 = γ 1 + γ 2 .
( γ 1 γ 2 ) 2 = γ 1 2 + γ 2 2 2 γ 1 γ 2 = I 1 2 4 J 2 .
ψ = ω ( 1 8 G ( I 1 2 4 J 2 ) + 1 8 K ( I 1 2 ) 2 ) d x
Φ = 1 A ω ( I 1 J 2 ) 2 d x , A = ω 1 d x
Φ = ( γ 1 γ 2 J ) 2 + 4 .
e rel = Γ ˜ E z F z 2 d s Γ ˜ F z 2 d s
q 1 = m ε + ( 1 m ) ε + , q 2 = ( m ε + ( 1 m ) ε + ) 1

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