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 Optical Society of America

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  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).
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2009

. 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,” Nat. Mater. 8(7), 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(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. Hu, X. Zhou, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. 46(3), 708–712 (2009).
[CrossRef]

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

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

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

2008

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(15), 11555–11567 (2008).
[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 Nanostruct. Fundam. Appl. 6(1), 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]

. 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

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

2006

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 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 J. Phys. 8, 248–1 – 248–20 (2006).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

2000

O. Sigmund, “A new class of extremal composites,” J. Mech. Phys. Solids 48(2), 397–428 (2000).
[CrossRef]

1998

B. Hassani, and E. Hinton, “A review of homogenization and topology optimization I-homogenization theory for media with periodic structure,” Comput. Struc. 69(6), 707–717 (1998).
[CrossRef]

1997

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio,” J. Micromech. Syst. 6(2), 99–106 (1997).
[CrossRef]

1996

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

1994

O. Sigmund, “Materials with prescribed constitutive parameters: an inverse homogenization problem,” Int. J. Solids Struct. 31(17), 2313–2329 (1994).
[CrossRef]

1987

R. Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science 235(4792), 1038–1040 (1987).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Bouwstra, S.

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio,” J. Micromech. Syst. 6(2), 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 J. Phys. 8, 248–1 – 248–20 (2006).
[CrossRef]

Brun, M.

. M. Brun, S. Guenneau and A. B. Movchan, “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(5912), 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(5912), 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 Nanostruct. Fundam. Appl. 6(1), 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(5801), 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,” Phys. Rev. 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,” Phys. Rev. 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,” Phys. Rev. 79, 033102–1 – 033102–4 (2009).

. M. Brun, S. Guenneau and A. B. Movchan, “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,” Comput. Struc. 69(6), 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,” Comput. Struc. 69(6), 707–717 (1998).
[CrossRef]

Hu, G.

J. Hu, X. Zhou, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. 46(3), 708–712 (2009).
[CrossRef]

Hu, J.

J. Hu, X. Zhou, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. 46(3), 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(5912), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Lakes, R.

R. Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science 235(4792), 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,” J. Micromech. Syst. 6(2), 99–106 (1997).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 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(5912), 366–369 (2009).
[CrossRef] [PubMed]

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, 248–1 – 248–20 (2006).
[CrossRef]

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(5912), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Movchan, A. B.

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

. M. Brun, S. Guenneau and A. B. Movchan, “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]

. 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(15), 11555–11567 (2008).
[CrossRef] [PubMed]

. 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]

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 Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (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(5801), 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]

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]

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(15), 11555–11567 (2008).
[CrossRef] [PubMed]

. 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 Nanostruct. Fundam. Appl. 6(1), 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 Maxwells equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 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, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

Schurig, D.

. 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. 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 Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Sigmund, O.

O. Sigmund, “A new class of extremal composites,” J. Mech. Phys. Solids 48(2), 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,” J. Micromech. Syst. 6(2), 99–106 (1997).
[CrossRef]

O. Sigmund, “Materials with prescribed constitutive parameters: an inverse homogenization problem,” Int. J. Solids Struct. 31(17), 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(5912), 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 Nanostruct. Fundam. Appl. 6(1), 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]

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(15), 11555–11567 (2008).
[CrossRef] [PubMed]

. 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]

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(5801), 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(21), 9794–9804 (2006).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef]

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(5801), 977–980 (2006).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
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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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. 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, 248–1 – 248–20 (2006).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhou, X.

J. Hu, X. Zhou, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. 46(3), 708–712 (2009).
[CrossRef]

Appl. Phys. Lett.

. 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. Brun, S. Guenneau and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903–1 – 061903–3 (2009).
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Comput. Mater. Sci.

J. Hu, X. Zhou, and G. Hu, “A numerical method for designing acoustic cloak with arbitrary shapes,” Comput. Mater. Sci. 46(3), 708–712 (2009).
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A. J. Ward, and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

New J. Phys.

. 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, 248–1 – 248–20 (2006).
[CrossRef]

Opt. Express

Photonics Nanostruct. Fundam. Appl.

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 Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[CrossRef]

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

Phys. Rev. Lett.

. 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]

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U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 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(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312(5781), 1780–1782 (2006).
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[CrossRef] [PubMed]

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

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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)

Equations on this page are rendered with MathJax. Learn more.

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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