Abstract

We analyze the wave propagation in two-dimensional bianisotropic media with the Finite Element Method (FEM). Starting from the Maxwell-Tellegen’s equations in bianisotropic media, we derive some system of coupled Partial Differential Equations (PDEs) for longitudinal electric and magnetic field components. These PDEs are implemented in FEM using a solid mechanics formulation. Perfectly Matched Layers (PMLs) are also discussed to model unbounded bianisotropic media. The PDEs and PMLs are then implemented in a finite element software, and transformation optics is further introduced to design some bianisotropic media with interesting functionalities, such as cloaks, concentrators and rotators. In addition, we propose a design of metamaterial with concentric layers made of homogeneous media with isotropic permittivity, permeability and magnetoelectric parameters that mimic the required effective anisotropic tensors of a bianisotropic cloak in the long wavelength limit (homogenization approach). Our numerical results show that transformation based electromagnetic metamaterials can be extended to bianisotropic media.

© 2016 Optical Society of America

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References

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  26. Y. O. Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to microstructured optical fibres,” COMPEL 27(1), 95–109 (2008).
    [Crossref]
  27. W. C. Chew and W. H. Weedon, “A 3D perfectly mathched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7(13), 599–604 (1994).
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2015 (1)

M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

2014 (1)

2013 (2)

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

2010 (2)

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Eletron. 16(2), 367–375 (2010).
[Crossref]

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18(11), 12027–12032 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (3)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s 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 cooridnate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[Crossref] [PubMed]

Y. O. Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to microstructured optical fibres,” COMPEL 27(1), 95–109 (2008).
[Crossref]

2007 (3)

Y. Huang, Y. J. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15(18), 11133–11141 (2007).
[Crossref] [PubMed]

S. Guenneau and F. Zolla, “Homogenization of 3D finite chiral photonic crystals,” Physica B: Condens. Matter 394(2), 145–147 (2007).
[Crossref]

H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[Crossref]

2006 (3)

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100(2), 024507 (2006).
[Crossref]

2004 (1)

E. Bossy, M. Talmant, and P. Laugier, “Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models,” J. Acoust. Soc. Am. 115(1), 2314–2324 (2004).
[Crossref] [PubMed]

2002 (1)

R. Marques, M. Francisco, and R. Rachid, “Role of bianisotropy in negative permeability and left-handed metamate-rials,” Phys. Rev. B 65(65), 144440 (2002).
[Crossref]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

1998 (2)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett. 8(6), 223–225 (1998).
[Crossref]

F. L. Teixeira and W. C. Chew, “Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves,” Microwave Opt. Technol. Lett. 17(4), 231–236 (1998).
[Crossref]

1997 (1)

F. L. Teixeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microw. Guided Wave Lett. 7(9), 285–287 (1997).
[Crossref]

1995 (1)

G. W. Milton and A. B. Movchan, “A correspondence between plane elasticity and the two-dimensional real and complex dielectric equations in anisotropic media,” Proc. R. Soc. A 450, 293–317 (1995).
[Crossref]

1994 (2)

W. C. Chew and W. H. Weedon, “A 3D perfectly mathched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

1968 (1)

M. Zlamal, “On the finite element method,” Numer. Math. 12(5), 394–409 (1968).
[Crossref]

1943 (1)

R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Amer. Math. Soc. 49, 1–23 (1943).
[Crossref]

Agha, Y. O.

Y. O. Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to microstructured optical fibres,” COMPEL 27(1), 95–109 (2008).
[Crossref]

Alexander, A.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

Bossy, E.

E. Bossy, M. Talmant, and P. Laugier, “Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models,” J. Acoust. Soc. Am. 115(1), 2314–2324 (2004).
[Crossref] [PubMed]

Buckmann, T.

M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

Chan, C. T.

H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[Crossref]

Chen, H. Y.

H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[Crossref]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett. 8(6), 223–225 (1998).
[Crossref]

F. L. Teixeira and W. C. Chew, “Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves,” Microwave Opt. Technol. Lett. 17(4), 231–236 (1998).
[Crossref]

F. L. Teixeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microw. Guided Wave Lett. 7(9), 285–287 (1997).
[Crossref]

W. C. Chew and W. H. Weedon, “A 3D perfectly mathched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Courant, R.

R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Amer. Math. Soc. 49, 1–23 (1943).
[Crossref]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s 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 cooridnate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[Crossref] [PubMed]

Didier, F.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

Enoch, S.

Feng, Y. J.

Francisco, M.

R. Marques, M. Francisco, and R. Rachid, “Role of bianisotropy in negative permeability and left-handed metamate-rials,” Phys. Rev. B 65(65), 144440 (2002).
[Crossref]

Freymann, G.

Gollub, J.

D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100(2), 024507 (2006).
[Crossref]

Gralak, B.

Y. Liu, B. Gralak, R. C. McPhedran, and S. Guenneau, “Finite frequency external cloaking with coplementary bianisotropic media,” Opt. Express 22(14), 17387–17402 (2014).
[Crossref] [PubMed]

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

Grzegorczyk, T. M.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials, 1st ed. (CRC Press, 2008).
[Crossref]

Guenneau, S.

Y. Liu, B. Gralak, R. C. McPhedran, and S. Guenneau, “Finite frequency external cloaking with coplementary bianisotropic media,” Opt. Express 22(14), 17387–17402 (2014).
[Crossref] [PubMed]

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18(11), 12027–12032 (2010).
[Crossref] [PubMed]

Y. O. Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to microstructured optical fibres,” COMPEL 27(1), 95–109 (2008).
[Crossref]

S. Guenneau and F. Zolla, “Homogenization of 3D finite chiral photonic crystals,” Physica B: Condens. Matter 394(2), 145–147 (2007).
[Crossref]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

Huang, Y.

Jiang, T.

Kadic, M.

M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18(11), 12027–12032 (2010).
[Crossref] [PubMed]

Kern, C.

M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

Kriegler, C. E.

Kuhlmey, B.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

Laugier, P.

E. Bossy, M. Talmant, and P. Laugier, “Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models,” J. Acoust. Soc. Am. 115(1), 2314–2324 (2004).
[Crossref] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Linden, S.

Liu, Y.

Y. Liu, B. Gralak, R. C. McPhedran, and S. Guenneau, “Finite frequency external cloaking with coplementary bianisotropic media,” Opt. Express 22(14), 17387–17402 (2014).
[Crossref] [PubMed]

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Marques, R.

R. Marques, M. Francisco, and R. Rachid, “Role of bianisotropy in negative permeability and left-handed metamate-rials,” Phys. Rev. B 65(65), 144440 (2002).
[Crossref]

McPhedran, R. C.

Milton, G. W.

G. W. Milton and A. B. Movchan, “A correspondence between plane elasticity and the two-dimensional real and complex dielectric equations in anisotropic media,” Proc. R. Soc. A 450, 293–317 (1995).
[Crossref]

Mock, J. J.

D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100(2), 024507 (2006).
[Crossref]

Movchan, A. B.

G. W. Milton and A. B. Movchan, “A correspondence between plane elasticity and the two-dimensional real and complex dielectric equations in anisotropic media,” Proc. R. Soc. A 450, 293–317 (1995).
[Crossref]

Nicolet, A.

Y. O. Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to microstructured optical fibres,” COMPEL 27(1), 95–109 (2008).
[Crossref]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

Padilla, W. J.

D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100(2), 024507 (2006).
[Crossref]

Pendry, J. B.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded cooridnate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[Crossref] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[Crossref]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Rachid, R.

R. Marques, M. Francisco, and R. Rachid, “Role of bianisotropy in negative permeability and left-handed metamate-rials,” Phys. Rev. B 65(65), 144440 (2002).
[Crossref]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s 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 cooridnate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[Crossref] [PubMed]

Ramakrishna, S. A.

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials, 1st ed. (CRC Press, 2008).
[Crossref]

Renversez, G.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

Rill, M. S.

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[Crossref]

Schittny, R.

M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

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 cooridnate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[Crossref] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
[Crossref]

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F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
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Shurig, D.

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Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
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M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded cooridnate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
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D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100(2), 024507 (2006).
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M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Eletron. 16(2), 367–375 (2010).
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S. Guenneau and F. Zolla, “Homogenization of 3D finite chiral photonic crystals,” Physica B: Condens. Matter 394(2), 145–147 (2007).
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F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
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Y. O. Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to microstructured optical fibres,” COMPEL 27(1), 95–109 (2008).
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IEEE J. Sel. Top. Quantum Eletron. (1)

C. E. Kriegler, M. S. Rill, S. Linden, and M. Wegener, “Bianisotropic photonic metamaterials,” IEEE J. Sel. Top. Quantum Eletron. 16(2), 367–375 (2010).
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IEEE Microw. Guided Wave Lett. (2)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett. 8(6), 223–225 (1998).
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E. Bossy, M. Talmant, and P. Laugier, “Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models,” J. Acoust. Soc. Am. 115(1), 2314–2324 (2004).
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D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator metamaterial,” J. Appl. Phys. 100(2), 024507 (2006).
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[Crossref]

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Opt. Express (3)

Opt. Lett. (1)

Photonics Nanostruct. Fundam. Appl. (1)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromangetic cloaks and concetrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008).
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M. Kadic, R. Schittny, T. Buckmann, C. Kern, and M. Wegener, “Hall-effect sign inversion in a realizable 3D metamaterial,” Phys. Rev. X 5, 021030 (2015).

Physica B: Condens. Matter (1)

S. Guenneau and F. Zolla, “Homogenization of 3D finite chiral photonic crystals,” Physica B: Condens. Matter 394(2), 145–147 (2007).
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Science (2)

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Other (3)

R. M. Walser, “Electromagnetic metamaterials,” Proc. SPIE4467, 1–15 (2001).
[Crossref]

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials, 1st ed. (CRC Press, 2008).
[Crossref]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, F. Didier, A. Alexander, and L. S. Sergio, Foundations of Photonic Crystal Fibres, 2nd ed. (Imperial College Press, 2012).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 (a) Schematic diagram of bianisotropic cloak with a L-shaped obstacle inside the cloaking region, and a point source (s-polarized with the electric field along z with f = 8.7 GHz) locates outside the shell. Plots of Re(Ez) with the same arbitrary units for the three following panels: (b) A reference point source in a pure matrix with parameters ε = ε0I3, µ = µ0I3 and ξ = ζ = 0.99/c0I3; (c) Same point source in the matrix with a presence of L-shaped obstacle, the parameters of obstacle are ε = (1 + 5 × i) ε0I3, µ = µ0I3 and ξ = ζ = 0.99/c0I3; (d) Same point source radiates in the presence of an L-shaped obstacle surrounded by an invisibility cloak.
Fig. 2
Fig. 2 (a) Schematic diagram of a bianisotropic concentrator; (b) Plot of Re(Ez) under an s-polarized incidence (amplitude normalized to unity) with frequency f = 8.7 GHz radiating from the above, the parameters in the region r′ ≤ R1, and the annulus R1 < r′ ≤ R3 are given by Eq. (52) with transformation matrix T x x 1 in Eq. (55).
Fig. 3
Fig. 3 (a) Schematic diagram of the bianisotropic rotator; (b) Plot of Re(Ez) under an s-polarized incidence (amplitude normalized to unity) with frequency f = 8.7 GHz radiating from the above, the parameters in the region r′ ≤ R1, annulus R1 < r′ ≤ R2 are given by Eq. (52) with transformation matrix T x x 1 in Eq. (58).
Fig. 4
Fig. 4 (a) Scattering of an infinite conducting obstacle (of radius 14 mm) by a point source at a frequency 5.7 GHz; (b) Scattering like in (a) when the obstacle is surrounded by a layered cloak; (c) Scattering like in (a) for a point source at a frequency 7.7 GHz; (d) Scattering like in (b) for a point source at a frequency 7.7 GHz. Color scale corresponds to Re(Ez) (the same arbitrary units have been used in the four panels).

Tables (1)

Tables Icon

Table 1 Parameters of the layered cloak. (ε = εrε0, µ = µrµ0, ξ = ξrξ0, ζ = ζrζ0.)

Equations (60)

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× E = i ω B , × H = i ω D ,
D = ε E + i ξ H , B = i ζ E + μ H ,
× E = ω ζ E + i ω μ H , × H = i ω ε E + ω ξ H .
ε = [ ε 11 ε 12 0 ε 21 ε 22 0 0 0 ε 33 ] , μ = [ μ 11 μ 12 0 μ 21 μ 22 0 0 0 μ 33 ] , ξ = [ ξ 11 ξ 12 0 ξ 21 ξ 22 0 0 0 ξ 33 ] , ζ = [ ζ 11 ζ 12 0 ζ 21 ζ 22 0 0 0 ζ 33 ] .
x 2 E 3 = ω ζ 11 E 1 + ω ζ 12 E 2 + i ω μ 11 H 1 + i ω μ 12 H 2 ,
x 1 E 3 = ω ζ 21 E 1 + ω ζ 22 E 2 + i ω μ 21 H 1 + i ω μ 22 H 2 ,
x 1 E 2 x 2 E 1 = ω ζ 33 E 3 + i ω μ 33 H 3 ,
x 2 H 3 = i ω ε 11 E 1 i ω ε 12 E 2 + ω ξ 11 H 1 + ω ξ 12 H 2 ,
x 1 H 3 = i ω ε 21 E 1 i ω ε 22 E 2 + ω ξ 21 H 1 + ω ξ 22 H 2 ,
x 1 H 2 x 2 H 1 = i ω ε 33 E 3 + ω ξ 33 H 3 .
[ x 1 E 3 x 2 E 3 ] = ω [ ζ 22 ζ 21 ζ 12 ζ 11 ] [ E 2 E 1 ] + i ω [ μ 22 μ 21 μ 12 μ 11 ] [ H 2 H 1 ] .
[ x 1 H 3 x 2 H 3 ] = i ω [ ε 22 ε 21 ε 12 ε 11 ] [ E 2 E 1 ] + ω [ ξ 22 ξ 21 ξ 12 ξ 11 ] [ H 2 H 1 ] .
ε T = [ ε 22 ε 21 ε 12 ε 11 ] , μ T = [ μ 22 μ 21 μ 12 μ 11 ] , ξ T = [ ξ 22 ξ 21 ξ 12 ξ 11 ] , ζ T = [ ζ 22 ζ 21 ζ 12 ζ 11 ] ,
A _ = [ x 1 x 2 ] , E _ = [ E 2 E 1 ] , H _ = [ H 2 H 1 ] ,
A _ E 3 = ω ζ T E _ + i ω μ T H _ ,
A _ H 3 = i ω ε T E _ + ω ξ T H _ .
E _ = ω 1 ζ T 1 A _ E 3 i ζ T 1 μ T H _ .
A _ H 3 = i ω ε T ( ω 1 ζ T 1 A _ E 3 i ζ T 1 μ T H _ ) + ω ξ T H _ = i ε T ζ T 1 A _ E 3 + ω ε T ζ T 1 μ T H _ + ω ξ T H _ .
H _ = B 1 ( A _ H 3 i ε T ζ T 1 A _ E 3 ) ,
B = ω ξ T + ω ε T ζ T 1 μ T .
[ x 1 x 2 ] H _ = [ x 1 x 2 ] ( B 1 A _ H 3 i B 1 ε T ζ T 1 A _ E 3 ) = [ x 1 x 2 ] [ i ω ( μ T + ζ T ε T 1 ξ T ) 1 ] A _ E 3 + [ x 1 x 2 ] [ 1 ω ( ξ T + ε T ζ T 1 μ T ) 1 ] A _ H 3 = i ω ε 33 E 3 + ω ξ 33 H 3 .
H _ = ω 1 ξ T 1 A _ H 3 i ξ T 1 ε T E _ ,
A _ E 3 = ω ζ T E _ + i ω μ T ( ω 1 ξ T 1 A _ H 3 i ξ T 1 ε T E _ ) = ω ζ T E _ + i μ T ξ T 1 A _ H 3 + ω μ T ξ T 1 ε T E _ ,
E _ = C 1 A _ E 3 i C 1 μ T ξ T 1 A _ H 3 ,
C = ω ζ T + ω μ T ξ T 1 ε T .
[ x 1 x 2 ] E _ = [ x 1 x 2 ] ( C 1 A _ E 3 i C 1 μ T ξ T 1 A _ H 3 ) = [ x 1 x 2 ] [ 1 ω ( ζ T + μ T ξ T 1 ε T ) 1 ] A _ E 3 [ x 1 x 2 ] [ i ω ( ε T + ξ T μ T 1 ζ T ) 1 ] A _ H 3 = ω ζ 33 E 3 + i ω μ 33 H 3 .
[ x 1 x 2 ] [ i ( μ T + ζ T ε T 1 ξ T ) 1 ] [ x 1 x 2 ] T E 3 + [ x 1 x 2 ] [ ( ξ T + ε T ζ T 1 μ T ) 1 ] [ x 1 x 2 ] T H 3 = i ω 2 ε 33 E 3 + ω 2 ξ 33 H 3 , [ x 1 x 2 ] [ ( ζ T + μ T ξ T 1 ε T ) 1 ] [ x 1 x 2 ] T E 3 + [ x 1 x 2 ] [ i ( ε T + ξ T μ T 1 ζ T ) 1 ] [ x 1 x 2 ] T H 3 = ω 2 ζ 33 E 3 + i ω 2 μ 33 H 3 .
( c : u α u + γ ) + a u + β u = f .
[ x 1 x 2 ] [ ( μ T + ζ T ε T 1 ξ T ) 1 ] [ x 1 x 2 ] T E 3 + [ x 1 x 2 ] [ i ( ξ T + ε T ζ T 1 μ T ) 1 ] [ x 1 x 2 ] T H 3 = ω 2 ε 33 E 3 + i ω 2 ξ 33 H 3 , [ x 1 x 2 ] [ i ( ζ T + μ T ξ T 1 ε T ) 1 ] [ x 1 x 2 ] T E 3 + [ x 1 x 2 ] [ ( ε T + ξ T μ T 1 ζ T ) 1 ] [ x 1 x 2 ] T H 3 = i ω 2 ζ 33 E 3 ω 2 μ 33 H 3 .
u = [ E 3 H 3 ] ,
c = [ c 11 c 12 c 21 c 22 ] , a = [ ω 2 ε 33 i ω 2 ξ 33 i ω 2 ζ 33 ω 2 μ 33 ] ,
c 11 = ( μ T + ζ T ε T 1 ξ T ) 1 , c 12 = i ( ξ T + ε T ζ T 1 μ T ) 1 , c 21 = i ( ζ T + μ T ξ T 1 ε T ) 1 , c 22 = ( ε T + ξ T μ T 1 ζ T ) 1 .
v = diag ( ν 11 , ν 22 , ν 33 ) , v = ε , μ , ξ , ζ .
c 11 = [ ε 22 μ 22 ε 22 ξ 22 ζ 22 0 0 ε 11 μ 11 ε 11 ξ 11 ζ 11 ] , c 12 = [ i ζ 22 μ 22 ε 22 ξ 22 ζ 22 0 0 i ζ 11 μ 11 ε 11 ξ 11 ζ 11 ] , c 21 = [ i ξ 22 μ 22 ε 22 ξ 22 ζ 22 0 0 i ξ 11 μ 11 ε 11 ξ 11 ζ 11 ] , c 22 = [ μ 22 μ 22 ε 22 ξ 22 ζ 22 0 0 μ 11 μ 11 ε 11 ξ 11 ζ 11 ] .
ε k k μ k k ξ k k ζ k k 0 , ( k = 1 , 2 ) .
u ˜ = 0 u s u ( u ) d u .
˜ = x ^ x ˜ + y ^ y ˜ + z ^ z ˜ = x ^ 1 s x x + y ^ 1 s y y + z ^ 1 s z z ,
˜ = S ¯ ¯ ,
S ¯ ¯ = x ^ x ^ ( 1 s x ) + y ^ y ^ ( 1 s y ) + z ^ z ^ ( 1 s z ) .
ν PML = ( det S ¯ ¯ ) 1 [ S ¯ ¯ ν S ¯ ¯ ] , ν = ε , μ , ξ , ζ .
PML x : s x = a b × i , s y = 1 , s z = 1 , PML y : s x = 1 , s y = a b × i , s z = 1 , PML x y : s x = a b × i , s y = a b × i , s z = 1 .
ν = diag ( ν x x L x x , ν y y L y y , ν z z L z z ) , ν = ε , μ , ξ , ζ .
L x x = s y s z s x , L y y = s x s z s y , L z z = s x s y s z .
x = r cos θ , y = r sin θ , z = z .
{ r = R 1 + r ( R 2 R 1 ) / R 2 , 0 r R 2 θ = θ , 0 < θ 2 π z = z .
J r r = ( r , θ , z ) ( r , θ , z ) = diag ( R 2 R 2 R 1 , 1 , 1 ) , 0 r R 2 .
J x x = J x r J r r J r x .
J x x = J x r J r r J r x = R ( θ ) diag ( R 2 R 2 R 1 , r r , 1 ) R ( θ ) .
T x x 1 = [ J x x T J x x / det ( J x x ) ] 1 .
T x x 1 = [ T 11 T 12 0 T 21 T 22 0 0 0 T 33 ] ,
T 11 = 1 R 1 r cos 2 θ + R 1 r R 1 sin 2 θ , T 12 = T 21 = R 1 ( R 1 2 r ) r ( r R 1 ) sin θ cos θ , T 22 = 1 R 1 r sin 2 θ + R 1 r R 1 cos 2 θ , T 33 = R 2 2 ( R 2 R 1 ) 2 r R 1 r .
ν = ν 0 T x x 1 , ν = ε , μ , ξ , ζ .
{ r = { R 1 R 2 r , 0 r R 2 R 3 R 1 R 3 R 2 r R 2 R 1 R 3 R 2 R 3 , R 2 < r R 3 θ = θ , 0 θ < 2 π z = z .
T x x 1 = { R ( θ ) diag ( 1 , 1 , R 2 2 R 1 2 ) R ( θ ) , 0 r R 2 R ( θ ) diag ( R 3 R 1 R 3 R 2 r r , R 3 R 2 R 3 R 1 r r , R 3 R 2 R 3 R 1 r r ) R ( θ ) , R 2 < r R 3
0 r R 1 : T 11 = T 22 = 1 , T 12 = T 21 = 0 , T 33 = R 2 2 R 1 2 . R 1 < r R 3 : T 11 = A cos 2 θ + B sin 2 θ ,   T 12 = T 21 = ( A B ) s i n θ cos θ ,   T 22 = A sin 2 θ + B cos 2 θ , T 33 = C .
A = 1 + R 2 R 1 R 3 R 2 R 3 r , B = r r + R 2 R 1 R 3 R 2 R 3 , C = ( R 3 R 2 R 3 R 1 ) 2 ( 1 + R 2 R 1 R 3 R 2 R 3 r ) .
r R 1 : r = r , θ = θ + θ 0 , z = z . R 1 < r R 2 : r = r , θ = θ + θ 0 f ( R 2 ) f ( r ) f ( R 2 ) f ( R 1 ) , z = z . r > R 2 : r = r , θ = θ , z = z .
R 1 < r R 2 : T 11 = 1 + r 2 D 2 sin 2 θ + 2 r D sin θ cos θ ,   T 12 = T 21 = r D ( s i n 2 θ c o s 2 θ ) r 2 D 2 sin θ cos θ ,   T 22 = 1 + r 2 D 2 cos 2 θ 2 r D sin θ cos θ , T 33 = 1 . r R 1 and r > R 2 : T x x 1 = I 3 .
× E eff = ω ζ eff E eff + i ω μ eff H eff , × H eff = i ω ε eff E eff + ω ξ eff H eff .
ν eff = diag ( < ν 1 > 1 , < ν > , < ν > ) , ν = ε , μ , χ , ζ .

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