Abstract

A conformal cubical transformation-based metamaterial invisibility cloak is presented and verified, in the near and the far field, by a rigorous full-wave numerical technique based on a higher-order, large-domain finite element method, employing large anisotropic, continuously inhomogeneous generalized hexahedral finite elements, with no need for discretization of the permittivity and permeability profiles of the cloak. The analysis requires about 30 times fewer unknowns than with commercial software. To our knowledge, this is the first conformal cubical cloak and the first full-wave computational characterization of such a structure with sharp edges. The presented methodology can also be used in development of conformal, transformation-based perfectly matched layers.

© 2012 Optical Society of America

Full Article  |  PDF Article

Corrections

Slobodan V. Savić, Branislav M. Notaroš, and Milan M. Ilić, "Conformal cubical 3D transformation-based metamaterial invisibility cloak: erratum," J. Opt. Soc. Am. A 32, 1474-1474 (2015)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-32-8-1474

References

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  1. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
    [CrossRef]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef]
  3. S. V. Savić, A. B. Manić, M. M. Ilić, and B. M. Notaroš, “Efficient higher order full-wave numerical analysis of 3-D cloaking structures,” Plasmonics, online first, doi: 10.1007/s11468-012-9410-0 (posted 08 July 2012).
    [CrossRef]
  4. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
    [CrossRef]
  5. F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
    [CrossRef]
  6. D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Antennas Propag. Mag. 52(1), 24–46 (2010).
    [CrossRef]
  7. J. Pendry, “Taking the wraps off cloaking,” Physics 2, 95 (2009).
    [CrossRef]
  8. H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008).
    [CrossRef]
  9. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16, 6134–6145 (2008).
    [CrossRef]
  10. 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,” Photon. Nanostr. Fundam. Appl. 6, 87–95 (2008).
    [CrossRef]
  11. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133–11141 (2007).
    [CrossRef]
  12. B. Zhang, H. Chen, and B.-I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express 16, 14655–14660 (2008).
    [CrossRef]
  13. 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]
  14. M. M. Ilić, A. Ž. Ilić, and B. M. Notaroš, “Continuously inhomogeneous higher order finite elements for 3-D electromagnetic analysis,” IEEE Trans. Antennas Propag. 57, 2798–2803 (2009).
    [CrossRef]
  15. A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
    [CrossRef]
  16. M. M. Ilić and B. M. Notaroš, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Microw. Theor. Tech. 51, 1026–1033 (2003).
    [CrossRef]
  17. M. Djordjević and B. M. Notaroš, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag. 52, 2118–2129 (2004).
    [CrossRef]
  18. M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
    [CrossRef]

2012 (1)

A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
[CrossRef]

2010 (1)

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Antennas Propag. Mag. 52(1), 24–46 (2010).
[CrossRef]

2009 (4)

J. Pendry, “Taking the wraps off cloaking,” Physics 2, 95 (2009).
[CrossRef]

F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
[CrossRef]

M. M. Ilić, A. Ž. Ilić, and B. M. Notaroš, “Continuously inhomogeneous higher order finite elements for 3-D electromagnetic analysis,” IEEE Trans. Antennas Propag. 57, 2798–2803 (2009).
[CrossRef]

M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
[CrossRef]

2008 (4)

2007 (2)

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
[CrossRef]

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

2006 (3)

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]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

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

2004 (1)

M. Djordjević and B. M. Notaroš, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag. 52, 2118–2129 (2004).
[CrossRef]

2003 (1)

M. M. Ilić and B. M. Notaroš, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Microw. Theor. Tech. 51, 1026–1033 (2003).
[CrossRef]

Chen, H.

B. Zhang, H. Chen, and B.-I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express 16, 14655–14660 (2008).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
[CrossRef]

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,” Photon. Nanostr. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Djordjevic, M.

M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
[CrossRef]

M. Djordjević and B. M. Notaroš, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag. 52, 2118–2129 (2004).
[CrossRef]

Feng, Y.

Huang, Y.

Ilic, A. Ž.

M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
[CrossRef]

M. M. Ilić, A. Ž. Ilić, and B. M. Notaroš, “Continuously inhomogeneous higher order finite elements for 3-D electromagnetic analysis,” IEEE Trans. Antennas Propag. 57, 2798–2803 (2009).
[CrossRef]

Ilic, M. M.

A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
[CrossRef]

M. M. Ilić, A. Ž. Ilić, and B. M. Notaroš, “Continuously inhomogeneous higher order finite elements for 3-D electromagnetic analysis,” IEEE Trans. Antennas Propag. 57, 2798–2803 (2009).
[CrossRef]

M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
[CrossRef]

M. M. Ilić and B. M. Notaroš, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Microw. Theor. Tech. 51, 1026–1033 (2003).
[CrossRef]

S. V. Savić, A. B. Manić, M. M. Ilić, and B. M. Notaroš, “Efficient higher order full-wave numerical analysis of 3-D cloaking structures,” Plasmonics, online first, doi: 10.1007/s11468-012-9410-0 (posted 08 July 2012).
[CrossRef]

Jiang, T.

Kattawar, G. W.

Kong, J. A.

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
[CrossRef]

Kwon, D.-H.

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Antennas Propag. Mag. 52(1), 24–46 (2010).
[CrossRef]

Li, L.-W.

F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
[CrossRef]

Liang, Y.

F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
[CrossRef]

Ma, H.

Manic, A. B.

A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
[CrossRef]

S. V. Savić, A. B. Manić, M. M. Ilić, and B. M. Notaroš, “Efficient higher order full-wave numerical analysis of 3-D cloaking structures,” Plasmonics, online first, doi: 10.1007/s11468-012-9410-0 (posted 08 July 2012).
[CrossRef]

Manic, S. B.

A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
[CrossRef]

Meng, F.-Y.

F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
[CrossRef]

Notaroš, B. M.

A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
[CrossRef]

M. M. Ilić, A. Ž. Ilić, and B. M. Notaroš, “Continuously inhomogeneous higher order finite elements for 3-D electromagnetic analysis,” IEEE Trans. Antennas Propag. 57, 2798–2803 (2009).
[CrossRef]

M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
[CrossRef]

M. Djordjević and B. M. Notaroš, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag. 52, 2118–2129 (2004).
[CrossRef]

M. M. Ilić and B. M. Notaroš, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Microw. Theor. Tech. 51, 1026–1033 (2003).
[CrossRef]

S. V. Savić, A. B. Manić, M. M. Ilić, and B. M. Notaroš, “Efficient higher order full-wave numerical analysis of 3-D cloaking structures,” Plasmonics, online first, doi: 10.1007/s11468-012-9410-0 (posted 08 July 2012).
[CrossRef]

Pendry, J.

J. Pendry, “Taking the wraps off cloaking,” Physics 2, 95 (2009).
[CrossRef]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[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,” Photon. Nanostr. 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]

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]

Popa, B.-I.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Qu, S.

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,” Photon. Nanostr. 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,” Photon. Nanostr. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Savic, S. V.

S. V. Savić, A. B. Manić, M. M. Ilić, and B. M. Notaroš, “Efficient higher order full-wave numerical analysis of 3-D cloaking structures,” Plasmonics, online first, doi: 10.1007/s11468-012-9410-0 (posted 08 July 2012).
[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,” Photon. Nanostr. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[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,” Photon. Nanostr. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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, 9794–9804 (2006).
[CrossRef]

Wang, J.

Werner, D. H.

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Antennas Propag. Mag. 52(1), 24–46 (2010).
[CrossRef]

Wu, B.-I.

B. Zhang, H. Chen, and B.-I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express 16, 14655–14660 (2008).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
[CrossRef]

Wu, Q.

F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
[CrossRef]

Xu, Z.

Yang, P.

You, Y.

Zhai, P.-W.

Zhang, B.

B. Zhang, H. Chen, and B.-I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express 16, 14655–14660 (2008).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
[CrossRef]

Appl. Phys. A (1)

F.-Y. Meng, Y. Liang, Q. Wu, and L.-W. Li, “Invisibility of a metamaterial cloak illuminated by spherical electromagnetic wave,” Appl. Phys. A 95, 881–888 (2009).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Antennas Propag. Mag. 52(1), 24–46 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

M. M. Ilić, A. Ž. Ilić, and B. M. Notaroš, “Continuously inhomogeneous higher order finite elements for 3-D electromagnetic analysis,” IEEE Trans. Antennas Propag. 57, 2798–2803 (2009).
[CrossRef]

M. Djordjević and B. M. Notaroš, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag. 52, 2118–2129 (2004).
[CrossRef]

M. M. Ilić, M. Djordjević, A. Ž. Ilić, and B. M. Notaroš, “Higher order hybrid FEM–MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag. 57, 1452–1460 (2009).
[CrossRef]

IEEE Trans. Microw. Theor. Tech. (1)

M. M. Ilić and B. M. Notaroš, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Microw. Theor. Tech. 51, 1026–1033 (2003).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaroš, “Large anisotropic inhomogeneous higher order hierarchical generalized hexahedral finite elements for 3-D electromagnetic modeling of scattering and waveguide structures,” Microw. Opt. Technol. Lett. 54, 1644–1649 (2012).
[CrossRef]

Opt. Express (5)

Photon. Nanostr. 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,” Photon. Nanostr. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Phys. Rev. E (1)

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903–063904 (2007).
[CrossRef]

Physics (1)

J. Pendry, “Taking the wraps off cloaking,” Physics 2, 95 (2009).
[CrossRef]

Science (1)

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

Other (1)

S. V. Savić, A. B. Manić, M. M. Ilić, and B. M. Notaroš, “Efficient higher order full-wave numerical analysis of 3-D cloaking structures,” Plasmonics, online first, doi: 10.1007/s11468-012-9410-0 (posted 08 July 2012).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Illustration of the effects of the coordinate transformation given in Eq. (1), which leads to the construction of the cubical cloak: (a) the original space with an air-filled homogeneous isotropic cube and (b) the transformed space with a void region and an anisotropic continuously inhomogeneous cubical-shell cloak composed of six pyramidal frusta. One sixth of the cloak constructed by transformation given in Eq. (1) is highlighted in both figures.

Fig. 2.
Fig. 2.

Generalized Lagrange-type parametric hexahedral finite element of geometrical orders Ku, Kv, and Kw, in the representation of the position vector r(u,v,w), 1u, v, w1 field-approximation orders Nu, Nv, and Nw, in the approximation of the electric field vector E(u,v,w), and material-representation orders Mu, Mv, and Mw, to model the material parameters ε¯¯r(u,v,w) and μ¯¯r(u,v,w).

Fig. 3.
Fig. 3.

FEM–MoM model of a PEC cubical scatterer with a cubical-shell cloak: the cloak is modeled using 24 large continuously inhomogeneous anisotropic finite elements with high-order polynomial field expansions in parametric coordinates.

Fig. 4.
Fig. 4.

Normalized backscattering cross section of the cloaked PEC cube with R2/R1=1.1 in Fig. 3, including lossless (original) and lossy cloaks and an uncloaked PEC cube (the cloak shell replaced by an air layer), obtained by the higher order full-wave rigorous FEM–MoM numerical analysis versus the normalized PEC cube side length (λ0 is the free-space wavelength). FEM–MoM results for the uncloaked PEC cubical scatterer (PEC cube with the air layer, analyzed using the same 24-element numerical model as the cloaked cube) are compared with WIPL-D results for a PEC scatterer. WIPL-D results for a homogeneous air-filled cube are also shown, as a reference for verification of the best numerical approximation of the zero backscatter from an empty cubical region of the same size as the original scatterer.

Fig. 5.
Fig. 5.

Normalized vertical–vertical (VV) scattering cross section of the cloaked (lossless and lossy, for different loss tangents) and uncloaked PEC cube in Fig. 3 (R2/R1=1.1, d/λ0=0.3, and d=2R1), obtained by the higher-order FEM–MoM technique. The results include WIPL-D solutions for the PEC cubical scatterer and for the air-filled cube, as specified in the caption to Fig. 4.

Fig. 6.
Fig. 6.

Normalized horizontal–horizontal (HH) scattering cross section of the cloaked (lossless and lossy) and uncloaked PEC cube in Fig. 3 (R2/R1=1.1 and d/λ0=0.3), simulated by the FEM–MoM. WIPL-D results are for structures specified in the caption to Fig. 4.

Fig. 7.
Fig. 7.

Near field (in z=0 plane) of the cloaked PEC cube in Fig. 3 (R2/R1=1.5 and d/λ0=0.66) excited by a uniform plane wave of magnitude E0=1V/m, incident from the direction defined by (a) θinc=90°, ϕinc=0° and (b) θinc=90°, ϕinc=45°.

Equations (4)

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

x=R1+R2R1R2x,y=y(R2R1R2+R1x),andz=z(R2R1R2+R1x),
J=[(R2R1)R200R1yx2(R2R1)R2+R1x0R1zx20(R2R1)R2+R1x].
ε¯¯r=Jε¯¯rJTdet(J),andμ¯¯r=Jμ¯¯rJTdet(J).
εr,xx=a(R1x)2/x2,εr,yy=a+aR12y2/x4,εr,zz=a+aR12z2/x4,εr,xy=εr,yx=aR1(R1x)y/x3,εr,xz=εr,zx=aR1(R1x)z/x3,andεr,yz=εr,zy=aR12yz/x4,

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