Abstract

Achieving negative permittivity and negative permeability signifies a key topic of research in the design of metamaterials. This paper introduces a level-set based topology optimization method, in which the interface between the vacuum and metal phases is implicitly expressed by the zero-level contour of a higher dimensional level-set function. Following a sensitivity analysis, the optimization maximizes the objective based on the normal direction of the level-set function and induced current flow, thereby generating the desirable patterns of current flow on metal surface. As a benchmark example, the U-shaped structure and its variations are obtained from the level-set topology optimization. Numerical examples demonstrate that both negative permittivity and negative permeability can be attained.

© 2010 OSA

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  10. N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
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  11. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
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  13. G. P. Steven, Q. Li, and Y. M. Xie, “Evolutionary topology and shape design for general physical field problems,” Comput. Mech. 26(2), 129–139 (2000).
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  14. S. W. Zhou and Q. Li, “The relation of constant mean curvature surfaces to multiphase composites with extremal thermal conductivity,” J. Phys. D Appl. Phys. 40(19), 6083–6093 (2007).
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    [CrossRef]
  19. S. W. Zhou and Q. Li, “A variational level set method for the topology optimization of steady-state Navier-Stokes flow,” J. Comput. Phys. 227(24), 10178–10195 (2008).
    [CrossRef]
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    [CrossRef]
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  23. W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
    [CrossRef]
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    [CrossRef]
  25. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306(5700), 1351–1353 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. G. Lubkowski, R. Schuhmann, and T. Weiland, “Extraction of effective metamaterial parameters by parameter fitting of dispersive models,” Microwave Optical Tech. Lett. 49(2), 285–288 (2007).
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    [CrossRef] [PubMed]

2009

J. A. Bossard, S. Yun, D. H. Werner, and T. S. Mayer, “Synthesizing low loss negative index metamaterial stacks for the mid-infrared using genetic algorithms,” Opt. Express 2009(17), 14771–14779 (2009).
[CrossRef]

2008

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

P. Y. Chen, C. H. Chen, H. Wang, J. H. Tsai, and W. X. Ni, “Synthesis design of artificial magnetic metamaterials using a genetic algorithm,” Opt. Express 16(17), 12806–12818 (2008).
[CrossRef] [PubMed]

S. W. Zhou and Q. Li, “A variational level set method for the topology optimization of steady-state Navier-Stokes flow,” J. Comput. Phys. 227(24), 10178–10195 (2008).
[CrossRef]

2007

S. W. Zhou and Q. Li, “The relation of constant mean curvature surfaces to multiphase composites with extremal thermal conductivity,” J. Phys. D Appl. Phys. 40(19), 6083–6093 (2007).
[CrossRef]

W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
[CrossRef]

G. Lubkowski, R. Schuhmann, and T. Weiland, “Extraction of effective metamaterial parameters by parameter fitting of dispersive models,” Microwave Optical Tech. Lett. 49(2), 285–288 (2007).
[CrossRef]

D. H. Kwon and D. H. Werner, “Low-index metamaterial designs in the visible spectrum,” Opt. Express 15(15), 9267–9272 (2007).
[CrossRef] [PubMed]

2006

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

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

2005

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

M. Burger and S. J. Osher, “A survey on level set methods for inverse problems and optimal design,” Eur. J. Appl. Math. 16(2), 263–301 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
[CrossRef]

2004

G. Allaire, F. Jouve, and A. M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194(1), 363–393 (2004).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

2003

M. Y. Wang, X. M. Wang, and D. M. Guo, “A level set method for structural topology optimization,” Comput. Methods Appl. Mech. Eng. 192(1-2), 227–246 (2003).
[CrossRef]

2000

G. P. Steven, Q. Li, and Y. M. Xie, “Evolutionary topology and shape design for general physical field problems,” Comput. Mech. 26(2), 129–139 (2000).
[CrossRef]

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

1999

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Q. Li, G. P. Steven, O. M. Querin, and Y. M. Xie, “Shape and topology design for heat conduction by evolutionary structural optimisation,” Int. J. Heat Mass Transfer 42(17), 3361–3371 (1999).
[CrossRef]

1993

Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. Struc. 49(5), 885–896 (1993).
[CrossRef]

1991

M. Zhou and G. I. N. Rozvany, “The COC algorithm. II: Topological, geometrical and generalized shape optimization,” Comput. Methods Appl. Mech. Eng. 89(1-3), 309–336 (1991).
[CrossRef]

1988

S. Osher and J. A. Sethian, “Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79(1), 12–49 (1988).
[CrossRef]

1968

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. USPEKI 10(4), 509–514 (1968).
[CrossRef]

1949

A. W. Maue, “On the formulation of a general scattering problem by means of an integral equation,” Z. Phys. 126, 601–618 (1949).
[CrossRef]

Allaire, G.

G. Allaire, F. Jouve, and A. M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194(1), 363–393 (2004).
[CrossRef]

Aronsson, M. T.

W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
[CrossRef]

Averitt, R. D.

W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
[CrossRef]

Bossard, J. A.

J. A. Bossard, S. Yun, D. H. Werner, and T. S. Mayer, “Synthesizing low loss negative index metamaterial stacks for the mid-infrared using genetic algorithms,” Opt. Express 2009(17), 14771–14779 (2009).
[CrossRef]

Burger, M.

M. Burger and S. J. Osher, “A survey on level set methods for inverse problems and optimal design,” Eur. J. Appl. Math. 16(2), 263–301 (2005).
[CrossRef]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Cai, W.

Chen, C. H.

Chen, P. Y.

Chettiar, U. K.

Dorn, O.

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

Drachev, V. P.

Enkrich, C.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Fu, L. W.

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Giessen, H.

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Guo, D. M.

M. Y. Wang, X. M. Wang, and D. M. Guo, “A level set method for structural topology optimization,” Comput. Methods Appl. Mech. Eng. 192(1-2), 227–246 (2003).
[CrossRef]

Guo, H. C.

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Highstrete, C.

W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Jouve, F.

G. Allaire, F. Jouve, and A. M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194(1), 363–393 (2004).
[CrossRef]

Kaiser, S.

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Kildishev, A. V.

Koschny, T.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Kwon, D. H.

Lee, M.

W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
[CrossRef]

Lesselier, D.

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006).
[CrossRef]

Li, Q.

S. W. Zhou and Q. Li, “A variational level set method for the topology optimization of steady-state Navier-Stokes flow,” J. Comput. Phys. 227(24), 10178–10195 (2008).
[CrossRef]

S. W. Zhou and Q. Li, “The relation of constant mean curvature surfaces to multiphase composites with extremal thermal conductivity,” J. Phys. D Appl. Phys. 40(19), 6083–6093 (2007).
[CrossRef]

G. P. Steven, Q. Li, and Y. M. Xie, “Evolutionary topology and shape design for general physical field problems,” Comput. Mech. 26(2), 129–139 (2000).
[CrossRef]

Q. Li, G. P. Steven, O. M. Querin, and Y. M. Xie, “Shape and topology design for heat conduction by evolutionary structural optimisation,” Int. J. Heat Mass Transfer 42(17), 3361–3371 (1999).
[CrossRef]

Linden, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Liu, N.

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Lubkowski, G.

G. Lubkowski, R. Schuhmann, and T. Weiland, “Extraction of effective metamaterial parameters by parameter fitting of dispersive models,” Microwave Optical Tech. Lett. 49(2), 285–288 (2007).
[CrossRef]

Maue, A. W.

A. W. Maue, “On the formulation of a general scattering problem by means of an integral equation,” Z. Phys. 126, 601–618 (1949).
[CrossRef]

Mayer, T. S.

J. A. Bossard, S. Yun, D. H. Werner, and T. S. Mayer, “Synthesizing low loss negative index metamaterial stacks for the mid-infrared using genetic algorithms,” Opt. Express 2009(17), 14771–14779 (2009).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Ni, W. X.

Osher, S.

S. Osher and J. A. Sethian, “Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79(1), 12–49 (1988).
[CrossRef]

Osher, S. J.

M. Burger and S. J. Osher, “A survey on level set methods for inverse problems and optimal design,” Eur. J. Appl. Math. 16(2), 263–301 (2005).
[CrossRef]

Padilla, W. J.

W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B 75, 041102 041101/041104 (2007).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, D. Schurig, 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]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Querin, O. M.

Q. Li, G. P. Steven, O. M. Querin, and Y. M. Xie, “Shape and topology design for heat conduction by evolutionary structural optimisation,” Int. J. Heat Mass Transfer 42(17), 3361–3371 (1999).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[CrossRef]

Rozvany, G. I. N.

M. Zhou and G. I. N. Rozvany, “The COC algorithm. II: Topological, geometrical and generalized shape optimization,” Comput. Methods Appl. Mech. Eng. 89(1-3), 309–336 (1991).
[CrossRef]

Sarychev, A. K.

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

Schuhmann, R.

G. Lubkowski, R. Schuhmann, and T. Weiland, “Extraction of effective metamaterial parameters by parameter fitting of dispersive models,” Microwave Optical Tech. Lett. 49(2), 285–288 (2007).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

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

Schweizer, H.

N. Liu, H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Sethian, J. A.

S. Osher and J. A. Sethian, “Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79(1), 12–49 (1988).
[CrossRef]

Shalaev, V. M.

Smith, D. R.

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

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Steven, G. P.

G. P. Steven, Q. Li, and Y. M. Xie, “Evolutionary topology and shape design for general physical field problems,” Comput. Mech. 26(2), 129–139 (2000).
[CrossRef]

Q. Li, G. P. Steven, O. M. Querin, and Y. M. Xie, “Shape and topology design for heat conduction by evolutionary structural optimisation,” Int. J. Heat Mass Transfer 42(17), 3361–3371 (1999).
[CrossRef]

Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. Struc. 49(5), 885–896 (1993).
[CrossRef]

Stewart, W. J.

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C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
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[CrossRef]

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[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
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[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

The zero-level contour (a) of a higher-dimensional level-set function (b).

Fig. 2
Fig. 2

The effective electromagnetic properties of different structures and their induced surface current flow: (a1)-(a3) structural configurations; (b1)-(b3) induced surface current flow; (c1)-(c3) effective permittivity; (d1)-(d3) effective permeability.

Fig. 3
Fig. 3

The current flow obtained by solving EFIE: (a) the adaptive triangular mesh for a SRR; (b) the current flow on a SRR; (c) the current flow on a circular ring without aperture.

Fig. 4
Fig. 4

The history of the objective function and evolution of the metal region (Example 1).

Fig. 5
Fig. 5

The effective electromagnetic properties for structures in initial and final stages: (a) the real part of permittivity (Re(ε)); (b) the imaginary part of permeability (Im(μ)). (Example 1)

Fig. 6
Fig. 6

The metal shapes in different stages for Example 2 starting from two identical circles: (a) structure 0; (b) structure 5; (c) structure 50; (d) structure 75.

Fig. 7
Fig. 7

The effective electromagnetic properties for structures in different stages: (a) the real part of permittivity (Re(ε)); (b) the imaginary part of permeability (Im(μ)). (Example 2)

Fig. 8
Fig. 8

The initial structure (a) and different final structures (b-c) with different control parameters (Example 3).

Fig. 9
Fig. 9

The effective electromagnetic properties for the structures in different design stages: (a) the real part of permittivity (Re(ε)); (b) the imaginary part of permeability (Im(μ)). (Example 3)

Equations (7)

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

n × E I = n × ( j ω A ( J s ) + V ( J s ) ) x Ω
φ t + V n | φ | = 0
max Ω F ( Ω , J s ) = 1 2 S ( J s , n ) Ω J s J s d x
d F = S ( J s , n ) < J s , w > = P ( w , J s ) = < v n , P ( w , J s ) / n > Ω
j ω μ 0 4 π v , Ω w G d s j 4 π ω ε 0 ( s w , G , s v ) = S ( J s , n ) Ω J s v d x
V n = Ω P ( w , J s ) / n d x
d F / Δ t = F ( J s ) Ω t 1 F ( J s ) Ω F ( J s ) Ω t 1 F ( J s ) Ω = Δ t Ω ( P ( w , J s ) / n ) 2 d x 0

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