## A level-set procedure for the design of electromagnetic metamaterials

Optics Express, Vol. 18, Issue 7, pp. 6693-6702 (2010)

http://dx.doi.org/10.1364/OE.18.006693

Acrobat PDF (307 KB)

### 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

## 1. Introduction

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

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

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

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

4. 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]

5. 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]

4. 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]

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

9. 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]

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). [CrossRef]

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). [CrossRef]

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

4. 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]

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]

## 2. Materials and methods

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

*V*are defined as

*ω*and

*λ*

_{0}refer to the frequency and wavelength of the incident wave, and

*ε*

_{0}and

*μ*

_{0}denote the permittivity and permeability of vacuum. It must be pointed out that, in this paper, we use EFIE rather than the Maxwell's equations to simplify the optimization process. Such a simplification is sensible because EFIE can be derived directly from the Maxwell's equations.

15. 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]

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

16. 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]

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

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

**47**(11), 2075–2084 (1999). [CrossRef]

5. 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]

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]

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]

**47**(11), 2075–2084 (1999). [CrossRef]

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]

26. 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]

**75**, 041102 041101/041104 (2007). [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]

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]

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). [CrossRef]

*S*is obtained from the projection of the normal direction of the 3D level-set function to the 2D metal domain Ω. This objective function tends to be zero for a nearly-symmetric pattern of current flow, such as those in Figs. 2 (b1) and (b3), because the signed weighting factor

*S*takes –1 and + 1, respectively, on the two vertical sides; thus the current intensity (the norm of current flow) is largely cancelled out. While for the circuit current flow in Fig. 2 (b2), this objective becomes highly positive as the direction of current flow keeps constant with respect to the normal direction (i.e.

*S*keeps + 1 all the time). The cost function (in both magnitude and sign) plays a key role on driving the topology optimization in terms of velocity (sensitivity) in the proposed level-set procedure. It is also a measure of the optimization extent. In fact, even though the sign of the current flow is desirably right, the metamaterial structure can continue evolving toward an optimum. The following examples showed that the magnitude of the current flow can increase hundreds of times throughout the optimization process. In some parts of the optimal structure, the current flow can attain rather great values.

## 3. Results and discussion

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). [CrossRef]

29. 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]

*th*iteration step). The optimization makes these two circles merge in an early stage, shaping two dents on bilateral edges (Fig. 6(b), structure 5). As the design progresses, the right dent gradually disappears, while the left dent deepens further (Fig. 6(c), structure 50). Finally, the growth of the left dent also shapes the metal region into a U-shaped structure (Fig. 6(d), structure 75), agreeing well with the previous benchmarking example.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. USPEKI |

4. | 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. |

5. | 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. |

6. | M. P. Bendsøe, and O. Sigmund, |

7. | D. H. Kwon and D. H. Werner, “Low-index metamaterial designs in the visible spectrum,” Opt. Express |

8. | 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 |

9. | 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 |

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. |

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. |

12. | 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 |

13. | G. P. Steven, Q. Li, and Y. M. Xie, “Evolutionary topology and shape design for general physical field problems,” Comput. Mech. |

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. |

15. | M. Zhou and G. I. N. Rozvany, “The COC algorithm. II: Topological, geometrical and generalized shape optimization,” Comput. Methods Appl. Mech. Eng. |

16. | S. Osher and J. A. Sethian, “Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. |

17. | G. Allaire, F. Jouve, and A. M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. |

18. | M. Y. Wang, X. M. Wang, and D. M. Guo, “A level set method for structural topology optimization,” Comput. Methods Appl. Mech. Eng. |

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. |

20. | M. Burger and S. J. Osher, “A survey on level set methods for inverse problems and optimal design,” Eur. J. Appl. Math. |

21. | O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. |

22. | A. W. Maue, “On the formulation of a general scattering problem by means of an integral equation,” Z. Phys. |

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 |

24. | Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. Struc. |

25. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science |

26. | 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. |

27. | G. Lubkowski, R. Schuhmann, and T. Weiland, “Extraction of effective metamaterial parameters by parameter fitting of dispersive models,” Microwave Optical Tech. Lett. |

28. | R. F. Harrington, |

29. | 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. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: October 15, 2009

Revised Manuscript: February 5, 2010

Manuscript Accepted: March 6, 2010

Published: March 17, 2010

**Citation**

Shiwei Zhou, Wei Li, Guangyong Sun, and Qing Li, "A level-set procedure for the design of electromagnetic metamaterials," Opt. Express **18**, 6693-6702 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6693

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### References

- 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]
- V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. USPEKI 10(4), 509–514 (1968). [CrossRef]
- 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]
- 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]
- M. P. Bendsøe, and O. Sigmund, Topology Optimisation: theory, methods, and applications (Springer, Berlin; New York, 2003).
- D. H. Kwon and D. H. Werner, “Low-index metamaterial designs in the visible spectrum,” Opt. Express 15(15), 9267–9272 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22(4), R67–R131 (2006). [CrossRef]
- A. W. Maue, “On the formulation of a general scattering problem by means of an integral equation,” Z. Phys. 126, 601–618 (1949). [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]
- Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. Struc. 49(5), 885–896 (1993). [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]
- 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]
- 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]
- R. F. Harrington, Field Computation by the Moment Methods (IEEE Press, New York, 1993).
- 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]

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