## Low-index metamaterial designs in the visible spectrum

Optics Express, Vol. 15, Issue 15, pp. 9267-9272 (2007)

http://dx.doi.org/10.1364/OE.15.009267

Acrobat PDF (132 KB)

### Abstract

Low-index metamaterial designs in the visible spectrum that are impedance matched to free space are presented. The unit cell of the periodic metamaterial design incorporates a magnetic resonator and silver meshes for respective control of the effective permeability and permittivity. A genetic algorithm is employed to optimize the metamaterial design to achieve a desired set of values for the index of refraction and the intrinsic impedance. Two example GA optimized designs are provided which target the important special cases of a zero and unity index of refraction.

© 2007 Optical Society of America

## 1. Introduction

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

2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977–980 (2006). [CrossRef] [PubMed]

*n*≤1), play a critical role in the ability to realize an electromagnetic cloak. The LIMs as defined here are bounded by two important special cases; namely the zero-index metamaterials (ZIMs) on one hand and on the other hand metamaterials which exhibit the “transparency” or “invisibility” condition when

*n*=1.

3. 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**, 3356–3358 (2005). [CrossRef]

4. S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Demonstration of metaldielectric negative-index metamaterials with improved performance at optical frequencies,” J. Opt. Soc. Am. B **23**, 434–438 (2006). [CrossRef]

5. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. **32**, 53–55 (2007). [CrossRef]

6. R.W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,”Phys. Rev. E **70**, 046608 (2004). [CrossRef]

7. B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B **20**, 2448–2453 (2003). [CrossRef]

8. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. **89**, 213902 (2002). [CrossRef] [PubMed]

10. D.-H. Kwon, L. Li, J. A. Bossard, M. G. Bray, and D. H. Werner, “Zero index metamaterials with checkerboard structure,” Electron. Lett. **43**, 319–320 (2007). [CrossRef]

7. B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B **20**, 2448–2453 (2003). [CrossRef]

## 2. Metamaterial architecture

*n*and the intrinsic impedance

*z*of a metamaterial be defined as

*n*=

*n*′+

*in*″ and

*z=z*′+

*iz*″, respectively, normalized by their corresponding values in free space. These quantities may be expressed in terms of the relative effective permittivity

*ε*=ε

*′*+

*iε*″ and the relative effective permeability

*μ*=

*μ*′+

*iμ*″ such that

*n*}≥0 and Re{

*z*}≥0. Equation (1) suggests that it is possible to realize a desired set of values for

*n*and

*z*by independently adjusting the values of

*ε*and

*µ*. This can be achieved by incorporating sub-structures into the unit cell of a metamaterial that have dominant effects on either

*ε*or

*µ*.

*p*both in the

*x*̂ and the

*ŷ*directions is illustrated in Fig. 1. A pair of square silver plates separated by an alumina (Al

_{2}O

_{3}) layer form a magnetic resonator. These periodic magnetic resonators are bounded on the top and bottom by a silver mesh. Each mesh is formed by silver strips running both in the ±

*x*̂ and the ±

*ŷ*directions with grid points located at the center of each magnetic resonator. The regions which are not occupied by either silver or alumina are filled with silica (SiO

_{2}). Finally, the entire metamaterial slab is placed on a thick glass substrate. The metamaterial slab is illuminated by a normally incident plane wave propagating in the

*x*̂ direction.

*µ*and

*ε*, respectively. At a given wavelength, the value of

*µ*will be strongly dependent upon the geometrical parameters of the magnetic resonator —

*w*,

*t*and

*d*. Similarly, the dimensional parameters of the silver mesh

*s*and

*t*will strongly affect the value of

_{m}*ε*. It is expected that the period

*p*will contribute to both

*µ*and

*ε*.

## 3. Optimization methodology

*n*and

*z*, six geometrical parameters (

*p, w, s, t, d*, and

*t*) in the LIM design need to be properly chosen to achieve the goal. In this study, a GA [11

_{m}11. R. L. Haupt and D. H. Werner, *Genetic Algorithms in Electromagnetics* (Wiley, Hoboken, NJ, 2007). [CrossRef]

12. S. Chakravarty, R. Mittra, and N. R. Williams, “Application of a micro-genetic algorithm (MGA) to the design of broad-band microwave absorbers using multiple frequency selective surface screens buried in dielectrics,” IEEE Trans. Antennas Propag. **53**, 284–296 (2002). [CrossRef]

13. J. L. Volakis, A. Chatterjee, and L. C. Kempel, *Finite Element Method for Electromagnetics* (IEEE Press, Piscataway, NJ, 1998). [CrossRef]

14. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

15. A. V. Kildishev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and V. M. Shalaev, “Negative refractive index in optics of metal-dielectric composites,” J. Opt. Soc. Am. B **23**, 423–433 (2006). [CrossRef]

*n*and

*z*of a homogeneous isotropic material slab of the same thickness. However, although widely used, there is a controversy as to whether the retrieved material parameters correspond to homogeneous, isotropic, and passive materials [16

16. T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E **68**, 065602(R) (2003). [CrossRef]

17. A. L. Efros, “Comment II on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E **70**, 048602 (2004). [CrossRef]

18. T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Reply to comments on “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E **70**, 048603 (2004). [CrossRef]

19. E. Saenz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. **101**, 114910 (2007). [CrossRef]

*ε*″ or

*µ*″ at times, which appears to contradict the passivity requirement [19

19. E. Saenz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. **101**, 114910 (2007). [CrossRef]

## 4. Numerical results

*n*→0) and a unity-index design (

*n*→1). In both of these cases, an additional condition was imposed that the metamaterial be impedance matched to free space (

*z*→1) over the range 400 nm≤λ≤800 nm. The ranges of the six geometrical parameters that define the GA optimization search space were constrained as follows: 133 nm≤

*p*≤400 nm, 0≤

*w*≤

*p*, 0≤

*s*≤

*p*, 20 nm≤

*t*≤60 nm, 20 nm≤

*d*≤100 nm, and 20 nm≤

*t*≤100 nm. The values of 1.445, 1.62, and 1.5 were used in the simulations to represent the refractive index of silica, alumina, and glass, respectively. In addition, the refractive index of silver was based on the measured values reported in [21

_{m}21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

### 4.1. Impedance-matched zero-index metamaterial

*n*→0) and the intrinsic impedance matched to free space (

*z*→1) is desired. An appropriate fitness function is chosen as

*f*=21.5 at

*λ*=0.71

*µ*m (edge of red light) with the effective material parameters given by

*n*=0.159+

*i*0.094 and

*z*=0.973−

*i*0.108. The optimized geometrical parameter values were found to be

*p*=345 nm,

*w*=153 nm,

*s*=307 nm,

*t*=39.1 nm,

*d*=94.9 nm, and

*t*=20.0 nm. The total thickness was found to be 213 nm. Figure 2 shows plots of the effective material parameters for the optimized geometry as well as the reflectance (

_{m}*R*), transmittance (

*T*), and absorbance (

*A*) spectra with respect to the free space wavelength. Figure 2(a) shows that a very low index of refraction is achieved during the transition from a positive-index material to a negative-index material as the wavelength is increased. Figure 2(b) shows that the transmittance

*T*reaches the maximum value of 68 % at the optimal wavelength of

*λ*=0.71

*µ*m. The reduced

*T*from the ideal 96 % (corresponding to that of a free space-glass interface) is attributed to an imperfect impedance match together with a non-zero value of

*n*″. A magnetic resonance typically accompanies enhanced absorption [16

16. T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E **68**, 065602(R) (2003). [CrossRef]

*A*=29 % is attributed to the magnetic resonance around

*λ*=0.72

*µ*m that is associated with the immediately-following negative index band.

### 4.2. Impedance-matched unity-index metamaterial

*n*→1) with the impedance matched to free space (

*z*→1). This corresponds to the metamaterial “transparency” condition. The GA fitness function in this case was chosen to be

*f*=117 at

*λ*=0.69

*µ*m. The corresponding effective material parameters for this design are given by

*n*=0.959+

*i*0.019 and

*z*=0.974-

*i*0.077. The values of the optimal geometrical parameters were found to be

*p*=345 nm,

*w*=153 nm,

*s*=38.3 nm,

*t*=23.2 nm,

*d*=77.1 nm, and

*t*=56.8 nm. The effective material parameters along with the three spectra (i.e.,

_{m}*R, T*, and

*A*) are plotted with respect to wavelength in Fig. 3. In contrast to the previous design example, no negative index band is observed in Fig. 3(a). No magnetic resonance occurs at or near the optimal wavelength

*λ*=0.69

*µ*m so that the small value of

*A*=9.5 % allows the corresponding value of

*T*to reach 88 %. Although the structure was optimized to maximize the fitness at a single visible wavelength, it is noted from Fig. 3(b) that the metamaterial is highly transmissive with the value of

*T*maintained above 70 % over a wide spectral range from 0.52 (green) to 0.71

*µ*m (red). This may be attributed to the nearly real values which are close to unity for both

*n*and

*z*over the indicated range. Moreover, this is in sharp contrast to the range from 0.74 to 0.89 µm, over which the almost purely imaginary intrinsic impedance dramatically reduces

*T*below 30 % and significantly increases

*R*without seriously affecting

*A*.

*λ*=0.55

*µ*m (green light). A GA optimization for this problem with the fitness function given in (3) leads to a maximum fitness

*f*=34.1 associated with the optimized parameters

*n*=1.127+

*i*0.030 and

*z*=1.023–

*i*0.109.

*z*̂ direction (i.e., in the direction of propagation) were considered, in contrast to numerous bulk-type metamaterial realizations reported in the GHz and microwave regimes. This is mainly due to (i) significantly higher losses associated with metals at optical wavelengths and (ii) fabrication difficulty with multi-layer structures including maintaining structural uniformity in the

*z*̂ direction using lithographic processes. The latest fabrication and characterization efforts for optical metamaterials involve designs having 1–3 layers [3

3. 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**, 3356–3358 (2005). [CrossRef]

5. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. **32**, 53–55 (2007). [CrossRef]

22. G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic meta-material,” Opt. Lett. **32**, 551–553 (2007). [CrossRef] [PubMed]

## 5. Conclusion

*n*and the intrinsic impedance

*z*simply by properly adjusting the geometrical parameters of the electric and magnetic sub-structures. This design flexibility allows these LIMs to be employed in a variety of applications such as perfect electric and magnetic mirrors, optically transparent metamaterials, and perhaps even as building blocks for a cloak of invisibility in the visible spectrum.

*λ*=0.71

*µ*m (red light), where a transmittance value of 68 % was achieved. For the design targeted at achieving a perfect transparency condition, a value of

*T*=88%was realized at

*λ*=0.69

*µ*m.

## Acknowledgments

## References and links

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

2. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

3. | 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. |

4. | S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Demonstration of metaldielectric negative-index metamaterials with improved performance at optical frequencies,” J. Opt. Soc. Am. B |

5. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. |

6. | R.W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,”Phys. Rev. E |

7. | B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B |

8. | S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. |

9. | M. A. Gingrich and D. H. Werner, “Synthesis of low/zero index of refraction metamaterials from frequency selective surfaces using genetic algorithms,” Electron. Lett. |

10. | D.-H. Kwon, L. Li, J. A. Bossard, M. G. Bray, and D. H. Werner, “Zero index metamaterials with checkerboard structure,” Electron. Lett. |

11. | R. L. Haupt and D. H. Werner, |

12. | S. Chakravarty, R. Mittra, and N. R. Williams, “Application of a micro-genetic algorithm (MGA) to the design of broad-band microwave absorbers using multiple frequency selective surface screens buried in dielectrics,” IEEE Trans. Antennas Propag. |

13. | J. L. Volakis, A. Chatterjee, and L. C. Kempel, |

14. | D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

15. | A. V. Kildishev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and V. M. Shalaev, “Negative refractive index in optics of metal-dielectric composites,” J. Opt. Soc. Am. B |

16. | T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

17. | A. L. Efros, “Comment II on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

18. | T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Reply to comments on “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

19. | E. Saenz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. |

20. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

21. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

22. | G. Dolling, M. Wegener, and S. Linden, “Realization of a three-functional-layer negative-index photonic meta-material,” Opt. Lett. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 16, 2007

Revised Manuscript: July 4, 2007

Manuscript Accepted: July 6, 2007

Published: July 12, 2007

**Citation**

Do-Hoon Kwon and Douglas H. Werner, "Low-index metamaterial designs in the visible spectrum," Opt. Express **15**, 9267-9272 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9267

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
- 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, 3356-3358 (2005). [CrossRef]
- S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, "Demonstration of metaldielectric negative-index metamaterials with improved performance at optical frequencies," J. Opt. Soc. Am. B 23, 434-438 (2006). [CrossRef]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007). [CrossRef]
- R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004). [CrossRef]
- B. T. Schwartz and R. Piestun, "Total external reflection from metamaterials with ultralow refractive index," J. Opt. Soc. Am. B 20, 2448-2453 (2003). [CrossRef]
- S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]
- M. A. Gingrich and D. H. Werner, "Synthesis of low/zero index of refraction metamaterials from frequency selective surfaces using genetic algorithms," Electron. Lett. 41, 1266-1267 (2005).
- D.-H. Kwon, L. Li, J. A. Bossard, M. G. Bray, and D. H. Werner, "Zero index metamaterials with checkerboard structure," Electron. Lett. 43, 319-320 (2007). [CrossRef]
- R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics (Wiley, Hoboken, N J, 2007). [CrossRef]
- S. Chakravarty, R. Mittra, and N. R. Williams, "Application of a micro-genetic algorithm (MGA) to the design of broad-band microwave absorbers using multiple frequency selective surface screens buried in dielectrics," IEEE Trans. Antennas Propag. 53, 284-296 (2002). [CrossRef]
- J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, Piscataway, NJ, 1998). [CrossRef]
- D. R. Smith, S. Schultz, P. Markoˇs, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002). [CrossRef]
- A. V. Kildishev,W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and V. M. Shalaev, "Negative refractive index in optics of metal-dielectric composites," J. Opt. Soc. Am. B 23, 423-433 (2006). [CrossRef]
- T. Koschny, P. Markoˇs, D. R. Smith, and C. M. Soukoulis, "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 68, 065602(R) (2003). [CrossRef]
- A. L. Efros, "Comment II on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 70, 048602 (2004). [CrossRef]
- T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, "Reply to comments on "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,"Phys. Rev. E 70, 048603 (2004). [CrossRef]
- E. Saenz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, "On the definition of effective permittivity and permeability for thin composite layers," J. Appl. Phys. 101, 114910 (2007). [CrossRef]
- L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Butterworth- Heinemann, Oxford, 1984).
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- G. Dolling, M. Wegener, and S. Linden, "Realization of a three-functional-layer negative-index photonic metamaterial," Opt. Lett. 32, 551-553 (2007). [CrossRef] [PubMed]

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