## Spatial dispersion of multilayer fishnet metamaterials |

Optics Express, Vol. 20, Issue 14, pp. 15100-15105 (2012)

http://dx.doi.org/10.1364/OE.20.015100

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### Abstract

We study the anisotropic properties of multilayer fishnet optical metamaterials and describe topological transitions between the elliptic and hyperbolic dispersion regimes. In contrast to other hyperbolic media, multilayer fishnet metamaterials may have negative components not only in the effective permittivity tensor but also in the effective permeability tensor, thus allowing the realization of magnetic hyperbolic and generalized indefinite media.

© 2012 OSA

## 1. Introduction

1. D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

*electric hyperbolic media*appear as effective media in the theories describing the averaged characteristics of alternating dielectric and metallic layers and lattices created by metallic wires, or plasmonic crystals of nanorods [2

2. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B **73**, 075103 (2006). [CrossRef]

6. L. M. Custodio, C. T. Sousa, J. Ventura, J. M. Teixeira, P. V. S. Marques, and J. P. Araujo, “Birefringence swap at the transition to hyperbolic dispersion in metamaterilas,” Phys. Rev. B **85**, 165408 (2012). [CrossRef]

2. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B **73**, 075103 (2006). [CrossRef]

3. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B **74**, 075103 (2006). [CrossRef]

7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science **336**, 205–209 (2012). [CrossRef] [PubMed]

*magnetic hyperbolic media*with a highly anisotropic

*μ*-tensor, here we suggest utilizing multilayer optical fishnet metamaterials with artificial magnetism [9

9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature **455**, 376–379 (2008). [CrossRef] [PubMed]

10. M. Beruete, M. Navarro-Cia, and M. Sorolla, “High numerical aperture and low-loss negative refraction based on the fishnet rich anisotropy,” Photonics Nanostruct. Fundam. Appl. **10**(3), 263–270 (2012). [CrossRef]

11. M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. **12**, 063037 (2010). [CrossRef]

## 2. Optical properties of multilayer fishnet metamaterials

_{2}) layers. Figs. 1(b) and 1(c) show the single-layer fishnet consisting of two Ag layers, separated by a MgF

_{2}layer and multilayer fishnet structures respectively. We numerically study their optical properties using CST Microwave Studio. For the permittivity of Ag we use experimental data from Ref. [12], while for MgF

_{2}we use fixed permittivity of 1.90. Without lack of generality we assume the electric vector

*E⃗*to be in the

*x*−

*z*coordinate plane.

_{2}layers. Figure 1(g) also shows the transmission versus wavelength and increasing number of layers. An important conclusion from these simulations is that the addition of more functional layers doesn’t decrease dramatically the overall transmission through the metamaterial. As such the multilayer fishnet has been favored as a good example of a bulk metamaterial [9

9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature **455**, 376–379 (2008). [CrossRef] [PubMed]

*k*by using the inverted Fresnel formula [13

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

*h*is the thickness of the structure,

*r*and

*t*are the complex reflection and transmission coefficients and

*m*is an integer branch number. The criteria for choosing the correct sign and branch number can be found in Refs. [13

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

*n*=

*k*/

*k*

_{0}, where

*k*

_{0}is the wavenumber in free space; and the figure of merit

*FOM*=

*Re*[

*n*]/

*Im*[

*n*]. Figures 1(e,f) show the calculated spectra for

*n*and

*FOM*for the single-layer and 22-layer structures. It is seen that for the 22-layer structure the spectral region with negative

*Re*[

*n*] becomes broader, absorption

*Im*[

*n*] becomes smaller and

*FOM*increases dramatically. Alternatively, Figs. 1(h,i) show

*n*and

*FOM*versus wavelength and number of layers. Similar to Ref. [15

15. J. Zhou, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B. **80**, 035109 (2009). [CrossRef]

*E*component at wavelength 1.25

_{x}*μ*m through the

*y*−

*z*plane of the 22 layer structure. Mode analysis of these fields confirm the values of

*k*and

*n*obtained by Eq. (1). However, to differentiate between the forward and backward-wave regimes at normal incidence, it is necessary to view the time variation of the fields, which is shown in Media 1. The left column in the Media 1 corresponds to a wavelength of 0.86 μm and positive values of

*Re*[

*k*] and

*Re*[

*n*]. As we can see, for this case the phase velocity inside the metamaterial has a forward direction. The right column of the Media 1 corresponds to a wavelength of 1.25 μm and negative values of

*Re*[

*k*] and

*Re*[

*n*]. In this case we can indeed observe a backward direction of phase velocity inside the structure, corresponding to the negative index of the metamaterial. The upper and middle rows correspond to electric

*E*and magnetic

_{x}*H*transverse components, respectively. The lower row corresponds to the longitudinal component of electric field

_{y}*E*, which represents the displacement current through the dielectric layers induced by the magnetic field. Importantly, the

_{z}*E*component is strong only in the negative-index spectral region and is a signature of the artificial magnetic response of the metamaterial. The effective circuits of current and displacement current cover several layers of the structure, which leads to the qualitative difference between multilayer and single-layer fishnet metamaterials [9

_{z}9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature **455**, 376–379 (2008). [CrossRef] [PubMed]

## 3. Response at oblique incidence

*x*−

*z*plane. The two cases of TE and TM polarizations are shown in Figs. 3(a,e). A snapshot of the

*E*component is shown in Fig. 2(b) for 1.20 μm wavelength. Media 2 demonstrates the propagation of the transverse component of the electric field through the structure for the cases of: (a,b) normal incidence at wavelength 1.25 μm; (c,d) TE polarization, 45° angle of incidence at wavelength 1.23 μm; and (e,f) TM polarization, 45° angle of incidence at wavelength 1.42 μm. Here we can directly observe the backward direction of phase velocity and negative refraction.

_{x}*k⃗*inside the metamaterial. For the

*k*component we use Eq. (1) while the boundary conditions ensure that

_{z}*k*and

_{x}*k*are continuous at the interface. Figure 3(c) shows dispersion

_{y}*Re*[

*k*] versus

_{z}*k*above the light cone for three different wavelengths for the TE-polarization. At wavelength 1.03 μm (red curve)

_{y}*Re*[

*k*] > 0 and the structure exhibits ordinary elliptic dispersion. At wavelength 1.17 μm (blue curve)

_{z}*Re*[

*k*] < 0 and the structure exhibits

_{z}*hyperbolic dispersion*[1

1. D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

18. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

*Re*[

*k*] = 0 and the structure is in a regime of topological transition between elliptic and hyperbolic dispersions [7

_{z}7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science **336**, 205–209 (2012). [CrossRef] [PubMed]

19. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. **97**, 157403 (2006). [CrossRef] [PubMed]

11. M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. **12**, 063037 (2010). [CrossRef]

*k*, but as

_{x}*k*increases it crosses through zero and becomes a forward-wave. This zero

_{x}*k*feature corresponds to excitation of the gap plasmon, which is responsible for the effective magnetic response of the metamaterial. As seen in the transmission plot [Fig. 3(f)] this mode has strong dispersion for TM incident polarization. The change of incident angle shifts this plasmon resonance, causing the negative index condition to be broken and resulting in a change of sign in

_{z}*k*. For the zero-index wavelength of 1.09 μm the dispersion exhibits an X-shaped contour, as shown by the green curve, while for the positive index wavelength of 1.03 μm the dispersion is twin V-shaped contours. In all these cases the contours are diverging for larger

_{z}*k*, indicating that this media should exhibit number of indefinite medium properties, such as high density of states and strong Purcell factor enhancement.

_{x}*n*

_{eff}cos(

*θ*) =

*k*/

_{z}*k*

_{0}, where

*θ*is the angle of refraction, and then define the real and the imaginary parts as [20]: where Here

*ϕ*is the angle of incidence.

*Re*[

*n*] for three different angles of incidence for the TE and TM polarizations, respectively. The red curve corresponds to

*Re*[

*n*] at normal incidence. The dashed green and blue curves correspond to

*Re*[

*n*] at oblique incidence with the sign chosen from the sign of

*k*to yield results consistent with the normal incidence case. The solid green and blue curves correspond to |

_{z}*Re*[

*n*]|. As one can see, for oblique incidence the

*Re*[

*n*] curves become discontinuous while the |

*Re*[

*n*]| curves are continuous. For the TE case, the break in

*Re*[

*n*] corresponds to the topological transition between elliptic and hyperbolic dispersion. For the case of normal incidence along a high symmetry direction, the sign of

*Re*[

*n*] defines whether the group and phase velocity are parallel or antiparallel. But for the case of oblique incidence, the phase and group velocities are not in general collinear, therefore the relationship between them cannot be adequately characterized using the sign of the refractive index. Thus even an angle-dependent scalar refractive index is not generally meaningful in such media [14].

## 4. Effective parameters

*ε*̂ and permeability

*μ*̂ having zero non-diagonal elements. All diagonal components are functions of frequency

*ω*due to the resonant behavior of the metamaterial, in addition, they are complex due to high losses. Our coordinate system is chosen in such a way that the main axes of the tensors coincide with

*x*,

*y*and

*z*coordinate axes. Furthermore

*μ*= 1 as the structure does not exhibit artificial magnetism for the

_{z}*H*component. As we consider

_{z}*E⃗*to be in the

*x*−

*z*plane, the components

*ε*and

_{y}*μ*do not contribute to the response in our simulations.

_{x}*k⃗*can be derived from Maxwell’s equations: For the TE case, the dispersion curves obtained with Eq. (4) fit numerical data if

*ε*= 0.45,

_{x}*μ*= 0.18 for the elliptical dispersion [Fig. 3(c), red curve] and

_{y}*ε*= −0.20,

_{x}*μ*= −0.48 for the hyperbolic dispersion [Fig. 3(c), blue curve]. For the TM case, we can describe the dispersion curves plotted in Fig. 3(g) with Eq. (4) only if we assume

_{y}*ε*to be dependent on

_{x}*k*. Therefore, to describe our numerical data the permittivity and permeability must take the form: Importantly, the hyperbolic dispersion observed in the negative index band has contributions from both of these tensors, in contrast to hyperbolic media reported to date where all components of

_{x}*μ*̂ are 1. Thus the multilayered fishnet is a

*general form of indefinite media*incorporating both the electric and magnetic responses. Similar to the earlier studies of the electric hyperbolic media [21

21. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B **84**, 115438 (2011). [CrossRef]

22. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B **84**, 045424 (2011). [CrossRef]

## 5. Conclusions

*ε*̂ and

*μ*̂ tensors having negative components. The multi-functional properties of multilayer fishnet metamaterials including their magnetic hyperbolic dispersion open a number of novel applications, including control and enhancement of spontaneous emission.

## Acknowledgments

## References and links

1. | D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

2. | P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B |

3. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B |

4. | M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. |

5. | A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B |

6. | L. M. Custodio, C. T. Sousa, J. Ventura, J. M. Teixeira, P. V. S. Marques, and J. P. Araujo, “Birefringence swap at the transition to hyperbolic dispersion in metamaterilas,” Phys. Rev. B |

7. | H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science |

8. | E. Narimanov and I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444v1. |

9. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature |

10. | M. Beruete, M. Navarro-Cia, and M. Sorolla, “High numerical aperture and low-loss negative refraction based on the fishnet rich anisotropy,” Photonics Nanostruct. Fundam. Appl. |

11. | M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. |

12. | E. D. Palik, |

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

14. | C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B |

15. | J. Zhou, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B. |

16. | A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Yu. S. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B |

17. | C. Garcia-Meca, J. Hurtado, J. Marti, A. Martinez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. |

18. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

19. | M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. |

20. | M. Born and E. Wolf, |

21. | A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B |

22. | A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B |

**OCIS Codes**

(260.2065) Physical optics : Effective medium theory

(350.3618) Other areas of optics : Left-handed materials

(160.3918) Materials : Metamaterials

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 11, 2012

Revised Manuscript: June 5, 2012

Manuscript Accepted: June 8, 2012

Published: June 20, 2012

**Citation**

Sergey S. Kruk, David A. Powell, Alexander Minovich, Dragomir N. Neshev, and Yuri S. Kivshar, "Spatial dispersion of multilayer fishnet metamaterials," Opt. Express **20**, 15100-15105 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15100

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### References

- D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003). [CrossRef] [PubMed]
- P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B73, 075103 (2006). [CrossRef]
- A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006). [CrossRef]
- M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett.94, 151105 (2009). [CrossRef]
- A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009). [CrossRef]
- L. M. Custodio, C. T. Sousa, J. Ventura, J. M. Teixeira, P. V. S. Marques, and J. P. Araujo, “Birefringence swap at the transition to hyperbolic dispersion in metamaterilas,” Phys. Rev. B85, 165408 (2012). [CrossRef]
- H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science336, 205–209 (2012). [CrossRef] [PubMed]
- E. Narimanov and I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444v1.
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455, 376–379 (2008). [CrossRef] [PubMed]
- M. Beruete, M. Navarro-Cia, and M. Sorolla, “High numerical aperture and low-loss negative refraction based on the fishnet rich anisotropy,” Photonics Nanostruct. Fundam. Appl.10(3), 263–270 (2012). [CrossRef]
- M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys.12, 063037 (2010). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002). [CrossRef]
- C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
- J. Zhou, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B.80, 035109 (2009). [CrossRef]
- A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Yu. S. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B81, 115109 (2010). [CrossRef]
- C. Garcia-Meca, J. Hurtado, J. Marti, A. Martinez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett.106, 083104 (2011). [CrossRef]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8247–8256 (2006). [CrossRef] [PubMed]
- M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1959).
- A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011). [CrossRef]
- A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011). [CrossRef]

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