- Email: [email protected]
Permeability enhancement of stratified metal dielectric metamaterial in optical regime
Available online at www.sciencedirect.com
Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328 www.elsevier.com/locate/photonics
Permeability enhancement of stratified metal dielectric metamaterial in optical regime Aunuddin Syabba Vioktalamo, Ryosuke Watanabe, Teruya Ishihara * Department of Physics, Tohoku University, 6-3 Aramaki-Aoba, Sendai 980-8578, Japan Received 31 December 2010; received in revised form 19 July 2011; accepted 12 August 2011 Available online 22 August 2011
Abstract Magnetic response of stratified metal dielectric metamaterial (SMDM) is demonstrated numerically and experimentally. One unit cell of SMDM has a sandwich unit cell consisting of alumina (60 nm)/silver (30 nm)/alumina (60 nm). A Mach–Zehnder interferometer is used to obtain phase information of transmittance and reflectance from which effective permeability is determined. The maximum permeability amounts to 20 and 17 for calculation and experiment, respectively. This huge resonance occurs when the magnetic field is concentrated at the metal layer, while the electric field has a node at the center. # 2011 Elsevier B.V. All rights reserved. Keywords: Permeability enhancement; Metamaterials; Optical magnetism
1. Introduction In typical metamaterial, an artificial structure unit is small enough compared to wavelength, which replaces the role of atom and molecule in conventional materials. Consequently, the composite can be regarded as an effective new medium and be well described by effective medium parameters [1]. Exotic electromagnetic response is attributed to the unusual inhomogeneous electromagnetic distribution in a metamaterial unit cell structure at particular frequencies. But special unit cell design is not the only way to realize an inhomogeneous electromagnetic distribution. Stratified metal dielectric metamaterial, hereafter referred to as SMDM, is an artificial metal dielectric composite that consists of metal sandwiched by dielectric layers. This
* Corresponding author. Tel.: +81 22 795 6420; fax: +81 22 795 6425. E-mail address: [email protected] (T. Ishihara). 1569-4410/$ – see front matter # 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2011.08.005
multilayer structure is predicted to have negative refraction and super resolution properties [2]. In such a structure, it has been shown that internal magnetic field is enhanced [3,4] and s-polarization Brewster angle was observed, which may suggest non-unity relative permeability [5,6], although permeability in the constituent materials is essentially identical with permeability in vacuum at optical frequency [7]. In this paper, we investigate optical response of SMDM numerically and experimentally and describe the response in terms of effective permeability. The frequency range we are interested in is just below the first photonic band gap due to periodicity of the sample. Usually the optical response in this frequency range is simply ascribed to the photonic crystal effect. The aim of this article is to go one further step to generalize the concept of effective medium to the structure with much larger unit cell size. Numerical calculation was done by following the standard retrieval procedure from complex transmission and reflection coefficients [8]. We regard the artificial structure to be a hypothetical
326
A.S. Vioktalamo et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328
ability can be determined by knowing the complex transmittance ˜t and complex reflectance r˜. By following the retrieval procedure [10], the set of optical properties of SMDM on a substrate with refractive index ns can be calculated. The first step is to express transmittance and reflectance in terms of 2 2 transfer matrix M, ˜t 1 ¼M ; 0 r˜ Fig. 1. Schematic structure of one unit cell SMDM consisting of Al2O3 (60 nm)/Ag (30 nm)/Al2O3 (60 nm). The light propagate in zdirection at normal incident, while magnetic field (H) and electric field (E) in x- and y- direction, respectively.
uniform material, which can be described in terms of effective permittivity and effective permeability. Mach– Zehnder interferometer was employed to obtain phase of our three-period SMDM. Huge resonance appears in both calculation and experiment. 2. Theory of SMDM One unit cell of SMDM is defined by Ag (30 nm) sandwiched by identical Al2O3 (60 nm) as shown in Fig. 1. It is noteworthy that this unit cell design does not have any resonant frequency due to the structure unlike a metal/dielectric/metal structure [9]. In SMDM with thickness d, effective permittivity and effective perme-
Impedance Z˜ and refractive index n˜ can be expressed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ r˜Þ2 ˜t2 Z˜ ¼ ; ð1 r˜Þ2 ns 2 ˜t2 ! c 1 r˜2 þ ns ˜t2 1 1 cos : n˜ ¼ ˜t ð1 þ ns Þ þ ðns 1Þ˜r vd ˜ As the ˜ ¼ n˜ Z. Then the permeability is obtained to be m optical constant of silver, we took it from literature [11], and for optical constant of alumina we used a constant (e = 2.7). 3. Experimental Three-period SMDM with symmetric unit cell is used in our experiment. The sample was grown on 0.46 mm sapphire substrate using the RF magnetron sputtering method, where the sputtering rate of Ag and
Fig. 2. Optical spectra of three periods SMDM. (a) Reflectance. (b) Transmittance. (c) Phase of Reflectance (wR). (d) Phase of Transmittance (wT). Open circles correspond to measured data, while a solid curve corresponds to calculation.
A.S. Vioktalamo et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328
327
from a halogen lamp instead of diode laser with various wavelengths, making very time efficient in measurement. The first walk-off prism splits incoming light into two arms based on polarization state. After rotating polarization 908 by using a half waveplate, the second walk-off prism recombines the light. Interference appears when a sample is placed at one of the arms due to the difference in optical length. As a compensator of the phase delay, the liquid crystal variable retarder is used. Phase measurement for reflection is carried out, using similar configuration by adding a beam splitter for the normal incident condition. Transmission and reflection data are measured in the same system by blocking the reference arm. Fig. 2 shows the reflectance, transmittance, phase of reflectance (wR) and phase of transmittance (wT) retrieved from this set up. Fig. 3. Numerical calculation of magnetic field (Hx) and electric field (Ey) distribution at resonance condition. The magnetic field is concentrated in the metal layer, while electric field diminishes at the center.
Al2O3 were 1 and 0.01 nm/s, respectively. The roughness of the surface of each layer estimated with an atomic force microscope was below 1 nm. The comparison with the normal calculation of the optical spectra shows that the thickness of Al2O3 was slightly smaller, 57 nm. A part of the substrate is left undeposited, which is used as a phase reference. Several interferometric characterization were operated to extract phase information in metamaterial [12– 16]. Basically, we employed a Mach–Zehnder interferometer [16] for phase measurement by using walk-off prism and retarder, except that we use a white light source
4. Results and discussion The microscopic electromagnetic distribution Hx and Ey at the permeability resonance is illustrated in Fig. 3. It was calculated for a sample with many unit cells in order to suppress the reflected wave from the interface. The magnetic field is strongly enhanced around the dielectric–metal–dielectric interfaces, while the electric field has a node inside the metal layer. Note that in microscopic field distribution, B-field is identical with H-field because permeability is identical to that of vacuum for each layer. The cusp-like features for H-field at the dielectric–metal interface is explained by Ampere-Maxwell equation with different sign of permittivity in Ag and alumina. The effective permeability is shown in Fig. 4. The open circles are experimental data, while the solid line
Fig. 4. Relative permeability as a function of wavelength. Solid curve and open circles are deduced from complex transmission and reflection spectra obtained from the transfer matrix calculation for the assumed structure and experiments, respectively. The huge resonance appearing at 472 nm is due to inhomogeneous field distribution in a unit cell. The dashed line corresponds to permeability in vacuum.
328
A.S. Vioktalamo et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328
is calculation. The real part of permeability is about two in the region between 500 and 600 nm and increases rapidly up to the first photonic band gap at 472 nm. The largest permeability is 17 for the experiment and 20 for the numerical calculation with the transfer matrix. The origin of permeability is due to inhomogeneity of the field distribution in the unit cell [1]. Due to the different nature of B-field and H-field, the inhomogeneous microscopic magnetic field distribution in the artificial structure results in the non-unity permeability. The resonant condition occurs when the half wavelength of the light in the structure matches to the unit cell size, when the microscopic magnetic field is concentrated in the metal layer as shown in Fig. 3. The structure in the experimental data of permeability around 525 nm, which corresponds to the dip in reflection spectrum, is not reproduced by the numerical calculation. This dip in the reflection spectrum corresponds to one of the Fabry–Perot interference resonance for three periods SMDM. Recently, we found that by introducing fluctuation in unit cell size, it was possible to reproduce the resonance around 525 nm, which will be reported in a separate paper. 5. Conclusion We have investigated the optical response of SMDM and by regarding the structure as an effective material, we showed that the structure can be considered as an effective material with large permeability just below the photonic band gap frequency, although the structure consists of non-magnetic materials. Investigation of SMDM is the first step to explore the novel type of periodic electromagnetic metamaterials. In spite of the lack of resonance in a unit cell, even such a simple structure gives artificial magnetic response originated from the periodicity in the sample. We can control the resonance by modifying the thickness of each slab, or choosing another type of metal–dielectric materials. Obviously additional structure in each layer enhances great possibility of functions and physics in metamaterials. Acknowledgement The authors are grateful to Dr. Seigo Ohno for fruitful discussion. This work was partly supported by Grant-in-Aid for Scientific Research on Innovative Areas ‘‘Electromagnetic Metamaterials’’ (22109005).
References [1] J.B. Pendry, A.J. Holden, D.J. Robbin, W.J. Steward, Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075–2084. [2] M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, D. Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M.G. Cappeddu, M. Fowler, J.W. Haus, Negative refraction and subwavelength focusing in the visible range using transparent metallo-dielectric stacks, Opt. Express 15 (2007) 508–523. [3] L. Zhou, W. Wen, C.T. Chan, P. Sheng, Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields, Phys. Rev. Lett. 94 (2005) 243905. [4] G. Shvets, Y.A. Urzhumov, Negative index meta-materials based on two-dimesional metallic structures, J. Opt. A: Pure Appl. Opt. 8 (2006) S122–S130. [5] Y. Tamayama, T. Nakanishi, K. Sugiyama, M. Kitano, Observation of Brewster’s effct for transverse-electric electromagnetic waves in metamaterials: experiment and theory, Phys. Rev. B 73 (2006) 193104. [6] R. Watanabe, M. Iwanaga, T. Ishihara, s-polarization Brewster’s angle of stratified metal–dielectric metamaterial in optical regime, Phys. Status Solidi B 245 (2008) 2696–2701. [7] V.M. Shalaev, Optical negative-index metamaterials, Nat. Photonics 1 (2007) 41–48. [8] D.R. Smith, S. Schultz, P. Markos, C.M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B 65 (2002) 195104. [9] H. Daninthe, S. Foteinopoulou, C.M. Soukoulis, Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials, Photonics Nanostruct. 4 (2006) 123–131. [10] A.V. Kildishev, W. Cai, U.K. Chettiar, H.K. Yuan, A.K. Sarychev, V.P. Drachev, V.M. Shalaev, Negative refractive index in optics of metal–dielectric composites, J. Opt. Soc. Am. B 23 (2006) 423–431. [11] P.B. Johnson, R.W. Christy, Optical constants of noble metals, Phys. Rev. B 6 (1972) 4370–4379. [12] L. Lepetit, G. Che´riaux, M. Joffre, Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectrocopy, J. Opt. Soc. Am. B 12 (1995) 2467–2474. [13] V.P. Drachev, W. Cai, U. Chettiar, H.K. Yuan, A.K. Sarychev, A.V. Kildishev, G. Klimeck, V.M. Shalaev, Experimental verification of an optical negative-index material, Laser Phys. Lett. 3 (2006) 49–55. [14] G. Dolling, C. Enkrich, M. Wegener, C.M. Soukoulis, S. Linden, Simultaneous negative phase and group velocity of light in a metamaterial, Science 312 (2006) 892–894. [15] E. Pshenay-Severin, F. Setzpfandt, C. Helgert, U. Hu¨bner, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tu¨nnermann, F. Ladered, T. Pertsch, Experimental determination of the dispersion relation of light in metamaterials by white-light interferometry, J. Opt. Soc. Am. B 27 (2010) 660–666. [16] X. Zhang, M. Davanco, K. Maller, T.W. Jarvis, C. Wu, C. Fietz, D. Korobkin, X. Li, G. Shvets, S.R. Forrest, Interferometric characterization of a sub-wavelegth near-infrared negative index metamaterial, Opt. Express 18 (2010) 17788–17794.
Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328 www.elsevier.com/locate/photonics
Permeability enhancement of stratified metal dielectric metamaterial in optical regime Aunuddin Syabba Vioktalamo, Ryosuke Watanabe, Teruya Ishihara * Department of Physics, Tohoku University, 6-3 Aramaki-Aoba, Sendai 980-8578, Japan Received 31 December 2010; received in revised form 19 July 2011; accepted 12 August 2011 Available online 22 August 2011
Abstract Magnetic response of stratified metal dielectric metamaterial (SMDM) is demonstrated numerically and experimentally. One unit cell of SMDM has a sandwich unit cell consisting of alumina (60 nm)/silver (30 nm)/alumina (60 nm). A Mach–Zehnder interferometer is used to obtain phase information of transmittance and reflectance from which effective permeability is determined. The maximum permeability amounts to 20 and 17 for calculation and experiment, respectively. This huge resonance occurs when the magnetic field is concentrated at the metal layer, while the electric field has a node at the center. # 2011 Elsevier B.V. All rights reserved. Keywords: Permeability enhancement; Metamaterials; Optical magnetism
1. Introduction In typical metamaterial, an artificial structure unit is small enough compared to wavelength, which replaces the role of atom and molecule in conventional materials. Consequently, the composite can be regarded as an effective new medium and be well described by effective medium parameters [1]. Exotic electromagnetic response is attributed to the unusual inhomogeneous electromagnetic distribution in a metamaterial unit cell structure at particular frequencies. But special unit cell design is not the only way to realize an inhomogeneous electromagnetic distribution. Stratified metal dielectric metamaterial, hereafter referred to as SMDM, is an artificial metal dielectric composite that consists of metal sandwiched by dielectric layers. This
* Corresponding author. Tel.: +81 22 795 6420; fax: +81 22 795 6425. E-mail address: [email protected] (T. Ishihara). 1569-4410/$ – see front matter # 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2011.08.005
multilayer structure is predicted to have negative refraction and super resolution properties [2]. In such a structure, it has been shown that internal magnetic field is enhanced [3,4] and s-polarization Brewster angle was observed, which may suggest non-unity relative permeability [5,6], although permeability in the constituent materials is essentially identical with permeability in vacuum at optical frequency [7]. In this paper, we investigate optical response of SMDM numerically and experimentally and describe the response in terms of effective permeability. The frequency range we are interested in is just below the first photonic band gap due to periodicity of the sample. Usually the optical response in this frequency range is simply ascribed to the photonic crystal effect. The aim of this article is to go one further step to generalize the concept of effective medium to the structure with much larger unit cell size. Numerical calculation was done by following the standard retrieval procedure from complex transmission and reflection coefficients [8]. We regard the artificial structure to be a hypothetical
326
A.S. Vioktalamo et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328
ability can be determined by knowing the complex transmittance ˜t and complex reflectance r˜. By following the retrieval procedure [10], the set of optical properties of SMDM on a substrate with refractive index ns can be calculated. The first step is to express transmittance and reflectance in terms of 2 2 transfer matrix M, ˜t 1 ¼M ; 0 r˜ Fig. 1. Schematic structure of one unit cell SMDM consisting of Al2O3 (60 nm)/Ag (30 nm)/Al2O3 (60 nm). The light propagate in zdirection at normal incident, while magnetic field (H) and electric field (E) in x- and y- direction, respectively.
uniform material, which can be described in terms of effective permittivity and effective permeability. Mach– Zehnder interferometer was employed to obtain phase of our three-period SMDM. Huge resonance appears in both calculation and experiment. 2. Theory of SMDM One unit cell of SMDM is defined by Ag (30 nm) sandwiched by identical Al2O3 (60 nm) as shown in Fig. 1. It is noteworthy that this unit cell design does not have any resonant frequency due to the structure unlike a metal/dielectric/metal structure [9]. In SMDM with thickness d, effective permittivity and effective perme-
Impedance Z˜ and refractive index n˜ can be expressed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ r˜Þ2 ˜t2 Z˜ ¼ ; ð1 r˜Þ2 ns 2 ˜t2 ! c 1 r˜2 þ ns ˜t2 1 1 cos : n˜ ¼ ˜t ð1 þ ns Þ þ ðns 1Þ˜r vd ˜ As the ˜ ¼ n˜ Z. Then the permeability is obtained to be m optical constant of silver, we took it from literature [11], and for optical constant of alumina we used a constant (e = 2.7). 3. Experimental Three-period SMDM with symmetric unit cell is used in our experiment. The sample was grown on 0.46 mm sapphire substrate using the RF magnetron sputtering method, where the sputtering rate of Ag and
Fig. 2. Optical spectra of three periods SMDM. (a) Reflectance. (b) Transmittance. (c) Phase of Reflectance (wR). (d) Phase of Transmittance (wT). Open circles correspond to measured data, while a solid curve corresponds to calculation.
A.S. Vioktalamo et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328
327
from a halogen lamp instead of diode laser with various wavelengths, making very time efficient in measurement. The first walk-off prism splits incoming light into two arms based on polarization state. After rotating polarization 908 by using a half waveplate, the second walk-off prism recombines the light. Interference appears when a sample is placed at one of the arms due to the difference in optical length. As a compensator of the phase delay, the liquid crystal variable retarder is used. Phase measurement for reflection is carried out, using similar configuration by adding a beam splitter for the normal incident condition. Transmission and reflection data are measured in the same system by blocking the reference arm. Fig. 2 shows the reflectance, transmittance, phase of reflectance (wR) and phase of transmittance (wT) retrieved from this set up. Fig. 3. Numerical calculation of magnetic field (Hx) and electric field (Ey) distribution at resonance condition. The magnetic field is concentrated in the metal layer, while electric field diminishes at the center.
Al2O3 were 1 and 0.01 nm/s, respectively. The roughness of the surface of each layer estimated with an atomic force microscope was below 1 nm. The comparison with the normal calculation of the optical spectra shows that the thickness of Al2O3 was slightly smaller, 57 nm. A part of the substrate is left undeposited, which is used as a phase reference. Several interferometric characterization were operated to extract phase information in metamaterial [12– 16]. Basically, we employed a Mach–Zehnder interferometer [16] for phase measurement by using walk-off prism and retarder, except that we use a white light source
4. Results and discussion The microscopic electromagnetic distribution Hx and Ey at the permeability resonance is illustrated in Fig. 3. It was calculated for a sample with many unit cells in order to suppress the reflected wave from the interface. The magnetic field is strongly enhanced around the dielectric–metal–dielectric interfaces, while the electric field has a node inside the metal layer. Note that in microscopic field distribution, B-field is identical with H-field because permeability is identical to that of vacuum for each layer. The cusp-like features for H-field at the dielectric–metal interface is explained by Ampere-Maxwell equation with different sign of permittivity in Ag and alumina. The effective permeability is shown in Fig. 4. The open circles are experimental data, while the solid line
Fig. 4. Relative permeability as a function of wavelength. Solid curve and open circles are deduced from complex transmission and reflection spectra obtained from the transfer matrix calculation for the assumed structure and experiments, respectively. The huge resonance appearing at 472 nm is due to inhomogeneous field distribution in a unit cell. The dashed line corresponds to permeability in vacuum.
328
A.S. Vioktalamo et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 325–328
is calculation. The real part of permeability is about two in the region between 500 and 600 nm and increases rapidly up to the first photonic band gap at 472 nm. The largest permeability is 17 for the experiment and 20 for the numerical calculation with the transfer matrix. The origin of permeability is due to inhomogeneity of the field distribution in the unit cell [1]. Due to the different nature of B-field and H-field, the inhomogeneous microscopic magnetic field distribution in the artificial structure results in the non-unity permeability. The resonant condition occurs when the half wavelength of the light in the structure matches to the unit cell size, when the microscopic magnetic field is concentrated in the metal layer as shown in Fig. 3. The structure in the experimental data of permeability around 525 nm, which corresponds to the dip in reflection spectrum, is not reproduced by the numerical calculation. This dip in the reflection spectrum corresponds to one of the Fabry–Perot interference resonance for three periods SMDM. Recently, we found that by introducing fluctuation in unit cell size, it was possible to reproduce the resonance around 525 nm, which will be reported in a separate paper. 5. Conclusion We have investigated the optical response of SMDM and by regarding the structure as an effective material, we showed that the structure can be considered as an effective material with large permeability just below the photonic band gap frequency, although the structure consists of non-magnetic materials. Investigation of SMDM is the first step to explore the novel type of periodic electromagnetic metamaterials. In spite of the lack of resonance in a unit cell, even such a simple structure gives artificial magnetic response originated from the periodicity in the sample. We can control the resonance by modifying the thickness of each slab, or choosing another type of metal–dielectric materials. Obviously additional structure in each layer enhances great possibility of functions and physics in metamaterials. Acknowledgement The authors are grateful to Dr. Seigo Ohno for fruitful discussion. This work was partly supported by Grant-in-Aid for Scientific Research on Innovative Areas ‘‘Electromagnetic Metamaterials’’ (22109005).
References [1] J.B. Pendry, A.J. Holden, D.J. Robbin, W.J. Steward, Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075–2084. [2] M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, D. Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M.G. Cappeddu, M. Fowler, J.W. Haus, Negative refraction and subwavelength focusing in the visible range using transparent metallo-dielectric stacks, Opt. Express 15 (2007) 508–523. [3] L. Zhou, W. Wen, C.T. Chan, P. Sheng, Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields, Phys. Rev. Lett. 94 (2005) 243905. [4] G. Shvets, Y.A. Urzhumov, Negative index meta-materials based on two-dimesional metallic structures, J. Opt. A: Pure Appl. Opt. 8 (2006) S122–S130. [5] Y. Tamayama, T. Nakanishi, K. Sugiyama, M. Kitano, Observation of Brewster’s effct for transverse-electric electromagnetic waves in metamaterials: experiment and theory, Phys. Rev. B 73 (2006) 193104. [6] R. Watanabe, M. Iwanaga, T. Ishihara, s-polarization Brewster’s angle of stratified metal–dielectric metamaterial in optical regime, Phys. Status Solidi B 245 (2008) 2696–2701. [7] V.M. Shalaev, Optical negative-index metamaterials, Nat. Photonics 1 (2007) 41–48. [8] D.R. Smith, S. Schultz, P. Markos, C.M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B 65 (2002) 195104. [9] H. Daninthe, S. Foteinopoulou, C.M. Soukoulis, Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials, Photonics Nanostruct. 4 (2006) 123–131. [10] A.V. Kildishev, W. Cai, U.K. Chettiar, H.K. Yuan, A.K. Sarychev, V.P. Drachev, V.M. Shalaev, Negative refractive index in optics of metal–dielectric composites, J. Opt. Soc. Am. B 23 (2006) 423–431. [11] P.B. Johnson, R.W. Christy, Optical constants of noble metals, Phys. Rev. B 6 (1972) 4370–4379. [12] L. Lepetit, G. Che´riaux, M. Joffre, Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectrocopy, J. Opt. Soc. Am. B 12 (1995) 2467–2474. [13] V.P. Drachev, W. Cai, U. Chettiar, H.K. Yuan, A.K. Sarychev, A.V. Kildishev, G. Klimeck, V.M. Shalaev, Experimental verification of an optical negative-index material, Laser Phys. Lett. 3 (2006) 49–55. [14] G. Dolling, C. Enkrich, M. Wegener, C.M. Soukoulis, S. Linden, Simultaneous negative phase and group velocity of light in a metamaterial, Science 312 (2006) 892–894. [15] E. Pshenay-Severin, F. Setzpfandt, C. Helgert, U. Hu¨bner, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tu¨nnermann, F. Ladered, T. Pertsch, Experimental determination of the dispersion relation of light in metamaterials by white-light interferometry, J. Opt. Soc. Am. B 27 (2010) 660–666. [16] X. Zhang, M. Davanco, K. Maller, T.W. Jarvis, C. Wu, C. Fietz, D. Korobkin, X. Li, G. Shvets, S.R. Forrest, Interferometric characterization of a sub-wavelegth near-infrared negative index metamaterial, Opt. Express 18 (2010) 17788–17794.