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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23511–23521
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Hybrid metal-dielectric ring resonators for homogenizable optical metamaterials with strong magnetic response at short wavelengths down to the ultraviolet range

Jianwei Tang and Sailing He  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23511-23521 (2013)
http://dx.doi.org/10.1364/OE.21.023511


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Abstract

We derive an analytical LC model from Maxwell's equations for the magnetic resonance of subwavelength ring resonators. Using the LC model, we revisit the scaling of split-ring resonators. Inspired by the LC model, we propose a hybrid metal-dielectric ring resonator mainly composed of high index dielectric material (e.g., TiO2) with some gaps filled with metal (e.g., Ag). The saturation frequency of magnetic response for the hybrid metal-dielectric ring resonator is much higher (up to the ultraviolet range) than that for split-ring resonators, and can be controlled by the metal fraction in the ring. The hybrid metal-dielectric ring resonator can also overcome the homogenization problem of all-dielectric magnetic resonators, and therefore can form homogenizable magnetic metamaterials at short wavelengths down to the ultraviolet range.

© 2013 OSA

1. Introduction

The magnetic response of a nature material is usually very weak, especially at optical frequencies, where the magnetic permeability (μ) is simply put as one. Non-unity or even zero or negative magnetic permeability is very charming, because it leads to unusual electromagnetic phenomena [1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values ofε and μ, ” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]

] and very exciting applications such as superlensing [2

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

], cloaking [3

3. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

,4

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

], electromagnetic energy absorbing [5

5. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed]

], tunneling [6

6. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]

], squeezing [6

6. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]

,7

7. Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielecric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81(8), 085117 (2010). [CrossRef]

] and radiation engineering [8

8. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailor ring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]

, 9

9. Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18(16), 16587–16593 (2010). [CrossRef] [PubMed]

]. In order to engineer the magnetic permeability, people developed magnetic metamaterials composed of subwavelength artificial magnetic resonators, which can be treated as effectively homogeneous media by homogenization. With the help of magnetic metamaterials, people have managed to achieve non-unity or even negative magnetic permeability from microwave frequencies to optical frequencies [10

10. 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]

29

29. M. Lorente-Crespo, L. Wang, R. Ortuño, C. García-Meca, Y. Ekinci, and A. Martínez, “Magnetic hot spots in closely spaced thick gold nanorings,” Nano Lett. 13(6), 2654–2661 (2013). [CrossRef] [PubMed]

]. Optical magnetism can be realized through metallic structures supporting magnetic plasmonic modes (i.e., can interact with magnetic fields), such as split-ring resonators (SRRs) [12

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

14

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

,24

24. B. Lahiri, S. G. McMeekin, A. Z. Khokhar, R. M. De La Rue, and N. P. Johnson, “Magnetic response of split ring resonators (srrs) at visible frequencies,” Opt. Express 18(3), 3210–3218 (2010). [CrossRef] [PubMed]

] and the like [15

15. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005). [CrossRef] [PubMed]

,17

17. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30(23), 3198–3200 (2005). [CrossRef] [PubMed]

,19

19. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). [CrossRef] [PubMed]

,23

23. U. K. Chettiar, S. Xiao, A. V. Kildishev, W. Cai, H. Yuan, V. P. Drachev, and V. M. Shalaev, “Optical metamagnetism and negative-index metamaterials,” MRS Bull. 33(10), 921–926 (2008). [CrossRef]

,26

26. J. Tang and S. He, “A novel structure for double negative NIMs towards UV spectrum with high FOM,” Opt. Express 18(24), 25256–25263 (2010). [CrossRef] [PubMed]

,29

29. M. Lorente-Crespo, L. Wang, R. Ortuño, C. García-Meca, Y. Ekinci, and A. Martínez, “Magnetic hot spots in closely spaced thick gold nanorings,” Nano Lett. 13(6), 2654–2661 (2013). [CrossRef] [PubMed]

]. However, the operation of metallic structures at optical frequencies is limited by the kinetic energy of the electrons in the metal, leading to saturation of the magnetic response when we push the operating frequency deeper into the optical frequency by size scaling [30

30. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). [CrossRef] [PubMed]

]. All-dielectric magnetic resonators have no saturation effect and have a much lower loss than metallic structures. They can be applied in microwave or terahertz ranges where large permittivity of several tens of ε0 can be easily achieved, but they present a challenging issue in the infrared and visible frequency range due to a relatively low refractive index of materials in this range [31

31. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009). [CrossRef]

]. Only recently has optical magnetism in the midinfrared been realized with tellurium dielectric cubic resonators [32

32. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). [CrossRef] [PubMed]

]. Optical magnetism in the visible range can also be realized using crystalline silicon, where a very specific fabrication method is needed in order to guarantee that the constituent silicon is crystalline silicon [33

33. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12(7), 3749–3755 (2012). [CrossRef] [PubMed]

,34

34. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. B. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012). [CrossRef] [PubMed]

]. Nevertheless, optical magnetism or optical magnetic dipole responses are not sufficient for magnetic metamaterials, which can be homogenized according to the conventional definition for a metamaterial. To form a homogenizable magnetic metamaterial, they have to satisfy the homogenization requirement [35

35. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009). [CrossRef] [PubMed]

]. If the refractive index of the constituent dielectric material is not large enough, the homogenization of all-dielectric resonators will be unreasonable. This limits the use of all-dielectric magnetic resonators as homogenizable magnetic metamaterials at short optical wavelengths due to the lack of dielectric materials of very high indices at short optical wavelength. Therefore, it is still necessary to develop elegant designs for realization of homogenizable magnetic metamaterials at short optical wavelengths, such as the blue, violet and ultraviolet regions.

In this paper, we aim to demonstrate a hybrid metal-dielectric ring resonator design for homogenizable magnetic metamaterials at short optical wavelengths down to the ultraviolet range. With this design, the saturation for magnetic response occurs at much higher frequencies (up to ultraviolet rang) as compared to SRR design while the resonators are of truly subwavelength size in contrast to all-dielectric design. We first use an LC model to describe magnetic ring resonators in section 2, where we classify ring resonators into three types: split-ring resonator, all-dielectric ring resonator and hybrid ring resonator. Then based on the LC model, we revisit the scaling of the split-ring resonator and analyze the origin of the saturation of the magnetic response in section 3. Unlike [30

30. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). [CrossRef] [PubMed]

], where the kinetic inductance Lk is derived from the kinetic energy of electrons, we derive the effective inductance from Maxwell's equations and the dispersive permittivity of a metal. From this, we find an effective inductance Le which is similar to Lk for frequencies far below the plasma frequency of the metal, but significantly larger than Lk for visible wavelengths, rendering the size scaling to short wavelengths even more difficult. Then in section 4 we propose a hybrid metal-dielectric ring resonator mainly composed of a high index dielectric material (e.g., TiO2) with some gaps filled with metal (e.g., Ag). Such a new magnetic metamaterial can operate at short wavelengths down to the ultraviolet range, and the operating principle is inspiring for more designs of short wavelength metamaterials. In section 5, we discuss the issue of effective medium homogenization. The homogenization problem of all-dielectric magnetic resonators is illustrated. The advantage of the hybrid ring resonator over all-dielectric magnetic resonators is interpreted on the homogenization issue.

2. Model description

The early artificial material exhibiting strong magnetic response is a type of ring resonator called split-ring resonator [36

36. 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]

], which later appears in many variations, with the key mechanism unchanged. The unit cell (a×a) of the ring resonator metamaterial explored in the present paper is shown in Fig. 1
Fig. 1 The cross-sectional schematic diagram of the two-dimensional ring resonator in a unit cell a×a. The radius r (r=d12b) and thickness b of the ring are r=0.3a and b=0.2a, respectively. The permittivity of the two θ1 (θ2) angle part of the ring is ε1 (ε2). The background is air.
. It is a two dimensional ring resonator with radius r=0.3a and ring thickness b=0.2a. Here the ring shape is fixed (as an example), but the size of the ring is determined by the value of a (used as the size scaling parameter hereafter). The ring is composed of two types of materials with permittivity ε1 and ε2, occupying two θ1 and two θ2 portions by angle (where θ1<θ2), respectively.

Based on the implication that the ring resonator has a well-defined magnetic dipole resonance, we can assume a uniform displacement current in the ring (i.e., for any angle position, the displacement current confined in the ring layer is the same). We also approximate the displacement current density at the middle (i.e., r=d12b) of the ring layer as the average displacement current density in the ring layer, so that we can express the displacement current as bDt, where Dt is the average displacement current density. Therefore the displacement current can be expressed as (assuming time-harmonic dependence eiωt)
I=bDt=iωbε1E1=iωbε2E2,
(1)
where E1 and E2 are the angular components of the electric field at the middle of the ring layer in materials with permittivity ε1 and ε2, respectively. According to Maxwell's equations, we have
2rθ1E1+2rθ2E2=iω(Φint+Φext),
(2)
where Φint=(1F)LgI is the magnetic flux produced by the induced displacement current, Lg is the effective magnetic inductance of the ring (which can be approximately evaluated as μ0FS), and Φext=μ0H0FS (S=a2is the area of the unit cell) is the external driving magnetic flux by incident magnetic field H0. The depolarizing field contributes to the (1F) factor of Φint where F=πr2/a2 is the filling factor. It accounts for the magnetic coupling between unit cells. Then we derive the following equation from Eqs. (1) and (2):
[1iωC+(1F)(iωLg)]I=iωΦext,
(3)
where C=(1C1+1C2)1 is the effective capacitance, which is the serial capacitance of C1[=(bε1)/(2rθ1)] and C2[=(bε2)/(2rθ2)]. As long as the real part of C is positive, the ring will exhibit a magnetic resonance when ω=1/LgC. A magnetic resonance is possible for any of the following three cases: (1) ε1>0,ε2>0; (2) ε1>0,ε2<0; (3) ε1<0,ε2>0. The all-dielectric magnetic resonators belong to the first case. The conventional metallic SRR at optical wavelength is of the second case, which will be studied in detail in section 3 to revisit the problem of saturation. The third case is a brand new type, which is very intriguing and significant owing to its high saturation frequency up to the ultraviolet range and in the mean time truly subwavelength resonator size. This case will be studied in detail in section 4.

3. Scaling of conventional split-ring resonator

To analyze the scaling problem of the conventional SRR at the optical wavelength, we use Ag as the metal material (ε2) and use air as the dielectric material (ε1). The permittivity of Ag can be described by the Drude model ε2=ε0(εωp2ω2+iγω) to fit the experimental data [37

37. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] in the frequency range 150 – 700 THz (i.e., wavelength range 2μm – 430nm). The fit parameters are ε=4, ωp=1.4×1016radandγ=1×1014rad. The permittivity ε2 is dispersive, and thus C2[=(bε2)/(2rθ2)] is also dispersive. The permittivity ε2 has a negative real part and a positive imaginary part; therefore C2 is a negative effective capacitance in parallel connection with a resistance, which is equivalent to a positive inductance Le in series connection with a resistance R, i.e., (iωC2)1=iωLe+R, where Le=Im(1/iωC2)/ω [solid line in Fig. 2(a)
Fig. 2 (a) The comparison between the effective inductance Le (solid line), the well-known kinetic inductance Lk (dashed line) and the geometrical inductance Lg (dotted line). (b) Determination of the magnetic resonance frequency (denoted by the circle) when dispersive effective inductance Le is used. The structure parameters are: a=100nm,r=30nm,b=20nm,θ1=20.
] and R=Re(1/iωC2). Then the effective magnetic permeability can be expressed as a resonant form
μeff=1F'ω2ω21(Lg+Le)C1+iωRLg+Le,
(4)
where F'=LgLg+LeF. The resonance frequency of the effective permeability can be determined by the intersection point of the line 1/(Lg+Le)C1 [solid line in Fig. 2(b)] and the frequency line [dotted line in Fig. 2(b)]. For frequencies far below the plasma frequency of the metal (e.g., below 100 THz), ε2 can be approximated as ε2=ε0(ωp2ω2+iγω). With this approximation, Le becomes the well-known kinetic inductance Lk=2rθ2bε0ωp2 [30

30. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). [CrossRef] [PubMed]

], which is non-dispersive as shown in Fig. 2(a) (dashed line), and R=2rθ2bε0ωp2γ. The resonance frequency can then be directly expressed as 1/(Lg+Lk)C1 [dashed line in Fig. 2(b)], and the quality factor can be expressed as Q=R1(Lg+Lk)/C1. For the 2D SRR shown in Fig. 1, when we scale all the geometrical parameters (a,b,r), we see easily that Le (equal to Lk), C1 and R remain constant, while Lg scales proportional to the scale factor as a2. For a silver SRR structure operating at optical frequencies, Le is comparable with or even larger than Lg. Thus reducing Lg by size scaling is not an effective method to push the resonance wavelength to the short optical wavelength. For frequencies not far from the plasma frequency of the metal, Le increases quickly when the wavelength decreases [cf. solid line in Fig. 2(a)], which makes it even more difficult to push the resonance wavelength to the shorter optical wavelength by size scaling. By comparing the dashed line with the intersection point denoted with a circle in Fig. 2(b), we can see that for the given structure parameters, the resonance frequency determined by the LC model with the approximation Le=Lk is overestimated by about 20 THz. For a resonator with higher resonance frequency, the overestimation would be even larger (as will be indicated below referring to Fig. 3
Fig. 3 The scaling of the magnetic resonance frequency of conventional SRRs as a function of the unit cell size a. The solid lines give the simulated results while the non-solid lines are calculated with our analytical LC model. The black, red, green and blue solid lines are simulated results for SRRs with θ1=10, θ1=20, θ1=30 and θ1=40, respectively. The black, green, red and blue dashed lines are the results calculated with the LC model for SRRs with θ1=10, θ1=20, θ1=30 and θ1=40, respectively. The red dash-dotted line is the LC model result with the approximation Le=Lk for SRRs with θ1=20.
). Some effort has been made to reduce Le by using a metal with a higher plasma frequency, e.g., aluminum [25

25. Y. Jeyaram, S. K. Jha, M. Agio, J. F. Löffler, and Y. Ekinci, “Magnetic metamaterials in the blue range using aluminum nanostructures,” Opt. Lett. 35(10), 1656–1658 (2010). [CrossRef] [PubMed]

]. However, the loss of aluminum is much higher than silver, which means that although magnetic resonances at short wavelengths can be achieved, they are very weak. For SRRs, another problem during scaling is that although the quality factor decreases slowly, F' decreases very quickly (due to the decrease of Lg and the increase ofLe), which reduces the amplitude of the magnetic resonance. Hence, the conventional metallic SRRs are not suitable for magnetic metamaterials at short optical wavelengths.

In Fig. 3, we show the scaling process of the SRR structures in terms of magnetic resonance frequency. From the simulation [38

38. All simulations in this paper were performed using finite-difference time-domain (FDTD) technique by the commercial software CST Microwave Studio.

] results (solid lines), we can see the saturation phenomenon when the scaling factor 1/a is large. SRRs with θ1=10 saturate at about 400 THz, whereas SRRs with θ1=20, θ1=30 and θ1=40 saturate at about 500 THz, 550 THz and 600 THz, respectively. This is because a larger θ1 results in a smallerC1, and thus a larger resonance frequency. Note that there are two assumptions [see the text before Eq. (1)] that lead to Eqs. (1) and (2), where we started the introduction of the LC model. The first assumption is that the displacement current in the ring is uniform. The second assumption is that the displacement current density at the middle of the ring layer may be taken as the average displacement current density in the ring layer. The first assumption mainly causes some quantitative error on the derived effective capacitance C1[=(bε1)/(2rθ1)]. This assumption is valid when all the materials forming the ring resonator are of large permittivity (with respect to the background), or the angle portion θ of the material of small permittivity is very small so that rθb. When there is a material of small permittivity and its angle portion is not very small, the electric field in this material will not be well confined and will have a large radial component, and consequently the assumption of uniform displacement current is not valid in this material. This leads to underestimation of the effective capacitance, and thus overestimation of the resonance frequency in the LC model, as has been indicated in [30

30. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). [CrossRef] [PubMed]

]. The second assumption may also cause some quantitative error, and mainly affects the determination of the effective radius reff of the ring resonator. The displacement current density at reff is the average displacement current density in the ring layer. As implied by Eq. (2), we have taken r(=d12b) as the effective radius reff. The effective radius reff may deviate a bit fromr. The possible value of reff is a value between (db) (the inner edge of the ring layer) and d (the outer edge of the ring layer). We find that the possible quantitative error (on the resonance frequency of the magnetic resonator) induced by the error on the effective radius (the deviation of reff from r) is much smaller than that induced by the error on the effective capacitance. Therefore, we carry out a quantitative correction only to the effective capacitance for the LC model, by simply adding a correction factor η to C1 as C1=η(bε1)/(2rθ1). The correction factor η is larger for SRRs with larger θ1. We use ηθ1=10=1.43, ηθ1=20=1.84,ηθ1=30=2.2and ηθ1=40=2.5for SRRs with θ1=10, θ1=20, θ1=30and θ1=40, respectively. We can see from Fig. 3 that the resonance frequencies calculated using the LC model (dashed lines) match very well with the simulation results (solid lines). Without the correction to the effective capacitance, the LC model would give larger resonance frequencies (but still well describe the scaling process and the saturation effect). In fact, we do not intend to formulate/construct a very quantitatively accurate theoretic model, which requires lots of detailed factors to be involved at the cost of losing some degree of simplicity and physical intuitiveness. Our LC model aims at revealing the resonance conditions and key factors that influence the magnetic resonance, and thus inspiring new designs with specific features of performance (e.g., magnetic metamaterials at short wavelengths as described in section 4).

We can find again in Fig. 3 the different results given by the LC model with (red dashed line) and without (red dash-dotted line) the approximation Le=Lk. The LC model with the approximation Le=Lk overestimates the resonance frequencies. When the scaling factor 1/a is small (e.g., <6μm1), the overestimation is not apparent, which means that in this region the dispersive inductance Le can be well approximated by the non-dispersive inductance Lk. However, when the scaling factor is larger, the overestimation is quite apparent, up to 60 THz. In this region, we have to use the dispersive inductanceLe.

4. Hybrid metal-dielectric ring resonator

In order to operate at a short wavelength down to the ultraviolet range, we propose a new type of ring resonator, of which ε2 is a high index dielectric material while ε1 is a metal (Ag). This is the case “ε1<0,ε2>0” described in section 2. In this paper we refer to this new type of ring resonator as HRR, short for Hybrid Ring Resonator. Note that the Drude model fit used for SRRs in section 3 is not accurate at shorter wavelengths down to the ultraviolet range. Therefore, we use another Drude model ε1=ε0(εωp2ω2+iγω)to fit the experimental data of Ag [37

37. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] for the HRR in the frequency range 700 - 930 THz (i.e., wavelength range 430 – 323 nm). The fit parameters are ε=8.5, ωp=1.7×1016rad andγ=1.4×1014rad. For demonstration, we assume ε2=6ε0, which is reasonable since low-loss dielectric materials with permittivity around this value at short wavelengths are available by using materials such as TiO2, SiC and ZnS from various deposition technologies. The effective capacitance C2=(bε2)/(2rθ2) is positive while C1=(bε1)/(2rθ1) is negative and equivalent to a positive inductance Le in series connection with a resistance R, i.e., (iωC1)1=iωLe+R, where Le=Im(1/iωC1)/ω and R=Re(1/iωC1). The effective permeability has the same form as Eq. (4)
μeff=1F'ω2ω21(Lg+Le)C2+iωRLg+Le,
(5)
where F'=LgLg+LeF. For frequencies far below ωp, we can approximate ε1 as ε0(ωp2ω2+iγω) so that Le=2rθ1bε0ωp2, R=2rθ1bε0ωp2γ and the resonance frequency can be expressed as 1/(Lg+Le)C2. We can now analyze why an HRR can operate at short wavelengths with a strong magnetic response. Firstly, Le is proportional to the length of the metal part, i.e., θ1 for HRRs and θ2 for SRRs. Therefore the Le of an HRR is much smaller than that of a conventional SRR and we can thus expect a much shorter resonance wavelength for an HRR. Secondly, we can expect a higher quality factor of resonance since the resistance R is also proportional to θ1. Thirdly, a smaller Le also gives a larger F', and thus a larger amplitude of magnetic resonance. Moreover, unlike conventional SRRs, HRRs will continue to work when the permittivity of the metal is too small to confine an electric field near plasma frequency. This is because the high index dielectric material plays the dominant role in confining the electric field inside the ring.

The scaling of HRRs is represented in Fig. 4(a)
Fig. 4 (a) The scaling of the simulated (solid lines) and LC model calculated (dashed lines) magnetic resonance frequency of HRRs as a function of the unit cell size a. (b,c) The effective permeability (retrieved from simulated S parameters) of HRRs with a=100nm [denoted by the vertical dotted line in panel (a)] and θ1=40,30,20,10,5, from left to right. The Drude model of Ag is used in (b), the experimental material data of Ag from [37] is used in (c).
for θ1=5 (black lines), 10 (red lines), 20 (green lines) and 40 (blue lines) by using our LC model (dashed lines) and simulation (solid lines). For each scaling curve (i.e., constant θ1), we still observe the saturation phenomenon as expected by our LC model. However, the saturation frequencies are much higher than those for conventional SRRs (cf. Figure 3) mainly due to the reduction of Le. The saturation frequency can be pushed higher by reducing the metal fraction in the ring resonator (i.e., decreasing θ1). Therefore, we can achieve a strong magnetic response at a higher frequency by reducing the metal fraction. To demonstrate this, we keep the scaling factor a constant as 100 nm [as indicated by the vertical dotted line in Fig. 4(a)] and change θ1 from 40 to 5. The effective permeability spectra (retrieved from simulated S parameters [39

39. 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(19), 195104 (2002). [CrossRef]

]) are plotted in Fig. 4(b). By decreasingθ1, the resonance frequency increases from 748 THz (i.e., 401 nm) to 899 THz (i.e., 334 nm). It is worthwhile to point out that according to the Drude model of Ag, the real part of the permittivity is only −0.55 at 899 THz (i.e., 334 nm) and is positive at frequencies higher than 928 THz (i.e., 323 nm). In fact, by decreasing θ1, strong magnetic resonances can be achieved as long as the permittivity of the metal is still negative. We can also see from Fig. 4(b) that, as expected by our analysis with the LC model, the amplitude and quality factor of the magnetic resonances remain quite large, only decreasing gently while being pushed to higher frequencies.

Until now we have simulated HRRs using the Drude model in order to equally compare the simulation results with the LC model results (the Drude model was used in the LC model in order to give analytical expressions and evidently draw the mechanism/principle). The Drude model fit for the real part is quite reliable, though the fit for the imaginary part is less accurate. However, the imaginary part only affects the amplitude and quality factor of the resonances. We yet again check the performance of the HRR by simulation using experimental material data of Ag [37

37. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

]. The simulated effective permeability spectra are shown in Fig. 4(c). As expected, the simulation results in Fig. 4(b) correspond well with that in Fig. 4(c), except for the differences on the resonance amplitude and quality factor. For resonances at frequencies lower than 840 THz [blue and orange lines in Figs. 4(b) and 4(c)], the simulation using the Drude model results in smaller amplitudes and lower quality factors, while for resonances at frequencies lower than 840 THz [red and black lines in Figs. 4(b) and 4(c)], the simulation using the Drude model results in larger amplitudes and higher quality factors. This agrees with the fact that the imaginary part of the Drude model is larger than the experimental data at frequencies lower than 840 THz while smaller at frequencies higher than 840 THz.

5. Discussion on effective medium homogenization

As fundamental elements of a metamaterial which can be viewed as an effectively homogeneous medium, the subwavelength magnetic resonators should satisfy the homogenization requirements λ/|Re(neff)|2a and λ2a [35

35. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009). [CrossRef] [PubMed]

]. The two requirements are independent and have different physical significances. The first condition guarantees that inside the metamaterial the phase advance across the unit cell is not significant so that the spatial dispersion effect is negligible. The second condition guarantees that the electromagnetic wave shining onto the surface of the metamaterial sees a homogeneous interface. The first condition is usually stronger than the second condition because usually |Re(neff)|1 for a conventional material. However, the first condition will be a weaker condition when |Re(neff)|<1, which may occur quite often for a metamaterial. Thus we utilized both the normalized effective wavelength λeff/a [where λeff=λ/|Re(neff)|] and the normalized vacuum wavelength λ/a to evaluate the reasonableness of the homogenization. The homogenization is more reasonable when the normalized wavelength is longer. A photonic band gap regime typically occurs around λeff/a=2. Note that neff is the effective refractive index retrieved from the S parameters, assuming the metamaterial as an effectively homogeneous medium [39

39. 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(19), 195104 (2002). [CrossRef]

]. In the magnetic resonance frequency region, the effective refractive index neff is strongly dispersive. |Re(neff)| changes quickly across the resonance frequency, from a larger value at the low-frequency side of the resonance to a smaller value (even to zero) at the high-frequency side of the resonance. Thus the normalized effective wavelength λeff/a also changes quickly with the frequency in the resonance region.

For SRRs and HRRs in the present paper, we plotted the minimum normalized effective wavelength (i.e., the worst homogenization condition in the resonance region) versus the scaling factor 1/a as dashed lines with solid symbols in Figs. 5(a)
Fig. 5 The normalized wavelength (left y-axis) and the minimum effective permeability (right y-axis) versus the scaling factor 1/a for SRRs (a) and HRRs (b). The solid lines with solid symbols are the normalized vacuum wavelengthλ/a, the dashed lines with solid symbols are the normalized effective wavelengthλeff/a, the solid lines with hollowed symbols are the minimum effective permeabilityμeffmin. Note that the right y-axis for μeffmin is in a reversed direction.
and 5(b). We also plotted the normalized vacuum wavelength λ/a as solid lines with solid symbols in Figs. 5(a) and 5(b). We found that both the normalized effective wavelength and the normalized vacuum wavelength increase monotonically when the resonators are scaled down (i.e., 1/a increases). When we scale down deep into the saturation region [see Fig. 3 for SRRs and Fig. 4(a) for HRRs, where the resonance wavelength decreases very slowly with 1/a], both the normalized effective wavelength and the normalized vacuum wavelength increase nearly linearly with 1/a. It is worthwhile to point out that although the saturation effect for SRRs and HRRs is unfavorable for pushing the operating wavelength to the short end by geometrical size scaling, it is favorable for increasing the normalized wavelength (i.e., more suitable for homogenization) by just shrinking the structure. However, the strength of the magnetic resonance decreases as a trade-off when the normalized wavelength increases. We plotted the achievable minimum effective magnetic permeability μeffmin versus 1/a as solid lines with hollowed symbols in Figs. 5(a) and 5(b). The achievable minimum effective permeability μeffmin indicates the strength of the magnetic resonance. Note that the right y-axis for μeffmin is in a reversed direction. We can observe the trade-off between the minimum effective permeability μeffmin and the normalized wavelength (i.e., if longer normalized wavelength is expected, the minimum effective permeability μeffmin achieved will be larger).

For all-dielectric magnetic resonators (e.g., the case “ε1>0,ε2>0” described in section 2), there is no saturation effect, i.e., the resonance wavelength decreases linearly as the structure is scaled down (i.e., λa). This means that, for all-dielectric magnetic resonators, the normalized vacuum wavelength λ/a is constant when we scale down the structure. Therefore, the normalized vacuum wavelength λ/a cannot be increased by shrinking the structure. In fact, the normalized vacuum wavelength λ/a is determined by the refractive index of the constituent material. For an all-dielectric magnetic resonator of a fixed size, it is well-known that the resonance wavelength λ increases with the refractive index of the constituent dielectric material. Thus the only effective way to increase the normalized vacuum wavelength λ/a of all-dielectric magnetic metamaterials is to utilize a dielectric material with a higher refractive index. Therefore, all-dielectric homogenizable metamaterials are challenging at short optical wavelengths due to the lack of dielectric materials of very high indices at short optical wavelengths. For all-dielectric resonators made of dielectric material with refractive index n<4, it is very difficult to form a magnetic metamaterial with a regime where λeff/a>2 (the minimal requirement for possible homogenization) in the magnetic resonance region, although they can support well defined magnetic dipole resonances [33

33. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12(7), 3749–3755 (2012). [CrossRef] [PubMed]

,34

34. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. B. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012). [CrossRef] [PubMed]

,40

40. A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19(6), 4815–4826 (2011). [CrossRef] [PubMed]

]. So far, no natural dielectric material with refractive index n>4 has been found in the NIR-vis-UV range, except crystalline silicon in the wavelength region 420~600nm (at wavelengths shorter than 420nm, the loss of crystalline silicon is too large). For all-dielectric magnetic metamaterials made of crystalline silicon, however, the normalized vacuum wavelength λ/a is still not large enough for good homogenization (which requires, for example, λ/a>4). For our HRR design (using a dielectric material of refractive index of only 2.45), both the normalized vacuum wavelength λ/a and the normalized effective wavelength λeff/a can easily get a value larger than 4 at optical frequencies [Fig. 5(b)].

6. Conclusion

We have proposed an LC model for ring resonators, based on which we summarized three cases of magnetic resonances. In the framework of our model, we revisited the scaling of conventional SRRs and proposed a hybrid metal-dielectric ring resonator with a much higher saturation frequency (up to the ultraviolet range) than that of SRRs and a truly subwavelength resonator size in contrast to all-dielectric magnetic resonators. The hybrid metal-dielectric ring resonators can form homogenizable magnetic metamaterials at short wavelengths, down to the ultraviolet range. Our model is physically intuitive and can provide some guidance for the design of magnetic metamaterials at optical frequencies. Our proposal for the hybrid ring resonator design is just a prototype, and variations and improvements are possible for better performance and ease of fabrication.

Acknowledgments

This work is partially supported by the National High Technology Research and Development Program (863 Program) of China (2012AA030402), the National Natural Science Foundation of China (60990322 and 61178062), the Swedish VR grant (621-2011-4620) and the Asian Office of Aerospace Research and Development (AOARD) (114045).

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values ofε and μ, ” Sov. Phys. Usp. 10, 509 (1968). [CrossRef]

2.

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

3.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

4.

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

5.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed]

6.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]

7.

Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielecric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81(8), 085117 (2010). [CrossRef]

8.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailor ring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]

9.

Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18(16), 16587–16593 (2010). [CrossRef] [PubMed]

10.

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]

11.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004). [CrossRef] [PubMed]

12.

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

13.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94(3), 037402 (2005). [CrossRef] [PubMed]

14.

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

15.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005). [CrossRef] [PubMed]

16.

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] [PubMed]

17.

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30(23), 3198–3200 (2005). [CrossRef] [PubMed]

18.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]

19.

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). [CrossRef] [PubMed]

20.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]

21.

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007). [CrossRef] [PubMed]

22.

J. Valentine, S. Zhang, Th. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]

23.

U. K. Chettiar, S. Xiao, A. V. Kildishev, W. Cai, H. Yuan, V. P. Drachev, and V. M. Shalaev, “Optical metamagnetism and negative-index metamaterials,” MRS Bull. 33(10), 921–926 (2008). [CrossRef]

24.

B. Lahiri, S. G. McMeekin, A. Z. Khokhar, R. M. De La Rue, and N. P. Johnson, “Magnetic response of split ring resonators (srrs) at visible frequencies,” Opt. Express 18(3), 3210–3218 (2010). [CrossRef] [PubMed]

25.

Y. Jeyaram, S. K. Jha, M. Agio, J. F. Löffler, and Y. Ekinci, “Magnetic metamaterials in the blue range using aluminum nanostructures,” Opt. Lett. 35(10), 1656–1658 (2010). [CrossRef] [PubMed]

26.

J. Tang and S. He, “A novel structure for double negative NIMs towards UV spectrum with high FOM,” Opt. Express 18(24), 25256–25263 (2010). [CrossRef] [PubMed]

27.

V. A. Fedotov, T. Uchino, and J. Y. Ou, “Low-loss plasmonic metamaterial based on epitaxial gold monocrystal film,” Opt. Express 20(9), 9545–9550 (2012). [CrossRef] [PubMed]

28.

Z. H. Jiang, S. Yun, L. Lin, J. A. Bossard, D. H. Werner, and T. S. Mayer, “Tailoring dispersion for broadband low-loss optical metamaterials using deep-subwavelength inclusions,” Sci. Rep. 3, 1571 (2013). [CrossRef] [PubMed]

29.

M. Lorente-Crespo, L. Wang, R. Ortuño, C. García-Meca, Y. Ekinci, and A. Martínez, “Magnetic hot spots in closely spaced thick gold nanorings,” Nano Lett. 13(6), 2654–2661 (2013). [CrossRef] [PubMed]

30.

J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95(22), 223902 (2005). [CrossRef] [PubMed]

31.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009). [CrossRef]

32.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). [CrossRef] [PubMed]

33.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12(7), 3749–3755 (2012). [CrossRef] [PubMed]

34.

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. B. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012). [CrossRef] [PubMed]

35.

R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009). [CrossRef] [PubMed]

36.

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]

37.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

38.

All simulations in this paper were performed using finite-difference time-domain (FDTD) technique by the commercial software CST Microwave Studio.

39.

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(19), 195104 (2002). [CrossRef]

40.

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19(6), 4815–4826 (2011). [CrossRef] [PubMed]

OCIS Codes
(160.4670) Materials : Optical materials
(260.7190) Physical optics : Ultraviolet
(260.2065) Physical optics : Effective medium theory
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: June 13, 2013
Revised Manuscript: September 16, 2013
Manuscript Accepted: September 16, 2013
Published: September 26, 2013

Citation
Jianwei Tang and Sailing He, "Hybrid metal-dielectric ring resonators for homogenizable optical metamaterials with strong magnetic response at short wavelengths down to the ultraviolet range," Opt. Express 21, 23511-23521 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23511


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References

  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values ofε and μ, ” Sov. Phys. Usp.10, 509 (1968). [CrossRef]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  3. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  4. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  5. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett.100(20), 207402 (2008). [CrossRef] [PubMed]
  6. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett.100(3), 033903 (2008). [CrossRef] [PubMed]
  7. Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielecric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B81(8), 085117 (2010). [CrossRef]
  8. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailor ring the radiation phase pattern,” Phys. Rev. B75(15), 155410 (2007). [CrossRef]
  9. Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express18(16), 16587–16593 (2010). [CrossRef] [PubMed]
  10. 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]
  11. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science303(5663), 1494–1496 (2004). [CrossRef] [PubMed]
  12. S. Linden, C. Enkrich, M. Wegener, J. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306(5700), 1351–1353 (2004). [CrossRef] [PubMed]
  13. S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett.94(3), 037402 (2005). [CrossRef] [PubMed]
  14. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95(20), 203901 (2005). [CrossRef] [PubMed]
  15. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature438(7066), 335–338 (2005). [CrossRef] [PubMed]
  16. 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] [PubMed]
  17. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005). [CrossRef] [PubMed]
  18. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett.95(13), 137404 (2005). [CrossRef] [PubMed]
  19. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express14(4), 1557–1567 (2006). [CrossRef] [PubMed]
  20. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics1(1), 41–48 (2007). [CrossRef]
  21. C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science315(5808), 47–49 (2007). [CrossRef] [PubMed]
  22. J. Valentine, S. Zhang, Th. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008). [CrossRef] [PubMed]
  23. U. K. Chettiar, S. Xiao, A. V. Kildishev, W. Cai, H. Yuan, V. P. Drachev, and V. M. Shalaev, “Optical metamagnetism and negative-index metamaterials,” MRS Bull.33(10), 921–926 (2008). [CrossRef]
  24. B. Lahiri, S. G. McMeekin, A. Z. Khokhar, R. M. De La Rue, and N. P. Johnson, “Magnetic response of split ring resonators (srrs) at visible frequencies,” Opt. Express18(3), 3210–3218 (2010). [CrossRef] [PubMed]
  25. Y. Jeyaram, S. K. Jha, M. Agio, J. F. Löffler, and Y. Ekinci, “Magnetic metamaterials in the blue range using aluminum nanostructures,” Opt. Lett.35(10), 1656–1658 (2010). [CrossRef] [PubMed]
  26. J. Tang and S. He, “A novel structure for double negative NIMs towards UV spectrum with high FOM,” Opt. Express18(24), 25256–25263 (2010). [CrossRef] [PubMed]
  27. V. A. Fedotov, T. Uchino, and J. Y. Ou, “Low-loss plasmonic metamaterial based on epitaxial gold monocrystal film,” Opt. Express20(9), 9545–9550 (2012). [CrossRef] [PubMed]
  28. Z. H. Jiang, S. Yun, L. Lin, J. A. Bossard, D. H. Werner, and T. S. Mayer, “Tailoring dispersion for broadband low-loss optical metamaterials using deep-subwavelength inclusions,” Sci. Rep.3, 1571 (2013). [CrossRef] [PubMed]
  29. M. Lorente-Crespo, L. Wang, R. Ortuño, C. García-Meca, Y. Ekinci, and A. Martínez, “Magnetic hot spots in closely spaced thick gold nanorings,” Nano Lett.13(6), 2654–2661 (2013). [CrossRef] [PubMed]
  30. J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett.95(22), 223902 (2005). [CrossRef] [PubMed]
  31. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today12(12), 60–69 (2009). [CrossRef]
  32. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett.108(9), 097402 (2012). [CrossRef] [PubMed]
  33. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12(7), 3749–3755 (2012). [CrossRef] [PubMed]
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