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A three-layered Ag−low-permittivity (LP)−high-permittivity (HP) nanoshell is proposed as a plausible meta-atom for building the three-dimensional isotropic negative refractive index metamaterials (NIMs). The overlap between the electric and magnetic responses of Ag−LP−HP nanoshell can be realized by designing the geometry of the particle, which can lead to the negative electric and magnetic polarizabilities. Then, the negative refractive index is found in the random arrangement of Ag−LP−HP nanoshells. Especially, the modulation of the middle LP layer can move the negative refractive index range into the visible region. Because the responses arise from the each meta-atom, the metamaterial is intrinsically isotropic and polarization independent. It is further found with the increase of the LP layer thickness that the negative refractive index range of the random arrangement shows a large blue-shift and becomes narrow. With the decrease of the filling fraction, the negative refractive index range shows a blue-shift and becomes narrow while the maximum of the negative refractive index decreases.
©2013 Optical Society of America
1. Introduction
Metamaterials are artificial materials with rationally designed meta-atoms, which can allow both electric and magnetic components of light to be coupled to meta-atoms. The electric and magnetic responses of the metamaterials can be varied and modified independently, which will lead to the unique new optical properties and exciting applications [1–4]. Negative refractive index metamaterials (NIMs) [5], typically consisting of noble metals and dielectrics, have attracted a significant amount of research attention due to their potential applications such as superlensing [6] and light storing [7]. The NIMs were first demonstrated experimentally at microwave frequencies and then have been fabricated to operate from the terahertz frequency region to the near-infrared spectral region [8, 9]. Recently, many attempts to achieve the NIMs at visible frequencies have been made for taking full advantage of NIM properties [10–13]. García-Meca et al. [11] have fabricated a multilayered fishnet NIMs exhibiting a negative refractive index in the visible spectral range, which achieves low losses and polarization independence at normal incidence. Rodríguez-Fortuño et al. [12] proposed a NIM composed of the array of deep-subwavelength coaxial plasmonic waveguides. They have found that the metamaterial losses can be easily predicted through the waveguide dispersion relation and pose a main drawback. Burgos et al. [13] reported a wide-angle NIM at visible frequencies composed of a single layer of coupled plasmonic coaxial waveguides, but the resulting negative refractive index is insensitive to both polarization and angle-of-incidence. The isotropy is also a specially desired feature for a NIM. For example, the superlensing critically requires a direction-independent negative refractive index. However, most of the previously proposed designs for the NIMs are restricted to operate under certain polarization and incidence conditions. The truly three-dimensional (3D) isotropic NIMs were seldom reported, especially operating in visible wavelengths. Kante et al. [14] reported a random monometatomic route to a 3D isotropic negative refractive index at about 1.7 μm based on ring-resonator symmetry and/or parity.
Plasmonics is another rapidly developing research area during the past few years [15–18]. Interplay between metamaterials and plasmons can exhibit some fantastic phenomenon and cause superior applications [19–21]. Metallodielectric layered nanoparticles are special interesting plasmonic structures and have been widely studied due to the promising applications in nano-electronics, biomedical imaging, nano-optical device, and optical sensing [22–25]. Recently, a simple two-layered nanoshell, consisting of a metal core coated with a high-refractive index dielectric shell, has been found to exhibit some special optical properties [26–28]. Liu et al. [26] have demonstrated the suppression of the backward scattering and enhancement of the forward directional scattering in metal core–dielectric shell nanoparticles. The azimuthally symmetric broadband unidirectional scattering can be achieved by superimposing electric and artificial magnetic dipolar responses. They further found the polarization independent Fano resonances in arrays of these nanoparticles [27]. Especially, Paniagua-Domínguez et al. [28] have reported a fully 3D isotropic NIM based on these metal core–dielectric shell nanoparticles. They found that the magnetic-dipole resonance of the dielectric shell with a high refractive index can move to the plasmon resonance wavelength of the metal core and the metamaterials present the negative refractive indexes in the range of 1.2–1.55μm. Even if the tuning of negative refractive index regime to the visible range can take place in the two-layered core–dielectric shell structure, the adjustability of two-layered structure in visible range is limited.
In this paper, we propose a three-layered Ag−low-permittivity (LP)−high-permittivity (HP) nanoshell. The electric and magnetic scattering spectra of the Ag−LP−HP nanoshells have been calculated by using Mie theory and the effective optical parameters of the random arrangement of Ag−LP−HP nanoshells have been obtained based on Lorenta-Lorenz theory. The Ag core can induce the electric responses while the HP shell provides the strong diamagnetic responses. The overlap between the electric and magnetic dipole modes can lead to the negative refractive index in the random arrangement of Ag−LP−HP nanoshells. The modulation of the LP layer can move the negative refractive index range into the visible region. The Ag−LP−HP nanoshell may be an effective candidate for building 3D isotropic NIMs in visible region. In addition, the influences of the middle LP layer of Ag−LP−HP nanoshell on the effective parameters of the NIMs have been studied in detail.
2. Electromagnetic scattering model
The composite meta-atom consists of a nanoshell with a metal core, a thin middle dielectric layer with low-permittivity, and a dielectric outer shell with high-permittivity. The geometry of the Ag−LP−HP nanoshell is depicted in Fig. 1 . The radius of the metal core is r1. The thicknesses of the LP layer and the outer HP shell are (r2 – r1) and (r3 – r2), respectively. The permittivities of the core, middle layer, outer shell and embedding medium are ε1, ε2, ε3 and ε4, respectively. For metallic core, ε1 has real and imaginary frequency-dependent components. When the size of metal core is smaller than the bulk electron mean free path, the electron surface scattering in the metal core surface becomes important. Thus, ε1 is usually accounted by replacing the ideal Drude part in the dielectric functions with a size dependent one, and can be expressed as [29]
where the background susceptibilities χ∞ arises from the electron polarizability and interband transition, ωp is the bulk plasma frequency. When size-dependent electron scattering is important, the collision frequency γ should be modified as [29]Where γf and Vf are the bulk collision frequency and the Fermi velocity of the metal core, respectively. The reduced electron mean free path a = 2r1. For simple Drude theory and isotropic scattering, A = 1. The parameters can be obtained from the bulk dielectric data for Ag [30].The plane wave scattering from an isotropic sphere was originally presented by Mie and extended to the more general case of concentric spherical shells [31]. For a polarized plane wave incident on a three-layered spherical particle, the electromagnetic waves are expanded to spherical partial waves using vector spherical harmonics and then Maxwell’s boundary conditions are applied to resolve the unknown expansion coefficients of the scattered and interior waves [32]. The obtained scattering efficiency Qsca can be expressed as [31]
The quantity is the wave number outside the particle and λ is the incident wavelength. The scattering coefficients al and bl represent the different electric and magnetic multipolar contributions, respectively.The Lorentz-Lorenz theory has been widely used in the composites of a cubic or random arrangement of dipolar particles [28, 33]. For a random arrangement of the meta-atoms, the effective permeability μeff and permittivity εeff can be related to the polarizabilities of the particles and the filling fraction f by using the Clausius-Mossotti equation
where N is the number of particles per unit volume and μ0 is the permeability of the vacuum. The magnetic and electric polarizabilities, i. e., αM and αE, are directly proportional to the scattering coefficients b1 and a1 [factor i(k3/6π)−1]. The refractive index of the metamaterial can be express as .3. Results and discussion
We start our researches from the core−shell Ag−Si nanoshells. In this paper, all of the systems we study are assumed to be in a vacuum. Figure 2(a) shows the scattering spectra of the Ag−Si nanoshell. The solid and dashed lines represent the contributions of the electric and magnetic terms, respectively. Here, the radii of Ag core and Si shell are fixed at 35 and 148.5 nm, respectively. Within the wavelength range of 800-2000 nm, the refractive index of Si shell can be considered as a constant n3 = (ε3/ε0)1/2 ≈3.4 [28]. It is observed that the electric and magnetic dipole peaks of the Ag−Si nanoshell both appear at about 1051 nm. The electric resonance modes arise from the plasmon resonances in the Ag core and the magnetic modes correspond to the combined electromagnetic resonances in the outer Si shell [28]. Figure 2(b) and 2(c) represent the electric and magnetic polarizabilities αE and αM for the Ag−Si nanoshells, respectively. The solid and dashed lines show the real and imaginary parts of the polarizabilities, respectively. The negative electric and magnetic polarizabilities can be obtained within the spectral range of 992-1047 nm. Such negative electric and magnetic polarizabilities should further lead to the negative permittivity and permeability for the metamaterials based on these Ag−Si nanoshells.
Fig. 2 (a) Scattering spectra, (b) electric polarizability, and (c) magnetic polarizability of the Ag−Si nanoshell. Here the radii of Ag core and Si shell r1 and r2 are fixed at 35 and 148.5 nm, respectively.
Figures 3(a) and 3(b) show the electric and magnetic contributions to the scattering spectra of the Ag−SiO2−Si nanoshells, respectively. The solid, dashed, dotted and dash-dot lines represent the scattering spectra for the nanoshells with the middle layer thicknesses (r2 – r1) of 0, 1, 3, and 5 nm, respectively. Here the radii of the Ag core and outer Si shell r1 and r3 are fixed at 35 and 148.5 nm respectively. The refractive index of SiO2 is 1.43. In Fig. 3(a), with the increase of (r2 – r1)-value, the electric dipole peak of the Ag−SiO2−Si nanoshell shows a distinct blue-shift from 1051 nm at r2 – r1 = 0 to 855 nm at r2 – r1 = 5 nm. The decreased permittivity of the medium around Ag core will increase the induced charges in the surface of the Ag core [34]. In this case, the restoring forces of the collective oscillation of conduction electrons become strong and hence the increase of the plasmon resonance energy of Ag−SiO2−Si nanoshells. Thus, the existence of the middle LP layer should greatly decrease the electric dipole wavelength of the Ag−SiO2−Si nanoshell. In addition, the increased thickness of the LP layer will increase the restoring forces for the plasmon resonances. With the increase of the LP layer thickness, the effect of the LP layer on the restoring force is enhanced while the influence of the outer HP shell on the restoring force becomes weak. Then the blue-shift of the electric dipole peak of Ag−SiO2−Si nanoshell happens. On the other hand, as shown in Fig. 3(b), the variation of the middle LP layer has no effect on the magnetic dipole peak of the Ag−SiO2−Si nanoshell. Therefore, the outer HP layer should be modified to realize the overlap between the electric and magnetic resonances.
Fig. 3 (a) The electric and (b) magnetic contributions to the scattering spectra of the Ag-SiO2-Si nanoshells. Here the radii of the Ag core and outer Si shell r1 and r3 are fixed at 35 and 148.5 nm respectively. The solid, dashed, dotted and dash-dot lines represent the scattering spectra for the nanoshells with the middle layer thicknesses (r2 – r1) of 0, 1, 3, and 5 nm, respectively.
For the wavelengths below 800 nm, the Si becomes absorptive and the imaginary part of the Si refractive index should be considered in the calculations. Figure 4 shows the dielectric permittivity of Si with wavelength below 800 nm [35, 36]. Figures 5(a) and 5(b) depict the contour plots of the electric and magnetic contributions to the scattering spectra of the Ag−SiO2−Si nanoshells as a function of r3-values, respectively. Figure 5(c) shows the contour plot for total scattering spectra of the Ag−SiO2−Si nanoshells as a function of r3-values. Here the radius of Ag core and the thickness of SiO2 layer are fixed at 35 and 5 nm, respectively. In Fig. 5(a), with the increase of r3-value, the electric dipole peak of the Ag−SiO2−Si nanoshell shows a red-shift from ~542 nm at r3 = 50 nm to ~800 nm at r3 = 125 nm. At the meantime, the magnetic dipole peak shows a red-shift from ~400 nm at r3 = 50 nm to 800 nm at r3 = 107 nm. Therefore, as shown in Fig. 5(c), an overlap between the electric and magnetic dipole resonances can be found within 680−720 nm and for r3~88−96 nm.
Fig. 5 Contour plots of the (a) electric and (b) magnetic contributions to the scattering spectra of the Ag-SiO2-Si nanoshells as a function of r3-values. (c) Total scattering spectra of the Ag-SiO2-Si nanoshells as a function of r3-values. Here the radius of Ag core and the thickness of SiO2 layer are fixed at 35 and 5 nm, respectively.
In Fig. 6(a) , the solid and dashed lines show the contributions of the electric and magnetic terms to the scattering spectra of the Ag−SiO2−Si nanoshells, respectively. Here the radii of the core, middle layer, and outer shell are fixed at 35, 40, and 91 nm, respectively. The electric and magnetic dipole peaks of the Ag−SiO2−Si nanoshell both appear at about ~695 nm. Figures 5(b) and 5(c) represent the variations of the electric and magnetic polarizabilities for Ag−SiO2−Si nanoshells as a function of wavelengths, respectively. The solid and dashed lines show the real and imaginary parts of the polarizabilities, respectively. It is found that the negative electric and magnetic polarizabilities can be obtained within the spectral range of 664−692 nm. We have further built a random arrangement model based on these Ag−SiO2−Si nanoshells and calculated the corresponding effective permittivity and permeability by using the Clausius-Mossotti equation. Figures 5(d)-5(f) show the variations of the effective permittivity, permeability and refractive index for the random arrangement models as a function of wavelengths. The solid and dashed lines depict the real and imaginary parts, respectively. Here the filling fraction f of the model is fixed at 0.5. It is observed that the negative real parts of the effective permittivities and permeabilities can be found within 681−711 nm and then the negative real parts of the refractive indexes also appear in 678−716 nm. In addition, we calculated the so-called figure of merit (FOM) [12], which means a standard loss measure defined as . The maximum FOM of this model reaches about 0.495, which corresponds to .
Fig. 6 (a) Scattering spectra of the Ag−SiO2−Si nanoshells. (b) Electric and (c) magnetic polarizabilities of the Ag−SiO2−Si nanoshell as a function of wavelengths. (d) Effective permittivity, (e) permeability and (f) refractive index of the random arrangement of Ag−SiO2−Si nanoshells as a function of wavelengths. Here r1, r2, and r3 are fixed at 35, 40, and 91 nm, respectively. The filling fraction f is fixed at 0.5.
Figure 7 shows the negative real part of the effective refractive index for the random arrangement of Ag−SiO2−Si nanoshells as a function of the wavelengths and the f-values. Here the radii of the inner core, middle layer, and outer shell are fixed at 35, 40, and 91 nm, respectively. It is found that the spectral range of the negative refractive index becomes broad and shows a red-shift with the increase of the filling fraction while the maximum of the negative refractive index will increase.
Fig. 7 Contour plot of the real part of the effective refractive index for the random arrangement as a function of the wavelengths and the f-values. Here r1, r2, and r3 are fixed at 35, 40, and 91 nm, respectively.
We further investigate the influence of the middle layer thickness of the Ag−SiO2−Si nanoshell on the effective refractive index of the random arrangement model. In Fig. 8(a) , the solid lines represent the variations of the real part of effective refractive index for the metamaterial and the dashed lines depict the corresponding FOMs. Here the radius of the inner core is fixed at 35 nm and the filling fraction f is fixed at 0.5. The black, blue, and red lines represent the results of the Ag−SiO2−Si nanoshells with the various LP layer thicknesses of 5, 8, and 10 nm, respectively, the corresponding radii of outer shell are 91, 80, and 76 nm. The negative refractive index ranges of the metamaterial with the middle layer thickness of 5, 8, and 10 nm appear at (678−716 nm), (599−630nm), and (568−594nm), respectively. It is clear that the negative refractive index range of the random arrangement shows a large blue-shift into visible region and becomes narrow in the frequency domain with the increase of the middle layer thickness. The blue-shift of the negative refractive index range is mainly due to the blue-shifts of the electric and magnetic dipole peaks of Ag−SiO2−Si nanoshells. With the increase of the LP layer thickness, the corresponding outer shell radius will be decreased for realizing the overlap between the electric and magnetic dipole peaks. The decreased HP shell thickness will reduce the phase retardation and hence the electric dipole peak becomes narrow. Then the negative refractive index range of the random arrangement becomes narrow in the frequency domain. With increasing the LP layer thickness, the maximum of the negative refractive index decreases from −0.807 at (r2 – r1) = 5 nm to −0.687 at (r2 – r1) = 10 nm while the FOM maximum decreases from 0.495 to 0.433. In addition, we discuss the influence of the permittivity of middle layer on the effective refractive index of the random arrangement. In Fig. 8(b), the black, blue, and red lines represent the results of the Ag−LP−Si nanoshells with the various middle layer thicknesses of 5, 8, and 10 nm, respectively, in which the corresponding radii of outer shell are 67, 60, and 59 nm. Here the permittivity of the LP layer is assumed as 1. The radius of the inner core is fixed at 35 nm and the filling fraction is 0.5. The negative refractive index ranges of the random arrangement with the LP layer thickness of 5, 8, and 10 nm appear at (519−539 nm), (463−473 nm), and (445−452 nm), respectively. The decreased permittivity of the LP layer will further blue-shift the negative refractive index range.
Fig. 8 Real part of the effective refractive index for the random arrangement of (a) Ag−SiO2−Si and (b) Ag−LP−Si (ε2 = 1) as a function of wavelengths. The dashed lines depict the corresponding FOMs. The black, blue, and red lines show the variations with the various middle layer thicknesses of 5, 8, and 10 nm, respectively. Here the radius of Ag core is fixed at 35 nm and the filling fraction f is fixed at 0.5.
4. Summary
The electric and magnetic scattering spectra of the Ag−LP−HP nanoshells have been calculated by using Mie theory. It is found that the inner Ag core can induce the electric responses while the HP shell provides the strong diamagnetic responses. Such phenomenon happens exactly as that reported in previous Ag-HP nanoshells [28]. The modulation of the middle LP layer can lead to a great blue-shift of the electric dipole mode in the Ag−LP−HP nanoshell. The overlap between the electric and magnetic dipole peaks can be realized within visible region by designing the geometry of the nanoshell. Then the negative electric and magnetic polarizabilities can be achieved, which leads to the negative refractive index in the random arrangement of Ag−LP−HP nanoshells. With the increase of the middle layer thickness, the negative refractive index range of the random arrangement of Ag−LP−HP nanoshells shows a large blue-shift and becomes narrow in the frequency domain. With the decrease of the filling fraction, the negative refractive index range becomes narrow in the frequency domain and shows a blue-shift while the maximum of the negative refractive index decreases. Because the responses arise from the each meta-atom, the metamaterial is intrinsically isotropic and polarization independent. We believe that the Ag−LP−HP nanoshell may be an effective meta-atom for building isotropic NIMs at visible frequencies. But the direct demonstrations for the NIM behaviors are necessary. Further investigations of the NIM behaviors in the different periodic arrays of Ag−LP−HP nanoshells should be carried out in the future work. Such 3D isotropic NIMs may be realized with current fabrication techniques based on the realistic materials [37–39]. In addition, the tunable Ag−LP−HP nanoshells may achieve the azimuthally symmetric unidirectional scattering in a very large spectral region, which can find applications in nanoantennas, photovoltaic devices, and nanoscale lasers.
Acknowledgments
This work was supported by the National Basic Research Program of China under Grant No. 2012CB921504, National Natural Science Foundation of China under Grant Nos. 11174113, 10904052, 11274171, 11074124 and 11104319, and Project Funded by the Priority Academic Program Development of Jiangsu higher education institutions.
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