Mie-resonator dieletric MM914

ISSN:1369 7021 © Elsevier Ltd 2009DECEMBER 2009 | VOLUME 12 | NUMBER 1260 REVIEW Mie resonance-based dielectric metamaterials Mie resonance-based dielectric metamaterials Metamaterials are artificial electromagnetic media structured on a scale much shorter than their operating wavelength. Under this condition they can be considered as homogeneous media whose electromagnetic properties rely mainly on the basic cell rather than periodic effects as it is the case for photonic crystal or more generally electromagnetic band gap material. Their basic cells are generally constituted of resonant inclusions which yield a π phaseshift in the material response above the resonant frequency. As a consequence, their effective permittivity and permeability can be negative either in separate or overlapping frequency bands, a unique and distinct property that is not observed in naturally occurring materials1 . For the latter condition they can be considered as double negative or negative index media hence opening the effective parameter space, so that new functionalities in the light scattering can be envisaged. Also recently, the achievement of near-zero or less than unity values of the effective permittivity and permeability was also recognized as one of the major goals of this research area. Great progress in electromagnetic metamaterials has been achieved for these unique physical properties and novel potential applications, such as negative refraction2, 3, perfect lens4, 5 and cloaking6-8 have been shown. They have been experimentally demonstrated in a frequency range from the radio frequencies9 to millimeter waves10, infrared wavelengths11-13, and visible optics14. Up to date, most of metamaterials are constructed with the use of sub-wavelength resonant metallic elements. For instance, the first left-handed Increasing attention on metamaterials has been paid due to their exciting physical behaviors and potential applications. While most of such artificial material structures developed so far are based on metallic resonant structures, Mie resonances of dielectric particles open a simpler and more versatile route for construction of isotropic metamaterials with higher operating frequencies. Here, we review the recent progresses of Mie resonance-based metamaterials by providing a description of the underlying mechanisms to realize negative permeability, negative permittivity and double negative media. We address some potential novel applications. Qian Zhao1,2, Ji Zhou1,*, Fuli Zhang3, Didier Lippens3 1 State Key Lab of New Ceramics and Fine Processing, Department of Materials Science and Engineering, Tsinghua University, Beijing, PRC 2 State Key Lab of Tribology, Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, PRC 3 Institut d’Electronique de Micro-électronique et de Nanotechnologie, UMR CNRS 8520, University of Lille 1, Villeneuve d’Ascq Cedex, France *E-mail: zhouji@mail.tsinghua.edu.cn MT1212p60_69.indd 60 11/11/2009 15:11:37 Mie resonance-based dielectric metamaterials REVIEW DECEMBER 2009 | VOLUME 12 | NUMBER 12 61 metamaterials (LHMs) with simultaneously negative permittivity and permeability have been fabricated by means of metallic split ring resonators (SRRs) and wires15, 16, for tailoring the magnetic and electric responses, respectively. Other metallic elements, such as Ω-shaped structures17, 18, U-shaped structures19, staplelike structures20, paired rods21, dendritic22 and fishnet structures23, are also successfully used to fabricate LHMs. Usually, the metallic constitutive elements have conductive loss and anisotropic electromagnetic responses. Several authors have suggested methods24-26 to construct isotropic metamaterials by combining metallic SRRs and wires. However it is difficult to fabricate the bulk arrays of complex geometry with submicron or nanoscale sizes in order to generate a negative permeability effect at infrared and optical frequencies27. Recently, another route based on the interaction between electromagnetic waves and dielectric particles28, 29 was proposed to achieve the electric or magnetic resonances. The Mie resonances of dielectric inclusions provide a novel mechanism for the creation of magnetic or electric resonance based on displacement currents, and offer a simpler and more versatile route for the fabrication of isotropic metamaterials operating at higher frequencies. The progress on metallic elements-based metamaterials has been reviewed in many articles30-32 and books33, 34. In this review, we focus on the scattering mechanisms based on Mie resonance in dielectric particles and describe the recent progress in these metamaterials including ferroelectric and polaritonic particles and their tunable behaviors. Mie resonance of particles From the viewpoint of scattering theory, all scattering objects can be represented by effective electric and/or magnetic polarizability densities. Light scattering by small (relative to the incident light wavelength) spherical particles is a fundamental topic in classical electrodynamics35, and is based upon the exact Mie solution of the diffraction problem36. The scattered field of a single isolated dielectric sphere with radius r0 and relative refractive index n can be decomposed into a multipole series with the 2m-pole term of the scattered electric field proportional to )()()()( )()()()( '' '' nxxxnxn nxxxnxna mmmm mmmm m ψξξψ ψψψψ − − = (1) whereas the 2m-pole term of the scattered magnetic field is proportional to )()()()( )()()()( '' '' nxxnxnx nxxnxnxb mmmm mmmm m ψξξψ ψψψψ − − = (2) where x=k0r0, k0 is the free-space wavenumber, and ψm(x) and ξm(x) are the Riccati-Bessel functions. The primes indicate derivation with respect to the arguments. The scattering coefficient am and bm are related to the electric and magnetic responses of the sphere, respectively. From the Mie theory we can calculate the electric and magnetic dipole coefficients, a1 and b1, respectively. From effective medium theory, we know that these are the multipole terms which contribute most significantly to the effective permittivity and Fig. 1 Electric and magnetic field distribution in a dielectric cube with the magnetic field polarized along the z axis and electric field polarized along the y axis. (a) Electric field in the plane z=0 near the first Mie resonance. (b) Magnetic field in the plane y=0 near the first Mie resonance. (c) Electric field in the plane z=0 near the second Mie resonance. (d) Magnetic field in the plane y=0 near the second Mie resonance. (Reprinted with permission from37. © 2008 American Physical Society.) (b)(a) (c) (d) MT1212p60_69.indd 61 11/11/2009 15:11:41 Administrator 高亮 REVIEW Mie resonance-based dielectric metamaterials DECEMBER 2009 | VOLUME 12 | NUMBER 1262 permeability of the particle composite. Since the magnetic response of a nonmagnetic particle is usually weak, it is important to strengthen the electromagnetic resonant behavior. For the lowest resonant frequencies of a1 and b1, the sphere exhibits electric and magnetic dipoles. This conclusion can be assessed from the electromagnetic intensity distributions in a high dielectric ceramic cube for a plane incident wave propagating along the x axis37 (Fig. 1). It can be seen that the electric or magnetic fields are mainly localized in the cubes. The azimuthal component of the displacement current inside each cube is greatly enhanced at the first Mie resonance (Fig. 1a) resulting in a large magnetic field along the z axis (Fig. 1b) which corresponds to the TE011 Mie resonance mode. At the second Mie resonance, the y component of the displacement current inside the cubes increases dramatically (Fig. 1c) and hence with a large magnetic field along the azimuth (Fig. 1d), which corresponds to the TM011 Mie resonance mode. These electric and magnetic dipole resonances act as artificial ‘atoms’ which form the basis of new optical materials. In a material made up of a collection of such resonant particles, their combined scattering response can act like a material with almost arbitrary values of effective permittivity and permeability. This idea can be verified by using the model proposed in 1947 by Lewin38 who considered the electromagnetic scattering properties of a composite material which was constituted of an array of lossless magnetodielectric spheres (ε2 and μ2) embedded in another background matrix (ε1 and μ1). The effective permittivity εeff and permeability μeff expressions based on Mie theory are as follows38. ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ − − + += f e e f eff v bF bF v )( 2)( 3 11 θ θεε (3) ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ − − + += f m m f eff v bF bF v )( 2)( 3 11 θ θμμ (4) where θθθθ θθθθ cossin)1( )cos(sin2)( 2 +− − =F (5) 21 εε=eb , 21 μμ=mb (6) The volume fraction of the spherical particles, 30 )( 3 4 p r v f π= , 2200 μεθ rk= , r0 and p are the particle radius and the lattice constant, respectively. Eq 5 shows that F(θ) is a resonant function and becomes negative above resonance in some range of θ, resulting in the negative permittivity or permeability for negative values of F(θ) with a magnitude on the order of unity as given by Eqs. (3) and (4). In Lewin’s model, the constitutive parameters were formulated only considering the spheres resonating either in the first or second resonant modes of the Mie series. This is because the higher order Mie resonances often occur at frequencies beyond the long wavelength limit, and thus the Clausius-Mossotti equation does not apply. Then, Jylha et al.39 improved those formulations by taking into account the electric polarizabilities of spheres operating in the magnetic resonant modes. Independent of Lewin’s model, O’Brien and Pendry28 showed that a negative effective permeability can be obtained in a two-dimensional array of ferroelectric rods with the magnetic field polarized along the axes of the rods. Although the underlying physics is the same, the authors used another method40 (transfer matrix method) to find the effective media values with similar conclusion. The aforementioned theoretical results show that a high permittivity microstructured medium can exhibit isotropic negative values of the effective permeability and permittivity. Mie resonance as a new route for metamaterials With the rapid development of LHMs, Lewin’s model was reconsidered and introduced into the realm of metamaterials to realize single and double negative media. Based on Lewin’s model, Holloway et al.29 numerically demonstrated the feasibility of achieving simultaneously negative εeff and μeff in the magnetodielectric sphere arrays for wavelengths where the electric and magnetic resonances are excited in the spheres. Eqs. (3) and (4) show that εeff and μeff depend on the permeability and permittivity of the dielectric inclusions and host medium, as well as the volume fraction of spheres of radius r0. Fig. 2 shows εeff and μeff calculated for υf = 0.5, ε1 = μ1 = 1, ε2 = 40, and Fig. 2 Calculated effective permittivity εeff and permeability μeff of the magnetodielectric sphere arrays composite. (Reprinted with permission from29. © 2003 IEEE.) MT1212p60_69.indd 62 11/11/2009 15:11:44 Mie resonance-based dielectric metamaterials REVIEW DECEMBER 2009 | VOLUME 12 | NUMBER 12 63 μ2 = 200 as a function of k0r0. There are two regions where both εeff and μeff become negative with the bandwidths becoming narrower by decreasing the volume fraction υf. From a practical point of view, many kinds of materials and shapes of particles have been proposed in the literature to fabricate negative or less than unity permeability and permittivity dielectric metamaterials operating in different frequency regions. The designing ideas are summarized in the following. Magnetic response The realization of abnormal permeability is a tricky issue for various metamaterials, because the magnetic response of materials is usually weak, especially at infrared or visible frequencies. Lewin’s model shows that two methods based on Mie resonance can be used to achieve a negative effective permeability effect. The first one is to use magnetodielectric inclusions with large values of permeability and permittivity. The second method is to use nonmagnetic dielectric particles with extremely large permittivity values. In the former case, two terms be and bm in Eqs. (3) and (4) contribute to the negative permeability, while only be plays an important role for the latter. The possibility to use magnetodielectric particles has been numerically demonstrated29. However a magnetodielectric particle with simultaneously large values of permittivity and permeability seems physically unattainable even in the microwave. Eq. (4) indicates that nonmagnetic spheres with very high permittivity values can also be employed to induce negative permeability and permittivity effects at the different Mie resonance frequencies. Polaritonic materials, such as ionic solids and polar semiconductors, or ferroelectric materials can provide the large permittivity. For ferroelectric materials, their extreme high permittivity values can be preserved at least up to millimeter wavelengths41. At infrared frequencies however a roll-off of the dielectric constant is observed so that they cannot be used in this frequency range. In contrast, the lattice resonance in polaritonic crystals can be exploited to tailor the permittivity resonance at infrared and optical frequencies. As a consequence, the polaritonic resonance of crystals could be potentially used in the infrared spectral region. The relative permittivity is42 ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − − + ∞= ω γω ω ω ω ε ω ε i T T L r 2 2 2 2 1 ) ( ) ( , (7) where ε(∞) is the high-frequency limit of the permittivity, ωT and ωL are the transverse and longitudinal optical phonon frequencies, and γ is the damping coefficient. Therefore, large values of εr(ω) can be achieved near the transverse phonon frequency. A three-dimensional dielectric composite consisting of an array of dielectric cubes [Ba0.5Sr0.5TiO3 (BST)] with a relative permittivity of 1600 in a Teflon substrate was fabricated (Fig. 3a) to demonstrate the feasibility of this full dielectric metamaterial route. On this basis, isotropic negative values of the effective permeability were experimentally demonstrated in the microwave region37. The dielectric cube side length l was 1.0 mm for a lattice constant of 2.5 mm. And the first TE and TM resonance modes, i.e., magnetic and electric resonances were determined by transmission measurements at 6.12 and 8.28 GHz respectively. Using smaller cubes (l =0.75 mm) with a magnetic resonance near 8.5 GHz, in conjunction with an electric response from metallic wires, researchers fabricated double negative media. The results of transmission measurements are displayed in Fig. 3b and the retrieved electromagnetic parameters (Fig. 3c) further verified the negative permeability near the first Mie resonance. A rectangular dielectric block43 and piezoelectric disks44 were also experimentally verified to design very low-loss magnetic metamaterials. Fig. 3 A three-dimensional dielectric composite with isotropic negative permeability. (a) Photograph of BST cubes arrayed in Teflon substrate. (b) Transmission for the dielectric cube array only (dashed line), wire array only (dotted line), and the combination of BST cubes and wires (solid line). Inset of Fig. 3(b) shows the measured and calculated transmission phase. (c) Retrieved refractive index and permeability (inset). (Reprinted with permission from37. © 2008 American Physical Society.) (b) (a) (c) MT1212p60_69.indd 63 11/11/2009 15:11:45 REVIEW Mie resonance-based dielectric metamaterials DECEMBER 2009 | VOLUME 12 | NUMBER 1264 Also a two-dimensional square lattice of dielectric disk inserted between two metallic parallel-plate waveguide (Fig. 4a) was also used to demonstrate negative refraction, where the macroscopic behavior of dielectric disk provides negative effective permeability and the waveguide plays a role in a TE cut-off waveguide leading to negative effective permittivity45. A dielectric block [(Zr,Sn)TiO4 ceramics, dielectric constant 36.7] with a thin metallic rod screwed inside was proposed to fabricate gradient metamaterial lenses and numerically demonstrated to deflect and focus the incident plane waves46. Dielectric-resonator-based composite right/left-handed transmission lines used to the design of leaky wave antenna can alleviate the significantly large conducting loss and increase the radiation efficiency47. Terahertz metamaterials based on the Mie resonance of polycrystalline TiO2 cubes-arrays on the Al2O3 substrate (Fig. 4b) were also experimentally demonstrated by Shibuya et al.48, in which the negative permeability and permittivity occur around 0.28 and 0.38 THz, respectively. A three-dimensional collection of polaritonic spheres was proposed to obtain isotropic negative effective permeability near the first Mie resonance at infrared frequencies49. The calculated effective parameters (LiTaO3 crystal) reported in this reference are shown in Fig. 5 and the isotropy of electromagnetic properties was also verified by a comparison with a full multiple scattering approach. Semiconductor nanoparticles, such as CuCl with a Z3 exciton line at 386.93 nm, Cu2O with a 2P exciton50, and GaP51 were also suggested to realize negative permeability within and below the optical region. Metal nanoclusters possessing large permittivity at visible frequency can also be used to obtain magnetic Mie resonance52, 53. Similar to spherical or cubic dielectric elements, the cylindrical particles can also generate the Mie electromagnetic resonances. The most important issue is to determine the resonant modes. For Fig. 5 The calculated effective relative permeability (a) and permittivity (b) of a collection of LiTaO3 spheres. (Reprinted with permission from49. © 2005 American Physical Society.) Fig. 4 Schematic of metamaterials based on Mie resonance. (a) A two-dimensional lattice structure for one dielectric-resonator in the cutoff microwave waveguide. (Reprinted with permission from45. © 2007 IEEE.) (b) Terahertz metamaterial consisting of polycrystalline TiO2 cubes-arrays on the Al2O3 substrate. (Reprinted with permission from48. © 2008 Metamaterials.) (b)(a) (a) (b) MT1212p60_69.indd 64 11/11/2009 15:11:47 Mie resonance-based dielectric metamaterials REVIEW DECEMBER 2009 | VOLUME 12 | NUMBER 12 65 a TE polarization (E field perpendicular to the rod axis), the lowest Mie resonance mode (TE0) corresponds to the magnetic response, resulting in negative permeability values. A two-dimensional array of ferroelectric rods28, 54 and polaritonic rods55 were proposed to obtain negative effective permeability56, which can be considered as the TE0 mode. For TM polarization (E field parallel to the rod axis), the lowest two Mie resonances modes, TM0 and TM1, correspond to the electric and magnetic responses, and consequently lead to negative permittivity and permeability, respectively. The electromagnetic field distribution displayed for the TM1 mode (Fig. 6) can further clarify the magnetic resonance mechanism57. The electric field is distributed along the ± z directions with opposite signs along the propagation direction (Fig. 6b), accounting for a strong circular displacement current (Fig. 6d