Metamaterial

A comparison of refraction in a left-handed metamaterial to that in a normal material

Negative-index metamaterials (NIM) are characterized by a negative index of refraction. Other terms for NIMs include "left-handed media", "media with a negative refractive index", and "backward-wave media".[4] NIMs where the negative index of refraction arises from simultaneously negative permittivity and negative permeability are also known as double negative metamaterials or double negative materials (DNG).[17]

Assuming a material well-approximated by a real permittivity and permeability, the relationship between permittivity ${\displaystyle \epsilon _{r}}$, permeability ${\displaystyle \mu _{r}}$ and refractive index n is given by ${\displaystyle \scriptstyle n=\pm {\sqrt {\epsilon _{\mathrm {r} }\mu _{\mathrm {r} }}}}$. All known non-metamaterial transparent materials (glass, water, ...) possess positive ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$. By convention the positive square root is used for n. However, some engineered metamaterials have ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}<0}$. Because the product ${\displaystyle \epsilon _{r}\mu _{r}}$ is positive, n is real. Under such circumstances, it is necessary to take the negative square root for n. When both ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$ are positive (negative), waves travel in the forward (backward) direction. Electromagnetic waves cannot propagate in materials with ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$ of opposite sign as the refractive index becomes imaginary. Such materials are opaque for electromagnetic radiation and examples include plasmonic materials such as metals (gold, silver, ...).

The foregoing considerations are simplistic for actual materials, which must have complex-valued ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$. The real parts of both ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$ do not have to be negative for a passive material to display negative refraction.[27][28] Indeed, a negative refractive index for circularly polarized waves can also arise from chirality.[29][30] Metamaterials with negative n have numerous interesting properties:[5][31]

• Snell's law (n₁sinθ₁ = n₂sinθ₂) still describes refraction, but as n₂ is negative, incident and refracted rays are on the same side of the surface normal at an interface of positive and negative index materials.
• Cherenkov radiation points the other way.
• The time-averaged Poynting vector is antiparallel to phase velocity. However, for waves (energy) to propagate, a –µ must be paired with a –ε in order to satisfy the wave number dependence on the material parameters ${\displaystyle kc=\omega {\sqrt {\mu \epsilon }}}$.

Negative index of refraction derives mathematically from the vector triplet E, H and k.[5]

For plane waves propagating in electromagnetic metamaterials, the electric field, magnetic field and wave vector follow a left-hand rule, the reverse of the behavior of conventional optical materials.

To date, only metamaterials exhibit a negative index of refraction.[4][31][32]

Single negative (SNG) metamaterials have either negative relative permittivity (εr) or negative relative permeability (µr), but not both.[17] They act as metamaterials when combined with a different, complementary SNG, jointly acting as a DNG.

Epsilon negative media (ENG) display a negative εr while µr is positive.[4][31][17] Many plasmas exhibit this characteristic. For example, noble metals such as gold or silver are ENG in the infrared and visible spectrums.

Mu-negative media (MNG) display a positive εr and negative µr.[4][31][17] Gyrotropic or gyromagnetic materials exhibit this characteristic. A gyrotropic material is one that has been altered by the presence of a quasistatic magnetic field, enabling a magneto-optic effect. A magneto-optic effect is a phenomenon in which an electromagnetic wave propagates through such a medium. In such a material, left- and right-rotating elliptical polarizations can propagate at different speeds. When light is transmitted through a layer of magneto-optic material, the result is called the Faraday effect: the polarization plane can be rotated, forming a Faraday rotator. The results of such a reflection are known as the magneto-optic Kerr effect (not to be confused with the nonlinear Kerr effect). Two gyrotropic materials with reversed rotation directions of the two principal polarizations are called optical isomers.

Joining a slab of ENG material and slab of MNG material resulted in properties such as resonances, anomalous tunneling, transparency and zero reflection. Like negative-index materials, SNGs are innately dispersive, so their εr, µr and refraction index n, are a function of frequency.[31]

Hyperbolic metamaterials (HMMs) behave as a metal for certain polarization or direction of light propagation and behave as a dielectric for the other due to the negative and positive permittivity tensor components, giving extreme anisotropy. The material's dispersion relation in wavevector space forms a hyperboloid and therefore it is called a hyperbolic metamaterial. The extreme anisotropy of HMMs leads to directional propagation of light within and on the surface.[33] HMMs have showed various potential applications, such as sensing, imaging, steering of optical signals, enhanced plasmon resonance effects.[34]

Electromagnetic bandgap metamaterials (EBG or EBM) control light propagation. This is accomplished either with photonic crystals (PC) or left-handed materials (LHM). PCs can prohibit light propagation altogether. Both classes can allow light to propagate in specific, designed directions and both can be designed with bandgaps at desired frequencies.[35][36] The period size of EBGs is an appreciable fraction of the wavelength, creating constructive and destructive interference.

PC are distinguished from sub-wavelength structures, such as tunable metamaterials, because the PC derives its properties from its bandgap characteristics. PCs are sized to match the wavelength of light, versus other metamaterials that expose sub-wavelength structure. Furthermore, PCs function by diffracting light. In contrast, metamaterial does not use diffraction.[37]

PCs have periodic inclusions that inhibit wave propagation due to the inclusions' destructive interference from scattering. The photonic bandgap property of PCs makes them the electromagnetic analog of electronic semi-conductor crystals.[38]

EBGs have the goal of creating high quality, low loss, periodic, dielectric structures. An EBG affects photons in the same way semiconductor materials affect electrons. PCs are the perfect bandgap material, because they allow no light propagation.[39] Each unit of the prescribed periodic structure acts like one atom, albeit of a much larger size.[4][39]

EBGs are designed to prevent the propagation of an allocated bandwidth of frequencies, for certain arrival angles and polarizations. Various geometries and structures have been proposed to fabricate EBG's special properties. In practice it is impossible to build a flawless EBG device.[4][5]

EBGs have been manufactured for frequencies ranging from a few gigahertz (GHz) to a few terahertz (THz), radio, microwave and mid-infrared frequency regions. EBG application developments include a transmission line, woodpiles made of square dielectric bars and several different types of low gain antennas.[4][5]

Double positive mediums (DPS) do occur in nature, such as naturally occurring dielectrics. Permittivity and magnetic permeability are both positive and wave propagation is in the forward direction. Artificial materials have been fabricated which combine DPS, ENG and MNG properties.[4][17]

Categorizing metamaterials into double or single negative, or double positive, normally assumes that the metamaterial has independent electric and magnetic responses described by ε and µ. However, in many cases, the electric field causes magnetic polarization, while the magnetic field induces electrical polarization, known as magnetoelectric coupling. Such media are denoted as bi-isotropic. Media that exhibit magnetoelectric coupling and that are anisotropic (which is the case for many metamaterial structures[40]), are referred to as bi-anisotropic.[41][42]

Four material parameters are intrinsic to magnetoelectric coupling of bi-isotropic media. They are the electric (E) and magnetic (H) field strengths, and electric (D) and magnetic (B) flux densities. These parameters are ε, µ, κ and χ or permittivity, permeability, strength of chirality, and the Tellegen parameter, respectively. In this type of media, material parameters do not vary with changes along a rotated coordinate system of measurements. In this sense they are invariant or scalar.[5]

The intrinsic magnetoelectric parameters, κ and χ, affect the phase of the wave. The effect of the chirality parameter is to split the refractive index. In isotropic media this results in wave propagation only if ε and µ have the same sign. In bi-isotropic media with χ assumed to be zero, and κ a non-zero value, different results appear. Either a backward wave or a forward wave can occur. Alternatively, two forward waves or two backward waves can occur, depending on the strength of the chirality parameter.

In the general case, the constitutive relations for bi-anisotropic materials read ${\displaystyle \mathbf {D} =\epsilon \mathbf {E} +\xi \mathbf {H} ,}$ ${\displaystyle \mathbf {B} =\zeta \mathbf {E} +\mu \mathbf {H} ,}$ where ${\displaystyle \epsilon }$ and ${\displaystyle \mu }$ are the permittivity and the permeability tensors, respectively, whereas ${\displaystyle \xi }$ and ${\displaystyle \zeta }$ are the two magneto-electric tensors. If the medium is reciprocal, permittivity and permeability are symmetric tensors, and ${\displaystyle \xi =-\zeta ^{T}=-i\kappa ^{T}}$, where ${\displaystyle \kappa }$ is the chiral tensor describing chiral electromagnetic and reciprocal magneto-electric response. The chiral tensor can be expressed as ${\displaystyle \kappa ={\tfrac {1}{3}}\operatorname {tr} (\kappa )I+N+J}$, where ${\displaystyle \operatorname {tr} (\kappa )}$ is the trace of ${\displaystyle \kappa }$, I is the identity matrix, N is a symmetric trace-free tensor, and J is an antisymmetric tensor. Such decomposition allows us to classify the reciprocal bianisotropic response and we can identify the following three main classes: (i) chiral media (${\displaystyle \operatorname {tr} (\kappa )\neq 0,N\neq 0,J=0}$), (ii) pseudochiral media (${\displaystyle \operatorname {tr} (\kappa )=0,N\neq 0,J=0}$), (iii) omega media (${\displaystyle \operatorname {tr} (\kappa )=0,N=0,J\neq 0}$).

Handedness of metamaterials is a potential source of confusion as the metamaterial literature includes two conflicting uses of the terms left- and right-handed. The first refers to one of the two circularly polarized waves that are the propagating modes in chiral media. The second relates to the triplet of electric field, magnetic field and Poynting vector that arise in negative refractive index media, which in most cases are not chiral.

Generally a chiral and/or bianisotropic electromagnetic response is a consequence of 3D geometrical chirality: 3D-chiral metamaterials are composed by embedding 3D-chiral structures in a host medium and they show chirality-related polarization effects such as optical activity and circular dichroism. The concept of 2D chirality also exists and a planar object is said to be chiral if it cannot be superposed onto its mirror image unless it is lifted from the plane. 2D-chiral metamaterials that are anisotropic and lossy have been observed to exhibit directionally asymmetric transmission (reflection, absorption) of circularly polarized waves due to circular conversion dichrosim.[43][44] On the other hand, bianisotropic response can arise from geometrical achiral structures possessing neither 2D nor 3D intrinsic chirality. Plum and colleagues investigated magneto-electric coupling due to extrinsic chirality, where the arrangement of a (achiral) structure together with the radiation wave vector is different from its mirror image, and observed large, tuneable linear optical activity,[45] nonlinear optical activity,[46] specular optical activity[47] and circular conversion dichroism.[48] Rizza et al.[49] suggested 1D chiral metamaterials where the effective chiral tensor is not vanishing if the system is geometrically one-dimensional chiral (the mirror image of the entire structure cannot be superposed onto it by using translations without rotations).

3D-chiral metamaterials are constructed from chiral materials or resonators in which the effective chirality parameter ${\displaystyle \kappa }$ is non-zero. Wave propagation properties in such chiral metamaterials demonstrate that negative refraction can be realized in metamaterials with a strong chirality and positive ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$.[50] [51] This is because the refractive index ${\displaystyle n}$ has distinct values for left and right circularly polarized waves, given by

${\displaystyle n=\pm {\sqrt {\epsilon _{r}\mu _{r}}}\pm \kappa }$

It can be seen that a negative index will occur for one polarization if ${\displaystyle \kappa }$ > ${\displaystyle {\sqrt {\epsilon _{r}\mu _{r}}}}$. In this case, it is not necessary that either or both ${\displaystyle \epsilon _{r}}$ and ${\displaystyle \mu _{r}}$ be negative for backward wave propagation.[5] A negative refractive index due to chirality was first observed simultaneously and independently by Plum et al.[29] and Zhang et al.[30] in 2009.

Frequency selective surface-based metamaterials block signals in one waveband and pass those at another waveband. They have become an alternative to fixed frequency metamaterials. They allow for optional changes of frequencies in a single medium, rather than the restrictive limitations of a fixed frequency response.[52]

These metamaterials use different parameters to achieve a negative index of refraction in materials that are not electromagnetic. Furthermore, "a new design for elastic metamaterials that can behave either as liquids or solids over a limited frequency range may enable new applications based on the control of acoustic, elastic and seismic waves."[53] They are also called mechanical metamaterials.

Acoustic metamaterials control, direct and manipulate sound in the form of sonic, infrasonic or ultrasonic waves in gases, liquids and solids. As with electromagnetic waves, sonic waves can exhibit negative refraction.[14]

Control of sound waves is mostly accomplished through the bulk modulus β, mass density ρ and chirality. The bulk modulus and density are analogs of permittivity and permeability in electromagnetic metamaterials. Related to this is the mechanics of sound wave propagation in a lattice structure.[54] Also materials have mass and intrinsic degrees of stiffness. Together, these form a resonant system and the mechanical (sonic) resonance may be excited by appropriate sonic frequencies (for example audible pulses).

Structural metamaterials provide properties such as crushability and light weight. Using projection micro-stereolithography, microlattices can be created using forms much like trusses and girders. Materials four orders of magnitude stiffer than conventional aerogel, but with the same density have been created. Such materials can withstand a load of at least 160,000 times their own weight by over-constraining the materials.[55][56]

A ceramic nanotruss metamaterial can be flattened and revert to its original state.[57]

Metamaterials may be fabricated that include some form of nonlinear media, whose properties change with the power of the incident wave. Nonlinear media are essential for nonlinear optics. Most optical materials have a relatively weak response, meaning that their properties change by only a small amount for large changes in the intensity of the electromagnetic field. The local electromagnetic fields of the inclusions in nonlinear metamaterials can be much larger than the average value of the field. Besides, remarkable nonlinear effects have been predicted and observed if the metamaterial effective dielectric permittivity is very small (epsilon-near-zero media).[58][59][60] In addition, exotic properties such as a negative refractive index, create opportunities to tailor the phase matching conditions that must be satisfied in any nonlinear optical structure.

In 2009, Marc Briane and Graeme Milton[61] proved mathematically that one can in principle invert the sign of a 3 materials based composite in 3D made out of only positive or negative sign Hall coefficient materials. Later in 2015 Muamer Kadic et al.[62] showed that a simple perforation of isotropic material can lead to its change of sign of the Hall coefficient. This theoretical claim was finally experimentally demonstrated by Christian Kern et al.[63]

In 2015, it was also demonstrated by Christian Kern et al. that an anisotropic perforation of a single material can lead to a yet more unusual effect namely the parallel Hall effect.[64] This means that the induced electric field inside a conducting media is no longer orthogonal to the current and the magnetic field but is actually parallel to the latest.

Terahertz metamaterials interact at terahertz frequencies, usually defined as 0.1 to 10 THz. Terahertz radiation lies at the far end of the infrared band, just after the end of the microwave band. This corresponds to millimeter and submillimeter wavelengths between the 3 mm (EHF band) and 0.03 mm (long-wavelength edge of far-infrared light).

Photonic metamaterial interact with optical frequencies (mid-infrared). The sub-wavelength period distinguishes them from photonic band gap structures.[65][66]

Tunable metamaterials allow arbitrary adjustments to frequency changes in the refractive index. A tunable metamaterial expands beyond the bandwidth limitations in left-handed materials by constructing various types of metamaterials.

Plasmonic metamaterials exploit surface plasmons, which are produced from the interaction of light with metal-dielectrics. Under specific conditions, the incident light couples with the surface plasmons to create self-sustaining, propagating electromagnetic waves or surface waves[67] known as surface plasmon polaritons. Bulk plasma oscillations make possible the effect of negative mass (density).[68][69]

Metamaterial antennas are a class of antennas that use metamaterials to improve performance.[13][17][72][73] Demonstrations showed that metamaterials could enhance an antenna's radiated power.[13][74] Materials that can attain negative permeability allow for properties such as small antenna size, high directivity and tunable frequency.[13][17]

A metamaterial absorber manipulates the loss components of metamaterials' permittivity and magnetic permeability, to absorb large amounts of electromagnetic radiation. This is a useful feature for photodetection[75][76] and solar photovoltaic applications.[77] Loss components are also relevant in applications of negative refractive index (photonic metamaterials, antenna systems) or transformation optics (metamaterial cloaking, celestial mechanics), but often are not utilized in these applications.

A superlens is a two or three-dimensional device that uses metamaterials, usually with negative refraction properties, to achieve resolution beyond the diffraction limit (ideally, infinite resolution). Such a behaviour is enabled by the capability of double-negative materials to yield negative phase velocity. The diffraction limit is inherent in conventional optical devices or lenses.[78][79]

Metamaterials are a potential basis for a practical cloaking device. The proof of principle was demonstrated on October 19, 2006. No practical cloaks are publicly known to exist.[80][81][82][83][84][85]

Conventionally, the RCS has been reduced either by Radar absorbent material (RAM) or by purpose shaping of the targets such that the scattered energy can be redirected away from the source. While RAMs have narrow frequency band functionality, purpose shaping limits the aerodynamic performance of the target. More recently, metamaterials or metasurfaces are synthesized that can redirect the scattered energy away from the source using either array theory[86][87][88][89] or generalized Snell's law.[90][91] This has led to aerodynamically favorable shapes for the targets with the reduced RCS.

Seismic metamaterials counteract the adverse effects of seismic waves on man-made structures.[10][92][93]

Metamaterials textured with nanoscale wrinkles could control sound or light signals, such as changing a material's color or improving ultrasound resolution. Uses include nondestructive material testing, medical diagnostics and sound suppression. The materials can be made through a high-precision, multi-layer deposition process. The thickness of each layer can be controlled within a fraction of a wavelength. The material is then compressed, creating precise wrinkles whose spacing can cause scattering of selected frequencies.[94][95]

The Multidisciplinary University Research Initiative (MURI) encompasses dozens of Universities and a few government organizations. Participating universities include UC Berkeley, UC Los Angeles, UC San Diego, Massachusetts Institute of Technology, and Imperial College in London. The sponsors are Office of Naval Research and the Defense Advanced Research Project Agency.[98]

MURI supports research that intersects more than one traditional science and engineering discipline to accelerate both research and translation to applications. As of 2009, 69 academic institutions were expected to participate in 41 research efforts.[99]

The Virtual Institute for Artificial Electromagnetic Materials and Metamaterials "Metamorphose VI AISBL" is an international association to promote artificial electromagnetic materials and metamaterials. It organizes scientific conferences, supports specialized journals, creates and manages research programs, provides training programs (including PhD and training programs for industrial partners); and technology transfer to European Industry.[100][101]