Almost all materials encountered in optics, such as glass or water, have positive values for both permittivity ε and permeability µ. However, metals such as silver and gold have negative permittivity at shorter wavelengths. A material such as a surface plasmon that has either (but not both) ε or µ negative is often opaque to electromagnetic radiation. However, anisotropic materials with only negative permittivity can produce negative refraction due to chirality.
Although the optical properties of a transparent material are fully specified by the parametersεr and µr, refractive index n is often used in practice, which can be determined from . All known non-metamaterial transparent materials possess positive εr and µr. By convention the positive square root is used for n.
However, some engineered metamaterials have εr < 0 and µr < 0. Because the product εrµr is positive, n is real. Under such circumstances, it is necessary to take the negative square root for n.Video representing negative refraction of light at uniform planar interface.
The foregoing considerations are simplistic for actual materials, which must have complex-valued εr and µr. The real parts of both εr and µr do not have to be negative for a passive material to display negative refraction. Metamaterials with negative n have numerous interesting properties:
- Snell's law (n1sinθ1 = n2sinθ2), but as n2 is negative, the rays will be refracted on the same side of the normal on entering the material.
- 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 .
For plane waves propagating in electromagnetic metamaterials, the electric field, magnetic field and wave vector follow a left-hand rule. This is a reversal of direction when compared to the behavior of conventional optical materials.
Negative refractive index is an important characteristic in metamaterial design and fabrication. As reverse-refraction media, these occur when both permittivity ε andpermeability µ are negative. Furthermore, this condition occurs mathematically from the vector triplet E, H and k.
In ordinary materials – solid, liquid, or gas; transparent or opaque; conductor or insulator – the conventional refractive indexdominates. This means that permittivity and permeability are both positive resulting in an ordinary index of refraction. However, metamaterials have the capability to exhibit a state where both permittivity and permeability are negative, resulting in an extraordinary,index of negative refraction.Negative index metamaterials and Negative refraction
In negative index metamaterials (NIM), both permittivity and permeability are negative resulting in a negative index of refraction. These are also known as Double Negative Metamaterials or double negative materials (DNG). Other terms for NIMs include "left-handed media", "media with a negative refractive index", and "backward-wave media".
In optical materials, if both permittivity ε and permeability µ are positive, wave propagation travels in the forward direction. If both ε and µ are negative, a backward wave is produced. If ε and µ have different polarities, waves do not propagate. Mathematically, quadrant II and quadrant IV have coordinates (0,0) in a coordinate plane where ε is the horizontal axis, and µ is the vertical axis.
Single negative (SNG) metamaterials have either negative relative permittivity (εr) or negative relative permeability (µr), but not both. They act as metamaterials in combination with a different, complementary SNG, jointly acting as a DNG.
Epsilon negative media (ENG) display a negative εr while µr is positive. Many plasmas exhibit this characteristic. For examplenoble metals such as gold or silver will exhibit this characteristic in the infrared and visible spectrums.
Mu-negative media (MNG) display a positive εr while µr is negative. 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, alter with changes in frequency.Electronic band structure and Coupled oscillation
Electromagnetic bandgap metamaterials (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. The period size of EBGs is an appreciable amount 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 operate as a sub-wavelength structure. Furthermore, photonic crystals function by diffracting light. In contrast, a permittivity and permeability define each metamaterial, which is derived from its sub-wavelength structure and does not use diffraction.
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.
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. Each unit of the prescribed periodic structure acts like one atom, albeit of a much larger size.
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.
EBGs have been manufactured for frequencies ranging from a few gigahertz (GHz) up 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.
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.
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 also anisotropic (which is the case for many metamaterial structures), are referred to as bi-anisotropic.
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 four material parameters are ε, µ, κ and χ or permittivity, permeability, strength of chirality, and the Tellegen parameter respectively. In this type of media, the material parameters do not vary with changes along a rotated coordinate system of measurements. In this sense they are invariant or scalar.
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 χassumed to be zero, and κ a non-zero value, different results appear. Both a backward wave and a forward wave can occur. Alternatively, two forward waves or two backward waves can occur, depending on the strength of the chirality parameter.
Chiral metamaterials are constructed from chiral in which the effective parameter k is non-zero. This 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.
Wave propagation properties in chiral metamaterials demonstrate that negative refraction can be realized in metamaterials with a strong chirality and positive ε and μ.  This is because the refractive index has distinct values for left and right, given by
It can be seen that a negative index will occur for one polarization if κ > √. In this case, it is not necessary that either or both εrand µr be negative for backward wave propagation.Tunable metamaterials#Frequency selective surface based metamaterials
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.
These are a type of metamaterial that uses 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." They are also calledmechanical metamaterials.
Acoustic metamaterials control, direct and manipulate sound in the form of sonic, infrasonic, orultrasonic waves in gases, liquids and solids. As with electromagnetic waves, sonic waves can exhibit negative refraction.
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. 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 audiblepulses).Nonlinear metamaterials
Metamaterials may also be fabricated which 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 a nonlinear metamaterials can be much larger than the average value of the field. In addition, exotic properties such as a negative refractive index, create opportunities to tailor the phase matching conditions, which must be satisfied in any nonlinear optical structure.Terahertz metamaterials
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 submillimeterwavelengths between the 3 mm (EHF band) and 0.03 mm (long-wavelength edge of far-infrared light).Photonic metamaterialsTunable metamaterials
Tunable metamaterials allow arbitrary adjustments to frequency changes in the refractive index. A tunable metamaterial encompasses the development of expanding beyond the bandwidth limitations in left-handed materials by constructing various types of metamaterials.Plasmonic 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 wavesknown as surface plasmon polaritons.
Metamaterials are under consideration for many applications. Metamaterial antennas are commercially available.
In 2007, one researcher stated that for metamaterial applications to be realized, energy loss must be reduced, materials must be extended into three-dimensional isotropicmaterials and production techniques must be industrialized.Metamaterial antennas
Metamaterial antennas are a class of antennas that use metamaterials to improve performance. Demonstrations have shown that metamaterials could enhance an antenna's radiated power. Materials that can attain negative permeability allow for properties such as an electrically small antenna size, high directivity and tunable operational frequency.Metamaterial absorber
A metamaterial absorber manipulates the loss components of metamaterials' permittivity and magnetic permeability, to absorb large amounts of electromagnetic radiation. Loss components are often noted in applications of negative refractive index (photonic metamaterials, antenna systems) or transformation optics (metamaterial cloaking, celestial mechanics), but often not utilized in these applications.SuperlensMetamaterial cloakingSeismic metamaterialsHistory of metamaterials
Seminal explorations of artificial materials for manipulating electromagnetic waves at the end of the 19th century. Some of the earliest structures that may be considered metamaterials date back to Jagadish Chandra Bose who in 1898 researched substances with chiral properties. Karl Ferdinand Lindman studied wave interaction with metallic helices as artificial chiral media in the early twentieth century. In the 1950s and 1960s, artificial dielectrics were studied for lightweight microwave antennas. Microwave radar absorbersmoved into the research arena in the 1980s and 1990s as applications for artificial chiral media.
Winston E. Kock developed materials that had similar characteristics to metamaterials in the late 1940s.
Materials that exhibited reversed physical characteristics were first described theoretically by Victor Veselago in 1967. He proved that substances with a negative index can transmit light. In such a material, he showed that the phase velocity would be anti-parallel to the direction of Poynting vector. This is contrary to wave propagation in naturally-occurring materials.
John Pendry was the first to identify a practical way to make a left-handed metamaterial, a material in which the right-hand rule is not followed. Such a material allows an electromagnetic wave to convey energy (have a group velocity) against its phase velocity. Pendry's idea was that metallic wires aligned along the direction of propagation could provide negative permittivity (ε < 0). Note however that natural materials (such as ferroelectrics) already displayed negative permittivity; the challenge was achieving negative permeability (µ < 0). In 1999 Pendry demonstrated that a split ring (C shape) with its axis placed along the direction of wave propagation could do so. In the same paper, he showed that a periodic array of wires and ring could give rise to a negative refractive index. Pendry also proposed a related negative-permeability design, the Swiss roll.
In the year 2000, Smith et al. reported the experimental demonstration of functioning electromagnetic metamaterials by horizontally stacking, periodically, split-ring resonators and thin wire structures. Later, a method was provided in 2002 to realize negative index metamaterials using artificial lumped-element loaded transmission lines in microstrip technology. In 2003, complex (both real and imaginary parts of) negative refractive index  and imaging by flat lens  using left handed metamaterials were demonstrated by the Northeastern University group. By 2007, research experiments which involved negative refractive index had been conducted by many groups. At microwave frequencies, the first real invisibility cloak was realized in 2006. However, only a very small object was imperfectly hidden.
The analogy is as follows: All materials are made of atoms, which are dipoles. These dipoles modify the light velocity by a factor n (the refractive index). The ring and wire units play the role of atomic dipoles: the wire acts as a ferroelectric atom, while the ring acts as an inductor L and the open section as a capacitor C. The ring as a whole therefore acts as an LC circuit. When the electromagnetic field passes through the ring, an induced current is created and the generated field is perpendicular to the magnetic field of the light. The magnetic resonance results in a negative permeability; the index is negative as well. (The lens is not truly flat, since the capacitance of the structure imposes a slope for the electric induction.)
Several (mathematical) material models which frequency response in DNGs. One of these is the Lorentz model. This describes electron motion in terms of a driven-damped, harmonic oscillator. The Debye relaxation model applies when the accelerationcomponent of the Lorentz mathematical model is small compared to the other components of the equation. The Drude model applies when the restoring force component is negligible and the coupling coefficient is generally the plasma frequency. Other component distinctions call for the use of one of these models, depending on its polarity, or purpose.
Three-dimensional composites of metal/non-metallic inclusions periodically/randomly embedded in a low permittivity matrix are usually modeled by analytical methods including mixing formulas and scattering-matrix based methods. The particle is modeled by either an electric dipole parallel to the electric field or a pair of crossed electric and magnetic dipoles parallel to the electric and magnetic fields, respectively, of the applied EM wave. These dipoles are the leading terms in the multipole series. They are the only existing ones for a homogeneous sphere, whose polarizability can be easily obtained from the Mie scattering coefficients. In general, this procedure is known as the "point-dipole approximation", which is a good approximation for metamaterials consisting of composites of electrically small spheres. Merits of these methods include low calculation cost and mathematical simplicity. 
Duke University has initiated an umbrella organization researching metamaterials under the banner "Novel Electromagnetic Materials" and became a leading research center. The center is a part of an international team, which includes California Institute of Technology,Harvard University, UCLA, Max Planck Institute of Germany, and the FOM Institute of the Netherlands. Six other groups are connected to this organization.
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, UK. The sponsors are Office of Naval Research and the Defense Advanced Research Project Agency.
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.
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.[6