A nonlinear metamaterial is an artificially constructed material that can exhibit properties not found in nature. Its response to

electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible ...

can be characterized by its

permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more i ...

and

material permeability Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geologica ...

. The product of the permittivity and permeability results in the

refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...

. Unlike natural materials, nonlinear metamaterials can produce a negative refractive index. These can also produce a more pronounced nonlinear response than naturally occurring materials. Nonlinear metamaterials are a

periodic Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values t ...

, nonlinear,

transmission medium A transmission medium is a system or substance that can mediate the propagation of signals for the purposes of telecommunication. Signals are typically imposed on a wave of some kind suitable for the chosen medium. For example, data can modula ...

. These are a type of

negative index metamaterial Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range. NIMs are constructed of periodic basic parts called unit cells, whi ...

where the

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

ity is available because the microscopic electric field of the inclusions can be larger than the macroscopic electric field of the electromagnetic (EM) source. This then becomes a useful tool which allows for enhancing the nonlinear behavior of the

metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...

. A dominant nonlinear response, however, can be derived from the hysteresis-type dependence of the material's magnetic permeability on the magnetic component of the incident

electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible ...

(light) propagating through the material. Furthermore, the hysteresis-type dependence of the magnetic permeability on the

field intensity In physics, field strength means the ''magnitude'' of a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E''). For example, an electromagnetic field results in both electric field strength and magnetic field strength. As ...

allows changing the material from left to right-handed and back.

Nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

media are essential for

nonlinear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typic ...

. However most

optical Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...

materials have a relatively weak nonlinear response, meaning that their properties only change by a small amount for large changes in intensity of the

electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...

. Nonlinear metamaterials can overcome this limitation, since the local fields of the resonant structures can be much larger than the average value of the field - in this respect metamaterials are similar to other composite media, such e.g. as random metal-dielectric composites, including fractal clusters and semicoutinouos/percolation metal films, where the areas with enhanced local light fields - “hot spots” - produce giant linear and non-linear optical responses.M.I. Stockman, V.M. Shalaev, M. Moskovits, R. Botet, T.F. George
Enhanced Raman scattering by fractal clusters: Scale-invariant theory
Physical Review B, v. 46, pp. 2821–2830 (1992)

Overview of metamaterials

Metamaterials are incarnations of materials first proposed by a Russian theorist,

Victor Veselago Victor Georgievich Veselago (13 June 1929, Ukraine – 15 September 2018)metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...

, are being developed in order to manipulate

in new ways. Optical and electromagnetic properties of natural materials are often altered through chemistry. With metamaterials optical and electromagnetic properties can be engineered through the geometry of its unit cells. The unit cells are materials that are ordered in geometric arrangements with dimensions that are fractions of the wavelength of the radiated

. By having the freedom to alter effects by adjusting the configurations and sizes of the unit cells, control over

and magnetic permeability can be achieved. These two parameters (or quantities) determine the propagation of electromagnetic waves in matter. Therefore, the achievable electromagnetic and optical effects can be extended. Optical properties can be expanded beyond the capabilities of lenses, mirrors, and other conventional materials. One of the effects most studied is the

negative index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

first proposed by Victor Veselago in 1967.

Negative index material Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range. NIMs are constructed of periodic basic parts called unit cells, whi ...

s, exhibit optical properties opposite to those of glass, air, and the other conventional materials. At the correct frequencies, the negative index materials refract electromagnetic waves in novel ways, to a zero index or negative index. Also, energy can propagate in the opposite direction which can result in compensation mechanisms, among other possibilities.

Interactions

Materials which scatter

light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...

or other

electromagnetic waves In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...

create a general

physical process Physical changes are changes affecting the form of a chemical substance, but not its chemical composition. Physical changes are used to separate mixtures into their component compounds, but can not usually be used to separate compounds into ch ...

where the different

frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is e ...

of light are forced to deviate from a straight trajectory. It is because, physically, the material is non-uniform at one, or more, or many places. Furthermore, the

optical science Atomic, molecular, and optical physics (AMO) is the study of matter-matter and light-matter interactions; at the scale of one or a few atoms and energy scales around several electron volts. The three areas are closely interrelated. AMO theory in ...

s make predictions about the path of light traversing through a material. When light deviates from its predicted (reflected) path, this also is considered

scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

. The split ring resonators which make up metamaterials are engineered to scatter light at

resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...

. Moreover, these resonant scattering elements are purposely designed at a uniform size throughout the material. This uniform size is much smaller than the

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...

of the

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...

of light

propagating Plant propagation is the process by which new plants grow from a variety of sources: seeds, cuttings, and other plant parts. Plant propagation can also refer to the man-made or natural dispersal of seeds. Propagation typically occurs as a step i ...

through the material. Since the repeating, scattering, resonant elements, which make up the

engineered Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

material are much smaller than the frequency of propagating light, metamaterials can now, also, be described in terms of

macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena ...

quantities. This description is simply another way to view metamaterials. And these are

electric permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...

, ''ε'' and magnetic permeability, ''μ''. Hence, by designing the individual,

geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

ly shaped unit of the material, called a cell, as the right kind of composite, it becomes a material with ''macroscopic properties that do not occur in nature''. Of particular interest regarding ''nonlinear metamaterials'', is the artificially induced macroscopic property known as

negative refractive index Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range. NIMs are constructed of periodic basic parts called unit cells, whi ...

. This effect is created by

Negative index metamaterials Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range. NIMs are constructed of periodic basic parts called unit cells, whi ...

(NIMs), which are

employed Employment is a relationship between two parties regulating the provision of paid labour services. Usually based on a contract, one party, the employer, which might be a corporation, a not-for-profit organization, a co-operative, or any othe ...

for use as ''nonlinear metamaterials''. In NIMs, nonlinear phenomena such as

second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...

and parametric amplification can take on highly unusual characteristics. Namely, the fact that the

wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...

and the

Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or ''power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...

of a wave propagating in a NIM are counter-directed alters the

phase-matching Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typic ...

conditions for the interacting waves, resulting in backward propagating waves as well as considerably changed Manley-Rowe relations and the distribution in space of the interacting fields' intensity.A.K. Popov and V.M.Shalaev
Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification
Applied Physics B, v. 84, pp. 131–37 (2006)

Non-linear properties of left-handed metamaterials

Previous studies of left-handed or

negative index metamaterials Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range. NIMs are constructed of periodic basic parts called unit cells, whi ...

were focused on the linear properties of the medium during

wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative ...

. In such cases, the view was that magnetic permeability and material permittivity are each not dependent on the intensity of the electromagnetic field. However, creating tunable structures requires knowledge of non-linear properties where the intensity of the

alters the permittivity, or permeability, or both, which in turn affects the range of transmission spectra or stop band spectra. Hence, the effective permeability is dependent on the

magnetic field intensity. As the field intensity is varied, switching between its positive and negative values can occur. Consequently, the material can switch from being left-handed to being right-handed, or vice versa. A composite structure consisting of a square lattice of the periodic arrays of conducting wires and split-ring resonators, produces an enhanced magnetic response. Without the correct magnetic response, it is not possible to produce a left-handed material.

Tunable split-ring resonators for nonlinear negative-index metamaterials

Variable capacitance diodes are incorporated into the split-ring cell producing a dynamic tunable system.

Reconfigurable refractive index (infrared)

Source radiation of near infrared

s are applied to a

system. The

index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

can be reconfigured to exhibit negative values, zero, or positive values.

SRR microwave nonlinear tunable metamaterial

Fabrication and

experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs wh ...

al studies of the properties of the ''first nonlinear tunable metamaterial'' were operating at

microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequency, frequencies between 300 MHz and 300 GHz respectively. Different sources define different fre ...

frequencies. Such a metamaterial was fabricated by modifying the properties of

SRRs The Holmes and Rahe stress scale () is a list of 43 stressful life events that can contribute to illness. The test works via a point accumulation score which then gives an assessment of risk. The American Institute of Stress for instance, regards ...

and introducing

varactor In electronics, a varicap diode, varactor diode, variable capacitance diode, variable reactance diode or tuning diode is a type of diode designed to exploit the voltage-dependent capacitance of a reverse-biased p–n junction. Applications Var ...

diodes in each SRR element of the composite structure such that the whole structure becomes dynamically tunable by varying the

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...

of the propagating

. In particular, the power dependent transmission of the left-handed and magnetic metamaterials at higher powers were demonstrated, as well as the generation of particular harmonics, as was

theoretical A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...

ly suggested earlier.

SRR microwave nonlinear magnetic metamaterials

The fabrication and experimental studies of the properties of the''nonlinear tunable magnetic metamaterial'' were operating at microwave frequencies. Varactor diodes are symmetrically introduced, which results in dynamic tunability for the whole structure. Since the magnetic component of the interaction determines the application, the power dependency is demonstrated. Nonlinearity-dependent enhancement or suppression of the transmission turns out to be dynamically tunable.

SRR microwave nonlinear electric metamaterials

A new type of nonlinear metamaterials, is proposed and designed, exhibiting a resonant electric response at microwave frequencies. By introducing a varactor diode as a nonlinear element within each resonator, the frequency of the electric mode stop band is shifted by changing the incident power without affecting the magnetic response. These elements could be combined with the previously developed nonlinear magnetic metamaterials in order to create negative index media with a control over both electric and magnetic nonlinearities. Nonlinear resonators are designed in a similar fashion. A strong nonlinear electric response is obtained.

Sub-diffraction limit for non-linear metamaterial lens

By covering a thin flat nonlinear lens on the sources, the sub-diffraction-limit observation can be achieved by measuring either the near-field distribution or the far-field radiation of the sources at the harmonic frequencies and calculating the IFT to obtain the sub-wavelength imaging. The higher order harmonics are used, the higher resolution is obtained.

Non-linear electric metamaterial

A new type of nonlinear metamaterial is designed, and analyzed with a dominant negative electric response. Introducing nonlinearity into the electric response makes it tunable while leaving the magnetic response unchanged. A nonlinear NIM containing tunable electric and magnetic elements, which can respond independently is possible.

EM field shielding by non-linear metamaterials

It is well known that over certain frequencies, typical metals can reflect electromagnetic (EM) fields and can thus be used as electromagnetic shielding materials. However, conventional ''linear'' LHMs cannot be used to shield electromagnetic fields. This is drastically modified when nonlinearity of the magnetic response is taken into account, creating a controllable shielding effect in LHMs, accompanied by a parametric reflection.

Meta-dimer metamaterial

A meta-dimer is composed of two spatially separated SRRs, with the two SRRs identical in each unit cell. The proximity of the SRRs in the dimer results in relatively strong coupling between them. A metamaterial comprising a large number of such metadimers can be utilized as an actively tunable medium at optical wavelengths. If either or both of the SRRs in the meta-dimer become nonlinear, the metamaterial itself acquires nonlinear properties. This can allow for nonlinear behavior, such as tunability in real time
Stereometamaterials
are also a type of meta-dimer.