Negative refraction
Encyclopedia
Negative refraction is the name for an electromagnetic
phenomenon where light
rays are refracted
at an interface
in the reverse sense to that normally expected. Such an effect can be obtained using a metamaterial
which has been designed to achieve a negative value for both (electric) permittivity
ε and (magnetic) permeability
μ, as in such cases the material can be assigned a negative refractive index
. Such materials are sometimes called "double negative" materials.
Negative refraction occurs at interfaces between materials at which one has an ordinary positive phase velocity
(i.e. a positive refractive index
), and the other has the more exotic negative phase velocity (refractive index
).
of opposition of wavevector and (Abraham) Poynting vector
, i.e. E×H; other common choices are opposition of wavevector to group velocity
, or to energy velocity.
The use of "phase velocity" in the naming convention as opposed to the perhaps more appropriate "wavevector" follows since phase velocity has the same sign as the wavevector.
A typical criterion used to determine Veselago NPV is that the dot product of the Poynting vector
and wavevector is negative (i.e. that
); however this definition is not covariant
. Whilst this restriction is rarely of practical significance, the criterion has nevertheless been generalized into a covariant form.
For plane waves propagating in a Veselago NPV medium, the electric field, magnetic field and wave vector follow a left-hand rule, rather than the usual right-hand rule. This gives rise to the name "left-handed (meta)materials". However, the terms left-handed and right-handed can also arise in the study of chiral media, so this terminology is best avoided.
Group velocity NPV has the interesting feature that one can use Kramers-Kronig relations to provide causality-based constraints on the potential for NPV, however these tend to be either rather restricted in nature, or rather difficult to use.
µ result in NPV. Since both ε and µ are in general complex, their real parts do not have to be negative for a passive (i.e. lossy
) material to display negative refraction. The most general Veselago criterion applying to ε and µ is that of Depine and Lakhtakia,
although other less general forms exist. The Depine-Lakhtakia criterion for negative phase velocity is
where
are the real valued parts of ε and µ, respectively. However, negative refraction (negative refractive index) and negative phase velocity can be distinct from each other, even in passive materials, but also in active materials.
Typically, the refractive index
n is determined using
, where by convention the positive square root is chosen for n. However, in NPV materials, we reverse that convention and pick the negative sign to mimic the fact that the wavevector (and hence phase velocity) are likewise reversed. Strictly speaking, the refractive index is a derived quantity telling us how the wavevector is related to the optical frequency and propagation direction of the light, thus the sign of n must be chosen to match the physical situation.
refracted on the same side of the normal on entering the material, as indicated in the diagram, and by a suitably general form of Snell's law
.
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
phenomenon where light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
rays are refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
at an interface
Interface (chemistry)
An interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...
in the reverse sense to that normally expected. Such an effect can be obtained using a metamaterial
Metamaterial
Metamaterials are artificial materials engineered to have properties that may not be found in nature. Metamaterials usually gain their properties from structure rather than composition, using small inhomogeneities to create effective macroscopic behavior....
which has been designed to achieve a negative value for both (electric) permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
ε and (magnetic) permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
μ, as in such cases the material can be assigned a negative refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
. Such materials are sometimes called "double negative" materials.
Negative refraction occurs at interfaces between materials at which one has an ordinary positive phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
(i.e. a positive refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
), and the other has the more exotic negative phase velocity (refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
).
Negative phase velocity
Negative phase velocity (NPV) is a property of light propagation in a medium. There are different definitions of NPV, the most common being Veselago's original proposalof opposition of wavevector and (Abraham) Poynting vector
Poynting vector
In physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector...
, i.e. E×H; other common choices are opposition of wavevector to group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
, or to energy velocity.
The use of "phase velocity" in the naming convention as opposed to the perhaps more appropriate "wavevector" follows since phase velocity has the same sign as the wavevector.
A typical criterion used to determine Veselago NPV is that the dot product of the Poynting vector
Poynting vector
In physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector...
and wavevector is negative (i.e. that
); however this definition is not covariantGeneral covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
. Whilst this restriction is rarely of practical significance, the criterion has nevertheless been generalized into a covariant form.
For plane waves propagating in a Veselago NPV medium, the electric field, magnetic field and wave vector follow a left-hand rule, rather than the usual right-hand rule. This gives rise to the name "left-handed (meta)materials". However, the terms left-handed and right-handed can also arise in the study of chiral media, so this terminology is best avoided.
Group velocity NPV has the interesting feature that one can use Kramers-Kronig relations to provide causality-based constraints on the potential for NPV, however these tend to be either rather restricted in nature, or rather difficult to use.
Negative refractive index
We can choose to avoid directly considering the Poynting vector and wavevector or a propagating light field, and consider instead the response of the materials directly: that is, we consider what values of permittivity ε and permeabilityPermeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
µ result in NPV. Since both ε and µ are in general complex, their real parts do not have to be negative for a passive (i.e. lossy
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...
) material to display negative refraction. The most general Veselago criterion applying to ε and µ is that of Depine and Lakhtakia,
although other less general forms exist. The Depine-Lakhtakia criterion for negative phase velocity is
where
are the real valued parts of ε and µ, respectively. However, negative refraction (negative refractive index) and negative phase velocity can be distinct from each other, even in passive materials, but also in active materials.Typically, the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
n is determined using
, where by convention the positive square root is chosen for n. However, in NPV materials, we reverse that convention and pick the negative sign to mimic the fact that the wavevector (and hence phase velocity) are likewise reversed. Strictly speaking, the refractive index is a derived quantity telling us how the wavevector is related to the optical frequency and propagation direction of the light, thus the sign of n must be chosen to match the physical situation.Refraction
The principle symptom of negative refraction is just that – light rays arerefracted on the same side of the normal on entering the material, as indicated in the diagram, and by a suitably general form of Snell's law
Snell's law
In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...
.
See also
- Acoustic metamaterialsAcoustic metamaterialsAcoustic metamaterials are artificially fabricated materials designed to control, direct, and manipulate sound in the form of sonic, infrasonic, or ultrasonic waves, as these might occur in gases, liquids, and solids. The hereditary line into acoustic metamaterials follows from theory and research...
- MetamaterialMetamaterialMetamaterials are artificial materials engineered to have properties that may not be found in nature. Metamaterials usually gain their properties from structure rather than composition, using small inhomogeneities to create effective macroscopic behavior....
- Negative index metamaterialsNegative index metamaterialsNegative index metamaterials or negative index materials are artificial structures where the refractive index has a negative value over some frequency range. This does not occur in any known natural materials, and thus is only achievable with engineered structures known as metamaterials...
- Metamaterial antennasMetamaterial antennasMetamaterial antennas are a class of antennas which use metamaterials to increase performance of miniaturized antenna systems. Their purpose, as with any electromagnetic antenna, is to launch energy into free space...
- Perfect lens
- Photonic metamaterialsPhotonic metamaterialsPhotonic metamaterials, also known as Optical metamaterials, are a type of electromagnetic metamaterial, which are designed to interact with optical frequencies which are terahertz , infrared , and eventually, visible wavelengths. As a type of metamaterial, the periodic structures are made up of...
- Photonic crystalPhotonic crystalPhotonic crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons...
- Seismic metamaterialsSeismic metamaterialsSeismic metamaterials, are metamaterials which are designed to counteract the adverse effects of seismic waves on artificial structures, which exist on or near the surface of the earth...
- Split-ring resonator
- Tunable metamaterialsTunable metamaterialsA tunable metamaterial is a metamaterial with a variable response to an incident electromagnetic wave. This includes remotely controlling how an incident electromagnetic wave interacts with a metamaterial. This means the capablitity to determine whether the EM wave is transmitted, reflected, or...
Electromagnetic interactions
- Bloch waveBloch waveA Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
- Casimir effectCasimir effectIn quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
- DielectricDielectricA dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
- ElectromagnetismElectromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
- EM radiation
- Electron mobilityElectron mobilityIn solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor, when pulled by an electric field. In semiconductors, there is an analogous quantity for holes, called hole mobility...
- Permeability (electromagnetism)Permeability (electromagnetism)In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
* - PermittivityPermittivityIn electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
* - WavenumberWavenumberIn the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
- Photo-demberPhoto-demberIn semiconductor physics, the photo-Dember effect consists in the formation of a charge dipole in the vicinity of a semiconductor surface after ultra-fast photo-generation of charge carriers...
- ImpedanceElectrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...