Spatial dispersion
   HOME

TheInfoList



OR:

In the
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
of
continuous media Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis Cauchy was the first to fo ...
, spatial dispersion is a phenomenon where material parameters such as
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 ...
or
conductivity Conductivity may refer to: *Electrical conductivity, a measure of a material's ability to conduct an electric current **Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution **Ionic conductivity (solid state), elec ...
have dependence on
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), ...
. Normally, such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials. Spatial dispersion can be compared to temporal dispersion, the latter often just called
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
. Temporal dispersion represents memory effects in systems, commonly seen in optics and electronics. Spatial dispersion on the other hand represents spreading effects and is usually significant only at microscopic length scales. Spatial dispersion contributes relatively small perturbations to optics, giving weak effects such as
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circul ...
. Spatial dispersion and temporal dispersion may occur in the same system.


Origin: nonlocal response

The origin of spatial dispersion is nonlocal response, where response to a force field appears at many locations, and can appear even in locations where the force is zero. This usually arises due to a spreading of effects by the hidden microscopic degrees of freedom. As an example, consider the current J(x,t) that is driven in response to an electric field E(x,t), which is varying in space (x) and time (t). Simplified laws such as
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
would say that these are directly proportional to each other, J = \sigma E, but this breaks down if the system has memory (temporal dispersion) or spreading (spatial dispersion). The most general
linear response A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information ...
is given by: :J(x,t) = \int_^ dx' \int_^ dt'\, \sigma(x, x', t, t') E(x', t'), where \sigma(x, x', t, t')dx'\,dt' is the ''nonlocal'' conductivity function. If the system is invariant in time (
time translation symmetry Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ( ...
) and invariant in space (space translation symmetry), then we can simplify because \sigma(x, x', t, t') = \sigma_(x-x', t-t') for some convolution kernel \sigma_. We can also consider
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
solutions for E and J like so: :J(x,t) = \operatorname(J_0 e^) :E(x,t) = \operatorname(E_0 e^) which yields a remarkably simple relationship between the two plane waves' complex amplitudes: :J_0 = \tilde\sigma(k,\omega) E_0. where the function \tilde\sigma(k,\omega) is given by a
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the space-time response function: :\tilde\sigma(k,\omega) = \int_^ dx'' \int_^ dt'' \, e^ \sigma_(x'', t''). The conductivity function \tilde\sigma(k,\omega) has spatial dispersion if it is dependent on the wavevector ''k''. This occurs if the spatial function \sigma_(x-x', t-t') is not pointlike (
delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
) response in ''x-x' ''.


Spatial dispersion in electromagnetism

In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, spatial dispersion plays a role in a few material effects such as
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circul ...
and
doppler broadening In atomic physics, Doppler broadening is broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting (or absorbing) particles result in different Dopp ...
. Spatial dispersion also plays an important role in the understanding of electromagnetic
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. ...
s. Most commonly, the spatial dispersion in
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'' is of interest.


Crystal optics

Inside crystals there may be a combination of spatial dispersion, temporal dispersion, and anisotropy.Agranovich & Ginzburg . Crystal Optics with Spatial Dispersion, and Excitons ed. 978-3-662-02408-9, 978-3-662-02406-5 The
constitutive relation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
for the polarization vector can be written as: : P_i(\vec k,\omega) = \sum_j (\epsilon_(\vec k, \omega) - \epsilon_0\delta_) E_j(\vec k, \omega), i.e., the permittivity is a wavevector- and frequency-dependent
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
. Considering
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, one can find the plane wave
normal mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s inside such crystals. These occur when the following relationship is satisfied for a nonzero electric field vector \vec E: : \omega^2 \mu_0 \epsilon(\vec k, \omega)\vec E - (\vec k \cdot \vec k) \vec E + (\vec k \cdot \vec E) \vec k = 0 . Spatial dispersion in \epsilon(\vec k, \omega) can lead to strange phenomena, such as the existence of multiple modes at the same frequency and wavevector direction, but with different wavevector magnitudes. Nearby crystal surfaces and boundaries, it is no longer valid to describe system response in terms of wavevectors. For a full description it is necessary to return to a full nonlocal response function (without translational symmetry), however the end effect can sometimes be described by "additional boundary conditions" (ABC's).


In isotropic media

In materials that have no relevant crystalline structure, spatial dispersion can be important. Although symmetry demands that the permittivity is isotropic for zero wavevector, this restriction does not apply for nonzero wavevector. The non-isotropic permittivity for nonzero wavevector leads to effects such as
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circul ...
in solutions of chiral molecules. In isotropic materials without optical activity, the permittivity tensor can be broken down to transverse and longitudinal components, referring to the response to electric fields either perpendicular or parallel to the wavevector. For frequencies nearby an absorption line (e.g., an
exciton An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. The ...
), spatial dispersion can play an important role.


Landau damping

In plasma physics, a wave can be collisionlessly damped by particles in the plasma whose velocity matches the wave's phase velocity. This is typically represented as a spatially dispersive loss in the plasma's permittivity.


Permittivity–permeability ambiguity at nonzero frequency

At nonzero frequencies, it is possible to represent all
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Di ...
s as time-varying polarizations. Moreover, since the electric and magnetic fields are directly related by \nabla \times E = -\partial B/\partial t, the
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Di ...
induced by a magnetic field can be represented instead as a polarization induced by the electric field, though with a highly dispersive relationship. What this means is that at nonzero frequency, any contribution to permeability ''μ'' can instead be alternatively represented by a spatially dispersive contribution to permittivity ''ε''. The values of the permeability and permittivity are different in this alternative representation, however this leads to no observable differences in real quantities such as electric field, magnetic flux density, magnetic moments, and current. As a result, it is most common at optical frequencies to set ''μ'' to the
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
''μ''0 and only consider a dispersive permittivity ''ε''. There is some discussion over whether this is appropriate in
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. ...
s where
effective medium approximations In materials science, effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. EMAs or EMTs are developed from averagi ...
for ''μ'' are used, and debate over the reality of "negative permeability" seen in
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, wh ...
.


Spatial dispersion in acoustics

In
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, especially in solids, spatial dispersion can be significant for wavelengths comparable to the lattice spacing, which typically occurs at very high frequencies (
gigahertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one h ...
and above). In solids, the difference in propagation for transverse acoustic modes and longitudinal acoustic modes of sound is due to a spatial dispersion in the
elasticity tensor In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
which relates stress and strain. For polar vibrations (
optical phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...
s), the distinction between longitudinal and transverse modes can be seen as a spatial dispersion in the restoring forces, from the "hidden" non-mechanical degree of freedom that is the electromagnetic field. Many electromagnetic wave effects from spatial dispersion find an analogue in acoustic waves. For example, there is acoustical activity — the rotation of the polarization plane of transverse sound waves — in chiral materials, analogous to optical activity.


References

{{Reflist Physical phenomena