Electromagnetic dispersion. Wave dispersion

2000

/

December

Dispersion of electromagnetic waves in layered and nonstationary media (exactly solvable models)

A.B. Schwarzburg a, b
A United Institute of High Temperatures RAS, st. Izhorskaya 13/19, Moscow, 127412, Russian Federation
b Institute of Space Research RAS, st. Profsoyuznaya 84/32, Moscow, 117997, Russian Federation

The propagation and reflection of electromagnetic waves in layered and nonstationary media is considered within the framework of a unified approach using exact analytical solutions of Maxwell's equations. With this approach, the spatial structure of wave fields in inhomogeneous media is represented as a function of the optical length of the path traveled by the wave (one-dimensional problem). These solutions reveal strong effects of both normal and anomalous wave dispersion in a given medium, depending on the gradient and curvature of the continuous smooth profile of the inhomogeneous dielectric constant ε( z). The effect of such nonlocal dispersion on wave reflection is represented using generalized Fresnel formulas. Exactly solvable models of the influence of the monotonic and oscillating dependence ε( t) on wave dispersion caused by the finite relaxation time of the dielectric constant.

Nowadays, quantitative knowledge of the electronic structure of atoms and molecules, as well as solids built from them, is based on experimental studies of the optical spectra of reflection, absorption and transmission and their quantum mechanical interpretation. The band structure and defectiveness of various types of solids (semiconductors, metals, ionic and atomic crystals, amorphous materials) are being studied very intensively. A comparison of the data obtained during these studies with theoretical calculations made it possible to reliably determine for a number of substances the structural features of energy bands and the magnitude of interband gaps (band gap E g) in the vicinity of the main points and directions of the first Brillouin zone. These results, in turn, make it possible to reliably interpret such macroscopic properties of solids as electrical conductivity and its temperature dependence, the refractive index and its dispersion, the color of crystals, glasses, ceramics, glass ceramics and its variation under radiation and thermal influences.

2.4.2.1. Dispersion of electromagnetic waves, refractive index

Dispersion is the phenomenon of the relationship between the refractive index of a substance, and, consequently, the phase velocity of wave propagation, and the wavelength (or frequency) of radiation. Thus, the transmission of visible light through a glass trihedral prism is accompanied by decomposition into a spectrum, with the violet short-wavelength part of the radiation being most strongly deflected (Fig. 2.4.2).

The dispersion is called normal if, with increasing frequency n(w), the refractive index n also increases dn/dn>0 (or dn/dl<0). Такой характер зависимости n от n наблюдается в тех областях спектра, где среда прозрачна для излучения. Например, силикатное стекло прозрачно для видимого света и обладает в этом интервале частот нормальной дисперсией.

Dispersion is called anomalous if, with increasing radiation frequency, the refractive index of the medium decreases (dn/dn<0 или dn/dl>0). Anomalous dispersion corresponds to frequencies corresponding to optical absorption bands; the physical content of the absorption phenomenon will be briefly discussed below. For example, for sodium silicate glass, the absorption bands correspond to the ultraviolet and infrared regions of the spectrum; quartz glass has normal dispersion in the ultraviolet and visible parts of the spectrum, and anomalous dispersion in the infrared.

The physical nature of normal and anomalous dispersion of electromagnetic waves becomes clear if we consider this phenomenon from the standpoint of classical electronic theory. Let us consider the simple case of normal incidence of a plane electromagnetic wave in the optical range on a flat boundary of a homogeneous dielectric. Electrons of matter bound to atoms under the influence of an alternating wave field of intensity perform forced oscillations with the same circular frequency w, but with a phase j that differs from the phase of the waves. Taking into account the possible damping of a wave in a medium with a natural frequency of electron oscillation w 0, the equation of forced transverse oscillations in the direction - the direction of propagation of a plane-polarized wave - has the form

(2.4.13)

known from the course of general physics (q and m are the charge and mass of the electron).

For the optical region w 0 » 10 15 s -1 , and the attenuation coefficient g can be determined in an ideal environment under the condition of a non-relativistic electron velocity (u<

(2.4.14)

At w 0 = 10 15 s -1 the value of g » 10 7 s -1 . Neglecting the relatively short stage of unsteady oscillations, let us consider a particular solution of the inhomogeneous equation (2.4.13) at the stage of steady oscillations. We look for a solution in the form

(2.4.15)

Then from equation (2.4.13) we obtain

or , where the oscillation amplitude is equal to

(2.4.16)

Here

Then the solution for coordinate (2.4.15) can be rewritten as

(2.4.17)

Thus, forced harmonic oscillations of the electron occur with amplitude A and are ahead in phase of oscillations in the incident wave by angle j. Near the resonance value w = w 0, the dependence of A and j on w/w 0 is of particular interest.

Rice. 2.4.3. Graphs of the amplitude (a) and phase (b) of electron oscillations near the resonant frequency (at g » 0.1w 0)

In real cases, g is usually less than g » 0.1 w 0, chosen for clarity in Fig. 2.4.3, the amplitude and phase change more sharply. If the light incident on the dielectric is not monochromatic, then near the resonance, at frequencies w®w 0, it is absorbed, and the electrons of the substance dissipate this energy in the volume. This is how absorption bands appear in the spectra. The linewidth of the absorption spectrum is determined by the formula

Wave propagation in dispersive media

Literature

The general form of a plane harmonic wave is determined by an equation of the form:

u (r , t ) = A exp(i  t  i kr ) = A exp(i ( t  k " r ) ( k " r )), ()

where k ( ) = k "( ) + ik "( ) the wave number is, generally speaking, complex. Its real part k "( ) = v f /  characterizes the dependence of the phase velocity of the wave on frequency, and the imaginary part k "( ) dependence of the wave amplitude attenuation coefficient on frequency. Dispersion, as a rule, is associated with the internal properties of the material environment; they are usually distinguished frequency (time) dispersion , when polarization in a dispersive medium depends on the field values ​​at previous moments of time (memory), andspatial dispersion , when the polarization at a given point depends on the field values ​​in a certain region (nonlocality).

Equation of the electromagnetic field in a medium with dispersion

In a medium with spatial and temporal dispersion, the material equations have an operator form

This provides for summation over repeating indices (Einstein's rule). This is the most general form of linear matter equations, taking into account nonlocality, delay and anisotropy. For a homogeneous and stationary medium, material characteristics,  and  should depend only on differences in coordinates and time R = r r 1,  = t t 1:

, (.)

, ()

. ()

Wave E (r, t ) can be represented as a 4-dimensional Fourier integral (expansion in plane harmonic waves)

, ()

. ()

Similarly we can define D(k, ), j(k,  ). Taking the Fourier transform of the form (5) from the right and left sides of equations (2), (3) and (4), we obtain, taking into account the well-known theorem on the convolution spectrum

, ()

where the dielectric constant tensor, the components of which depend, in the general case, on both the frequency and the wave vector, has the form

. (.)

Similar relationships are obtained for i j (k, ) and  i j (k, ).

Frequency dispersion of dielectric constant

When taking into account only frequency dispersion, material equations (7) take the form:

D j (r, ) =  i j () E i (r, ), ()

. ()

For an isotropic medium, the tensor i j ( ) turns into a scalar, respectively

D (r, ) =  () E (r, ), . ()

Because receptivity ( ) real value, then

 ( ) =  "( ) + i  "( ),  "(  ) =  "( ),  "(  ) =  "( ). ()

In exactly the same way we get

j (r, ) =  () E (r, ), . ()

A comprehensive dielectric permeability

. ()

Integrating relation (11) by parts and taking into account that ( ) = 0, it can be shown that

Taking into account formula (14), Maxwell’s equations (1.16) (1.19) for complex amplitudes take the form

. ()

It is taken into account here that 4  = i 4  div ( E )/  = div (D ) = div ( E ). Accordingly, complex polarization and total current are often introduced

. ()

Kramers Kronig relation

Let us write complex permeability (14) taking into account relations (11) (13) in the form

, ()

where  ( ) Heaviside function, ( < 0) = 0,  (  0) = 1. Но  ( < 0) =  ( < 0) = 0, поэтому  ( )  ( ) =  ( ),  ( )  ( ) =  ( ). Hence,

where  ( ) Fourier transform of the Heaviside function,

. ()

Thus, or

. ()

It's similarly easy to get

. ()

Note that the integrals in relations (19) and (20) are taken in the leading value. Now, taking into account relations (17), (19) and (20), we obtain:

Equating the imaginary and real parts on the right and left sides of this equality, we obtain the Kramers Kronig relations

, ()

, ()

establishing a universal connection between the real and imaginary parts of complex permeability. From the Kramers Kronig relations (21), (22) it follows that the dispersive medium is an absorbing medium.

Dispersion during the propagation of an electromagnetic wave in a dielectric

Let P = N p = Ne r volumetric polarization of the medium, where N volumetric density of molecules, r offset. Vibrations of molecules under the influence of an external electric field are described by the Drude-Lorentz model (harmonic oscillator), corresponding to the vibrations of an electron in a molecule. The vibration equation of one molecule (dipole) has the form

where m effective electron mass, 0 frequency of normal vibrations, m coefficient describing attenuation (radiation losses), E d = E + 4  P /3 electric field acting on a dipole in a homogeneous dielectric under the influence of an external field E.

If the external field changes according to the harmonic law E (t) = E exp ( i  t ), then for the complex polarization amplitude we obtain the algebraic equation

or

Since D =  E = E + 4  P, then

. ()

Marked here. Another form of relation (23):

. ()

From formula (23) it follows that when   0 . In gases where the density of molecules is low, it can be assumed that then

From here, by virtue of formula (1.31) for the refractive and absorption indices, we obtain, taking into account that tg ( ) =  "/  "<< 1:

The graph of these dependencies is shown in Fig. 1. Note that when   0 anomalous dispersion dn / d  is obtained < 0, то есть фазовая скорость волны возрастает с частотой.

Dispersion in a medium with free charges

Examples of a medium with free charges are metal and plasma. When an electromagnetic wave propagates in such a medium, heavy ions can be considered motionless, and for electrons the equation of motion can be written in the form

Unlike a dielectric, there is no restoring force here, since the electrons are considered free, and frequency of collisions of electrons with ions. In harmonic mode at E = E exp ( i  t ) we get:

Then

, ()

where plasma, or Langmuir frequency.

It is natural to determine the conductivity of such a medium through the imaginary part of the permeability:

. ()

In metal <<  ,  p <<  ,  ( )   0 = const ,  ( ) is purely imaginary, the field in the medium exists only in a skin layer of thickness d  (kn) -1<<  , R  1.

In a rarefied plasma ~ (10 3 ... 10 4 ) s -1 and at  >>  permeability  ( ) is purely real, that is

– ()

dispersion equation , its graph is shown in Fig. Note that when

 >  p refractive index n real and the wave propagates freely, and when <  p refractive index n imaginary, that is, the wave is reflected from the plasma boundary.

Finally, with  =  p we get n = 0, that is,  = 0, which means D =  E = 0. Accordingly, by virtue of Maxwell’s equations (1.16) and (1.19) rot H = 0, div H = 0, that is, H = const . In this case, it follows from equation (1.17) that rot E = 0, that is

E = grad  potential field. Consequently, the existence of longitudinal ( plasma ) waves.

Waves in media with spatial dispersion

When taking into account both spatial and temporal dispersion, the electromagnetic field equation for plane waves has the form (7) with material equations of the form (8):

Accordingly, for plane harmonic waves at = 1 Maxwell’s equations (15) taking into account relation (1.25) take the form:

Let us multiply the second of relations (28) on the left vectorially by k and, taking into account the first relation, we get:

In tensor notation, taking into account relation (7), this means

Here, as before, we mean summation over a repeating index, in this case over j.

Nontrivial solutions to the system of equations (29) exist when its determinant is equal to zero

This condition implicitly specifies the dispersion law(k ). To obtain an explicit form, it is necessary to calculate the dielectric constant tensor.

Let us consider the case of weak dispersion, when ka<< 1, где а characteristic size of the inhomogeneity of the medium. Then we can assume that i j (R ,  ) is nonzero only for | R |< a . The exponential factor in equation (8) changes noticeably only when | R | ~ 2  / k =  >> a , that is, the exponential can be expanded into a power series R:

exp ( i kR ) = 1 ik l x l k l k m x l x m /2 + ... , l , m = 1, 2, 3.

Substituting this expansion into equation (8), we obtain

Since with weak dispersion integration over R in equation (30) is satisfied in a region of order size a 3, then

Let's introduce the vector n = k  / c and rewrite equation (30) as:

, ()

where indicated.

Since all components i j susceptibility tensor are real values, then from equation (8) follows the Hermitian conjugacy property of the dielectric constant tensor. For a medium with a center of symmetry, the dielectric constant tensor is also symmetric: i j (k ,  ) =  j i (k ,  ) =  i j ( k ,  ), while the expansion i j (k ,  ) by k contains only even powers k . Such environments are called optically inactive or non-gyrotropic.

Optically active There can only be a medium without a center of symmetry. This environment is called gyrotropic and is described by the asymmetric dielectric constant tensor i j (k ,  ) =  j i ( k ,  ) =  * j i (k ,  ).

For an isotropic gyrotropic medium, the tensor i j ( ) is a scalar,

 i j ( ) =  ( )  i j , and antisymmetric tensors of the second rank i j l n l and g i j l n l in relation (31) pseudoscalars, that is i j l ( ) =  ( ) e i j l , g i j l ( ) = g ( ) e i j l , where e i j l unit completely antisymmetric tensor of the third rank. Then from relation (31) we obtain for weak dispersion ( a<<  ):

 i j (k ,  ) =  ( )  i j i  ( ) e i j l n l .

Substituting this expression into equation (29), we obtain:

or in coordinate form, directing the axis z along vector k,

Here n = n z, k = k z =  n / c.

From the third equation of the system it follows that Ez = 0, that is, a transverse wave (to a first approximation for a weakly gyrotropic medium). The condition for the existence of nontrivial solutions of the first and second equations of the system is that the determinant is equal to zero: [ n 2  ( )] 2  2 ( ) n 2 = 0. Since a<<  , то и

 2 /4 <<  , поэтому

. ()

Two values ​​n 2 correspond to two waves with right and left circular polarization, from relation (1.38) it follows that. In this case, as follows from relation (32), the phase velocities of these waves are different, which leads to a rotation of the plane of polarization of a linearly polarized wave when propagating in a gyrotropic medium (Faraday effect).

Propagation of a wave packet in a dispersive medium

The information carrier (signal) in electronics is a modulated wave. The propagation of a plane wave in a dispersive medium is described by an equation of the form:

, ()

For electromagnetic waves in a medium with time dispersion, the operator L has the form:

Let the dispersive medium occupy the half-space z > 0 and the input signal is given at its boundary u (t, z = 0) = u 0 (t ) with frequency spectrum

. ()

Since a linear medium satisfies the principle of superposition, then

. ()

Substituting relation (35) into equation (33), we can find the dispersion law k ( ), which will be determined by the type of operatorL(u). On the other hand, substituting relation (34) into equation (35), we obtain

. ()

Let the signal at the input of the medium be a narrow-band process, or a wave packetu0 (t) = A0 (t) exp(– i 0 t), | dA0 (t)/ dt| <<  0 A0 (t), that is, the signal is an MMA process. If <<  0 , WhereF( 0   ) = 0,7 F( 0 ), That

()

and wave packet (36) can be written in the formu(z, t) = A(z, t) exp(i(k0 z –  0 t)), Where

. ()

As a first approximation, dispersion theories are limited to linear expansion. Then the inner integral over in equation (38) turns into a delta function:

u(z, t) = A0 (t – zdk/ d )exp(i(k0 z –  0 t)), ()

which corresponds to the propagation of a wave packet without distortion withgroupspeed

vgr = [ dk( 0 )/ d ] -1 . ()

From relation (39) it is clear that the group velocity is the speed of propagation of the envelope (amplitude)A(z, t) wave packet, that is, the speed of transmission of energy and information in the wave. Indeed, in the first approximation of dispersion theory, the amplitude of the wave packet satisfies the first-order equation:

. ()

Multiplying equation (41) byA* and adding it to the complex conjugate of equation (41), multiplied byA, we get

,

that is, the energy of the wave packet propagates with group velocity.

It's not hard to see that

.

In the region of anomalous dispersion ( 1 <  0 <  2 , rice. 1) possible case

dn/ d < 0, что соответствует vgr > c, but at the same time there is such a strong attenuation that neither the MMA method itself nor the first approximation of the dispersion theory are applicable.

The wave packet propagates without distortion only in the first order of dispersion theory. Taking into account the quadratic term in expansion (37), we obtain integral (38) in the form:

. ()

Here it is indicated = t – z/ vgr, k" = d2 k( 0 )/ d 2 = d(1/ vgr)/ d – dispersiongroupspeed. By direct substitution it can be shown that the amplitude of the wave packetA(z, t) of the form (42) satisfies the diffusion equation

()

with imaginary diffusion coefficientD = – id2 k( 0 )/ d 2 = – id(1/ vgr)/ d .

Note that even if the dispersion is very weak and the signal spectrum very narrow, so that within its limits the third term in expansion (37) is much smaller than the second, that is d2 k( 0 )/ d 2 << dk( 0 )/ d , then at some distance from the entrance to the medium the distortion of the pulse shape becomes quite large. Let a pulse be generated at the entrance to the mediumA0 (t) duration And. Opening the brackets in the exponent in relation (42), we obtain:

.

The integration variable here varies within the order of magnitude And, so if (far zone), then we can put it, then the integral will take the form of the Fourier transform:

,

where is the spectrum of the input pulse, .

Thus, a pulse in a medium with linear group velocity dispersion in the far zone turns intospectrona pulse whose envelope follows the spectrum of the input pulse. With further propagation, the shape of the pulse does not change, but its duration increases while the amplitude decreases.

From equation (43) some useful conservation laws for the wave packet can be obtained. If we integrate over time the expression

A* L(A) + AL(A* ), where, we obtain the law of conservation of energy:

.

If we integrate over time the expressionL(A)  A* /  – L(A* )  A/  = 0, then we obtain the second conservation law:

.

Having integrated equation (43) itself over time, we obtain the third conservation law:

.

When deriving all conservation laws, it was taken into account thatA( ) = dA( )/ d = 0.

Electromagnetic field energy in a dispersive medium

In the presence of losses, the law of conservation of electromagnetic energy (1.33) takes the form:

 W/  t + divS + Q = 0, ()

WhereSPointing vector of the form (1.34),Qpower of heat losses, which lead to a decrease in wave amplitude over time. Let's consider quasi-monochromatic MMA waves.

()

Using the expression for the divergence of the vector product and Maxwell’s equation (1.16), (1.17), we obtain:

.

Substituting here expression (45) for MMA fields and averaging it over the period of oscillations of the electromagnetic fieldT = 2  /  , which destroys rapidly oscillating componentsexp(2i 0 t) Andexp(2 i 0 t), we get:

. ()

We will consider a non-magnetic medium with = 1, that isB0 = H0 , and use a material equation of the form (2), connecting the vectorsDAndEto obtain a connection between slowly varying amplitudes of fields of the form (45) for the case of a homogeneous and isotropic medium without spatial dispersion

.

In a slightly dispersive environment ( ) almost delta function, that is, during the polarization delay, the field almost does not change and can be expanded in powers , taking into account only the first two terms:

.

Note that the value in square brackets, as follows from relation (11), is equal to the dielectric constant of the medium at the frequency 0 , That's why

.

For a narrowband process, the derivative D0 /  twith the same accuracy has the form

 D0 /  t =  ( 0 )  E0 /  t+ ... . Then relation (46) takes the form:

()

For a purely monochromatic wave of constant amplitude dW/ dt = 0, then from equations (44) and (47) we obtain:

. ()

If we neglect dissipation, that is, we put in equation (44)Q= 0, and in equation (47) due to relation (48) " = 0, then we get:

,

whence it follows for the average energy density of the electromagnetic field

. ()

Literature

Belikov B.S. Solving problems in physics. M.: Higher. school, 2007. 256 p.

Volkenshtein V.S. Collection of problems for the general course of physics. M.: Nauka, 2008. 464 p.

Gevorkyan R.G. General physics course: Proc. manual for universities. Ed. 3rd, revised M.: Higher. school, 2007. 598 p.

Detlaf A.A., Physics course: Textbook. manual for universities M.: Higher. school, 2008 608 p.,

Irodov I.E. Problems in general physics 2nd ed. reworked M.: Nauka, 2007.-416 p.

Kikoin I.K., Kitaygorodsky A.I. Introduction to Physics. M.: Nauka, 2008. 685 p.

Rybakov G.I. Collection of problems in general physics. M.: Higher. school, 2009.-159p.

Rymkevich P.A. Textbook for engineering - economics. specialist. Universities. M.: Higher. school, 2007. 552 p.

Savelyev I.V. Collection of questions and tasks 2nd ed. reworked M.: Nauka, 2007.-288 p.

10. Sivukhin D.V. General physics course. Thermodynamics and molecules. physics M.: Nauka, 2009. 551 p.

11. Trofimova T.I. Physics course M.: Higher. school, 2007. 432 p. .

12. Firgang E.V. Guide to solving problems in the course of general physics. M.: Higher. school, 2008.-350s

13. Chertov A.G. Physics problem book with examples of problem solving and reference materials. For universities. Under. ed. A.G. Chertova M.: Higher. school, 2007.-510p.

14. Shepel V.V. Grabovsky R.I. Physics course Textbook for universities. Ed. 3rd, revised M.: Higher. school, 2008. - 614 p.

15. Shubin A.S. Course of general physics M.: Higher. school, 2008. 575 p.

WAVE DISPERSION

WAVE DISPERSION, division of a single wave into waves of different lengths. This is due to the fact that the REFRACTIVE INDEX of the medium is different for different wavelengths. This happens with any electromagnetic radiation, but is most noticeable at visible wavelengths, where a beam of light is broken down into its component colors. Dispersion can be observed when a beam of light passes through a refractive medium, such as a glass PRISM, resulting in a SPECTRUM. Each color has a different wavelength, so the prism deflects different color components of the beam at different angles. Red (longer wavelength) deviates less than violet (shorter wavelength). Dispersion can cause chromatic aberration in lenses. see alsoREFRACTION.

Scientific and technical encyclopedic dictionary.

See what "WAVE DISPERSION" is in other dictionaries:

A wave is a change in the state of a medium (perturbation) that propagates in this medium and carries energy with it. In other words: “...waves or waves are the spatial alternation of maxima and minima of any... ... Wikipedia that changes over time

- (sound speed dispersion), dependence of phase velocity harmonic. sound. waves from their frequency. D. z. may be due to physical the environment, and the presence of foreign inclusions in it and the presence of body boundaries, in addition to the avuk. wave… … Physical encyclopedia

The dependence of the refractive index n in VA on the frequency n (wavelength l) of light or the dependence of the phase speed of light waves on their frequency. Consequence D. s. decomposition into a spectrum of a beam of white light when passing through a prism (see SPECTRA... ... Physical encyclopedia

Changes in the state of the environment (disturbances) that propagate in this environment and carry with them energy. The most important and common types of waves are elastic waves, waves on the surface of a liquid, and electromagnetic waves. Special cases of elastic V.... ... Physical encyclopedia

Wave dispersion, dependence of the phase velocity of harmonic waves on their frequency. D. is determined by the physical properties of the medium in which the waves propagate. For example, in a vacuum, electromagnetic waves propagate without dispersion, in... ... Great Soviet Encyclopedia

Modern encyclopedia

Dispersion- (from the Latin dispersio scattering) of waves, the dependence of the speed of propagation of waves in a substance on the wavelength (frequency). Dispersion is determined by the physical properties of the medium in which the waves propagate. For example, in a vacuum... ...

- (from Latin dispersio scattering), dependence of the phase velocity vf harmonic. waves from its frequency w. The simplest example is D.v. in linear homogeneous media, characterized by the so-called. dispers. equation (dispersion law); it connects frequency and... Physical encyclopedia

DISPERSION- DISPERSION, a change in the refractive index depending on the light wavelength I. The result of D. is, for example. the decomposition of white light into a spectrum when passing through a prism. For colorless, transparent substances in the visible part of the spectrum, the change ... Great Medical Encyclopedia

Waves- Waves: a single wave; b wave train; c infinite sine wave; l wavelength. WAVES, changes in the state of a medium (disturbances) propagating in this medium and carrying energy with them. The main property of all waves, regardless of their... ... Illustrated Encyclopedic Dictionary

Books

• University course of general physics physics. Optics, Aleshkevich Viktor Alexandrovich. The main feature of the textbook is a multi-level concept of presentation of the most important experimental facts and fundamentals of the theory of physical phenomena, taking into account modern scientific achievements. The book includes...

Until now, when discussing the dielectric properties of a substance, we assumed that the value of induction is determined by the values ​​of the electric field strength at the same point in space, although (in the presence of dispersion) not only at the same time, but also at all previous moments of time. This assumption is not always true. In general, the value depends on the values ​​in some region of space around the point. The linear relationship between D and E is then written in the form that generalizes expression (77.3):

it is presented here immediately in a form that also applies to an anisotropic medium. Such a non-local connection is a manifestation, as they say, of spatial dispersion (in this regard, the usual dispersion discussed in § 77 is called time or frequency dispersion). For monochromatic field components, the dependence of which on t is given by the factors , this relationship takes the form

Let us immediately note that in most cases, spatial dispersion plays a much smaller role than temporal dispersion. The fact is that for ordinary dielectrics the kernel of the integral operator decreases significantly already at distances large only in comparison with the atomic dimensions a. Meanwhile, macroscopic fields averaged over physically infinitesimal volume elements should, by definition, change little over distances. As a first approximation, we can then remove the sign of the integral over (103.1), as a result of which we return to (77.3). In such cases, spatial dispersion may only appear as small corrections. But these corrections, as we will see, can lead to qualitatively new physical phenomena and therefore be significant.

Another situation may occur in conducting media (metals, electrolyte solutions, plasma): the movement of free current carriers leads to nonlocality extending over distances that can be large compared to atomic sizes. In such cases, significant spatial dispersion can already occur within the framework of macroscopic theory.

A manifestation of spatial dispersion is the Doppler broadening of the absorption line in the gas. If a stationary atom has an absorption line with a negligibly small width at a frequency, then for a moving atom this frequency shifts, due to the Doppler effect, by an amount, where v is the speed of the atom. This leads to the appearance of a line of width in the absorption spectrum of the gas as a whole, where is the average thermal velocity of the atoms. In turn, such broadening means that the dielectric constant of the gas has a significant spatial dispersion at .

In connection with the form of notation (103.1), the following remark must be made. No considerations of symmetry (spatial or temporal) can exclude the possibility of electrical polarization of a dielectric in an alternating inhomogeneous magnetic field. In this regard, the question may arise as to whether the right side of equality (103.1) or (103.2) should be supplemented with a term with a magnetic field. In reality, however, this is not necessary. The fact is that fields E and B cannot be considered completely independent. They are related to each other (in the monochromatic case) by the equation. Due to this equality, the dependence of D on B can be considered as a dependence on the spatial derivatives of E, i.e., as one of the manifestations of nonlocality.

When taking spatial dispersion into account, it seems appropriate, without detracting from the degree of generality of the theory, to write Maxwell’s equations in the form

(103,3)

without introducing, along with the average magnetic field strength, another value H.

Instead, all terms arising from the averaging of microscopic currents are assumed to be included in the definition of D. The previously used division of the average current into two parts according to (79.3) is, generally speaking, not unique. In the absence of spatial dispersion, it is fixed by the condition that P be an electric polarization locally related to E. In the absence of such a connection, it is more convenient to assume that

which is what the representation of Maxwell’s equations in the form (103.3-4) corresponds to.

The tensor components - the kernels of the integral operator in (103.2) - satisfy the symmetry relations

This follows from the same reasoning that was carried out in § 96 for the tensor. The only difference is that the permutation of the indices a, b in the generalized susceptibilities, which means a permutation of both the tensor indices t, k and the points, now leads to a permutation of the corresponding arguments in the functions.

Below we will consider an unbounded macroscopically homogeneous medium. In this case, the kernel of the integral operator in (103.1) or (103.2) depends only on the difference . It is then advisable to expand the functions D and E into the Fourier integral not only in time, but also in coordinates, reducing them to a set of plane waves, the dependence of which on and t is given by the factor For such waves, the relationship between D and E takes the form

In this description, spatial dispersion is reduced to the appearance of the dependence of the dielectric constant tensor on the wave vector.

“Wavelength” defines the distances over which the field changes significantly. We can therefore say that spatial dispersion is an expression of the dependence of the macroscopic properties of a substance on the spatial inhomogeneity of the electromagnetic field, just as frequency dispersion expresses the dependence of the temporal change in the field. When the field tends to be homogeneous, accordingly it tends to ordinary permeability.

From definition (103.8) it is clear that

Relation generalizing (77.7). Symmetry (103.6), expressed in terms of functions, now gives

where the parameter is written out explicitly - the external magnetic field, if any. If the medium has an inversion center, the components are even functions of the vector k; the axial vector does not change during inversion and therefore equality (103.10) reduces to

Spatial dispersion does not affect the derivation of formula (96.5) for energy dissipation. Therefore, the condition for the absence of absorption is still expressed by the Hermitianity of the tensor.

In the presence of spatial dispersion, the dielectric constant is a tensor (not a scalar) even in an isotropic medium: the preferred direction is created by the wave vector. If the medium is not only isotropic, but also has a center of inversion, the tensor can be composed only of the components of the vector k and the unit tensor (in the absence of a center of symmetry, a term with a unit antisymmetric tensor may also become possible; see § 104). The general form of such a tensor can be written as

where they depend only on the absolute value of the wave vector (and on ). If the intensity E is directed along the wave vector, then the induction if then

Accordingly, the quantities are called longitudinal and transverse permeability. When expression (103.12) should tend to a value independent of the direction of; it is therefore clear that