category: Physics


? 1. Constitutive equations of the electromagnetic field in a dispersive medium
? 2. Dispersion. Vector volume density of polarization
? 3. Dependence of the refractive index and absorption on the frequency


?????The most important characteristic of a linear distributed system is the dispersion relation, which connects the wave number and frequency of monochromatic wave. It can be written as, or in implicit form.
?????When a plane wave is described by one (generally speaking, integrodifferential) equation, the dispersion obtained, looking for his decision in the form. In the simplest case, the process of wave propagation is described by
In this case the wave number associated with the frequency of linear dependence, or, where the wave velocity is constant. However, even with allowance for dissipative processes, the behavior of waves is described by more complicated equations. Dispersion is also complicated. For sound waves in a viscous heat-conducting medium and electromagnetic waves in a medium with a conductivity of the following relations between the wave number and frequency:
In more general cases, the frequency can be a complex manner dependent real and imaginary parts of the wave number:
The real part describes the frequency dependence of the phase velocity of wave propagation, and the imaginary part - dependence of wave attenuation on frequency.
?????In many cases, the wave process is not convenient to describe one type of wave equation, and the system of coupled integrodifferential equations. Here - a matrix operator acting on a column vector. As an example, for acoustic waves can serve as a set of variables (the vibrational rate, increment of density, pressure, temperature), and for electromagnetic waves - the components of the electric and magnetic fields and electric displacement and magnetic induction. In this case, the formal scheme of finding the dispersion law is as follows. We are looking for solution in the form of:
The solution will be nontrivial only if. Hence we obtain the desired dependence. The presence of multiple roots of the dispersion equation means that the system can describe several types of eigenmodes (modes) of the medium.
?????Frequency dispersion leads to a change in patterns of distribution nonmonochromatic waves. Indeed, the different spectral components in a dispersive medium have different velocities and damping coefficients:
Because of phase velocity dispersion in the distribution of change phase relations between the spectral components. Consequently, changes the result of their interference: form nonmonochromatic wave is distorted. The dispersion of the absorption coefficient leads to the transformation of the frequency spectrum of waves and further distortion of the pulse shape.
???? 1. Constitutive equations of the electromagnetic field in a dispersive medium.
?????Dispersion effects are often manifested during the propagation of electromagnetic waves. We show how the original equations are modified taking into account these properties. The system of Maxwell's equations retain their form. The properties of the medium should be taken into account in the constitutive relations:
For static and slowly changing fields, you can write
where - constant, ie, the values and at some point in the medium and at some point in time determined by the values and at the same point and at the same time.
?????With the rapid change of the field due to the inertia of internal movements and the presence of spatial dependence of the observed microstructure of the medium polarization on the field, operating in other locations and other points in time. It should be borne in mind that because of the causality condition polarization and, consequently, the induction depend on fields, operated only in the previous time.
?????The above can be written mathematically, representing the material equations in the general integral form:
?????, (1.1)
?????, (1.2)
?????. (1.3)
For double-occurring indices here and everywhere in the future expected summation.
?????Expressions (1.1) - (1.3) represent the most common functional form of record of constitutive equations for a linear medium. This recording is taken into account the possibility of manifestation of nonlocality, delay and anisotropic properties of the medium.
?????In the particular case, if the medium is homogeneous in space and does not change with time their properties, physical characteristics, must depend only on the differences of coordinates and time. Then
?????, (1.4)
?????, (1.5)
?????. (1.6)
Relationship between the electric displacement and magnetic induction fields and the polarization of the medium is determined by the relations
?????. (1.7)
Therefore, constitutive equations can also be written in the form of
?????, (1.8)
where - the susceptibility tensor environment. A similar expression can be written to.
?????To further analyze conveniently expanded in plane waves:
After the usual transition to the Fourier representation in the expressions for and obtain a simple dependence
?????, (1.9)
?????, (1.9)
?????. (1.10)
?????It is seen that the components of the dielectric tensor in the general case depend on the frequency and the wave vector of the wave.
?????A similar conclusion can be drawn for the magnetic permeability and conductivity.
?????Thus, the dispersion of the propagation of electromagnetic waves can be manifested in two ways - as frequency (due to dependence on the frequency) and how the spatial (due to the dependence of these parameters on the same wave vector). Frequency dispersion is significant, if the frequency of electromagnetic waves is close to the natural frequencies of oscillations in the medium. Spatial dispersion also becomes noticeable when the wavelength is comparable with some characteristic dimensions.
?????For electromagnetic waves in most cases, even in the optical range, the characteristic size (where - is the wavelength in the medium:) and spatial dispersion can be neglected. However, in a magnetoactive plasma are the resonance region, in which the parameter is already significant in the radio. Furthermore, in total disregard of values that contain a small ratio does not take into account some phenomena arising in the propagation of electromagnetic waves in various media. Thus, spatial dispersion in the plasma can explain the appearance of running the plasma waves. Spatial dispersion is the main reason (and not an amendment), causing the appearance of natural optical activity and optical anisotropy of cubic crystals. If you are not interested in these special cases, that when considering the frequency dispersion of the spatial dispersion can be neglected.
?????When taking into account only the frequency dispersion of the material equation (1.9) has the form
?????. (1.11)
In contrast to (1.9) is taken not components of plane-wave field, but only time harmonic. Permittivity for waves with frequency - this is a tensor, which in the case of an isotropic medium becomes a scalar:
(remember that - the actual value). From (1.12) implies that the function is complex:
?????, (1.13)
?????, (1.14)
ie is an even function, as well - odd. The above also applies to:
?????. (1.15)
If a nondispersive medium the permittivity - a purely reactive parameter and conductivity - purely active, then in a dispersive medium, this difference is lost. With increasing frequency up to values close to eigenmodes environment, the difference in the properties of dielectrics and conductors are gradually disappearing. Thus, the presence in the environment of the imaginary part of permittivity from a macroscopic point of view is indistinguishable from the existence of conductivity - both leads to heat release. Therefore, the electrical properties of matter can be characterized by a single quantity - the complex permittivity
?????, (1.16)
?????You can set the limiting form of the permittivity at high frequencies. In the limit we have
and the dielectric constant is given by (1.6), (1.12), tends to one.
?????This property of the permittivity follows from a simple physical examination. If, when the wave frequency is large compared with the natural frequencies of oscillations of electrons in atoms of matter, electrons can be considered free. The equation of motion of free electrons under the action of a harmonic field and the solution of this equation have the form
Here - electron mass and charge. We do not consider the force acting on the charge of the magnetic field, as seen nonrelativistic case (). Medium polarization (dipole moment per unit volume, containing electrons) is
?????. (1.17)
When we get from (1.17) the previous result: and. Range of applicability of formula (1.17) for environments where there are no free electrons, lies in the range of the far ultraviolet region for the lightest elements.
?????In view of (1.16) Maxwell's equations for complex amplitudes take the form
?????, (1.18)
?????. (1.18)
Let us explain the derivation of the equation. From the continuity equation with harmonic time dependence follows that
Substituting this relation in the Maxwell equation, we write it in the form
Given the definition, we obtain the equation.
?????Thus, for high-monochromatic fields instead of the permittivity and conductivity is convenient to introduce the complex dielectric constant, combining these two concepts. Physically, this means that the current in the medium to high-frequency fields inappropriate to consider as the sum of the conduction current and displacement current. Instead, enter the total current
?????, (1.19)
where - a complex vector polarization of the medium.
???? 2. Dispersion. Vector volume density of polarization.
?????Consider the simple physical models of dispersive media. It is clear that simple models reflecting the real properties of the medium can be constructed in a few cases. Nevertheless, they are very important for understanding the physics and deserve detailed discussion.
?????To find the dependence on frequency (dispersion relation) is necessary to solve the problem of the interaction of electromagnetic waves in a medium with the existing charges.
?????All the modern theory of dispersion into account the molecular structure of matter and consider the molecules as dynamic systems with their own frequencies. Molecular systems obey the laws of quantum mechanics. However, the results of the classical theory of dispersion in many cases lead to qualitatively correct expression for the refractive index and absorption as a function of frequency.
?????Dielectrics conventionally divided into two types - non-polar and polar. In the molecules of nonpolar dielectrics charges of the electrons exactly compensate the charges of the nuclei, and the centers of negative and positive charges coincide. In this case, in the absence of the electromagnetic field of the molecule does not possess a dipole moment. Under the influence of the wave is shifted electrons (ions at the same time can be regarded as fixed, since their mass is large compared with the mass of electrons) but each molecule is polarized - acquires a dipole moment. If the dielectric is homogeneous and in a unit volume contains the same molecule, the vector of the volume density of polarization.
?????To determine the vector is necessary to solve the equation of motion of electrons in the molecule under the action of the wave field and find the displacement of electrons as a function of the field. In the classical dispersion theory description of the motion of electrons in a molecule based on the model of Drude - Lorentz, according to which the molecule is represented as one or more of linear harmonic oscillators, corresponding to the normal fluctuations of the electrons in the molecule. Consider the equation of motion of the oscillator:
?????. (2.1)
Here - the effective mass - damping constant with the dimensions of frequency - the resonant angular frequency of the normal fluctuations - the field acting on the dipole. For dense media the local field in a homogeneous dielectric is different from the average macroscopic field in the medium on the magnitude and equal
Note that the last equality is valid for an isotropic medium and for cubic crystals.
?????When a harmonic time dependence of the field equation (2.1) we obtain the following relation:
It is convenient to express:
?????. (2.2)
Given that, from (2.2) we find
?????, (2.3)
Separating in (2.3) the real and imaginary parts, we obtain
Here we introduce the notation. In the case of low frequencies, satisfying the condition, we get the expression for the static permittivity
For solid and liquid dielectrics can significantly exceed unity.
?????In gases, the density of polarized molecules is usually low. At the same time and we can assume that differs little from unity. Therefore
?????. (2.4)
???? 3. Dependence of the refractive index and absorption of the frequency.
?????From (2.4) taking into account formulas
for the refractive index and absorption obtain
?????. (3.1)
?????Let us find out how dependent refractive indices and absorption of frequency. If the condition is satisfied, ie if the frequency of the waves far from the resonance (or), then
?????, (3.2)
ie, the refractive index differs little from unity. When the quantity, it increases with increasing frequency. If the value is negative, also increases with approaching to unity (Fig. 1). Rate of absorption in this frequency range is small. Near the resonance the refractive index decreases with increasing frequency. Assuming exact resonance, when it becomes unity, and the absorption reaches its maximum value. Frequency region in which the refractive index decreases with increasing frequency, called the domain of anomalous dispersion, here we have a growth of the phase velocity.
?????In the case when the molecule is modeled by a set of oscillators of various types, with different resonant frequencies for the permittivity can obtain an expression which generalizes (2.3):
?????. (3.3)
Here - the volume density of oscillators with frequency.
?????If we calculate the dipole moment per unit volume, using the methods of quantum mechanics, then we obtain a formula similar to (3.3), with the only difference being that replaced it on, where - is the oscillator strength for the transition frequency. The summation is over all allowed dipole transitions.
?????Formulas (2.3) and (3.3) are obtained for the model of independent atoms, but they give quite a correct phenomenological description of any system, the absorption spectrum which represents a set of discrete lines.
?????We discussed the model, which gives the dispersion law for dielectrics whose molecules only acquire a dipole moment in an external field. But the molecules of polar dielectrics (eg, water) have a dipole moment and in the absence of the field. Mechanism of polarization of the dielectric is reduced to the orienting action of the wave field.
?????Let the dipole moment of a molecule is. In the absence of wave vectors due to thermal motion oriented randomly. If the wave propagates in the medium, each elementary dipole acquires a component parallel to the vector. Consequently, it becomes non-zero dipole moment per unit volume:
?????. (3.4)
In this expression - the angle between the vectors and - a random parameter, the angle brackets denote averaging over an ensemble of molecules. To calculate we use the statistical distribution law of Boltzmann
Here - the potential energy of the molecule in an electric field; erg / K - Boltzmann constant - a constant determined by the normalization condition
?????. (3.5)
We are not interested here nonlinear effects, we feel the energy of orientation is small compared with the energy of thermal motion:. In this approximation, from (3.5) we have. Carrying out the averaging in equation (3.4), we obtain
?????. (3.6)
If, then the expansion in powers of the nonlinear terms appear.
?????Until now it was assumed that the reorientation of the dipole moment follows the changes in the field of electromagnetic wave. In fact, there zapazdyvanie, consideration of which allows us to describe the effects of frequency dispersion in the signal propagation in a medium with randomly oriented dipole molecules.
?????We believe, following Debye, that when you turn on at the time of the wave polarization in a given point in space varies according to the law
?????. (3.7)
Here - static (at) susceptibility. When taking into account only the frequency dispersion for an isotropic medium of the formula (1.8) we obtain
?????. (3.8)
It is easy to verify relation (3.7) (3.8) for
?????. (3.9)
?????, (3.10)
where - the static dielectric constant. Function, and hence the loss of energy have a maximum at. The relaxation time, for example, in water vapor is of the order, and "resonance" absorption is possible in the millimeter range of electromagnetic waves.
?????When the variance (3.10) is negligible. Thus, the propagation of waves in the centimeter range and longer in the troposphere, which is a mixture of molecules of air (oxygen, nitrogen, etc.) and water vapor, you can use the formula
?????. (3.11)
Here - volumetric concentration of molecules of air and steam. It is assumed that the field in the medium is the wave field, and collisions can be neglected. The natural frequencies of molecules of the gases in the air, lie in the area> 15 GHz (cm). Therefore (3.11) for cm. However, optical and millimeter wavelengths are resonant absorption of waves. Therefore, for the purposes of radio communication in the troposphere in this range is necessary to choose a "window of transparency", ie use frequencies that do not coincide with the eigenfrequencies of the environment.


?????Summing up, it should be noted that the dispersion of electromagnetic waves can be divided into frequency (due to dependence on the frequency) and spatial (due to the dependence of these parameters on the same wave vector). As already mentioned, the frequency dispersion is significant, if the frequency of electromagnetic waves is close to the natural frequencies of oscillations in the medium. Spatial dispersion also becomes noticeable when the wavelength is comparable with some characteristic dimensions.
?????When using dielectrics in alternating electromagnetic fields need to know the natural frequencies of molecular vibrations of the dielectric material to determine the nature of dependence of the refractive index and absorption (and other parameters) on the frequency and avoid (if necessary) the resonant absorption of electromagnetic waves.
?????A characteristic feature of insulators is the need to separate the phenomenon of dispersion for polar and nonpolar molecules, due to the presence (absence) of the dipole moment in the absence of an external electromagnetic field at the polar (nonpolar) dielectric.


Vinogradova MB, Rudenko OV, Sukhorukov AP "Wave Theory". Moscow "Nauka", 1990


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