rus
Issue #3/2016
A.Shvartsburg
Displacement Currents In The Non – Conducting Dielectrics: The "Ugly Duckling" Becomes A Swan
Displacement Currents In The Non – Conducting Dielectrics: The "Ugly Duckling" Becomes A Swan
The key words in the title of this article shall be owed to two geniuses of the century before last: the term "dielectric" has been suggested by M. Faraday; "displacement current" has been introduced by D. Maxwell. The fate of these innovations was developing in different ways: at the end of XIX century the electric motors, lighting systems and emerging telephony has been using habitual conduction currents in metals; non-conductive dielectrics has got a modest but a useful role of insulators. In contrast, the "displacement current" seemed to be a mathematical phantom, inserted in one of Maxwell’s equations, in order to provide the continuity of AC line in a non-conductive medium. However, in actual electrical circuits this phantom was observed long before Maxwell, therefore a direct question "What is displaced in a displacement current?" had a long history.
Теги: displacement currents metatronics non- conducting dielectrics метатроника непроводящие диэлектрики токи смещения
The birth of the paradigm
In physics textbooks, the English name "Thomson" is found in many sections: J.J. Thomson, legendary "J.–J.", the discoverer of the electron; his son, J.P. Thomson, the pioneer of electron diffraction, both were the Nobel laureates; moreover, W. Thomson, one of the "fathers – founder’ of the first transatlantic telegraph, the creator of the absolute temperature scale, awarded for his discoveries the title "Baron Kelvin"; namely this baron is remembered, while we speak about the low temperature, kelvins and millikelvins. At the beginning of his career W. Thomson, who became a professor at the age 22, has created the first device generating the oscillating electric current in the circuit (fig. 1) comprising a battery , a capacitor (capacitance ) and an inductance coil (self-inductance ). When the capacitor was discharging, the electric current passing through the coil was exciting the magnetic field, and then the drop of magnetic field excited the current, providing the recharge of capacitor; energy of the system was migrating periodically from the condenser to the coil and back. This "electric pendulum" has become known as the "Thomson oscillating circuit", the oscillation period is determined by formula
. (1)
One can be delighted with the ingenuity of young professor who created the first oscillating circuit, one of the key elements of the future radio technique, back in those days (1853) when the electrical researches have been conducted on the primitive devices looking like fancy mahogany furniture (fig. 2).
However, at the time of appearance the novelty has revealed a paradox: in contrast to the currents usually flowing through the conductors, AC somehow worked through the empty clearance between the capacitor plates, so that the gap has been formed in the current line of the conductor. If the gap was not empty, for example, filled with non-conductive dielectric, the current oscillations in the circuit still occurred. On the other hand, the current continuity, understood by analogy with fluid flow in a pipe, seemed obvious: the gap was not possible, but there it was! A few years later the conflict between the theory and the experiment has been solved by a brilliant Maxwell’s assumption which confirmed the symmetry of the electric and magnetic fields: similarly to change in the magnetic field inducing voltage in the conductor (electromagnetic induction), time-varying electric induction , as Maxwell has suggested, excites the magnetic field . Such excitement, according to Ampere law, can be attributed to some imaginary current, which is not related to the motion of charges and occurs in the place where the induction changes in time, but – what is important – induction can vary both in conductive and non-conductive medium, and even, in particular, in vacuum, where the value is equal to the electric field . Designating the usual current in a conductor as "conduction current", Maxwell has named his innovation as "displacement current", and has introduced the formula for the calculation of this current density
; (2)
The new concept has allowed Maxwell to write a basic system of equations of electrodynamics and to predict the existence of a special type of oscillations – electromagnetic waves, whose energy, similar to Thomson circuit, is transforming periodically from electric to the magnetic energy and vice versa, and the speed of these waves was equal to the speed of light in vacuum.
It took some time for the perceptions of displacement currents and electromagnetic waves to have been accepted by the physical community, Maxwell himself died before the triumph of Maxwell’s equations. Several years have passed, and Heinrich Hertz, experimenting with sparks in the arresters, demonstrated the propagation of electromagnetic waves through the air separating the arresters. Moreover, in another experiment, these waves were propagating through the dielectric layer, a large concrete slab. To explain the propagation of electromagnetic waves in a non-conductive medium it was necessary to admit the existence of displacement current. In the particular case of dielectric dipole, this current can be visually represented as dipole oscillations in general, the hypothesis of displacement current in vacuum has appealed to the model of mechanical displacement of particles of imaginary "world ether", invisible and omnipresent, filled with elastic "magnetic field tubes", whose oscillations have been perceived in this model as the electromagnetic waves. Years later, Maxwell’s equations give work to tens of thousands of engineers and physicists for almost a century and a half. Thus, the arguments about the mysterious "world ether" have been remained in the dusty archives of science with the advent of the theory of relativity, meanwhile the electromagnetics of displacement currents has recently got a second wind.
Non-magnetic magnets
At first appear, the displacement current concept seemed merely a convenient handy tool intended to ensure the continuity of the current lines. Indeed, this new object of electrodynamics, although called "current", had little to do with the established laws of the conduction currents, formed the physical basis of electrical engineering. Thus, in contrast to conduction current, displacement current, has proved to be: a) proportional not to the electric field , but to the velocity of change of the field (2), i. e. displacement current does not obey Ohm’s law; b) proportional not to the conductivity of the medium, but to its dielectric permittivity , which determines the electrical induction in equation (2): in an isotropic dielectric ; c) not suitable for electric heating: while changing the polarization state of the dielectric medium, alternating electric field stimulates the interaction of polarized molecules of the medium, resulting usually in a slight heating of the insulator.
The only thing that seemed to make the displacement current and conduction current kin is that they both can excite alternating magnetic field, however, the properties of the excited fields vary widely. The differences can be seen in the simple examples of the scattering of linearly polarized electromagnetic waves on the metal and dielectric spheres. The problem of metal sphere which is not penetratable for the waves has been considered at the rise of radio physics at the beginning of the last century. A qualitative picture of scattering seems to be simple: under the influence of a periodic electric field (fig. 3), positive and negative charges are alternatively induced on the surfaces of hemispheres, creating electric dipole directed along . At the same time, periodic magnetic field induces circular surface currents , forming magnetic dipole whose moment is directed along . Radiation of these dipoles forms scattered waves in the far field, and standing waves "pinned" to the surface of the sphere in the near field.
In contrast with this habitual picture, the waves, incident on a non-conductive dielectric sphere penetrate partially into the sphere, thus exciting a complex spatial structure of the displacement currents. Under certain ratios between the radius of the sphere, dielectric permittivity and the length of the incident wave , corresponding to the resonant scattering, significant magnetic field are induced both in the sphere and in the near field, converting the dielectric sphere consisting of a non-magnetic material, to a magnetic structure element. Furthermore, by controlling the spatial structure of displacement currents inside the sphere, it is possible to control the flow of dissipated energy, i. e. it is possible to use the dielectric sphere for the reradiation of waves in a given direction [1].
Moreover, the interest to displacement currents is intensified due to their resonant properties. These properties are conveniently illustrated by comparing the electromotive forces, induced by the same alternating magnetic flux in two identical thin rings, one of which is made from conductor, and the other one – from non-polar dielectric (fig. 4). When the flux is piercing through the ring planes, each of them is induced with the vortex electric field , proportional, as is known, to the velocity of flux change ; is induced in each ring; however, the similarity of induction effects in the rings is ended here: the current in the conductive ring is proportional to , meanwhile in the non-conducting one – to derivative of induction (2), i. e. . These currents in their turn induce secondary magnetic fluxes . Resonant frequency of displacement current oscillations occurs in the non-conductive ring [2]; the effect resembles a well-known "Thomson oscillating circuit" (see fig. 1), but, unlike its predecessor, the oscillator in fig. 4 is excited with displacement currents and does not contain any wires or metallic elements with resistive losses. As the frequency approaches to , the amplitude of vibrations of the induced flux through the non-conductive ring grows, and an unusual spectral range, where the directions of inducing and induced flows are opposed, comes into play. While presenting the link of and in the form , we can note that in the frequency regionт , parameter this condition corresponds to a negative magnetic susceptibility of the circular element [2].
Circular displacement currents in the dielectric described herein form the magnetic dipoles. Magnetic fields of more complex structures can be created during motion of the polarized dielectric. Thus, the magnetic quadrupole described back in 1904 in the article of professor of Moscow University A. Eichenwald "The magnetic action of bodies moving in the electric field" [3], is created by rotation of the circular rubber disc between the plates of a charged capacitor (fig. 5). The charges of opposite sign induced on the surface of a rotating disc form surface currents flowing, respectively, clockwise and counter-clockwise and generating the quadrupole magnetic field ; at a constant speed of disc rotation, this field is manifested in the experiments on the deflection of the compass needle.
Since the displacement currents are generated due to rapid changes in the inductance (2), the role of these currents in different frequency ranges vary greatly: radio devices designed for frequencies up to hundreds of GHz, were based on the conduction currents and the displacement currents were viewed as an annoying interference; in the optoelectronic systems of THz range, the contribution of displacement currents to the formation of electromagnetic fields was increasing, and for the fields of visible and IR spectrum, this contribution could be decisive. A new, previously unheard physical phenomenon has been introduced into the electrodynamics – the concentration of alternating magnetic fields in the non-magnetic bodies, magnetization of transparent non-conductive dielectrics with transmitted light, "optical magnetism" [4]. A dream came into existence: may be these rapid changes of fields can open the way to solution of a grandiose problem – creation of a high-speed optical computer?
"Metatronics" –nanoelectronics of metamaterials
In the transition from GHz to optical range, decrease of wavelength influenced the decrease in the size of transmitting and receiving devices. Initially such miniaturization was based on the circuits developed in radio engineering: microwave waveguides have become the examples for the first optical fibers, optical cavities also had their microwave "ancestors", and a "borderline" scientific branch – radio optics came into play. However, while trying to create optical elements with dimensions of the order of microns or less compatible with the wavelength of light, such concepts transfer was stalled. In the visible and infrared range, the optical circuits cannot be created like reduced copies of radio engineering devices. To design such nanoscale elements new principles, new materials and new technologies were in need; microelectronics was followed by nanoelectronics.
The new principles of nanoelectronics are embodied in the development of optical resonant circuits consisting of the elements with dimensions of tens to hundreds of nanometers. Key properties of these elements can be easily visualized for a cylindrical nanoparticle whose length l and diameter d are considerably smaller than the light wavelength and the dielectric permittivity of particle material is described by some value :
• In the simplest case, when the electric field of wave s directed along the cylinder axis, it is possible to evaluate the voltage between the ends of the cylinder = and displacement current flowing between the ends = , where is the butt-end area, and is displacement current density (2). In this elementary model, one can find the optical impedance Z from the relation connecting current I and voltage U in AC circuits: ; substituting the values I and U, we get where is capacity of a plane capacitor: and the value > 0. The obtained value Z coincides with the well known expression for "capacitance" of Thomson circuit. In actual nanoparticles, the optical impedance depends on the shape, size and orientation of the particle with respect to the wave electric component and value.
• Displacement current, unlike the conduction current can flow out from the current-carrying elements to an another insulator or to the air; such redistribution of displacement currents, defined by the ratio of dielectric permittivity between the adjacent elements, contributes to the concentration of these currents in the region of large values . Thus, a thin dielectric layer with a low value surrounding a nanoparticle can be viewed as an insulator, preventing the leakage of displacement current from the nanoparticles. On the other hand, the media with >> 1, for example, ferroelectrics, play the role of guides for displacement currents; such a conductor may be presented as a layered fiber-like structure (fig. 6); note that, the optical fiber is also a guide system for optical range displacement currents.
• If the dielectric permittivity depends on the frequency of the wave , the frequency range with < 0 can arise; thus, in metals, the values are negative for frequencies lower than the frequency of collective oscillations of metal plasma – "plasma frequency" ; for the nanoparticles of gold and silver , so that values in the visible and infrared ranges are negative. For these nanoparticles, optical impedance is associated with the "inductive resistance". The use of plasma oscillations in thin metal films for processing optical signals is currently studied by a new branch of electronics – "plasmonics".
Optical analogues of RF circuits occupying nanoscale volumes allow to combine the properties of nanocondenser, nanoinductor and nanoresistor in one element. Furthermore, these elements allow to create in a certain frequency range the magnetic induction , directed oppositely to the magnetic field ; in the ratio this corresponds to values , i. e. element with negative magnetic permeability [5]. The media with do not exist in the nature, and man-made media with this property form one of the classes of new composite materials, so called metamaterials. It is the use of metamaterials that has opened a new path for the indoctrination of microwave – like circuits to the nanoelectronics. One of today’s pioneers in this way, professor N. Engheta (Pennsylvania University, USA), had suggested the short synthetic name for this newly shaping part of nanoelectronics – "metatronics’ [6].
Metatronics had shown how to synthesize materials for the unusual nanostructures and how to design their properties, but a new problem had arisen – how to build these structures?
Art of subwavelength miniatures
An "assembler of apparatus’ for metatronics devices deals with miniature parts with the dimensions ranging from tens to hundreds of nanometers, which is less than the wavelength of light, so that we can speak about the of subwavelength optical elements, for example, [6], about the of resonant circuit, formed by of the glass sphere with a radius of 20 nm (capacity), covered by with a thin silver shell (inductance), or a resonator for IR waves in the form of a metal horseshoe with the dimensions of 300–400 nm and 20 nm thick (one might say "nanohorseshoe!"), deposited on a quartz substrate. These elements are used to construct three-dimensional periodic structures, similar to artificial crystals; the periods of such structures are also measured in hundreds of nanometers. Unlike microwave devices, the role of an annoying interference is played by the conduction currents – they create resistive losses.
An extensive network of displacement currents circulating in the space grating of nanoelements generates a contradictory situation: from one hand, these currents, as already noted, tend to flow out from the current-carrying structures. This leakage violates the interaction of adjacent elements of the grating, so that the assembler has to "keep the distance" between the elements, ensuring compatibility of all elements, forming a three-dimensional nanostructure. From the other hand, the trend to miniaturization of the system requires the reduction of this distance. Competition of both trends stimulates the optimization of optical and geometrical parameters of the nanostructure. While creating and, in particular, while reproducing these spatial compositions, special methods of lithography, magnetron sputtering, electron and ion beams blur the line between the art of a technologist and the technique of a jeweler, constituting often a "know-how" of the manufacturer.
Step beyond the horizon
The establishment of nanoelectronics has turned electrodynamics of displacement currents into the rapidly developing field of applied optics. This development has highlighted the new and yet purely academic problems; one of them is followed directly from formula (2), connecting the displacement current generation with the rapid variations of electric induction in the medium. So far, we discussed the changes caused by changes in the electric field the value being constant; however, there is another mechanism of generation of such current – time-dependent dielectric permittivity described in the definition (2) by derivative ; if the speed of variations is large enough, then noticeable displacement current providing the electromagnetic radiation, can be generated even in a constant field . But how one can create a rapid variations of ?
The literature describes the mechanism of generation of eigenmodes in a cavity filled with a dielectric having a time-varying value ; such changes may be caused by non-linear modulation of the dielectric permittivity by the pump wave [7]. Modulation of results in changes of the cavity eigenfrequencies. In particular, if the modulation frequency is equal to double eigenfrequency frequency of the cavity, there is a well known effect of parametric generation. Another mechanism for generating displacement currents is associated with the propagation of electromagnetic field in a dielectric medium, especially in plasma with rapidly varying polarization. If the polarization variations are stipulated by the movement of the boundaries of the medium, the spectrum of the emitted wave is enriched by additional Doppler harmonics. These combined problems are typical to a newly shaping branch of electromagnetics – optics of nonstationary media. Today, such quick variations seem to be somewhat exotic, but the temporal intervals between the discovery of effect and its use are reduced too fast, and – who knows? – perhaps tomorrow the effects associated with variable polarization will become an another front edge of optics.
In physics textbooks, the English name "Thomson" is found in many sections: J.J. Thomson, legendary "J.–J.", the discoverer of the electron; his son, J.P. Thomson, the pioneer of electron diffraction, both were the Nobel laureates; moreover, W. Thomson, one of the "fathers – founder’ of the first transatlantic telegraph, the creator of the absolute temperature scale, awarded for his discoveries the title "Baron Kelvin"; namely this baron is remembered, while we speak about the low temperature, kelvins and millikelvins. At the beginning of his career W. Thomson, who became a professor at the age 22, has created the first device generating the oscillating electric current in the circuit (fig. 1) comprising a battery , a capacitor (capacitance ) and an inductance coil (self-inductance ). When the capacitor was discharging, the electric current passing through the coil was exciting the magnetic field, and then the drop of magnetic field excited the current, providing the recharge of capacitor; energy of the system was migrating periodically from the condenser to the coil and back. This "electric pendulum" has become known as the "Thomson oscillating circuit", the oscillation period is determined by formula
. (1)
One can be delighted with the ingenuity of young professor who created the first oscillating circuit, one of the key elements of the future radio technique, back in those days (1853) when the electrical researches have been conducted on the primitive devices looking like fancy mahogany furniture (fig. 2).
However, at the time of appearance the novelty has revealed a paradox: in contrast to the currents usually flowing through the conductors, AC somehow worked through the empty clearance between the capacitor plates, so that the gap has been formed in the current line of the conductor. If the gap was not empty, for example, filled with non-conductive dielectric, the current oscillations in the circuit still occurred. On the other hand, the current continuity, understood by analogy with fluid flow in a pipe, seemed obvious: the gap was not possible, but there it was! A few years later the conflict between the theory and the experiment has been solved by a brilliant Maxwell’s assumption which confirmed the symmetry of the electric and magnetic fields: similarly to change in the magnetic field inducing voltage in the conductor (electromagnetic induction), time-varying electric induction , as Maxwell has suggested, excites the magnetic field . Such excitement, according to Ampere law, can be attributed to some imaginary current, which is not related to the motion of charges and occurs in the place where the induction changes in time, but – what is important – induction can vary both in conductive and non-conductive medium, and even, in particular, in vacuum, where the value is equal to the electric field . Designating the usual current in a conductor as "conduction current", Maxwell has named his innovation as "displacement current", and has introduced the formula for the calculation of this current density
; (2)
The new concept has allowed Maxwell to write a basic system of equations of electrodynamics and to predict the existence of a special type of oscillations – electromagnetic waves, whose energy, similar to Thomson circuit, is transforming periodically from electric to the magnetic energy and vice versa, and the speed of these waves was equal to the speed of light in vacuum.
It took some time for the perceptions of displacement currents and electromagnetic waves to have been accepted by the physical community, Maxwell himself died before the triumph of Maxwell’s equations. Several years have passed, and Heinrich Hertz, experimenting with sparks in the arresters, demonstrated the propagation of electromagnetic waves through the air separating the arresters. Moreover, in another experiment, these waves were propagating through the dielectric layer, a large concrete slab. To explain the propagation of electromagnetic waves in a non-conductive medium it was necessary to admit the existence of displacement current. In the particular case of dielectric dipole, this current can be visually represented as dipole oscillations in general, the hypothesis of displacement current in vacuum has appealed to the model of mechanical displacement of particles of imaginary "world ether", invisible and omnipresent, filled with elastic "magnetic field tubes", whose oscillations have been perceived in this model as the electromagnetic waves. Years later, Maxwell’s equations give work to tens of thousands of engineers and physicists for almost a century and a half. Thus, the arguments about the mysterious "world ether" have been remained in the dusty archives of science with the advent of the theory of relativity, meanwhile the electromagnetics of displacement currents has recently got a second wind.
Non-magnetic magnets
At first appear, the displacement current concept seemed merely a convenient handy tool intended to ensure the continuity of the current lines. Indeed, this new object of electrodynamics, although called "current", had little to do with the established laws of the conduction currents, formed the physical basis of electrical engineering. Thus, in contrast to conduction current, displacement current, has proved to be: a) proportional not to the electric field , but to the velocity of change of the field (2), i. e. displacement current does not obey Ohm’s law; b) proportional not to the conductivity of the medium, but to its dielectric permittivity , which determines the electrical induction in equation (2): in an isotropic dielectric ; c) not suitable for electric heating: while changing the polarization state of the dielectric medium, alternating electric field stimulates the interaction of polarized molecules of the medium, resulting usually in a slight heating of the insulator.
The only thing that seemed to make the displacement current and conduction current kin is that they both can excite alternating magnetic field, however, the properties of the excited fields vary widely. The differences can be seen in the simple examples of the scattering of linearly polarized electromagnetic waves on the metal and dielectric spheres. The problem of metal sphere which is not penetratable for the waves has been considered at the rise of radio physics at the beginning of the last century. A qualitative picture of scattering seems to be simple: under the influence of a periodic electric field (fig. 3), positive and negative charges are alternatively induced on the surfaces of hemispheres, creating electric dipole directed along . At the same time, periodic magnetic field induces circular surface currents , forming magnetic dipole whose moment is directed along . Radiation of these dipoles forms scattered waves in the far field, and standing waves "pinned" to the surface of the sphere in the near field.
In contrast with this habitual picture, the waves, incident on a non-conductive dielectric sphere penetrate partially into the sphere, thus exciting a complex spatial structure of the displacement currents. Under certain ratios between the radius of the sphere, dielectric permittivity and the length of the incident wave , corresponding to the resonant scattering, significant magnetic field are induced both in the sphere and in the near field, converting the dielectric sphere consisting of a non-magnetic material, to a magnetic structure element. Furthermore, by controlling the spatial structure of displacement currents inside the sphere, it is possible to control the flow of dissipated energy, i. e. it is possible to use the dielectric sphere for the reradiation of waves in a given direction [1].
Moreover, the interest to displacement currents is intensified due to their resonant properties. These properties are conveniently illustrated by comparing the electromotive forces, induced by the same alternating magnetic flux in two identical thin rings, one of which is made from conductor, and the other one – from non-polar dielectric (fig. 4). When the flux is piercing through the ring planes, each of them is induced with the vortex electric field , proportional, as is known, to the velocity of flux change ; is induced in each ring; however, the similarity of induction effects in the rings is ended here: the current in the conductive ring is proportional to , meanwhile in the non-conducting one – to derivative of induction (2), i. e. . These currents in their turn induce secondary magnetic fluxes . Resonant frequency of displacement current oscillations occurs in the non-conductive ring [2]; the effect resembles a well-known "Thomson oscillating circuit" (see fig. 1), but, unlike its predecessor, the oscillator in fig. 4 is excited with displacement currents and does not contain any wires or metallic elements with resistive losses. As the frequency approaches to , the amplitude of vibrations of the induced flux through the non-conductive ring grows, and an unusual spectral range, where the directions of inducing and induced flows are opposed, comes into play. While presenting the link of and in the form , we can note that in the frequency regionт , parameter this condition corresponds to a negative magnetic susceptibility of the circular element [2].
Circular displacement currents in the dielectric described herein form the magnetic dipoles. Magnetic fields of more complex structures can be created during motion of the polarized dielectric. Thus, the magnetic quadrupole described back in 1904 in the article of professor of Moscow University A. Eichenwald "The magnetic action of bodies moving in the electric field" [3], is created by rotation of the circular rubber disc between the plates of a charged capacitor (fig. 5). The charges of opposite sign induced on the surface of a rotating disc form surface currents flowing, respectively, clockwise and counter-clockwise and generating the quadrupole magnetic field ; at a constant speed of disc rotation, this field is manifested in the experiments on the deflection of the compass needle.
Since the displacement currents are generated due to rapid changes in the inductance (2), the role of these currents in different frequency ranges vary greatly: radio devices designed for frequencies up to hundreds of GHz, were based on the conduction currents and the displacement currents were viewed as an annoying interference; in the optoelectronic systems of THz range, the contribution of displacement currents to the formation of electromagnetic fields was increasing, and for the fields of visible and IR spectrum, this contribution could be decisive. A new, previously unheard physical phenomenon has been introduced into the electrodynamics – the concentration of alternating magnetic fields in the non-magnetic bodies, magnetization of transparent non-conductive dielectrics with transmitted light, "optical magnetism" [4]. A dream came into existence: may be these rapid changes of fields can open the way to solution of a grandiose problem – creation of a high-speed optical computer?
"Metatronics" –nanoelectronics of metamaterials
In the transition from GHz to optical range, decrease of wavelength influenced the decrease in the size of transmitting and receiving devices. Initially such miniaturization was based on the circuits developed in radio engineering: microwave waveguides have become the examples for the first optical fibers, optical cavities also had their microwave "ancestors", and a "borderline" scientific branch – radio optics came into play. However, while trying to create optical elements with dimensions of the order of microns or less compatible with the wavelength of light, such concepts transfer was stalled. In the visible and infrared range, the optical circuits cannot be created like reduced copies of radio engineering devices. To design such nanoscale elements new principles, new materials and new technologies were in need; microelectronics was followed by nanoelectronics.
The new principles of nanoelectronics are embodied in the development of optical resonant circuits consisting of the elements with dimensions of tens to hundreds of nanometers. Key properties of these elements can be easily visualized for a cylindrical nanoparticle whose length l and diameter d are considerably smaller than the light wavelength and the dielectric permittivity of particle material is described by some value :
• In the simplest case, when the electric field of wave s directed along the cylinder axis, it is possible to evaluate the voltage between the ends of the cylinder = and displacement current flowing between the ends = , where is the butt-end area, and is displacement current density (2). In this elementary model, one can find the optical impedance Z from the relation connecting current I and voltage U in AC circuits: ; substituting the values I and U, we get where is capacity of a plane capacitor: and the value > 0. The obtained value Z coincides with the well known expression for "capacitance" of Thomson circuit. In actual nanoparticles, the optical impedance depends on the shape, size and orientation of the particle with respect to the wave electric component and value.
• Displacement current, unlike the conduction current can flow out from the current-carrying elements to an another insulator or to the air; such redistribution of displacement currents, defined by the ratio of dielectric permittivity between the adjacent elements, contributes to the concentration of these currents in the region of large values . Thus, a thin dielectric layer with a low value surrounding a nanoparticle can be viewed as an insulator, preventing the leakage of displacement current from the nanoparticles. On the other hand, the media with >> 1, for example, ferroelectrics, play the role of guides for displacement currents; such a conductor may be presented as a layered fiber-like structure (fig. 6); note that, the optical fiber is also a guide system for optical range displacement currents.
• If the dielectric permittivity depends on the frequency of the wave , the frequency range with < 0 can arise; thus, in metals, the values are negative for frequencies lower than the frequency of collective oscillations of metal plasma – "plasma frequency" ; for the nanoparticles of gold and silver , so that values in the visible and infrared ranges are negative. For these nanoparticles, optical impedance is associated with the "inductive resistance". The use of plasma oscillations in thin metal films for processing optical signals is currently studied by a new branch of electronics – "plasmonics".
Optical analogues of RF circuits occupying nanoscale volumes allow to combine the properties of nanocondenser, nanoinductor and nanoresistor in one element. Furthermore, these elements allow to create in a certain frequency range the magnetic induction , directed oppositely to the magnetic field ; in the ratio this corresponds to values , i. e. element with negative magnetic permeability [5]. The media with do not exist in the nature, and man-made media with this property form one of the classes of new composite materials, so called metamaterials. It is the use of metamaterials that has opened a new path for the indoctrination of microwave – like circuits to the nanoelectronics. One of today’s pioneers in this way, professor N. Engheta (Pennsylvania University, USA), had suggested the short synthetic name for this newly shaping part of nanoelectronics – "metatronics’ [6].
Metatronics had shown how to synthesize materials for the unusual nanostructures and how to design their properties, but a new problem had arisen – how to build these structures?
Art of subwavelength miniatures
An "assembler of apparatus’ for metatronics devices deals with miniature parts with the dimensions ranging from tens to hundreds of nanometers, which is less than the wavelength of light, so that we can speak about the of subwavelength optical elements, for example, [6], about the of resonant circuit, formed by of the glass sphere with a radius of 20 nm (capacity), covered by with a thin silver shell (inductance), or a resonator for IR waves in the form of a metal horseshoe with the dimensions of 300–400 nm and 20 nm thick (one might say "nanohorseshoe!"), deposited on a quartz substrate. These elements are used to construct three-dimensional periodic structures, similar to artificial crystals; the periods of such structures are also measured in hundreds of nanometers. Unlike microwave devices, the role of an annoying interference is played by the conduction currents – they create resistive losses.
An extensive network of displacement currents circulating in the space grating of nanoelements generates a contradictory situation: from one hand, these currents, as already noted, tend to flow out from the current-carrying structures. This leakage violates the interaction of adjacent elements of the grating, so that the assembler has to "keep the distance" between the elements, ensuring compatibility of all elements, forming a three-dimensional nanostructure. From the other hand, the trend to miniaturization of the system requires the reduction of this distance. Competition of both trends stimulates the optimization of optical and geometrical parameters of the nanostructure. While creating and, in particular, while reproducing these spatial compositions, special methods of lithography, magnetron sputtering, electron and ion beams blur the line between the art of a technologist and the technique of a jeweler, constituting often a "know-how" of the manufacturer.
Step beyond the horizon
The establishment of nanoelectronics has turned electrodynamics of displacement currents into the rapidly developing field of applied optics. This development has highlighted the new and yet purely academic problems; one of them is followed directly from formula (2), connecting the displacement current generation with the rapid variations of electric induction in the medium. So far, we discussed the changes caused by changes in the electric field the value being constant; however, there is another mechanism of generation of such current – time-dependent dielectric permittivity described in the definition (2) by derivative ; if the speed of variations is large enough, then noticeable displacement current providing the electromagnetic radiation, can be generated even in a constant field . But how one can create a rapid variations of ?
The literature describes the mechanism of generation of eigenmodes in a cavity filled with a dielectric having a time-varying value ; such changes may be caused by non-linear modulation of the dielectric permittivity by the pump wave [7]. Modulation of results in changes of the cavity eigenfrequencies. In particular, if the modulation frequency is equal to double eigenfrequency frequency of the cavity, there is a well known effect of parametric generation. Another mechanism for generating displacement currents is associated with the propagation of electromagnetic field in a dielectric medium, especially in plasma with rapidly varying polarization. If the polarization variations are stipulated by the movement of the boundaries of the medium, the spectrum of the emitted wave is enriched by additional Doppler harmonics. These combined problems are typical to a newly shaping branch of electromagnetics – optics of nonstationary media. Today, such quick variations seem to be somewhat exotic, but the temporal intervals between the discovery of effect and its use are reduced too fast, and – who knows? – perhaps tomorrow the effects associated with variable polarization will become an another front edge of optics.
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