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The traditional physical model describing the propagation of surface electromagnetic waves (SEW) is based on the concept of a "sharp boundary" separating two homogeneous media; However the boundary between the air and the gradient dielectric without free carriers is shown to support the propagation of a new type of SEW. Spectrum of this new SEW and the spatial distribution of its electric field in the depth of transition layer of gradient dielectric are considered.
Теги: gradient dielectric surface electromagnetic waves (sew) градиентный диэлектрик поверхностные электромагнитные волны (пэв)
WHAT DO A KITE, A WHISTLE AND A CANDLE HAVE IN COMMON?
At the end of 1901, Guglielmo Marconi was preparing for his star hour – the transmission of a radio signal across the Atlantic, from England to Canada. Skeptics predicted a failure: how can a directed radio wave circumnavigate a giant mount formed by the convex water surface of the Earth between England and Canada? But Marconi who had already established transhorizon radio communication between Buckingham Palace and the yacht of Prince of Wales, cruised near London, was adamant: at the fourth attempt the signal from his antenna – 200-meter wire, taken up in the air by a kite – was accepted on the other side of the Atlantic Ocean; only three dots (the letter "S" in Morse code) but it was the Triumph, it was the Beginning.
When blare of the trumpets went down, the first radio physicists had to acknowledge the existence of unusual electromagnetic waves running along the interface of the conducting medium (sea water is a good conductor!), enveloping this interface so that the wave fields are "pressed" to the interface surface. The results obtained have quickly became the property of the theorists who noticed that something similar has already been known, however, for waves of a different nature, the so-called "whispering gallery waves". This effect was familiar to the builders of medieval castles: the words that were quietly said "into the wall" in one location of a large hall could be heard, pressing your ear against the wall of the hall away from the speaker. The famous Rayleigh, one of the founders of wave mechanics, had not only derived the equations describing the propagation of sound along the elastic surface, but also had performed a beautiful demonstration of this effect in the St. Paul’s Cathedral in London: by blowing into the whistle near the walls of the cathedral, he extinguished the candle leaning against the wall away from the sound source. As in the case of a radio wave, the signal propagated along the interface of the media (air – wall), sharply weakening at a distance from this interface.
The similarity in the discoveries made with the help of a kite, a whistle and a candle led to the emergence of a new chapter in physics – the physics of surface waves localized near the interface of two media. The new concept also penetrated into quantum mechanics: in 1932, I. E. Tamm showed that there exist special surface states or surface levels of electrons in the crystals (Tamm levels): the wave functions of such states, in contrast to the wave functions of bulk electrons, attenuate near the surface of the crystal. The discovery of Tamm levels gave the concept of surface waves a generality, even versatility, the family of surface waves in different media began to expand rapidly: electromagnetic waves on the surface of a dielectric cylinder (Sommerfeld waves), and magnetic (magnetoelastic surface waves), and piezoelectric (Bleustein-Gulyaev waves). The abbreviation SEW has appeared – surface electromagnetic waves.
TWO-DIMENSIONAL OPTICS
OF FLAT SURFACES
In describing the SEWs, the "sharp interface" model is usually used in the form of a geometric surface separating two homogeneous media with different values of electromagnetic parameters; when crossing the interface, the values of these parameters change abruptly. A simple model of this system is a flat interface between a non-magnetic medium with free electrons (metal or semiconductor) and air. Speaking, to be specific, about metal, we can note several well-known properties of SEWs:
1a. The field of the wave traveling in the direction y (Fig. 1 a), contains components Ez, Ey, Hx (TM polarization); the TE-polarized surface wave in this system is impossible;
2a. Surface wave at the air – metal interface is possible provided ε (ω) < –1;
3a. When moving away from the interface of the medium in both directions, all the components of the wave field decrease monotonically according to exponential laws;
4a. Spectrum of surface wave frequencies ω at the interface between air and metal with a plasma frequency Ωp is bounded above: .
5a. The absorption of a wave in a medium is determined by collisions of free electrons with a crystalline lattice.
The essential features of this model are connected with the presence of a solid-state plasma in the metal, along the interface of which the surface wave propagates, and the approach of a very thin plasma-air transition layer with the thickness much smaller than the wavelength (sharp interface). Surface waves, described by this model, are used today in the development of light modulators, optical transistors, tunable sensors and many other electro-optical devices. Based on the concepts of coupled oscillations of the electromagnetic field and solid-state plasma in optoelectronics, a large direction of applied physics, plasmonics, has been established.
The second wind in the physics of surface waves came with the appearance of artificial media – metamaterials. In contrast to plasmonics, which uses natural media, below we will talk about metamaterials based on dielectrics, where there are no free carriers [1]. This new trend in the physics of surface waves is neither associated with any solid-state plasma, nor with its uniformity, nor with a sharp interface of the medium, has appeared thanks to the success of nanotechnology, allowing you to create an optically inhomogeneous dielectric films and coatings with the refractive index n distributed within these media according to a given law; such materials are called gradient coatings. In order to manufacture gradient coatings, many techniques have been developed; the one most frequently used is magnetron sputtering of a mixture of components on a transparent substrate. Thus, to create a nanofilm containing silicon dioxide SiO2 and niobium Nb2O3, the simultaneous operation of two magnetrons is used [2]: one sputter SiO2, and another Nb2O3. The substrate moves between the magnetrons, and the rate of its motion determines the fraction of each sputtered material and the spatial profile of the refractive index n, which depends on the ratio of these fractions. In the simplest case, when the value n for a dielectric depends only on one coordinate z, perpendicular to its surface (the plane z = 0, one can consider a simple dependence
n2 (z) = n02 U2 (z);
. (1)
Here n0 is the value of the refractive index at the boundary z = 0; dimensionless quantity g > 1 and the characteristic length L are the free parameters of the model (1); these parameters determine the structure of the wave field in the gradient dielectric; profile U2 (z) (1) shown in Fig. 2, illustrates the decrease in the refractive index from the value n0 at the interface of the medium (U = 1) and its "saturation" in the depth of the environment (z >> L): n2 (z >> L) = n02 (1 – g–1).
The surface wave in the transition layer shown in Fig. 1, is described by exact analytic solutions of the Maxwell equations [2].
The properties of this new wave fundamentally differ from the known properties of waves on the sharp interface, noted above in p. 1a‑5a. To emphasize these differences, the characteristics of a new wave are listed below in p. 1b – 5b; pairwise comparison of pp. 1a and 1b, 2a and 2b, etc. stresses on the opposite tendencies of the corresponding characteristics:
1b. The field of a wave traveling in the direction y (Fig. 1b), contains components Hz, Hy, Ex (TE polarization);
2b. The surface wave at the air – dielectric interface is possible under the condition n2 (z) > 0, i. e., the wave field is localized near the interface of the transparent media;
3b. When moving away from the boundary into the interior of the medium, the wave field transverse components Ex and Hz (Fig.1b) vary nonmonotonically (Fig.3): the electric component Ex begins to grow, reaching a maximum at the point zm at a distance of the order of the characteristic scale of the inhomogeneity L, and then decreases; the magnetic component Hz, proportional to Ex is similarly distributed; thus, the energy flow along the interface of the media z = 0 reaches its maximum not at the interface, but inside the dielectric at the point zm. Comparison of curves 1 and 3 shows that the increase in the frequency of the wave contributes to the field enhancement at the distribution maximum Ex. The longitudinal component of the magnetic field Hy at the same point zm passes through zero and changes direction.
These properties underline the differences in the polarization and spatial structure of the surface wave in the gradient dielectric from the known surface waves in homogeneous metals and semiconductors; but even more significant differences arise when the spectra of these waves are compared (see below). More specifically, considering the shift in the maximum of the field, this wave could be called not a surface wave, but a "subsurface" one, but as the British logic Ockham taught back in the 13th century, "entities are not to be multiplied without necessity".
VISIBLE SEWS – FROM BLUE TO RED ONES
The physical basis of the surface wave in the gradient dielectric are coupled oscillations of the electromagnetic field and polarization of the dielectric medium; since a dielectric is considered without free electrons, the plasma frequency Ωp vanishes, and the restriction of the spectrum (3a) disappears. This implies the important features of new SEWs:
4b. Surface waves in the medium (1) exist in a wide spectrum of frequencies, and are determined by the gradient parameters L and g.
Thus, selecting technologically controlled parameters of the gradient layer L and g, it is possible to ensure the propagation of surface waves in a wide range of frequencies, including (which is important!) the whole visible part of the spectrum. It is this case that is shown in Fig. 4, illustrating at the selected value of L = 140 nm, the refractive index of surface waves n (λ) in the range of colors from red (λ = 0,75 µm) to blue (λ = 0,38 µm). Estimating the phase velocity of these waves Vph = с / n, where c is the speed of light, "sublight" values can be found from Fig. 4 Vph = 0,6–0,7 с.
5b. Absorption in a dielectric without free carriers is small, the loss tangent is of the order of 10–3.
The spectrum in Fig. 4 is represented as a refractive index n from the dimensionless parameter ky L. In this representation, the spectrum can be used for analysis of surface waves in gradient dielectric of model (1) with other values of the characteristic length L. Thus, it was noted above that in the case of L = 140 nm, the wave with kyL = 1.92 (λ = 0.75 µ) corresponds to the value of n = 2.1; the same value n at longer length of L, e. g., L = 200 nm will correspond to a longer wavelength λ = 1.06 µ. Thus, the spectrum shown in Fig. 4 can be used for analysis surface waves both visible and IR range.
The studies of a new family of SEWs have been recently begun. Such fast color broadband SEWs can be of interest in optoelectronics; moreover, a surface wave with a magnetic component perpendicular to the interface of media, is of particular interest for the analysis of magnetic micro-objects. However, the first examples of SEWs in gradient dielectric nanostructures are limited to TE-polarization waves. The question naturally arises: whether this structure is also able to support the propagation of TM-polarized waves? This new SEW branch would provide additional light flows control capabilities, and its interference with the TE-polarized wave would enhance the prospects of mixing colors on the surface of the gradient dielectric. The studies are ongoing, the survey is open…
At the end of 1901, Guglielmo Marconi was preparing for his star hour – the transmission of a radio signal across the Atlantic, from England to Canada. Skeptics predicted a failure: how can a directed radio wave circumnavigate a giant mount formed by the convex water surface of the Earth between England and Canada? But Marconi who had already established transhorizon radio communication between Buckingham Palace and the yacht of Prince of Wales, cruised near London, was adamant: at the fourth attempt the signal from his antenna – 200-meter wire, taken up in the air by a kite – was accepted on the other side of the Atlantic Ocean; only three dots (the letter "S" in Morse code) but it was the Triumph, it was the Beginning.
When blare of the trumpets went down, the first radio physicists had to acknowledge the existence of unusual electromagnetic waves running along the interface of the conducting medium (sea water is a good conductor!), enveloping this interface so that the wave fields are "pressed" to the interface surface. The results obtained have quickly became the property of the theorists who noticed that something similar has already been known, however, for waves of a different nature, the so-called "whispering gallery waves". This effect was familiar to the builders of medieval castles: the words that were quietly said "into the wall" in one location of a large hall could be heard, pressing your ear against the wall of the hall away from the speaker. The famous Rayleigh, one of the founders of wave mechanics, had not only derived the equations describing the propagation of sound along the elastic surface, but also had performed a beautiful demonstration of this effect in the St. Paul’s Cathedral in London: by blowing into the whistle near the walls of the cathedral, he extinguished the candle leaning against the wall away from the sound source. As in the case of a radio wave, the signal propagated along the interface of the media (air – wall), sharply weakening at a distance from this interface.
The similarity in the discoveries made with the help of a kite, a whistle and a candle led to the emergence of a new chapter in physics – the physics of surface waves localized near the interface of two media. The new concept also penetrated into quantum mechanics: in 1932, I. E. Tamm showed that there exist special surface states or surface levels of electrons in the crystals (Tamm levels): the wave functions of such states, in contrast to the wave functions of bulk electrons, attenuate near the surface of the crystal. The discovery of Tamm levels gave the concept of surface waves a generality, even versatility, the family of surface waves in different media began to expand rapidly: electromagnetic waves on the surface of a dielectric cylinder (Sommerfeld waves), and magnetic (magnetoelastic surface waves), and piezoelectric (Bleustein-Gulyaev waves). The abbreviation SEW has appeared – surface electromagnetic waves.
TWO-DIMENSIONAL OPTICS
OF FLAT SURFACES
In describing the SEWs, the "sharp interface" model is usually used in the form of a geometric surface separating two homogeneous media with different values of electromagnetic parameters; when crossing the interface, the values of these parameters change abruptly. A simple model of this system is a flat interface between a non-magnetic medium with free electrons (metal or semiconductor) and air. Speaking, to be specific, about metal, we can note several well-known properties of SEWs:
1a. The field of the wave traveling in the direction y (Fig. 1 a), contains components Ez, Ey, Hx (TM polarization); the TE-polarized surface wave in this system is impossible;
2a. Surface wave at the air – metal interface is possible provided ε (ω) < –1;
3a. When moving away from the interface of the medium in both directions, all the components of the wave field decrease monotonically according to exponential laws;
4a. Spectrum of surface wave frequencies ω at the interface between air and metal with a plasma frequency Ωp is bounded above: .
5a. The absorption of a wave in a medium is determined by collisions of free electrons with a crystalline lattice.
The essential features of this model are connected with the presence of a solid-state plasma in the metal, along the interface of which the surface wave propagates, and the approach of a very thin plasma-air transition layer with the thickness much smaller than the wavelength (sharp interface). Surface waves, described by this model, are used today in the development of light modulators, optical transistors, tunable sensors and many other electro-optical devices. Based on the concepts of coupled oscillations of the electromagnetic field and solid-state plasma in optoelectronics, a large direction of applied physics, plasmonics, has been established.
The second wind in the physics of surface waves came with the appearance of artificial media – metamaterials. In contrast to plasmonics, which uses natural media, below we will talk about metamaterials based on dielectrics, where there are no free carriers [1]. This new trend in the physics of surface waves is neither associated with any solid-state plasma, nor with its uniformity, nor with a sharp interface of the medium, has appeared thanks to the success of nanotechnology, allowing you to create an optically inhomogeneous dielectric films and coatings with the refractive index n distributed within these media according to a given law; such materials are called gradient coatings. In order to manufacture gradient coatings, many techniques have been developed; the one most frequently used is magnetron sputtering of a mixture of components on a transparent substrate. Thus, to create a nanofilm containing silicon dioxide SiO2 and niobium Nb2O3, the simultaneous operation of two magnetrons is used [2]: one sputter SiO2, and another Nb2O3. The substrate moves between the magnetrons, and the rate of its motion determines the fraction of each sputtered material and the spatial profile of the refractive index n, which depends on the ratio of these fractions. In the simplest case, when the value n for a dielectric depends only on one coordinate z, perpendicular to its surface (the plane z = 0, one can consider a simple dependence
n2 (z) = n02 U2 (z);
. (1)
Here n0 is the value of the refractive index at the boundary z = 0; dimensionless quantity g > 1 and the characteristic length L are the free parameters of the model (1); these parameters determine the structure of the wave field in the gradient dielectric; profile U2 (z) (1) shown in Fig. 2, illustrates the decrease in the refractive index from the value n0 at the interface of the medium (U = 1) and its "saturation" in the depth of the environment (z >> L): n2 (z >> L) = n02 (1 – g–1).
The surface wave in the transition layer shown in Fig. 1, is described by exact analytic solutions of the Maxwell equations [2].
The properties of this new wave fundamentally differ from the known properties of waves on the sharp interface, noted above in p. 1a‑5a. To emphasize these differences, the characteristics of a new wave are listed below in p. 1b – 5b; pairwise comparison of pp. 1a and 1b, 2a and 2b, etc. stresses on the opposite tendencies of the corresponding characteristics:
1b. The field of a wave traveling in the direction y (Fig. 1b), contains components Hz, Hy, Ex (TE polarization);
2b. The surface wave at the air – dielectric interface is possible under the condition n2 (z) > 0, i. e., the wave field is localized near the interface of the transparent media;
3b. When moving away from the boundary into the interior of the medium, the wave field transverse components Ex and Hz (Fig.1b) vary nonmonotonically (Fig.3): the electric component Ex begins to grow, reaching a maximum at the point zm at a distance of the order of the characteristic scale of the inhomogeneity L, and then decreases; the magnetic component Hz, proportional to Ex is similarly distributed; thus, the energy flow along the interface of the media z = 0 reaches its maximum not at the interface, but inside the dielectric at the point zm. Comparison of curves 1 and 3 shows that the increase in the frequency of the wave contributes to the field enhancement at the distribution maximum Ex. The longitudinal component of the magnetic field Hy at the same point zm passes through zero and changes direction.
These properties underline the differences in the polarization and spatial structure of the surface wave in the gradient dielectric from the known surface waves in homogeneous metals and semiconductors; but even more significant differences arise when the spectra of these waves are compared (see below). More specifically, considering the shift in the maximum of the field, this wave could be called not a surface wave, but a "subsurface" one, but as the British logic Ockham taught back in the 13th century, "entities are not to be multiplied without necessity".
VISIBLE SEWS – FROM BLUE TO RED ONES
The physical basis of the surface wave in the gradient dielectric are coupled oscillations of the electromagnetic field and polarization of the dielectric medium; since a dielectric is considered without free electrons, the plasma frequency Ωp vanishes, and the restriction of the spectrum (3a) disappears. This implies the important features of new SEWs:
4b. Surface waves in the medium (1) exist in a wide spectrum of frequencies, and are determined by the gradient parameters L and g.
Thus, selecting technologically controlled parameters of the gradient layer L and g, it is possible to ensure the propagation of surface waves in a wide range of frequencies, including (which is important!) the whole visible part of the spectrum. It is this case that is shown in Fig. 4, illustrating at the selected value of L = 140 nm, the refractive index of surface waves n (λ) in the range of colors from red (λ = 0,75 µm) to blue (λ = 0,38 µm). Estimating the phase velocity of these waves Vph = с / n, where c is the speed of light, "sublight" values can be found from Fig. 4 Vph = 0,6–0,7 с.
5b. Absorption in a dielectric without free carriers is small, the loss tangent is of the order of 10–3.
The spectrum in Fig. 4 is represented as a refractive index n from the dimensionless parameter ky L. In this representation, the spectrum can be used for analysis of surface waves in gradient dielectric of model (1) with other values of the characteristic length L. Thus, it was noted above that in the case of L = 140 nm, the wave with kyL = 1.92 (λ = 0.75 µ) corresponds to the value of n = 2.1; the same value n at longer length of L, e. g., L = 200 nm will correspond to a longer wavelength λ = 1.06 µ. Thus, the spectrum shown in Fig. 4 can be used for analysis surface waves both visible and IR range.
The studies of a new family of SEWs have been recently begun. Such fast color broadband SEWs can be of interest in optoelectronics; moreover, a surface wave with a magnetic component perpendicular to the interface of media, is of particular interest for the analysis of magnetic micro-objects. However, the first examples of SEWs in gradient dielectric nanostructures are limited to TE-polarization waves. The question naturally arises: whether this structure is also able to support the propagation of TM-polarized waves? This new SEW branch would provide additional light flows control capabilities, and its interference with the TE-polarized wave would enhance the prospects of mixing colors on the surface of the gradient dielectric. The studies are ongoing, the survey is open…
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