The abbreviation "TIR" in optics, as opposed to the army regulations, means "the total internal reflection" – the phenomenon which is known from the school course in physics: light ray, which is obliquely incident from the medium 1 with higher refractive index to the medium 2 with lower refractive index , will be reflected back into the medium 1 if the incidence angle is higher than so-called critical angle . The value of this angle is determined on the basis of the ratio between refractive indices of the medium 1 and medium 2. Physics historians note that this amazing effect was known by Johannes Kepler who not only discovered three laws of planetary motion but also had great interest in optics improving the astronomical tools.
Three centuries has passed and the TIR effect drew attention of researchers again. Electromagnetic light theory has occurred, and Maxwell equations and concepts concerning the light waves have become scientific practice. Using the new ideas A. A. Eikhenwald, professor of the Moscow University, theoretically showed  that in case of TIR occurrence at the media interface the light field does not stop, penetrating into the reflecting medium it decays exponentially. The energy of penetrating wave monotonically decreases at the distances of the order of light wavelength and this decay is not connected with the wave absorption in any way. Eichenwald results showed that the effect has wave nature and cannot be described in the common language of geometrical optics – on the basis of the light rays. This conclusion was soon graphically confirmed in the experiment of L. I. Mandelstam and P. Seleni.
In this experiment the lower bound of glass prism was immersed into the liquid in which the fluorescent material has been dissolved (Fig. 1). The light incident through the prism on the liquid bound at the angle, which is higher than the critical angle, experiences TIR effect at the bound. However, the part of light flux penetrating into the thin layer of bound liquid causes its fluorescent glow. The fluorescence color differs from the light of incident radiation and the glow of bound layer gives opportunity to observe the effect. This experiment proved the other Eikhenwald prediction: during decay the light flux penetrated into the liquid at low but finite depth which is commensurable with the wavelength and, what is the most unexpected, the decay was not connected with the wave absorption. Such partial light penetration through the non-transparent barrier received the short name "FTIR" ("frustrated TIR").
The Russian Physical and Chemical Society Magazine published Eikhenwald paper in 1909. During those years against the background of swift development of the other branch of electromagnetism – radio engineering – the concept of FTIR, which defines the conventional law of light refraction more precisely, could appear to be refined but unpractical theory. But, "there is no more practical thing than good theory!" – said the American Edward Condon, one of spectroscopy pioneers. And indeed, twenty years have not passed and the turn of FTIR theory came.
G. Gamov: from FTIR of de Broglie Waves to Tunneling of Particles
The second wind in the theory of FTIR was given by G. A. Gamov. Twenty years old graduate of Leningrad University was sent to Leipzig for training, as we would say today, and there he started studying the "hot" task, which has arisen in the laboratory of Rutherford, the discoverer of atomic nucleus, in the group of the youngest (26 years old!) German professor W. Heisenberg: it was known that the radiation occurring during the radioactive decay of uranium atomic nuclei contains the particles of two types, which Rutherford called "alpha" and "beta". The nature of this radiation was also known; in particular, the nucleus of helium atom consisting of two neutrons and two protons (the name "proton" was also suggested by Rutherford!) was identified in alpha particle. However, the dark spot appeared in this harmonious picture: leaving the mother nucleus alpha particle had to break the potential barrier, which was created by the nuclear attraction forces. Calculations showed that the particle work concerning the barrier breakage appeared to be greater than the energy of the particle itself. And again the temptation to declare about the possible violation of energy conservation law has occurred, this time – in microworld…
While searching the solution Gamov paid attention to the outward similarity of recently suggested Schrödinger equation, which described the motion of atomic objects through the potential barrier, and wave equation, which described the light transmission through the layer of non-transparent material. Presenting formally the motion of atomic object with the help of the waves of special type, so-called wave functions, it was possible to see analogy between the penetration of alpha particles through the potential barrier and leakage of electromagnetic waves through the non-transparent layer during the performance of FTIR. Notwithstanding the mystery of alpha decay, the concept of the total internal reflection "was in the air" revealing itself in the other discovery, which was also made in England in 1924, – reflection of the radio waves emitted by ground transmitter from the layer of ionized gas surrounding the Earth at the height of 90–100 km (at that time ionosphere was not spoken of, the familiar phrase was "Heaviside layer"). One step separated the analogy of equations from the analogy of solutions and this step was made: in 1928 Gamov formula occurred ; as opposed to the conventional mechanics, it expressed the exponentially small but finite probability of particle transit through the barrier, in other words, the probability of atomic nucleus decay. This effect does not infringe on the energy conservation law: in the determination of the pulse of quantum particle p and coordinate x there are always "uncertainties" and which are connected with the Planck’s constant on the basis of the fundamental Heisenberg "uncertainty principle": , where the "uncertainty" of pulse of the particle transiting through the barrier allows the "uncertainty" of the coordinate behind the barrier.
The work of Gamov introduced the new fundamental concept into the physics language – tunneling, the concept which is common for the wave fields of different physical nature. During the tunneling processes, the wave fields vary aperiodically in space, the key concept "wavelength" does not occur and the wave phase does not change. For more than half of century this concept was associated with exponentially small transmission of the fluxes of particles and waves tunneling through non-transparent barriers. The new life of this concept, which put the conventional "smallness’ to an end, started with the occurrence of the special type of artificial materials, so-called metamaterials, and new methods of creation of miniature nano-sized optoelectronic systems, so-called nanotechnologies, in optics and electronics.
Nonlocal Nano-Optics of Dielectric Metamaterials: When Fresnel Formulas don’t describe the Light Reflection?
Nano-optics of dielectric metamaterials studies the interaction of light with thin nano-sized films and coatings which consist of artificial dielectrics. Only one issue connected with the specific class of nanofilms, which can be obtained on the basis of the magnetron deposition of tantalum and silicon atoms in the oxygen atmosphere to the quartz substrate, is marked out from today’s "hot" area in this article. During the deposition process, the substrate moves according to the certain law, which controls the percentage of formed oxides Ta2O5 and SiO2 in deposition layers; refractive indices of these oxides are different so that such motion, which is controlled by the computer program, automatically provides the spatial distribution of refractive index in the film and effects caused by this inhomogeneity. In electrodynamics it is spoken of the nonlocal effects when the medium response on the electromagnetic field at the given point depends not only on the field at this point but also on the field values in some area surrounding this point. The occurrence of gradient transparent nanofilms, the refractive index of which is smoothly modulated in space at the distances of the order or at the distances which are even smaller than , drew attention to the profound nonlocal effects in the reflection, transition and dispersion of waves in the gradient transparent nanostructures. Such effects are easily observed in simple one-dimensional task when the refractive index of flat film n is modulated in the direction z, which is perpendicular to the film bounds: ; where is the value on the film surface = 0 on which the light is incident, is dimensionless function, which describes the distribution of refractive index (profile n) inside the film. Example of such distribution given by the function
is shown in Fig. 2а, combination of signs s1 = –1, s2 = +1 and s1=+1, s2 = –1 in (1) corresponds to the convex and concave profiles . Values L1 and L2, which have length dimensionality, are free parameters of the model (1) connected with the layer thickness d and maximum (minimum) of the profile . Often, used model of "Rayleigh profile"  can be considered as the special case of more flexible distribution (1) corresponding to the threshold . Sequentially depositing the layers (1) the periodic gradient nanostructure can be obtained; its adjacent layers are shown in Fig. 2b. It should be noted that the technology of deposition on the moving substrate allows obtaining the various profiles but use of the model (1) is especially convenient because within the framework of this model the wave fields in gradient media are described on the basis of the exact analytical solutions expressed through the elementary functions. Specifically this model was used in "Fotron-Auto Ltd" Company (Moscow) during the works connected with the deposition of gradient dielectric nanostructures, measurement and estimation of their spectrums of reflection and transmission in the visible and IR bands; herewith, the profile (1) was formed as the variable profile with the film thickness (d = 140 nm) by the content ratio of oxides Ta2O5 and SiO2 , so that the values n are maximal on the film surface and minimal in its central plane . For the simplicity, only cases of normal radiation incidence on the bound 0 are considered here.
The spectrum of transmission of the periodical nanostructure , which contains 11 such gradient layers, is shown by the curve 1 in Fig. 3. This spectrum was obtained during the experiment; however, it turned out to be impossible to calculate the values using the conventional Fresnel formulas: these formulas describe the light transmission through the homogeneous plate which is characterized by the discontinuities of the refractive index n on the plate bounds. Fresnel formulas were derived for the light almost two hundred years ago; with the lapse of time, the analogs of these expressions for other wave tasks have occurred – for example, for the reflection of waves from impedance discontinuities in electromagnetic  and acoustic  systems; these formulas work today even for the estimation of nano-optical structures based on the alternation of homogeneous layers with high and low values of the refractive index n experiencing the discontinuities at layer bounds. As opposed to it, the profile n at the bounds of gradient films making the studied nanostructure is continuous (see Fig. 2a) but the gradient of this profile at layer bounds experiences the discontinuity. In order to find the reflection and transmission of such structures, it was required to construct the generalization of Fresnel formulas for the transparent gradient media. Theoretical analysis of this task on the basis of precise analytical solutions of Maxwell equations for the profile n (1), which are correct for any wavelength, showed that along with the discontinuities of refractive index there are two more effects which are typical for inhomogeneous media and form the spectra of gradient multi-layer transmission:
1. Reflection from the gradient discontinuity in the continuous profile n(z). The curve 2 in Fig. 3 represents the transmission spectrum estimated with the help of discovered generalized Fresnel formulas for the same periodical nanostructure for which the spectrum (curve 1) was measured experimentally. Discrepancy of the values for the curves 1 and 2 does not exceed 2–3%, and this precision gives the reasons to apply obtained results for the "construction" of spectra of transmission and other gradient nanostructures which are designated for the operation in other spectral ranges characterized by the different number of layers m and other dimensions d. Analysis of such spectra reveals the unexpected effect of artificial dispersion of gradient layer. Saving the review of this effect and its consequences for later, one more mechanism of wave reflection in inhomogeneous dielectric should be noted here.
2. Reflection from curvature discontinuity in the continuous profile n(z). This situation is shown in Fig. 4a which represents two different distributions in the transition layer between the homogeneous media with the refractive indices and . Both distributions at the layer bounds are continuous; gradients of these distributions at the layer bounds vanish concurring with the zero value of gradient in homogeneous layer. Thus, there are no discontinuities of the refractive index n and discontinuity of gradients n at the layer bounds. The profiles themselves consist of concave and convex arcs represented by the parts of the curves shown in Fig. 1; these arcs contact inside the layer smoothly (the gradient of each profile at the arc contact point changes continuously). Transition from the concave to the convex part of smooth profile is characterized by the discontinuity of profile curvature. The spectrums of reflection from the profiles 1 and 2 (Fig. 4b) show the noticeable difference which is stipulated by the additional reflection at the curvature discontinuity point. This effect forms the physical basis for the nondestructive inspection of graded distribution of refractive index inside the transitional layer.
These results are stipulated by the complex phase structure of interfering forward and backward waves conditioned by the discontinuities of profile gradient and curvature in gradient film. When the inhomogeneity of the model (1) disappears () the specified reflection mechanisms also disappear and obtained formulas  turn into the classic Fresnel formulas.
Gradient Photonic Barriers: Artificial Dispersion and Reflectionless Light Tunneling
The subwave transparent film with specifically formed distribution of the refractive index n, which controls the photon flux, is graphically called "gradient photon barrier". Important property of such barrier consists in the artificial dispersion: as opposed to the natural dispersion connected with the local dependence n from the wave frequency , the dispersion of gradient medium is nonlocal effect which is determined on the basis of the spatial distribution n. Thus, the profile , which is represented by the curve 1 in Fig. 2a, is characterized by plasma-like dispersion: the plasma frequency , which divides the spectral intervals of propagating ( > , wave numbers are valid) and tunneling ( < , imaginary wave numbers) fields, is determined in the plasma of free carriers; this frequency depends on the density of free carriers. It was repeatedly emphasized above that nanofilms, which are considered here, are created from dielectrics without free carriers and so there is no plasma frequency for them. However, for the films without free carriers the distribution n (Fig. 2а, curve 1) determines the threshold frequency :
depending only on the distribution parameters and ; similarly to the plasma frequency the frequency divides the spectral intervals which correspond to the valid and imaginary values of wave numbers for the fields inside the gradient photon barrier. In order to "feel" order of the value it should be noted that, for example, for the profiles shown in Fig. 3 the frequency corresponds to the length of the wave from near IR band: = = 1320 nm. The frequency is the characteristic of gradient barrier: if the inhomogeneity n decays , then .
Using the frequency it is convenient to represent the spectrums of transmission of periodical structures, which contain gradient nanofilms (1), in the form of generalized dependency on the dimensionless parameter , so that the areas correspond to the interval of valid (imaginary) wave numbers. Calculations of such spectrums given in Fig. 5 confirmed the expected effects of artificial dispersion, which are shown for one nanostructure in Fig. 4: in the area of high frequencies the nonlocal transmission dispersion is high and the value varies considerably depending on the structure thickness; in the area of low frequencies – on the contrary, the dispersion is insignificant and transmission of the wave with imaginary wave numbers is high and almost continuous. Spectrums in Fig. 5 have "universal" character: the coefficient of transmission through the periodical structure with the set parameters and for each value u retains its value at any frequencies and film thicknesses bound by the condition
On the surface, these results seemed strange, especially the statement of efficient energy transfer by the wave fields with imaginary wave numbers, in other words, the statement of transfer under the conditions of FTIR. In order to check this effect the special experiment was carried out by Yu. A. Obod and O. D. Volpian – researchers from "Fotron – Auto Ltd" Company in Moscow: the transmission spectra in visible and IR ranges for two different periodical nanostructures containing 7 and 11 gradient nanofilms respectively (Fig. 6) were measured. The measurements showed:
narrow deep dip and extreme frequency dispersion of transmission of visible and near IR ranges (= 1255 nm);
wide-band plateau corresponding to the weak dispersion and high transmission in the area where the radiation propagates through the barrier under the conditions of FTIR and is described by the fields with imaginary wave numbers;
what is especially interesting – the noted high transmission in the area of FTIR practically does not depend on the thickness of photon barrier, in other words, on the number of films m: difference in the values and does not exceed 2–3%.
Described effects are stipulated by the peculiar interference of forward and backward tunneling waves which are aperiodic in space and harmonic in time; the backward waves reflecting from both borders of flat parallel substrate make contribution into the interference picture and the transmission coefficient is 92–95% in this case. If the wedge-shaped substrate, as opposed to such reflector, is used where the waves reflected from the back border do not return to the interference area the discrete "windows of transparency" occur in the transmission spectrum where the transmission coefficient under the conditions of FTIR reaches 100% (Fig. 7).
These results show the principal difference between the efficient transfer of radiation fluxes in gradient structures under the conditions of FTIR and exponential decay of these fluxes during tunneling through the homogeneous non-transparent layers. As opposed to the considered simple case of normal radiation incidence on the bound of gradient layer, in case of oblique radiation incidence on such layer the conditions for the occurrence of FTIR considerably vary for - and - polarized waves and the consideration becomes significantly complex . For the simplicity, one more simple case – the propagation of waves along the bound of gradient layer is discussed below.
Gradient Optics of Surface Waves of Visible and IR Ranges
The concept of surface electromagnetic waves at the bound of conducting medium was introduced into the scientific practice by A. Sommerfeld in 1899 and earlier the similar situation for acoustic waves at the bound of elastic solid was discussed by Rayleigh . The propagation of surface electromagnetic waves (SEW) along the sharp bound of media with free carriers – metals, semiconductors and even salt water – has been the study subject in electronics, radio physics and geophysics for a long time already. However, the gradual changes of medium permittivity in the subwave layer near the bound of division of dielectrics without free carriers (gradient near-surface layer which is called sometimes "metasurphace") can completely reconstruct the conventional SEW picture cancelling a number of prohibitions which restrict the conditions of SEWs existence.
As opposed to the analysis of the fields propagating along the gradient n, the fields propagating across the gradient n or along the bounds of such dielectrics are considered here. Emphasizing the peculiarity of such fields it is convenient to compare them with the surface waves at the sharp bound of division of two homogeneous media with free carriers. Polarization and spectrum of such waves travelling in the direction y along the bound (plane z = 0) are well known :
existence of the surface wave at the bound of homogeneous media is possible if the permittivity of these media meet the following condition: ; it means that at least in one of bounding media the condition must be met and this condition is typical for the metal or dielectric plasma with free carriers, for example;
the wave field contains the components , and ( polarization); polarized surface wave (components , and ) on such bound surface is impossible;
the spectrum of surface wave frequencies at the bound of divisions of air and plasma with the plasma frequency is limited from above: .
Traditional medium for the excitation of such waves is the plasma of free carriers in metals and semiconductors.
To the contrary the other family of SEWs can occur in the transparent gradient layer near the surface of transparent dielectric without free carriers the permittivity of which decreases from the medium surface into the depth, so that .
These waves can be easily considered within the framework of exactly solvable model of the spatial distribution of refractive index of dielectric medium, :
where L is arbitrary spatial scale, g is arbitrary dimensionless parameter. Dimensionless function U(z) (Fig. 8) describes the "saturation" n(z) in the medium depth :
= . (5)
Medium parameters L and g determine the threshold frequency , which depends on the gradient n in the near-surface layer
Along with the conditions of SEWs excitation, the frequency limits the spectral interval in which the considered SEWs exist. In the area of low frequencies the wave field propagates along the dielectric bound and decays on both sides from this bound. The frequency in dielectric without free carriers reminds the frequency limiting the SEW spectrum in electron plasma; however, as opposed to the plasma, the threshold frequency , which is controlled by the method of gradient layer fabrication, can be created in the medium without free carriers in the previously set spectral interval.
Optimization of the parameters and typical thickness of near-surface layer L will allow the formation of S-polarized surface waves in narrow spectral intervals in the dielectrics with fixed values of refractive index far from the surface. Thus, in the dielectric, which is characterized by the parameters = 1,42, = 2, = 50 nm, the spectral interval for specified surface waves exists in the near IR range, which is limited by the frequencies = 2,13 · 1015 rad/s (= 884,5 nm) and = 2,016 · 1015 rad/s ( = 935,5 nm); the interval width is 50.5 nm. However, with the same values and the decrease of layer thickness (L = 30 nm) results in the formation of narrow spectral band in the visible range = 3,55 · 1015 rad/s ( = 530,7 nm) and = 3,36 · 1015 rad/s ( = 560,7 nm), so that the interval width becomes narrower to = 30nm.
The transitional layer with finite thickness is required for the existence of surface waves on the dielectric surface; in case of widening of this layer the critical frequency (6) decreases to zero and the considered branch of wave spectrum disappears. The finite values L determine the typical peculiarities of such waves on the gradient surface:
The critical frequency (the upper bound of the spectrum of surface modes) determined by the profile of refractive index of dielectric near the surface characterizes the artificial dielectric dispersion only in the thin near-surface layer.
Capabilities of selection of critical frequency determined by the nonlocal dispersion of near-surface layer make it possible to widen the spectrum of the frequencies of surface waves in short-wave and long-wave parts of the spectrum.
Loss connected with the decay of surface waves can be optimized by the selection of the gradient material, absorption bands of which are located far from used frequencies of these waves.
Selection of distribution parameters (3) allows decreasing the decay of surface wave at the bound "medium-air" at the expense of "displacement" of the wave field from the dissipative medium into air.
Three-dimensional periodical structure which is localized near the surface of gradient dielectric is of interest; it can be created by virtue of the interference of two surface electromagnetic waves with the same frequency . Thickness of the layer where the surface mode of medium-wave IR band is localized is about 0.1–1 µm and is controlled by the value determined on the basis of the technology of surface layer deposition.
It should be emphasized that, as opposed to the traditional consideration of SEWs on the surfaces of the materials with free carriers, the specified effects in gradient dielectrics without carriers considerably widen the spectral range of SEW existence as well as circle of the materials which are prospective for the creation of new SEW systems.
From Light Tunneling to Sound Tunneling?
The typical frequencies (2) and (6) in the dielectric layer without free electrons remind plasma frequencies in metals and semiconductors but (and this is the fundamental difference!) the frequencies and can be formed in any previously set spectral range. This freedom of choice reveals the opportunities to optimize the parameters of gradient media in respect to the needed frequency interval in different parts of the spectrum of electromagnetic radiation – from light to radio waves . Furthermore, the effects of artificial dispersion for the waves with different physical nature in gradient media are often described by the similar solutions of wave equations confirming the explicit expression of one of founders of the statistical physics, J. W. Gibbs, which is used in epigraph. The fresh example of such "percolation" of the concept of artificial nonlocal dispersion between different areas of wave physics is the analysis of acoustic dispersion of the solid body with controlled distributions of density and elastic parameters. Such dispersion can result in a number of the effects of "gradient acoustics’ which are analogous to the effects of gradient optics and, in particular, in the unexpected effect of sound tunneling in inhomogeneous solid materials . Nowadays, these developments are in the beginning of their way representing the "thought-provoking information", as well-known movie hero said.