Artificial dispersion of all-dielectric gradient nanostructures: frequency-selective interfaces and tunneling-assisted broadband antireflection coatings
Perspectives of optical circuits in nanoscales for the visible and infrared spectral ranges have captured recently the attention of optical community. This attention was inspired by the first successes in design of optical analogies of microwave circuitry elements. Just as circuits at the microwave domain involve the elements, which are smaller than the wavelengths of operation, the nanostructures with the subwavelength dimensions for the optical wavelengths were designed. The physical fundamentals of these optical circuits are based on the electromagnetics of all-dielectric metamaterial structures, operating mainly with the displacement currents. The miniaturized nanometer-scale metamaterial elements with displacement currents were shown to possess the inductance and capacitance, behaving as the nanoinductors and nanocapacitors . The “epsilon-near-zero” and “epsilon-very-large” dielectric composites can provide the platform for elaboration of nanoresistors for lumped optical circuits [2, 3]. Extension of the radio technique concepts to the optical domain paved the way to the generation of magnetic fields at optical frequencies by means of resonant excitement of non-magnetic dielectric nanocylinders  and nanospheres  by light. Note, that all these elements can be constructed from spatially homogeneous dielectric composites, characterized by some fixed value, both positive or negative, of dielectric permittivity. A substantial part of the general interest to all-dielectric nanostructures is stipulated by the progress in physics and technology of inhomogeneous thin films, widely used as optical filters , antireflection coatings , and transition layers between two media with different refractive indices . Traditionally, the design of these devices was based on the multilayer structures with steeply alternating high nh and low nl refraction indices.
Gradient nanostructures, fabricated from the dielectrics without free carriers, possess the strong nonlocal heterogeneity-induced dispersion, determined by the shape, gradient, and curvature of refractive index inside this structure, controlled by the technology of fabrication . The attention is given below to the controlled distribution of refractive index n(z) along the direction z across the plate gradient dielectric nanofilm. Formation of artificial plasma-like dispersion and characteristic frequency Ω in these nanofilms proves to be feasible for some profiles of n(z); this frequency Ω separates the spectral ranges, characterized by real and imaginary values of wave vectors in a transparent lossless nanofilm. The wave energy in the low frequency range ω ≤ Ω is transmitted through these “photonic barriers” in the tunneling regimes by means of waves with the imaginary wave numbers (evanescent modes).
This paper is aimed on the unusual transparency properties of gradient all-dielectric periodical nanostructures stipulated by the evanescent and antievanescent modes. The theory of effective radiation transport through these structures is developed in Sec. II; its experimental verifications, illustrating the frequency-selective and antireflection properties of gradient nanofilms, are presented in Sec. III. The obtained results give the chance to revisit in Sec. IV the well known Hartman paradox16 in the framework of unusual amplitude-phase spectra of waves, transmitted through the gradient photonic barrier. Some applications of these phenomena for the design of optical circuitry are noted in the Conclusion (Sec. V).
II. EFFECTIVE TRANSPORT OF RADIATION SUPPORTED BY THE EVANESCENT MODES IN THE GRADIENT PERIODICAL NANOSTRUCTURES
We recall here some results exposed in a paper . Consider a simple problem of normal incidence of linearly polarized EM wave with the components Ex and Hy propagating in z-direction, incident on the interface z = 0 of a lossless dielectric film characterized by continuous distribution of refractive index n(z) = n0U(z) across the film. Here, n0 is the value of refractive index of material on the interface z = 0. Expressing the field components Ex and Hy through the vector-potential 
one can reduce the set of Maxwell equations, related to this geometry, to one equation governing the generating function W,
So far, as this equation will be used for analysis of wave fields in the nanolayers with thickness comparable or less han the wavelengths, any suppositions concerning smallness or slowness of variations of fields or media are invalid, and the exact analytical solutions of Eq. (2) are in need. The large family of exact solutions, related to different distributions U(z), is presented in Ref. 14; we will use one of these profiles, providing the diversity of such solutions
The flexibility of profile (3) is connected with the interplay of two free parameters L1 and L2 having the dimension of length; these spatial scales L1 and L2 are linked with the layer’s thickness d and the minimum value of refractive index nm
Solutions of Eq. (2) with distribution U(z) (3) describe waves with both real and imaginary wave numbers, corresponding, respectively, to propagating and evanescent modes. The spectral ranges related to these regimes are separated by some characteristic frequency Ω, dependent upon the shape and size of profile U(z),
The frequencies ω ≥ Ω (ω ≤ Ω) fall to the propagating (evanescent) spectral ranges. The attention will be focused below on the low frequency range ω ≤ Ω.Omitting for simplicity the factor exp(–iωt) and introducing the new variable η, one can present the solution of Eq. (2), describing the field with frequency ω inside the layer as a result of interference of forward (evanescent) and backward (antievanescent) waves,
Substitution of this function Ψ to the equalities (1) brings the components of EM field inside the gradient layer. Parameter Q, describing the contribution of the backward wave to the entire field inside the layer, has to be found from the continuity conditions on the layer’s boundaries. It is essential that, in the gradient dielectric, the waves with imaginary wave numbers arise in the transparent medium with the real positive value of refractive index; herein, the profile of refractive index (3) provides the plasma-like dispersion of dielectric layer without free carriers; characteristic frequency Ω plays the role of plasma frequency.
Let us consider the low frequency wave field (ω ≤ Ω, u ≥ 1) for the periodical nanostructure, containing m ≥ 2 similar adjacent gradient nanofilms with thickness d, deposited on a homogeneous transparent dielectric halfspace with refractive index n. Attributing the number m = 1 to the first gradient layer at the far side of this structure and supposing the incidence of radiation from the air on the m-th layer, one can write the standard continuity conditions for the field components on the boundary between m-th and (m – 1)th layers
Application of the same approach to all the adjacent layers of the gradient nanostructure brings the explicit expression for it’s complex transmission coefficient Tm,
Parameters Λm in (10) are linked by the chain of recursive relations, obtained from the continuity conditions on the interfaces between the m-th and (m – 1)th layers, where m > 2
The value of parameter Λ1 in the first factor in the product (10), calculated, unlike (8), from the continuity conditions on the boundary between the first layer and the transparent substrate with thickness h and refractive index n, is
While using expression (12), one has to distinguish two different shapes of substrate, ensuring the different directions of wave, reflected from the far side of substrate:
• If the substrate’s interfaces are parallel, the backward and forward waves in the substrate are interfering; in this case, the value Λ0 = 1, following from the continuity conditions, is
• If the substrate’s interfaces are not parallel (wedgeshaped substrate), the backward wave reflected from the back side of substrate does not interfere with the forward wave in the substrate; in this case, Λ0 = 1.
The expressions for coefficients Tm for the case of propagating waves (ω ≥ Ω, u ≤ 1) follow directly from (10)–(13) due to the replacements
Formulae (7) and (14) point out the peculiar effect of heterogeneity-induced dispersion, arising due to the spatial distribution of refractive index in both cases ω ≤ Ω и ω ≥ Ω; it is remarkable that this plasma-like dispersion appears in the transparent dielectric medium without free carriers.
It is remarkable that unlike the usual reflection on the discontinuities of refractive index, described by the classical Fresnel formulae, expressions (10)–(11) are based on the reflection and transmission of waves caused by the discontinuities of gradient of refractive index on the boundaries between m-th and (m – 1)th films. These non-Fresnel spectra, visualizing the drastic influence of artificial dispersion on the transmittance of periodical gradient nanostructures, containing several similar adjacent layers (Fig. 1), are examined below.
All forthcoming computational and experimental results are obtained for the periodical nanostructures, consisting from the adjacent gradient nanofilms with the same profiles of refractive index (3). The transmittance spectra |Tm|2 for waves traversing these gradient multilayer nanostructures are presented here for the visible and infrared spectral ranges.
Note that the exactly solvable models of gradient barriers, examined above, are based on continuous distributions of the dielectric permittivity ε(z). However, these distributions, owing to technological conditions, are actually formed by plane layers of different geometrical and optical thicknesses fabricated by means of magnetron sputtering from Si and Ta targets in the oxygen environment (for details see the problem 2 in Sec. III). The discretized structure of ε(z) is illustrated in Fig. 1: to mimic the continuous distribution of ε(z) across each nanofilm, the geometrical thicknesses of these layers are decreasing from 10 to 12nm near by the minimum of ε(z) down to 5–7 nm at the area of its rapid variation close to the nanofilm boundaries .
The good agreement of transmittance spectra, calculated in the framework of continuous model, with the experimental data (Figs. 2–6) shows the applicability of “continuous” model to the discussed problem of gradient optics.
III. GRADIENT ANTIREFLECTION AND FREQUENCY-SELECTIVE
To visualize the unusual effects of interference of evanescent and antievanescent waves in the gradient nanostructures, let us consider first the transmittance spectra |Tm|2 of periodical structures deposited on a wedge-shaped substrate, characterized by the value Λ0=1 in Eq. (12). The results of computations, carried out by means of formulae for complex transmission coefficients Tm (10)–(12), is shown in Fig. 2. To broaden the applicability of these graphs, the frequency dependences of |Tm|2 are presented via the dimensionless variable u (7). If the parameters of gradient film n0 and y are fixed, the coefficient |Tm(u)|2 remains its value for any given value u for the waves with different frequencies ω traversing the films (3) with different thicknesses d, linked by the fixed value of dimensionless parameter υ
Thus, for the structures, depicted in Fig. 2, the value of parameter υ for any value of variable u is υ = 0,7u–1. Herein, the complete transparency (transmission coefficient |Tm(u)|2 = 1) can be provided, e.g., for the nanostructure, containing 10 films, in a case u = 1,4 for the frequency ω = 1,5 ∙ 1015 rad/s (λ = 1256nm) as well as for the frequency ω = 1015 rad/s (λ = 1884nm), if the thicknesses of films are 100 and 150nm, respectively. These generalized spectra may become useful for optimization of thickness of the nanostructure.
Investigating the energy transport by evanescent waves, one has to stress out that neither the forward nor the backward waves describe the flow of tunneling electromagnetic energy; this flow is determined by the interference of both these evanescent waves, presented by the entire function ψ (6). Unlike the structures deposited on the wedge-shaped substrate, considered above, the more complicated phenomena in structures, deposited on the flat substrate (13), are considered below; the interference of propagating waves in the transparent substrate have to be taken into account now. The tunneling regime for this case is examined below for the structures consisting of adjacent gradient nanofilms; the thickness of each film was 140 nm, the refractive index n(z) was varying according to profile (3) from its maximum n0 = 2 to its minimum nmin = 1,5 (Fig. 1); all these nanostructures were deposited on the quartz homogeneous flat lossless substrate with thickness h = 2 mm and refractive index n = 1.52. To illustrate some salient features of effective wave energy transport through the gradient periodical nanostructure in the tunneling-assisted regime, the following computational and experimental problems for these structures are considered below:
1. The interference phenomena for evanescent and antievanescent waves in the single layer and multilayer gradient nanostructures and their influence on the spectra of |Tm|2 ; these spectra were calculated here for the structures containing 1, 5, and 11 nanofilms. Contrary to Fig. 2, we will consider the case, when the wave, reflected from the back side of the substrate, makes the contribution to the wave field inside the substrate. The results of computations, carried out by means of formulae for complex transmission coefficients Tm (10)–(13), are shown in Fig. 3. The transmittance spectra are presented here via the dimensionless frequency-related variable ln(u); the logarithmic scale permits to compress the graphs in order to compare the broadband spectra of |Tm|2, including both propagating and evanescent spectral ranges.
2. Experimental verification of this model by means of measurements of spectrum |Tm|2 for the abovementioned structures; these structures were fabricated by means of magnetron sputtering of Ta and Si components through the oxygen atmosphere on the movable quartz substrate . The needed profile of refractive index n(z) was provided by the variable spatial distribution of oxides Ta2O5 and SiO2 in the sputtered film, controlled due to prescribed motion of substrate . The |Tm|2 spectrum for the same conditions was calculated too; the theoretical and experimental transmittance spectra are presented on Fig. 4. To ensure the comparison of these data, both spectra are presented here, unlike the Fig. 2, as the functions of free space wavelength k. The theoretical and experimental curves are fitted well in the visible range, the discrepancy in the infrared range does not exceed 2%–3%.
3. The unexpected dependence of transmittance of gradient nanostructure in the Frustrated Total Internal Reflection (FTIR) regime upon the entire thickness of structure; to reveal this dependence, the experimental measurements of transmittance of two structures containing 7 and 11 nanolayers, respectively, were carried out. The difference between these spectra reaches 10% at the visible range; meanwhile, at the infrared range, the graphs of |Tm|2 are distinguished weakly.
The effective tunneling-assisted energy transport through the periodical gradient nanostructure is formed due to interplay of partial evanescent waves arising at the discontinuities of gradient of refractive index on the boundaries of adjoined nanofilms. Inspection of Figures 2–5 permits to stress out some unusual features of this transport:
a. The discrete series of frequencies providing the complete transmittance (|Tm|2 = 1) of periodical nanostructures in the tunneling-assisted regime under the conditions depicted in Fig. 2. To compare and contrast these peculiar “transparency windows” with the standard transmittance spectrum of waves tunneling through the dispersive medium, it is convenient to recall the abovementioned analogy between the low frequency ω < Ω waves propagation through the gradient film (3) and the tunneling of electromagnetic waves with frequency x through the homogeneous plasma layer with plasma frequency ωp (ω < ωp). The complex transmission coefficient of such film with thickness h deposited on the dielectric wedge-shaped substrate with refractive index n may be written as T = |T| exp (iφt), the values of amplitude |T| and phase φt are
The quantity p was determined in (7); meanwhile, the value ne is coincided with its definition in (7) in a case n0 = 1 due to replacement Ω → ωp; and . Expression (16) describes the monotonic decrease of transmission spectra of tunneling waves |Tm|2 due to diminution of their frequency ω. To the contrary, the diminution of frequency of waves tunneling through the periodical structure results in oscillations of |T|2 with the peak values |T|2 = 1; this tendency is displayed in Fig. 2 for the opaque structures containing both 10 and 20 adjacent films.
b. Strong frequency dependence of transmittance (Fig. 3), determining the frequency-selective properties of nanocoatings in the visible and near infrared spectral ranges, where the dispersion of widely used optical materials is usually insignificant.
c. Deep and narrow minima (Figs. 4 and 5) near by the blue edge of the visible range of transmission spectrum (|T|2 ≤ 5%).
d. Broadband weakly dispersive plateau near by the red edge of visible range and adjoining part of infrared range of transmittance spectra |T|2 (Fig. 4), inherent to the multilayer structures and characterized by high and almost invariable transmittance (|T|2 ≈ 85–95%).
e. Weak dependence of the transmittance of nanostructure in the FTIR regime upon its thickness: thus, the difference in transmittance of nanostructures, containing 7 and 11 gradient nanolayers, is about 1%–3% (Fig. 5); so far, as the characteristic frequency X (5) for the spectra shown in Figs. 4 and 5 relates to the wavelength λ = 1320nm, these spectra can exemplify the effective tunneling-assisted transport of IR radiation through the gradient nanostructure.
These properties (a)–(c) illustrate the possibility to design the frequency-selective all-dielectric radiant metasurfaces, characterized by strong artificial dispersion in a needed spectral range; the properties (d) and (e) can provide the potential for creation of a new family of broadband tunneling-assisted antireflection coatings for the visible and near infrared spectral ranges.
IV. HARTMAN PARADOX REVISITED
Expressions (10)–(13), determining the complex transmission coefficient Tm for the periodical gradient nanostructure, can be written in a form Tm = |Tm| exp (iφt); till now, we were considering only the effects, connected with the square of modulus |Tm|2; meanwhile, the analysis of phase shift φt of wave tunneling through the nanostructure gives rise to the controversial conclusions, concerning the possibilities of superluminal tunneling.20 The phase of wave tunneling through the opaque layer is known to be formed by the phase discontinuities at the layer’s boundaries.21 In a case of homogeneous thick plasma layer ph >> 1, the phase φt (16) tends to the saturation φt → φs, the constant value φs is independent upon the layer width h; herein, the tunneling phase time τp determined as 
is decreasing; thus, substitution of phase φt from (16) to (17) shows that in a case ω << ωp, the time τp is decreasing as exp (–2ωph/c). The fact that the tunneling time for the opaque barrier does not depend on the barrier width gave rise to the paradoxical conclusion (Hartman paradox) that, for the thick barriers, the tunneling speed can become arbitrarily large, exceeding the vacuum light speed c [16,20]. The attempts to record this superluminality in optical tunneling phenomena clash with the contradictive demands: to ensure the saturation of phase φt, one has to use the thick opaque barrier (ph >> 1); however, the fulfillment of this condition impedes the registration of transmitted waves due to the exponential decay of energy flow (16), traversing the opaque barrier .
It is remarkable that this contradiction does not arise for the tunneling effects in the gradient structures under discussion: these structures are shown to provide the weak attenuation of energy flow even for the thick multilayer barriers (Fig. 5). The phase spectra of transmitted waves, calculated for the same conditions, which were used for the measurements of transmittance spectra (Fig. 3), are shown in Fig. 6. Like the amplitude spectra (Fig. 3), the phase spectra are presented in Fig. 6 by means of dimensionless variable u; this presentation ensures the possibility to use the same spectral curves for the different frequencies ω and different thicknesses of nanofilms d, linked by the condition (15). Let us stress out some features of these spectra, useful for the analysis of Hartman paradox for periodical structures formed by m gradient nanofilms, deposited on the transparent substrate:
a. increase of the amount of films m results in the growth of steepness of the phase spectral curves at the same frequency intervals;
b. the spectra of φt become more gently sloping in the area of lesser frequencies, i.e., for the larger frequencies of variable u;
c. (c) the phase shift for the given frequency is rising due to the increase of m, i.e., due to the increase of structure’s thickness h = md, from m = 1 to m = 11 films.
To use these phase spectra for the analysis of superluminality in the tunneling effects, it is convenient to compare the phase time τp (17) with the “light time” τ0, determined as the structure thickness h divided by c. Presentation of time τp via the dimensionless parameter u brings the result
Taking the values of derivatives from the curves at Fig. 6, one can see that in all the spectral range 1 ≤ u ≤ 3 the ratio τp/τ0 remains superluminal: 2 < τp / τ0 < 4; this inequality permits to generalize the concept of superluminality of phase times for the tunneling effects in the gradient structures. Moreover, the almost constant high transmittance in the tunneling regime (Fig. 5) can improve the experimental measurements of speed of tunneling pulse, based on the displacement of pulse peak after tunneling ; using of gradient nanostructures can weaken a “pulse reshaping” process, in which the opaque medium attenuates preferentially the later parts of the incident pulse, and so, the output peak proves to be shifted towards the earlier times. It has to be noted that, in all these experiments, induced by Hartman paradox, the tunneling speed is not a signal speed, and the Einstein causality is not violated.
Generalization of the concept of tunneling-assisted effective transport of radiation supported by evanescent modes in the transparent heterogeneous media is developed. Strong artificial heterogeneity-induced dispersion of transparent gradient all-dielectric lossless nanofilms, determined by the geometry of continuous concave refractive index profiles n(z), is shown to form the spectral range of waves with purely imaginary wave numbers. Interference of evanescent and antievanescent modes in this spectral range provides the weakly attenuated tunneling regime for energy transport through the periodical gradient nanostructures. The general expressions for complex transmission coefficients for the discussed nanostructures, containing an arbitrary amount of adjacent gradient nanofilms, were derived in the framework of exactly solvable model of profile nрzЮ. The experimental measurements of transmittance of structures consisting of 1, 5, and 11 sub-wavelength nanofilms were carried out in the visible and near infrared spectral ranges, and the difference between computational and experimental spectra is shown to be limited by few percents. Some unusual features of radiation tunneling through the all-dielectric gradient nanostructures, including the strong dispersion and weak transmittance nearby the red edge of visible range, almost constant high transmittance in the near infrared range and insignificant dependence of tunneling energy flow upon the multilayer structure thickness, were visualized. Possibility of using of gradient nano-optical structures for the analysis of Hartman paradox is noted. The all-dielectric metamaterials provide a basis for dispersion engineering of media with desired spatial dispersion; the perspectives to employ the tunneling-assisted phenomena in gradient metamaterial nanostructures for elaboration of new miniaturized optical dispersive elements and broadband antireflection coatings are considered.
We appreciate Professor V. G. Veselago for the useful discussions. A.S. thanks Professor N. Engheta, N. Silin, and L. Vazquez for their interest to these researches. This work was supported by the Direction of Scientific/Technical Programs, Project No. 14.579.21.0066, and Far Eastern Federal University, Project No. 14-08-2/3-20.
1 R&D Company "Fotron-Auto," Novodanilovskaya quay 8, Moscow 117105, Russia
2 Joint Institute for High Temperatures Russian Academy of Sciences, Izhorskaya Str., 13/2, Moscow 127412, Russia; Institute of Space Researches Russian Academy of Sciences, Profsouznaya Str. 84/32, Moscow 117997, Russia; Far Eastern Federal University, 8 Sukhanova Str., Vladivostok 690950, Russia
3 M.F.Stelmakh Research Center "Pole," Vvedenskogo Str. 3, Moscow 117342, Russia.