domain: phys spi

text:
Alexander Zharov
email: alexander.zharov@univ-lorraine.fr

Vanessa Fierro

Alain Celzard
email: alain.celzard@univ-lorraine.fr

Resonant absorption in an inhomogeneous disordered metamaterial: first-principles simulation

In this study, we perform first-principles simulations of the resonant excitation of plasma-like oscillations in a two-dimensional inhomogeneous disordered metamaterial. The oscillations are initiated by an oblique incidence of a linearly polarized electromagnetic wave. The conditions for resonant excitation are satisfied near the point where the real part of the effective permittivity of the metamaterial changes sign from positive to negative and crosses zero. First, the problem was analyzed in the framework of the effective medium approximation, which predicts a resonant growth of the electric field and associated resonant absorption near the interface between positive and negative permittivity. It was shown that an array of point dipoles can sustain plasma-like waves near the zero of the effective permittivity. In order to trace the appearance of such a mode at the microscopic scale, full-scale first-principles simulations of a two-dimensional metamaterial with a randomly generated distribution and a linearly increasing average concentration of meta-atoms were performed. The results of the simulations were compared to the predictions of the effective medium approximation. The influence of the fluctuations in the meta-atom concentration on the excitation of the field oscillations and the resonant absorption was quantified.

I. INTRODUCTION

In modern electromagnetics, studies on metamaterials (MMs) occupy an important place as they pave the way for the control of electromagnetic fields at micro-and nanoscale. Since the first studies [START_REF] Smith | Composite medium with simultaneously negative permeability and permittivity[END_REF][START_REF] Shelby | Experimental verification of a negative index of refraction[END_REF] and until today, a variety of types of MMs have been proposed and fabricated, such as linear and non-linear negative-index (or left-handed) MMs [START_REF] Linden | Magnetic response of metamaterials at 100 terahertz[END_REF][START_REF] Zhang | Midinfrared resonant magnetic nanostructures exhibiting a negative permeability[END_REF][START_REF] Lezec | Negative refraction at visible frequencies[END_REF][START_REF] Minovich | Liquid crystal based nonlinear fishnet metamaterials[END_REF][START_REF] Zharov | Nonlinear properties of left-handed metamaterials[END_REF][START_REF] Lapine | Metamaterials with conformational nonlinearity[END_REF], tunable non-linear and elastic MMs [START_REF] Lapine | Magnetoelastic metamaterials[END_REF][START_REF] Slobozhanyuk | Flexible helices for nonlinear metamaterials[END_REF], plasmonic MMs [START_REF] Bames | Surface plasmon subwavelength optics[END_REF][START_REF] Shalaev | Nanophotonics with Surface Plasmons[END_REF][START_REF] Zharov | Control of Light Transmission in a Plasmonic Liquid Metacrystal[END_REF], and artificial hyperbolic media [START_REF] Noginov | Bulk photonic metamaterial with hyperbolic dispersion[END_REF][START_REF] Guo | Applications of hyperbolic metamaterial substrates[END_REF][START_REF] Poddubny | Hyperbolic metamaterials[END_REF]. Inspired by their very special properties, which are not found in natural materials, MMs have found a wide range of applications, including invisibility cloaking [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF][START_REF] Edwards | Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials[END_REF], superlensing [START_REF] Pendry | Negative refraction makes a perfect lens[END_REF][START_REF] Belov | Subwavelength microwave imaging using an array of parallel conducting wires as a lens[END_REF], transformation optics [START_REF] Pendry | Controlling electromagnetic fields[END_REF][START_REF] Leonhardt | Optical conformal mapping[END_REF], and enhancement of chirality and optical activity [START_REF] Tang | Optical chirality and its interaction with matter[END_REF][START_REF] Plum | Optical activity in extrinsically chiral metamaterial[END_REF].

Historically, much attention has been paid to MMs with periodically ordered structural elements, called meta-atoms. Various methods have been formulated to quantify the propagation of electromagnetic waves in such systems [START_REF] Weber | Propagation of optical excitations by dipolar interactions in metal nanoparticle chains[END_REF][START_REF] Park | Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation[END_REF][START_REF] Bankov | Eigenmodes of a 2D array of coupled slot lines in the impedance approximation[END_REF][START_REF] Simovski | Wire metamaterials: physics and applications[END_REF][START_REF] Tihon | Numerically stable eigenmode extraction in 3D periodic metamaterials[END_REF][START_REF] Gorkunov | The role of spatial dispersion near zero points of the dielectric function of cubic and uniaxial crystals[END_REF]. Notably, a variety of homogenization techniques has been developed to simplify the macroscopic description of MMs [START_REF] Sihvola | Electromagnetic Mixing Formulae and Applications[END_REF][START_REF] Liu | First-principle homogenization of magnetodielectric metamaterial arrays[END_REF][START_REF] Silveirinha | Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters[END_REF][START_REF] Gorkunov | Effective magnetic properties of a composite material with circular conductive elements[END_REF][START_REF] Simovski | On electromagnetic characterization and homogenization of nanostructured metamaterials[END_REF]. At the same time, a large class of MMs, including liquid MMs [START_REF] Zhang | Metafluidic metamaterial: a review[END_REF][START_REF] Urzhumov | Plasmonic nanoclusters: a path towards negative-index metafluids[END_REF][START_REF] Zharov | Liquid metacrystals[END_REF], does not suggest periodicity, implying random variations in the parameters of meta-atoms or their spatial distribution. The properties of such disordered MMs have been extensively studied [START_REF] Asatryan | Transmission and Anderson localization in dispersive metamaterials[END_REF][START_REF] Gorkunov | Effect of microscopic disorder on magnetic properties of metamaterials[END_REF][START_REF] Bahcivan | Phase diffusion and random walk interpretation of electromagnetic scattering[END_REF]. Different effective medium approximations for random MMs have been elaborated [START_REF] Mcphedran | Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders[END_REF][START_REF] Guérin | Effective-medium theory for finite-size aggregates[END_REF][START_REF] Papasimakis | Cooperative field localization and excitation eigenmodes in disordered metamaterials[END_REF]. Yet, the examination of the effects beyond the effective medium approximation is attractive from a theoretical point of view.

A particular case of interest is the analysis of electromagnetic systems that show a singular behavior within the effective medium approximations. An example of such a system can be an MM whose permittivity varies continuously. Indeed, near the zero-index transition point, i.e., the point where the effective refractive index crosses zero, anomalous field enhancement and resonant absorption have been predicted [START_REF] Gibson | Resonant absorption in transition metamaterials: parametric study[END_REF][START_REF] Gibson | Transition metamaterials with spatially separated zeros[END_REF]. Similar effects are observed in inhomogeneous plasmas where the permittivity changes sign [START_REF] Denisov | On a singularity of the field of an electromagnetic wave propagated in an inhomogeneous plasma[END_REF].

In this study, we perform first-principles simulations of a resonant excitation of plasma-like oscillations near the transition point in an inhomogeneous disordered two-dimensional (2D) MM. The MM is modeled as an array of randomly distributed meta-atoms, each of which is represented by a 2D ball (disc) with a Drude permittivity. The resonant mode is excited by an oblique incidence of a linearly polarized plane electromagnetic wave on a layer of MM with a linearly increasing average concentration of meta-atoms. As meta-atoms possess a negative polarizability in a certain frequency range above the plasmon resonance, the effective medium theory predicts the emergence of negative effective permittivity at sufficiently high meta-atom concentrations. As a result, with a linear increase in concentration, at a certain point the real part of the effective permittivity changes sign and crosses zero. This leads to the appearance of a resonant collective excitation near the transition point along with the associated resonant absorption of the incident electromagnetic wave.

The paper is organized as follows: In Section II, we define the studied system and analyze it in the effective medium approximation in order to establish a reference for the following simulations. In Section III, we describe the simulations performed for several realizations of the randomly distributed meta-atoms, as well as for the effective medium with the parameters obtained by homogenizing the randomly generated systems. In Section IV, we analyze the results of the simulations, compare them with the effective medium approximation, and discuss the effects beyond the homogenization theory. Finally, the conclusion section summarizes the results.

II. THEORETICAL EVALUATION

The system studied is an inhomogeneous MM with a random distribution of meta-atoms whose concentration is lin-early dependent on a single coordinate. As this system will subsequently be a subject of full-scale numerical simulations, we consider a two-dimensional problem to remain within the limits of the available computational power. This has no qualitative effect and thus no loss of generality. The geometry of the problem is shown in Fig. 1(a). A linearly polarized electromagnetic wave with E x and E y components of the electric field E and H z component of the magnetic field H is obliquely incident on a slab of inhomogeneous MM. The MM is composed of randomly distributed meta-atoms with an average concentration that increases linearly along the x-axis. The meta-atoms are represented by circles and are therefore isotropic in two dimensions. We assume that the meta-atom material has a Drude relative permittivity:

ε MA (ω) = 1 - ω 2 p ω 2 + iγω , (1)
where ω p is the plasma frequency, ω is the frequency, γ is the damping factor, and i is the imaginary unit. Thus, the polarizability per unit length of such particles in the xy transverse plane can be written as follows:

α MA (ω) = 1 4π 2S 0 ε MA (ω) -ε m ε MA (ω) + ε m , (2)
where S 0 = πr 2 0 is the area of a meta-atom of radius r 0 and ε m is the relative permittivity of the surrounding medium. At frequencies higher than the surface plasmon frequency, E x and E y components of the electric field E and H z component of the magnetic field H is incident on a MM with randomly distributed meta-atoms with linearly increasing concentration in the x-direction. The dashed box shows the area occupied by the MM. When randomly seeded, the meta-atoms were prevented from overlapping and touching. The inset schematically shows the zoomed area enclosed by the red rectangle. (b) Qualitative plots of the meta-atom concentration (blue, upper subplot) and the effective permittivity (red, lower subplot). When the concentration reaches some critical value n c , the real part of the effective permittivity crosses zero and becomes negative.

ω sp = ω p / √ 1 + ε m ,
First, let us consider the problem within the continuous medium approximation. Assuming that the radii of the meta-atoms are much smaller than the wavelength, and that the number of meta-atoms on the wavelength scale is much larger than 1, i.e., λ n 1/2 1, where n is the 2D concentration and λ is the wavelength, the meta-atom distribution can be treated as a continuous medium with an effective relative permittivity. As so far we are interested only in qualitative effects, we will use the simplest homogenization technique, namely the Maxwell Garnett (MG) mixing formula, which in 2D takes the form:

ε eff (ω, x) = ε m 1 + ρ (x) π 2 ε MA (ω)-ε m ε MA (ω)+ε m 1 -ρ (x) π 2 ε MA (ω)-ε m ε MA (ω)+ε m , (3)
where ρ (x) = n (x) S 0 is the average filling fraction of metaatoms. Evidently, this formula does not account for the finite size of the meta-atoms, nor for any retardation effects. As follows from Eq. ( 3), in the frequency range where the polarizability is negative, the effective permittivity may also become negative at sufficiently high filling fractions. We will focus on this particular case, when, due to the increasing concentration of meta-atoms along the x-axis, the effective permittivity becomes negative at some point. The problem stated above has a strong resemblance to that of electromagnetic wave propagation in an inhomogeneous plasma, which was solved elsewhere [START_REF] Denisov | On a singularity of the field of an electromagnetic wave propagated in an inhomogeneous plasma[END_REF]. It is indeed clear that the fields in these problems are governed by similar equations:

d 2 u dx 2 - 1 ε eff (x) dε eff (x) dx du dx + k 2 0 ε eff (x) -q 2 u = 0, ( 4
)
where k 0 = ω/c is the wavenumber in vacuum, c is the speed of light; q = sin θ 0 , where θ 0 is the value of the incidence angle. The function u defines the magnetic and electric fields as follows:

H z = u (x) e -iωt+ik 0 qy , (5)
E x = 1 ik 0 ε eff (x) ∂ H z ∂ y , (6)
E y = - 1 ik 0 ε eff (x) ∂ H z ∂ x . ( 7
)
It can be shown that u (x) remains finite at the point x 0 where ε eff crosses zero, which leads to the appearance of singularities in both E x and E y . Namely, E x diverges as 1/ (xx 0 ), while E y has a logarithmic divergence. Evidently, in a lossy material, the fields do not tend to infinity and grow to some maximum value, which is defined by the losses. It should be noted that the resonant growth of the electric field only takes place at the oblique incidence of the electromagnetic wave. Indeed, according to Eq. ( 6), the most divergent component of the electric field, E x , is proportional to q = sin θ 0 , which disappears at normal incidence, θ 0 = 0. Moreover, as shown in [START_REF] Denisov | On a singularity of the field of an electromagnetic wave propagated in an inhomogeneous plasma[END_REF] and references therein, the singular logarithmic term in E y is also proportional to q, and also disappears at normal incidence. As a result, at normal incidence, the resonant increase of the field does not appear. At the same time, at sufficiently high values of the incidence angle θ 0 , the maximum of the electric field is suppressed because of the exponential decay of u(x) between the reflection point defined by ε eff (x)q 2 = 0 and the point of singularity, ε eff (x) = 0. Thus, for further calculations, we have arbitrarily set the value of the incident angle to π/7. Examples of the electric field distribution are shown in Fig. 2.

(

( The associated total resonant absorption W can be calculated by integrating the electromagnetic power loss density, 1/2Re (jE * ), as follows:

W = 1 2 Re (jE * ) dx = 1 2 Re (σ ) |E| 2 dx, ( 8
)
where j is the current density and σ is the effective electrical conductivity of the medium. The integration can be performed by substituting ( 6) and ( 7) into (8) and approximating the real part of the permittivity as a linear function crossing zero. As a result, the total absorption W can be estimated as follows:

W = π 2k 2 0 Re (σ ) Re (dε eff /dx) Im (ε eff ) ∂ H z ∂ y (x = x 0 ) 2 . (9)
Given that the real part of the conductivity can be expressed as Re (σ ) = ωIm (ε eff ), we obtain that Im (ε eff ) in Eq. ( 9) cancels out . Therefore, the resonant absorption does not depend on the imaginary part of the permittivity, which means that even though the losses may have a different origin or value, the total absorption remains the same as long as the other parameters are similar. Such a behavior is common for resonant systems, as changing the losses simultaneously changes the width and the height of the resonant peak while keeping the integral Eq. ( 8) unchanged.

Although the problem statements for plasma and MM are seemingly very similar, the microscopic mechanisms underlying the enhancement of the electric field are different. In plasma, charge separation is possible and is responsible for the increase of the macroscopic electric field near the surface where the real part of the permittivity crosses zero. Indeed, when the real part of the permittivity is zero, the electrostatic plasma oscillations can be resonantly excited, leading to the rapid growth of the electric field. The width of this resonance and the amplitude of the field oscillations are defined by the losses.

On the other hand, in a MM, charge separation is only possible within the meta-atoms, which are assumed small compared to any other scale of the problem. In other words, the MM considered here can be generally treated as an array of point dipoles, which do not allow for macroscopic charge separation. Therefore, the macroscopic increase in the electric field must be attributed to the excitation of a collective dipole mode, which may propagate in the MM. To understand this, consider a 2D MM with randomly distributed meta-atoms. Assuming that the excitation propagates along the x axis as a longitudinal wave, it will induce the polarization of meta-atoms in the x direction with the dipole moment p(r m ) = xp exp (ikx m ), where r m = (x m , y m ) are the coordinates of the mth meta-atom, x represents the unit vector along the x axis, p is the dipole moment value, and k is the wavevector.

A dipole positioned at an observation point r = (x, y) is subject to the electric field produced by all the other dipoles, which can be expressed by the following sum:

E (r) = ∑ m 4δ r m (p(r m )δ r m ) δ r 4 m - 2p(r m ) δ r 2 m , (10)
where δ r m = rr m is the vector connecting the mth dipole and the observation point. Substituting the expression for the dipole moments, the electric field takes the form:

E (r) = xE (r) = xe ikx ∑ m pe -ikδ r m cos φ m 4 cos 2 φ m -2 δ r 2 m , (11)
where φ m is the angle between p(r m ) and δ r m . As we are interested in the field averaged over the spatial variations of the dipole distribution, the summation can be approximated by integration, leading to:

E (r) = 2pne ikx ∞ l δ rdδ r 2π 0 dφ e -ikδ r cos φ 2 cos 2 φ -1 δ r 2 , ( 12
)
where n is again the concentration of meta-atoms, i.e., the number of meta-atoms per unit area, and l = n -1/2 is the average distance between the meta-atoms, which plays the role of cut-off distance. The integration thus gives:

E (r) = -4π pne ikx J 1 (k l) k l , (13)
where J 1 (z) is the first-order Bessel function.

Finally, the self-consistency restriction requires that p(r) = α MA (ω) E (r), which provides us with the condition for the existence of a special eigenmode, a self-correlated longitudinal dipole wave analogous to plasma wave:

1 + ρ ε MA (ω) -ε m ε MA (ω) + ε m J 1 (k l) πk l = 0. ( 14
)
It can be seen that the propagation of such waves is governed solely by spatial dispersion, while otherwise they reduce to simultaneous dipole oscillations in the entire space. Nevertheless, in the non-dispersive limit k l → 0, Eq. ( 14) reduces to 1 + ρ π 2 ε MA (ω)-ε m ε MA (ω)+ε m = 0, which reads exactly as ε eff = 0. Thus, the increase in the electric field near the transition point is attributed to the resonant excitation of plasma-like longitudinal waves sustained by the dipole oscillations.

III. SIMULATION

The simulations were performed using COMSOL Multiphysics (Electromagnetic Waves, Frequency Domain solver). The system was modeled by randomly seeding 2485 circular meta-atoms with linearly increasing concentration on a rectangle with an aspect ratio of 2.43. When randomly seeded, the particles were prevented from overlapping and touching. The permittivity of the meta-atoms was simulated by the Drude model. The angle of incidence was set to π/7. FIG. 3. Electric field distribution in the MM system shown in Fig. 1 illuminated by an electromagnetic wave at different frequencies. The wave polarization and orientation of axes are the same as in Fig. 1. The frequencies were set to ω = 0.773ω p ; 0.785ω p ; 0.798ω p ; 0.810ω p ; 0.823ω p ; 0.836ω p , as indicated in the upper left corner of each sub-image. The incidence angle was set to π/7. The number of meta-atoms is 2485. The meta-atom distribution was randomly generated with a concentration increasing linearly in the x-direction; the meta-atoms were prevented from overlapping and touching. The color bar represents the absolute value of the electric field normalized by the incident wave amplitude. It should be noted that the color scale is limited to 50 for ease of reading the figure, while the actual maximum of the normalized electric field reaches approximately 250.

Figure 3 shows the electric field distribution at different frequencies. It can be seen that at each frequency there is an area of maximum average field, which corresponds to the satisfaction of condition [START_REF] Noginov | Bulk photonic metamaterial with hyperbolic dispersion[END_REF].

The dependence of the field distribution on the real-time frequency change is available in the Supplemental Material [48].

Figure 4 shows the frequency dependence of the transmission and reflection coefficients averaged over 5 random realizations and the absorption as a function of the damping factor at different frequencies. The irregular behavior of the reflection and transmission in Fig. 4(a) is evidently a consequence of the random character of the considered system. The absorption, Fig. 4(b), depends weakly on the damping factor of the meta-atom material and varies between 0.6 and 0.8 depending on the frequency.

It is interesting to trace the similarities between the excited modes in the homogenized and random systems with similar average parameters. The comparison between the simulations of the MG approximation and the random system at different frequencies is shown in Fig. 5. FIG. 5. Comparison between the field distributions calculated for the random system (a1,a2,a3,a4) and the MG approximation (b1,b2,b3,b4). The frequencies were set to ω = 0.792ω p (a1,b1), ω = 0.804ω p (a2,b4), ω = 0.823ω p (a3,b3), and ω = 0.842ω p (a4,b4). The other parameters are similar to those in Fig. 3.

Figure 6 shows the comparison of the averaged electric field distributions for MG approximation and random distribution. The averaging was performed over the entire y-coordinate and with Gaussian smoothing function over the x-coordinate.

Finally, to verify that the simulation results do not qualitatively depend on a particular realization of the meta-atom dis-FIG. 6. Comparison between the averaged field distributions calculated for the random system (blue solid line) and the MG approximation (red dot-dashed line) for four different frequencies, ω = 0.792ω p (a), ω = 0.804ω p (b), ω = 0.823ω p (c) and ω = 0.842ω p (d). The averaging was performed over the entire y-coordinate and with Gaussian smoothing function over the x-coordinate. The other parameters are similar to those in Fig. 5. tribution, several random distributions generated by the same randomization procedure were simulated. The comparison of these simulations is shown in Fig. 7. The averaged field distributions in the random systems shown in Fig. 7 are presented in Fig. 8.

IV. DISCUSSION

The problem considered here is of interest as it sheds light on how collective modes appear in random MMs without turning to effective medium approximations. First, as can be seen in Fig. 3, with the increase of the electromagnetic wave frequency, the resonant region moves to the areas of higher meta-atom concentration. This observation is in agreement with the condition Eq. ( 14) with the permittivity given by Eq. ( 1). Furthermore, Figs. 5 and6 demonstrate similar trends between the resonance excitations in the random and homogenized systems. Second, although the resonant region is clearly visible, it is not very pronounced, unlike one might have expected from Fig. 2. However, it is clear that in a system constituted by discrete particles, field cannot be localized in a region smaller than the interparticle distance. Moreover, as the considered field enhancement is maintained by a collective excitation of dipoles, a significant number of dipoles is required to sustain it. The comparison presented in Fig. 5 shows that randomness plays a crucial role in the widening of the resonant region.

The widening of the resonant area is evidently attributed to the breakdown of the continuous medium approximation. Indeed, when the localization of the field reaches certain intrinsic scales of the material, the continuous medium approximation is no longer applicable. An evident intrinsic scale is the average interparticle distance. However, for the random distribution of meta-atoms, the resonant area is much wider than the average distance between the meta-atoms. It turns out that the factor that defines the size of the resonance region in this case is the fluctuations of the meta-atom concentration. Indeed, let us write the meta-atom filling fraction in the following form:

ρ (x, y) = ρ (x) + δ ρ (x, y) , (15)
where ρ (x) is the average filling fraction, which grows linearly in the x-direction, and δ ρ (x, y) is the local deviation of the filling fraction from the average. Therefore, according to Eq. ( 3), the effective permittivity can also be expressed in a similar form:

ε eff (x, y) = ε eff (x) + δ ε eff (x, y) , (16)
where ε eff (x) is the average effective permittivity and δ ε eff (x, y) is its deviation from the average. Using Eq. ( 6) and ( 7), the expression for the squared electric field amplitude can be written as:

|E| 2 = |E x | 2 + E y 2 = 1 k 2 0 |ε eff (x, y) | 2 ∂ H z ∂ x 2 + ∂ H z ∂ y 2 . ( 17
)
As we are interested in the general properties of the random system, the expression for the electric field amplitude, Eq. ( 17), should be averaged over the random realizations of the distribution. Thus, the average amplitude will take the form:

|E| 2 = 1 k 2 0 |ε eff (x, y) | 2 ∂ H z ∂ x 2 + ∂ H z ∂ y 2 , ( 18
)
where the angular brackets .. represent the averaging. It is easy to see that the amplitude is maximal when the averaged permittivity is minimal. The averaged square of the effective permittivity can be rewritten as follows:

|ε eff (x, y) | 2 = (Reε eff (x)) 2 +(Imε eff (x)) 2 + (δ ε eff (x, y)) 2 . ( 19
)
It takes its minimal value when Reε eff (x 0 ) 2 = 0, which corresponds to the condition Eq. ( 14). Thus, the maximum value of the field in the resonance region as well as its size are determined by the imaginary part of the effective permittivity and the average absolute value of the deviation of the effective permittivity from its mean. It is noteworthy that the average deviation of the permittivity, i.e., the fluctuations of the metaatom concentration, plays a similar role to that of the imaginary part of the effective permittivity. In other words, the meta-atom density fluctuations operate as additional losses, decreasing the maximum field value in the resonance region and increasing the size of this region.

At the same time, the resonant absorption in the disordered system is still defined by Eq. ( 9) with Re (σ ) = ωIm (ε eff ). Thus, although the concentration fluctuations act like losses on the height and width of the resonance, they do not contribute to the absorption. As a result, the imaginary part of the permittivity does not cancel, and the following factor appears in the expression for the total losses:

W ∝ Imε eff (x) Imε eff (x) 2 + (δ ε eff (x, y)) 2 . ( 20
)
Therefore, fluctuations in the meta-atom concentration reduce the total resonant absorption.

To estimate the widening of the resonant area, it is possible to find the standard deviation of the permittivity. If the positions of the meta-atoms are assumed to be uncorrelated, the fluctuations in the number of meta-atoms in a given area S is √ N, where N = ρS/S 0 is the average number of particles in this area. Thus, the filling fraction distribution can be assumed Gaussian with the distribution function f (ρ) = A exp -(ρ-ρ) 2 ρ S S 0 . The standard deviation of the permittivity can be found as follows:

δ ε 2 = ∞ 0 f (ρ)(ε eff (ρ) -ε eff ) 2 dρ, (21)
where ε eff (ρ) is given by Eq. ( 3). Applied to the considered system, Eq. ( 21) allows estimating the widening of the resonance region as Imε 2 + δ ε 2 /Imε, which gives the ratio of the width of the resonant region in the random system to that in the homogenized system. The comparison between the widening of the resonance region calculated using Eq. ( 21) and that obtained using first-principles simulations is shown in Fig. 9. Despite the evaluative nature of the estimates, they are quite consistent with the numerical calculations.

FIG. 9. Widening of the resonance region due to fluctuations in the meta-atom concentration. The theoretical estimate obtained by calculating the standard deviation of the permittivity, Eq. ( 21), is represented by the blue line; the widening obtained by direct simulation of the random system is represented by the red circles.

As the relative fluctuations in the number of meta-atoms are proportional to the inverse square root of the number of meta-atoms itself, δ N/ N ∝ 1/ √ N, the larger the number of meta-atoms is in a given area, the smaller are the relative fluctuations. In other words, if the number of meta-atoms is increased while keeping the filling fraction constant, the role of fluctuations decreases. Finally, in the effective medium limit N → ∞, the influence of the fluctuations tends to zero.

Another distinctive feature of the considered system, directly related to the concentration fluctuations, is the appearance of the hot-spot pattern outside the resonant region. Two examples of such hot spots are shown in Fig. 10.

Such a picture can be explained by the fact that, due to random fluctuations in the concentration, the condition Eq. ( 14) can be satisfied at some random spots sufficiently far from the line where the real part of the average permittivity crosses zero. As a result, at these spots, resonant field excitation takes place.

From a practical point of view, systems similar to the one considered here might be advantageous as wide-range electromagnetic absorbers. Indeed, the position of the resonance region where the absorption takes place is determined by condition Eq. ( 14), which links the frequency and the local concentration. As a result, with the change of frequency in a certain range, the resonance region changes its position in the layer of continuous variation of meta-atom concentration, thus constantly providing resonant absorption. According to Fig. 4(a), the considered layer provides significant suppression of both reflected and transmitted radiation in a fairly wide range of FIG. 10. Appearance of hot spots in a random distribution of metaatoms at two different frequencies, ω = 0.773ω p and ω = 0.798ω p . The hot spots are marked by with white circles. The light-gray dashed line represents the line at which the condition Eq. ( 14) is satisfied. The other parameters are similar to those in Fig. 3. frequencies. In particular, reflection is suppressed by 10 -18 dB, while transmission is suppressed by 4 -11 dB. It should be noted that the thickness of the MM layer is smaller than the wavelength, ranging between 0.68 and 0.85 of the wavelength depending on the frequency. As shown in Eq. ( 20), the absorption can be improved by reducing the fluctuations in the effective relative permittivity, i.e., by increasing the number of meta-atoms while simultaneously decreasing their size.

V. CONCLUSION

In this study, first-principles simulations of an inhomogeneous 2D MM with a random distribution of meta-atoms were performed. In particular, a resonant excitation of plasma-like oscillations by an oblique incidence of a linearly polarized electromagnetic wave was demonstrated. Within the effective medium approximation, such an excitation appears near the surface where the real part of the effective permittivity crosses zero. It has been shown that although the effective medium approximation becomes inadequate near the transition point, it still produces qualitatively reasonable predictions about the existence and position of the resonant mode. While the MG approximation fails to produce the width of the resonant area, analysis of the statistical properties of the system allows for accurate estimates. It has been shown that fluctuations in the meta-atom concentration in a random distribution act as additional losses. The simulations showed that the randomness of the MM causes electric field hot spots to appear outside the resonant region. The fluctuations in the meta-atom concentration can lead to an accidental satisfaction of the resonant excitation conditions of plasma-like oscillations at some point, producing hot spots.

VI. ACKNOWLEDGEMENTS

This study was supported by the French PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE and TALiSMAN project, funded by ERDF (2019-000214).

  the polarizability becomes negative (see Fig. 1(b)).

FIG. 1 .

 1 FIG. 1. (a) Schematic picture of the simulated system. An electromagnetic wave with E x and E y components of the electric field E and H z component of the magnetic field H is incident on a MM with randomly distributed meta-atoms with linearly increasing concentration in the x-direction. The dashed box shows the area occupied by the MM. When randomly seeded, the meta-atoms were prevented from overlapping and touching. The inset schematically shows the zoomed area enclosed by the red rectangle. (b) Qualitative plots of the meta-atom concentration (blue, upper subplot) and the effective permittivity (red, lower subplot). When the concentration reaches some critical value n c , the real part of the effective permittivity crosses zero and becomes negative.

FIG. 4 .

 4 FIG. 4. (a) Reflection (blue solid line) and transmission (red dotdashed line) coefficients of the MM layer as a function of frequency averaged over 5 random realizations. Dashed lines represent reflection and transmission of particular random realizations. (b) Absorption as a function of the damping factor γ at 3 different frequencies.

FIG. 7 .

 7 FIG. 7. Comparison between 3 different realizations of random distribution at two different frequencies, ω = 0.785ω p (a1,a2,a3) and ω = 0.842ω p (b1, b2, b3). The other parameters are similar to those in Fig. 3.

FIG. 8 .

 8 FIG. 8. Comparison between averaged field distributions calculated for the 3 realizations of random systems shown in Fig. 7.


text:
Alexander Zharov
email: alexander.zharov@univ-lorraine.fr

Vanessa Fierro

Alain Celzard
email: alain.celzard@univ-lorraine.fr

Resonant absorption of electromagnetic waves by an induced inhomogeneity in a liquid metamaterial

In this study, we consider the scattering of a plane electromagnetic wave on a cylindrically inhomogeneous distribution of meta-atoms in a liquid metamaterial. The distribution inhomogeneity appears naturally as a result of the application of an electric potential to an electrode immersed in the liquid metamaterial, which leads to the appearance of inhomogeneity in the effective permittivity of such a medium. Consequently, in the considered type of metamaterial, the concentration-dependent effective permittivity can become negative in some region of space. In the vicinity of the interface between positive and negative permittivity, the behavior of the electromagnetic wave becomes singular, giving rise to effects such as resonant absorption, resonant scattering, and shielding of the central region.These effects can be controlled in real time by changing the applied electric potential and thus rearranging the liquid metamaterial.

I. INTRODUCTION

Since the first studies [START_REF] Smith | Composite medium with simultaneously negative permeability and permittivity[END_REF][START_REF] Shelby | Experimental verification of a negative index of refraction[END_REF], the primary motivation for developing and manufacturing metamaterials (MMs) has been the ability to control electromagnetic fields at micro-and nanoscale. In general, the peculiar electromagnetic properties of MMs are conditioned by the special design of their structural elements, called meta-atoms. During the years of MM research, different types of MMs have been studied and fabricated, such as linear and non-linear negative-index (or left-handed) MMs [START_REF] Linden | Magnetic response of metamaterials at 100 terahertz[END_REF][START_REF] Zhang | Midinfrared resonant magnetic nanostructures exhibiting a negative permeability[END_REF][START_REF] Lezec | Negative refraction at visible frequencies[END_REF][START_REF] Minovich | Liquid crystal based nonlinear fishnet metamaterials[END_REF][START_REF] Zharov | Nonlinear properties of left-handed metamaterials[END_REF][START_REF] Lapine | Metamaterials with conformational nonlinearity[END_REF], tunable non-linear, elastic, and disordered MMs [START_REF] Lapine | Magnetoelastic metamaterials[END_REF][START_REF] Slobozhanyuk | Flexible helices for nonlinear metamaterials[END_REF][START_REF] Liu | Bendable disordered metamaterials for broadband terahertz invisibility[END_REF][START_REF] Boddeti | Spectral domain inverse design for accelerating nanocomposite metamaterials discovery[END_REF], plasmonic MMs [START_REF] Bames | Surface plasmon subwavelength optics[END_REF][START_REF] Shalaev | Nanophotonics with Surface Plasmons[END_REF], and so-called artificial hyperbolic media [START_REF] Noginov | Bulk photonic metamaterial with hyperbolic dispersion[END_REF][START_REF] Guo | Applications of hyperbolic metamaterial substrates[END_REF][START_REF] Poddubny | Hyperbolic metamaterials[END_REF]. Depending on the metaatom design, these materials can exhibit their distinctive properties at frequencies ranging from microwaves to optical. As a result, MMs have found applications in invisibility cloaking [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF][START_REF] Edwards | Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials[END_REF], superlensing [START_REF] Pendry | Negative refraction makes a perfect lens[END_REF][START_REF] Belov | Subwavelength microwave imaging using an array of parallel conducting wires as a lens[END_REF], transformation optics [START_REF] Pendry | Controlling electromagnetic fields[END_REF][START_REF] Leonhardt | Optical conformal mapping[END_REF], and enhancement of chirality and optical activity [START_REF] Tang | Optical chirality and its interaction with matter[END_REF][START_REF] Plum | Optical activity in extrinsiccally chiral metamaterial[END_REF].

Amongst others, liquid MMs [START_REF] Zhang | Metafluidic metamaterial: a review[END_REF] are distinguished by their high tunability and reconfigurability. In particular, nanocluster plasmonic negative index metafluids [START_REF] Urzhumov | Plasmonic nanoclusters: a path towards negative-index metafluids[END_REF][START_REF] Fruhnert | Towards negative index self-assembled metamaterials[END_REF], colloidal MMs [START_REF] Golovin | Electrically reconfigurable optical metamaterial based on colloidal dispersion of metal nano-rods in dielectric fluid[END_REF], hybrid liquid crystal-based colloids [START_REF] Liu | Self-Alignment of Plasmonic Gold Nanorods in Reconfigurable Anisotropic Fluids for Tunable Bulk Metamaterial Applications[END_REF], and liquid metacrystals [START_REF] Zharov | Liquid metacrystals[END_REF][START_REF] Zharov | Spontaneous reorientations of meta-atoms and electromagnetic spatial solitons in a liquid metacrystal[END_REF][START_REF] Liu | Tunable meta-liquid crystals[END_REF][START_REF] Zharov | Control of Light Transmission in a Plasmonic Liquid Metacrystal[END_REF] have been studied theoretically and experimentally. The simplest liquid MM can be imagined as an array of identical isotropic particles suspended in a viscous liquid. If the size of the particles is well below the wavelength, then such a suspension can be treated as an effective continuous medium with effective macroscopic permittivity and permeability. These macroscopic effective parameters, however, are defined by the microscopic properties of each meta-atom. Even the most obvious meta-atom design, a homogeneous sphere, can lead to non-trivial behavior. For instance, the electrical polarizability of metallic particles can become negative above the plasmon resonance, while in dielectric particles, either the magnetic or the electrical polarizability can be negative due to Mie resonances. As a result, when the meta-atom concentration is sufficient, the effective permittivity (or permeability) can become negative. Moreover, by using the spe-cial meta-atom design, negative permittivity and permeability can be achieved simultaneously [START_REF] Zharov | All-dielectric bulk isotropic double-negative metamaterials[END_REF].

A particular case of interest is the behavior of an electromagnetic wave in an MM with continuously varied permittivity, especially near the zero-index transition point, i.e., the point where the refractive index crosses zero. MMs with linear index transition have been studied in [START_REF] Boddeti | Spectral domain inverse design for accelerating nanocomposite metamaterials discovery[END_REF][START_REF] Gibson | Resonant absorption in transition metamaterials: parametric study[END_REF][START_REF] Gibson | Transition metamaterials with spatially separated zeros[END_REF]. Notably, anomalous field enhancement and resonant absorption in the vicinity of the transition point were predicted. Similar effects are observed in inhomogeneous plasmas with the permittivity changes sign [START_REF] Denisov | On a singularity of the field of an electromagnetic wave propagated in an inhomogeneous plasma[END_REF].

In this study, we analyze the scattering of a plane electromagnetic wave on a cylindrically symmetric inhomogeneity of a liquid MM. The inhomogeneity appears naturally as a result of the electrophoretic pull of meta-atoms in the region of stronger electric field, which is controlled by applying an electric potential to a metallic electrode immersed in the MM. The meta-atoms are treated as spherical particles made of a Drude material, thus having a negative polarizability in a certain frequency range. Therefore, in the region of the highest meta-atom concentration, the effective permittivity can become negative, which leads to a singular behavior of the electromagnetic field near the transition from positive to negative permittivity. This gives rise to effects such as resonant absorption, resonant scattering, and shielding of the central region.

II. MODEL DESCRIPTION

As a model problem, we consider a transparent container filled with a liquid MM with a cylindrical electrode immersed in it, as shown in Fig. 1. We assume that the size of the container is much larger than any other scale in the problem. In particular, in what follows, we will consider the scattering of a plane electromagnetic wave on the metamaterial inhomogeneity induced by the immersed electrode, and we will not take into account the influence of the outer boundary of the container. In other words, we will neglect the scattering on the external surfaces of the container and focus solely on its inner part. The MM is regarded as a suspension of subwavelength spherical particles, the meta-atoms, which possess isotropic negative polarizability in a certain frequency range. The metaatoms are considered non-interacting, i.e., we assume that they do not tend to agglomerate and we neglect the dipoledipole interaction between them. As long as the conditions of the effective medium approximation are fulfilled, the exact nature of the meta-atoms is not relevant here; for example, the meta-atoms can be metallic or dielectric spheres. In the first case, the negative polarizability can be achieved at frequencies above the plasmon resonance, while in the second case the negative polarizability can be caused by Mie resonances. We assume that the meta-atom size is much smaller than the wavelength λ , and the concentration of meta-atoms n satisfies the condition n 1/3 λ 1, i.e., the number of particles in a wavelength scale is much greater than one. Thus, the MM can be treated as an effective continuous medium with an effective relative permittivity ε eff . In general, ε eff can be calculated as follows [START_REF] Malasi | From Mie to Fresnel through effective medium approximation with multipole contributions[END_REF]:

ε eff = ε h - 3νε h 2q 3 [∑ N j=1 i(2 j + 1)(a j + b j )] 1 + ν 2q 3 [∑ N j=1 i(2 j + 1)(a j + b j )] (1)
where ε h is the relative permittivity of the host medium, q = 2π √ ε h λ -1 is the size parameter of a meta-atom, ν = nV 0 is the volume fraction of meta-atoms; n is the concentration of meta-atoms and V 0 is the volume of a single meta-atom, i is the imaginary unit, a j and b j are the Mie coefficients of j th order, and N is the cutoff number. However, in the present work, we will restrict ourselves to considering a particular case of metaatoms made of a Drude material near the plasmon resonance. In this case, the electric dipole Mie coefficient is dominant over the other multipoles, i.e., a 1 b 1 , a 2 ... Therefore, Eq. (1) can be simplified to the well-known Maxwell Garnett approximation:

ε eff = ε h 1 + 3ν ε MA -ε h ε MA (1 -ν) + ε h (2 + ν) , (2)
where ε MA is the relative permittivity of the meta-atoms. When no voltage is applied to the immersed electrode, i.e., when the potential of the electrode is equal to that which is remote from the electrode and the electric field is zero, the meta-atoms in thermal equilibrium are distributed uniformly in space. This corresponds to a uniform concentration and, consequently, to a homogeneous effective permittivity. On the other hand, the non-zero potential u applied to the immersed electrode induces an electric charge in it and, therefore, gives rise to the cylindrically symmetric static electric field in the vicinity of the electrode, which can be written as:

E (ρ) = ρ 0 ql -1 2πε 0 ε(ρ) 1 ρ = ρ 0 u log (l/r i )ε(ρ) 1 ρ , (3)
where E (ρ) is the electric field, ρ 0 is the unit vector in the radial direction in the cylindrical coordinate system (ρ, φ , z), ρ is the corresponding radial coordinate, q is the electric charge of the electrode, ε 0 is the permittivity of vacuum, ε(ρ) is the static relative permittivity of the medium at the coordinate ρ, u is the applied potential measured with respect to zero potential at infinity, and l and r i are the length and the radius of the electrode. The electric charge q induced by the voltage u was found to take the form q = Cu ≈ 2πε 0 lu/ log (l/r i ), where C is the capacitance of an isolated conductive cylinder taken in the limit of an infinite length-to-radius ratio [START_REF] Butler | Capacitance of a finite-length conducting cylindrical tube[END_REF]. Equation ( 3) is valid for ρ l, i.e., when the electrode can be treated locally as an infinite cylinder. Thus, the problem will be further treated as two-dimensional in the plane perpendicular to the cylinder axis. The electric field induces dipole moments in meta-atoms, which, in turn, interact with the electric field, producing the electrophoretic force F EF :

F EF = (p∇) E = ρ 0 α 0 E ∂ E ∂ ρ , ( 4
)
where p is the dipole moment of a meta-atom, α 0 is the polarizability of a meta-atom, and E is the magnitude of the electric field at the meta-atom position. Substituting Eq. (3) in Eq. ( 4), we obtain:

F EF = -ρ 0 α 0 u log (l/r i )ε(ρ) 2 1 ρ 3 . (5)
Evidently, the electrophoretic force attracts the meta-atoms to the central electrode, making their concentration inhomogeneous. It should be noted that the force does not depend on the sign of the potential difference or the direction of the electric field vector. The particles are always attracted to the region where the field is the highest. The equilibrium distribution of meta-atoms can be found by applying the Gibbs formula:

f (ρ) = A 0 exp - U (ρ) k B T , (6)
where f (ρ) is the meta-atom distribution function, U (ρ) is the potential energy of a meta-atom in the field of the electrophoretic force given by Eq. ( 5), k B is the Boltzmann constant, and T is the temperature. The normalization constant A 0 is defined from the normalization condition:

V f (ρ) d 3 r = N, (7)
where V is the volume of the container and N is the total number of meta-atoms. With this normalization, the distribution function readily provides the local concentration n (ρ) as a function of ρ: n (ρ) = f (ρ). The potential energy U (ρ) can be calculated as follows:

U (ρ) = -pE = -α 0 u log (l/r i )ε(ρ) 2 1 ρ 2 . (8)
Thus, the distribution function and, therefore, the radial dependence of the meta-atom concentration, takes the form:

f (ρ) = n (ρ) = A 0 exp α 0 k B T u log (l/r i )ε(ρ) 2 1 ρ 2 . (9)
The evident complication arises from the fact that the local relative dielectric permittivity ε(ρ) is itself affected by the redistribution of meta-atoms governed by Eq. ( 9). Indeed, the local permittivity in the considered statement of the problem is simply the effective relative permittivity ε eff defined by Eq. ( 2), which contains the volume fraction ν, which, in turn, is specified by the local meta-atom concentration. Naively, the local effective relative permittivity could be approximated by its unperturbed value. In such a case, the applicability conditions have to be supplemented by an additional restriction on the maximum concentration, limiting the deviation of the effective permittivity from its unperturbed value due to the change in the meta-atom concentration. On the other hand, this restriction may contradict the need to obtain a negative effective permittivity at high frequency. Indeed, due to the frequency dispersion, for some types of meta-atoms, it is possible that the perturbation of the static permittivity is small, while its high-frequency variation is not. However, for metaatoms made of a material with Drude permittivity, which we will explicitly consider in this paper, the meta-atom contribution to the effective permittivity at zero frequency is never less than that at any finite frequency. Therefore, the metaatom distribution was found self-consistently using numerical methods.

It should be noted that the derived distribution is only valid when the concentration does not reach the close-packing values. Otherwise, an interparticle interaction term must be introduced into the energy, and a more complicated problem has to be solved to derive the distribution.

In the considerations above, the gravitation effects on the redistribution of meta-atoms have been neglected. To find the applicability conditions of such an approximation, the influences of gravity and electric field on the meta-atoms have to be compared. This can be done by comparing the potential energy given by Eq. ( 8) with the gravitational potential energy U G = m MA gl, where m MA is the mass of a meta-atom, g = 9.8 m/s 2 is the free-fall acceleration, and l, being the length of the electrode, represents the characteristic scale of the system. The straightforward estimation shows that if we limit the voltage to a maximum of 600 V, the maximum radius of the electrode when the gravitation can be neglected is r i ≈ 100 µm.

Distribution (9) may be rewritten in a more convenient way as

f (ρ) = n (ρ) = A 0 exp u u 0 2 1 (ε eff (ρ)/ε h ) 2 1 (ρ/r i ) 2 , ( 10
) where u 2 0 = k B Tr 2 i α -1 0 ε 2 h log 2 (l/r i )
is some characteristic voltage of the system. By definition, it determines the voltage at which the electrophoretic force predominates over the thermal fluctuations. For instance, in a system at room temperature, 300 K, with the electrode radius of 1 µm and 10-nmradius meta-atoms, the characteristic voltage is u 0 ≈ 39.5 V. On the other hand, the form of Eq. ( 10) determines the scaling properties of the meta-atom distribution. Indeed, as far as the ratio u/u 0 and the host permittivity ε h are fixed, the metaatom distribution over the dimensionless coordinate ρ/r i remains the same, no matter what the particular parameters of the system are.

The radial dependence of the effective permittivity ε eff (ρ) was obtained as a self-consistent solution of Eqs. ( 10) and ( 2) with the normalization condition Eq. ( 7). The possible effective permittivity profiles at different voltages are shown in Fig. 2. For definitiveness, we supposed that the meta-atoms are spheres made of a Drude material whose permittivity is given by ε MA (ω) = 1ω 2 p / ω 2 + iγω , where ω is the electromagnetic wave frequency, ω p is the plasma frequency, and γ is the damping factor, which defines the losses in the metaatoms material. We will focus on the cases where the real part of the permittivity changes sign and becomes negative at some point. Evidently, the region with negative real part of the effective permittivity appears when the voltage reaches some threshold value, which is determined by the parameters of the system. Consider a plane electromagnetic wave with one component of magnetic field, H z , and two components of electric field, E ρ and E φ , scattering on the inner electrode and the inhomogeneity of permittivity induced by it. Maxwell's equations allow us to derive the equation governing H z as follows:

ρ 2 ∂ 2 H z ∂ ρ 2 +ρ 1 -ρ ε eff (ρ) ε eff (ρ) ∂ H z ∂ ρ + ∂ 2 H z ∂ φ 2 +k 2 0 ε eff (ρ) ρ 2 H z = 0, (11)
where ε eff (ρ) = dε eff (ρ) /dρ is the derivative of the effective permittivity ε eff (ρ) and k 0 = ω/c is the free-space wavevector; ω is the frequency of the electromagnetic wave and c is the speed of light. The components of the electric field can be found as follows:

E ρ = 1 ik 0 ε eff (ρ) 1 ρ ∂ H z ∂ φ , (12)
E φ = - 1 ik 0 ε eff (ρ) ∂ H z ∂ ρ , ( 13
)
where i is the imaginary unit. Equations (11-13) must be completed by the boundary condition on the central electrode. Assuming that the electrode is a perfect electrical conductor, the boundary condition would state the vanishing of the tangential component of the electric field at the electrode surface, i.e., E φ | ρ=r i = 0. However, in the case where the real part of the permittivity crosses zero, the electric field decreases exponentially in the region of negative permittivity. As a result, near the electrode surface, the electric field is very close to zero, and scattering from the electrode itself is negligibly small. In other words, the boundary condition is automatically satisfied in this case, because the electric field (including its tangential component) is already zero at the electrode surface.

After substituting

H z (ρ) = ∑ ∞ m=-∞ H (m) z (ρ) exp (imφ )
, where m is an integer, and changing the variable as ξ = k 0 ρ, Eq. ( 11) takes the form:

d 2 H (m) z dξ 2 + 1 ξ - ε eff (ξ ) ε eff (ξ ) dH (m) z dξ + ε eff (ξ ) - m 2 ξ 2 H (m) z = 0 (14)
with the electric field components expressed as:

E (m) ρ = 1 ε eff (ξ ) m ξ H (m) z , (15)
E (m) φ = - 1 iε eff (ξ ) ∂ H (m) z ∂ ξ . (16)
Equation ( 14) does not allow for an analytical solution for an arbitrary ε eff (ξ ). However, certain field characteristics can be found.

First, it can be seen that Eq. ( 14) has a singular point ξ 0 , at which the effective permittivity becomes zero, ε eff (ξ 0 ) = 0. The two linearly independent solutions of Eq. ( 14) in the vicinity of this point can be found as:

H (m) z ∝ ε eff (ξ ) ξ M κ m ,1 κ m 2 ξ -ξ 0 ξ 0 , ε eff (ξ ) ξ W κ m ,1 κ m 2 ξ -ξ 0 ξ 0 , (17)
where κ m = 2 √ 4m 2 -1 and M κ,µ (a) and W κ,µ (a) represent the Whittaker functions of the arbitrary arguments κ, µ, and a. These solutions suggest that the magnetic field remains finite in the vicinity of the singular point. As a result, the radial component of the electric field is singular at ξ 0 for all harmonics except m = 0, since

E (m) ρ (ξ 0 ) ∝ H (m) z (ξ 0 ) /ε eff (ξ 0 ) → ∞.
The behavior of the tangential component of the electric field turns out to be logarithmically divergent. Such a singularity is induced by a plasma-like resonance in the region where ε eff = 0. Evidently, when the meta-atoms are lossy, i.e., when the imaginary part of the effective permittivity is not zero, the electric field does not tend to infinity, but grows to some value defined by the losses. Consequently, when the imaginary part of the effective permittivity is small, i.e., when Imε eff Reε eff , the overall losses in the resonant region do not depend on the actual value of Imε eff . Indeed, the local electromagnetic loss density is proportional to Imε eff |E| 2 , the maximum of |E| 2 is proportional to (Imε eff ) -2 , and the width of the resonance is proportional to Imε eff . Therefore, the total electromagnetic losses, defined by the integral of the loss density over the resonance region, do not depend on Imε eff . It should be emphasized that this resonance is a collective effect, which is not directly related to the plasmon resonance in a single particle. The considered resonance appears as a result of the crossing to zero of the effective permittivity and does not rely on any specific physical mechanism responsible for this crossing.

To analyze the scattering, it is useful to introduce the transverse impedance Z ⊥ = E ⊥ /H ⊥ ≡ E φ /H z . For each harmonic m, the impedance is governed by the following equation:

dZ (m) ⊥ dξ + 1 ξ Z (m) ⊥ + iε eff (ξ ) Z (m) ⊥ 2 -Z 2 0 (m, ξ ) = 0, (18)
where

Z 2 0 (m, ξ ) = ε -2 eff (ξ ) ε eff (ξ ) -m 2 /ξ 2 .
First, by knowing the solutions of Eq. ( 18), it is easy to find the scattering coefficients for each m and, therefore, the crosssection. The solutions of Eq. ( 18) can be found numerically, applying the boundary condition Z (m) ⊥ = 0 at the surface of the electrode, which corresponds to the vanishing of the tangential component of the electric field. Then, by fitting them to the analytical decomposition of the plane and scattered waves in the appropriate region, the reflection coefficients for each m can be calculated. Finally, by summing all the reflected waves, the total scattering cross-section can be found.

Second, Eq. ( 18) allows us to analyze the electromagnetic absorption in the resonant region near ξ 0 where Reε eff (ξ 0 ) = 0. The real part of the transverse impedance reveals a step behavior as a function of ξ in the vicinity of ξ 0 . The magnitude of this step, Re∆Z (m) ⊥ , corresponds to the electromagnetic absorption density in a mode m. Since the real part of the impedance represents the energy flux density, the magnitude of its step corresponds to the decrease in the energy flux due to absorption, i.e., electromagnetic losses. By analyzing Eq. ( 18), it can be concluded that Re∆Z (m) ⊥ ∝ m 2 /ε eff (ξ 0 ) and does not depend on the imaginary part of the effective permittivity. Figure 3 shows the dependence of the total scattering cross-section and absorption cross-section on the applied voltage and the angular scattering patterns for different voltages.

In a particular realization of the proposed system, all the restrictions described above have to be taken into account. Thus, as a model system, let us introduce the following parameters. c,d). The frequencies were chosen to be close to the plasmon resonance frequency, which is 3 -1/2 ω p ≈ 0.577ω p .

The central metallic electrode has a radius of 1 µm. The metaatoms are represented by doped semiconductor nanospheres of 10-nm radius with the plasma frequency of 100 THz. The meta-atoms are suspended in, e.g., ethanol, which is transparent in the IR and has a refraction coefficient of ∼ 1.4 in this frequency band. The unperturbed volume fraction was set to ν = 0.04, which corresponds to the unperturbed meta-atom concentration of ∼ 10 15 cm -3 . The characteristic voltage u 0 for such a system at room temperature is u 0 ≈ 39.5 V. The system is illuminated by a plane TM-polarized electromagnetic wave at a frequency of 0.7ω p , 70 THz. The corresponding wavelength in this medium is ∼ 3µm, which is much larger than the meta-atom size. The specified parameters were used for the simulations presented in the following section.

III. SIMULATIONS

We performed full-wave simulations of the scattering of an electromagnetic wave on the permittivity distribution defined by Eqs. ( 2) and ( 9) using COMSOL Multiphysics software. For the simulations, the Electromagnetic Waves, Frequency Domain interface of the COMSOL RF Module was used. The permittivity distributions were calculated using custom software developed in Matlab and imported into COMSOL to set the permittivity distributions.

The color plots of the effective permittivity, electromagnetic loss, and radial and tangential field component distributions, as well as the absorption patterns for different applied potentials, are shown in Fig. 4. As can be seen, the resonant region appears at some threshold voltage and grows with the increase of the voltage. The radial distributions of the radial and tangential fields are shown in Fig. 5. It is noteworthy that the radial component of the electric field has a stronger resonant peak compared to the tangential component. Such a behavior is in accordance with the theoretical analysis: the radial component has a power divergence, while the tangential has a logarithmic divergence. Figure 6 shows the dependence of the electromagnetic field absorption cross-section σ in the resonant region as a function of the applied voltage for different values of the intrinsic metaatom loss. As expected, when the intrinsic loss is not too high, the resonant absorption is almost independent of its value. FIG. 6. Electromagnetic absorption cross-section as a function of voltage for different values of the normalized damping factor γ/ω p (a), and color plots of the absorption in the plane transverse to the cylinder axis (b). For visibility, one quarter of the (circular) resonant region is shown. In (b), the normalized damping factor was set to 0.0025 (i), 0.005 (ii), 0.01 (iii), and 0.02 (iv). The frequency was set to ω = 0.64ω p . Other parameters are similar to those in Fig. 4.

Figure 7 shows the energy flux in the scattering region. Evidently, the highest energy flux is in the resonance region; its direction is along the surface, where Reε eff = 0. Such a behavior is an indication of the excitation of the surface wave at the interface between the regions with Reε eff > 0 and Reε eff < 0.

IV. DISCUSSION

The considered scattering problem may be of interest in several aspects. In particular, the interaction between the electromagnetic wave and the system is controlled by a simple variation of the electric potential of the immersed electrode. Below some threshold value, the scattering almost does not depend on the voltage, being just the scattering off the central cylindrical electrode. However, when the voltage exceeds this threshold, the scattering characteristics change drastically.

First, the total scattering cross-section starts to grow with the increase of the voltage up to a certain maximum. This can be partially attributed to the growth of the scattering region. Indeed, the scatterer is no longer the central electrode, but the surface surrounding the volume whose meta-atom concentration is high enough to make the permittivity negative. However, because of the resonant nature of the scattering, the cross-section grows faster than the actual scatterer size. Namely, below the threshold, the cross-section roughly equals the diameter of the central electrode. On the other hand, above the threshold, the scattering cross-section becomes significantly larger than the scattering region diameter. For instance, at the frequency of 0.7ω p , the diameter of the scattering area reaches 3.36r i , while the total cross-section reaches 40r i ; at the frequency of 0.64ω p , the diameter of the scattering area reaches 5.1r i , and the total cross-section grows as high as 50r i .

At the same time, although the total cross-section increases when the voltage threshold is reached, the backscattering is suppressed. As can be seen in Figs. 3 (b) and (d), while at zero voltage the preferential scattering direction is backwards (with respect to the wave propagation), with the increase of the voltage the backscattering is reduced. Moreover, for certain parameter combinations,e.g. at the frequency of 0.7ω p and the dimensionless voltage of u/u 0 = 9, or at the frequency of 0.64ω p and the dimensionless voltage of u/u 0 = 14, the backscattering is almost entirely eliminated.

Another feature of the considered scattering mechanism is the associated resonant absorption. It takes place in a thin layer nearby the surface, at which the real part of the effective permittivity is zero. A similar effect has been studied in inhomogeneous plasma [START_REF] Denisov | On a singularity of the field of an electromagnetic wave propagated in an inhomogeneous plasma[END_REF]. Resonant absorption appears at the voltage threshold, similarly to resonant scattering. However, while the scattering cross-section is a continuous function of the voltage, the absorption presents a discontinuous step at the threshold. It can be understood as the immediate appearance of finite absorption as soon as the real part of the effective permittivity crosses zero. Since the absorption takes place in a very narrow region, a very small increase in voltage is needed for the resonant region to fit between the Reε eff = 0 surface and the surface of the central electrode. The width of the resonant region is governed by the damping factor of the meta-atom material. Evidently, the larger the damping is, the wider becomes the resonant region. At the same time, the maximum field amplitude and, therefore, the maximum loss depend inversely on damping: the larger the damping, the smaller the field maximum. These two effects compensate each other keeping the total loss constant as long as the damping is not too large. This can be seen in Fig. 6. Indeed, in Fig. 6 (a), the energy loss plots follow almost the same line for different values of damping, confirming the suggested property. In Fig. 6 (b), the variations of the loss region width and loss intensity depending on the damping factor are clearly visible.

The narrowness of the resonant region can be exploited for local electromagnetic heating. Clearly, as a major part of the electromagnetic power is absorbed in a thin layer near Reε eff = 0, this layer inevitably becomes a heat source. Moreover, because of the independence of the losses from the damping factor, the heat generation in the layer does not depend on the damping factor either, and, more broadly, on the nature of the loss in the meta-atoms.

Finally, the region limited by the surface Reε eff = 0, including the central electrode itself, turns out to be effectively shielded. Namely, the electromagnetic wave can hardly penetrate the region such that Reε eff < 0, which makes it isolated from the electromagnetic radiation.

It should be noted that all of the listed effects can be controlled by the application of a voltage to the central electrode. By varying the voltage, the resonant absorption can be turned off or on, its intensity can be altered; the backscattering can be minimized, the total scattering can be manipulated; the central region can be shielded.

As was already mentioned, the proposed system is scalable in the sense that as long as the homogenization applicability conditions are satisfied, the parameters of the system, such as the meta-atom size and the negative polarizability mechanism, the electromagnetic wave frequency, and the overall size of the system, may vary significantly. In the previous sections, we have considered the system designed for 70-THz infrared radiation. At the same time, the system can be scaled for either higher or lower frequencies by varying the sizes and choosing the meta-atom materials. For instance, the meta-atoms can be made of highly doped GaAs with the dopant concentration of 6 × 10 16 cm -3 and the corresponding plasma frequency of 14.9 THz. As a result, a suspension of such nanoparticles with the radius of 600 nm and the volume fraction of 0.08 placed in a 0.5-mm-diameter transparent cylindrical vessel with a 6µm-thick copper electrode immersed in its center would provide an alternative realization of the proposed system. The corresponding electromagnetic frequency band, within which the effects described above would be prominent, would lie between 6.34 and 14.9 THz. Conversely, the system can be scaled for higher-frequency band by choosing metal particles of a few nanometers in diameter as meta-atoms and a wire of 100 nanometers in diameter as the electrode. For instance, for Ag nanoparticles with the plasma frequency of 2200 THz, the applicable frequency band would lie between 1500 and 2000 THz.

One possible realization of a device based on the effect described above can be an array of cylindrical cells, each of which is represented by the proposed system. By manipulating the voltages applied to each cell, one could control the transparency, absorption, and backscattering of such a panel. Another possible application is mode manipulation in a waveguide or a resonator. Again, by simply applying the voltage, the Q-factor can be drastically decreased, thus providing controlled damping.

V. CONCLUSION

We have considered the scattering of a plane electromagnetic wave on the cylindrically symmetric inhomogeneous distribution of a liquid MM induced by a static electric field, which can be generated by applying an electric potential to an electrode immersed in the liquid MM. We have shown that by changing the voltage, it is possible to manipulate the scattering of this wave in different ways. For instance, the scattering cross-section can be varied over a wide range. By tuning the voltage, backscattering can be suppressed. In addition, in the scattering region, resonant absorption enhanced by the excitation of a surface wave takes place. The proposed controlled scattering mechanism benefits from the particular simplicity of all its constituents. For instance, the meta-atoms can be any spherical particles that can provide negative polarizability at a certain frequency, i.e., metallic or high-index dielectric spheres. The host liquid and the outer boundaries just have to be transparent for the electromagnetic wave at that frequency. Thus, a realization of the proposed mechanism may be used for controllable shielding, controllable backscattering suppression, or manipulation with scattering.

Funding. This study was supported by the French PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE and TALiSMAN project, funded by ERDF (2019-000214).

Disclosures. The authors declare no conflicts of interest. Data availability. No data were generated or analyzed in the presented research.

Supplemental document. See Visualizations 1-4 for supporting content.

FIG. 1 .

 1 FIG. 1. (a) Schematic picture of the considered system. The system is illuminated by a plane electromagnetic wave. The outer boundary of the container is schematically represented by the external cylinder. However, it is assumed that it has no impact on the electromagnetic wave propagation. (b) Meta-atom distribution without applied voltage. (c) Meta-atom distribution with applied voltage.

FIG. 2 .

 2 FIG. 2. Real (a) and imaginary (b) parts of the possible effective permittivity dependences on the normalized distance from the cylinder center for different dimensionless voltages, u/u 0 = 1, 3, 6, 9, and 14. The calculations were made for the host permittivity ε h = 1.4. The meta-atom material was meant to have Drude permittivity; the frequency was set to 0.7ω p . The unperturbed meta-atom volume fraction was set to 0.04.

FIG. 3 .

 3 FIG. 3. Scattering and absorption cross-sections (a, c) and scattering diagrams (b, d) at different dimensionless voltages, u/u 0 . The parameters are the same as in Fig. 2. The frequency was set to ω = 0.7ω p in (a, b) and ω = 0.64ω p in (c, d). The frequencies were chosen to be close to the plasmon resonance frequency, which is 3 -1/2 ω p ≈ 0.577ω p .

FIG. 4 .

 4 FIG. 4. Effective permittivity (a), electromagnetic absorption (b), tangential (c) and radial (d) components of the electric field, and electromagnetic absorption at different applied potentials (e). In (a), the dashed line represents the surface at which effective permittivity becomes zero, ε eff = 0. In (a-d), the voltage is set to 335 V. In (e), the voltages are set to 118 -533 V (u/u 0 = 3.0 -13.5), as noted in the lower right corner of each sub-image. The frequency is set to ω = 0.7ω p . The radius of the central metallic electrode was set to 1 µm; the meta-atom radius was set to 10 nm; the plasma frequency of meta-atoms was set to 100 THz. The refraction index of the host liquid was set to 1.4. The unperturbed volume fraction of meta-atoms was set to ν = 0.04. The temperature was set to 300 K. The arrows represent the direction of the wave propagation.

FIG. 5 .

 5 FIG. 5. Radial distribution of radial E ρ (a) and tangential E φ (b) components of the electric field at different voltages. The parameters are similar to those in Fig. 4.

FIG. 7 .

 7 FIG. 7. Energy flux (Poynting vector field) in the scattering region. The voltage was set to 0 V (below the threshold) (a) and 158 V (above the threshold) (b). The other parameters are similar to those in Fig. 4. The arrows represent the Poynting vector. The lengths of the arrows are proportional to the logarithm of the Poynting vector moduli.