WAVES IN PLASMAS

Subject and Purpose. Th e solid-state structures involving metasurfaces can be used to eff ectively control some of the basic properties of electromagnetic waves, like amplitude, phase and polarization. Th e present work is aimed at analyzing the new eff ects that may appear during incidence of p-polarized electromagnetic waves upon a solid-state structure involving a uniaxial plasmonic metasurface, a dielectric interlayer, and a layer of metal. Methods and Methodology. Th e conditions suitable for identifying the eff ects that result from the refl ection of a p-polarized electro-magnetic wave incident upon a solid-state structure of the above described type have been sought for via numerical simulation. Th at has allowed fi nding the magnitudes of the essential parameters, such as angles of incidence and frequencies of the electromagnetic waves, as well as thicknesses of the dielectric interlayer, that could stipulate appearance of novel electromagnetic eff ects. Results. It has been shown that the solid-state structure involving a uniaxial plasmonic metasurface, a dielectric interlayer, and a layer of metal is capable, under certain conditions, to fully absorb an incident electromagnetic wave of p-polarization. Moreover, a new eff ect has been predicted, specifi cally that of full conversion of the incident p-polarized electromagnetic wave into a refl ected wave of s-polariza-tion. Th e necessary condition is that the plane of incidence of the electromagnetic wave were at an acute angle to the principal symmetry axis of the plasmonic metasurface. Conclusions. Th e solid-state structures of the type involving a uniaxial plasmonic metasurface, a dielectric interlayer, and a layer of metal are characterized by unique refl ective properties. Th ey are capable of fully absorbing, under certain conditions, the p-polarized electromagnetic waves incident upon them. Such structures can be used for creating optical and nanoelectronic devices of new types.


Introduction
Metamaterials and metasurfaces have of late been attracting ever more attention because of their special qualities [1][2][3][4][5][6][7].Th ey permit eff ectively controlling all the basic properties of electro-magnetic waves, like amplitude, phase and polarization [3,4], hence can be used for creating optical and nanoelectronic devices of novel types. Of particular interest are the eff ects accompanying electromagnetic wave refl ection from metamaterials and metasurfaces. One of the most interesting among them concerns conversion of the electromagnetic wave's polarization for the wave incident upon a metamaterial overlying a metal substrate [3].
In the present work, we have studied the new effect of non-refl ective incidence of p-polarized electromagnetic waves upon a solid-state structure composed of a uniaxial plasmonic metasurface, a dielectric interlayer, and a layer of metal. Th e eff ect owes to the destructive interference of direct and reverse electromagnetic waves superimposed in the solid-state structure. As has been shown, such behavior can only occur when the plane of incidence of the p-polarized electromagnetic wave is either parallel or perpendicular to the principal axis of the plasmonic metasurface. Th e specifi c wave frequencies and dielectric layer thicknesses have been identifi ed, with which the non-refl ective incidence eff ect can occur for p-polarized electromagnetic waves.
Moreover, we have predicted a new eff ect, namely that of full conversion of an incident p-polarized electromagnetic wave into a refl ected wave of s-polarization. Th e eff ect has been shown to take place when the plane of incidence of the p-polarized wave makes an acute angle with the principal axis of the plasmonic metasurface. Th e conditions have been found that can ensure full conversion of the incident electromagnetic wave of p-polarization into an s-polarized refl ected one.

Problem formulation
Consider the spatial domain z 0 which is occupied by a dielectric material of permittivity 1 .
ε A dielectric interlayer of permittivity 2 ε (occupying the space 0 z d ) is placed on top of a perfectly conducting metal substrate (area z d). Th e uniaxial plasmonic metasurface lying within the plane z 0 is represented by a two-dimensional array of conductive ellipsoids (Fig. 1) [5][6][7].
We will assume the electric fi eld vector of the p-polarized electromagnetic wave to lie within the plane making an angle j with the principal symmetry axis of the plasmonic metasurface. Th e electromagnetic wave of frequency w is incident upon the metasurface at an angle q.
Th e electromagnetic properties of the solid-state structure under consideration will be described in terms of the eff ective conductivity tensor of the uniaxial plasmonic metasurface [5][6][7]. Th e diagonal components of the tensor, normalized by c / 4 p where c is the speed of light, are of the form , , , .
Th e indices " | | " and "  " here relate to the directions along and across the principal axis of the plasmonic metasurface; , Ω  and , γ  are, respectively, the resonant frequencies and half-widths of the resonance lines; , A  stand for oscillator strengths, and , ∞ σ  are background conductivities. Finally, , σ  and , σ  denote the real and the imaginary parts, respectively, of the corresponding conductivity tensor components. To carry out the calculations, we have assumed , , and 1.2 Ω [5]. Within the reference frame which is rotated by an angle j with respect to the principal axis of the plasmonic metasurface (note the plane of incidence of the electromagnetic wave to be XZ), the conductivity tensor of the metasurface can be written as [5][6][7] xx xy yx yy ϕ σ σ σ σ σ .
Note that the presence at j 0 and j 90 of the non-zero off -diagonal conductivity tensor components xy σ and yx σ results in the appearance of refl ected electromagnetic waves of s-polarization. Hence, upon incidence on the uniaxial plasmonic metasurface of a p-polarized electromagnetic wave, the refl ected wave obtains all the fi eld components, and generally is characterized by an elliptical polarization. In the coordinate system selected, the electromagnetic fi eld of the p-polarized electromagnetic wave has the components as follows: ,0, , Now, let us write down the non-zero tangential components of the electromagnetic fi elds for each medium of the solid-state structure in question. Th e multiplier exp( ) x ik x t ω will be omitted; the subscript "p" relates to p-polarized, and subscript "s" to s-polarized waves.
Medium1 (area z 0): E stand for the amplitudes of the direct ( ) and reverse (-) p-or s-polarized waves in the layer of dielectric permittivity 2 ε . To determine the pp r and ps r it is necessary to use the boundary conditions at z 0 and z d.
Th e roots of Eq. (3) defi ne the frequencies at which the eff ect of non-refl ective incidence of p-polarized electromagnetic waves on the uniaxial plasmonic metasurface is produced. Th e functions , ( ) σ ω  are symmetric, positively defi ned functions relative the resonant frequencies Ω  and Ω . Accordingly, Eq. (3) demonstrates two roots, arranged symmetrically with respect to the frequencies Ω  and Ω . Since the magnitude of 1 k decreases at higher values of q, while the highest value of , ( ) σ ω Another thing which can be concluded from Fig. 2 is that for every value of q there are two values of frequency which follow from Eq. (3). Accordingly, there are two values of d, obtainable from Eq. (4), that correspond to this pair of frequencies. Th e vertical dashed line in Fig. 2 relates to the case of , m θ θ and the circular sign marks the common. With account of 2 tg( ) k δ being a periodic function, we have only sought for those dependences of ( ) δ θ for which the magnitude of d is the lowest.
It should be noted that the value of the dielectric permittivity 2 ε does not suggest any limitations on the non-refl ective incidence eff ect for p-polarized electromagnetic waves at the uniaxial plasmonic metasurface. Fig. 3 shows the 2 ( ) δ ε dependences for the case p R 0, with j 0 and q 45 . Th e dependences have been plotted for parameter values 1 ω ≈ 0.976 (solid curve) and 2 ω ≈ 1.025 (dashed curve). It can be seen from Fig. 3, that both branches of the 2 ( ) δ ε dependency are monotonically decreasing functions.
Shown in Fig. 4 are ( ) p R ω dependences for 2 ε 2.0 and q 45 , with j 0 (solid lines) and j 90 (dashed lines), wherefrom we can conclude that with every value of j the magnitude of p R becomes zero at two frequencies. However, the eff ect is observable at diff erent thicknesses of the dielectric layer d. Th us, the eff ect of refl ectionless incidence of p-polarized electromagnetic waves takes place at two frequencies and with "properly" selected magnitudes of the dielectric layer thickness.

Full conversion of p-polarized into s-polarized electromagnetic waves
As can be seen from Eq. (2), the conditions j 0 and j 90 ( 0) yx σ provide for appearance of an s-polarized refl ected electromagnetic wave ( 0). ps r So, the electromagnetic wave refl ected from a uniaxial plasmonic metasurface is a sum of p-polarized and s-polarized components.
Meanwhile, it has been found that with some values of the wave frequency and thicknesses of the dielectric layer (dependent on 2 ε , j and q) the refl ected p-polarized waves may vanish. Accordingly, under such conditions we observe the case of full conversion of the incident p-polarized electromagnetic wave into an s-polarized one. Note that with j 30 the frequencies of a bright full conversion eff ect for the ppolarized electromagnetic wave are localized near the resonant frequency Ω  1.0 Th us, an incident electromagnetic wave of p-polarization gets transformed into an s-polarized wave for two pairs of (w, d) magnitudes. Non-refl ective incidence of p-polarized electromagnetic waves on the solid-state structure "uniaxial... At higher values of j the frequencies where the full conversion eff ect occurs for the p-polarized and s-polarized electromagnetic waves also get higher, shift ing toward the other resonant frequency, i.e. W^ 1.2. Th is situation is illustrated in Fig. 6 for d ≈ 0.5 (panel (a)) and d ≈ 2.0 (panel (b)) with the parameters like j 60 , q 45 , and 2 ε 2.0. It can be concluded from this plot that at w ≈ 1.17 (panel (a)) and w ≈ 1.24 (panel (b)) the p-polarized electromagnetic wave is absent among the refl ected. So, at these frequencies we observe a complete conversion of a p-polarized into an s-polarized electromagnetic wave.

Conclusions
It has been shown that the solid-state structure involving a uniaxial plasmonic metasurface, a dielectric interlayer, and a layer of metal is capable, under certain conditions, to fully absorb an incident electromagnetic wave of p-polarization. To achieve that, the plane of incidence of the p-polarized electromagnetic wave should be either parallel or perpendicular to the principal axis of the plasmonic metasurface. Th en the eff ect of a non-refl ective incidence of the p-polarized electromagnetic wave is observable at two frequencies, conditioned by a proper choice of the dielectric layer's thickness. As has been found, the eff ect of non-refl ective incidence of p-polarized electromagnetic waves is not aff ected by any limitations associated with the value of the dielectric layer's permittivity.
Also, a new eff ect has been predicted, specifi cally that of complete conversion of the incident p-polarized electromagnetic wave into a refl ected wave of s-polarization. Th e necessary condition is that the plane of incidence of the electromagnetic wave were at an acute angle to the principal symmetry axis of the plasmonic metasurface, (j 0 , j 90 ). In addition, the wave frequencies and thicknesses of the dielectric layer, which the eff ect takes place for, have been found. Th e dependence upon the angle j between the plane of wave's incidence and the principal symmetry axis of the plasmonic metasurface has also been studied with respect to its importance for the full conversion of a p-polarized electromagnetic wave into an s-polarized electromagnetic wave.
Th e new eff ects discovered can be used both for improving the characteristics of existing optical and nanoelectronic devices, and for creating new equipment with unique properties.