Perfect tunneling of obliquely-incident wave through a structure with a double-negative layer

Optics Communications 369 (2016) 164–170

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Perfect tunneling of obliquely-incident wave through a structure with a double-negative layer S.A. Afanas’ev n, D.I. Sementsov, Y.V. Yakimov Department of Radiophysics and Electronics, Ulyanovsk State University, 432017 Ulyanovsk, Russian Federation

art ic l e i nf o

a b s t r a c t

Article history: Received 31 December 2015 Received in revised form 22 February 2016 Accepted 23 February 2016 Available online 1 March 2016

The oblique incidence of TE-polarized plane electromagnetic wave on a three-layered lossless structure containing the layer of double-negative medium is discussed. The resonant values of the angle of incidence are obtained, for which the perfect tunneling of electromagnetic power through the structure can be achieved. The results of exact numerical analysis are compared with approximate solution based on the model of symmetrical slab waveguide. & 2016 Elsevier B.V. All rights reserved.

Keywords: Metamaterial Double-negative medium Perfect tunneling Symmetrical slab waveguide

1. Introduction The media transparent for electromagnetic (EM) radiation may be either double-positive (DPS) or double-negative (DNG) depending on the sign of its permittivity ε and permeability μ . Conventional DPS (or “right-handed”) media with ε, μ > 0 have positive refractive index n > 0. DNG media with simultaneously negative ε and μ and n < 0 were introduced hypothetically by Veselago [1], who called them “left-handed”. The artificial composite DNG media called “metamaterials” were first demonstrated in 2000 [2,3]. They exhibit unusual electromagnetic properties if compared with DPS media. Among them, the remarkable transmission properties of layered structures containing DNG layers are of great interest [3–5]. Also there are single-negative (SNG) media, in which only one of the two material constants is negative. They can be epsilonnegative (ENG) with ε < 0, μ > 0 or my-negative (MNG) when ε > 0, μ < 0. In contrary to DPS and DNG media, the SNG media are opaque even if they are lossless. The matter is that they support only evanescent waves because of purely imaginary wavevector. However, evanescent waves can transfer EM power through a slab of SNG material of a finite thickness. The EM field in the slab is the result of the interference of two evanescent waves decaying at opposite directions. As it is known, these two waves give rise to the non-decaying energy flux, providing the power transmission through the slab, i.e. the EM tunneling. Moreover, the transmission n

Corresponding author. E-mail address: [email protected] (S.A. Afanas’ev).

http://dx.doi.org/10.1016/j.optcom.2016.02.053 0030-4018/& 2016 Elsevier B.V. All rights reserved.

may be total, i.e. the transmittance coefficient may reach unity. This effect is called the “perfect” or “complete” tunneling. It was predicted in [6] for a system of alternating ENG and MNG layers with matching parameters. In general, the perfect tunneling may be reached for any type of waves in various structures supporting evanescent waves [7–9]. Commonly, the total transmission is connected with resonance effects such as, for example, the excitation of surface waves. Up to date, a lot of various lossless layered structures containing SNG layers have been demonstrated to realize the perfect tunneling [10–15]. Yet in [6], the possibility of complete tunneling has been shown also for multilayer structures with alternating DPS and DNG layers. Since then a few works have appeared considering the use of DNG metamaterials in tunneling systems [5,16–20]. In [5,16,17] the perfect tunneling has been considered in the system based on a hollow rectangular waveguide that included three sections. The central section, operating below cutoff, was empty. The input and output sections filled with DPS dielectric having ε > 1, μ = 1 were above cutoff. The propagative fundamental TE10 mode existing in the first section transformed into the evanescent mode in the second section of the waveguide. Due to the presence of reflected evanescent mode decaying at opposite direction, some power could tunnel to the third section. A multilayer structure that included m layers of DNG metamaterial spaced by air gaps was placed in the central section. In [5,16,17], only the case of “perfect” DNG metamaterial with ε = μ = − 1 ( n = − 1) has been discussed. These waveguide structures were called in [17] “the mth-order structures”, where m is the number of DNG layers. Under certain conditions, the tunneling of power through these waveguide structures could be perfect. As it has been shown in

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[5,16,17], the perfect tunneling is accompanied by the resonant amplification of evanescent waves at the metamaterial – air interfaces. Another way to obtain evanescent waves in layered structures is provided by the phenomenon of total internal reflection. If an EM wave is incident on a slab from optically dense medium at the angle of incidence which exceeds the critical angle of total reflection, the evanescent waves decaying in opposite directions are excited in the slab and the tunneling of EM power is observed. So, there is a possibility to obtain the evanescent waves in air gap sandwiched between two media with n > 1. In [21], the anomalous Goos–Hänchen effect has been studied in a five-layer structure containing a slab of DNG metamaterial. The middle DNG layer was separated by two gap slabs from optically dense bounding medium. The oblique incidence of a Gaussian beam of transverse-electric (TE) polarization at the angle larger than critical angle of total reflection was considered. At these circumstances, the waves in gap slabs were evanescent, resulting in the excitation of either surface waves at the surfaces of DNG layer or leaky guided waves supported by DNG slab. If the tangential component of beam wavevector was the same as the propagation constant of one of the guided modes of DNG slab, the large values of Goos–Hänchen effect were observed. The arrangement of layers in the five-layered structure studied in [21] is similar to the waveguide structure of first order ( m = 1) from [5,16,17]. In this connection, we may suppose the possibility of perfect tunneling for such structures under oblique incidence of EM wave in the case of total reflection. Also, one may expect that the condition of total transmission may be associated with the guidance conditions of a DNG slab waveguide. Indeed, if the guidance conditions are matched for a symmetric slab waveguide, the distribution of EM field amplitude across the slab has a symmetrical shape, what is the distinctive feature of perfect tunneling. In this paper we consider a lossless layered structure DPS– air
DNG–air–DPS which is analogous to the structure from [21]. In contrast to [21], our purpose is to determine the conditions of total transmission under the oblique incidence of a plane TE-polarized EM wave from the first semi-indefinite DPS medium. We vary in a wide enough range the angle of incidence, the refractive index of the middle DNG layer as well as the ratio of thicknesses of DNG layer and air gaps. The results obtained by exact numerical solution of EM boundary problem are compared with the known solution [22–24] for guided modes of a symmetric DNG-slab waveguide.

coordinate system. We imply that the layers are infinite in the directions of x- and y-axes. In the direction of z-axis, the structure is bounded by two semi-infinite dielectric DPS media (regions 1 ( z < 0) and 5 ( z > z3)), having equal material parameters ε1 > 0, μ1 > 0 and positive refractive index n1 = (ε1μ1)1/2 . The central layer (region 3, z1 < z < z2) of thickness L3 is DNG medium with ε3 < 0, μ3 < 0 and negative refractive index n3 = − (ε3 μ3 )1/2. The regions 2 ( 0 < z < z1) and 4 ( z2 < z < z3) are two air gaps of equal thickness L2 = L 4 having parameters ε2 = μ2 = 1 and n2 = 1. A plane monochromatic wave of frequency ω with the electric vector polarized along y-axis is incident on the structure from region 1 at the angle of incidence θ . The x-component h of its wavevector is

2. Basic relationships

ginary). The first case is realized under the condition h2 < k 02 εj μj when γj = βj and the plus sign of the radical in (2) should be chosen for air in the regions 2 and 4 and minus sign for the DNG medium in the region 3. The condition for evanescent waves is γj is determined by h2 > k 02 εj μj , when the quantity of

The layered structure which is under our investigation is depicted schematically in Fig. 1 with the axes of attached Cartesian

(1)

h = k 0 n1 sin θ ,

where k 0 = ω/c – wavenumber for vacuum (air) and c is the light velocity for vacuum. The quantity h has the same value in each region, while the transversal z-component of the wavevector depends on the material parameters of a medium and is, in general, complex:

γj = ±

k 02 εj μj − h2 = βj + iαj,

(2)

where j = 1...5 is a number of the region. The electric field of incident wave in the region 1 may be written as +

E1y(z ) = A exp (iγ1z ),

(3)

where A is the field amplitude and γ1 = k 0 n1 cos θ has a positive real value (in the case of lossless DPS medium). Besides the incident wave, there is the reflected wave in the region 1, with the field determined as −

E1y(z ) = rA exp ( − iγ1z ),

(4)

where r is the complex amplitude coefficient of reflection. Let us note that the fields (3) and (4) have the phase factor exp (ihx − iωt ), common to them and to all fields in each region of the structure. In what follows we shall omit this factor. The electric fields in regions 2–4 are also the superposition of two waves propagating in opposite directions along z-axis: ±

±

E jy(z ) = aj A exp ( ± iγj z ), j = 2, 3, 4,

(5)

± aj

where are complex amplitude coefficients. The upper signs in (5) and below correspond to the waves propagating forward and backward to z-axis. For lossless media in the regions 2–4 these waves may be non-decaying ( γj is real) or evanescent ( γj is ima-

γj = iαj = i h2 − k 02 εj μj for j = 2, 3, 4 . Here, the sign of the radical is always plus, corresponding to decaying of field amplitudes along the direction of wave propagation. There is only a single transmitted wave in the region 5, which electric field differs from the field of the incident wave (1) by the amplitude coefficient of transmission t : +

E5y(z ) = tA exp (iγ1z ).

(6)

Accordingly to Maxwell equations, the magnetic field components may be obtained from the electric fields (3)–(6) with the help of expressions ±

Fig. 1. Geometry of the problem.

±

H jx = ± ζj E jy;

±

±

H jz = ξj E jy,

(7)

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where ζj = − γj/(ωμ0 μj ), ξj = h/(ωμ0 μj ), μ0 is vacuum permeability, j = 1... 5 for forward propagating and j = 1... 4 for backward propagating waves. To solve the boundary problem, the standard boundary conditions are written. They imply that the tangential components Ejy and Hjx of electric and magnetic fields are continuous at the interfaces z = zj − 1: +

−

+

−

E jy + E jy = E(j + 1) y + E(j + 1) y, +

−

+

−

H jx + H jx = H(j + 1) x + H(j + 1) x ,

j = 1, 2, 3, 4 ,

(8)

where z0 = 0, the field components Ejy and Hjx are taken from (3)– (7), and E5−y = H5−x = 0. Solving numerically the set (8) of eight equations, we obtain the values of complex amplitude coefficients r , t and a±j ( j = 2, 3, 4 ) in the expressions (3)–(7). Then, the components of Poynting vector (i.e. energy flux density) may be calculated. The time-averaged Poynting vector is determined by expression

1 Re ⎡⎣ Ej , H*j ⎤⎦, j = 1... 5. 2

Sj =

(9)

It has the transversal z-component, which provides the transmission of EM power through the structure, and the in-plane xcomponent:

Sjz = −

1 Re Ejy H *jx , 2

(

)

Sjx =

1 Re Ejy H *jz . 2

(

)

(10)

In the regions 1–4, the components of wave fields in (9) and (10) represent the sum of corresponding components of forward and backward propagating waves. Substitution of wave fields (3)– (7) into (10) leads to expressions of energy flux components for all regions 1–5. In the region 1, the components of Poynting vector are determined as:

1 ζ1 A2 1 − r 2 ; 2 1 = ξ1A2 1 + r 2 + 2 r cos (2γj z − argr ) . 2

S1z = S1x

( (

)

)

(11)

For regions 2–4, two cases should be distinguished. For propagating (non-decaying) waves with real propagation constant γj = βj the Poynting vector components are

Sjz =

Sjx =

2 ⎛ 2⎞ 1 ζj A2 ⎜ a+j − a−j ⎟, ⎝ ⎠ 2

(12)

1 1 ζ1 t 2 A2 = ζ1 TA2 , 2 2

S5x =

1 1 ξ1 t 2 A2 = ξ1TA2 , 2 2

(14)

where T = t 2 is the coefficient of EM power transmission, i.e. the transmittance of the structure.

3. Guided modes of a DNG slab The in-plane transfer of EM power (along x-axis) makes it possible to represent the given structure as a symmetrical fivelayer waveguide, where the leaky guided wave is excited. At definite conditions, the central layer (region 3) may be regarded as a symmetric DNG slab waveguide. The condition for excitation of guided waves in the slab is the decaying of wave fields in surrounding media. For given five-layer structure it is possible, if the angle of incidence θ exceeds the critical angle of total internal reflection θcr = arcsin (1/n1). Then the waves in regions 2 and 4 are evanescent and the tunneling of incident wave through the structure is observed. The EM waves guided by a DNG slab were discussed in [22–24]. The solution of this problem can be found using the standard graphical method [26]. Let us consider a slab of thickness L having material parameters ε, μ < 0 and surrounding by vacuum. The slab is assumed to be infinite in the directions of x- and y-axes. We will discuss TE guided modes propagating along the x-axis, with the electric vector polarized along y-axis. The guidance condition calls for transverse wavenumber in vacuum being imaginary:

γ0 = ± iα0 = ± i h2 − k 02 ,

(15)

where h is the longitudinal (in-plane) wavenumber and h > k 0 ; the sign of the radical should be chosen so that the waves decay outside the slab. The transversal wavenumber γ = β + iα in the lossless DNG slab may be real for volume modes

γ=β=

k 02 εμ − h2 , h2 < k 02 εμ

γ = iα = i h2 − k 02 εμ , h2 > k 02 εμ .

while for evanescent waves with γj = iαj we have

(13)

Sjx =

S5z =

(16)

or imaginary for surface modes

⎞ 2 1 2 ⎛⎜ + 2 ξj A a j + a−j + 2 a+j a−j cos (2βj z + arga+j − arga−j ) ⎟, ⎝ ⎠ 2

Sjz = − ζj a+j a−j A2 sin (arga+j − arga−j ),

interference of two evanescent waves decaying in opposite directions (so-called “interference flux” or “tunneling flux” [25]). The in-plane energy flux Sjx , existing in each layer of the structure, includes in both cases three summands: the fluxes corresponding to the forward and backward propagating waves and the interference flux. The components of Poynting vector in region 5 consist of a single energy flux corresponding to the transmitted wave:

2 1 2 ⎛⎜ + 2 ξj A a j exp ( − 2αj z ) + a−j exp (2αj z ) ⎝ 2 ⎞ + 2 a+j a−j cos (arga+j − arga−j ) ⎟. ⎠

It is seen that, in contrary with the in-plane x-component, the expressions (12) and (13) for transversal energy flux have significantly different structure. In the first case (12), the total value of Sjz is obtained by subtraction of energy fluxes of forward and backward propagating waves. In the second case (13), evanescent waves with imaginary propagation constants do not transfer EM power along z-axis, and non-decaying energy flux appears due to

(17)

The dispersion relation for guided waves is obtained by the appliance of standard boundary conditions, which are similar to those used above in Section 2. The difference is that here we have only two interfaces and, therefore, four equations for field components. In the first case of volume modes the dispersion relation takes the following form [22, 23]:

α0 L =

⎛ βL 1 Nπ ⎞ ⎟, (βL ) tan ⎜ − ⎝ 2 μ 2 ⎠

(18)

where N is a positive integer. Normalized dependencies (18) of α0 on β are shown in Fig. 2 by solid lines for the first three volume modes (the order N of the mode is subscribed in parentheses). The shape of these dispersion curves at fixed frequency is determined by the values of slab permeability μ and its thickness L . For Fig. 2 the values of μ = − 1 and L = λ 0/2 have been chosen, where λ 0 = 2πc /ω is the wavelength for vacuum. The dispersion relations for surface waves ( γ = iα ) are written

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of intersection points in Fig. 2: 2

sin θ = ( 1/n1) 1 + ( α0/k 0 ) ≥ sin θcr.

(23)

The condition (23) is satisfied if

α0 < k 02 n12 − 1 .

(24)

In the following section, this supposition will be verified by numerical analysis for transmission properties of given structure in the “tunneling” region of values of the angle θ ( θ > θcr ).

4. Numerical results and discussion Fig. 2. Graphical solution of dispersion relations for DNG slab waveguide with μ = − 1 and L = λ 0/2 .

as [22, 23]

α0 L = − (αL/μ) tanh (αL ),

(19)

α0 L = − (αL/μ) coth (αL ).

(20)

Normalized dependencies (19) and (20) of α0 on α are shown by solid lines on the left-handed half-plane of the plot in Fig. 2, as it was made in [22, 24]. It should be noticed that the curve for dependence (19) continues the curve for volume mode with N = 1. By that reason, we denominate them both as “first order mode”. The surface mode given by relation (20) we denominate as “zeroth order mode”. The longitudinal wavenumber h has no discontinuities at the surfaces of the slab. Using this condition, we obtain from (15) and (16) the following relation:

(

)

α02 + β 2 = k 02 n2 − 1 ,

(21)

For calculations we use the next constant parameters of the structure: n1 = 2 (which corresponds to θcr = 30∘ ); the thickness of air gaps is L2 = L 4 = 0.5λ 0 . The refractive index n3 of DNG layer is varied as well as its thickness, given by expression L3 = ηL2. Thus, we introduce the dimensionless parameter η , which defines the ratio of thicknesses L3 and L2. For simplicity, the permeability of DNG layer is assumed to be μ = − 1, so that the value of n3 is varied only due to its permittivity ε3. In Fig. 3a, the examples are given of the dependencies of transmittance T = t 2 on the angle of incidence θ for different values of parameter η and refractive index n3 from the region n3 > 1. Excepting the green line ( η = 1.5, n3 = 1.2), we see in “tunneling” region θ > θcr the pronounced resonance peaks against the background of low levels of transmittance T . The resonant values of T (if the structure is lossless) reach unity, i.e. we have perfect tunneling. The closer is the angle θ to its critical value, the wider are the peaks. An additional peak with T = 1 may be observed in the region of transparency θ < θcr . In Fig. 3b, the dependencies T (θ ) are represented for the case of n3 ≤ 1. Here we may observe two resonant peaks in “tunneling”

where n = − (εμ)1/2 is negative refractive index of the DNG slab. Similar relation between α0 and α follows from (15) and (17):

(

)

α02 − α 2 = k 02 n2 − 1 .

(22)

The dependences (21) and (22) are plotted by dashed lines in Fig. 2 for different values of refractive index n. The intersections of solid and dashed lines give the solutions of systems of (Eqs. (18) and (21), 19) and (22), (20) and (22), corresponding to the volume and surface TE modes of a DNG slab waveguide. As it follows from Fig. 2, the mode of zeroth order can be ex(0) cited in DNG slab waveguides with nmin ≤ n < 1, where the value (0) of n = nmin corresponds to the tangency of curves (20) и (22) (see the dashed curve A). For all other values of n from mentioned interval, there are two points of intersection (the curve B). There are no solutions at n = − 1 (the dashed straight line С). In the (1) , there is only one solution for the mode of interval 1 < n ≤ nmax first order with either imaginary (D) or real (E) value of γ . The (1) curve F corresponds to the “critical” value n = nmax , above which

this mode can not be excited. For the mode with N = 2 we have (2) (2) two solutions in the interval nmin (the curves G and < n < nmax H correspond to the boundary values of this interval) and one (2) (2) solution at n = nmin and n ≥ nmax . In the following, we show that the volume and surface guided modes of DNG slab can be excited in the region 3 of the structure described in Section 1. We suppose that this situation can be achieved if the value of the in-plane wavenumber (1) of the incident wave equals the longitudinal wavenumber of certain guided wave. Using (15), one may obtain the relation between corresponding values of the angle of incidence θ and the ordinates α0/k 0

Fig. 3. The examples of dependencies T (θ ) in the regions n3 > 1 (a) and n3 ≤ 1 (b). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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Fig. 4. The dependencies θ res ( n3 ) for (a) η ¼ 0.5, 1 and (b) 1.5, 2. The order of resonance is subscribed in parentheses. Dashed lines correspond to the approximation of DNG slab waveguide. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

region (blue line). The special case is realized for η = 2 and n3 = − 1 (red line). The matter is that the case of L3 = 2L2 represents one of the known conditions of perfect tunneling through waveguide structures of mth order from [5,17], which, in general, is written as L3 = L2 + L 4 . The additional condition is the matching of material parameters of DNG layer and adjacent media: ε3/ε2 = μ3 /μ2 = − 1. For air ε3 = μ3 = − 1, so that at n3 = − 1 the perfect tunneling is observed for each value of θ . It is clear that the account of losses for DNG medium leads to a decrease in peak values of transmittance, i.e. the transmission is not really perfect. But, according to our estimate, the resonant peaks of transmittance remain well expressed up to the value of the imaginary part of refractive index n3 of about Imn3 ≃ 0.01. To understand the behavior of dependencies represented in Fig. 3, the resonant values θres of the angle of incidence are plotted against the refractive index n3 of DNG layer at fixed values of parameter η (see Fig. 4). In Fig. 4a, the dependencies θres ( n3 ) are plotted for structures with η = 0.5 (red lines) and 1 (blue lines). For comparison, the results of graphical solution for guided waves of DNG slab waveguide are shown by dashed lines (the angle of incidence, required for exciting of certain guided wave, is determined by (23)). The plot looks as a set of branches. In parentheses “the order of resonance” is subscribed, which is assigned to each of these branches. Moreover, each branch is associated with certain guided wave which are obtained using the model of DNG slab waveguide. In fact, the order of resonance equals the order of corresponding guided mode. For all curves in Fig. 4a almost complete agreement is observed between exact and approximate solutions for the angles of incidence exceeding ≈40∘ . The resonances of zeroth order exist in the region n3 < 1 provided that the quantity n3 exceeds some minimal value. At η = 0.5, it equals n3 = 0.537 (obtained at θ = 54∘ ), which is exactly

(0) the value nmin , determined above graphically for guided mode of (0) zeroth order. As it mentioned above, in the region nmin ≤ n <1 the dispersion relations for this mode give two solutions. However, for the considered structure two resonant peaks may be obtained in sufficiently narrow interval of n3. For example, at η = 0.5 this interval is 0.537 < n3 < 0.585 (at n3 = 0.585 the condition (24) ceases to be satisfied). In region n3 > 0.585, as the value of n3 increasing, the single peak shifts to smaller angles. Approaching to the critical angle θcr , the mismatch with the model of slab waveguide becomes more pronounced. According to this model, the limit value of the angle of incidence, required for exciting of zeroth order mode, is θ = θcr = 30∘ at n3 = 1. For our five-layer structure the peak at θ = 30∘ is obtained yet at n3 = 0.76, then the maximum of transmittance is shifted to the region of transparency. One may suppose that with an increase of parameter η the difference between exact numerical solution and its approximation must become more sufficient. Really, starting with η ≈ 1, the first maximum with T = 1 appears in the region of transparency. So that, at η = 1 the minimal value of n3 for perfect tunneling is 0.875 at θ = 26∘ < θcr . With an increase of n3 this maximum splits into two peaks, and one of them is shifted to the larger values of θ , moving to the “tunneling” region. The approximation gives no corresponding solutions up to n3 = 0.90 at θ = 40∘ . This value of (0) (the tangency point of curves n3 coincides with the value of nmin (20) и (22) in Fig. 2, giving the angle θ = 36∘ ). But starting from n3 = 0.90, the resonant angles θres are exactly the same as the angles corresponding to the “upper” intersection point of curves (20) and (22) on the coordinate plane ( α , α0 ). However, the “lower” solution is never realized, since the second peak with T = 1 remains in the region of transparency. At n3 > 1 for the resonance of first order there may be only one resonance peak in the ”tunneling” region, shifting to the smaller

S.A. Afanas’ev et al. / Optics Communications 369 (2016) 164–170

angle of incidence with the increase of n3 . In addition, at η = 0.5 there is the resonance of zeroth order in the interval 1.55 < n3 < 2 at small angles of incidence. In the interval 2.05 < n3 < 2.55 we have two maxima of first order in the region of transparency. In the interval 2.15 < n3 < 2.30 at η = 1 three maxima of second order are found, and one of them is located in “tunneling” region. Now we shall note briefly the characteristic features of perfect tunneling for structures with η > 1, when thickness of DNG layer exceeds the thicknesses of air gaps. In Fig. 4b, the dependencies θres ( n3 ) are shown for resonances of order 0, 1 and 2 at η = 1.5 (red lines) and 2 (blue lines). In “tunneling” region, the resonances of zeroth and first order are localized in sufficiently narrow region of n3 values near n3 = − 1. The dependence θres ( n3 ) in “tunneling” region is monotonically decreasing for the resonance of first order and monotonically increasing for all others. As before, the approximate solution (dashed lines) is almost the same as the exact (for the angles θ far enough from critical value θcr ). In Fig. 5, z-coordinate distribution of electric field amplitude Ey is shown in the limits of DNG layer at η = 1 for the values of n3, corresponding to the resonances of order 0, 1, 2 and 3. Two values of the angle of incidence have been chosen: θ = 45∘ from the “tunneling” region for the plot (a) and θ = 15∘ from the region of transparency for the plot (b). As one can see, in conditions of perfect tunneling (see Fig. 5a), the electric field is distributed symmetrically relative to the plane z = (z1 + z2 ) /2. The number of minima (“nodes”) of amplitude equals to the order of resonance (the minimum for the zeroth order resonance is significantly less pronounced). Such character of field distribution is typical for guided modes of symmetrical slab waveguides, both DPS and DNG [22,26]. But there is the significant difference: the amplitude at “nodes” is small, but does not reach zero. The same can be said about the magnetic field amplitude, which provides the power transmission through the structure (along z-axis), so that the guided modes of the structure are leaky. In the region of transparency (see Fig. 5b) the character of field distribution at resonance conditions, in general, remains unchanged. The number of minima remains equal to the order of resonance (however, the minima are noticeably less deep). The clear difference is in behavior of field amplitude at the surfaces of DNG layer. For the angles θ > θcr it is like a surface wave with maxima at the surfaces (in the case of imaginary values of γ3 it is really a surface wave). In the region of transparency ( θ < θcr ) there are minima of amplitude at the surfaces. In conclusion, it should be noticed that in all discussed cases the energy flux along x-axis in the limits of DNG layer is negative, i.e. the wave guided by DNG layer always is a backward wave, which must be necessarily for media with negative refractive index.

5. Conclusions The performed analysis shows that choosing the parameters of the DNG layer (refractive index and thickness) the perfect tunneling of incident power through the structure can be reached. This resonant effect is connected with the excitation of volume and surface backward waves guided by the DNG layer. If the values of the angle of incidence are far enough from the critical angle of total internal reflection, the resonant angles may be easily obtained by the solution of standard waveguide problem of waves guided by symmetrical slab waveguide.

169

Fig. 5. Electric field amplitude distribution along z-axis in the limits of DNG layer for resonances of order 0, 1, 2 and 3 at (a) η = 1, θ = 45∘ and (b) θ = 15∘ .

Acknowledgments This work was supported by the Ministry of Education and Science of the Russian Federation (projects Nos. 3.175.2014K and 14.Z50.31.0015).

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