ISSN 1068–3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2013, Vol. 48, No. 6, pp. 276–284. © Allerton Press, Inc., 2013. Original Russian Text © A.H. Gevorgyan, M.S. Rafayelyan, 2013, published in Izvestiya NAN Armenii, Fizika, 2013, Vol. 48, No. 6, pp. 407–419.
Light Propagation in Anisotropic Metamaterials. I. Dispersion Surfaces A. H. Gevorgyan* and M. S. Rafayelyan Yerevan State University, Yerevan, Armenia *
[email protected] Received July 24, 2013
Abstract⎯The peculiarities of solutions of the wave dispersion equation in an unbounded indefinite medium for arbitrarily oriented optical axes are considered. All possible dispersion surfaces arising in such media are described, and the conditions of their existence are obtained. It is shown that only some specific pairs of surfaces are possible. The dispersion equation for the boundary problem is obtained, as well as the dependence of dispersion curves on the orientation of optical axis is investigated. The nonreciprocity of the wave refraction is shown. DOI: 10.3103/S1068337213060042 Keywords: anisotropic metamaterials, light propagation, dispersion equation, nonreciprocity
1. INTRODUCTION Metamaterial is a system of artificial structural elements constructed for achieving useful and/or unusual electromagnetic properties. They exhibit linear and nonlinear optical properties such as negative refraction, inverse Doppler effect, propagation of energy of electromagnetic wave in the direction opposite to wave vector, and so on [1–5]. We can also note the optical Tamm state of such media, a huge increase of the density of optical states of molecules or a quantum dot placed in such a medium, considerable increase in time of fluorescence of molecules on the surface of such a medium, a huge rise of infrared radiation of a heated body in the presence of a layer of such a medium, and so on. They find such noticeable applications as perfect lenses [6], invisible masking [7, 8], perfect absorbers [9], and so on. Although negative refraction is most easily detected in isotropic metamaterials, it may be observed in also anisotropic ones. In latter case there is no need to require that all elements of tensors of dielectric and magnetic permittivities take negative values [10]. Moreover, all their elements may also be positively definite [11]. Studies of such anisotropic metamaterials are currently of great interest [12–23]. However, in literature are mainly considered the cases where principal directions of dielectric and magnetic permittivities are either parallel or perpendicular to the interface of media. Work [16] studies peculiarities of superluminal propagation of light in anisotropic metamaterial at arbitrary orientation of optical axis in the plane of light incidence. Omnidirectional total transmission and possibility of existence of a negative Brewster angle on the interface between isotropic and anisotropic media with arbitrary orientation of optical axis in case μˆ = Iˆ ( μˆ being the tensor of magnetic permeability and Iˆ the unit matrix) is investigated in work [18]. Work [21] studies the possibility of total reflection from the interface between isotropic medium and anisotropic metamaterial with arbitrary orientation of optical axis again at μˆ = Iˆ . Work [15] classifies dispersion equations for anisotropic metamaterials. We study in the present work peculiarities of solutions of dispersion equation of wave in unbounded indefinite medium in a system of coordinate oriented arbitrarily with respect to direction of the optical axis.
2. UNBOUNDED MEDIA. DISPERSION SURFACES In this Section we derive the Fresnel equation for the medium described above and consider which surfaces of wave normals may arise and finally which pairs of surfaces appear. These questions are especially important for optics, in particular, crystal optics. We assume that the wavelength is much longer than the typical length of structural elements of the metamaterial, so that the medium may be 276
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considered as continuous and characterized by the matrices of dielectric and magnetic permittivities. Furthermore, we assume also that the tensors of dielectric and magnetic permittivities, εˆ 0 and μˆ 0 , can be diagonalized in one and the same coordinate system and these tensors have in corresponding system of coordinate the form
⎛ ε1 0 ˆε0 = ⎜⎜ 0 ε2 ⎜0 0 ⎝
0⎞ ⎟ 0 ⎟, ε3 ⎟⎠
⎛ μ1 0 ⎜ μˆ 0 = ⎜ 0 μ 2 ⎜0 0 ⎝
0⎞ ⎟ 0 ⎟. μ3 ⎟⎠
(1)
Principal values of dielectric and magnetic permittivities are complex numbers, εi = ε 'i + iε ''i and μ i = μ 'i + iμ''i , i = 1,2,3 (complex form is chosen as exp(–iωt), hence media with positive imaginary parts of the wave vector are absorbing, while those with negative imaginary parts are amplifying). Real parts of dielectric and magnetic permittivities can be both positive and negative. We will assume below that the medium is uniaxial, i.e., ε 2 = ε 3 ≠ ε1 and μ 2 = μ 3 ≠ μ1 and the optical axis lies in the xz-plane and makes with the x-axis an angle φ; thus we have −1 εˆ = Tˆ [ y , φ] εˆ 0Tˆ [ y, φ] ,
(2)
−1 μˆ = Τˆ [ y, φ] μˆ 0 Τˆ [ y, φ] ,
(3)
where Tˆ [ y , φ] is the matrix of rotation by angle φ around the y-axis; this matrix has the form ⎡cos φ 0 − sin φ ⎤ ˆΤ [ y, φ] = ⎢ 0 1 0 ⎥⎥ . (4) ⎢ ⎢⎣ sin φ 0 cos φ ⎥⎦ From Maxwell equation for the field of a plane monochromatic wave we obtain for the wave vector the following dispersion equation:
( n (1 − δ ) + μ 2
ε
m
)(
)
ε m ( δε2 − 1) (1 − δμ ) + δε ξ n2 (1 − δμ ) + μ m ε m ( δμ2 − 1) (1 − δε ) + δμ ξ = 0,
(5)
where n 2 = nx2 + ny2 + nz2 , ξ = 2 ( nx cos φ + nz sin φ ) , εm = ( ε1 + ε2 ) / 2 , μ m = ( μ1 + μ 2 ) / 2 , the wave vector λ λ λ k x , ny = k y , nz = k z , δμ = ( μ1 − μ 2 ) / ( μ1 + μ 2 ) , and λ is components kx,y,z are defined by nx = 2π 2π 2π the 2
wavelength of light in vacuum. Dispersion equation (5) is equivalent to the following two equations: n 2 (1 − δε ) + μ m ε m ( δε2 − 1) (1 − δμ ) + δε ξ = 0 ,
(5.1)
n 2 (1 − δμ ) + μ m ε m ( δμ2 − 1) (1 − δ ε ) + δμ ξ = 0 .
(5.2)
Here (5.1) is the dispersion equation for the electric mode and (5.2) is that for the magnetic mode. Let us write the dispersion equation (5.1) in the canonical form. For this purpose we represent nx , ny and nz in the form nx = a n 1+ n2 + n3 , n y = n1 + b n2 , nz = n1 + n2 + c n3 ,
(6)
where
a = 1, b = 2
1 + δ ε sin 2φ 1 + δ ε cos 2φ + δε sin 2φ . , c= δε − 1 −1 + δε cos 2φ − δε sin 2φ
Then (5.1) acquires the form n12 n22 n32 + + = 1, λ1 λ 2 λ 3
(7)
where
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λ1 = ε m μ m ( 3 − δε + 2δε sin 2φ ) ( δ 2ε − 1) (1 − δμ ) ,
λ 2 = 2ε m μ m (1 + δε sin 2φ ) (1 − δμ ) ,
(8)
λ 3 = 2ε m μ m (1 + δε sin 2φ )( 3 − δε + 2δε sin 2φ )(1 + δε ) (1 − δμ ) .
The same algebra may be performed also for dispersion equation (5.2) of the magnetic mode. Dispersion surfaces describe dependence of refraction of electromagnetic wave in medium on the direction of wave propagation. Plane electromagnetic waves travelling inside the medium can produce, depending on parameters λ1 , λ 2 , and λ 3 , dispersion surfaces in the form of ellipsoids of revolution, oneor two-sheet hyperboloids, etc. Furthermore, depending on orientation of optical axis, intersections of these surfaces with specific planes (in particular, the plane of incidence) can produce circles, ellipses, hyperboloids, or straight lines. Below we consider specific cases. I. If in relation (7) the quantities λ1 , λ 2 , λ 3 are positive, i.e.,
⎧ f = ε m μ m ( δε − 1) ( δμ − 1) > 0, ⎪⎪ ⎨ g = ( δμ + 1)( 3 − δμ + 2δμ sin 2φ ) > 0, ⎪ ⎪⎩h = ( δμ − 1) (1 + δμ sin 2φ) < 0, then the dispersion surface of the electric mode is an ellipsoid of revolution with semiaxes directed along n1 , n 2 , and n3 : n1 = ( nˆ x + nˆ z )(1 + δε sin 2φ ) + ( nˆ x − nˆ z ) δε cos 2φ + nˆ y (1 − δε ) , n 2 = ( nˆ x + n z − 2nˆ y ) (1 + δε sin 2φ ) + ( nˆ x − nˆ z ) δε cos 2φ , n 3 = nˆ x − nˆ z ,
where nˆ x , nˆ y , and nˆ z are unit vectors of respective axes. II. If λ1 < 0, λ 2 < 0, and λ 3 < 0 (i.e., at f < 0, g > 0, and h < 0), then the mode represented by equation (5.1), is evanescent. III. If one of parameters λ1 , λ 2 , λ 3 is negative, while the others are positive (i.e., at f < 0 and g < 0 or at f < 0 and h > 0), then the dispersion surface is a one-sheet hyperboloid with semiaxes along directions n1 , n 2 , and n3 . IV. If one of parameters λ1 , λ 2 , λ 3 is positive, while the others are negative (i.e., at f > 0 and g < 0 or at f > 0 and h > 0), then the dispersion surface is a two-sheet hyperboloid with semiaxes along directions n1 , n 2 , and n3 . V. If δε = 1 , the dispersion surface is a plane, since in this case we obtain from equation (5.1) nx cos φ + nz sin φ = 0 .
We should note that at δ ε = −1 the dispersion equation acquires the form n y2 + ( nz cos φ − nx sin φ ) = 0 . This means that at n y = 0 the dispersion surface is a straight line nz = nx tg φ . In opposite case the mode is evanescent. Emphasize that a plane appears only at δε = 1 and the straight line only at δ ε = −1 . Figure 1 shows possible pairs of dispersion surfaces (one for electric mode and one for magnetic mode) at different parameters of problem. They are determined from dispersion equation (5). We note that in general case it is impossible to obtain the following pairs: ellipsoid of revolution with one-sheet hyperboloid, one-sheet hyperboloid with two-sheet hyperboloid, and one evanescent mode with two-sheet hyperboloid. This is natural, since it may be proved analytically that λ1λ 2 λ 3 λ 4 λ 5 λ 6 > 0 , i.e., there exists only even number of negative λ i . Here λ 4 , λ 5 , and λ 6 are corresponding coefficients of dispersion equation for the magnetic mode written in canonical form. They are obtained from λ1 , λ 2 , and λ 3 by replacements δε → δμ and μ m → ε m in (8). We note also that if dispersion surface of one of modes is a straight line, the other one is either a conical surface shown in Fig. 1e (which in special cases becomes either a plane or a line) or evanescent. Indeed, at δε = 1 (a special case) dispersion equation for the magnetic mode has apparently the form pnx2 + qn y2 + rnz2 + snx nz = 0 , where p, q, r, and s are some parameters characterizing the medium. This means that the corresponding dispersion surface is either conical or the mode is evanescent. But if the dispersion surface of one of modes is a straight line, the other one may be ellipsoid of 2
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(b)
0 nz
279
(c)
ny 0
0 ny
0 nz
0 nz
0 ny
0 nx
0 nx
0 nx
(e)
(d)
0 nx
ny 0
0 nz
ny
0 nz
0 0 nx
Fig. 1. Dispersion surfaces for different parameters of medium: (a) ε1 = 2.5, ε2 = 1.5, φ = π/3, µ1 = 1.7, µ2 = 2.9; (b) ε1 = 3.1, ε2 = 2.5, φ = π/4, µ1 = −1.3, µ2 = 2.2; (c) ε1 =1.2, ε2 = −1.5, φ = π/5, µ1 =1.3, µ2 = −1.1; d) ε1 = −22, ε2 = 3, φ = π/4, µ1 =1.3, µ2 = −2.2; (e) ε1 = 2.5, ε2 = 0, φ = π/4, µ1 = −0.9, µ2 = 3.7.
revolution, one- or two-sheet hyperboloid, or a plane (Fig. 2). At last, if both surfaces are straight lines, they coincide necessarily. 3. REFLECTION FROM HALF-SPACE Let us consider reflection and refraction of light on the interface between isotropic non-absorbing medium and anisotropic metamaterial. Geometry of problem is shown in Fig. 3. The medium occupies half-space z ≥ 0 , i.e., the interface coincides with the xz-plane ( xyz is the lab system). Electromagnetic wave at the frequency ω is incident from medium 1 onto the considered half-space (medium 2) at angle α. Medium 1 is homogeneous and isotropic with parameters εl and μl (dielectric and magnetic permittivities of the medium). According to Berreman technique of 4×4 matrices, the system of Maxwell equations may be represented in the following form: ∂ ψ = −ik0 Δˆ ψ , (9) ∂z where the column-vector ψ and Berreman 4×4 matrix Δˆ have the form ψ = ⎡ Ex ⎣
Hy
Ey
T
−H x ⎤ , ⎦
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(b) 0 ny
nz 0
nz 0 0 ny 0 nx
0 nx
(c)
ny 0
(d)
ny 0
0 nz
nz 0
0 nx
0 nx
Fig. 2. Dispersion surfaces in case where one of them is a straight line: (a) ε1 = 0, ε2 = 0.7, φ = π/3, µ1 = 1.2, µ2 = 1.1; (b) ε1 = 0, ε2 = 0.7, φ = π/3, µ1 = 1.2, µ2 = −1.5; (c) ε1 = 0, ε2 = 0.7, φ = π/3, µ1 = −1.2, µ2 = 1.5; (d) ε1 = 0, ε2 = 0.7, φ = π/3, µ1 =1.5, µ2 = 0.
⎛ ε31 nx ⎜ − ε33 ⎜ ⎜ ε ε − ε2 ⎜ 11 33 13 ⎜ ε33 Δˆ = ⎜ ⎜ 0 ⎜ ⎜ ⎜ ε 23ε31 ⎜ ε 21 − ε33 ⎝
μˆ 22 − −
μ 223 nx2 − μ33 ε 33 ε13 nx ε33
μ μ μ12 − 13 32 μ33 ⎛ μ32 ε 23 ⎞ − ⎜ ⎟ nx ⎝ μ33 ε33 ⎠
⎛ μ 23 ε32 ⎞ − ⎜ ⎟ nx ⎝ μ33 ε33 ⎠ ε12 ε 33 − ε13ε 32 ε33 μ13 nx μ33 ε 22 −
ε 223 nx2 − ε33 μ33
μ 23μ31 ⎞ ⎟ μ33 ⎟ ⎟ ⎟ 0 ⎟ ⎟, 2 μ13 ⎟ μ11 − μ33 ⎟ ⎟ ⎟ μ31 nx ⎟ μ33 ⎠
μ 21 −
(11)
With the following notations: n x = ε l μ l sin α , ε11 = ε m (1 + δε cos 2φ ) , ε 22 = ε m (1 − δε ) , ε13 = ε 31 = ε m δ ε sin 2φ ,
ε33 = ε m (1 − δε cos 2φ ) , ε12 = ε 21 = 0 , ε 23 = ε 32 = 0 , μ 22 = μ m (1 − δμ ) , μ33 = μ m (1 − δμ cos 2φ ) , μ11 = μ m (1 + δμ cos 2φ ) , μ12 = μ 21 = 0 , μ 23 = μ 32 = 0 .
Since for homogeneous anisotropic medium Δˆ is constant and independent of z, the z-dependence of fields may be represented in the form JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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Fig. 3. Geometry of the problem.
Ψ ( z ) = Ψ j ( 0 ) exp(ik jz z ), j = 1, 2,3, 4 .
(12)
By substituting (12) into (9), we obtain matrix equation for eigenvalues, ⎛ ˆ ωˆ⎞ ⎜ kz I − Δ ⎟ Ψ ( 0) = 0 , c ⎠ ⎝ ˆ where I is unit matrix. Eigenvalues of this equation are roots of fourth-order algebraic equation,
( n (1 − δ ) + μ 2
ε
m
)
ε m ( δε2 − 1) (1 − δμ ) + δε ξ ( n 2 (1 − δμ ) +
)
(13)
(14)
+ μm ε m ( δμ2 − 1) (1 − δε ) + δμ ξ = 0,
where n2 = nx2 + nz2 . Note that the dispersion equation is not biquadratic as it is the case for isotropic or anisotropic media, where optical axis is perpendicular (or parallel) to interface of media. Solutions of equation (14) have the form ± B − nx δμ sin 2φ ± A − nx δε sin 2φ , nz 3,4 = , (15) nz1,2 = 1 − δε cos 2φ 1 − δμ cos 2φ where A= В=
( ( δ − 1) ( n (δ
2 ε
2 μ
) cos 2φ ) ) .
− 1) nx2 + ε m μ m ( δμ − 1) (1 − δε cos 2φ ) , 2 x
+ ε mμ m ( δ ε − 1) (1 − δμ
(16)
Difference of magnitudes of forward and backward travelling waves determined by solutions (15) (i.e., difference of absolute values of n1z and n2 z and similarly n3z and n4 z ) is evidence of non-interaction of waves in considered system: n ( k , ω) ≠ n ( −k , ω) .
(17)
Proceed now to study of peculiarities of dependence of z-components of the phase and group velocities on the orientation of optical axis with respect to interface. Figure 4 shows dependence of niz on angle φ. It may be observed in the figure that very different situations are possible, specifically, situations where niz are positive for two modes and negative for other two modes, positive for three modes and negative for one mode, positive for all four modes, and negative for all four modes. This means that the system has phase non-reciprocity. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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Fig. 4. Dependence of phase velocity on orientation of optical axis with respect to boundary surface for ε1 = −1.5, ε2 = 0.5, µ1 = 1.5, µ2 = −1, α = π/3, εl = 1, µl = 1.
For x- and z-components of group velocities of electric modes we have A Vgz1 = −c , Vgz 2 = −Vgz1 , ε m μ m ( δμ − 1) ( δε2 − 1) Vgx1 = c
nx (1 − δ2ε ) − δε A
ε m μ m ( δμ − 1) ( δε2 − 1) (1 − δε cos 2φ )
, Vgx 2 = Vgx1 ,
while x- and z-components of group velocities of magnetic modes are obtained by replacements δε → δμ and μ m → ε m . Figure 5 demonstrates the dependence of Vgiz on angle φ. It is seen that these curves are symmetric with respect to φ-axis. The situation where three Vgiz -components have a sign opposite to that of the fourth component is impossible. Note that the explicit form of matrix Δˆ and solution of dispersion equation allows constructing the transfer-matrix Pˆ (d), which in its turn allows studying peculiarities of inhomogeneous anisotropic media and multilayer structures. This is proposed to be done in our subsequent work. We pass now to detailed analysis of dispersion equation (14). We study the dependence of dispersion curves on the orientation of optical axis. Dispersion equation (14) is equivalent to the following two equations: n 2 (1 − δε ) + μ m ε m ( δε2 − 1) (1 − δμ ) + δε ξ = 0 ,
(18)
n 2 (1 − δμ ) + μ m ε m ( δμ2 − 1) (1 − δ ε ) + δμ ξ = 0 .
(19)
Performing corresponding rotation in the xz-plane we a obtain canonical form of equation (18): n12 n22 + = 1, λ1 λ 2
(20)
Fig. 5. Dependence of the group velocity on the angle of orientation of optical axis with respect to boundary surface for ε1 = 2.5, ε2 = 1.5, µ1 = −0.5, µ2 = 1, α = π/3, εl = 1, µl = 1. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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where ⎧λ1 = μ m ε m (1 − δμ ) (1 − δε2 ) (1 + δε cos 2φ ) , n1 = nx (1 + δε cos 2φ ) + nz δε sin 2φ , ⎪ (21) ⎨ 1 1 cos 2 , n n . λ = μ ε − δ + δ φ = ( ) ( ) ⎪⎩ 2 m m μ ε z 2 Dispersion equation for the magnetic mode has similar form, but in this case in expressions for λ1 , λ 2 and n1 , n2 in (21) the following replacements should be done: δε → δμ , μ m → ε m . Dispersion surfaces (20) may also be systemized by means of analysis of signs of quantities λ1 and λ 2 . In general the following cases can be distinguished: I. If λ1 and λ 2 are positive, i.e.,
⎧⎪μ m ε m (1 − δμ ) (1 + δε cos 2φ ) > 0, (22) ⎨ 2 − δ > 1 0, ⎪⎩ ε sections of dispersion surfaces by the plane of incidence are ellipses with semi-axes along the directions n1 = nˆ x (1 + δε cos 2φ ) + nˆ z δε sin 2φ and n 2 = nˆ z . II. If λ1 and λ 2 are negative, i.e., ⎧⎪μ m ε m (1 − δμ ) (1 + δε cos 2φ ) < 0, ⎨ 2 ⎪⎩1 − δε < 0,
(23)
the modes given by equation (18) are evanescent. In this case anisotropic finite-thickness layer can become an ideal mirror and light incident at such a layer is totally reflected at arbitrary angles of incidence and polarization. Hence, such a layer is an omnidirectional reflector. III. If λ1 and λ 2 are of opposite signs, i.e.,
δε2 > 1 ,
(24)
dispersion surfaces are hyperboloids with semi-axes along the directions n1 and n 2 . IV. If δ ε = ±1 , dispersion curves are straight lines (note that the lines arise in only this case). At last, we proceed to study the dependence of dispersion curves on parameter φ characterizing the orientation of optical axis. Figure 6a shows the dependence of dispersion curves on the angle φ,for those parameters of problem for which the dispersion curve is an ellipse. If the angle of rotation of optical axis φ equals kπ/2 (with k an integer), semi-axes of ellipse are directed along nˆ x and nˆ z , which is also observed in Fig. 6a. At other values of this angle ellipse semi-axes are shifted from directions nˆ x и nˆ z , which also is seen in Fig. 6a. Figure 6b demonstrates the dependence of dispersion curves on angle φ at all those parameters of problem at which the dispersion curve is a hyperbola. Figure shows that at variation of angle φ the (b)
(a)
0
nx
0 nz
π 2
nz 0 0 0 nx
−
π 2
φ, rad
π 2
0 π − 2
φ, rad
Fig. 6. Dependence of the dispersion curves on the angle of orientation of optical axis for different parameters of medium: (a) ε1 = 1.2, ε2 = 3.8, µ1 = 1.5, µ2 = 1.1; (b) ε1 = 1.2, ε2 = −0.7, µ1 = 1.5, µ2 = 1.2. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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dispersion curves (hyperbolas) rotate in the nx nz plane and at certain values of this angle the axes nˆ x and nˆ z become asymptotes. Note that the results described above are valid for also the magnetic mode (it is sufficient to perform the following replacements: δε → δμ , μ m → ε m ). 4. CONCLUSION We studied the peculiarities of dispersion surfaces for anisotropic metamaterials with dielectric and magnetic anisotropies. It was shown that at different orientations of optical axis the following pairs of dispersion surfaces can arise: two ellipsoids of revolution, ellipsoid of revolution and two-sheet hyperboloid, two two-sheet hyperboloids, two one-sheet hyperboloids, plane and conical surface. If one of dispersion surfaces becomes a straight line, it may arise together with any of above-listed types (but not conical). In other cases modes are evanescent. We examined the dependence of dispersion curves on orientation of optical axis. As it is apparent in the canonical form of the dispersion equation (20), (21), it is impossible to transform a curve from ellipse to hyperbola and vice versa by changing the orientation of optical axis (in the plane of incidence). The change in orientation of optical axis leads only to changes in magnitudes and to rotation of axes of these curves. ACKNOWLEDGMENTS We express our gratitude to M.Z. Harutyunyan, S.G. Rafayelyan, and V.A. Arzumanyan for valuable discussions. The work was supported by the grant 13A-1c34 of State Committee of Science of MES of the Republic of Armenia. REFERENCES 1. Veselago, V.G., Sov. Phys. Usp., 1968, vol. 10, p. 509. 2. Smith, D.R., Padilla, W.J., et al., Phys. Rev. Lett., 2000, vol. 84, p. 4184. 3. Shelby, R.A., Smith, D.R., and Schultz. S., Science, 2001, vol. 292, p. 77. 4. Shalaev, V.M., Nature Photonics, 2007, vol. 1, p. 41. 5. Lee, S.H., Park, C.M., Seo, Y.M., and Kim, C.K., Phys. Rev. B, 2010, vol. 81, p. 241102. 6. Pendry, J.B., Schurig, D., and Smith D.R., Science, 2006, vol. 312, p. 1780. 7. Alu, A. and Engheta, N., Phys. Rev. E, 2005, vol. 72, p. 016623. 8. Leonhardt, U., Science, 2006, vol. 312, p. 1777. 9. Landy, N.I., Sajuyigbe, S., et al., Phys. Rev. Lett., 2008, vol. 100, p. 207402. 10. Lindell, I.V., Tretyakov, S.A., et al., Microw. Opt. Technol. Lett., 2001, vol. 31, p. 129. 11. Silin, R.A., Neobychnye zakony prelomleniya (Unusual Laws of Refraction), Moscow: FAZIS, 1999. 12. Smith, D.R. and Schurig. D., Phys. Rev. Lett., 2003, vol. 90, p. 077405. 13. Belov, P.A., Microw. Opt. Technol. Lett., 2003, vol. 37, p. 259. 14. Shen, N.H., Wang, Q., Chen, J., et al., Phys. Rev. B, 2005, vol. 72, p. 153104. 15. Depine, R.A., Inchaussandague, M.E., and Lakhtakia, A., J. Opt. Soc. Amer. A, 2006, vol. 23, p. 949. 16. Luo, H., Hu, W., Shu, W., Li, F., and Ren, Z., Europhys. Lett., 2006, vol. 74, p. 1081. 17. Jen, Y.-J., Lakhtakia, A., Yu, C.-W., and Lin, C.-T., Eur. J. Phys., 2009, vol. 30, p. 1381. 18. Chen, H., Xu, Sh., and Li, J., Opt. Express, 2009, vol. 17, p. 19791. 19. Liu, H., Lv, Q., Luo, H., et al., J. Opt A, Pure Appl. Opt., 2009, vol. 11, p. 105103. 20. Markel, V.A. and Schotland, J.C., J. Opt., 2010, vol. 12, p. 01510. 21. Xiang, Y., Dai, X. and Wen, S., Opt. Commun., 2007, vol. 274, p. 248. 22. Yonghua, L., Pei, W., Peijun, Y., Jianping, X., and Hai, M., Opt. Commun., 2005, vol. 246, p. 429. 23. Lekner, J., JOSA A, 1993, vol. 10, p. 2059.
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