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Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media

Transformation-media designed by standard transformation-optics (TO) approaches, based on real-valued coordinate-mapping, cannot exhibit single-negative (SNG) character unless such character is already possessed by the domain that is being transformed. In this paper, we show that, for a given field polarisation, pseudo-SNG transformation media can be obtained by transforming a domain featuring double positive (or double-negative) character, via complex analytic continuation of the coordinate transformation rules. Moreover, we apply this concept to the TO-based interpretation of phenomena analogous to the tunnelling effects observable in bi-layers made of complementary epsilon-negative (ENG) and mu-negative (MNG) media, and explore their possible TO-inspired extensions and generalizations.


Introduction and background
Transformation optics (TO) (see e.g., [1][2][3][4][5]) is a powerful and systematic approach to the synthesis of metamaterials with broad field-manipulation capabilities. Its technological viability is rapidly getting established, in view of the formidable advances in the fabrication of artificial materials and metamaterials with controllable anisotropy and spatial inhomogeneity properties. The reader is referred to [6][7][8][9][10][11][12][13][14] for a sparse sampling of recent representative applications, ranging from the celebrated "invisibility cloaking" [7], to hyper/super-lensing scenarios [8][9][10][11], and electromagnetic (EM) analogous of relativistic effects [12] and celestial-mechanical phenomena [13][14]. Standard TO is based on the design of a suitable real-valued coordinate transformation, which produces the desired field-behaviour in a curved-metric fictitious space. Thanks to the formal invariance of Maxwell equations, this behaviour may be translated to a flat-metric physical space filled up with an anisotropic, spatially-inhomogeneous "transformation medium" [1][2][3][4][5]. The constitutive parameters of such transformation medium are systematically derived from those of the fictitious space via the Jacobian matrix of the transformation [see, e.g., equation (1) below], and this imposes some restrictions on the material properties that can be obtained by transforming a given scenario into another. For instance, in the standard TO approach, the reciprocity character of the domain that is being transformed cannot be changed [15]. Moreover, the signs of the constitutive-tensor eigenvalues can either be all preserved or all flipped [15]. Thus, e.g., transforming a domain featuring a double positive (DPS) character, one can only obtain a DPS or a double negative (DNG) transformation medium (and vice-versa).
The latter case is obtained via coordinate folding, and allows (though with some caveats [16]) the TO interpretation of interesting effects such as, e.g., superlensing [10][11] and anti-cloaking [17][18]. To sum up, important material properties, such as non-reciprocity and single-negative (SNG) character, cannot be generated via standard TO, unless already possessed by the domain that is being transformed. In order to overcome these limitations, some possible generalizations have been proposed. For instance, in [15] an extension based on "triple spacetime metamaterials" has been proposed, which allows geometrical interpretation of non-reciprocal and indefinite media. In [19], as a possible extension of a general class of transformation slabs, we speculated that, for a given field polarisation, SNG transformation media could be obtained by transforming a domain featuring DPS (or DNG) character, via complex analytic continuation of the coordinate transformation. In this paper, we further elaborate on this concept, and address its application to the TO interpretation of the tunnelling effects observed in [20] in connection with bi-layers of (homogeneous, isotropic) epsilon-negative (ENG) and munegative (MNG) slabs, and their further generalization to (inhomogeneous, anisotropic) complementary media [21,22]. Our results reproduce as special cases those in [20][21][22], and suggest other TO-inspired broad tunnelling conditions. Accordingly, the rest of the paper is laid out as follows. In Section 2, we introduce the problem scenario and formulation. In Section 3, we outline the main analytical derivations, starting from the field calculation, and proceeding with the study of total-transmission conditions. In Section 4, we provide a physical interpretation of the results, and illustrate their relationship with those in [20][21]. In Section 5, we illustrate some novel representative results, not amenable to those in [20][21]. Finally, in Section 6, we provide some brief conclusions and hints to future research directions related to these findings.

Problem formulation
In [19], in connection with a general class of (DPS or DNG) transformation slabs, we have theoretically speculated that SNG transformation media could be generated by transforming a domain featuring DPS or DNG character, via a suitable analytical continuation of the coordinate transformation in the complex plane. From a physical viewpoint, one intuitively expects such coordinate transformation to exhibit an in-plane purely imaginary character, in order to map a propagating field solution in the fictitious "above cut-off" space into an evanescent "below cut-off" one in the transformed (SNG) domain. In this framework, while a single SNG transformation slab would be of limited interest (being inherently opaque), it may be of interest to verify that paired (ENG-MNG) transformation-media configurations may support interesting tunnelling phenomena under proper matching conditions, in analogy to what was shown in [20][21]. Accordingly, as illustrated in figure 1(a), we consider a transformation bi-layer which occupies the region x d < in a physical space ( ) , , x y z , characterised by the relative permittivity and permeability tensors from a fictitious (vacuum) space ( ) , , x y z ′ ′ ′ . In (2), and henceforth, 0 a α ≠ are real scaling parameters,  (2)]. We point out that the transformation in (2) is neither the only one, nor the most general in order to achieve the sought SNG character, and that it was chosen in view of its close resemblance to the (real) transformation considered in [19], so as to exploit (with some minor variations) our previous analytical derivations. As shown in [19], the constitutive tensors in (1) are real and symmetric, and hence they can be diagonalised, thereby providing a more insightful interpretation. Accordingly, by indicating with ( ) , , z α α ξ υ the principal reference systems (constituted by the orthogonal eigenvectors), and with α ξ Λ and α υ Λ the in-plane eigenvalues, the diagonalised forms can be written as (see [19] for details) Therefore, depending on field polarisation and on the sign of the scaling parameters a α , the corresponding transformation media may effectively behave as either ENG or MNG, as compactly summarised in table 1. However, since they exhibit by construction α α ε µ = , the terms "ENG," "MNG," and "SNG" may appear not entirely appropriate, and throughout the paper we shall refer to them as "pseudo-ENG" (P-ENG), "pseudo-MNG" (P-MNG), and "pseudo-SNG" (P-SNG).
In what follows, generalising the studies in [20][21], we analyse the transmission properties of bi-layers made of paired P-ENG/P-MNG transformation media.

Field calculation
As a first step, we calculate analytically the EM response for (unit-amplitude) plane-wave incidence, and, without losing in generality, we focus on the TM-polarised case (z-directed magnetic field). In the fictitious (vacuum) space, no reflection takes place, and the total field coincides with the incident one, which can be written as where, for a propagating wave with incidence angle i θ with respect to the x-axis (see figure 1), the x-and ydomain wave numbers are given by denoting the vacuum wave number (and 0 λ the corresponding wavelength).
In the standard TO approaches, the field in the physical space would be obtained by straightforward coordinate mapping [via (2)] of the fictitious-space expression in (5). However, in view of the discontinuity of the coordinate transformation in (2) at the interfaces 0, = ± , such mapping is not straightforward here, and special care is needed. Specifically, the physical-space field can be written as a superposition of forward-and backward-propagating coordinate-mapped [via (2)] plane waves, which can be compactly written as is assumed. In (7), the amplitude coefficients A α ± , 0,1, 2,3 α = , as well as the wave numbers x k α and y k α , 1, 2,3 α = , need to be computed by enforcing the boundary conditions. First, we note that 0 1 A + = , consistently with the assumed unit-amplitude excitation [cf. (5)], whereas 3 0 A − = in order to fulfil the radiation condition. Next, by enforcing the phase-matching conditions at the interfaces x d = ± and 0 x = , we obtain which yield all the unknown wave numbers in (7)  tanh tanh tanh tanh exp 2 , tanh tanh tanh tanh

Total transmission conditions
Based on the analytical solution above, we can now derive the conditions for total transmission, by zeroing the reflection coefficient in (10) In general, the above equation needs to be solved numerically, and, for a given bi-layer configuration, its solutions [if any, and provided they are not roots of the denominator in (10)  can be verified that their Wronskian is generally nonzero, unless both functions vanish 1 . Recalling (11) (and that, by assumption, 0 a α ≠ and 0 u α > ɺ ), this implies the following necessary condition: From (8) and (9), it is evident that, by enforcing , 0 so that the total-transmission condition in (14) can be rewritten as 10 20 tanh tanh 0 To sum up, the conditions 1 Alternatively, it can be verified that the product of two hyperbolic tangents (of generally nonzero argument) in (14) cannot be exactly represented in terms of the functions 0 χ and 12 χ (which do not contain any exponential term). turn out to be necessary and sufficient in order to obtain total transmission for any value of 0 y k , as it can also be verified that they do not imply the vanishing of the denominator in (10). Note that the conditions in (21) depend on the scaling parameters a α and on the boundary values of the functions u α and their derivatives, but are independent of the actual form of these functions as well as of the functions v α . Moreover, while they may be interpreted as angle-independent conditions, they implicitly depend on frequency, in view of the unavoidable temporal-dispersion effects in passive SNG media.
Recalling the assumed positive character of the derivatives u α ɺ , which implies that ( ) ( ) and thus, depending on the field polarisation (see table 1), we obtain two possible configurations corresponding to the P-ENG/P-MNG and P-MNG/P-ENG pairings. Under the total-transmission condition, the total phase accumulated by the field transmitted through the bilayer can be written as and therefore, at normal incidence ( ) 0 0 y k = for arbitrary functions v α , or at arbitrary incidence for − − , the field will undergo a complete tunnelling through the bi-layer, without any phase delay. The bi-layer thus behaves as an EM "nullity," in close resemblance with the case of conjugate matched (homogeneous, isotropic) ENG-MNG pairs in [20], and its (inhomogeneous, anisotropic) complementary-media generalization in [21,22] (see also the discussion in Sec. 4 below). More in general, an arbitrary aperture field distribution located at a source-plane along the y-axis.

Special cases
In order to understand the physical mechanisms underlying the rather general class of transparent SNG transformation bi-layers identified by the conditions in (21), it is insightful to start by explicating their relation with the homogeneous, isotropic ENG-MNG bi-layers investigated in [20], and their further (inhomogeneous, anisotropic) generalizations within the framework on complementary media [21,22]. In [20], it was shown that complete tunnelling (with zero phase delay) could be achieved by pairing two slabs (of thickness 1 d and 2 d ) made of homogenous, isotropic ENG and MNG materials. In particular, under the conjugate-matching condition: such tunnelling effects would occur for any value of the incident wave number, as long as the above condition (24) remains satisfied. Such conditions were further generalised in [21,22] to the case of anisotropic, inhomogeneous (possibly lossy) materials satisfying the complementary-media condition, which, in our chosen reference system, can be expressed as It can easily be verified that our general class in (21) reduces to the complementary-media case in (25) for which, in turn, yields the homogeneous, isotropic case in (24) for ( ) In this framework, we point out that some seeming restrictions in our class [e.g., the fact that α α ε µ = , cf. (1) ] are only due to our original choice of transforming a vacuum fictitious space, and can accordingly be overcome by considering a more general filling (not necessarily homogeneous and isotropic, and possibly DNG).

Complex-mapping-induced surface modes
While the complementary-media arguments above provide an interesting interpretation of some special cases, the conditions in (21) define a broad class of configurations that do not generally satisfy the symmetry conditions in (24) or (25). A deeper physical insight can be gained by looking at the analytical structure of the field inside the bi-layer region. As anticipated, one intuitively expects the in-plane purely-imaginary character of the coordinate transformation in (2) to map a propagating field [in the vacuum fictitious space, cf. (5)] into an evanescent one in the P-SNG transformed domain. However, when pairing two conjugate SNG transformation slabs, an exponentially-growing behaviour may also occur. Specifically, by particularising the general expression in (7) under the total-transmission conditions in (21), we obtain We note that the field exhibits unit amplitude at the interfaces x d = ± (as expected, in order to match the incident and totally-transmitted fields), while its magnitude inside the bi-layer region is always greater than unity, exponentially growing towards the interface 0 x = , where it reaches its peak. Thus, the complexmapping in (2), under the total-transmission conditions in (21) In the complementary-media case [20][21], the presence of surface modes has been observed and associated with tunnelling phenomena. Our complex-mapping arguments above provide an alternative TO-based interpretation, and extend the association between surface modes and tunnelling phenomena beyond the complementary-media scenario.

Representative numerical results
As a representative example, we consider a configuration featuring a homogeneous, isotropic MNG slab ( ) paired with an inhomogeneous, anisotropic P-ENG transformation slab ( ) where the polynomial coefficients in 2 u have been chosen so as to satisfy the total-transmission conditions in (21), as well as to ensure that ( ) Figure 3 illustrates the distributions of the relative-permittivity principal components of the P-ENG transformation slab, from which the inhomogeneous, anisotropic (and, obviously, negative) character is evident. We highlight that the bi-layer configuration defined by (29) and (30) does not satisfy the symmetry conditions in (24) or (25), and thus its EM response cannot be explained within the complementary-media framework. We also investigated the effects of the unavoidable material losses in the above illustrated tunnelling phenomena (see [20,22] for the complementary-media case). With reference to the parameter configurations in figures 4 and 5, figure 6 shows the transmission coefficient magnitude as a function of the material loss-tangent tanδ (assumed identical for the two constituents of the bilayer). The lossy configuration was obtained as a perturbation of the lossless one above, by multiplying all the negative constitutive parameters ( ) , so as to ensure the positivity of their imaginary parts, and hence the passivity of the media. As it can be observed, the tunnelling effects turn out to be rather robust up to moderate (loss-tangent=0.01) levels of losses. Being inherently a resonant phenomenon, however, significant metamaterial losses may significantly affect the tunnelling, as confirmed in Fig. 6. Analogous to the case of complementary SNG pairs [20], the loss effect is predicted to be more sensitive to thicker complementary layers.

Conclusions
In this paper, we have presented a TO-based interpretation of tunnelling phenomena in bi-layers made of paired P-ENG/P-MNG media. Our proposed interpretation, which relies on a complex coordinate transformation, includes the (homogeneous, isotropic) scenario in [20] and its (inhomogeneous, anisotropic) complementary-media extensions in [21,22], and suggests other interesting generalizations, thereby providing further evidence of the versatility and unifying character of TO. Current and future research are aimed at the exploration of different TO-inspired scenarios, such as bi-layers made of P-SNG and DPS media, for which interesting tunnelling effects may occur at given frequencies and incidence angles.