https://doi.org/10.1051/epjam/2016003
Research Article
Degeneratebandedge engineering inspired by nonlocal transformation optics
Waves Group, Department of Engineering, University of Sannio, Corso Garibaldi 107, 82100
Benevento, Italy
^{*} email: vgaldi@unisannio.it
Received:
18
March
2016
Accepted:
30
May
2016
Published online: 5 July 2016
We address the engineering of degeneratebandedge effects in nonlocal metamaterials. Our approach, inspired by nonlocaltransformationoptics concepts, is based on the approximation of analyticallyderived nonlocal constitutive “blueprints”. We illustrate the synthesis procedure, and present and validate a possible implementation based on multilayered metamaterials featuring anisotropic constituents. We also elucidate the physical mechanisms underlying our approach and proposed configuration, and highlight the substantial differences with respect to other examples available in the topical literature.
Key words: Dispersion engineering / Optical nonlocality / Transformation optics / Degenerate band edge
© M. Moccia et al., Published by EDP Sciences, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Nonlocal lightmatter interactions [1, 2] are becoming increasingly relevant in a broad variety of application scenarios including dispersion engineering [3], ultrafast nonlinear optical response [4], artificial magnetism [5], additional extraordinary waves [6], enhanced spontaneous emission [7, 8], Diracpoint conical dispersion [9], scattering suppression [10], topological transitions [11], and lightbased analog signal processing [12].
In a series of recent investigations [13, 14], we have been concerned with possible extensions of the wellestablished transformationoptics (TO) framework [15, 16] so as to systematically engineer metamaterials exhibiting desired nonlocal effects. As opposed to the conventional TO formulation, which exploits the forminvariant properties of Helmholtz or Maxwell’s equations with respect to spatial (and, hence, inherently local) coordinate transformations, the basic idea underlying our approach is to apply the coordinatetransformation machinery in the wavevectorfrequency phasespace, accessed via spatial Fourier transform. In such phase space, nonlocal effects naturally manifest themselves in terms of wavevectordependent constitutive properties, and can be associated with nonlinear wavevector transformations. In a series of application examples, we illustrated the insightful correspondences between typical nonlocal effects and the geometrical/analytical characteristics of the associated transformations, which indicate that the powerful geometricallydriven design, typical of conventional TO, is retained by our approach, with the actual synthesis problem reduced to the suitable approximation of analyticallyderived constitutive “blueprints”. For instance, we showed that (non)reciprocal effects and the appearance of additional extraordinary waves [6] are directly related to the (non)centersymmetry and multivaluedness, respectively, of the transformations [13]. Moreover, we derived the analytical properties of classes of transformations that could induce stationary points in the dispersion diagram, and we also showed that frequencydependent wavevector transformations enable a finer tailoring in the phase space, thereby opening up the possibility to engineer complex spatiotemporal dispersion effects such as Diracpoint conical singularities [14].
In this paper, as a further illustration of the potential of the above approach, we apply it to the engineering of degenerate band edge (DBE) effects. Such exotic dispersion effects, first studied by Figotin, and Vitebskiy [17–21], are eliciting a growing attention (see, e.g., [22–30] and references therein) in view of their potential relevance to diverse applications including slow light, solidstate lasers, quantumcascade lasers, sensors, optical delay lines, travelingwave tubes, distributed solidstate amplifiers, and switches. Here, inspired by our nonlocal TO approach, we investigate an alternative metamaterialbased design, which is amenable to a multilayered implementation.
Accordingly, the rest of the paper is structured as follows. In Section 2, we introduce the problem statement and our general design strategy. In Section 3, we outline the synthesis procedure, from the derivation of the ideal constitutive blueprints to the actual multilayered implementation. In Section 4, we illustrate some representative examples, and we validate them via fullwave numerical simulations, by ascertaining the emergence of typical DBE physical “footprints”. Finally, in Section 5, we draw some brief conclusions, and provide few hints for future research.
2 Problem statement and geometry
In a series of influential studies [17–21], Figotin and Vitebskiy investigated the theoretical implications of stationary points in a dispersion relationship. Assuming a timeharmonic exp(−iωt) dependence, and given a general dispersion law ω(β) relating the angular frequency ω and the propagation constant β, a νth order stationary point at ω = ω_{0} is characterized by(1)with β_{0} = β(ω_{0}). This implies that, within a neighborhood of such point, the dispersion law behaves as(2)
The most trivial example of stationary point is the socalled regular band edge, corresponding to ν = 2 in equations (1) and (2), and naturally occurring in periodic structures such as photonic crystals. Less common and more subtle are higherorder cases such as ν = 3 and ν = 4. The former (ν = 3) is associated with the socalled “frozenmodes”, which can occur, e.g., in suitably engineered photonic crystals containing anisotropic material constituents. The latter (ν = 4) constitutes the DBE case of specific interest here, and differs fundamentally from the regular band edge case above, mainly due to the critical contribution of degenerate evanescent modes. The reader is referred to [17–21] for a thorough discussion of the theoretical details and related physical implications.
Here, we limit ourselves to emphasize that DBEs may give rise to giant slowwave resonances which, by comparison with the regularbandedge counterparts, tend to exhibit much larger energy accumulation and, consequently, much stronger lightmatter interactions. This may lead, for instance, to giant gain enhancement [30].
DBEs can be obtained in photonic crystals containing uniaxial media with suitably tilted optical axes, as an effect of the coupling between two modes with different polarizations [17–21, 30]. Alternatively, they can also be engineered in optical fibers with multiple gratings [22, 24, 27], coupled periodic waveguides [23, 25, 26, 28], and periodicallyloaded circular waveguides [29].
Here, we explore a different configuration based on a nonlocal metamaterial. As schematically illustrated in Figure 1a, we consider a 2D scenario (with geometry and field quantities independent of y) where a transversemagneticpolarized planewave (with ydirected magnetic field) propagates in a homogeneous space entirely filled by a nonlocal, anisotropic medium. Such medium is assumed as nonmagnetic, and is characterized by a relativepermittivity tensor which depends on the wavevector k. Here and henceforth, the tilde ~ identifies wavevectordependent quantities.
Figure 1.
(a) Problem schematic in the assumed 2D (x, z) reference system (with geometry and field quantities independent of y). We consider a transverselymagneticpolarized planewave (with wavevector k) propagating in a homogenous space filled up by a nonlocal, nonmagnetic, anisotropic medium characterized by a relativepermittivity constitutive tensor . (b) Qualitative example of ideal dispersion surface [cf. Eq. (3)] pertaining to the assumed constitutive blueprints [cf. Eqs. (4) and (6), with ]. (c) Corresponding dispersion diagram (k_{x} = 0 cut), with the blackcross marker indicating the DBEtype stationary point. 
By focusing on a simple uniaxial anisotropy, the arising planewave dispersion relationship can be simply written in terms of the relevant components ( and ) as(3)where k_{x} and k_{z} denote the x and zdomain wavenumbers, respectively, and c is the speed of light in vacuum.
In what follows, inspired by our previously introduced nonlocal TO approach, we derive the ideal blueprints for the nonlocal medium so as to induce a DBE stationary point in the dispersion relationship (3). Moreover, we also address the synthesis of a multilayered metamaterial implementation that suitably approximates (within a neighborhood of the DBE stationary point) such ideal blueprints. Though still based on a periodic multilayer, our implementation is substantially different from those in references [17–21].
3 Synthesis procedure
3.1 Ideal constitutive blueprints
In reference [14], within the framework of nonlocal TO, we derived a class of wavevector transformations that could induce stationary points (of arbitrary order) in the dispersion relationship. In particular, we explored in detail the case ν = 3 in equations (1) and (2), corresponding to the “frozenmode” regime. A similar approach could be applied to deal with the DBE (ν = 4) case of interest here. For brevity, we do not repeat here the analytical developments, which can be found in reference [14]. In fact, as discussed in reference [14], while providing a systematic, constructive derivation of the required material blueprints, our approach is not the only possible one, as the problem does not admit a unique solution. Moreover, for the particular 2D, coordinateseparable and nonmagnetic structure of the arising constitutive blueprints, their interpretation is rather straightforward.
Paralleling [14], we consider the following class of coordinateseparable material blueprints (henceforth identified by the superscript “BP”):(4)where ε_{zz0} is a constant, and is a polynomial function. It can be readily verified, from equation (3), that(5)with denoting spectral functions. Therefore, by assuming propagation along the zdirection (i.e., k_{x} = 0, k_{z} = β), the desired DBE condition in equation (1) (with ν = 4) can be attained by choosing(6)where γ is a nonzero real constant, and k_{0} = ω_{0}/c = 2π/λ_{0} denotes the wavenumber in vacuum at the design angular frequency (with λ_{0} denoting the corresponding wavelength). Moreover, is an arbitrary function that vanishes (together with its first four derivatives) at k_{z} = β_{0},(7)with r_{m} denoting arbitrary coefficients.
As anticipated, the particularly simple analytical structure of the constitutive blueprints (4) [together with Eq. (6)] makes it possible to readily verify, by simple inspection, that the desired DBE conditions are fulfilled. In particular, via straightforward algebra, the coefficient γ in equation (6) can be directly related to the nonzero derivative term in the dispersion equation [cf. Eq. (1)],(8)
Figure 1b qualitatively shows the dispersion surface [cf. Eq. (3)] pertaining to the above class of constitutive blueprints (with ). As it can be observed, the quartic nature of the dispersion relationship gives rise to two modal branches, which degenerate at the design angular frequency, thereby yielding the sought DBEtype stationary point in the dispersion diagram, as more clearly visible in the k_{x} = 0 cut shown in Figure 1c.
It is worth highlighting that the class of constitutive blueprints in equations (4) and (6) is idealized, and does not necessarily fulfill the physical feasibility conditions (e.g., causality) in the entire (k, ω) phase space (see also the discussion in Ref. [14]). Likewise, the dispersion characteristics in Figures 1b and 1c should only be intended as conceptual illustrations, since explicit temporal dispersion (i.e., explicit ωdependence) is neglected.
3.2 Multilayered metamaterial synthesis
Having determined a class of ideal constitutive blueprints that yields the desired dispersion effects, as in reference [14], the synthesis problem can be formulated as finding a physical metamaterial structure whose (nonlocal) effective constitutive parameters suitably approximate the targeted blueprints. It is important to stress that, in our problem, such approximation is needed only within a limited region of the phase space, i.e., in the vicinity of the DBE stationarypoint (k_{x} = 0, k_{z} = β_{0}, ω = ω_{0}). As discussed in reference [14], this generally renders the synthesis problem tractable, even though the assumed blueprints may appear as unphysical if extended to the entire phase space (see also the discussion above).
In view of the uniaxial character and analytical (rational) structure of the assumed blueprints in equations (4) and (6), multilayered metamaterials represent a particularly suited implementation. In the recent technical literature, several approaches have been proposed for the derivation of nonlocal effective models of metallodielectric multilayered metamaterials (see, e.g., [31–35]). Here, we utilize a generalized version of the approach proposed by Elser et al. [31], which determines the approximate nonlocal effective model in such a way the corresponding dispersion relationship matches the Taylor expansion of the exact one (around a desired phasespace point) up to a given order. First, we note that, in view of equation (6), the constitutive blueprints (4) that we need to approximate contain odd powers of k_{z}. The implied symmetry breaking (in the zdirection) could be physically induced in various ways, e.g., by considering uniaxiallyanisotropic material constituents with tilted optical axes and/or gyrotropic (nonreciprocal) materials. Here, we consider this latter case, which allows us to directly exploit the results in reference [14]. Accordingly, we consider a generic periodic multilayer, stacked along the xdirection, and with a unitcell composed of nonmagnetic, homogeneous, isotropic or gyrotropic layers. Hence, the pth material constituent can be generally characterized by a relative permittivity tensor(9)with κ_{p} = 0 yielding the isotropic case. It can be shown (see [14] for details) that, in the vicinity of the DBE stationarypoint of interest k_{x} = 0, k_{z} = β_{0}, ω = ω_{0}, the above class of metamaterials can be described in terms of the nonlocal effective parameters(10)where d is the multilayer period (unitcell thickness), the array symbolically denotes the set of geometrical and constitutive parameters of the multilayer (with the explicit dependence on ω_{0} and β_{0} omitted for notational simplicity), and the functions χ_{l} are chosen so that arising dispersion equation matches the Taylor series (about k_{x} = 0, k_{z} = β_{0}, ω = ω_{0}) of the exact one up to the secondorder in k_{x} and fourthorder in k_{z}.
The synthesis problem can thus be cast as finding an “optimal” multilayer parameter set , so that the nonlocal effective model (10) approximately matches the desired constitutive blueprints [cf. Eqs. (4) and (6)] in the vicinity of the DBE stationary point. To further relax the synthesis problem, we only set and leave as degrees of freedom the parameters γ and β_{0} in the blueprints, whereas, though not directly relevant in the assumed incidence conditions, the parameter ε_{zz0} in equation (4) is obtained by simple matching at k_{x} = 0,(11)
We are thus led to an optimization problem, i.e., finding the multilayer parameter set as well as the blueprints degrees of freedom γ and β_{0} that minimize a suitable error functional,(12)with k_{zj} and w_{j}, j = 1, …, J denoting discrete k_{z}samples around k_{z} = β_{0} and positive weight coefficients, respectively.
For the minimization of the error functional (12), we found that a combination of a standard differentialevolution algorithm and the NelderMeald method (see Appendix for details), while not necessarily guaranteeing the convergence to the global minimum, usually provided reasonably good results.
4 Representative results
As an illustrative example of application, we consider a multilayered metamaterial implementation with a fourlayer unitcell, containing three isotropic layers and a gyrotropic one (see the inset in Figure 2b). Since we are only interested in a very narrow frequency range (ω ≈ ω_{0}), we neglect the material dispersion of the constituents. Moreover, for a better illustration of the phenomena, we also neglect material losses.
Figure 2.
(a), (b) Effective constitutive parameters at the design angular frequency ω = ω_{0}. Reddashed curves represent the optimized blueprints, obtained from equations (4) and (6), with , , and β_{0} = 0.46k_{0}. Bluesolid curves pertain to the nonlocal effective model of the synthesized multilayered metamaterial [fourlayer unit cell shown in the inset of panel (b)] with ε_{1} = 7.089, given in equation (9) (with ε_{2} = −1.104 and κ_{2} = 0.0365), ε_{3} = 3.35, ε_{4} = −1.957, d_{1} = 0.252λ_{0}, d_{2} = 0.0815λ_{0}, d_{3} = 0.102λ_{0}, d_{4} = 0.0648λ_{0}. The parameter matching is enforced at k_{x} = 0 and within the (yellowshaded) range 0.9β_{0} ≤ k_{z} ≤ 1.1β_{0}. 
Figure 2 illustrates the synthesis results (in terms of nonlocal effective constitutive parameters) obtained by choosing, in the error functional (12), J = 21 k_{z}samples equispaced within the interval [0.9β_{0}, 1.1β_{0}], with uniform weights (w_{j} = 1).
More specifically, for the optimized parameters given in the caption, Figure 2a compares the blueprint and synthesized components, over the relevant spectral range. A fairly good agreement is observed, especially within the (yellowshaded) range [0.9β_{0}, 1.1β_{0}] where the parameter matching was directly enforced. Though not directly relevant for the assumed k_{x} = 0 propagation direction, Figure 2b shows the corresponding components, which are exactly matched only at k_{x} = 0 [cf. Eq. (11)], but do not substantially differ over a reasonably wide k_{x}range.
From the optimized multilayer parameters given in Figure 2 caption, we note the presence of both positive and negative values of the permittivities. This is expectable, as strong nonlocal effects in multilayer configurations typically stem from the excitation and coupling of surfaceplasmon polaritons (SPPs) supported at the interfaces separating the negative and positivepermittivity layers [6–12].
For the optimized parameter configuration, Figure 3 shows the dispersion diagram for k_{x} = 0, numerically computed via a rigorous transfermatrixbased approach (see [14] for details), in the vicinity of the design angular frequency ω_{0}, from which the attained DBE condition (at k_{z} = β_{0} = 0.46k_{0}) is clearly visible. Also visible is the nonreciprocal character, manifested in the centersymmetry breaking; albeit not strictly required, this is a direct consequence of our design strategy relying on a gyrotropic material constituent (see the discussion in Sect. 3). The inset shows a magnified detail around the DBE stationary point, which highlights the good local agreement with the blueprint prescription.
Figure 3.
Numericallycomputed dispersion diagram (bluesolid curve) for the synthesized multilayered metamaterial (with parameters as in Figure 2 caption), for k_{x} = 0 in the vicinity of the design angular frequency ω_{0}. The inset shows a magnified view around the DBE stationary point, compared with the blueprint prediction (reddashed curve). 
Figure 4 shows instead a few representative (numericallycomputed) equifrequency contours pertaining to the synthesized multilayered metamaterial. It can be observed that, as the angular frequency approaches the design value ω_{0}, the equifrequency contours tend to shrink and loose their centersymmetric character, and eventually degenerate to the DBE point.
Figure 4.
Numericallycomputed equifrequency contours for the synthesized multilayered metamaterial (with parameters as in Figure 2 caption), at angular frequencies ω = 0.9ω_{0}, ω = 0.99ω_{0}, ω = 0.995ω_{0}, ω = 0.999ω_{0} (red, green, blue, magenta curves, respectively). At ω = ω_{0}, the equifrequency contour degenerates to the DBE stationary point (indicated with a blackcross marker). 
As a further, independent validation of the above synthesis, we now consider a halfspace made of the synthesized multilayered metamaterial under transversemagnetic planewave illumination (at the design angular frequency), normallyimpinging along the positive zdirection from a vacuum halfspace. Figure 5 shows a finiteelement computed (see Appendix for details) magnetic field map, over the multilayer unitcell, which exhibits the typical physical “footprints” associated with the DBE regime [17–21]. As it can be observed, the field is transmitted (with small reflection) in the metamaterial halfspace, where it gets converted into an extended mode with growing amplitude. At a distance of about ~5λ_{0} from the vacuummetamaterial interface, the amplitude reaches a maximum value that is over a factor 1000 larger than the incident one, and then it starts decreasing.
Figure 5.
Finiteelementcomputed magneticfield map, at the design angular frequency ω_{0}, over a unit cell of the synthesized multilayered metamaterial (with parameters as in Figure 2 caption). A halfspace configuration is assumed, with planewave excitation normally impinging from vacuum. The falsecolorscale map indicates the field magnitude normalized with respect to the incident one. The field map is superimposed on the unitcell (shown in trasparency). 
Figure 6 shows instead the magnetic energy density (normalized with respect to the incident one, and averaged across the unit cell)(13)as a function of the angular frequency, at a distance z_{0} = 5.25λ_{0} from the vacuummetamaterial interface (where the resonant transmitted field is maximum, cf. Figure 5). As it can be observed, this quantity is strongly peaked at the design angular frequency ω_{0}, reaching a value that is over five orders of magnitude larger that the incident one, and then rapidly decreasing to negligible values for ω > ω_{0} (i.e., in the bandgap). As evidenced in the magnified detail displayed in the inset, this is qualitatively consistent with the singular behavior predicted theoretically [17–21](14)
Figure 6.
Finiteelementcomputed normalized magnetic density energy averaged over a unit cell [cf. Eq. (13)] of the synthesized multilayered metamaterial (with parameters as in Figure 2), at z = 5.25λ_{0}, as a function of the angular frequency (blue solid curves). The inset shows a magnified detail around the design angular frequency, and compares it with the theoretically predicted behavior [reddashed curve, cf. Eq. (14)]. 
The above results clearly indicate that our designed multilayered metamaterial exhibits the typical physical footprints that characterize a DBEtype giant slowwave resonance. Within this framework, a few remarks are in order. First, though still based on a multilayered structure containing anisotropic constituents, our proposed configuration differs substantially from those in references [17–21]. By comparison with these, our configuration is 2D, as we do not rely on modes with different polarizations. Moreover, we assume a propagation direction that is parallel, rather than orthogonal, to the layer interfaces. Accordingly, the underlying physical mechanism is also different and, in our case, it essentially exploits the strong nonlocality arising from the excitation and coupling of SPP modes at the interfaces separating the negative and positivepermittivity layers. Finally, although the main focus of this prototype study was on a simple proofofconcept demonstration of the phenomenology, and we did not place special emphasis on applicationoriented aspects, we did make sure that the material parameters were constrained within realistic bounds. In particular, the positive relativepermittivity values considered in our synthesis range from nearly 3 to about 7, while those of the negativepermittivity and gyrotropic materials are consistent with configurations already considered in the literature (see, e.g., [9, 36]).
5 Conclusions
In conclusion, this prototype study has shown the possibility to engineer DBEtype giant slowwave resonances by harnessing the strong nonlocal effects that can occur in multilayered metamaterials featuring positive and negativepermittivity constituents. Starting from ideal nonlocal constitutive blueprints (inspired by our previouslydeveloped nonlocal TO extension), we have synthesized a multilayered metamaterial that (as verified via fullwave numerical simulations) exhibited the typical DBE physical footprints.
Our results may provide an alternative route for the engineering of DBE effects, and provide further evidence of the applicability and versatility of nonlocalTO concepts.
Current and future investigations are aimed at exploring alternative implementations, not relying on gyrotropic (nonreciprocal) materials, and/or directly exploiting the material dispersion effects. Also of great interest is the use of gain materials, for loss compensation and/or possible applications to lasing.
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Appendix
Details on the synthesis and numerical simulations
The minimization of the error functional in equation (12) was carried out by means of a Pythonbased implementation of the differentialevolution optimization algorithm available in the SciPy optimization library [37]. More specifically, we considered a population size of 100 elements (with random initialization), the “best1bin” strategy, a dithering of the mutation constant within the interval (0.5, 1), and a recombination constant equal to 0.6. We also found that slight improvements could be obtained by “polishing” the best element of the final population via the NelderMeald method. With reference to the specific example presented, the constitutive parameters of layer #2 (gyrotropic medium) were fixed to values already utilized in the literature [36]; for the remaining layers, the relative permittivities were assumed to vary with the realistic ranges 1 ≤ ε_{1,3} ≤ 11 and −10 ≤ ε_{4} ≤ −0.1. For the layer thicknesses, a variation range 0 < d_{k} ≤ 0.3λ_{0}, k = 1, …, 4 was assumed. Moreover, to avoid propagation of higherorder Bragg modes, the constraint d < λ_{0}/2 was enforced for the total unitcell thickness; this was implemented by adding a penalty term in the error functional (12) (equal to 100 if the constraint was not satisfied, and zero otherwise). For the remaining two optimization parameters β_{0} and γ, the variation ranges −2k_{0} ≤ β_{0} ≤ 2k_{0} and were set.
The field map (Figure 5) and magnetic energy density (Figure 6) were computed by means of the finiteelementbased commercial software package COMSOL Multiphysics [38]. More specifically, a single unitcell was considered, extending 10λ_{0} along the zdirection and terminated by periodic boundary conditions along x. The structure was excited by a plane wave impinging from a 2λ_{0}long section of vacuum. In order to simulate a semiinfinite structure, the other side was terminated by a 60λ_{0}long lossy section (not shown in Figure 5) with a linearlyincreasing loss profile. The structure was discretized with a maximum mesh size of λ_{0}/100 (resulting into about 6.3 million degrees of freedom), and the MUMPS direct solver (with default parameters) was utilized.
Cite this article as: Moccia M, Castaldi G & Galdi V: Degeneratebandedge engineering inspired by nonlocal transformation optics. EPJ Appl. Metamat. 2016, 3, 2.
All Figures
Figure 1.
(a) Problem schematic in the assumed 2D (x, z) reference system (with geometry and field quantities independent of y). We consider a transverselymagneticpolarized planewave (with wavevector k) propagating in a homogenous space filled up by a nonlocal, nonmagnetic, anisotropic medium characterized by a relativepermittivity constitutive tensor . (b) Qualitative example of ideal dispersion surface [cf. Eq. (3)] pertaining to the assumed constitutive blueprints [cf. Eqs. (4) and (6), with ]. (c) Corresponding dispersion diagram (k_{x} = 0 cut), with the blackcross marker indicating the DBEtype stationary point. 

In the text 
Figure 2.
(a), (b) Effective constitutive parameters at the design angular frequency ω = ω_{0}. Reddashed curves represent the optimized blueprints, obtained from equations (4) and (6), with , , and β_{0} = 0.46k_{0}. Bluesolid curves pertain to the nonlocal effective model of the synthesized multilayered metamaterial [fourlayer unit cell shown in the inset of panel (b)] with ε_{1} = 7.089, given in equation (9) (with ε_{2} = −1.104 and κ_{2} = 0.0365), ε_{3} = 3.35, ε_{4} = −1.957, d_{1} = 0.252λ_{0}, d_{2} = 0.0815λ_{0}, d_{3} = 0.102λ_{0}, d_{4} = 0.0648λ_{0}. The parameter matching is enforced at k_{x} = 0 and within the (yellowshaded) range 0.9β_{0} ≤ k_{z} ≤ 1.1β_{0}. 

In the text 
Figure 3.
Numericallycomputed dispersion diagram (bluesolid curve) for the synthesized multilayered metamaterial (with parameters as in Figure 2 caption), for k_{x} = 0 in the vicinity of the design angular frequency ω_{0}. The inset shows a magnified view around the DBE stationary point, compared with the blueprint prediction (reddashed curve). 

In the text 
Figure 4.
Numericallycomputed equifrequency contours for the synthesized multilayered metamaterial (with parameters as in Figure 2 caption), at angular frequencies ω = 0.9ω_{0}, ω = 0.99ω_{0}, ω = 0.995ω_{0}, ω = 0.999ω_{0} (red, green, blue, magenta curves, respectively). At ω = ω_{0}, the equifrequency contour degenerates to the DBE stationary point (indicated with a blackcross marker). 

In the text 
Figure 5.
Finiteelementcomputed magneticfield map, at the design angular frequency ω_{0}, over a unit cell of the synthesized multilayered metamaterial (with parameters as in Figure 2 caption). A halfspace configuration is assumed, with planewave excitation normally impinging from vacuum. The falsecolorscale map indicates the field magnitude normalized with respect to the incident one. The field map is superimposed on the unitcell (shown in trasparency). 

In the text 
Figure 6.
Finiteelementcomputed normalized magnetic density energy averaged over a unit cell [cf. Eq. (13)] of the synthesized multilayered metamaterial (with parameters as in Figure 2), at z = 5.25λ_{0}, as a function of the angular frequency (blue solid curves). The inset shows a magnified detail around the design angular frequency, and compares it with the theoretically predicted behavior [reddashed curve, cf. Eq. (14)]. 

In the text 