Nonlocal dispersion anomalies of Dyakonov-like surface waves at hyperbolic media interfaces

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ScienceDirect Photonics and Nanostructures – Fundamentals and Applications 18 (2016) 16–22

Nonlocal dispersion anomalies of Dyakonov-like surface waves at hyperbolic media interfaces夽 Juan J. Miret a , J. Aitor Sorní b , Mahin Naserpour b,c,∗ , Abbas Ghasempour Ardakani c , Carlos J. Zapata-Rodríguez b,∗ b

a Department of Optics, Pharmacology and Anatomy, University of Alicante, P.O. Box 99, Alicante, Spain Department of Optics and Optometry and Vision Science, University of Valencia, Dr. Moliner 50, Burjassot 46100, Spain c Department of Physics, College of Science, Shiraz University, Shiraz 71454, Iran

Received 9 July 2015; received in revised form 26 October 2015; accepted 3 December 2015 Available online 15 December 2015

Abstract Dyakonov-like surface waves (DSWs) propagating obliquely on an anisotropic nanostructure have been theoretically proved in a few cases including 2D photonic crystals and metal-insulator (MI) layered metamaterials. Up to now, the long-wavelength approximation has been employed in order to obtain effective parameters to be introduced in the Dyakonov equation, which is largely restricted to material inhomogeneities of a few nanometers when including metallic elements. Here, we explore DSWs propagating obliquely at the interface between an insulator and a hyperbolic metamaterial, the latter consisting of a 1D MI bandgap grating using realistic slab sizes. We found unexpected favorable conditions for the existence of such surface waves. The finite element method is used to investigate the peculiarities of this new family of DSWs. © 2015 Elsevier B.V. All rights reserved. Keywords: Surface waves; Hyperbolic metamaterials; Electromagnetic optics

1. Introduction Surface plasmon polaritons (SPPs) are electromagnetic (EM) excitations coupled to surface collective oscillations of free electrons in a metal, which form twodimensional EM modes propagating along MI interfaces and exponentially decay into neighboring media [1,2]. The profile of the metal surface determines the characteristics of SPP modes existing at flat and curved surfaces, including SPP modes of complex particle arrays and 夽

The article belongs to the special section Metamaterials. Corresponding authors. Tel.: +34 963543805. E-mail addresses: [email protected] (M. Naserpour), [email protected] (C.J. Zapata-Rodríguez). ∗

http://dx.doi.org/10.1016/j.photonics.2015.12.001 1569-4410/© 2015 Elsevier B.V. All rights reserved.

metal nanostructures [3]. SPP modes, whose propagation length is limited by ineluctable EM absorption in metals, induce unique plasmonic phenomena like EM field enhancement often resulting together in extreme light concentration [4,5]. Plasmonic nanostructures have also been considered in guiding electromagnetic waves by introducing solid-state materials [6,7]. Up to now, several architectures have been developed for efficiently guiding of electromagnetic surface waves, such as the channel plasmon polaritons including V-grooves [8], wedges [9], and strips [10], and the chain of closely spaced metal nanoparticles [11] to mention an few. The presence of a periodic distribution of holes and any other kind of carved subwavelength features in a metal surface may play a relevant role in fascinating phenomena like

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hyperlensing [12] and extraordinary optical transmission [13,14]. The description of SPP modes assisted by a 1D periodic arrangement of surface irregularities is well established when the flux of light is directed along the two perpendicular axes of symmetry [15,16]. However, oblique propagation is far less known. For instance, it has been predicted the existence of bound waves propagating at arbitrary angles on the boundary of a MI structure [17–20]. In the later case, Dyakonov-like surface waves [21] are prescribed on the anisotropic metamaterial implementing a simple homogenization theory based on the long-wavelength approximation (LWA). This was also applied in 2D photonic crystals [22]. Finally, hyperbolic metamaterials (HMMs) exhibiting principal permittivities with opposite sign may play a relevant role in the excitation of DSWs with unprecedented spatial-spectrum range and EM confinement [23–25]. Recently we demonstrated that the presence of metallic nanoelements leads to nonlocal effects and dissipation effects which reshape the propagation dynamics of the surface signal [26]. In this work, we examine extraordinary favorable conditions which may appear in bandgap MI layered media for the existence of DSWs. Engineering secondary bands by tuning the plasmonic-crystal geometry induces a controlled optical anisotropy, which is clearly dissimilar to the prescribed hyperbolic regime that is developed by the LWA, however, assisting the presence of DSWs on the interface between such HMM and an insulator. Moreover, the hyperbolic band remains passive for moderate and high band stops. Numerical simulations validate the efficient guiding of surface waves in a broad spatial-frequency band. Contrarily to ordinary hyperbolic DSWs, our numerical experiments show tight EM confinement around the metallic nanoplates and modal beating when the fields penetrate and attenuate into the HMM. 2. Dyakonov-like surface waves in hyperbolic metamaterials

17

(a)

(b)

SiO2

y x z

Ge Ag z x

d

m

wm wd

surface wave

Fig. 1. (a) Geometry of the Ag-Ge hyperbolic metamaterial that is covered by SiO2 . (b) Front view of a single cell of the semi-infinite MI lattice placed under the isotropic medium.

λ0 = 1550 nm respectively [27,28]; silver permittivity is set as m = −100.8 + i8.2 [29]. In order to observe DSWs, further considerations are typically taken into account. Within the LWA, where the metal width remains far below the wavelength (wm  λ0 ), the EM wave can pass through the metallic walls of the grating with neglecting field attenuation (the penetration depth of silver in the visible and near-infrared is 24 nm approximately) leading to the homogenization of the field [30]. Under these conditions, the grating naturally behaves like a uniaxial medium whose optic axis is oriented normally to the MI layers, that is the z-axis. The relative permittivity along such a direction, || , and in the transverse direction, ⊥ , may be given in terms of m , d and f [31], as illustrated in Fig. 2. As a result, the surface wave will travel on the interface between a uniaxial homogenized medium of permittivity ¯ = ⊥ (xx + yy) + || zz,

(1)

200

Є⊥ Є

150 100 50 0

We first consider a surface wave propagating on a silver grating of period  where the metal filling factor is given by the parameter f = wm /. As shown in Fig. 1, wm represents the width of a metal layer, which is oriented parallel to the xy-plane. We also assume that the interlayer space is loaded with a transparent material of relative permittivity d . The environment medium, which is set above the metallic grating, will be characterized by a dielectric constant . In our numerical simulations we will consider germanium and silicon dioxide with permittivities d = 18.3 and  = 2.08 at a wavelength

−50 −100

0.0

0.2

0.4

0.6

metal filling factor

0.8

1.0

Fig. 2. Dielectric functions of the uniaxial effective medium composed of Ag and Ge at a wavelength λ0 = 1550 nm, when we disregard losses in the metal (we set m = −100.8). The permittivities ⊥ and || were obtained by altering the metal filling factor, f. Within the range 0.154 < f < 0.846 (shaded area), the metamaterial shows hyperbolic dispersion with ⊥ < 0 and || > 0.

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and an isotropic nonabsorbing material of permittivity . When a bounded field propagates with a wave vector (ky , kz ), the field in the environment medium is evanescent following a variation of the form exp(− κ|x|), where the decay rate is given by  κ = ky2 + kz2 − , (2) in units of the wavenumber k0 = 2π/λ0 (ky and kz are also expressed in units of k0 ). In the homogenized anisotropic metamaterial, ordinary and extraordinary waves will contribute to the formation of the surface wave, exhibiting decay rates given by κo and κe , respectively, both depending on the effective permittivities ⊥ and || . For a detailed description of the electromagnetic fields characterizing the DSWs, see Ref. [32]. Finally, the dispersion equation of this surface EM signal follows the well-known Dyakonov equation [21]   (κ + κe ) (κ + κo ) (κo + ⊥ κe ) = || −  ( − ⊥ ) κo , (3) Let us consider the following numerical example. An Ag grating is loaded by Ge (and surrounded by SiO2 ) in such a way that if the metal filling factor were f = 0.25, the effective permittivities of the anisotropic metamaterial would yield ⊥ = −11.48 + i2.05 and || = 25.96 + i0.14 at a wavelength of λ0 = 1550 nm. Losses in silver lead to complex permittivities, however, it might be used m = −100.8 disregarding its imaginary part, thus giving ⊥ = −11.48 and || = 25.97 (approaching Re ⊥ and Re || respectively) for the numerical evaluation of the wave vector (ky , kz ). Though Dyakonov originally considered dielectric materials with all-positive permittivities, in principle we might find solutions of Eq. (3) when dealing with type I hyperbolic metamaterials exhibiting ⊥ < 0 [17]. In Fig. 3 we show the solutions of the Dyakonov equation (3) for our Ag-Ge HMM of f = 0.25 and some other filling factors. The DSW dispersion curve describes an incomplete hyperbolic curve, finding an endpoint under the condition κe = 0, where the extraordinary wave breaks its confinement in the vicinities of the isotropic-uniaxial interface [23]. 3. Numerical simulations and discussion Let us consider a realistic MI structure consisting of Ag layers of wm = 40 nm interspersed between Ge layers of wd = 120 nm, thus maintaining a metal filling factor of f = 0.25 as analyzed above. Since wm clearly surpasses the penetration depth of silver, we expect that nonlocal effects became evident [33]. In Fig. 4 we represent the dispersion equation of the 1D periodic

20 -20

15 kz 0

0.3

0

20 -20 -20

10

0 0

ky

5

0.2

kx

20 -20

0

0.2

5

0 0

5

10

15

20

normalized spatial frequency ky Fig. 3. Isofrequency curves of (incomplete) hyperbolic DSWs, which are solutions of Eq. (3), for different metal filling factors ranging from 0.175 to 0.325 (steps of 0.025). Inset: Dispersion in the bulk HMM for f = 0.25.

nanostructure for transverse electric (TE) and transverse magnetic (TM) polarized waves, calculated by using the well-known 2 × 2 matrix formulation and the Floquet theorem described for instance in Ref. [31]. For the sake of clarity, we considered real values of m in order to deal with real-valued spatial frequencies  kt = kx2 + ky2 (4)

4

Bloch wavenumber kz

18

elliptic

2

hyperbolic

0

-2

TE TM

-4 0

2

4

6

8

transverse frequency kt Fig. 4. Isofrequency contour plots of TE and TM Bloch modes for the HMM for wm = 40 nm and wd = 120 nm, at a wavelength λ0 = 1550 nm, proving the existence of a band gap (shaded region) and the extinction of the hyperbolic regime beyond kt = 5.8. For the sake of clarity we disregarded Ag losses by setting m = −100.8.

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and Bloch wavenumber kz . Despite the material is composed of MI nanolayers with subwavelength features, a bandgap occurs in the range spanning from kt = 1.49 to kt = 5.01. A first TM band with hyperbolic-like characteristics dominates at high in-plane frequencies kt ; however, the extinction of the hyperbolic regime is evident beyond kt = 5.8. Additionally a second band emerges for TM Bloch modes, which exhibits a moderate anisotropy, demonstrating near-elliptical dispersion curves (with positive effective permittivities) and positive birefringence, as stated in the original work by Dyakonov for extraordinary waves [21]. Furthermore, TE modal dispersion is roughly isotropic, as happens with ordinary waves. We conclude that satisfactory conditions may be found near the second TM band for the existence of Dyakonov-like surface waves. A semi-analytical evaluation for the in-plane wave vector (ky , kz ) of the surface wave may be carried out by properly estimating the parameters || and ⊥ as a preceding step to use the Dyakonov equation (3). From the dispersion curve shown in Fig. 4 we determined the positive permittivities || = 3.33 and ⊥ = 1.13 which allow us to approach the dispersion equation as kt2 + kz2 = ⊥ for TE-polarized waves, whereas the elliptic band characteristic of TM-polarized waves approximately remains kt2 /|| + kz2 /⊥ = 1. Evanescent wave propagation along the x direction, that is associated with positive values of κo and κe , both determined by the imaginary part of kx in the dispersion equations given above, occurs for a moderate but sufficiently large in-plane wave vector at a given orientation, provided that Bloch modes of the hyperbolic band could not be excited simultaneously. Note that ⊥ <  < || when considering SiO2 as isotropic semi-infinite medium. As a consequence, extraordinary favorable conditions appear in our bandgap MI layered medium for the existence of DSWs. We point out that surface modes on the interface of 1D photonic crystals which lie within the bandgaps have been previously reported [34,35]. As a distinctive feature, periodicity of those structures is oriented normally to the interface between the isotropic material and the periodic medium. In addition, Bloch surface waves are either TE- or TM-polarized electromagnetic waves, which may be coupled through a glass prism by means of a Kretschmann-Raether configuration [36]. Note that the decay of the mode along the propagation direction is caused by absorption by the material, surface scattering and leakage into the prism. Importantly, they may implement photonic and chemical functionalities at the same time, being mostly useful for efficient light-emitting devices and sensing applications [37]. In fact, excitation of pure DSWs by frustrated total internal

19

3 1

Im(kDSW) 2

0 0.5

1.0

2.0

1.5

2.5

ky

Re(kz) 1

0

Re(kDSW) TM Bloch TE Bloch Isotropic 0

1

2

3

spatial frequency ky Fig. 5. Graphical representation of the dispersion curve for DSWs, drawn in purple-colored lines. As a reference we also include solutions of the dispersion equation at kx = 0 in SiO2 (green solid line) and in the Ag-Ge periodic nanostructure for TE (dashed red line) and TM polarized modes (solid blue line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

reflection illumination has also been reported elsewhere [38]. Similarly our DSWs might be excited by such optical procedure. In that regard, the hyperbolic band remains passive for moderate and high bandwidths, as too happens in our numerical example, due to the critical momentum mismatch with the surface wave and consequently negligible wave coupling to these highfrequency Bloch modes [39,40]. In order to numerically obtain the dispersion curves and wave fields associated with DSWs, we will follow the same computational procedure followed in Refs. [23,26]. We employ a finite element method that is implemented in COMSOL Multiphysics software. Our problem can be largely simplified in two dimensions provided that we fix the spectral component ky of the surface wave vector and subsequently we simulate the physics-based problem to estimate the complex-valued propagation constant kz of the mode (also denoted as kDSW ) confined near the SiO2 -superlattice interface. In Fig. 5 we plot the propagation constant kz of DSWs in terms of the in-plane spatial frequency ky characterizing the obliquity of the surface wave propagation. As a reference we also include solutions of the dispersion equation where kx = 0 in SiO2 (green solid line) and the Ag-Ge superlattice (solid-blue and dashed-red lines), indicating EM fields with zero decay rate within the isotropic medium and the HMM (for TE and TM modes) respectively. As a result of utilizing a lossy anisotropic

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(a)

ky = 2.4

magnetic field |Hx| (a.u.)

(b)

ky = 1.8 (c)

ky = 1.2 (d)

ky = 0.6 -5

-2.5

0

2.5

5

Fig. 6. Absolute value of the magnetic field Hx , as it varies along the x-axis, for the particular case of a transverse spatial frequency (a) ky = 2.4, (b) ky = 1.8, (c) ky = 1.2 and (d) ky = 0.6. The field profile is taken by including the bottom corner of the Ag slab.

metamaterial, surface waves cannot propagate indefinitely and DSWs decay with a propagation length given by [18,41] LDSW =

1 . 2Im(kz )

(5)

Dyakonov-like surface waves with higher Im(kz ) (shorter LDSW ) occur at the highest values of the spatial frequency ky . The isofrequency curve kx = 0 for TE and TM polarized waves in the periodic nanostructure and the curve associated with DSWs come close in this spectral region. Short propagation lengths of DSWs are then attributed to the evanescent fields with low decay rates and thus deeply penetrating into the absorbing MI compound. As concluding remark, material losses have an explicit impact by means of the Ag-grating interaction with the slowly attenuating wave inside the metamaterial. For the particular case where the transverse spatial frequency ky = 2.4, the propagation constant (along the optic axis) of the surface mode yields kz = 0.14 + i0.70, for which we represent in Fig. 6(a) the absolute value of Hx as evaluated along the Ag-Ge undermost interface. The short propagation length of such surface wave,

LDSW = 180 nm, is attributed to the long evanescent field penetrating deep into the metallic compound, reaching a few microns. Close to the main peak slightly shifted to the anisotropic metamaterial, as expected, there occur irregular bursts originated by an interferencial activity that give rise to short but intense high-frequency oscillations in the EM field. In order to reduce the influence of metal losses and thus to boost the propagation length of the DSWs, we will red-shift the spatial frequency ky [26]. The latter implies that the propagation direction of the DSW will rotate toward the z-axis which marks the orientation of the lattice periodicity. In Fig. 6 we also plot |Hx | for (b) ky = 1.8, (c) 1.2 and (d) 0.6, resulting in a DSW propagation constant kz = 0.62 + i0.29, 1.02 + i0.16, and 1.28 + i0.04, respectively. It is evident that a decrease of the imaginary part of kz is governed by the imbalance of the transverse wave field that is transfered toward the semi-infinite isotropic material. Concurrently, the dispersion curve of the DSWs approaches the isofrequency curve kx = 0 for the isotropic medium, as seen in Fig. 5, disclosing a drop of the decay rate in SiO2 . Dissimilarly to what occurs in DSWs propagating between two transparent dielectrics, wave oscillations modulate the evanescent tail of the waveform inside the anisotropic metamaterial, as illustrated in Fig. 6. In order to gain a broad insight of this phenomenon, in Fig. 7 we plot the contours of the magnetic field Hx in the full unit cell of the MI lattice for different spatial frequencies ky . The field is enhanced in the Ag-Ge interfaces, showing damped harmonic oscillations as long as we move far from the SiO2 interface. The origin of such phenomenon seems to lie on the interference of two quasi-guided modes which are confined in each germanium nanofilm. Therefore, we observe a beating of the excited TMz evanescent modes in the Ag-Ge-Ag nanoguide. We point out that beating of TM modes in wide metal-insulatormetal structures have been analyzed elsewhere [42]. In our case, however, such modal waveforms exhibit a complex propagation constant, the imaginary part of which is governed primarily by the bound character of the field near the SiO2 interface rather than metal dissipation. As concluding remark, there exists a strong subwavelength localization in the Ag-SiO2 interface, which is mainly carried out by the z-component of the magnetic field (not shown in Fig. 7). Such a near-resonant plasmonic effect serves for achieving extreme intensity peaks. Nanoscale optical waveguide achieving small mode field size, which is based on the phenomenon of near-resonant SPP confined at a MI interface has been reported elsewhere [43].

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Fig. 7. Distribution of the magnetic field |Hx | in the xz-plane, taking a single unit cell of the MI lattice, for DSWs with spatial frequency (a) ky = 2.4, (b) ky = 1.8, (c) ky = 1.2 and (d) ky = 0.6. The length of the white bar is 200 nm.

4. Conclusions

References

By employing the finite-element method, we calculated the dispersion relation of wave fields bound in the interface between a metal-insulator lattice and an isotropic environment medium, exhibiting oblique direction of propagation. It has been demonstrated that the advent of Dyakonov-like surface waves proved in certain hyperbolic regimes can be reversed out of the longwavelength regime. In the presence of a multiple band structure, the dispersion curves of a secondary band match the appropriate positive birefringence that might be found in all-dielectric materials, thus mimicking the conventional conditions for the existence of Dyakonov surface waves. In the case of oblique propagation with high angular deviation with respect to the direction of periodicity of the MI lattice, the “center of mass” of the DSW peak significantly shifts toward the 1D metamaterial, leading to limited propagation lengths caused by metal losses. However, in the case of low angular deflection, the surface wave sparsely sinks into the metallic array. The non-homogenized decaying field inside the metamaterial presents a beating phenomenon. This effect is due to the excitation of two evanescent modes confined in the silver-cladded Ge waveguide. Our results confirm that excitation of Dyakonov-like surface waves are conceivable in unexplored regimes, in our case found by simply manipulating the geometry of periodic nanostructures.

[1] A.V. Zayats, I.I. Smolyaninov, A.A. Maradudin, Nano-optics of surface plasmon polaritons, Phys. Rep. 408 (2005) 131–314. [2] J.M. Pitarke, V.M. Silkin, E.V. Chulkov, P.M. Echenique, Theory of surface plasmons and surface-plasmon polaritons, Rep. Prog. Phys. 70 (2007) 1–87. [3] Y.B. Zheng, T.J. Huang, Surface plasmons of metal nanostructure arrays. from nanoengineering to active plasmonics, J. Assoc. Lab. Autom. 13 (2008) 215–226. [4] S. Lal, S. Link, N.J. Halas, Nano-optics from sensing to waveguiding, Nat. Photon. 1 (2007) 641–648. [5] J. Schuller, E. Barnard, W. Cai, Y.C. Jun, J.S. White, M.L. Brongersma, Plasmonics for extreme light concentration and manipulation, Nat. Mater. 9 (2010) 193–204. [6] E. Ozbay, Plasmonics: merging photonics and electronics at nanoscale dimensions, Science 311 (2006) 189–193. [7] Z. Han, S.I. Bozhevolnyi, Radiation guiding with surface plasmon polaritons, Rep. Prog. Phys. 76 (2013) 016402. [8] S.I. Bozhevolnyi, V.S. Volkov, E. Devaux, J.-Y. Laluet, T.W. Ebbesen, Channel plasmon subwavelength waveguide components including interferometers and ring resonators, Nature 440 (2006) 508–511. [9] E. Moreno, S.G. Rodrigo, S.I. Bozhevolnyi, L. Martín-Moreno, F.J. García-Vidal, Guiding and focusing of electromagnetic fields with wedge plasmon polaritons, Phys. Rev. Lett. 100 (2008) 023901. [10] D. Gacemi, J. Mangeney, R. Colombelli, A. Degiron, Subwavelength metallic waveguides as a tool for extreme confinement of THz surface waves, Sci. Rep. 3 (2013) 1369. [11] S.A. Maier, P.G. Kik, H.A. Atwater, S. Meltzer, E. Harel, B.E. Koel, A.A.G. Requicha, Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides, Nat. Mater. 2 (2003) 229–232. [12] I.I. Smolyaninov, Y.-J. Hung, C.C. Davis, Magnifying superlens in the visible frequency range, Science 315 (2007) 1699–1701. [13] T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, P.A. Wolf, Extraordinary optical transmission through sub-wavelength hole arrays, Nature (London) 391 (1998) 667–669. [14] L. Martín-Moreno, F.J. García-Vidal, H.J. Lezec, K.M. Pellerin, T. Thio, J.B. Pendry, T.W. Ebbesen, Theory of extraordinary optical transmission through subwavelength hole arrays, Phys. Rev. Lett. 86 (2001) 1114–1117.

Acknowledgments This research was funded by the Spanish Ministry of Economy and Competitiveness under the project TEC2013-50416-EXP.

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J.J. Miret et al. / Photonics and Nanostructures – Fundamentals and Applications 18 (2016) 16–22

[15] W.L. Barnes, T.W. Preist, S.C. Kitson, J.R. Sambles, Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings, Phys. Rev. B 54 (1996) 6227. [16] J.B. Pendry, L. Martín-Moreno, F.J. Garcia-Vidal, Mimicking surface plasmons with structured surfaces, Science 305 (2004) 847–848. [17] Z. Jacob, E.E. Narimanov, Optical hyperspace for plasmons: Dyakonov states in metamaterials, Appl. Phys. Lett. 93 (2008) 221109. [18] S.M. Vukovi´c, J.J. Miret, C.J. Zapata-Rodríguez, Z. Jak˘si´c, Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric, Phys. Scr. T149 (2012) 014041. [19] O. Takayama, D. Artigas, L. Torner, Practical dyakonons, Opt. Lett. 37 (2012) 4311–4313. [20] J.A. Sorni, M. Naserpour, C.J. Zapata-Rodríguez, J.J. Miret, Dyakonov surface waves in lossy metamaterials, Opt. Commun. 355 (2015) 251–255. [21] M.I. D’yakonov, New type of electromagnetic wave propagating at an interface, Sov. Phys. JETP 67 (1988) 714–716. [22] D. Artigas, L. Torner, Dyakonov surface waves in photonic metamaterials, Phys. Rev. Lett. 94 (2005) 013901. [23] C.J. Zapata-Rodríguez, J.J. Miret, S. Vukovi´c, M.R. Beli´c, Engineered surface waves in hyperbolic metamaterials, Opt. Express 21 (2013) 19113–19127. [24] Y. Xiang, J. Guo, X. Dai, S. Wen, D. Tang, Engineered surface Bloch waves in graphene-based hyperbolic metamaterials, Opt. Express 22 (2014) 3054–3062. [25] E. Cojocaru, Comparative analysis of Dyakonov hybrid surface waves at dielectric-elliptic and dielectric-hyperbolic media interfaces, J. Opt. Soc. Am. B 31 (2014) 2558–2564. [26] C.J. Zapata-Rodríguez, J.J. Miret, J.A. Sorni, S. Vukovi´c, Propagation of dyakonon wave-packets at the boundary of metallodielectric lattices, IEEE J. Sel. Top. Quantum Electron. 19 (2013) 4601408. [27] D. Aspnes, A. Studna, Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV, Phys. Rev. B 27 (1983) 985–1009. [28] E.D. Palik, G. Ghosh, The Electronic Handbook of Optical Constants of Solids, Academic, New York, 1999. [29] A.D. Raki´c, A.B. Djuri˘sic, J.M. Elazar, M.L. Majewski, Optical properties of metallic films for vertical-cavity optoelectronic devices, Appl. Opt. 37 (1998) 5271–5283.

[30] E. Popov, S. Enoch, Mystery of the double limit in homogenization of finitely or perfectly conducting periodic structures, Opt. Lett. 32 (2007) 3441–3443. [31] P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988. [32] O. Takayama, L.-C. Crasovan, S.K. Johansen, D. Mihalache, D. Artigas, L. Torner, Dyakonov surface waves: a review, Electromagnetics 28 (2008) 126–145. [33] J. Elser, V.A. Podolskiy, I. Salakhutdinov, I. Avrutsky, Nonlocal effects in effective-medium response of nanolayered metamaterials, Appl. Phys. Lett. 90 (2007) 191109. [34] P. Yeh, A. Yariv, A.Y. Cho, Optical surface waves in periodic layered media, Appl. Phys. Lett. 32 (1978) 104–105. [35] J.A. Gaspar-Armenta, F. Villa, Photonic surface-wave excitation: photonic crystal-metal interface, J. Opt. Soc. Am. B 20 (2003) 2349–2354. [36] M. Ballarini, F. Frascella, F. Michelotti, G. Digregorio, P. Rivolo, V. Paeder, V. Musi, F. Giorgis, E. Descrovi, Bloch surface waves-controlled emission of organic dyes grafted on a one-dimensional photonic crystal, Appl. Phys. Lett. 99 (2011) 043302. [37] M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J.E. Sipe, Strong modification of light emission from a dye monolayer via Bloch surface waves, Opt. Lett. 34 (2009) 2318–2320. [38] O. Takayama, L. Crasovan, D. Artigas, L. Torner, Observation of Dyakonov surface waves, Phys. Rev. Lett. 102 (2009) 043903. [39] A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, Engineered optical nonlocality in nanostructured metamaterials, Phys. Rev. B 84 (2011) 045424. [40] C.J. Zapata-Rodríguez, D. Pastor, J.J. Miret, S. Vukovi´c, Uniaxial epsilon-near-zero metamaterials: from superlensing to double refraction, J. Nanophoton. 8 (2014) 083895. [41] R. Warmbier, G.S. Manyali, A. Quandt, Surface plasmon polaritons in lossy uniaxial anisotropic materials, Phys. Rev. B 85 (2012) 085442. [42] Y. Kou, X. Chen, Multimode interference demultiplexers and splitters in metal-insulator-metal waveguides, Opt. Express 19 (2011) 6042–6047. [43] M. Yan, L. Thylén, M. Qiu, D. Parekh, Feasibility study of nanoscaled optical waveguide based on near-resonant surface plasmon polariton, Opt. Express 16 (2008) 7499–7507.