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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 16 — Aug. 12, 2013
  • pp: 19113–19127
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Engineered surface waves in hyperbolic metamaterials

Carlos J. Zapata-Rodríguez, Juan J. Miret, Slobodan Vuković, and Milivoj R. Belić  »View Author Affiliations


Optics Express, Vol. 21, Issue 16, pp. 19113-19127 (2013)
http://dx.doi.org/10.1364/OE.21.019113


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Abstract

We analyzed surface-wave propagation that takes place at the boundary between a semi-infinite dielectric and a multilayered metamaterial, the latter with indefinite permittivity and cut normally to the layers. Known hyperbolization of the dispersion curve is discussed within distinct spectral regimes, including the role of the surrounding material. Hybridization of surface waves enable tighter confinement near the interface in comparison with pure-TM surface-plasmon polaritons. We demonstrate that the effective-medium approach deviates severely in practical implementations. By using the finite-element method, we predict the existence of long-range oblique surface waves.

© 2013 OSA

1. Introduction

Artificial nanostructured materials can support electromagnetic modes that do not propagate in conventional media making them attractive for photonic devices with capabilities from nanoscale waveguiding to invisibility [1

1. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007) [CrossRef] .

, 2

2. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007) [CrossRef] .

]. The availability of metamaterials may lead to enhanced electromagnetic properties such as chirality, absorption, and anisotropy [3

3. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006) [CrossRef] .

5

5. J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B 83, 165107 (2011) [CrossRef] .

]. Engineered spatial dispersion is established as an essential route for diffraction management and subwavelength imaging [6

6. M. Conforti, M. Guasoni, and C. D. Angelis, “Subwavelength diffraction management,” Opt. Lett. 33, 2662–2664 (2008) [PubMed] .

, 7

7. C. J. Zapata-Rodríguez, D. Pastor, M. T. Caballero, and J. J. Miret, “Diffraction-managed superlensing using plasmonic lattices,” Opt. Commun. 285, 3358–3362 (2012) [CrossRef] .

]. In particular, a giant birefringence also creates proper conditions for excitation of nonresonant hybrid surface waves with potential application in nanosensing [8

8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008) [CrossRef] .

, 9

9. C. J. Zapata-Rodríguez, J. J. Miret, J. A. Sorni, and S. Vuković, “Propagation of dyakonon wave-packets at the boundary of metallodielectric lattices,” IEEE J. Sel. Top. Quant. Electron. 19, 4601408 (2013) [CrossRef] .

].

In its pioneering paper, Dyakonov theoretically demonstrated the existence of nondissipative surface waves at the boundary of a dielectric material and a transparent uniaxial medium [10

10. M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

]. However, the first experimental observation of Dyakonov surface waves came into sight more than 20 years later, most specially caused by a weak coupling with external sources [11

11. O. Takayama, L.-C. Crasovan, S. K. Johansen, D. Mihalache, D. Artigas, and L. Torner, “Dyakonov surface waves: A review,” Electromagnetics 28, 126–145 (2008) [CrossRef] .

, 12

12. O. Takayama, L. Crasovan, D. Artigas, and L. Torner, “Observation of Dyakonov surface waves,” Phys. Rev. Lett. 102, 043903 (2009) [CrossRef] [PubMed] .

]. Indeed Dyakonov-like surface waves (DSWs) also emerge in the case that a biaxial crystal [13

13. D. B. Walker, E. N. Glytsis, and T. K. Gaylord, “Surface mode at isotropic-uniaxial and isotropic-biaxial interfaces,” J. Opt. Soc. Am. A 15, 248–260 (1998) [CrossRef] .

, 14

14. M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010) [CrossRef] .

] or a structurally chiral material [15

15. J. Gao, A. Lakhtakia, J. A. Polo Jr., and M. Lei, “Dyakonov-Tamm wave guided by a twist defect in a structurally chiral material,” J. Opt. Soc. Am. A 26, 1615–1621 (2009) [CrossRef] .

, 16

16. J. Gao, A. Lakhtakia, and M. Lei, “Dyakonov-Tamm waves guided by the interface between two structurally chiral materials that differ only in handedness,” Phys. Rev. A 81, 013801 (2010) [CrossRef] .

] takes the place of the uniaxial medium. The case of metal-dielectric (MD) multilayered media is specially convenient since small filling fractions of the metallic inclusions enable metamaterials with an enormous birefringence, thus enhancing density of DSWs and relaxing their prominent directivity [17

17. O. Takayama, D. Artigas, and L. Torner, “Practical dyakonons,” Opt. Lett. 37, 4311–4313 (2012) [CrossRef] [PubMed] .

19

19. J. J. Miret, C. J. Zapata-Rodríguez, Z. Jaks̆ić, S. M. Vuković, and M. R. Belić, “Substantial enlargement of angular existence range for Dyakonov-like surface waves at semi-infinite metal-dielectric superlattice,” J. Nanophoton. 6, 063525 (2012) [CrossRef] .

]. We also include the relative ease of nanofabrication, bulk three-dimensional response, and broadband nonresonant tunability.

For near-infrared and visible wavelengths, nanolayered MD compounds behave like plasmonic crystals enabling a simplified description of the medium by using the long-wavelength approximation, which involves an homogenization of the structured metamaterial [20

20. S. M. Rytov, “Electromagnetic properties of layered media,” Sov. Phys. JETP 2, 466 (1956).

22

22. S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterial and metal-dielectric superlattices,” Appl. Phys. Lett 95, 041902 (2009) [CrossRef] .

]. Under certain conditions, the second-rank tensor denoting permittivity in the medium include elements of opposite sign, leading to extremely anisotropic metamaterials [23

23. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003) [CrossRef] [PubMed] .

, 24

24. I. I. Smolyaninov, E. Hwang, and E. Narimanov, “Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions,” Phys. Rev. B 85, 235122 (2012) [CrossRef] .

]. This class of nanostructured media with hyperbolic dispersion are promising metamaterials with a plethora of practical applications from biosensing to fluorescence engineering [25

25. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Advances in OptoElectronics 2012, ID 452502 (2012) [CrossRef] .

]. In this context, Jacob et al. showed for the first time the existence of DSWs when considering anisotropic media with indefinite permittivity [8

8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008) [CrossRef] .

]. This study was devoted mainly to surface waves enabling subdiffraction imaging in magnifying superlenses [26

26. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007) [CrossRef] [PubMed] .

], where hyperbolic DSWs exist at the interface of a metal and an all-dielectric birefringent metamaterial. When considering hyperbolic media, however, the authors provided only an elusive analysis of DSWs.

In this paper we retake the task and we perform a thorough analysis of DSWs taking place in semi-infinite MD lattices showing hyperbolic dispersion. In the first part of our study, our approach puts emphasis on the effective-medium approximation (EMA). Under these conditions, different regimes are found and they are thoroughly analyzed. These regimes include DSWs with non-hyperbolic dispersion. Validation of our results is carried out when put into practice using numerical simulations based on the finite-element method (FEM). The major points of our interest are nonlocal effects caused by the finite size of the layers and dissipative effects due to ohmic losses in the metals. Finally, the main conclusions are outlined.

2. The hyperbolic regimes of the plasmonic crystal

The system under analysis is depicted in Fig. 1. An isotropic material of dielectric constant ε is set in the semi-space x > 0. Filling the complementary space, x < 0, we consider a periodic bilayered structure made of two materials alternatively stacked along the z axis. Specifically, a transparent material of dielectric constant εd and slab width wd is followed by a metallic layer, the latter characterized by the permittivity εm and the width wm. For simplicity, we assume that dielectric materials are nondispersive; indeed we set ε = 1 and εd = 2.25 in our numerical simulations. Furthermore, if a Drude metal is included, its permittivity may be written as
εm=11Ω2,
(1)
neglecting damping. Note that frequencies in Eq. (1) are expressed in units of the plasma frequency, Ω = ω/ωp.

Fig. 1 Schematic arrangement under study, consisting of a semi-infinite MD lattice (x < 0) and an isotropic material (x > 0). In the numerical simulations, the periodic structure includes a Drude metal and a dielectric with εd = 2.25.

Form anisotropy of this type of plasmonic devices may be modelled in a simple way by employing average estimates of the dyadic permittivity ε̿[27

27. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

]. Under the conditions in which the EMA can be used (see details in Appendix A), the plasmonic lattice behaves like a uniaxial crystal whose optical axis is normal to the layers (the z axis in Fig. 1). The relative permittivity is set as ε̿ = ε(xx+yy)+ε||zz. We point out that the engineered anisotropy of the 1D lattice is modulated by the filling factor of the metal, f, but also by its strong dispersive character [28

28. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

].

In Fig. 2(a) we represent permittivities ε|| and ε of the plasmonic crystal shown in Fig. 1 for a wide range of frequencies. In practical terms, the filling factor of the metal governs the dissipative effects in the metamaterial, thus low values of f are of great convenience. Indeed f = 1/4 in our numerical simulations. For low frequencies, Ω ≪ 1, we come near the following expressions: εm < 0 and 0 < ε||εd/(1− f). Therefore, propagating TEz modes (Ez = 0) cannot exist in the bulk crystal since it behaves like a metal in these circumstances. On the other hand, TMz waves (Bz = 0) propagate following a spatial dispersion curve,
kx2+ky2ε||+kz2ε=Ω2,
(2)
where spatial frequencies are normalized to the wavenumber kp = ωp/c. Note that Eq. (2) denotes an hyperboloid of one sheet (see Fig. 2(b) at Ω = 0.20). Furthermore, the hyperbolic dispersion exists up to a frequency
Ω1=11+εd(1f)/f,
(3)
for which ε = 0; in our numerical example it occurs at Ω1 = 0.359. For slightly higher frequencies, both ε|| and ε are positive and Eq. (2) becomes an ellipsoid of revolution; Figure 2(b) at Ω = 0.45 is associated with such a case. Since its minor semi-axis is Ωε, the periodic multilayer simulates an anisotropic medium with positive birefringence. Raising the frequency even more, ε|| diverges at
Ω2=11+εdf/(1f),
(4)
leading to the so-called canalization regime [3

3. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006) [CrossRef] .

]. For the plasmonic lattice in Fig. 1 it happens at Ω2 = 0.756. In general, Ω1 < Ω2 provided that f < 1/2. Beyond Ω2, Eq. (2) turns to an hyperboloidal shape. In the range Ω2 < Ω < 1, however, the dispersion curve has two sheets. Figure 2(b) illustrate this case at Ω = 0.80. Note that the upper limit of this hyperbolic band is determined by the condition ε|| = 0, or equivalently εm = 0, occurring at the plasma frequency. To conclude, ε< (1 − f)εd in this spectral range.

Fig. 2 (a) Variation of relative permittivities ε|| and ε as a function of normalized frequency Ω, for the plasmonic crystal of Fig. 1. Here we assume that f = 1/4. (b) Plot of Eq. (2) in the kykz plane for extraordinary waves (TMz modes) for the three different cases that we come across in the range Ω < 1. Solid line corresponds to kx = 0 and shaded regions are associated with harmonic waves with kx > 0 (non-evanescent fields).

3. Dyakonov-like surface waves

In the isotropic medium, we consider a superposition of TEx (Ex = 0) and TMx (Bx = 0) space-harmonic waves whose wave vectors have the same real components ky and kz in the plane x = 0. These fields are evanescent in the isotropic medium, in direct proportion to exp(−κx), being the attenuation constant
κ=ky2+kz2εΩ2
(5)
in units of kp. On the other side of the boundary, the ordinary wave (o-wave) and extraordinary wave (e-wave) existing in the effective-uniaxial medium also decay exponentially with rates given by
κo=ky2+kz2εΩ2
(6)
and
κe=ky2+kz2ε||/εε||Ω2,
(7)
respectively. Using the appropriate boundary conditions, Dyakonov derived the following equation [10

10. M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

]
(κ+κe)(κ+κo)(εκo+εκe)=(ε||ε)(εε)Ω2κo,
(8)
which provides the spectral map of wave vectors kD.

In the special case of the surface wave propagation perpendicular to the optical axis (kz = 0), Eq. (8) reveals the following solution:
εκo+εκ=0.
(9)
In the case: ε < 0 and ε < |ε|, this equation has the well-known solution:
ky=Ωεεε+ε,
(10)
which resembles the dispersion of conventional surface plasmon polaritons [29

29. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

]. Indeed here we have purely TMx polarized waves, as expected. It is worth noting that no solutions in the form of surface waves can be found from the Eq. (8) in the case of propagation parallel to the optical axis (ky = 0) for hyperbolic metamaterials: εε|| < 0. That means that there is a threshold value of ky for the existence of surface waves. However, for the frequencies and the filling factors when both ε < 0; ε|| < 0, the solutions of Eq. (8) appear in the form of Bloch surface waves [22

22. S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterial and metal-dielectric superlattices,” Appl. Phys. Lett 95, 041902 (2009) [CrossRef] .

], i.e. for ky = 0.

4. DSWs in hyperbolic media

4.1. Low index of refraction n

First we analyze the case when ε < ε|| in the range Ω < Ω1, where in addition ε < 0 provided that f < 1/2. Note that this is satisfied if ε < εd/(1 − f). In the effective-uniaxial medium, o-waves are purely evanescent, and it is easy to see that κ < κo and also κe < κo. Under these circumstances, all brackets in Eq. (8) are positive provided that
εκo+εκe>0.
(11)
By the way, even though Eq. (11) may be satisfied if −ε < ε, in this case we cannot find a stationary solution of Maxwell’s equations satisfying Dyakonov’s equation (8). This happens within the spectral band Ω0 < Ω < Ω1, where
Ω0=11+ε/f+εd(1f)/f.
(12)
Note that Ω0 = 0.292 in our numerical simulation. For instance, in the limiting case ε = −ε, the unique solution of Eq. (8) is found for ky → ∞ and kz = 0, as can be deduced straightforwardly from Eq. (10).

In Fig. 3(a) and (b) we illustrate the dispersion equation of DSWs for two different frequencies in the range 0 < Ω < Ω0. In these cases, DSW dispersion curve approaches a hyperbola. Contrarily to what is shown in Fig. 3(b), we find a bandgap around kz = 0 in (a). In general terms it occurs if Ω < 0.271, whose limiting frequency is determined by the condition
1ε||1ε=1ε.
(13)
In this sense we point out that hybrid solutions near kz = 0 are additionally constrained to the condition kyΩε|| [see also Eq. (10)], which is a necessary condition for κe to exhibit real and positive values. Finally, a case similar to that shown in Fig. 3(b) was first reported in Fig. 5(b) from Ref. [8

8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008) [CrossRef] .

], though some discrepancies are evident.

Fig. 3 Graphical representation of Eq. (8), drawn in solid line, providing spatial dispersion of DSWs arising in the arrangement of Fig. 1, at different frequencies: (a) Ω = 0.20, (b) Ω = 0.28, and (c) Ω = 0.85. Here, the metamaterial is characterized by f = 0.25 and the isotropic medium is air. As a reference we also include equations κ = 0 (dotted line) and κe = 0 (dashed line). (d) Dispersion equation for DSWs as given in (c) but ranged over the region of interest. Points A, B, C, and SP are used in Fig. 4.

In order to determine the asymptotes of the hyperbolic-like DSW dispersion curve, we consider the quasi-static regime (Ω → 0) since |kD| = kD ≫ Ω. Under this approximation, κ = kD, κo = kD, and κe = ΘkD, where
Θ=1(1ε||ε)sin2θ,
(14)
being ky = kD cosθ, and kz = kD sinθ. Note that 0 ≤ Θ ≤ 1. By inserting all these approximations into Eq. (8), and performing the limit kD → ∞, we attain the equation ε + εΘ = 0 straighforwardly. The latter equation indicates that hyperbolic-like solutions of Dyakonov’s equation may be found provided that ε < 0 and additionally ε < −ε, occurring in the range Ω < Ω0. In this case, the asymptotes follow the equation kz = ky tanθD, where
tan2θD=ε2ε2ε||ε+ε2.
(15)
These asymptotes establish a canalization regime leading to a collective directional propagation of DSW beams [8

8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008) [CrossRef] .

, 9

9. C. J. Zapata-Rodríguez, J. J. Miret, J. A. Sorni, and S. Vuković, “Propagation of dyakonon wave-packets at the boundary of metallodielectric lattices,” IEEE J. Sel. Top. Quant. Electron. 19, 4601408 (2013) [CrossRef] .

, 28

28. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

]. At this point it is necessary to remind that the asymptotes of the e-waves dispersion curve, in the kykz-plane, have slopes given by tan2 θe = −ε/ε||. As a consequence θD < θe, as illustrated in Fig. 3(b), and in the limit Ω → 0 (ε → −∞) we obtain θD→θe.

Moving into the high-frequency band Ω2 < Ω < 1, we now find that ε|| < 0 < ε. The plot shown in Fig. 3(c) corresponds to this case. In a similar way found in Fig. 3(a) and (b), note the relevant proximity of DSW dispersion curve to κe = 0. Opposedly it crosses the e-wave hyperbolic curve at two different points, where solutions of Dyakonov’s equation begin and end respectively. In comparison, the angular range of DSWs turns to be significantly low. Apparently the z-component of kD tends to approach Ωε caused by the simultaneous dominance of o- and e-waves. In general, an slight increase of the refractive index in the isotropic medium pushes the wave vector kD to higher values, leading to an enormous shortening in the dispersion curve of the surface waves. As a consequence, high-n materials give rise to adverse conditions for the excitation of DSWs in the neighborhood of the plasma frequency.

Figure 4 shows the magnetic field for the points A, B, and C, all highlighted in Fig. 3. Also we represent the z-component of the field B that is associated with the point SP appearing in Fig. 3(b), and that corresponds to a surface plasmon (Bx = 0). The wave field is tightly confined near the surface x = 0, in a few units of 1/kp, for the cases A and B. Such a wave localization is even stronger than the confinement of the surface plasmon appearing at Ω = 0.28 (at ky = 0.625). This is caused by the large in-plane wavenumber of the DSW, being kD = 1.44 and 1.03 for the points A and B, respectively. Exceptionally, the lowest confinement is produced at Ω = 0.85 when making the choice C, in spite of considering a DSW with large wave vector kD = [0, 0.2, 1.07]. In this case, the interplay of slowly-decaying o- and e-waves counts against localization of the surface wave.

Fig. 4 Variation of the magnetic field (a) |Bx| and (b) |Bz| along the x-axis for the points A, B, and C highlighted in Fig. 3. The field is normalized to unity at its maximum absolute value. We include the point SP associated with TMx surface waves.
Fig. 5 (a) Solutions of Eq. (8) at a frequency Ω = 0.10 for an isotropic medium of permittivity ε = 10 and a layered metamaterial composed of a Drude metal and a dielectric of εd = 2.25. Curves κ = 0 and κe = 0 are also drawn in dotted and dashed lines, respectively. Profile of the magnetic field (b) |Bx| and (c) |Bz| along the x-axis for the point D shown in (a), including the point SP associated with a TMx surface wave.

4.2. High index of refraction n

In our numerical simulations we used a dielectric material with ε = 10, leading to Ω0 = 0.145. In Fig. 5(a) we illustrate the dispersion equation of DSWs at Ω = 0.1. Here DSW curve also approaches a hyperbola. In our instance, however, on-axis bandgaps are not found even at lower frequencies, and Eq. (8) provides solutions for every real value of kz. Figs. 5(b) and 5(c) show the profile of the magnetic field along the x axis for two different points (D and SP) of the dispersion curve. Once again, hybrid surface waves (case D) exhibit a tighter confinement near the boundary x = 0 than that offered by the solution of Eq. (8) and that is attributed to surface plasmons with pure TMx polarization (case SP).

Finally by assuming ε < ε at a given frequency of the spectral window Ω2 < Ω < 1, we have not found solutions of Eq. (8). As discussed in Sec. 4.1, high values of ε goes in prejudice of the appearance of hybrid surface waves.

5. Validity of the effective-medium approximation

Fig. 6 Exact dispersion curves of TMz modes in a Drude-metal/dielectric compound for different widths of the metallic layer, starting from wm → 0 (dashed line) and including higher widths at a constant rate of 1/10kp (solid lines). For an isotropic medium of permittivity ε = 1, frequencies are: (a) Ω = 0.20, (b) Ω = 0.28, and (c) Ω = 0.85. For ε = 10 we represent TMz modal curves at (d) Ω = 0.10. Also the curve κ = 0 is included in dotted lines.

6. Analysis of a practical case

Dissipation in metallic elements is a relevant issue of plasmonic devices. Taking into account ohmic losses, permittivities in Eqs. (1)(18) become complex valued. Consequently Dyakonov’s equation (8) provides complex values of the wave vector kD. This procedure has been discussed recently by Vuković et al in [18

18. S. M. Vuković, J. J. Miret, C. J. Zapata-Rodríguez, and Z. Jaks̆ić, “Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric,” Phys. Scripta T149, 014041 (2012) [CrossRef] .

]. Nevertheless, practical implementations are out of the long-wavelength limit, thus we also direct our efforts toward nonlocal effects. Numerical techniques to solve Maxwell’s equations seem to be convenient tools in order to provide a conclusive characterization of DSWs at the boundary of realistic metallodielectric lattices.

In order to tackle this problem, we evaluate numerically the value of ky for a given Bloch wavenumber kz. Since the imaginary part of εm is not neglected anymore, ky becomes complex. This means that the DSW cannot propagate indefinitely; Im(kx) denotes the attenuation factor of the surface wave along the metallic-film edges. In our numerical simulations we consider a dissipative DSW propagating on the side of a Ag-PMMA lattice at a wavelength of λ = 560 nm (normalized frequency Ω = 0.28), where the surrounding isotropic medium is air. Accordingly ε = 1, εd = 2.25, and εm = −11.7 + 0.83i (being Ωp = 12.0 rad/fs) [33

33. E. D. Palik and G. Ghosh, The Electronic Handbook of Optical Constants of Solids (Academic, 1999).

]. Bearing in mind a practical setting in the plasmonic lattice with current nanotechnology, we apply wm = 9 nm and f = 0.25.

Our computational approach lies on the FEM by means of COMSOL Multiphysics software. Thus given kz = 0.25, which is associated with the point B in Fig. 3(b), we finally estimate the complex propagation constant: ky = 0.70 + 0.06i. We point out that Eq. (8) predicts a value ky = 0.85 + 0.24i; in addition we obtain ky = 1.00 by neglecting losses in Dyakonov’s equation, as shown in Fig. 3(b). Our numerical experiment proves a “red shift” in the propagation constant caused by nonlocal effects. Furthermore, Im(kx) decreases sharply (roughly by a ratio 1/4) in comparison with EMA estimates. This major result enable DSWs propagating along distances significantly longer than those predicted by the long-wavelength approach.

Figure 7 shows the magnetic field B of the DSW in the xz-plane. The calculated pattern in one cell reveals the effects of retardation clearly. Along the z axis, an abrupt variation of B is evident inside the nanostructured material, in contrast with assumptions involving the effective-medium approach. The wave field cannot penetrate in the metal completely, and it is confined not only on the silver-air interface but also in the Ag-PMMA boundaries near x = 0. Indeed, from FEM simulations, the ratio of max|Bx| over max|Bz| yields 0.80, considerably higher that its value on the basis of the EMA (equal to 0.37). This proves a field enhancement on the walls of the metallic films and inside the dielectric nanolayers, minimizing dissipative effects in the lossy metamaterial. Finally, the distributed field along the x axis is analogous in all cases, also by comparing with EMA-based results.

Fig. 7 Contour plots of the magnetic fields (a) |Bx| and (b) |Bz| in the xz-plane, computed using FEM. The hyperbolic metamaterial is set on the left, for which only one period is represented. Also we graph the fields along (1) the center of the dielectric layer, (2) the center of the metallic slab, and (3) a plane containing the Ag-PMMA interface.

7. Conclusions

We showed that excitation of DSWs at the boundary of an isotropic dielectric and a hyperbolic metamaterial enable distinct regimes of propagation. At low frequencies, DSWs exhibit the well-known hyperbolic-like dispersion [8

8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008) [CrossRef] .

]. Exclusively, this occurs provided that 0 < ε < −ε. Importantly its vertex is placed on the ky axis, coinciding with a non-hybrid TMx surface wave. If ε||1ε1<ε1 is satisfied, a bandgap around kz = 0 arises in the DSW dispersion curve. However, these bandgaps disabling negative refraction would disappear by using an isotropic material with a sufficiently-high index of refraction, n.

We point out that the wave field of DSWs keeps tightly confined near the surface x = 0. Such a wave localization is even stronger than the confinement of the (TMx) surface plasmon appearing at kz = 0.

In the last part of this paper we analyzed retardation effects and dissipation effects caused by the finite width and ohmic losses of the metallic nanolayers, respectively. As a practical implementation we considered a dissipative DSW propagating at the boundary of an Ag-PMMA lattice and air. Our FEM-based numerical experiment proves a “red shift” in the propagation constant caused by nonlocal effects. Furthermore, the field is enhanced on the walls of the metallic films and inside the dielectric nanolayers, minimising dissipative effects in the lossy metamaterial. As a consequence, DSWs propagate along distances significantly longer than those predicted by the long-wavelength approach.

Finally, let us point out that the properties of the resulting bound states change rapidly with the refractive index of the surrounding medium, suggesting potential applications in chemical and biological sensing and nanoimaging.

Appendix A: Effective-medium approximation

Appendix B: Derivation of the Dyakonov’s equation

This appendix shows the procedure to derive the dispersion equation for the wave vector kD of the DSW. We start with the analytical expressions of the electric and magnetic fields involved in the boundary problem. In the isotropic medium, the bimodal electric field may be written in the complete form as E(r,t) = E(x) f (y,z,t), where
f(y,z,t)=exp(ikDriΩt),
(20)
and
E(x)=(ATEa1+ATMa2)exp(κx)
(21)
in x > 0. Here time- and space-coordinates are normalized to the inverse of ωp and kp, respectively. In addition ATE and ATM are complex-valued amplitudes, and the vectors
a1=[0,kz,ky]
(22)
a2=[ky2+kz2,ikyκ,ikzκ].
(23)
Note that the polarization vector a1 lies on the plane x = 0 where the DSW propagates, but it is perpendicular to its direction of propagation, kD. For the magnetic field we apply the Faraday’s law of induction, iΩcB = ∇ × E. In this case B(r,t) = B(x) f (y,z,t), where
B(x)=1Ωc(εΩ2ATMa1ATEa2)exp(κx).
(24)
From Eqs. (21) and (24) it is clear that ATE and ATM represent amplitudes of the TEx and TMx modes, respectively.

On the other side of the boundary, the dependence along the x direction of the multimodal electric field in x < 0 may be written as
E(x)=Aoboexp(κox)+Aeceexp(κex).
(25)
Here Ao and Ae stand for the amplitudes of the o-wave and the e-wave, respectively. Also we define the vectors
bo,e=[ky,iκo,e,0],
(26)
co,e=[iκo,ekz,kykz,kz2εΩ2].
(27)
Note that the vectors bo,e are perpendicular to the optical axis. The part of the magnetic field which provides its variation normally to the isotropic-uniaxial interface is set now as
B(x)=1Ωc[εΩ2Aebeexp(κex)Aocoexp(κ0x)].
(28)
As it is well-known, Eqs. (25) and (28) verify that the o-wave corresponds to a TEz mode and the e-wave is associated with a TMz mode.

Acknowledgments

This research was funded by the Spanish Ministry of Economy and Competitiveness under the project TEC2011-29120-C05-01, by the Serbian Ministry of Education and Science under the project III 45016, and by the Qatar National Research Fund under the project NPRP 09-462-1-074. CJZR gratefully acknowledges a financial support from the Generalitat Valenciana (grant BEST/2012/060).

References and links

1.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007) [CrossRef] .

2.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007) [CrossRef] .

3.

P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006) [CrossRef] .

4.

E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen, “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90, 223113 (2007) [CrossRef] .

5.

J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B 83, 165107 (2011) [CrossRef] .

6.

M. Conforti, M. Guasoni, and C. D. Angelis, “Subwavelength diffraction management,” Opt. Lett. 33, 2662–2664 (2008) [PubMed] .

7.

C. J. Zapata-Rodríguez, D. Pastor, M. T. Caballero, and J. J. Miret, “Diffraction-managed superlensing using plasmonic lattices,” Opt. Commun. 285, 3358–3362 (2012) [CrossRef] .

8.

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008) [CrossRef] .

9.

C. J. Zapata-Rodríguez, J. J. Miret, J. A. Sorni, and S. Vuković, “Propagation of dyakonon wave-packets at the boundary of metallodielectric lattices,” IEEE J. Sel. Top. Quant. Electron. 19, 4601408 (2013) [CrossRef] .

10.

M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

11.

O. Takayama, L.-C. Crasovan, S. K. Johansen, D. Mihalache, D. Artigas, and L. Torner, “Dyakonov surface waves: A review,” Electromagnetics 28, 126–145 (2008) [CrossRef] .

12.

O. Takayama, L. Crasovan, D. Artigas, and L. Torner, “Observation of Dyakonov surface waves,” Phys. Rev. Lett. 102, 043903 (2009) [CrossRef] [PubMed] .

13.

D. B. Walker, E. N. Glytsis, and T. K. Gaylord, “Surface mode at isotropic-uniaxial and isotropic-biaxial interfaces,” J. Opt. Soc. Am. A 15, 248–260 (1998) [CrossRef] .

14.

M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010) [CrossRef] .

15.

J. Gao, A. Lakhtakia, J. A. Polo Jr., and M. Lei, “Dyakonov-Tamm wave guided by a twist defect in a structurally chiral material,” J. Opt. Soc. Am. A 26, 1615–1621 (2009) [CrossRef] .

16.

J. Gao, A. Lakhtakia, and M. Lei, “Dyakonov-Tamm waves guided by the interface between two structurally chiral materials that differ only in handedness,” Phys. Rev. A 81, 013801 (2010) [CrossRef] .

17.

O. Takayama, D. Artigas, and L. Torner, “Practical dyakonons,” Opt. Lett. 37, 4311–4313 (2012) [CrossRef] [PubMed] .

18.

S. M. Vuković, J. J. Miret, C. J. Zapata-Rodríguez, and Z. Jaks̆ić, “Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric,” Phys. Scripta T149, 014041 (2012) [CrossRef] .

19.

J. J. Miret, C. J. Zapata-Rodríguez, Z. Jaks̆ić, S. M. Vuković, and M. R. Belić, “Substantial enlargement of angular existence range for Dyakonov-like surface waves at semi-infinite metal-dielectric superlattice,” J. Nanophoton. 6, 063525 (2012) [CrossRef] .

20.

S. M. Rytov, “Electromagnetic properties of layered media,” Sov. Phys. JETP 2, 466 (1956).

21.

A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977) [CrossRef] .

22.

S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterial and metal-dielectric superlattices,” Appl. Phys. Lett 95, 041902 (2009) [CrossRef] .

23.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003) [CrossRef] [PubMed] .

24.

I. I. Smolyaninov, E. Hwang, and E. Narimanov, “Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions,” Phys. Rev. B 85, 235122 (2012) [CrossRef] .

25.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Advances in OptoElectronics 2012, ID 452502 (2012) [CrossRef] .

26.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699–1701 (2007) [CrossRef] [PubMed] .

27.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

28.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

29.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

30.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007) [CrossRef] .

31.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011) [CrossRef] .

32.

A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express 21, 1593–1598 (2013) [CrossRef] [PubMed] .

33.

E. D. Palik and G. Ghosh, The Electronic Handbook of Optical Constants of Solids (Academic, 1999).

34.

E. Popov and S. Enoch, “Mystery of the double limit in homogenization of finitely or perfectly conducting periodic structures,” Opt. Lett. 32, 3441–3443 (2007) [CrossRef] [PubMed] .

35.

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011) [CrossRef] .

36.

P. Chaturvedi, W. Wu, V. J. Logeeswaran, Z. Yu, M. S. Islam, S. Y. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett. 96, 043102 (2010) [CrossRef] .

37.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336, 205–209 (2012) [CrossRef] [PubMed] .

OCIS Codes
(260.2065) Physical optics : Effective medium theory
(160.4236) Materials : Nanomaterials

ToC Category:
Metamaterials

History
Original Manuscript: April 5, 2013
Revised Manuscript: May 26, 2013
Manuscript Accepted: June 14, 2013
Published: August 5, 2013

Citation
Carlos J. Zapata-Rodríguez, Juan J. Miret, Slobodan Vuković, and Milivoj R. Belić, "Engineered surface waves in hyperbolic metamaterials," Opt. Express 21, 19113-19127 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-19113


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References

  1. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon.1, 224–227 (2007). [CrossRef]
  2. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon.1, 641–648 (2007). [CrossRef]
  3. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B73, 113110 (2006). [CrossRef]
  4. E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen, “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett.90, 223113 (2007). [CrossRef]
  5. J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B83, 165107 (2011). [CrossRef]
  6. M. Conforti, M. Guasoni, and C. D. Angelis, “Subwavelength diffraction management,” Opt. Lett.33, 2662–2664 (2008). [PubMed]
  7. C. J. Zapata-Rodríguez, D. Pastor, M. T. Caballero, and J. J. Miret, “Diffraction-managed superlensing using plasmonic lattices,” Opt. Commun.285, 3358–3362 (2012). [CrossRef]
  8. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett.93, 221109 (2008). [CrossRef]
  9. C. J. Zapata-Rodríguez, J. J. Miret, J. A. Sorni, and S. Vuković, “Propagation of dyakonon wave-packets at the boundary of metallodielectric lattices,” IEEE J. Sel. Top. Quant. Electron.19, 4601408 (2013). [CrossRef]
  10. M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP67, 714–716 (1988).
  11. O. Takayama, L.-C. Crasovan, S. K. Johansen, D. Mihalache, D. Artigas, and L. Torner, “Dyakonov surface waves: A review,” Electromagnetics28, 126–145 (2008). [CrossRef]
  12. O. Takayama, L. Crasovan, D. Artigas, and L. Torner, “Observation of Dyakonov surface waves,” Phys. Rev. Lett.102, 043903 (2009). [CrossRef] [PubMed]
  13. D. B. Walker, E. N. Glytsis, and T. K. Gaylord, “Surface mode at isotropic-uniaxial and isotropic-biaxial interfaces,” J. Opt. Soc. Am. A15, 248–260 (1998). [CrossRef]
  14. M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B81, 115335 (2010). [CrossRef]
  15. J. Gao, A. Lakhtakia, J. A. Polo, and M. Lei, “Dyakonov-Tamm wave guided by a twist defect in a structurally chiral material,” J. Opt. Soc. Am. A26, 1615–1621 (2009). [CrossRef]
  16. J. Gao, A. Lakhtakia, and M. Lei, “Dyakonov-Tamm waves guided by the interface between two structurally chiral materials that differ only in handedness,” Phys. Rev. A81, 013801 (2010). [CrossRef]
  17. O. Takayama, D. Artigas, and L. Torner, “Practical dyakonons,” Opt. Lett.37, 4311–4313 (2012). [CrossRef] [PubMed]
  18. S. M. Vuković, J. J. Miret, C. J. Zapata-Rodríguez, and Z. Jaks̆ić, “Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric,” Phys. ScriptaT149, 014041 (2012). [CrossRef]
  19. J. J. Miret, C. J. Zapata-Rodríguez, Z. Jaks̆ić, S. M. Vuković, and M. R. Belić, “Substantial enlargement of angular existence range for Dyakonov-like surface waves at semi-infinite metal-dielectric superlattice,” J. Nanophoton.6, 063525 (2012). [CrossRef]
  20. S. M. Rytov, “Electromagnetic properties of layered media,” Sov. Phys. JETP2, 466 (1956).
  21. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am.67, 438–448 (1977). [CrossRef]
  22. S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterial and metal-dielectric superlattices,” Appl. Phys. Lett95, 041902 (2009). [CrossRef]
  23. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003). [CrossRef] [PubMed]
  24. I. I. Smolyaninov, E. Hwang, and E. Narimanov, “Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions,” Phys. Rev. B85, 235122 (2012). [CrossRef]
  25. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Advances in OptoElectronics2012, ID 452502 (2012). [CrossRef]
  26. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science315, 1699–1701 (2007). [CrossRef] [PubMed]
  27. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  28. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006). [CrossRef]
  29. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  30. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007). [CrossRef]
  31. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011). [CrossRef]
  32. A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express21, 1593–1598 (2013). [CrossRef] [PubMed]
  33. E. D. Palik and G. Ghosh, The Electronic Handbook of Optical Constants of Solids (Academic, 1999).
  34. E. Popov and S. Enoch, “Mystery of the double limit in homogenization of finitely or perfectly conducting periodic structures,” Opt. Lett.32, 3441–3443 (2007). [CrossRef] [PubMed]
  35. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011). [CrossRef]
  36. P. Chaturvedi, W. Wu, V. J. Logeeswaran, Z. Yu, M. S. Islam, S. Y. Wang, R. S. Williams, and N. X. Fang, “A smooth optical superlens,” Appl. Phys. Lett.96, 043102 (2010). [CrossRef]
  37. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science336, 205–209 (2012). [CrossRef] [PubMed]

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