## Engineered surface waves in hyperbolic metamaterials |

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

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

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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. 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] .

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] .

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] .

## 2. The hyperbolic regimes of the plasmonic crystal

*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

*ε*and slab width

_{d}*w*is followed by a metallic layer, the latter characterized by the permittivity

_{d}*ε*and the width

_{m}*w*. For simplicity, we assume that dielectric materials are nondispersive; indeed we set

_{m}*ε*= 1 and

*ε*= 2.25 in our numerical simulations. Furthermore, if a Drude metal is included, its permittivity may be written as neglecting damping. Note that frequencies in Eq. (1) are expressed in units of the plasma frequency, Ω =

_{d}*ω/ω*.

_{p}*ε̿*[27]. 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] .

*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:

*ε*

_{⊥}≈

*fε*< 0 and 0 <

_{m}*ε*

_{||}≈

*ε*/(1−

_{d}*f*). Therefore, propagating TE

*modes (*

^{z}*E*= 0) cannot exist in the bulk crystal since it behaves like a metal in these circumstances. On the other hand, TM

_{z}*waves (*

^{z}*B*= 0) propagate following a spatial dispersion curve, where spatial frequencies are normalized to the wavenumber

_{z}*k*=

_{p}*ω*/

_{p}*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 for which

*ε*

_{⊥}= 0; in our numerical example it occurs at Ω

_{1}= 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

*ε*

_{||}diverges at 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] .

_{2}= 0.756. In general, Ω

_{1}< Ω

_{2}provided that

*f*< 1/2. Beyond Ω

_{2}, Eq. (2) turns to an hyperboloidal shape. In the range Ω

_{2}< 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

*ε*= 0, occurring at the plasma frequency. To conclude,

_{m}*ε*

_{⊥}< (1 −

*f*)

*ε*in this spectral range.

_{d}## 3. Dyakonov-like surface waves

*x*= 0, their amplitudes decay as |

*x*| → ∞, and ultimately they must satisfy Maxwell’s equations. For that purpose we follow Dyakonov by considering a modal treatment of our problem [10], also shown in Appendix B. This is a simplified procedure that is based on the characterization of the plasmonic lattice as a uniaxial crystal, enabling to establish the diffraction equation that gives the in-plane wave vector

**k**

*= [0,*

_{D}*k*,

_{y}*k*] of the DSW.

_{z}*(*

^{x}*E*= 0) and TM

_{x}*(*

^{x}*B*= 0) space-harmonic waves whose wave vectors have the same real components

_{x}*k*and

_{y}*k*in the plane

_{z}*x*= 0. These fields are evanescent in the isotropic medium, in direct proportion to exp(−

*κx*), being the attenuation constant in units of

*k*. On the other side of the boundary, the ordinary wave (

_{p}*o*-wave) and extraordinary wave (

*e*-wave) existing in the effective-uniaxial medium also decay exponentially with rates given by and respectively. Using the appropriate boundary conditions, Dyakonov derived the following equation [10] which provides the spectral map of wave vectors

**k**

*.*

_{D}*k*= 0), Eq. (8) reveals the following solution: In the case:

_{z}*ε*

_{⊥}< 0 and

*ε*< |

*ε*

_{⊥}|, this equation has the well-known solution: which resembles the dispersion of conventional surface plasmon polaritons [29]. Indeed here we have purely TM

*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 (*

^{x}*k*= 0) for hyperbolic metamaterials:

_{y}*ε*

_{⊥}

*ε*

_{||}< 0. That means that there is a threshold value of

*k*for the existence of surface waves. However, for the frequencies and the filling factors when both

_{y}*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] .

*k*= 0.

_{y}## 4. DSWs in hyperbolic media

*ε*

_{⊥}

*ε*

_{||}< 0. In this case, DSWs may be found in different regimes, which depend not only on the elements of

*ε̿*characterizing the metamaterial, but also that of the surrounding isotropic material

*ε*. Next we describe distinct configurations governing DSWs, first subject to a low value of the refractive index

*ε*<

*ε*

_{||}(

*ε*<

*ε*

_{⊥}) for low frequencies (in the neighborhood of the plasma frequency), and latter focused on a high index of refraction.

### 4.1. Low index of refraction n

*ε*<

*ε*

_{||}in the range Ω < Ω

_{1}, where in addition

*ε*

_{⊥}< 0 provided that

*f*< 1/2. Note that this is satisfied if

*ε*<

*ε*/(1 −

_{d}*f*). In the effective-uniaxial medium,

*o*-waves are purely evanescent, and it is easy to see that

*κ*<

*κ*and also

_{o}*κ*<

_{e}*κ*. Under these circumstances, all brackets in Eq. (8) are positive provided that By the way, even though Eq. (11) may be satisfied if −

_{o}*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 Ω

_{0}< Ω < Ω

_{1}, where Note that Ω

_{0}= 0.292 in our numerical simulation. For instance, in the limiting case

*ε*= −

*ε*

_{⊥}, the unique solution of Eq. (8) is found for

*k*→ ∞ and

_{y}*k*= 0, as can be deduced straightforwardly from Eq. (10).

_{z}_{0}. In these cases, DSW dispersion curve approaches a hyperbola. Contrarily to what is shown in Fig. 3(b), we find a bandgap around

*k*= 0 in (a). In general terms it occurs if Ω < 0.271, whose limiting frequency is determined by the condition In this sense we point out that hybrid solutions near

_{z}*k*= 0 are additionally constrained to the condition

_{z}*κ*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

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

**k**

*| =*

_{D}*k*≫ Ω. Under this approximation,

_{D}*κ*=

*k*,

_{D}*κ*=

_{o}*k*, and

_{D}*κ*= Θ

_{e}*k*, where being

_{D}*k*=

_{y}*k*cos

_{D}*θ*, and

*k*=

_{z}*k*sin

_{D}*θ*. Note that 0 ≤ Θ ≤ 1. By inserting all these approximations into Eq. (8), and performing the limit

*k*→ ∞, we attain the equation

_{D}*ε*+

*ε*

_{⊥}Θ = 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 < Ω

_{0}. In this case, the asymptotes follow the equation

*k*=

_{z}*k*tan

_{y}*θ*, where These asymptotes establish a canalization regime leading to a collective directional propagation of DSW beams [8

_{D}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] .

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] .

*e*-waves dispersion curve, in the

*k*-plane, have slopes given by tan2

_{y}k_{z}*θ*= −

_{e}*ε*

_{⊥}/

*ε*

_{||}. As a consequence

*θ*<

_{D}*θ*, as illustrated in Fig. 3(b), and in the limit Ω → 0 (

_{e}*ε*

_{⊥}→ −∞) we obtain

*θ*.

_{D}→θ_{e}_{2}< 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

*κ*= 0. Opposedly it crosses the

_{e}*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

**k**

*tends to approach*

_{D}*o*- and

*e*-waves. In general, an slight increase of the refractive index in the isotropic medium pushes the wave vector

**k**

*to higher values, leading to an enormous shortening in the dispersion curve of the surface waves. As a consequence, high-*

_{D}*n*materials give rise to adverse conditions for the excitation of DSWs in the neighborhood of the plasma frequency.

*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 (

*B*= 0). The wave field is tightly confined near the surface

_{x}*x*= 0, in a few units of 1/

*k*, for the cases

_{p}*A*and

*B*. Such a wave localization is even stronger than the confinement of the surface plasmon appearing at Ω = 0.28 (at

*k*= 0.625). This is caused by the large in-plane wavenumber of the DSW, being

_{y}*k*= 1.44 and 1.03 for the points

_{D}*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

**k**

*= [0, 0.2, 1.07]. In this case, the interplay of slowly-decaying*

_{D}*o*- and

*e*-waves counts against localization of the surface wave.

### 4.2. High index of refraction n

*ε*

_{||}<

*ε*in the spectral domain Ω < Ω

_{1}. Therefore, the curve

*κ*= 0 characterizing the isotropic medium crosses the TM

*dispersion curve*

^{z}*κ*= 0 of the uniaxial metamaterial. Note that such a curve crossing is mandatory by considering materials with all-positive permittivities, alike the pioneer paper by Dyakonov [10, 13

_{e}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] .

*ε*

_{⊥}<

*ε*, occurring at Ω

_{0}< Ω < Ω

_{1}. Nevertheless, there are some distinct features which are worthy to mention.

*ε*= 10, leading to Ω

_{0}= 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

*k*. Figs. 5(b) and 5(c) show the profile of the magnetic field along the

_{z}*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 TM

*polarization (case*

^{x}*SP*).

*ε*

_{⊥}<

*ε*at a given frequency of the spectral window Ω

_{2}< 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

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] .

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] .

*-polarized beam into negatively and positively refracted parts, within the favourable frequency range. Here, however, we deal with hyperbolic metamaterials that require frequency ranges out of the double refraction range. Confining ourselves to the wave fields of TM*

^{z}*polarization which approach*

^{z}*e*-wave in the regime of validity of EMA, the exact map of wave vectors

**k**

*characterizing Bloch waves is calculated via: where*

_{D}*φ*=

_{q}*k*, and

_{qz}w_{q}*q*=

*d*) and Drude metal (

*q*=

*m*).

*x*< 0, considering different widths

*w*of the metal but maintaining its filling factor,

_{m}*f*= 0.25; ohmic losses in the metal are neglected once again. We use the same configurations appearing in Figs. 3(a)–3(c) and Fig. 5(a). In the numerical simulations, we set

*k*= 0 in Eq. (16) accordingly. We observe that the EMA is extremely accurate for

_{x}*w*= 0.1 (in units of 1/

_{m}*k*) in the region of interest. However, deviations among the contours are evident for wider metallic layers. Apparently, Eq. (16) is in good agreement with the EMA in the vicinity of

_{p}*k*= 0, but rising

_{z}*w*makes the dispersion curve to sheer in direction to the

_{m}*z*axis. As a general rule, given a value of the on-axis frequency

*k*, Eq. (16) yields lower values of

_{z}*k*. As shown in the next section, this is also the cause of a spectral shift of

_{y}**k**

*along the same direction.*

_{D}**k**

*at the interface*

_{D}*x*= 0, as seen above, but also put additional conditions on this boundary. Since the introduction of an effective permittivity requires some kind of field averaging normally to the metal-dielectric layers, the excitation of evanescent fields in the isotropic medium would be fundamentally governed by the value of

**k**

*which determines the attenuation constant*

_{D}*κ*given in Eq. (5). However, spatial dispersion also leads to strong field oscillations across the system [30

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] .

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] .

*M̿*) = 0 are not valid anymore. Such a strong variation of the field is set on the scale of a single layer. Consequently, evanescent fields with spatial frequencies much higher than

*k*will participate vigorously in the isotropic medium extremely near the boundary. As we will see in the FEM simulations appearing in the next section, predominance of these high-frequency components of the field lies crucially by the edge of the metallic layers adjoining the isotropic medium.

_{D}## 6. Analysis of a practical case

**k**

*. This procedure has been discussed recently by Vuković et al in [18*

_{D}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] .

*k*for a given Bloch wavenumber

_{y}*k*. Since the imaginary part of

_{z}*ε*is not neglected anymore,

_{m}*k*becomes complex. This means that the DSW cannot propagate indefinitely; Im(

_{y}*k*) 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

_{x}*λ*= 560 nm (normalized frequency Ω = 0.28), where the surrounding isotropic medium is air. Accordingly

*ε*= 1,

*ε*= 2.25, and

_{d}*ε*= −11.7 + 0.83

_{m}*i*(being Ω

*= 12.0 rad/fs) [33]. Bearing in mind a practical setting in the plasmonic lattice with current nanotechnology, we apply*

_{p}*w*= 9 nm and

_{m}*f*= 0.25.

*k*= 0.25, which is associated with the point

_{z}*B*in Fig. 3(b), we finally estimate the complex propagation constant:

*k*= 0.70 + 0.06

_{y}*i*. We point out that Eq. (8) predicts a value

*k*= 0.85 + 0.24

_{y}*i*; in addition we obtain

*k*= 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(

_{y}*k*) 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.

_{x}**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|

*B*| over max|

_{x}*B*| 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

_{z}*x*axis is analogous in all cases, also by comparing with EMA-based results.

## 7. Conclusions

**93**, 221109 (2008) [CrossRef] .

*ε*< −

*ε*

_{⊥}. Importantly its vertex is placed on the

*k*axis, coinciding with a non-hybrid TM

_{y}*surface wave. If*

^{x}*k*= 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,

_{z}*n*.

*ε*

_{||}< 0 <

*ε*

_{⊥}(two-sheet hyperboloidal dispersion). DSWs dispersion curve crosses the

*e*-wave hyperbolic curve at two different points, where solutions of Dyakonov’s equation begin and end. In comparison with the previous case, the spectral range of DSWs turns to be significantly low. Furthermore, high-

*n*materials give rise to adverse conditions for the excitation of DSWs.

*x*= 0. Such a wave localization is even stronger than the confinement of the (TM

*) surface plasmon appearing at*

^{x}*k*= 0.

_{z}## Appendix A: Effective-medium approximation

*L*[22

_{D}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] .

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] .

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] .

*L*may be roughly estimated by the inverse of

_{D}*k*, that is deeply subwavelength in practice. In reference to this point, recent development of microfabrication technology makes it possible to create such subwavelength structures [36

_{p}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] .

*ε̿*=

*ε*

_{⊥}(

**xx**+

**yy**) +

*ε*

_{||}

**zz**, where gives the permittivity along the optical axis, and corresponds to the permittivity in the transversal direction. In the previous equations, denotes the filling factor of the metal.

## Appendix B: Derivation of the Dyakonov’s equation

**k**

*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*

_{D}**E**(

**r**

*,t*) =

**E**(

*x*)

*f*(

*y,z,t*), where and in

*x*> 0. Here time- and space-coordinates are normalized to the inverse of

*ω*and

_{p}*k*, respectively. In addition

_{p}*A*and

_{TE}*A*are complex-valued amplitudes, and the vectors Note that the polarization vector

_{TM}**a**

_{1}lies on the plane

*x*= 0 where the DSW propagates, but it is perpendicular to its direction of propagation,

**k**

*. For the magnetic field we apply the Faraday’s law of induction,*

_{D}*i*Ω

*c*

**B**= ∇ ×

**E**. In this case

**B**(

**r**

*,t*) =

**B**(

*x*)

*f*(

*y,z,t*), where From Eqs. (21) and (24) it is clear that

*A*and

_{TE}*A*represent amplitudes of the TE

_{TM}*and TM*

^{x}*modes, respectively.*

^{x}*x*direction of the multimodal electric field in

*x*< 0 may be written as Here

*A*and

_{o}*A*stand for the amplitudes of the

_{e}*o*-wave and the

*e*-wave, respectively. Also we define the vectors Note that the vectors

**b**

*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 As it is well-known, Eqs. (25) and (28) verify that the*

_{o,e}*o*-wave corresponds to a TE

*mode and the*

^{z}*e*-wave is associated with a TM

*mode.*

^{z}*x*= 0, that is, continuity of the

*y*- and

*z*-components of the fields

**E**and

**B**at the planar interface. This problem may be set in matrix form as where the vector

**A**= [

*A*,

_{TE}*A*,

_{TM}*A*,

_{o}*A*] includes the amplitudes of the four modes that the DSW integrates. In addition, the 4 × 4 matrix Nontrivial solution of Eq. (29) may be found if the determinant of

_{e}*M̿*vanishes. In this way, Dyakonov derived equation (8), which provides the spectral map of wave vectors

**k**

*.*

_{D}## Acknowledgments

## References and links

1. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. |

2. | S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. |

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 |

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. |

5. | J. Hao, L. Zhou, and M. Qiu, “Nearly total absorption of light and heat generation by plasmonic metamaterials,” Phys. Rev. B |

6. | M. Conforti, M. Guasoni, and C. D. Angelis, “Subwavelength diffraction management,” Opt. Lett. |

7. | C. J. Zapata-Rodríguez, D. Pastor, M. T. Caballero, and J. J. Miret, “Diffraction-managed superlensing using plasmonic lattices,” Opt. Commun. |

8. | Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. |

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. |

10. | M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP |

11. | O. Takayama, L.-C. Crasovan, S. K. Johansen, D. Mihalache, D. Artigas, and L. Torner, “Dyakonov surface waves: A review,” Electromagnetics |

12. | O. Takayama, L. Crasovan, D. Artigas, and L. Torner, “Observation of Dyakonov surface waves,” Phys. Rev. Lett. |

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 |

14. | M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B |

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 |

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 |

17. | O. Takayama, D. Artigas, and L. Torner, “Practical dyakonons,” Opt. Lett. |

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 |

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. |

20. | S. M. Rytov, “Electromagnetic properties of layered media,” Sov. Phys. JETP |

21. | A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. |

22. | S. M. Vukovic, I. V. Shadrivov, and Y. S. Kivshar, “Surface Bloch waves in metamaterial and metal-dielectric superlattices,” Appl. Phys. Lett |

23. | D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

24. | I. I. Smolyaninov, E. Hwang, and E. Narimanov, “Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions,” Phys. Rev. B |

25. | Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Advances in OptoElectronics |

26. | I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science |

27. | P. Yeh, |

28. | B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B |

29. | S. A. Maier, |

30. | J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. |

31. | A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B |

32. | A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express |

33. | E. D. Palik and G. Ghosh, |

34. | E. Popov and S. Enoch, “Mystery of the double limit in homogenization of finitely or perfectly conducting periodic structures,” Opt. Lett. |

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 |

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. |

37. | H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science |

**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

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