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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 7337–7348
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Near-perfect absorption in epsilon-near-zero structures with hyperbolic dispersion

Klaus Halterman and J. Merle Elson  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 7337-7348 (2014)
http://dx.doi.org/10.1364/OE.22.007337


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Abstract

We investigate the interaction of polarized electromagnetic waves with hyperbolic metamaterial structures, whereby the in-plane permittivity component εx is opposite in sign to the normal component εz. We find that when the thickness of the metamaterial is smaller than the wavelength of the incident wave, hyperbolic metamaterials can absorb significantly higher amounts of electromagnetic energy compared to their conventional counterparts. We also demonstrate that for wavelengths leading to ℜ(εz) ≈ 0, near-perfect absorption arises and persists over a range of frequencies and subwavelength structure thicknesses.

© 2014 Optical Society of America

1. Introduction

With recent advances in nanoscale fabrication of metal-dielectric multilayers and arrays of rods, hybrid structures can now be created that absorb a substantial portion of incident electromagnetic (EM) radiation. In conventional approaches, strong absorption was achieved by utilizing materials that had either high loss or large thickness. Nowadays, with the advent of metamaterials, absorbing structures can be created that harness plasmonic excitations or implement high impedance components [1

1. D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47(11), 2059–2074 (1999). [CrossRef]

] that have extreme values [2

2. M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials”, Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]

6

6. S. Molesky, C. J. Dewalt, and Z. Jacob, “High temperature epsilon-near-zero and epsilon-near-pole metamaterial emitters for thermophotovoltaics,” Opt. Express 21(S1), A96–A110 (2013). [CrossRef] [PubMed]

] of the permittivity ε or permeability μ. In close connection with these developments, there has also been a substantial amount of research lately involving anisotropic metamaterials, where now ε and μ are tensors that have in general differing components along the three coordinate axes. An important type of anisotropic metamaterial is one whose corresponding orthogonal tensor components are of opposite sign, sometimes referred to as indefinite media [7

7. D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003). [CrossRef]

]. When such structures are described by a diagonal tensor, the corresponding dispersion relation permits wavevectors that lie within a hyperbolic isofrequency surface, and hence such a material is also called a hyperbolic metamaterial (HMM). The inclusion of HMM elements in many designs can be beneficial due to their inherent nonresonant character, thus limiting loss effects [8

8. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012). [CrossRef]

].

The earliest HMM construct involving bilayers of anisotropic media was discussed in the context of bandpass spatial filters with tunable cutoffs [7

7. D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003). [CrossRef]

]. For wavelengths λ in the visible spectrum, an effective HMM was modeled using arrays of metallic nanowires [9

9. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008). [CrossRef] [PubMed]

] spaced apart distances much smaller than λ, thus avoiding the usual problems associated with resonances. Periodic arrays of carbon nanotubes [10

10. I. Nefedov and S. Tretyakov, “Ultrabroadband electromagnetically indenite medium formed by aligned carbon nanotubes,” Phys. Rev. B 84(11), 113410 (2011). [CrossRef]

] have been shown to exhibit HMM characteristics in the THz spectral range. Other possibilities involve metal-dielectric layers: The inclusion of active media in metal-dielectric multilayers can result in improved HMM-based imaging devices [11

11. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011). [CrossRef]

]. For certain layer configurations, nonlocal effects [12

12. S. Savoia, G. Castaldi, and V. Galdi, “Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices,” Phys. Rev. B 87, 235116 (2013). [CrossRef]

], which depending on geometry [13

13. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]

], can limit the number of accessible photonic states [14

14. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012). [CrossRef]

]. The absorption in thin films has been shown experimentally to be enhanced when in contact with a multilayered HMM substrate [15

15. T. Tumkur, L. Gu, J. Kitur, E. Narimanov, and M. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100, 161103 (2012). [CrossRef]

]. Rather than using metallic components, tunable graphene can switch between a hyperbolic and conventional material via a gate voltage [16

16. F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013). [CrossRef]

]. The HMM dispersion can be tuned in gyromagnetic/dielectric [17

17. W. Li, Z. Liu, X. Zhang, and X. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 1611084 (2012). [CrossRef]

] and semiconductor/dielectric structures [18

18. C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. 37, 3345–3347 (2012). [CrossRef]

]. Slabs of semiconductors can also exhibit tunability by photogenerating a grating via variations in the carrier density caused by two incident beams, revealing a hyperbolic character [19

19. C. Rizza, A. Ciattoni, L. Columbo, M. Brambilla, and F. Prati, “Terahertz optically tunable dielectric metamaterials without microfabrication,” Opt. Lett. 38, 1307 (2013). [CrossRef] [PubMed]

].

Increased absorption can also be achieved by incorporating a grating with the HMM, so that by introducing surface corrugations, or grooves, light can diffract and generate a broad spectrum of wave vectors into the HMM layer. These wavevectors can couple via surface modes [20

20. W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A 24, 530 (2007). [CrossRef]

] due to the impedance mismatch at the various openings. Grating lines were patterned above a layered Au/TiO2 HMM structure, creating a “hypergrating” capable of exciting both surface and bulk plasmons [21

21. K.V. Sreekanth, A. De Luca, and G. Strangi, “Experimental demonstration of surface and bulk plasmon polaritons in hypergratings,” Sci. Rep. 3, 3291 (2013). [CrossRef] [PubMed]

]. By judiciously designing the materials below the grating, it can be possible to absorb a considerable fraction of the diffracted EM field. Indeed, a HMM comprised of arrays of silver nanowires was experimentally shown to reduce the reflectance by introducing surface corrugations [22

22. E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6. [CrossRef]

]. Spherical nanoparticles deposited on planar HMM structures also resulted in reduced reflectance due to the increased density of photonic states [23

23. T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101, 091105 (2012). [CrossRef]

].

In this paper we show that near-perfect absorption of EM radiation can arise in a simple HMM structure adjacent to a metal. We investigate a range of frequencies where the permittivity components perpendicular and parallel to the interfaces are of opposite sign. We consider two possibilities: when the HMM dispersion relation is of type-1 or type-2, which for our geometry corresponds to εx > 0, εz < 0 or εx < 0, εz > 0 respectively (see Fig. 1). We show that for those λ leading to the real part of the permittivity component perpendicular to the interfaces (εz) nearly vanishing, an intricate balance between material loss and structure thickness (τ) yields a broad range of incident angles θ and τ in which nearly the entire EM wave is absorbed. These findings are absent in conventional anisotropic “elliptical” structures.

Fig. 1 (a) Schematic of the hyperbolic metamaterial configuration: The HMM layer of thickness τ is bordered by a semi-infinite superstrate and substrate. The permittivites εi (i=1,2,3) are in general anisotropic. The incident field is polarized in the xz plane at an angle θ. (b) Dispersion contours for a type-2 HMM where εx<0, and εz>0 and (c) for a type-1 HMM with εx>0, and εz<0.

2. Methods

We assume that the incident EM wave propagates with wave vector in the xz plane with polarization (Ex,Ez,By) (p-polarized) or (Ey,Bx,Bz) (s-polarized). Once the wave enters the anisotropic medium, its polarization state can then split into linear combinations of both TE and TM polarizations [24

24. J. M. Zhao, Y. Chen, and Y. J. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure,” Appl. Phys. Lett. 92, 071114 (2008). [CrossRef]

]. Consider an unbounded diagonally anisotropic medium described by homogeneous parameters (εx, εy, εz) and (μx,μy,μz), where it is always possible to choose principal coordinate axes so that the permittivity and permeability are diagonal. Assuming a harmonic time dependence, exp(−iωt), for the EM fields, Maxwell’s equations give the corresponding wave equations for the electric field components Ex and Ey:
2Exz2+[(ωc)2εxμy(εxεz)kx2]Ex=0,
(1)
2Eyz2+[(ωc)2εyμx(μxμz)kx2]Ey=0.
(2)
Equations (1) and (2) illustrate that the wave equations are different for Ex and Ey, resulting in two different wave vectors. In this work, we focus exclusively on p-polarization from which the nature of the HMM dispersion can be qualitatively understood. From Eq. (1), k^z2=εxμy(εx/εz)k^x2 (the caret symbol signifies normalization by ω/c). For this discussion we assume real valued material parameters and positive μy. Focusing on εx>0, we consider two scenarios (a) εz>0 and (b) εz<0, yielding the respective dispersion relations k^z2/(εxμy)+k^x2/(εzμy)=1 and k^z2/(εxμy)k^x2/(|εz|μy)=1. Thus the isofrequency contours are (a) ellipses and (b) hyperbola (see e.g., Fig. 1(c) when ky = 0). Moreover, for the ellipsoidal case, as x increases there will be a frequency cutoff since k^z2 eventually becomes negative. On the other hand, for the hyperbolic case, when x increases, there is no cutoff since k^z2 remains positive. If εx < 0 and εz > 0, we then have the possibility of a connected hyperbola (see Fig. 2(b)).

Fig. 2 Absorption as a function of incident angle θ. The superstrate is air, and the HMM layer is supported by a perfectly conducting substrate. In (a) and (b) a range of HMM widths τ are studied (legend units are in microns). In (a) ℜ(ε2x) > 0 and ℜ(ε2z) < 0 (type-1 HMM), and in (b) ℜ(ε2x) < 0, and ℜ(ε2z) > 0 (type-2 HMM). For both panels (a) and (b), λλz so that ℜ(ε2z) ≈ 0. Panels (c) and (d) show the effects of varying λ for both the type-1 and type-2 cases respectively. For those cases τ is fixed at 0.16 μm. For normal incidence (θ = 0°), there is generally little absorption (high reflectance). Remarkably, for a range of HMM widths and wavelengths there are strong absorption peaks spanning a broad range of θ.

To determine the absorbed EM energy, it is convenient to first determine the Fresnel reflection coefficient, r. The corresponding reflectance R is then given by R = |r|2. For a p-polarized plane wave incident at an angle θ relative to the normal of a planar layer of thickness τ, we find,
r=β[(k^z1εx2k^z2εx1)(k^z2εx3+k^z3εx2)eiϕ2+(k^z1εx2+k^z2εx1)(k^z2εx3k^z3εx2)eiϕ2(k^z1εx2k^z2εx1)(k^z2εx3k^z3εx2)eiϕ2+(k^z1εx2+k^z2εx1)(k^z2εx3+k^z3εx2)eiϕ2],
(3)
where the semi-infinite substrate and superstrate are in general anisotropic (see Fig. 1). The details can be found in Sec. 4. We define,
β=exp(2iϕ3),
(4)
where
ϕj(ω/c)k^zjτ,
(5)
and
k^zj2εxjμyj(εxj/εzj)k^x2.
(6)
The index j labels the regions 1,2 or 3 (see Fig. 1). In all cases below, the incident beam is in vacuum (region 3) so that x = sin θ, which is conserved across the interface. The frequency dispersion in the HMM takes the Drude-like form: εz2 = a + ib, where a = 1 − α2/[1+(αf)2], and b = α3f/[1 + (αf)2]. Here, αλ/λz, f = 0.02, and the characteristic wavelength, λz = 1.6μm. When discussing the two types of HMM, the permittivity parallel to the interface is described using εx2 = ±4 + 0.1i for type-1 (+) and type-2 (−). The wavelength range considered here, where the system exhibits HMM behavior is consistent with experimental work involving HMM semiconductor hybrids [25

25. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109, (23)8834–8838 (2012). [CrossRef] [PubMed]

].

When the surrounding media is air and the central layer is a diagonally anisotropic HMM, setting the numerator of (3) to zero leads to a set of conditions on the wavevector components that results in a complete absence of reflection (R = 0):
k^x2=εz2(εx2μy2εz2εx21);k^z22=εx22(εz2μy21εz2εx21)
(7)
k^x2=εz2μy2(εz2εx2)(nλ2τ)2;k^z22=(nλ2τ)2,
(8)
where n is an integer. The x in Eq. (7) corresponds to the classic Brewster angle condition for isotropic media: k^z2εx22=k^z22, and the x in Eq. (8) corresponds to a standing wave condition in the z-direction. In either case, when Eqs. (7) or (8) is satisfied, a minimum in R arises. Under the Brewster angle condition in Eq. (7), a simple rearrangement shows that k^x2(εx2εz21)=εx2μy2. This implies that if we choose εx2 = μy2 and εx2εz2 = 1, then we should have R = 0 for any value of x = sin θ. This choice of anisotropic material parameters is similar to the perfectly matched layer (PML) approach to eliminating unwanted reflection from absorbing computational domain boundaries, especially in time-domain [26

26. J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]

,27

27. Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. 43, 1460–1463, (1995). [CrossRef]

] and frequency-domain algorithms [28

28. J. Merle Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express 9, 461–475 (2001). [CrossRef] [PubMed]

,29

29. J. M. Elson, Proc. SPIE 4780, Surface Scattering and Diffraction for Advanced Metrology II, 32, (October1, 2002).

]. Note that such a PML medium is somewhat artificial since εx2=εz21 implies sources in region 2. Nonetheless such a concept is successful for absorbing layers designed to simulate an infinite computational domain.

We now illustrate the important case of a HMM backed by a perfectly conducting metal, and the near-perfect absorption that can arise. As is appropriate for HMM structures, we also consider the regime where all materials are nonmagnetic (μ = 1). The reflection coefficient in Eq. (3) then becomes,
r=e2iϕ[(k^z2+k^zεx2)eiϕ2(k^z2k^zεx2)eiϕ2(k^z2k^zεx2)eiϕ2(k^z2+k^zεx2)eiϕ2],
(9)
where ϕ ≡ (ω/c)zτ, and z = cos θ. It is readily verified that for lossless media, Eq. (9) yields perfect reflection (|r|2=1) as expected. In the absence of transmission, the absorption, A, is simply written as A = 1 − R.

3. Results

For the case of vacuum superstrate and substrate, Eq. (3) reveals that when sinϕ2 = 0, then R = 0. If on the other hand, both substrate and superstrate are perfectly conducting, then setting the denominator of (3) equal to zero also yields sin ϕ2 = 0, which coincides with the dispersion relation for guided waves in an HMM layer. Equation (8) shows that when n = 0, k^x2=εz2μy2 and k^z2=0, corresponding to a TEM mode which is essentially a plane wave confined to propagate in the x-direction. Thus if ϕ2 = z2τ̂ = nπ, this assertion is valid if ϕ2 << nπ. If however εz2/εx2 < 0, Eq. (8) reveals that there is no guided mode cutoff for k^x2.

To present a global view of the parameter space in which our anisotropic structure can absorb unusually large portions of incident energy, we present in Figs. 3(a) and (b), 2-D density plots that map the absorption versus λ and θ. The HMM thickness is fixed at τ=0.16μm, as in Figs. 2(c) and (d). In Fig. 3(a) εx2 = (4,0.1), so that the HMM region where ℜ(εz2) < 0 corresponds to λ > λz (recall that λz=1.6μm). Similarly for (b), εx2 = (−4, 0.1), and thus the HMM region there corresponds to λ < 1.6μm. Figs. 4(a) and (b) are slices from Figs. 3(a) and (b). In Fig. 4(a) near-perfect absorption occurs at θ = 65° for both HMM types. For λ = 1.7μm, ℜ(εx2) = 4 and ℜ(εz2) = −0.128 corresponding to a Type-1 HMM. For λ = 1.5μm, ℜ(εx2) = −4 and ℜ(εz2) = 0.121, corresponding to a Type-2 HMM. In Fig. 4(b), the Type-1 absorption peak occurs at λ = 1.66μm, where ℜ(εz2) = −0.076, and the Type-2 case peaks at λ = 1.55μm, where ℜ(εz2) = 0.062.

Fig. 3 Density plots showing absorption as a function of incident wavelength λ and angle θ. Bright regions correspond to high absorption. The HMM thickness in both plots is τ = 0.16μm. The characteristic wavelength, λz = 1.6μm separates the HMM regions according to (a) type-1: εx2 > 0 and εz2 < 0 for λ > 1.6μm, and (b) type 2: εx2 < 0, and εz2 > 0 for λ < 1.6μm. Thus we find that when the metamaterial is effectively hyperbolic, absorption can be strongly enhanced.
Fig. 4 Absorption as a function of incident angle θ (a) and wavelength λ (b) extracted from the high absorption regions of the density plots in Fig. 3(a) and (b).

Further insight into this anomalous absorption can be gained from studying the balance of energy [31

31. See, for example, J. D. Jackson, Classical Electrodynamics, 3 (Wiley and Sons, 1998).

]. For our structure and material parameters, it suffices to compute,
4πcVdvEJ*=Vdv(E×H*)iωcVdv[εx2*|Ex2|2+εz2*|Ez2|2|Hy2|2].
(12)
Since we have incorporated the conductive part of the HMM into the dielectric response, the J term is absent. In all of the near-perfect absorption examples investigated here, evaluation of Eq. (12) confirmed that the net energy flow into the HMM volume, V, is converted into heat.

To explore further the behavior of the energy flow, we present in Fig. 5 the average power P in the HMM as a function of θ for the two cases in Fig. 4(a). Thus, panel (a) is for λ = 1.7μm (type-1 HMM), and panel (b) corresponds to λ = 1.5μm (type-2 HMM). The average power along the x and z directions, Px2 and Pz2, is found from averaging the corresponding components of the Poynting vector over the HMM region (see Eqs. (29)). It is evident that the direction of energy flow depends on the sign of εz2 (or equivalently whether λ is above or below λz). The component of P normal to the interfaces (Pz2) must always have the same sign on both sides of the interface [32

32. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef] [PubMed]

]. Its direction parallel to the interface (Px2) however can be negative if the HMM is of type-1, as seen in Fig. 5(a), and is clearly opposite in direction to kx, which is always positive. This manifestation of “negative refraction” was discussed in the context of uniaxially anisotropic media [33

33. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett 31, 129 (2001). [CrossRef]

], certain nanowire structures [9

9. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008). [CrossRef] [PubMed]

], and observed in ZnO-based multilayers [25

25. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109, (23)8834–8838 (2012). [CrossRef] [PubMed]

]. Comparing the peaks in panels (a) and (b) with Fig. 4(a), we see the correlation with the angles where near-perfect absorption occurs and those where |Pz2| is maximal.

Fig. 5 The power P in the HMM, normalized by the incident power in the z-direction, and plotted as a function of θ. In (a) λ = 1.7 μm (type-1 HMM) and in (b) λ = 1.5 μm (type-2 HMM). The material parameters are the same as in Fig. 4(c). Panel (a) reveals that energy flow parallel to the interface (Px2) in the type-1 HMM is negative, which is opposite that of the vacuum region containing the incident beam.

4. Conclusion

In conclusion, we have investigated the absorption properties of both type-1 and type-2 hyperbolic metamaterials. We found that HMMs can absorb significantly higher amounts of electromagnetic energy compared to their conventional counterparts, where ℜ(εx2) and ℜ(εz2) are both of the same sign. Our results show that the incident beam can couple to the HMM structure without recourse for a second compensating layer. We also revealed that the condition ℜ(εz) ≈ 0 leads to near-perfect absorption over a range of frequencies, angles of incidence, and subwavelength structure thicknesses, making the proposed structures experimentally achievable. Alternate methods exist to achieve perfect absorption, including periodic layers of silver and conventional dielectrics that depending on the direction of incident wave propagation and loss, can exhibit anisotropic behavior that cancels the reflected and transmitted waves simultaneously [34

34. J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, and J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. 35, 16 (2010). [CrossRef] [PubMed]

]. Our HMM with metallic backing is a different configuration in which no energy can be transmitted, and the inherently finite width of the structure means that there are no Bloch wave excitations. Arrays of metal-dielectric films can serve as an effective HMM waveguide taper, resulting in light localization and enhanced absorption [35

35. H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow Trapping in Hyperbolic Metamaterial Waveguide,” Sci. Rep. 3, 1249 (2013). [CrossRef] [PubMed]

], however, the modes responsible for “slow-light” are very sensitive to the presence of loss [36

36. W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric eld on a slab waveguide made of indenite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010). [CrossRef]

].

When the incident wavelength results in the dielectric response of the metamaterial possessing a nearly vanishing component of the permittivity, contributions from nonlinear effects and/or spatial dispersions can become important. Nonlinear effects can in this case generate interesting phenomena such as two-peaked or flat solitons [37

37. C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A 83, 053805 (2011). [CrossRef]

], as well as additional venues for second- and third-harmonic generation [38

38. M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84, 063826 (2011). [CrossRef]

], and guided waves whose Poynting vector undergoes localized reversal [39

39. A. Ciattoni, C. Rizza, and E. Palange, “Transverse power flow reversing of guided waves in extreme nonlinear metamaterials,” Opt. Lett. 18, 11911 (2010).

]. Since the nonlinear part of the dielectric response can now be of the same order as the (small) linear part, the transmissivity can exhibit directional hysteresis behavior [40

40. A. Ciattoni, C. Rizza, and E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. 35, 2130 (2010). [CrossRef] [PubMed]

]. Spatial dispersion can moreover lead to the appearance of additional EM waves, as was reported for nanorods [41

41. R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A.V. Zayats, and V.A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009). [CrossRef] [PubMed]

]. For metal-dielectric structures, nonlocality arising from the excitation of surface plasmons can also lead to significant corrections [42

42. 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]

] to conventional effective medium theories [43

43. D. J. Bergman, “The dielectric constant of a composite material - a problem in classical physics,” Phys. Rep. 43, 377–407 (1978). [CrossRef]

].

Appendix: Poynting’s Theorem

In this section we present the details on how the EM fields are straightforwardly calculated for determining the reflectance and energy flow in HMM structures. We have considered in this paper diagonally anisotropic HMM layers (εx, εz, μy). We also assume that EM wave propagation and polarization is in the x-z plane. The wave equation for Ex is thus,
2Exz2+[(ωc)2εxμy(εxεz)kx2]Ex=0.
(13)
Taking into account that ▿·D = ▿·B = 0, this yields the electric field solutions in their respective media as,
E1=[A{x^+z^(k^xεx1k^z1εz1)}eikz1z]eikxx,
(14)
E2=[G{x^z^(k^xεx2k^z2εz2)}eikz2z+F{x^+z^(k^xεx2k^z2εz2)}eikz2z]eikxx,
(15)
E3=[C{x^z^(k^xεx3k^z3εz3)}eikz3z+I{x^+z^(k^xεx3k^z3εz3)}eikz3z]eikxx.
(16)
Similarly, the components of the magnetic field are written,
H1=y^(εx1k^z1)Aeikz1zeikxx,
(17)
H2=y^(εx2k^z2)[Geikz2zFeikz2z]eikxx,
(18)
H3=y^(εx3k^z3)[Ceikz3zIeikz3z]eikxx.
(19)
The I terms represent the incident field. The quantities zj and ϕj are defined in Eqs. (5) and (6). Utilizing matching boundary conditions for the tangential components of the electric and magnetic fields permits calculation of the coefficients,
A=4eiϕ3k^z1k^z2εx2εx3𝒢eiϕ2+𝒢++eiϕ2;C=β[𝒢+eiϕ2+𝒢+eiϕ2𝒢eiϕ2+𝒢++eiϕ2],
(20)
=2eiϕ3k^z2εx3𝒢+𝒢eiϕ2+𝒢++eiϕ2;𝒢=2eiϕ3k^z2εx3𝒢𝒢eiϕ2+𝒢++eiϕ2,
(21)
were β is given in Eq. (4). We also define,
±=k^z3εx2±k^z2εx3,
(22)
𝒢±=k^z2εx1±k^z1εx2,
(23)
where εxj describe the media for regions j = 1, 2, 3, and kzj is defined in Eq. (6). The caret symbol signifies that wavenumber components kx and kzj have been normalized to ω/c. In general, x can be any value, but for the case of an incident plane wave in vacuum, x = sin θ.

For time-harmonic fields, consider now the integral,
124πcVdvEJ*=12Vdv(E×H*)iω2cVdv[ED*H*B],
(24)
where we have used,
×H=4πcJiωcD;×E=iωcB.
(25)
The media are diagonally anisotropic with D = ε · E and B = μ · H. Inserting Eqs. (15) and (18) into Eq. (24) yields the following energy conservation relationships,
iω2c0τdzE2xD2x*=iε2x*2[|G|2(e2τ(k2z)12(k^2z))+|F|2(e2τ(k2z)12(k^2z))+2{GF*(e2iτ(k2z)12i(k^2z))}],
(26)
iω2c0τdzE2xD2x*=iε2z*2|k^xε2xk^2zε2z|2[|G|2(e2τ(k2z)12(k^2z))+|F|2(e2τ(k2z)12(k^2z))2{GF*(e2iτ(k2z)12i(k^2z))}],
(27)
iω2c0τdzH2y*B2y=iμ2y2|ε2xk^2z|2[|G|2(e2τ(k2z)12(k^2z))+|F|2(e2τ(k2z)12(k^2z))2{GF*(e2iτ(k2z)12i(k^2z))}],
(28)
where the x and y integrations over V are omitted.

Finally, the time-averaged Poynting vector in V is S = E × H*/2, giving the result,
S2x(z)=(k^xεz2)|εx2k^z2|2[|G|2e2z(kz2)+|F|2e2z(kz2)2(GF*e2iz(kz2))],
(29)
S2z(z)=(εx2k^z2)*[|G|2e2z(kz2)+|F|2e2z(kz2)2i(GF*e2iz(kz2))].
(30)

References and links

1.

D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47(11), 2059–2074 (1999). [CrossRef]

2.

M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials”, Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]

3.

K. Halterman and S. Feng, “Resonant transmission of electromagnetic fields through subwavelength zero-ε slits,” Phys. Rev. A 78, 021805 (2008). [CrossRef]

4.

K. Halterman, S. Feng, and V. C. Nguyen, “Controlled leaky wave radiation from anisotropic epsilon near zero metamaterials,” Phys. Rev. B 84, 075162 (2011). [CrossRef]

5.

S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012). [CrossRef]

6.

S. Molesky, C. J. Dewalt, and Z. Jacob, “High temperature epsilon-near-zero and epsilon-near-pole metamaterial emitters for thermophotovoltaics,” Opt. Express 21(S1), A96–A110 (2013). [CrossRef] [PubMed]

7.

D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003). [CrossRef]

8.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012). [CrossRef]

9.

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008). [CrossRef] [PubMed]

10.

I. Nefedov and S. Tretyakov, “Ultrabroadband electromagnetically indenite medium formed by aligned carbon nanotubes,” Phys. Rev. B 84(11), 113410 (2011). [CrossRef]

11.

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011). [CrossRef]

12.

S. Savoia, G. Castaldi, and V. Galdi, “Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices,” Phys. Rev. B 87, 235116 (2013). [CrossRef]

13.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]

14.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012). [CrossRef]

15.

T. Tumkur, L. Gu, J. Kitur, E. Narimanov, and M. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100, 161103 (2012). [CrossRef]

16.

F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013). [CrossRef]

17.

W. Li, Z. Liu, X. Zhang, and X. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 1611084 (2012). [CrossRef]

18.

C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. 37, 3345–3347 (2012). [CrossRef]

19.

C. Rizza, A. Ciattoni, L. Columbo, M. Brambilla, and F. Prati, “Terahertz optically tunable dielectric metamaterials without microfabrication,” Opt. Lett. 38, 1307 (2013). [CrossRef] [PubMed]

20.

W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A 24, 530 (2007). [CrossRef]

21.

K.V. Sreekanth, A. De Luca, and G. Strangi, “Experimental demonstration of surface and bulk plasmon polaritons in hypergratings,” Sci. Rep. 3, 3291 (2013). [CrossRef] [PubMed]

22.

E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6. [CrossRef]

23.

T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101, 091105 (2012). [CrossRef]

24.

J. M. Zhao, Y. Chen, and Y. J. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure,” Appl. Phys. Lett. 92, 071114 (2008). [CrossRef]

25.

G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109, (23)8834–8838 (2012). [CrossRef] [PubMed]

26.

J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]

27.

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. 43, 1460–1463, (1995). [CrossRef]

28.

J. Merle Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express 9, 461–475 (2001). [CrossRef] [PubMed]

29.

J. M. Elson, Proc. SPIE 4780, Surface Scattering and Diffraction for Advanced Metrology II, 32, (October1, 2002).

30.

Y. Jin, S. Xiao, N.A. Mortensen, and S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express 19, 11114 (2011). [CrossRef] [PubMed]

31.

See, for example, J. D. Jackson, Classical Electrodynamics, 3 (Wiley and Sons, 1998).

32.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef] [PubMed]

33.

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett 31, 129 (2001). [CrossRef]

34.

J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, and J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. 35, 16 (2010). [CrossRef] [PubMed]

35.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow Trapping in Hyperbolic Metamaterial Waveguide,” Sci. Rep. 3, 1249 (2013). [CrossRef] [PubMed]

36.

W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric eld on a slab waveguide made of indenite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010). [CrossRef]

37.

C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A 83, 053805 (2011). [CrossRef]

38.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84, 063826 (2011). [CrossRef]

39.

A. Ciattoni, C. Rizza, and E. Palange, “Transverse power flow reversing of guided waves in extreme nonlinear metamaterials,” Opt. Lett. 18, 11911 (2010).

40.

A. Ciattoni, C. Rizza, and E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. 35, 2130 (2010). [CrossRef] [PubMed]

41.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A.V. Zayats, and V.A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009). [CrossRef] [PubMed]

42.

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]

43.

D. J. Bergman, “The dielectric constant of a composite material - a problem in classical physics,” Phys. Rep. 43, 377–407 (1978). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: January 10, 2014
Revised Manuscript: February 28, 2014
Manuscript Accepted: March 4, 2014
Published: March 21, 2014

Citation
Klaus Halterman and J. Merle Elson, "Near-perfect absorption in epsilon-near-zero structures with hyperbolic dispersion," Opt. Express 22, 7337-7348 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7337


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References

  1. D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47(11), 2059–2074 (1999). [CrossRef]
  2. M. G. Silveirinha, N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials”, Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]
  3. K. Halterman, S. Feng, “Resonant transmission of electromagnetic fields through subwavelength zero-ε slits,” Phys. Rev. A 78, 021805 (2008). [CrossRef]
  4. K. Halterman, S. Feng, V. C. Nguyen, “Controlled leaky wave radiation from anisotropic epsilon near zero metamaterials,” Phys. Rev. B 84, 075162 (2011). [CrossRef]
  5. S. Feng, K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012). [CrossRef]
  6. S. Molesky, C. J. Dewalt, Z. Jacob, “High temperature epsilon-near-zero and epsilon-near-pole metamaterial emitters for thermophotovoltaics,” Opt. Express 21(S1), A96–A110 (2013). [CrossRef] [PubMed]
  7. D. Schurig, D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. 82, 2215–2217 (2003). [CrossRef]
  8. C. L. Cortes, W. Newman, S. Molesky, Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012). [CrossRef]
  9. Y. Liu, G. Bartal, X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008). [CrossRef] [PubMed]
  10. I. Nefedov, S. Tretyakov, “Ultrabroadband electromagnetically indenite medium formed by aligned carbon nanotubes,” Phys. Rev. B 84(11), 113410 (2011). [CrossRef]
  11. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011). [CrossRef]
  12. S. Savoia, G. Castaldi, V. Galdi, “Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices,” Phys. Rev. B 87, 235116 (2013). [CrossRef]
  13. O. Kidwai, S. V. Zhukovsky, J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]
  14. W. Yan, M. Wubs, N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012). [CrossRef]
  15. T. Tumkur, L. Gu, J. Kitur, E. Narimanov, M. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100, 161103 (2012). [CrossRef]
  16. F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013). [CrossRef]
  17. W. Li, Z. Liu, X. Zhang, X. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 1611084 (2012). [CrossRef]
  18. C. Rizza, A. Ciattoni, E. Spinozzi, L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. 37, 3345–3347 (2012). [CrossRef]
  19. C. Rizza, A. Ciattoni, L. Columbo, M. Brambilla, F. Prati, “Terahertz optically tunable dielectric metamaterials without microfabrication,” Opt. Lett. 38, 1307 (2013). [CrossRef] [PubMed]
  20. W. Yan, L. Shen, L. Ran, J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A 24, 530 (2007). [CrossRef]
  21. K.V. Sreekanth, A. De Luca, G. Strangi, “Experimental demonstration of surface and bulk plasmon polaritons in hypergratings,” Sci. Rep. 3, 3291 (2013). [CrossRef] [PubMed]
  22. E. Narimanov, M. A. Noginov, H. Li, Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6. [CrossRef]
  23. T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101, 091105 (2012). [CrossRef]
  24. J. M. Zhao, Y. Chen, Y. J. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure,” Appl. Phys. Lett. 92, 071114 (2008). [CrossRef]
  25. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. 109, (23)8834–8838 (2012). [CrossRef] [PubMed]
  26. J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]
  27. Z. Sacks, D. M. Kingsland, R. Lee, J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. 43, 1460–1463, (1995). [CrossRef]
  28. J. Merle Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express 9, 461–475 (2001). [CrossRef] [PubMed]
  29. J. M. Elson, Proc. SPIE 4780, Surface Scattering and Diffraction for Advanced Metrology II, 32, (October1, 2002).
  30. Y. Jin, S. Xiao, N.A. Mortensen, S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express 19, 11114 (2011). [CrossRef] [PubMed]
  31. See, for example, J. D. Jackson, Classical Electrodynamics, 3 (Wiley and Sons, 1998).
  32. D. R. Smith, D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef] [PubMed]
  33. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, S. Ilvonen, “BW media media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett 31, 129 (2001). [CrossRef]
  34. J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. 35, 16 (2010). [CrossRef] [PubMed]
  35. H. Hu, D. Ji, X. Zeng, K. Liu, Q. Gan, “Rainbow Trapping in Hyperbolic Metamaterial Waveguide,” Sci. Rep. 3, 1249 (2013). [CrossRef] [PubMed]
  36. W. T. Lu, S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric eld on a slab waveguide made of indenite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010). [CrossRef]
  37. C. Rizza, A. Ciattoni, E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A 83, 053805 (2011). [CrossRef]
  38. M. A. Vincenti, D. de Ceglia, A. Ciattoni, M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84, 063826 (2011). [CrossRef]
  39. A. Ciattoni, C. Rizza, E. Palange, “Transverse power flow reversing of guided waves in extreme nonlinear metamaterials,” Opt. Lett. 18, 11911 (2010).
  40. A. Ciattoni, C. Rizza, E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. 35, 2130 (2010). [CrossRef] [PubMed]
  41. R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A.V. Zayats, V.A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009). [CrossRef] [PubMed]
  42. A. A. Orlov, P. M. Voroshilov, P. A. Belov, Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]
  43. D. J. Bergman, “The dielectric constant of a composite material - a problem in classical physics,” Phys. Rep. 43, 377–407 (1978). [CrossRef]

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