Abstract

We investigate electromagnetic wave propagation in multilayered metal-dielectric hyperbolic metamaterials (HMMs). We demonstrate that high-k propagating waves in HMMs are volume plasmon polaritons. The volume plasmon polariton band is formed by coupling of short-range surface plasmon polariton excitations in the individual metal layers.

© 2013 Optical Society of America

1. Introduction

Hyperbolic metamaterials (HMMs) are subwavelength structures such as rod arrays [14] or multilayers [5, 6] made to imitate “indefinite media,” a special case of extreme anisotropy where diagonal elements of the permittivity tensor have different signs (e.g., εx = εy < 0 and εz > 0) [7]. Such media are unusual because their dispersion relation ω2/c2=(kx2+ky2)/εz+kz2/εx,y is hyperbolic rather than elliptical [Fig. 1(a)]. In the idealization that this relation holds for all kx,y,z, different signs of εx,y and εz result in propagating solutions for waves with infinitely large wave vectors, i.e., k2εx,y,zω2/c2. Such waves (the “high-k modes” [59]) are evanescent in isotropic media but become propagating in indefinite media. As a result, the photonic density of states (DOS) becomes unbounded, giving rise to new physical phenomena such as broadband spontaneous emission enhancement [5, 6, 10], anomalously high heat-transfer capabilities [11], and optical simulation of metric signature transitions [12]. Practical applications, e.g., near-field [1315] and far-field subwavelength imaging (hyperlensing) [8, 16] and highly absorptive “darker than black” coatings that benefit from surface roughness [17], were also proposed.

 figure: Fig. 1

Fig. 1 (a) Isofrequency surfaces in the dispersion relation (kx2+ky2)/εz+kz2/εx,y=ω2/c2 for conventional anisotropic medium (εx,y,z > 0) and indefinite medium (εx,y < 0 and εz > 0). (b) An infinite multilayer HMM with wave vector decomposition used in the paper (which is schematic in that w and k can be complex). (c) Comparison between the actual multilayer dispersion relation [Eq. (4)] for dm + dd =13.7 and 27.4 nm with the effective-medium dispersion relation for ρ = 0.17. The materials have εm = −17.2 (gold [18]) and εd = 2.59 (alumina [19]) for λ = 715 nm [5]. (d) Isofrequency surfaces for the actual dispersion relation (bottom) vs. effective medium approximation (top).

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However, attributing the physics of HMMs to high-k modes raises a few concerns. First, even though structural elements of an HMM, e.g., layer thicknesses in Fig. 1(b), may be much smaller than the vacuum wavelength at the frequency of interest, the wavelength of high-k modes, 2π/k, will eventually become even smaller. Therefore any real structure will have a range of k where it is no longer “subwavelength” and can no longer be described by the effective medium approach [2, 5, 6], displaying pronounced optical nonlocality effects [20, 21]. Hence, all conclusions about unbounded DOS in real HMMs become unreliable, and significant deviations from effective-medium predictions are expected in real HMMs [2124], even when optical nonlocalities are taken into account in the homogenization theory [25].

Furthermore, the effective index approach does not explain the transition from a conventional material on a smaller (microscopic) scale to an HMM on a larger scale, and fails to provide any insight into the physical nature of high-k modes other than to assert that they must be bulk propagating waves. Such insight would be crucial to explore the extent to which these modes can be used as information carriers. Since the structures involved are metal-dielectric, it is strongly believed that high-k modes are plasmonic in nature. So various groups call them multilayer plasmons[26], Bloch plasmon polaritons[27], or volume plasmon polaritons (VPPs) [28], while other groups relate them to leaky waves in anisotropic waveguides [13, 29, 30] or to surface polaritons [4, 31]. In metal-dielectric multilayers, it is stipulated that these VPPs arise from surface plasmon polaritons (SPPs) at layer interfaces [14, 32]. However, the direct connection between an SPP at a single interface and a VPP in the multilayer HMM is still to be drawn, and the prevailing coupling mechanism between the SPPs is still not quite clarified [33].

In this paper, we provide a direct proof that a band of bulk propagating waves (VPPs) with large wave vectors in periodic multilayer HMMs originates from coupling of SPPs in the individual metal layers by virtue of the Bloch theorem. When the reflection coefficient of metal layers is replaced by a simple pole expansion that only preserves short-range SPP (SRSPP) excitations, correct dispersion relation of an HMM is recovered throughout its entire 1D Brillouin zone. We conclude that “high-k waves” in HMMs can be physically understood as “Bloch VPPs”, and can be seen as a distinct type of plasmonic excitations.

The paper is organized as follows. In Section 2, we review the basics of wave propagation in metal-dielectric multilayers, and discuss the dispersion relation for bulk waves in multilayer HMMs. In Section 3, we employ the pole expansion formalism to preserve only the SRSPPs in the metal layers, and show that bulk VPPs originate from surface-bound SRSPPs in the metal layers via the Bloch theorem. Finally, in Section 4 we summarize.

2. High-k bulk propagating waves in HMMs

We consider an infinite periodic metal-dielectric HMM in Fig. 1(b) where metal layers with permittivity εm and thickness dm alternate with dielectric layers with permittivity εd and thickness dd. The dispersion relation can be determined by the standard transfer matrix approach [34]. The transfer matrix for one period of the structure can be written as

M1=1Tm[Tm2Rm2RmRm1][eiwddd00eiwddd].
The reflection and transmission coefficients of a metal layer are given by the Airy formulas,
Rm=rdm+tdmrmdtmde2iwmdm[1rmd2e2iwmdm]1,Tm=tdmtmdeiwmdm[1rmd2e2iwmdm]1.
In Eqs. (1)(2), wj = [εjω2/c2κ2]1/2 (j = m, d) is the z-directed component of the wave vector in metal or dielectric. (To fix the definition, we take Imwj ≥ 0; if Imwj = 0 we take Rewj ≥ 0.) The reflection and transmission coefficients rij and tij at the (i|j) interface are given by standard Fresnel formulas for p-polarized light,
rmd=wmεdwdεmwmεd+wdεm,rdm=wdεmwmεdwdεm+wmεd,tmd=2wmεmεdwmεd+wdεm,tdm=2wdεdεmwdεm+wmεd.
Assuming for now that the metal is lossless, we can conveniently label the plane wave components by a real tangential component κ lying in the xy plane, as it does not vary across layers. Since the multilayer is periodic in z, it supports propagating Bloch waves with the wave vector k2=kB2+κ2. The relation between kB and κ is given by the Bloch theorem [25, 32, 33]:
TrM12=cos[kB(dm+dd)]=cos(wmdm)cos(wddd)[(η+η1)/2]sin(wmdm)sin(wddd),
where η = (εmwd)/(εdwm). As shown earlier (see, e.g., [26]), expanding Eq. (4) up to (wjdj)2 reduces it to the hyperbolic dispersion equation ω2/c2=kB2/εx,y+κ2/εz, where εx,y = ρεm + (1 − ρ)εd < 0 and εz1=ρεm1+(1ρ)εd1>0 are the effective permittivity tensor components, and ρdm/(dm + dd) is the metal filling fraction.

Figure 1(c) compares this approximate dispersion relation with the exact result of Eq. (4). As expected [25, 26], we see that VPPs in multilayer HMMs are indeed Bloch waves in a periodic metal-dielectric multilayer, approaching a hyperbolic dispersion relation for smaller κ but deviating from it as κ increases. The cut-off in the range of possible κ, κmax ∝ 1/(dm + dd) [24], puts a limit on the photonic DOS in real multilayers [see Fig. 1(d)]. For κκmax, where the VPPs reach wavelengths as small as the layer thicknesses, the multilayer behaves like a photonic crystal rather than like a homogeneous effective medium, displaying a band structure. The hyperbolic dispersion relation comes from the local curvature of the propagation band near the -point of the 1D Brillouin zone. In the HMM regime, the high-k waves are backward-propagating [15, 21, 30].

The condition where the effective-medium approximation is valid, wjdj ≪ 1, can be regarded as a special case of a “subwavelength condition” for HMMs, implying that neither the phase nor the amplitude of a propagating wave should vary significantly across the thickness of any layer in the structure. Unlike conventional media where the subwavelength condition can simply be dj/λ ≪ 1 due to the elliptical dispersion relation restricting the range of possible κ, the “HMM subwavelength condition” breaks down for some large κ no matter how thin the layers are.

3. Pole expansion for surface plasmon polaritons in metal layers

Bloch waves in crystals are typically formed from elementary excitations in single elements (“atoms”) that couple in periodic systems to form bands. We can thus understand the underlying physics of HMMs by identifying the nature of the elementary excitations behind VPPs. Although it has been generally understood that such excitations in metal-dielectric multilayers must be plasmonic in nature [21, 26, 27], a direct connection between SPPs at individual metal-dielectric interfaces and VPPs in HMMs can be established. To do so, we replace physical metal layers in an HMM by ”fictitious” layers that only support one plasmonic mode, and compare the wave propagation properties in this fictitious structure with those in the original HMM.

From Eq. (1) we see that the Fresnel coefficients Rm and Tm play a central role in the dispersion relation, so the excitations we seek to characterize must be identified in these coefficients. Poles in the Fresnel coefficients signal the presence of guided wave-type excitations in multilayers. For example, SPPs at a single metal-dielectric interface are seen as poles in rdm and tdm [see Eq. (3)], but their wave number κmd, as given by the expression wm(κmd)εd + wd(κmd)εm = 0, is on the order of (ω/c)εd at optical frequencies. What we must clarify is how the wave vectors of VPPs in HMMs become enormously large (κκmd).

In thin metal layers, comprising two metal-dielectric interfaces, SPPs are known to couple either symmetrically or antisymmetrically, producing long-range SPPs (LRSPPs) or short-range SPPs (SRSPPs), respectively. Both are described by the pole condition in Eqs. (2),

1rmd2e2iwmdm=0
Asymmetric coupling in SRSPPs makes their wave vectors larger than for LRSPPs. Assuming that κ2εm,dω2/c2, we solve Eq. (5) to obtain a limit for the SRSPP wave number for dm → 0: κsp ≈ (ln r)/dm, r = (εmεd)/(εm + εd). We see that κsp becomes very large as dm decreases, confirming that the involvement of SRSPP in the VPP formation is a well-justified conjecture.

To confirm the validity of this conjecture and to see that SRSPPs are crucial in the dispersion relation of HMMs, we replace the expressions for Tm and Rm in Eq. (1) by their pole expansion around the SRSPP pole at κsp: Tm = τm/(κκsp), τm ≈ (r−1r)/(2dm). Note that as κ → 0, Tm → −τm/κsp ≈ 1, in agreement with the complete transmission of a very thin ( dmcεm/ω) metal film. The corresponding pole expansion for Rm would give −τm/(κκsp). However, to stay consistent with vanishing reflectivity for a very thin metal film at normal incidence we add a constant to this expression

Rm=τm/(κκsp)(τm/κsp),Tm2Rm2=2τm2/[κsp(κκsp)](τm/κsp)2.
Substituting Eq. (6) in Eq. (1), instead of Eq. (4) we get
cos[kB(dm+dd)][1(κκsp)/(2τm)]eiwddd+[(κκsp)/(2τm)]eiwddd.

In Fig. 2(a), it can be seen that Eq. (7) provides a good approximation for Eq. (4). The agreement remains reasonable for ρ up to 0.5, getting slightly better for smaller ρ, and persists for lossy metals. Interestingly, omitting the second terms in Eqs. (6) [using just Rm = −τm/(κκsp)] can also achieve a decent agreement between Eqs. (7) and (4) in the range of larger /ω with κ near κsp, which is the region of most VPPs [Fig. 2(b)]. However, adding the correction term drastically improves the agreement near kB = 0 where /ω is smaller, making the pole approximation accurate throughout the entire Brillouin zone.

 figure: Fig. 2

Fig. 2 Comparison between Eq. (4) (solid) and the coupled-SRSPP approximation given by Eq. (7) (dashed) for dm + dd =13.7 nm and ρ = 0.17, 0.25, and 0.5 for (a) the pole expansion as in Eq. (6) and (b) the pole expansion without the added constant terms (the vertical marks showing the location of κspc/ω for the respective filling fractions); (c) same as (a) for ρ = 0.12 (the effective medium approximation is indistinguishable from the inner branch of the exact relation). The materials are the same as in Fig. 1(c).

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It is also remarkable that the pole expansion (6) continues to give a correct dispersion relation for smaller ρ < εd/(εm +εd), where 0 < εx,y < εz, and the structure is no longer an HMM but an extremely anisotropic medium. As seen in Fig. 2(c), the exact dispersion relation has two branches [21]. Only the inner, low-k, forwrard-wave branch is recovered by the effective medium approximation, which may lead to an erroneous conclusion that high-k waves disappear for smaller filling fractions. In fact, high-k waves are still contained in the other (outer) branch, which gets pushed to higher κ, outside of the validity of the condition wmdm ≪ 1. This backward-wave branch is correctly recovered by the pole expansion, indicating (in line with the results of [21]) that VPPs in metal-dielectric multilayers also exist when the HMM requirements are not met.

For higher metal filling fractions, ρ > 0.5, it would be expected that the dispersion relation should be more accurately recovered by the pole expansion for a thin metal-dielectric-metal gap waveguide rather than a metal layer [33]. The corresponding expressions can be obtained from Eqs. (6)(7) with the subscripts m and d interchanged. However, the agreement with the exact Eq. (4) is only slightly better than Eq. (7) for 0.5 < ρ < −εm/(εm +εd) ≈ 0.8, so Eq. (7) remains a good approximation throughout the HMM parameter range.

Finally, note that there is surely more to the Fresnel coefficients Rm and Tm than SRSPPs. For example, LRSPPs are signaled by the presence of poles in those coefficients as well. However, the width of the LRSPP poles is much narrower in κ, and they do not have any influence on the dispersion relations at large /ω.

4. Conclusions

We have investigated electromagnetic wave propagation in multilayered metal-dielectric hyperbolic metamaterials (HMMs). Starting with the assumption that the only excitation supported by a thin metal film is an SRSPP, and coupling these excitations together through dielectric layers, we have reproduced the correct dispersion relation of “high-k waves” in a multilayer HMM throughout its 1D Brillouin zone. Hence, propagating waves in metal-dielectric multilayers, including HMMs, can indeed be understood as bulk plasmonic waves or VPPs arising from the coupling between SRSPPs in the constituent metal layers. In analogy with 2D arrays of plasmonic nanoparticles, where localized particle plasmons couple to form surface waves called Bloch SPPs [35], we can understand high-k propagating waves in HMMs as Bloch VPPs.

Acknowledgments

The authors thank V. Babicheva, E. Narimanov and M. Liscidini for helpful suggestions. This work was supported by the Natural Sciences and Engineering Research Council of Canada. One of us (S.V. Z.) wishes to acknowledge financial support from the People Programme (Marie Curie Actions) of the European Union’s 7th Framework Programme FP7-PEOPLE-2011-IIF under REA grant agreement No. 302009 (Project HyPHONE).

References and links

1. M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009) [CrossRef]  .

2. M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010) [CrossRef]   [PubMed]  .

3. A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Genanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20(13), 14663–14682 (2012) [CrossRef]   [PubMed]  .

4. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011) [CrossRef]  .

5. Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010) [CrossRef]  .

6. Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012) [CrossRef]  .

7. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004) [CrossRef]  .

8. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006) [CrossRef]   [PubMed]  .

9. A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011) [CrossRef]  .

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

11. E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

12. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010) [CrossRef]   [PubMed]  .

13. A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012) [CrossRef]   [PubMed]  .

14. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006) [CrossRef]  .

15. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009) [CrossRef]  .

16. S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

17. 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.

18. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006) [CrossRef]   [PubMed]  .

19. T. S. Eriksson, A. Hjortsberg, G. A. Niklasson, and C. G. Granqvist, “Infrared optical properties of evaporated alumina films,” Appl. Opt. 20(15), 2742–2746 (1981) [CrossRef]   [PubMed]  .

20. A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011) [CrossRef]  .

21. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011) [CrossRef]  .

22. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011) [CrossRef]   [PubMed]  .

23. I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012) [CrossRef]  .

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

25. 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]  .

26. J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006) [CrossRef]  .

27. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007) [CrossRef]  .

28. S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013) [CrossRef]  .

29. E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009) [CrossRef]   [PubMed]  .

30. H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010) [CrossRef]  .

31. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001) [CrossRef]  .

32. S. Feng, J. M. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express 13(11), 4113–4124 (2005) [CrossRef]   [PubMed]  .

33. G. Rosenblatt and M. Orenstein, “Competing coupled gaps and slabs for plasmonic metamaterial analysis,” Opt. Express 19(21), 20372–20385 (2011) [CrossRef]   [PubMed]  .

34. A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

35. T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009) [CrossRef]  .

References

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  1. M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
    [Crossref]
  2. M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
    [Crossref] [PubMed]
  3. A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Genanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20(13), 14663–14682 (2012).
    [Crossref] [PubMed]
  4. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
    [Crossref]
  5. Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
    [Crossref]
  6. Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
    [Crossref]
  7. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
    [Crossref]
  8. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [Crossref] [PubMed]
  9. A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
    [Crossref]
  10. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
    [Crossref]
  11. E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).
  12. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
    [Crossref] [PubMed]
  13. A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012).
    [Crossref] [PubMed]
  14. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
    [Crossref]
  15. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
    [Crossref]
  16. S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).
  17. 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.
  18. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
    [Crossref] [PubMed]
  19. T. S. Eriksson, A. Hjortsberg, G. A. Niklasson, and C. G. Granqvist, “Infrared optical properties of evaporated alumina films,” Appl. Opt. 20(15), 2742–2746 (1981).
    [Crossref] [PubMed]
  20. A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
    [Crossref]
  21. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
    [Crossref]
  22. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
    [Crossref] [PubMed]
  23. I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
    [Crossref]
  24. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
    [Crossref]
  25. 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]
  26. J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
    [Crossref]
  27. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
    [Crossref]
  28. S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
    [Crossref]
  29. E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
    [Crossref] [PubMed]
  30. H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
    [Crossref]
  31. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
    [Crossref]
  32. S. Feng, J. M. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express 13(11), 4113–4124 (2005).
    [Crossref] [PubMed]
  33. G. Rosenblatt and M. Orenstein, “Competing coupled gaps and slabs for plasmonic metamaterial analysis,” Opt. Express 19(21), 20372–20385 (2011).
    [Crossref] [PubMed]
  34. A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).
  35. T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
    [Crossref]

2013 (1)

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

2012 (6)

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

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

A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Genanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20(13), 14663–14682 (2012).
[Crossref] [PubMed]

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

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

A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012).
[Crossref] [PubMed]

2011 (7)

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
[Crossref]

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]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

G. Rosenblatt and M. Orenstein, “Competing coupled gaps and slabs for plasmonic metamaterial analysis,” Opt. Express 19(21), 20372–20385 (2011).
[Crossref] [PubMed]

2010 (4)

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

2009 (4)

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
[Crossref] [PubMed]

2007 (1)

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

2006 (4)

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

2005 (1)

2004 (1)

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

2003 (1)

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

2001 (1)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
[Crossref]

1981 (1)

Acosta, M. F.

Alekseyev, L. V.

Avrutsky, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Barnakov, Y.

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.

Barnakov, Yu. A.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Basharin, A. A.

Belov, P.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Belov, P. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Boltasseva, A.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Bonner, C. E.

Caglayan, H.

Cakmak, A. O.

Capolino, F.

Chebykin, A.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Colak, E.

Cortes, C. L.

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

Drachev, V. P.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

Dryden, D.

Economou, E. N.

Elser, J.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Elson, J. M.

Eriksson, T. S.

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

Feng, S.

Foteinopoulou, S.

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
[Crossref]

García de Abajo, J.

Genanakis, G.

Granqvist, C. G.

Gray, S.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Han, S.

Hjortsberg, A.

Iorsh, I.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

Ishii, S.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

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]

Jacob, Z.

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

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Kafesaki, M.

Katsarakis, N.

Kidwai, O.

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

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

Kildishev, A. V.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

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]

Kim, J.-Y.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Kivshar, Yu.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Kivshar, Yu. S.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Kolinko, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Lee, S.

Lee, T.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Li, H.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

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.

Liu, H.

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

Maria, J.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Maslovski, S.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Mavidis, Ch.

Mayy, M.

Merino, R. I.

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Mock, J. J.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Molesky, S.

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

Naik, G.V.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Narimanov, E.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

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.

Narimanov, E. E.

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

Narimanov, E.E.

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Nataraj, G.

Newman, W.

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

Ni, X.

Niklasson, G. A.

Noginov, M. A.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

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.

Nuzzo, R.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Orenstein, M.

Orera, V. M.

Orlov, A.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Orlov, A. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

Overfelt, P. L.

Ozbay, E.

Pendry, J.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Poddubny, A.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

Poddubny, A. N.

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Podolskiy, V.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Ramakrishna, S.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Reyes-Coronado, A.

Rogers, J.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Rosenblatt, G.

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
[Crossref]

Rye, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Salakhutdinov, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Schilling, J.

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
[Crossref]

Schurig, D.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Shalaev, V. M.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

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]

Shalaev, V.M.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Sipe, J. E.

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

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

Smith, D. R.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Smolyaninov, I. I.

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

Soukoulis, C. M.

Stewart, M.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Stewart, W.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Thoreson, M. D.

Truong, T.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Tumkur, T.

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Villa, A. D.

Voroshilov, P. M.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

Vozianova, A.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Webb, K. J.

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

Wiltshire, M.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Yao, J.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

Zhu, G.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Zhukovsky, S. V.

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

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Appl. Phys. Lett. (3)

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

J. Chem. Phys. (1)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

J. Mod. Opt. (1)

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

J. Opt. (1)

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

J. Phys.: Condens. Matter (1)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
[Crossref]

Laser Photon. Rev. (1)

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

Nanotechnol. (1)

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Opt. Express (7)

S. Feng, J. M. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express 13(11), 4113–4124 (2005).
[Crossref] [PubMed]

G. Rosenblatt and M. Orenstein, “Competing coupled gaps and slabs for plasmonic metamaterial analysis,” Opt. Express 19(21), 20372–20385 (2011).
[Crossref] [PubMed]

E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
[Crossref] [PubMed]

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]

A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012).
[Crossref] [PubMed]

A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Genanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20(13), 14663–14682 (2012).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Lett. A (1)

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

Phys. Rev. A (2)

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

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Phys. Rev. B (7)

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
[Crossref]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Phys. Rev. E (1)

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
[Crossref]

Phys. Rev. Lett. (1)

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

Other (3)

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.

E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

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Figures (2)

Fig. 1
Fig. 1 (a) Isofrequency surfaces in the dispersion relation ( k x 2 + k y 2 ) / ε z + k z 2 / ε x , y = ω 2 / c 2 for conventional anisotropic medium (εx,y,z > 0) and indefinite medium (εx,y < 0 and εz > 0). (b) An infinite multilayer HMM with wave vector decomposition used in the paper (which is schematic in that w and k can be complex). (c) Comparison between the actual multilayer dispersion relation [Eq. (4)] for dm + dd =13.7 and 27.4 nm with the effective-medium dispersion relation for ρ = 0.17. The materials have εm = −17.2 (gold [18]) and εd = 2.59 (alumina [19]) for λ = 715 nm [5]. (d) Isofrequency surfaces for the actual dispersion relation (bottom) vs. effective medium approximation (top).
Fig. 2
Fig. 2 Comparison between Eq. (4) (solid) and the coupled-SRSPP approximation given by Eq. (7) (dashed) for dm + dd =13.7 nm and ρ = 0.17, 0.25, and 0.5 for (a) the pole expansion as in Eq. (6) and (b) the pole expansion without the added constant terms (the vertical marks showing the location of κspc/ω for the respective filling fractions); (c) same as (a) for ρ = 0.12 (the effective medium approximation is indistinguishable from the inner branch of the exact relation). The materials are the same as in Fig. 1(c).

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

M 1 = 1 T m [ T m 2 R m 2 R m R m 1 ] [ e i w d d d 0 0 e i w d d d ] .
R m = r d m + t d m r m d t m d e 2 i w m d m [ 1 r m d 2 e 2 i w m d m ] 1 , T m = t d m t m d e i w m d m [ 1 r m d 2 e 2 i w m d m ] 1 .
r m d = w m ε d w d ε m w m ε d + w d ε m , r d m = w d ε m w m ε d w d ε m + w m ε d , t m d = 2 w m ε m ε d w m ε d + w d ε m , t d m = 2 w d ε d ε m w d ε m + w m ε d .
Tr M 1 2 = cos [ k B ( d m + d d ) ] = cos ( w m d m ) cos ( w d d d ) [ ( η + η 1 ) / 2 ] sin ( w m d m ) sin ( w d d d ) ,
1 r m d 2 e 2 i w m d m = 0
R m = τ m / ( κ κ s p ) ( τ m / κ s p ) , T m 2 R m 2 = 2 τ m 2 / [ κ s p ( κ κ s p ) ] ( τ m / κ s p ) 2 .
cos [ k B ( d m + d d ) ] [ 1 ( κ κ s p ) / ( 2 τ m ) ] e i w d d d + [ ( κ κ s p ) / ( 2 τ m ) ] e i w d d d .

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