Abstract

We show how to achieve a giant permittivity combined with negligible losses in both the visible and the near-IR for composites made of alternating layers of plasmonic and gain materials as the electric field is directed normally to the layers. The effects of nonlocality are taken into account that makes the method quite realistic. Solving the dispersion equation for eigenmodes of an infinite layered composite, we show that both propagating and nonpropagating modes can be excited, that leads to the realization of a giant nonlocal permittivity. Both phase and group velocities for the propagating eigenmode have been calculated showing that slow light can be achieved in the system under study. The results obtained open new possibilities for designing nanolaser, slow-light, superresolution imaging devices, etc.

© 2015 Optical Society of America

1. Introduction

There is a growing interest in the development of metamaterials (MMs) with very large permittivity (or, equivalently, refractive index) in the optical range. At low frequencies, in the microwave and far-IR ranges, such materials are well known (for a recent review, outlining various physical mechanisms that give rise to a giant permittivity, see Lunkenheimer et al. [1]). A peak refractive index of above 20 and the permittivity of above 500 have been experimentally realized at frequencies slightly below 1 THz [2]. The refractive index of above 5.5 with relatively low losses has been predicted at frequencies of 50 – 80 THz [3]. The implementation of the proposed structure, however, is not easy, as it requires sophisticated techniques and equipment.

In the near-IR, MMs with a giant permittivity (”epsilon-near-pole” MMs) could be of interest for thermophotovoltaics [4]. The implementation of such MMs at higher frequencies would offer new possibilities, such as the development of slow-light devices [5], super-resolution imaging and nanolithography techniques, where the resolution is known to scale inversely with the refractive index. The design of nanoscale laser devices is directly related to the availability of transparent (lossless) high refractive index materials. So, it was proposed to use the dipolar permittivity peak of an assembly of strongly prolate metal nanospheroids embedded in a gain medium [6]. But even after taking into account necessary corrections for the effective permittivity [7], such an approach looks hardly feasible due to obvious fabrication difficulties and tolerance constraints. In addition, the applicability of the local permittivity formalism for gain media becomes questionable in the considered case [8]. Conceptually different (cavity-free) mechanism of nanolasing involves the use of stopped light that allows to provide local feedback [9]. Anyway, one can state that in spite of much effort that has been directed toward designing nanoscale laser devices, much remains undone yet in this field [10, 11].

As was shown by Pendry and Ramakrishna, a series of thin slices of equal thicknesses alternating between ε = 1 and ε = −1 would have the infinite effective permittivity as the electric field is normal to the slices and zero as it is parallel to them, that gives rise to the so-called perfect lens with high optical resolution [12]. Similarly, imaging with subwavelength resolution was predicted in the visible with the use of metal-dielectric nanolayers operating in the so-called canalization regime [13, 14]. The losses, however, can significantly deteriorate the quality of the image [15]. It is also of interest to note that due to the formal analogy between light propagation in vacuous curved spacetime (spacetime subjected to gravitational fields) and in certain nongomogeneous media, MMs with high permittivity and low losses can be considered in terms of the curved spacetime [16].

One solution to the issue of losses is dealing with all-dielectric materials. So the realization of zero-index MMs with negligible losses in the near-IR was recently reported [17]. Another recipe is to incorporate gain into the nanostructure design to compensate losses. The simplest nanostructure could consist of alternative layers of plasmonic and gain materials [18]. If so, the plasmonic and gain domains are spatially separated that validates the use of the local permittivity formalism [8]. This approach was developed further by many authors (see, e.g., Refs. [19–21]). Although the full loss compensation is not an easy task, the partial compensation has already been demonstrated in experiments [22–24]. Nevertheless, to our knowledge, no previous study addressed the possibility of loss compensation to develop MMs with a giant permittivity.

This study aims at designing MMs with (i) simple and feasible geometry, (ii) negligibly small losses, and (iii) extremely high permittivity. First, in Sec.II we show, within the framework of the local approximation for the effective permittivity, that the full loss compensation combined with a giant permittivity is achievable for 1d nanostructures when dealing with dye molecules incorporated into a dielectric host with low refractive index. Furthermore, by solving exact dispersion equation, in Sec. III we go beyond the scope of the local approximation and take into account possible effects of nonlocality. Although in periodic nanocomposites, as the wave vector is limited within the first Brillouine zone and hence the effective permittivity is limited, too, its giant values are achievable as the wave vector is directed along the interfaces and hence the composite is homogeneous in this direction. In Sec.IV we discuss the issue of the phase and group velocities in MMs under study. Finally, in Sec.V we give our concluding remarks, in particular, limitations and possible generalization of our approach, as well as a short conclusion.

2. Local approximation

For convenient illustration, we invoke the formalism of the bounds on allowed values of the effective permittivity εeff of a two-component composite and use the fact that the extreme values of the effective permittivity are achievable when the composite consists of alternative parallel layers made of constituent media [25–27]. One of such extreme values of εeff (the so-called lower Wiener bound), which occurs when the electric field is normal to the layers, is

εlb=εeff=[fε1+f2ε2]1,
where ε1 = ε′1 + iε″1, ε2 = ε′2 + iε″2 are the constituent permittivities and f, f2 = 1 − f are the volume fractions of the constituents 1 and 2, respectively. Another bound (the upper Wiener bound), which occurs when the electric field is parallel to the layers, is
εub=εeff=fε1+f2ε2.

It is just the lower Wiener bound that makes it possible to realize a giant permittivity [27]; that is why in the following we consider its behavior in more detail. The 1d geometry, for which the lower Wiener bound applies, is shown in Fig. 1 [28].

 

Fig. 1 Sketh of the unit cell under consideration.

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On the complex plane, the lower Wiener bound is the circular arc which passes through ε1 and ε2, while its continuation passes through the origin [26]. After imposing the condition Imεlb = 0, one has:

f=f0=ε2|ε1|2ε2|ε1|2ε1|ε2|2.
Suppose that the component 1 is plasmonic and passive, i.e., ε′1 < 0 and ε″1 > 0. If the component 2 is also passive, i.e., ε″2 > 0, then Imεlb > 0 too, otherwise f0 takes nonphysical values f0 < 0 or f0 > 1. In other words, the condition given by Eq. (3) cannot be satisfied.

Let us now consider the case when the component 2 is dielectric and active, i.e., ε′2 > 0 and Im ε″2 < 0. As is easy to check, then a value of f0 exists satisfying Eq. (1) such that 0 < f0 < 1. After substituting it into Eq. (1), one has the corresponding (real) effective permittivity

εeff0εeff(f0)=ε2|ε1|2ε1|ε2|2ε2ε1ε1ε2.
As it follows from Eq. (4), εeff0 becomes singular as
ε1/ε1=ε2/ε2.
This condition has a simple geometrical interpretation: all three points, ε1, ε2, and the origin lie on a straight line on the complex plane (the arc degenerates into the straight line). So, as the lower Wiener bound (arc) degenerates into the straight line, its intersection with the axis of Reε on the complex plane (the condition of the lossless effective permittivity) approaches infinity.

For illustration, in Fig. 2 we show the lower Wiener bound εlb on the complex plane for ε1 = −6 + 0.2i (this corresponds approximately to the permittivity of silver at λ = 430 nm) [29], ε2 = 2.2 − 0.03i, ε2 = 2.2 − 0.04i, ε2 = 2.2 − 0.05i, and ε2 = 2.2 − 0.15i, i.e., we keep ε1 and ε′2 constant and change ε″2. Thus, the component 2 can be considered as a dye-doped dielectric with the imaginary part of permittivity dependent on dye concentration or on pump power (pumping should not exceed the lasing threshold [8]). In this case ε″1/ε′1 = −0.0333, while ε″2/ε″1 take the values −0.0136, −0.0181, and −0.0227 at ε″2 = −0.03, −0.04, and −0.05, respectively. As we see, when ε″2 drops, εeff0 grows: as ε″2ε″1ε′2/ε′1, εeff0. After crossing the singularity, εeff0 changes its sign and then grows again until it reaches a peak. This happens when εeff0/ε2=0, that yields the equation

ε1|ε2|2ε2|ε1|22ε2(ε2ε1ε1ε2)=0.
Its solution is of the form
ε2max=ε1ε21(ε1ε21)2+ε22|ε1|2ε21
with ε21 = ε′2/ε′1. After some algebra, the peak (negative) value of εeff0 is found to be
εeff0maxεeff0(ε2max)=ε22ε22+2ε1ε2|ε1|2ε2ε1.

According to Eq. (4), εeff0 can vary in wide limits, from −∞ and up to +∞, but the interval [ εeff0max, ε′2] must be excluded. Thus, the zero value of εeff0 is inaccessible (it is accessible when dealing with the upper Wiener bound, but this is outside the scope of the present paper). Indeed, if ε″1 > 0 and ε″2 < 0, the numerator in Eq. (4) is always negative. Besides, large (negative) values of εeff0, close to εeff0max, are also inaccessible, because they can occur only at unreasonably large negative ε″2.

 

Fig. 2 The lower Wiener bound (solid curves) computed for a two-component multilayered composite with ε1 = −6 + 0.2i and different values of ε2: ε2A=2.20.03i (blue curve), ε2B=2.20.04i (green curve), ε2C=2.20.05i (red curve), and ε2D=2.20.15i (orange curve). The arrows show the values of εeff0: εeff0=7.88,12.05,19.795, and −13.86 at ε″2 =−0.03, −0.04, −0.05, and −0.15, respectively. The dashed lines show the upper Wiener bound.

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In this example, we have taken ε′2 = 2.2 as the typical value of the dielectric permittivity. If so, according to Eq. (5), the above singularity should occur at ε″2 ≈ −0.073. This is less than the minimal value of ε″2, which is known to be of the order of −0.04 for most dyes, but exceeds that for semiconductor quantum wells or quantum dots, which is known to be of the order of −0.3 [21]. To obtain the effective permittivity as high as possible, the real part of the dielectric permittivity (ε′2) should be as low as possible. In this connection we note that fluoropolymers and alkali fluorides are appropriate for this purpose. So, Teflon AF 2400 has the lowest refractive index (1.29–1.31) and respectively the lowest permittivity (1.66–1.72) in the visible [30], while such compounds as LiF and NaF have only slightly higher permittivities [31].

3. Nonlocal homogenization

The above consideration is accurate as long as nonlocal effects are weak. An accurate homogenization technique, which takes into account nonlocalty, is based on Floquet representation; it introduces a single generalized permittivity tensor that properly describes all the polarization effects, including artificial magnetism, additional waves, and magnetoelectric coupling [32,33]. When dealing with in-plane propagation (k = 0) and TM-polarization, only one diagonal component of the tensor is enough to describe the nonlocal response; it may be introduced as [20]

ε˜eff(k)=k2/k02,
where k0 = ω/c and k is the in-plane eigenwave number which should satisfy the Rytov’s dispersion equation [34]
cos(k1d1)cos(k2d2)γsin(k1d1)sin(k2d2)=1
with γ=12(ε1k2ε2k1+ε2k1ε1k2), k1,2=k02ε1,2k2 and d1,2 the layers’ thickness. The high accuracy of this approach has been recently verified numerically using a spatial harmonic analysis method [20].

It is interesting to assess how accurate the local approximation is. To do this, we solve Eq. (10) for k, taking f = f0 as in Eq. (3), at ε1 = −20 + 0.45i (this corresponds approximately to silver at λ = 660 nm) and ε′2 = 1.82 (Teflon). After that, we calculate ε˜eff using Eq. (4) (local approximation) and Eq. (9) for two different values of the metal layer thickness d1 and two eigenmodes of the lowest order. The nonlocal effective permittivity now becomes complex; its real and imaginary parts are shown in Fig. 3. Naturally, as d1 → 0, the local approximation is good enough. At the same time, when Reε˜eff take large values, the local approximation becomes inaccurate even at such thick metal layers as few tens of nanometers. Thus, nonlocality decreases the effective permittivity. In addition, the condition of zero losses, Eq. (3), breaks down. As can be seen, the modes that should be lossless in the local approximation, become either lossy or amplified when nonlocality is taken into account.

 

Fig. 3 The nonlocal effective permittivity vs ε″2 calculated with the use of Eqs. (9) and (3) for two eigenmodes for d1 = 20 nm (blue dotted curves) and d1 = 40 nm (red dashed curves). The black curve shows the effective permittivity calculated with the use of Eq. (4) (local approximation).

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Although Eq. (10) has an infinite number of solutions which characterize corresponding eigenmodes [21,35], only few of them can provide Imε˜eff=0; those are the modes with either Rek = 0 or Imk = 0. If Rek = 0, then the corresponding mode is either amplified (the electric field grows exponentially) or evanescent (the electric field decays exponentially), depending on the sign of Imk. It is obvious that if Imk = 0 and Rek ≠ 0, then we are dealing with a lossless propagating mode, which can be identified as a volume plasmon polariton [36]. After solving Eq. (10), in Figs. 4 and 5 we plot four branches of both the real and the imaginary parts of the eigenwave number (i.e., four modes supported by the structure) as a function of fd1/(d1 + d2) with d1 = 40 nm (k0d1 = 0.38) at ε″2 = −0.04 and ε″2 = −0.05, respectively. At ε″2 = −0.04, the condition Imk = 0 can be satisfied, that characterizes the propagating mode. At ε″2 = −0.05, only the condition Rek = 0 can be satisfied at two different f values for one of the four shown eigenmodes.

 

Fig. 4 The real and imaginary parts of k vs f for four eigenmodes at k0d1 = 0.38, ε1 = −20 + 0.45i, and ε2 = 1.82 − 0.04i. The arrow shows the position of zero of Imk.

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Fig. 5 The same as in Fig. 4 but at ε2 = 1.82 − 0.05i. The arrows show the positions of zero of Rek.

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Amplified and evanescent modes are non-propagating, and hence are of little interest for this study. We note only that (i) the presence of amplified modes can result in instability, either absolute or convective [21], (ii) formally, nonpropagating modes correspond to negative ε˜eff, and (iii) as these modes have zero phase velocity, the regime of negative phase velocity [37] can be realized at slightly different parameters. The values of ε˜eff vs ε″2, calculated for the propagating mode, i.e., at the points where Imk = 0, are plotted in Fig. 6 for different values of k0d1. These results evidence that both the local and nonlocal effective permittivities can take giant values, remaining lossless. This happens when the imaginary part of the permittivity of gain dielectric approaches its ”critical” value, which, in accordance with Eq. (5), is ε″2 = ε″1ε′2/ε′1 (about −0.04095 for specific parameters, used in our calculations). This agrees with the fact that, in the lossless case, the nonlocal permittivity can have singularities, as noticed earlier [38]. The values of f = f0, at which Imε˜eff=0, depend on the layer thicknesses. An increase in the unit cell thickness, that corresponds to stronger nonlocality, results in larger values of f0. At the same time, the dependence of ε˜eff(f0) on the unit cell thickness is weak. For example, at k0d1 = 0.38 (d1 = 40 nm), f0 = 0.954 and ε˜eff=114.8, while at k0d1 = 0.76 (d1 = 80 nm), f0 = 0.975 and ε˜eff=104.4. This seems to be important from a practical point of view.

 

Fig. 6 The effective permittivity vs ε″2 calculated according to Eq. (9) for the propagating eigenmode at different values of the thickness d1.

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4. Phase and group velocities

It is obvious that as the refractive index is high, the phase velocity vp = ω/k becomes small. Here we show that the group velocity vg = dω/dk, which characterizes the velocity at which a pulse (wave packet) propagates, can be also small, that gives rise to slow-light structures [5,39]. To do this, we differentiate Eq. (10) with respect to k0 and obtain

cot(k2df2)(fk1+γf2k2)+cot(k1df)(f2k2+γfk1)+γd=0,
where k1,2=(2k0ε1,2+k02ε1,22kc/vg)/k1,2, ε1,2=dε1,2/dk0=c(dε1,2/dω), c is the the light speed in vacuum, γ=12(c1ε1+c2ε2+c3k2+c4k1), and c1=k2/(k1ε2)k1ε2/(k2ε12), c2=k1/(k2ε1)k2ε1/(k1ε22), c3=ε1/(ε2k1)ε2k1/(ε1k22), c4=ε2/(ε1k2)ε1k2/(ε2k12).

To solve Eq. (11) for vg, we should first evaluate the derivatives 1/dk0 and 2/dk0. Our estimation for silver at λ = 660 nm, based on the data by Johnson and Christy [29], is 1/dk0 ≃ 5.047 × 103 + 137i. To find 2/dk0, we adopt the Lorentz model for the dye-doped dielectric permittivity of the form

ε2=ε+σω02ω2iΔωω.
Then, at the emission frequency ω = ω0, the above model provides ε2=ε+iσΔωω0, and the sought derivative, evaluated at ω0, can be written as
dε2dk0=cdε2dω=σcΔωω0(2Δω+iω0)=cε2(2Δω+iω0)=ε2k0(2δ+i)
with δ = Δω0. Assuming that the wavelength linewidth for typical dyes is of the order of 10 – 50 nm, the parameter δ at λ = 660 nm can be estimated as δ ∼ 0.015 − 0.075.

Next, we solve Eq. (10) at d = 40 nm for the propagating eigenmode at different values of ε″2 and find f and Reε˜eff at which the condition Imε˜eff=0 meets. After that, we substitute the obtained values of f and ε˜eff into Eq. (11) and solve it for vg. Doing so, we estimate the lower bound for the group velocity (vgl), taking δ = 0.015, as well as the upper bound (vgu), taking δ = 0.075 (smaller δ results in a higher dispersion and hence smaller group velocity). Our results for the phase and group velocities are shown in Fig. 7. Both velocities tend to zero as ε″2 approaches its ”critical” value, allowing to realize the stopped-light effect.

 

Fig. 7 The phase and group velocities vs ε″2 for the propagating eigenmode.

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The group velocity, obtained in such a way, is, generally speaking, complex-valued. The imaginary part of vg is responsible for pulse reshaping. However, in our case it is small as compared to its real part, and we do not show Imvg here. For example, at ε″2 = −0.404, Imvgu/Revgu ≃ −0.012 and Imvgl/Revgl ≃ −0.0165. It should be also noted that if the loss is sufficiently small, Revg becomes numerically equivalent to the energy velocity [40].

5. Concluding remarks

As we have shown in Sec. 3, the local approximation becomes highly inaccurate for lattices under study in the high-permittivity regime, when nonlocal effects cannot be neglected. Our subsequent consideration is much more accurate. However, it is necessary to keep in mind, that it does not take into account intrinsic nonlocality of the metal layers. Due to the quantum many-body properties of the electron gas, important nonlocal corrections can occur in noble-metal nanostructures with characteristic dimension of the order of 1 – 10 nm [41]. We also want to mention the recently predicted effect of tilting [38]. It consists in the emergence of nonzero off-diagonal components of the nonlocal permittivity tensor, when the tensor can be transformed to a diagonal form only after a rotation of the coordinate system. However, this effect vanishes, in particular, at k = 0 [38], and hence is beyond the scope of the present work.

Furthermore, as the metal film thickness becomes comparable to or smaller than the mean free path of the conduction electrons (about 52 nm for silver [42]), the film permittivity (first of all, its imaginary part) becomes thickness-dependent and anisotropic. This dependence can be considered in terms of both surface scattering and grain boundary scattering models [43]. The resistivity of an evaporated silver film increases sharply as its thickness becomes below the mean free path, and only slightly depends on the thickness above it [43]. Therefore, dealing with very thin metal films, we cannot use the bulk metal permittivity, but should increase its imaginary part. To avoid this detrimental effect, which hampers the loss compensation, we should increase the metal film thickness.

In the above example, high values of ε˜eff (above 50) can be achieved at f0 ∼ 0.93 − 0.98. If so, the thickness of dielectric layers cannot exceed few nanometers. Experimental realization of devices with such thin layers can involve difficulties. To avoid them, it is essential to increase ε′1 that, in turn, decreases the corresponding f0 and hence allows one to increase d2.

The above results have been obtained for the infinite lattices. If the number of layers is finite, more sophisticated consideration may be required due to some complications, which can occur for finite 1d systems [44–46].

In this work we have dealt with 1d MMs. It would be of interest to consider possible extensions of our approach to higher dimensions. In this connection, we would like to note the following. Because the composite under study is homogeneous in the direction of the eigenwave propagation, k is not restricted by the geometry [47]. This means that the effective permittivity can be extremely large, and the formally introduced effective modal wavelength may become of the order of interatomic distances. One can expect that similar effect can be also realized for 2d MMs, e.g., for wire media, for the wave vector direction along the wires. However, 3d MMs are inhomogeneous in any direction. If so, the largest wavenumber and permittivity are limited by values of the order of π/d and (πc/ωd)2, respectively, where d is the lattice period.

To conclude, we have shown that a giant lossless permittivity (more exactly, normal component of the permittivity tensor) can be achieved at optical frequencies for 1d composites made of alternating metallic and gain dielectric layers. The local approximation, as well as nonlocal effects have been considered. The above permittivity can be associated either with propagating or nonpropagating eigenmodes. Extremely low phase and group velocities can be achieved in MMs under study. Limitations, related to the realization of the method, have been discussed. Our results open new possibilities for the design of various nanoscale devices and techniques, e.g., nanolaser and slow-light devices, subwavelength imaging and nanolithography.

Acknowledgments

V.U.N. acknowledges support from the Ministry of Science and Technology, Taiwan, Grants No. 103-2112-M-001-007 and 104-2923-M-001-001-MY3.

References and links

1. P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

2. M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011). [CrossRef]   [PubMed]  

3. J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009). [CrossRef]  

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

5. R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009). [CrossRef]  

6. N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007). [CrossRef]  

7. A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007). [CrossRef]  

8. A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012). [CrossRef]  

9. T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014). [CrossRef]  

10. P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012). [CrossRef]  

11. Q. Gu, J.S.T. Smalley, M.P. Nezhad, A. Simic, J.H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Subwavelength semiconductor lasers for dense chip-scale integration,” Adv. Opt. Photon. 6, 1–56 (2014). [CrossRef]  

12. J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003). [CrossRef]  

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

14. D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008). [CrossRef]  

15. P.A. Belov, “Subwavelength imaging by extremely anisotropic media,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S.A. Maier, eds. (Wiley, 2013). [CrossRef]  

16. T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011). [CrossRef]  

17. P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

18. S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003). [CrossRef]  

19. M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009). [CrossRef]  

20. 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, 25242–25254 (2011). [CrossRef]  

21. R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013). [CrossRef]  

22. M.A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C.E. Small, B.A. Ritzo, V.P. Drachev, and V.M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31, 3022–3024 (2006). [CrossRef]   [PubMed]  

23. C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011). [CrossRef]  

24. P. Berini, “Loss compensation and amplification of surface plasmon polaritons,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S. A. Maier, eds. (Wiley, 2013). [CrossRef]  

25. D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980). [CrossRef]  

26. G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981). [CrossRef]  

27. A.V. Concharenko, E.F. Venger, and A.O. Pinchuk, “Homogenization of quasi-1d metamaterials and the problem of extended bandwidth,” Opt. Express 22, 2429–2442 (2014). [CrossRef]  

28. To avoid confusion, it should be noted that Wiener dealt with the conductivity. Because the conductivity is proportional to the imaginary part of the permittivity, his lower bound (for the real conductivity) in fact corresponds to the upper bound for the permittivity.

29. P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979). [CrossRef]  

30. W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991). [CrossRef]  

31. H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976). [CrossRef]  

32. A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011). [CrossRef]  

33. V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009). [CrossRef]  

34. S.M. Rytov, “Electromagnetic properties of laminated medium,” Sov. Phys. JETP 2, 466–475 (1956).

35. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]  

36. S.V. Zhukovsky, O. Kidwai, and J.E. Sipe, “Physical nature of volume plasmon polaritons in hyperbolic meta-materials,” Opt. Express 21, 14982–14987 (2013). [CrossRef]   [PubMed]  

37. T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009). [CrossRef]  

38. A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012). [CrossRef]  

39. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008). [CrossRef]  

40. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

41. N.A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Photon. Nanostr. - Fundam. Appl. 11, 303–309 (2013). [CrossRef]  

42. F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952). [CrossRef]  

43. W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004). [CrossRef]  

44. A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002). [CrossRef]  

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

46. R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007). [CrossRef]  

47. R.L. Chern, “Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials,” Opt. Express 21, 16514–16527 (2013). [CrossRef]   [PubMed]  

References

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  1. P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).
  2. M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
    [Crossref] [PubMed]
  3. J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
    [Crossref]
  4. S. Molesky, C.J. Dewalt, and Z. Jacob, “High temperature epsilon-near-zero and epsilon-near-pole metamaterial emitters for thermophotovoltaics,” Opt. Express 21, A96–A110 (2013).
    [Crossref] [PubMed]
  5. R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009).
    [Crossref]
  6. N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007).
    [Crossref]
  7. A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007).
    [Crossref]
  8. A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
    [Crossref]
  9. T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
    [Crossref]
  10. P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
    [Crossref]
  11. Q. Gu, J.S.T. Smalley, M.P. Nezhad, A. Simic, J.H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Subwavelength semiconductor lasers for dense chip-scale integration,” Adv. Opt. Photon. 6, 1–56 (2014).
    [Crossref]
  12. J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
    [Crossref]
  13. P.A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
    [Crossref]
  14. D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
    [Crossref]
  15. P.A. Belov, “Subwavelength imaging by extremely anisotropic media,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S.A. Maier, eds. (Wiley, 2013).
    [Crossref]
  16. T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
    [Crossref]
  17. P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).
  18. S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
    [Crossref]
  19. M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
    [Crossref]
  20. 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, 25242–25254 (2011).
    [Crossref]
  21. R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
    [Crossref]
  22. M.A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C.E. Small, B.A. Ritzo, V.P. Drachev, and V.M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31, 3022–3024 (2006).
    [Crossref] [PubMed]
  23. C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
    [Crossref]
  24. P. Berini, “Loss compensation and amplification of surface plasmon polaritons,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S. A. Maier, eds. (Wiley, 2013).
    [Crossref]
  25. D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
    [Crossref]
  26. G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
    [Crossref]
  27. A.V. Concharenko, E.F. Venger, and A.O. Pinchuk, “Homogenization of quasi-1d metamaterials and the problem of extended bandwidth,” Opt. Express 22, 2429–2442 (2014).
    [Crossref]
  28. To avoid confusion, it should be noted that Wiener dealt with the conductivity. Because the conductivity is proportional to the imaginary part of the permittivity, his lower bound (for the real conductivity) in fact corresponds to the upper bound for the permittivity.
  29. P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
    [Crossref]
  30. W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
    [Crossref]
  31. H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
    [Crossref]
  32. A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
    [Crossref]
  33. V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
    [Crossref]
  34. S.M. Rytov, “Electromagnetic properties of laminated medium,” Sov. Phys. JETP 2, 466–475 (1956).
  35. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
    [Crossref]
  36. S.V. Zhukovsky, O. Kidwai, and J.E. Sipe, “Physical nature of volume plasmon polaritons in hyperbolic meta-materials,” Opt. Express 21, 14982–14987 (2013).
    [Crossref] [PubMed]
  37. T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
    [Crossref]
  38. A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
    [Crossref]
  39. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
    [Crossref]
  40. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  41. N.A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Photon. Nanostr. - Fundam. Appl. 11, 303–309 (2013).
    [Crossref]
  42. F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
    [Crossref]
  43. W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
    [Crossref]
  44. A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
    [Crossref]
  45. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolsky, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
    [Crossref]
  46. R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
    [Crossref]
  47. R.L. Chern, “Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials,” Opt. Express 21, 16514–16527 (2013).
    [Crossref] [PubMed]

2014 (3)

2013 (5)

2012 (3)

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

2011 (5)

T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
[Crossref]

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (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, 25242–25254 (2011).
[Crossref]

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

2010 (1)

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

2009 (5)

J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009).
[Crossref]

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
[Crossref]

T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

2008 (2)

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

2007 (5)

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

R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
[Crossref]

N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007).
[Crossref]

A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007).
[Crossref]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

2006 (2)

M.A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C.E. Small, B.A. Ritzo, V.P. Drachev, and V.M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31, 3022–3024 (2006).
[Crossref] [PubMed]

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

2004 (1)

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

2003 (2)

S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[Crossref]

J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
[Crossref]

2002 (1)

A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
[Crossref]

1991 (1)

W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
[Crossref]

1981 (1)

G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
[Crossref]

1980 (1)

D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

1979 (1)

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
[Crossref]

1976 (1)

H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
[Crossref]

1956 (1)

S.M. Rytov, “Electromagnetic properties of laminated medium,” Sov. Phys. JETP 2, 466–475 (1956).

1952 (1)

F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
[Crossref]

Adegoke, J.

Agranovich, V.M.

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
[Crossref]

Akozbek, N.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Alu, A.

A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[Crossref]

Anantha Ramakrishna, S.

S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[Crossref]

J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
[Crossref]

Anderson, Z.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Avrutsky, I.

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

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

Bahoura, M.

Belov, P.A.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

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

P.A. Belov, “Subwavelength imaging by extremely anisotropic media,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S.A. Maier, eds. (Wiley, 2013).
[Crossref]

Bergman, D.J.

D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

Berini, P.

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

P. Berini, “Loss compensation and amplification of surface plasmon polaritons,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S. A. Maier, eds. (Wiley, 2013).
[Crossref]

Bloemer, M.J.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Bondarenko, O.

Boyd, R.W.

R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009).
[Crossref]

Briggs, D.P.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Brijs, B.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Brongersma, S.H.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Cappeddu, M.G.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Centini, M.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Chebykin, A.V.

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Chern, R.L.

Choi, M.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Christy, R.W.

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
[Crossref]

Ciattori, A.

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

Concharenko, A.V.

D’Orazio, A.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

de Ceglia, D.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

De Leon, I.

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

Dewalt, C.J.

Di Falco, A.

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

Dorofeenko, A.V.

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

Drachev, V.P.

Ebbinghaus, S.G.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Elser, J.

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

Fainman, Y.

Fan, S.

J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Fedorov, S.V.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

Froyen, L.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Gartstein, Yu.N.

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
[Crossref]

Goncharenko, A.V.

A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007).
[Crossref]

Gorkunov, M.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Gralak, B.

R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
[Crossref]

Groh, W.

W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
[Crossref]

Gu, Q.

Hamm, J.M.

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
[Crossref]

Han, S.

Hao, Y.

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

Haus, J.W.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Hess, O.

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
[Crossref]

Ishii, S.

Jacob, Z.

Johnson, P.B.

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
[Crossref]

Kang, K.Y.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Kang, S.B.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Katz, M.

Kidwai, O.

Kildishev, A.V.

Kim, Y.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Kivshar, Y.S.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

Kivshar, Yu.S.

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Kravchenko, I.I.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Krohns, S.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Kwak, M.H.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Ladisa, A.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

Lakhtakia, A.

T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
[Crossref]

T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

Lawandy, N.M.

N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007).
[Crossref]

Lee, J.H.

Lee, S.

Lee, S.H.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Lee, Y.H.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Li, H.

H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
[Crossref]

Lisyansky, A.A.

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

Loidl, A.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Lomakin, V.

Lunkenheimer, P.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Mackay, T.

T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

Mackay, T.G.

T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
[Crossref]

Maex, K.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Merzlikin, A.V.

A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
[Crossref]

Milton, G.W.

G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
[Crossref]

Min, B.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Mizrahi, A.

Moitra, P.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Molesky, S.

Mortensen, N.A.

N.A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Photon. Nanostr. - Fundam. Appl. 11, 303–309 (2013).
[Crossref]

Nezhad, M.P.

Ni, X.

Noginov, M.A.

Orlov, A.A.

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Page, A.F.

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
[Crossref]

Palmans, R.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Park, N.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Pendry, J.B.

J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
[Crossref]

S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[Crossref]

Pickering, T.

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
[Crossref]

Pierre, R.

R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
[Crossref]

Pinchuk, A.O.

Podivilov, E.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Podolsky, V.

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

Pukhov, A.A.

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

Reller, A.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Reynolds, F.W.

F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
[Crossref]

Richard, O.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Riegg, S.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Ritzo, B.A.

Rizza, C.

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

Rondinone, V.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

Rosanov, N.N.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

Rytov, S.M.

S.M. Rytov, “Electromagnetic properties of laminated medium,” Sov. Phys. JETP 2, 466–475 (1956).

Salakhutdinov, I.

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

Savelev, R.S.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

Scalora, M.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Shadrivov, I.V.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

Shalaev, V.M.

Shen, J.T.

J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Shin, J.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Simic, A.

Simovski, C.R.

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Sipe, J.E.

Slutsky, B.

Small, C.E.

Smalley, J.S.T.

Stilwell, G.R.

F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
[Crossref]

Sturman, B.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Sukhorukov, A.A.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

Thoreson, M.D.

Valentine, J.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Venger, E.F.

Vincenti, M.A.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Vinogradov, A.P.

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
[Crossref]

Wuestner, S.

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
[Crossref]

Yang, Y.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Zhang, W.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Zhu, G.

Zhukovsky, S.V.

Zimmermann, A.

W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
[Crossref]

Zyablovsky, A.A.

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (3)

N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007).
[Crossref]

A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007).
[Crossref]

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

Eur. Phys. J. Special Topics (1)

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

J. Appl. Phys. (1)

G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
[Crossref]

J. Exp. Theor. Phys. (1)

A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
[Crossref]

J. Mod. Opt. (2)

R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
[Crossref]

R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009).
[Crossref]

J. Phys. Chem. Ref. Data (1)

H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
[Crossref]

Macromolecules (1)

W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
[Crossref]

Metamaterials (1)

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
[Crossref]

Microel. Eng. (1)

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Nature (1)

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Nature Commun. (1)

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
[Crossref]

Nature Photon. (1)

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

Natute Photon. (1)

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Photon. Nanostr. - Fundam. Appl. (1)

N.A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Photon. Nanostr. - Fundam. Appl. 11, 303–309 (2013).
[Crossref]

Phys. Rev. (1)

F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
[Crossref]

Phys. Rev. A (2)

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Phys. Rev. B (10)

T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
[Crossref]

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

S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[Crossref]

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[Crossref]

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
[Crossref]

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

T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Phys. Rev. Lett. (2)

D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Physica B (1)

J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
[Crossref]

Physics Uspekhi (1)

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

Sov. Phys. JETP (1)

S.M. Rytov, “Electromagnetic properties of laminated medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (5)

To avoid confusion, it should be noted that Wiener dealt with the conductivity. Because the conductivity is proportional to the imaginary part of the permittivity, his lower bound (for the real conductivity) in fact corresponds to the upper bound for the permittivity.

P. Berini, “Loss compensation and amplification of surface plasmon polaritons,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S. A. Maier, eds. (Wiley, 2013).
[Crossref]

P.A. Belov, “Subwavelength imaging by extremely anisotropic media,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S.A. Maier, eds. (Wiley, 2013).
[Crossref]

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

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Figures (7)

Fig. 1
Fig. 1 Sketh of the unit cell under consideration.
Fig. 2
Fig. 2 The lower Wiener bound (solid curves) computed for a two-component multilayered composite with ε1 = −6 + 0.2i and different values of ε2: ε 2 A = 2.2 0.03 i (blue curve), ε 2 B = 2.2 0.04 i (green curve), ε 2 C = 2.2 0.05 i (red curve), and ε 2 D = 2.2 0.15 i (orange curve). The arrows show the values of ε eff 0: ε eff 0 = 7.88 , 12.05 , 19.795, and −13.86 at ε″2 =−0.03, −0.04, −0.05, and −0.15, respectively. The dashed lines show the upper Wiener bound.
Fig. 3
Fig. 3 The nonlocal effective permittivity vs ε″2 calculated with the use of Eqs. (9) and (3) for two eigenmodes for d1 = 20 nm (blue dotted curves) and d1 = 40 nm (red dashed curves). The black curve shows the effective permittivity calculated with the use of Eq. (4) (local approximation).
Fig. 4
Fig. 4 The real and imaginary parts of k vs f for four eigenmodes at k0d1 = 0.38, ε1 = −20 + 0.45i, and ε2 = 1.82 − 0.04i. The arrow shows the position of zero of Imk.
Fig. 5
Fig. 5 The same as in Fig. 4 but at ε2 = 1.82 − 0.05i. The arrows show the positions of zero of Rek.
Fig. 6
Fig. 6 The effective permittivity vs ε″2 calculated according to Eq. (9) for the propagating eigenmode at different values of the thickness d1.
Fig. 7
Fig. 7 The phase and group velocities vs ε″2 for the propagating eigenmode.

Equations (13)

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ε lb = ε eff = [ f ε 1 + f 2 ε 2 ] 1 ,
ε ub = ε eff = f ε 1 + f 2 ε 2 .
f = f 0 = ε 2 | ε 1 | 2 ε 2 | ε 1 | 2 ε 1 | ε 2 | 2 .
ε eff 0 ε eff ( f 0 ) = ε 2 | ε 1 | 2 ε 1 | ε 2 | 2 ε 2 ε 1 ε 1 ε 2 .
ε 1 / ε 1 = ε 2 / ε 2 .
ε 1 | ε 2 | 2 ε 2 | ε 1 | 2 2 ε 2 ( ε 2 ε 1 ε 1 ε 2 ) = 0 .
ε 2 max = ε 1 ε 21 ( ε 1 ε 21 ) 2 + ε 2 2 | ε 1 | 2 ε 21
ε eff 0 max ε eff 0 ( ε 2 max ) = ε 2 2 ε 2 2 + 2 ε 1 ε 2 | ε 1 | 2 ε 2 ε 1 .
ε ˜ eff ( k ) = k 2 / k 0 2 ,
cos ( k 1 d 1 ) cos ( k 2 d 2 ) γ sin ( k 1 d 1 ) sin ( k 2 d 2 ) = 1
cot ( k 2 d f 2 ) ( f k 1 + γ f 2 k 2 ) + cot ( k 1 d f ) ( f 2 k 2 + γ f k 1 ) + γ d = 0 ,
ε 2 = ε + σ ω 0 2 ω 2 i Δ ω ω .
d ε 2 d k 0 = c d ε 2 d ω = σ c Δ ω ω 0 ( 2 Δ ω + i ω 0 ) = c ε 2 ( 2 Δ ω + i ω 0 ) = ε 2 k 0 ( 2 δ + i )

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