## 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 *ε _{eff}* 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

*ε*(the so-called lower Wiener bound), which occurs when the electric field is normal to the layers, is

_{eff}*ε*

_{1}=

*ε′*

_{1}+

*iε″*

_{1},

*ε*

_{2}=

*ε′*

_{2}+

*iε″*

_{2}are the constituent permittivities and

*f*,

*f*

_{2}= 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

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].

On the complex plane, the lower Wiener bound is the circular arc which passes through *ε*_{1} and *ε*_{2}, while its continuation passes through the origin [26]. After imposing the condition Im*ε _{lb}* = 0, one has:

*ε′*

_{1}< 0 and

*ε″*

_{1}> 0. If the component 2 is also passive, i.e.,

*ε″*

_{2}> 0, then Im

*ε*> 0 too, otherwise

_{lb}*f*

_{0}takes nonphysical values

*f*

_{0}< 0 or

*f*

_{0}> 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 *f*_{0} exists satisfying Eq. (1) such that 0 < *f*_{0} < 1. After substituting it into Eq. (1), one has the corresponding (real) effective permittivity

*ε*

_{1},

*ε*

_{2}, 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 Re

*ε*on the complex plane (the condition of the lossless effective permittivity) approaches infinity.

For illustration, in Fig. 2 we show the lower Wiener bound *ε _{lb}* on the complex plane for

*ε*

_{1}= −6 + 0.2

*i*(this corresponds approximately to the permittivity of silver at

*λ*= 430 nm) [29],

*ε*

_{2}= 2.2 − 0.03

*i*,

*ε*

_{2}= 2.2 − 0.04

*i*,

*ε*

_{2}= 2.2 − 0.05

*i*, and

*ε*

_{2}= 2.2 − 0.15

*i*, i.e., we keep

*ε*

_{1}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, ${\epsilon}_{\mathit{eff}}^{0}$ grows: as

*ε″*

_{2}→

*ε″*

_{1}

*ε′*

_{2}/

*ε′*

_{1}, ${\epsilon}_{\mathit{eff}}^{0}\to \infty $. After crossing the singularity, ${\epsilon}_{\mathit{eff}}^{0}$ changes its sign and then grows again until it reaches a peak. This happens when $\partial {\epsilon}_{\mathit{eff}}^{0}/\partial {\epsilon}_{2}^{\u2033}=0$, that yields the equation

*ε*

_{21}=

*ε′*

_{2}/

*ε′*

_{1}. After some algebra, the peak (negative) value of ${\epsilon}_{\mathit{eff}}^{0}$ is found to be

According to Eq. (4),
${\epsilon}_{\mathit{eff}}^{0}$ can vary in wide limits, from −∞ and up to +∞, but the interval [
${\epsilon}_{\mathit{eff}}^{0\mathit{max}}$, *ε′*_{2}] must be excluded. Thus, the zero value of
${\epsilon}_{\mathit{eff}}^{0}$ 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
${\epsilon}_{\mathit{eff}}^{0}$, close to
${\epsilon}_{\mathit{eff}}^{0\mathit{max}}$, are also inaccessible, because they can occur only at unreasonably large negative *ε″*_{2}.

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]

*k*

_{0}=

*ω/c*and

*k*

_{‖}is the in-plane eigenwave number which should satisfy the Rytov’s dispersion equation [34]

*d*

_{1,2}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* = *f*_{0} as in Eq. (3), at *ε*_{1} = −20 + 0.45*i* (this corresponds approximately to silver at *λ* = 660 nm) and *ε′*_{2} = 1.82 (Teflon). After that, we calculate
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$ using Eq. (4) (local approximation) and Eq. (9) for two different values of the metal layer thickness *d*_{1} 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 *d*_{1} → 0, the local approximation is good enough. At the same time, when
$\text{Re}{\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$ 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.

Although Eq. (10) has an infinite number of solutions which characterize corresponding eigenmodes [21,35], only few of them can provide
$\text{Im}{\tilde{\epsilon}}_{\mathit{eff}}^{\perp}=0$; those are the modes with either Re*k*_{‖} = 0 or Im*k*_{‖} = 0. If Re*k*_{‖} = 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 Im*k*_{‖}. It is obvious that if Im*k*_{‖} = 0 and Re*k*_{‖} ≠ 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 *f* ≡ *d*_{1}/(*d*_{1} + *d*_{2}) with *d*_{1} = 40 nm (*k*_{0}*d*_{1} = 0.38) at *ε″*_{2} = −0.04 and *ε″*_{2} = −0.05, respectively. At *ε″*_{2} = −0.04, the condition Im*k*_{‖} = 0 can be satisfied, that characterizes the propagating mode. At *ε″*_{2} = −0.05, only the condition Re*k*_{‖} = 0 can be satisfied at two different *f* values for one of the four shown eigenmodes.

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
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$, 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
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$ vs *ε″*_{2}, calculated for the propagating mode, i.e., at the points where Im*k*_{‖} = 0, are plotted in Fig. 6 for different values of *k*_{0}*d*_{1}. 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″*_{1}*ε′*_{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* = *f*_{0}, at which
$\text{Im}{\tilde{\epsilon}}_{\mathit{eff}}^{\perp}=0$, depend on the layer thicknesses. An increase in the unit cell thickness, that corresponds to stronger nonlocality, results in larger values of *f*_{0}. At the same time, the dependence of
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}({f}_{0})$ on the unit cell thickness is weak. For example, at *k*_{0}*d*_{1} = 0.38 (*d*_{1} = 40 nm), *f*_{0} = 0.954 and
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}=114.8$, while at *k*_{0}*d*_{1} = 0.76 (*d*_{1} = 80 nm), *f*_{0} = 0.975 and
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}=104.4$. This seems to be important from a practical point of view.

## 4. Phase and group velocities

It is obvious that as the refractive index is high, the phase velocity *v _{p}* =

*ω/k*

_{‖}becomes small. Here we show that the group velocity

*v*=

_{g}*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

*k*

_{0}and obtain

*c*is the the light speed in vacuum, ${\gamma}^{\u2605}=-\frac{1}{2}\left({c}_{1}{\epsilon}_{1}^{\u2605}+{c}_{2}{\epsilon}_{2}^{\u2605}+{c}_{3}{k}_{2}^{\u2605}+{c}_{4}{k}_{1}^{\u2605}\right)$, and ${c}_{1}={k}_{2}/\left({k}_{1}{\epsilon}_{2}\right)-{k}_{1}{\epsilon}_{2}/\left({k}_{2}{\epsilon}_{1}^{2}\right)$, ${c}_{2}={k}_{1}/\left({k}_{2}{\epsilon}_{1}\right)-{k}_{2}{\epsilon}_{1}/\left({k}_{1}{\epsilon}_{2}^{2}\right)$, ${c}_{3}={\epsilon}_{1}/\left({\epsilon}_{2}{k}_{1}\right)-{\epsilon}_{2}{k}_{1}/\left({\epsilon}_{1}{k}_{2}^{2}\right)$, ${c}_{4}={\epsilon}_{2}/\left({\epsilon}_{1}{k}_{2}\right)-{\epsilon}_{1}{k}_{2}/\left({\epsilon}_{2}{k}_{1}^{2}\right)$.

To solve Eq. (11) for *v _{g}*, we should first evaluate the derivatives

*dε*

_{1}/

*dk*

_{0}and

*dε*

_{2}/

*dk*

_{0}. Our estimation for silver at

*λ*= 660 nm, based on the data by Johnson and Christy [29], is

*dε*

_{1}/

*dk*

_{0}≃ 5.047 × 10

^{3}+ 137

*i*. To find

*dε*

_{2}/

*dk*

_{0}, we adopt the Lorentz model for the dye-doped dielectric permittivity of the form

*ω*=

*ω*

_{0}, the above model provides ${\epsilon}_{2}={\epsilon}_{\infty}+i\frac{\sigma}{{\mathrm{\Delta}}_{\omega}{\omega}_{0}}$, and the sought derivative, evaluated at

*ω*

_{0}, can be written as

*δ*= Δ

_{ω}/ω_{0}. 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
$\text{Re}{\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$ at which the condition
$\text{Im}{\tilde{\epsilon}}_{\mathit{eff}}^{\perp}=0$ meets. After that, we substitute the obtained values of *f* and
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$ into Eq. (11) and solve it for *v _{g}*. Doing so, we estimate the lower bound for the group velocity (

*v*), taking

_{gl}*δ*= 0.015, as well as the upper bound (

*v*), taking

_{gu}*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.

The group velocity, obtained in such a way, is, generally speaking, complex-valued. The imaginary part of *v _{g}* is responsible for pulse reshaping. However, in our case it is small as compared to its real part, and we do not show Im

*v*here. For example, at

_{g}*ε″*

_{2}= −0.404, Im

*v*/Re

_{gu}*v*≃ −0.012 and Im

_{gu}*v*/Re

_{gl}*v*≃ −0.0165. It should be also noted that if the loss is sufficiently small, Re

_{gl}*v*becomes numerically equivalent to the energy velocity [40].

_{g}## 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
${\tilde{\epsilon}}_{\mathit{eff}}^{\perp}$ (above 50) can be achieved at *f*_{0} ∼ 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 *f*_{0} and hence allows one to increase *d*_{2}.

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.

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