Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited]

: We report on the possibility of adopting active gain materials (specifically, made of fluorescent dyes) to mitigate the losses in a 3D periodic array of dielectric-core metallic-shell nanospheres. We find the modes with complex wavenumber in the structure, and describe the composite material in terms of homogenized effective permittivity, comparing results from modal analysis and Maxwell Garnett theory. We then design two metamaterials in which the epsilon-near-zero frequency region overlaps with the emission band of the adopted gain media, and we show that metamaterials with effective parameters with low losses are feasible, thanks to the gain materials. Even though fluorescent dyes embedded in the nanoshells’ dielectric cores are employed in this study, the formulation provided is general, and could account for the usage of other active materials, such as semiconductors and quantum dots.


Introduction
Metamaterials have been proposed for several innovative applications and have allowed, for example, the design of "perfect lenses" [1], and invisibility cloaks [2]. In general, however, plasmonic-based metamaterial losses at optical frequencies have been found to be significantly large, and thus have limited the application scenario. However, at infrared, ultraviolet and optical frequencies, the use of plasmonics mixed with active photonic materials has been found to be promising due to the fact that the gain experienced through the emission of a gain medium is capable of counteracting the high attenuation experienced by the electromagnetic wave due to the presence of the metal. This may indeed lead to loss-mitigated metamaterials, enabling effective permeability or low permittivity parameters at optical frequencies.
One of the key points is designing the metamaterial such that the frequency region of interest overlaps with the emission spectrum of the adopted gain medium. Different gain sources, optically pumped, could be adopted for this purpose: fluorescent dyes (e.g., Rhodamine, Fluorescein, Coumarin), semiconductor materials and quantum dots (e.g., InGaAs-GaAs quantum dots), rare earth materials (e.g., erbium).
It has been reported that the usage of the gain medium with metamaterials can provide a larger effective gain than when used alone, due to the strong local field enhancement inside the metamaterials [3,4].
Positive net gain (i.e., the gain is larger than the losses) has been shown to be possible over macroscopic distances in a dielectric-metal-dielectric plasmonic waveguide, where the gain has been provided by an optically pumped layer of fluorescent conjugated polymer (known to have very large emission cross sections) adjacent to the metal surface [5]. Also, a direct measurement of gain in propagating plasmons using the long-range surface plasmonpolariton supported by a symmetric metal strip waveguide that incorporates optically pumped dye molecules in solution as the gain medium has been shown [6]. Furthermore, optical loss compensation effects have been recently experimentally observed in [7], and [8], where Coumarin C500 and Rhodamine 6G fluorescent dyes were encapsulated into the dielectric shell of randomly dispersed nanoshell particles.
Effective parameters of metamaterials made of nanoshells with active gain materials embedded in the dielectric core, designed to operate in the visible range of the spectrum between 400 nm and 700 nm, have been simulated in [9] by artificially setting the imaginary part of the dielectric core to fixed ideal loss/gain conditions, i.e., realistic gain materials have not been considered. A detailed analysis observing the effects of the gain value in the nanoshells' core and of the density of the inclusions has been provided in [9] to investigate the tunability of such metamaterials. For 3D periodic arrays, the authors of [9] concluded that the effective permittivity can be engineered to assume both positive and negative values by selecting appropriately the lattice period and the gain value in the core of the nanoshells. In this paper we confirm the results found in [9], and moreover we show complex modes in the 3D lattice and utilize realistic parameters for the gain medium to analyze feasibility. Loss compensation of the intrinsic losses of metals at optical frequencies by using gain materials has also been proposed in [10][11][12][13][14]. In [15], it has been shown that metallic nanoparticles (nanoshells and nanorods) influence the properties of adjacent fluorophores; in that paper, the authors have shown an improvement in the quantum yield (defined here in Sec. 2.3) of the fluorophore IR800 showing the potential for contrast enhancement in fluorescence-based bioimaging. Similarly, in [16], Ruby dyes were incorporated into the dielectric core of randomly dispersed nanoshell particles, and an emission enhancement has been observed with respect to the case in absence of the metallic shell.
A computational approach including rate equations has been presented in [17] and references therein, allowing for a self-consistent treatment of a split ring resonator (SRR) array with a gain layer underneath, showing numerically that the magnetic losses of the SRR can be compensated by the gain. Rate equations have also been used in [18]. A review regarding the management of loss and gain in metamaterials has been presented in [19], and references therein.
In this paper, we provide the analysis of a loss-compensated metamaterial at optical frequencies through optical pumping. In particular, we analyze a 3D periodic array of dielectric-core metallic-shell nanospheres, assuming fluorescent dyes encapsulated into the core of each spherical nanoparticle. Each nanoshell is modeled as a single electric dipole and by its polarizability, using the single dipole approximation (SDA) [20-22] and the metal permittivity is described by the Drude model. We compute the modes following the procedure described in [21,22]. Then, also by using Maxwell Garnett homogenization theory [23,24], we compute the relative effective permittivity eff ε . Three interesting frequency regions can be outlined depending on its value: (i) one where eff ε is rather large and positive; (ii) one where eff ε is rather large and negative; and (iii) one where eff ε is close to zero (either positive or negative), also called the epsilon-near-zero (ENZ) frequency region, which has been proposed as a viable way for a number of applications including cloaking, tunneling, high directivity radiators, optic nanocircuits, etc, as reported for example in [25] and references therein. Certainly, high losses hinder the interesting properties in such frequency regions, and loss mitigation mechanisms are inherently required to overcome this issue. In this paper, we are interested in showing a formulation for loss compensation and then specifically reducing losses in the ENZ frequency region. Therefore, we design metamaterials such that the effective ENZ region overlaps with the emission spectrum of the considered dyes, and we observe that loss-compensation is feasible. Notice however that the analysis here reported does not limit the usage of gain materials to overcome the losses in other frequency regions. The structure of the paper is as follows. Mode analysis, Maxwell Garnett theory and modeling of the active gain material are introduced in Sec. 2. Then, in Sec. 3, we use two different fluorescent dyes (Rhodamine 6G and Rhodamine 800) to mitigate the losses for two particular metamaterials' designs. Conclusions are reported in Sec. 4.

Simulation model
The structure under analysis is the 3D periodic array of dielectric-core metallic-shell nanospheres reported in Fig. 1. We analyze two cases, first shells made of silver in Fig. 1(a), and then shells made of gold in Fig. 1(b). According to the experimental results in [26], gold is more lossy than silver at optical frequencies: our purpose is then to show that we can design loss-compensated metamaterials by using fluorescent dyes. The monochromatic time harmonic convention, ( )

Modal analysis for periodic arrays of plasmonic nanoshells
We model each nanoshell as a single electric dipole at optical frequencies. As such, for a plasmonic spherical particle the induced electric dipole moment is where ee α is the electric polarizability of the nanoshell, loc E is the local field produced by all the nanoshells of the array except the considered nanoshell plus the external incident field to the array, and bold letters refer to vector quantities. According to the Clausius-Mossotti approximation, the electric polarizability of a nanoshell is [20,23,31,32] where h ε is the relative permittivity of the host medium (which can be vacuum, glass, water, or any other solvent), 0 ε is the absolute permittivity of free space, The last imaginary term in Eq. (2) has been introduced to account for particle radiation [20,21]. According to Mie theory, instead, the polarizability of a nanoshell is [20] and are the core and shell relative refractive indexes. Notice that a prime in Eqs. (3) and (4) refers to the first derivative of the function with respect to its argument.
In this paper we consider dielectric-core metallic-shell particles, with 1 r ε ε = and 2 m ε ε = , where r ε is the relative permittivity of the chosen dielectric material and the metal permittivity m ε is described through the Drude model as where p ω is the plasma angular frequency, γ the damping term, and ε ∞ is a "high frequency" permittivity determined to match experimental data in the visible region. Consider now a 3D periodic array of nanoshells, immersed in a homogeneous background, with relative permittivity h ε , for which each nanoshell is placed at positions 0 n n = + r r d , where 1 2 3 , , 0, 1, 2,..., n n n n ≡ = ± ± is a triple index, and A k . In the following, we assume that the modes travel along the z direction with wavenumber z k (for the sake of brevity modes with oblique propagation direction are not considered in this feasibility study).

Effective parameters
In general, Maxwell Garnett theory [23,24] can be applied to retrieve the effective parameters of a composite medium as where z k is the wavenumber of the "dominant" mode (assuming there is one) computed from mode analysis.

Modeling of the gain material
We assume to model the gain material made of fluorescent dye molecules as a four level atomic system [17,36,37], as also proposed in [38,39], with occupation density According to chapter 2 in [36], the total displacement might be written as , r ε ε ε = + + = + D r E r P r P r E r P r (11) where ( ) r P r is the polarization contribution due to the dielectric medium hosting the gain material, and ( ) e P r is the polarization contribution due to the dispersed gain material itself, from which the effective absolute permittivity of the gain medium is The model shown in Eqs. ( ) e P r could be expressed in two slightly different ways; to avoid ambiguities and misunderstandings, we report in Eqs. (10) and (11) the expressions we used.
Under stationary regime assumption (constant electron densities in any state) with time harmonic polarization, the population inversion is where we have assumed that the electric field is small enough to neglect nonlinear saturation terms (as also discussed in [

Results of epsilon near zero composite materials with mitigated loss
Two cases are assumed, the first one made by silver shells (Fig. 1(a)), the second one made by gold shells (Fig. 1(b)), thus exhibiting resonance at lower frequency.

Case with silver shells
We assume that Rhodamine 6G (R6G) dye molecules are into the dielectric core as in Fig.  1  In this section, we adopt the structure in Fig. 1(a). The outer shell radius is 2    As stated in Sec. 2.2, by using Eq. (8), it is possible to compute the effective refractive index of the homogenized array from the wavenumber information reported in Fig. 2. Then, we observed by using Eq. (7) with the magnetic polarizability of a nanoshell that the effective