Control of gain/absorption in tunable hyperbolic metamaterials

Bartosz Janaszek; Anna Tyszka-Zawadzka; Paweł Szczepański

doi:10.1364/OE.25.013153

1. Introduction

Hyperbolic metamaterials are uniaxial structures characterized by the permittivity tensor of the diagonal form [ε] = diag(ε_||,ε_||,ε_⊥). The principal components have the opposite signs ε_||>0, ε_⊥<0 (Type I), ε_||<0, ε_⊥>0 (Type II) [1], which leads to unclosed (hyperbolic-type) surface of dispersion in space of wave vectors, known as hyperbolic dispersion. Moreover, this kind of structures can exhibit Epsilon-Near-Zero (ENZ) behavior, i.e., effective permittivity achieving value near zero [2], which can be utilize in harmonic generation [3,4], perfect absorption [5], solitons, freezing light [6], supercoupling [7] and shaping radiation pattern of the source [8]. Further applications can be found when it comes to Type I and Type II hyperbolic dispersion offering existence of unique metamaterial states with large magnitude wavevectors (the high-k states) which are evanescent and decay exponentially in conventional media [1,9]. This feature gives rise to many device applications, such as hyperlenses [10], superlenses [9], enhanced light emission [11–13], quantum information systems [14], biochemical sensing [15], plasmon waveguiding [16], and many others as e.g., broadband super absorbers and thermal emitters [9,17].

In particular, the important consequence of unbounded dispersion of Type I and Type II HMM is a divergent photonic density of states resulting in an enhancement in the spontaneous emission of dipole emitters placed in the vicinity of HMM leading to a metamaterial-based broadband Purcell effect [18], in contrast to conventional methods based on optical resonators [19], photonic crystals cavities [20] and plasmonic aperture [21], which are all bandwidth limited. This concept has been theoretically and experimentally investigated by many research groups, and they have observed a large modification (including decay rate and pattern) of spontaneous emission of emitters (organic dyes and quantum dots) localized nearby HMM or embedded inside HMM [13].

Generally, gain-enhanced metamaterial structures have been intensively discussed in means of enhancing active imaging, ultrafast nonlinearities as well as cavity-free nanolasing [22,23]. However, studies concerning HMM structures composed of active material with optical gain were limited only to loss compensation in bulk [24] and waveguide [25] structures. Recently, it has been experimentally demonstrated that active HMM structure, i.e., gold nanorod array coated in thin film of optical gain medium composed of Rhodamine 101 doped polyvinyl alcohol (PVA), can be utilized to enhance lasing action, which could serve as a platform to achieve broadband coherent photon sources [26].

On the other hand, Tunable Hyperbolic Metamaterials (THMM) based on stimulus-sensitive functional materials allowing for active control of structure’s properties by external perturbations, i.e., electric/magnetic field or temperature, have recently attracted a widespread attention of the researchers. In this case, possibility of controlling dispersion shape of such structures gives rise to new potential applications. Especially, main efforts have been put on graphene/dielectric multilayer nanostructures showing potential for designing efficient absorbers [27–31], photonic switches [32,33], optical modulators based on reflection [34] as well as stopped light effect [35], all controllable by external electric field. Recently it has been shown that, in this kind of structures the control of dispersion regimes, i.e., Type I HMM, elliptic, Type II HMM, using external biasing is possible [36]. Moreover, the manufacturing THMM structures is possible by use of functional materials, such as semiconductors [37], conductive oxides [38], liquid crystals [39], phase-change materials [40] or gyromagnetics [41].

In this paper we propose a special class of THMM structure based on bilayers consisting of a material possessing optical gain/absorption and voltage-sensitive graphene providing tunability of complete structure. The presented analysis is focused on effects resulting from resonant electromagnetic response of complete THMM structure. In particular, it has been revealed that the change of dispersion from Type I HMM to elliptic, accompanied by sharp resonance transition of effective permittivity, leads not only to significant increase of effective gain/absorption, but also narrowing of effective bandwidth with respect to constituent gain/absorptive material. Furthermore, the wavelength of the gain/absorption maximum can be selected by proper voltage biasing. It has been also shown that the other transitions, i.e., elliptic → Type II HMM and elliptic → Type I HMM, offer two opposite effects: gain/absorption enhancement and electromagnetic transparency, respectively, both controlled by external voltage. All of considered effects are anisotropic and sensitive for wave propagation direction.

2. Theory

The single unit cell of considered THMM structure is composed of a graphene layer and a gain/absorption layer with thicknesses t_graph, t_g/ab and complex permittivities ε_graph, ε_g/ab, respectively, where their imaginary parts contribute to loss (Im(ɛ) > 0) or gain (Im(ɛ) < 0), see Fig. 1. Moreover, dispersion of both permittivities has been taken into account. Without losing the generality, the gain\absorptive material is characterized by well-defined permittivity of thulium-doped silicate glass. The considered structure consists of 6 monolayers of graphene (t_graph = 2.1 nm) and 10 nm layer of thulium doped silicate glass. What is worth noting, the thicknesses of layers have been selected to ensure existence of all three dispersion regimes [36].

Fig. 1 Scheme of considered structure.

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According to [42], L.A. Falkovsky and A. A. Varlamov, the graphene’s effective permittivity can be written as:

ε_{g r a p h} = 1 - j \frac{σ (ω, μ_{c})}{ω ε_{o} t_{g r a p h}},

where ɛ₀ is the vacuum permittivity and σ is the conductivity of a single-layer graphene. Moreover, frequency and chemical potential dependent conductivity of a monolayer graphene can be given by Kubo formula [42]:

σ (ω, μ_{c}) = - \frac{j 4 π q^{2} k_{B} T}{h^{2} (ω - j 2 τ)} [\frac{μ_{c}}{k_{B} T} + 2 \ln (e^{- μ_{} / k_{B} T} + 1)] + - \frac{j 4 π q^{2} (ω - j 2 τ)}{h^{2}} \int_{0}^{\infty} \frac{f_{D} (- ξ) - f_{D} (ξ)}{{(ω - j 2 τ)}^{2} - 16 (\frac{π ξ}{h})} d ξ,

where f_D(ξ) is the Fermi-Dirac function: f_D(ξ) = [exp(ξ-μ_C/k_BT) + 1]⁻¹, μ_c is the chemical potential, T is temperature, k_B and h are Boltzmann and Planck’s constant, ω is the angular frequency of the incident wave, and τ is the phenomenological scattering rate, which we set equal 0.1 meV. Moreover, conductivity of multilayer graphene (up to 6 monolayers; N_graph ≤ 6) can be described by σ_ml = σ∙N_graph, where N_graph is the number of graphene layers [43]. Furthermore, chemical potential of graphene can be changed by applying external voltage, accordingly to following formula [32]:

| μ_{c} | = ℏ υ_{F} \sqrt{π | a_{0} (V_{g} - V_{d i r a c}) |},

where ħ is Dirac constant, υ_F is the Fermi velocity of Dirac fermions in graphene (∼10⁶ m/s), a₀ = 9 × 10¹⁶ m⁻¹V⁻¹, V_dirac is offset bias which reflects graphene’s doping and/or its impurities. This description of conductivity is consistent to the assumption that the electronic band structure of a graphene sheet is not affected by the neighboring graphene sheets [43]. It is worth noting that, more accurate model of electrical tuning for 6-layer graphene would change the relationship between voltage and chemical potential, but will not influence obtained functionality. In our analysis we assumed pristinity of graphene monolayers, i.e., V_dirac = 0. Moreover, maximal value of biasing voltage does not exceed V_gmax = 2 V to ensure elimination of tunneling between graphene layers separated by 10 nm thick layer of thulium-doped glass material [44].

In our analysis we assume that the real part of permittivity of silica glass lowly-doped with thulium ions (1000 ppm) is determined by pure silica glass and described by empirical Sellmeier equation [45]:

Re (ε_{g / a b} (λ)) \approx 1 + \frac{a_{1} λ^{2}}{λ^{2} - l_{1}^{2}} + \frac{a_{2} λ^{2}}{λ^{2} - l_{2}^{2}} + \frac{a_{3} λ^{2}}{λ^{2} - l_{3}^{2}}

where,

a_{1}

= 012212122

l_{1}

= 0.0684043,

a_{2}

= 0.4079426,

l_{2}

= 0.1162414,

a_{3}

= 0.8974794,

l_{3}

= 9.896161 are empirical coefficients describing dispersion of fused silica [45]. Moreover, the imaginary part of permittivity, responsible for gain or absorption, is determined by resonances of thulium ions and can be calculated on the basis of the McCumber theory [46]:

g (λ) = N \cdot σ_{e m} (λ),

α (λ) = - N \cdot σ_{a b s} (λ),

where N is ion density and equals to 1000 ppm, which is typical value for thulium doped media corresponding to 3.1∙10²⁵ ions/m³ [47]. and σ_em/abs is emission/absorption cross-section described for each optical transition by following formula [48]:

σ_{e m / a b s} (λ) = \sum_{k = 1}^{4} α_{k} \exp (- 2 {(\frac{λ - λ_{k}}{Δ λ_{k}})}^{2}) .

Our analysis is limited only to ${}^{3}F_{4} \to {}^{3}H_{6}$ (emission) and ${}^{3}H_{6} \to {}^{3}F_{4}$ (absorption) transitions covering spectral range from 1.5 μm to 2 μm and described by appropriate coefficients (see Table 1 in [48], S. D Jackson and T. A. King). Then, the imaginary part of permittivity is calculated by employing following relation:

Im (ε_{g / a b}) = g (λ) \cdot λ \cdot \sqrt{Re (ε_{g / a b})} .

In general, above assumptions are justified by the fact that the imaginary part of pure silica glass permittivity is negligible (in order of 10⁻⁸) in considered wavelength range, i.e., 1.5 ÷ 2 µm [45]. Moreover, the correction to real part of permittivity resulting from imaginary part appearing due to gain or losses introduced by thulium ions, and estimated with the help of Kramers-Kronig relation, is also insignificant (it is in order of 10⁻⁶).

Electromagnetic response of considered complete multilayer structure can be predicted on the basis of properties of permittivity tensor. In our analysis the components ɛ_⊥ (i.e., ɛ_zz – Fig. 1), ɛ_|| (i.e., ɛ_xx = ɛ_yy – Fig. 1) of the effective diagonal permittivity tensor are obtained using Effective Medium Theory:

ε_{| |} = \frac{t_{g r a p h} ε_{g r a p h} + t_{g / a b} ε_{g / a b}}{t_{g r a p h} + t_{g / a b}},

ε_{⊥} = \frac{ε_{g r a p h} ε_{g / a b} (t_{g r a p h} + t_{g / a b})}{t_{g r a p h} ε_{g / a b} + t_{g / a b} ε_{g r a p h}},

where t_g is the thickness of graphene layer and ε_g is the graphene’s permittivity. It is assumed that the thickness of gain material/absorber layers are deep subwavelength but thick enough to avoid interaction between graphene layers [44].

Finally, the effective gain and absorption for waves propagating in complete THMM structure has been calculated with use of permittivity-based model:

g_{| |, ⊥} (λ) = \frac{Im [ε_{| |, ⊥} (λ)] \cdot λ}{\sqrt{Re [| ε_{| |, ⊥} (λ) |]}}

α_{| |, ⊥} (λ) = - \frac{Im [ε_{| |, ⊥} (λ)] \cdot λ}{\sqrt{Re [| ε_{| |, ⊥} (λ) |]},} .

where g_||/α_|| and g_⊥/α_⊥ describe effective gain/absorption encountered by waves propagating in “x-y” plane and along “z” axis, respectively. The characteristics revealing effective gain and absorption of complete structure are presented in next section.

3. Results and discussion

Figure 2 illustrates the real part of permittivity tensor components ɛ_|| (red curve) and ɛ_⊥ (blue curve) plotted versus wavelength with no biasing voltage V_g = 0 V. As we can see, for certain wavelength ranges different dispersion regimes, i.e., elliptic dispersion (ED), Type I and Type II HMM, can be obtained.

Fig. 2 Real part of effective permittivity tensor components $ε$ _|| (red curve) and $ε$ _⊥ (blue curve) plotted vs. wavelength for biasing voltage V_g = 0 V, where $λ_{0}^{(1, 3)}$ and $λ_{0}^{(2)}$ denote the resonance wavelengths of $ε$ _⊥ and $ε$ _||, respectively.

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Transitions between certain regimes are characterized by resonance wavelengths (i.e., the wavelengths for which respective component of the effective diagonal tensor equaled zero) denoted by $λ_{0}^{(1)}$ (Type I HMM → ED; ɛ_⊥ = 0, ɛ_||>0), $λ_{0}^{(2)}$ (ED → Type II HMM; ɛ_|| = 0, ɛ_⊥>0) and $λ_{0}^{(3)}$ (ED → Type I HMM; ɛ_⊥ = 0, ɛ_||>0). It is worth noting, that these resonance wavelengths can be shifted by changing the biasing voltage [36].The further analysis concerns voltage-controlled effective gain/absorption properties of complete THMM structure. First, we focus on effective gain and absorption observed for the vicinity of the transition from Type I HMM to elliptic dispersion regime with resonance wavelength $λ_{0}^{(1)}$ , see Fig. 2.

Figures 3(a) and 3(b) illustrate the spectral dependencies of effective gain g_⊥ (encountered by waves propagating along “z” axis, i.e., perpendicularly to the structure layers - blue curves) and g_|| (encountered by waves propagating in “x-y” plane - red curves) for different values of biasing voltage V_g, as well as spectral dependency of gain of constituent standalone active material (green curve).

Fig. 3 Spectral dependencies of effective gain/absorption (g_⊥, g_|| / α_⊥, α_||) corresponding to resonance wavelengths $λ_{0}^{(1)}$ for different values of biasing voltage as well as gain/absorption (g/α) of constituent material: (a) g_⊥ (left axis; blue curves) for V_g = 0,3,8 mV and g (right axis; solid green curve); (b) g_|| (left axis; red curves) for V_g = 0,3,8 mV and g (right axis; solid green curve); (c) α_⊥ (left axis; blue curves) for V_g = 11,13,15 mV and α (right axis; dashed green curve); (d) α_|| (left axis; red curves) for V_g = 11,13,15 mV and α (right axis; dashed green curve).

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First of all, we can notice strong anisotropy of gain, i.e., g_⊥ achieves much higher values than g_||. The effective gain g_|| is approximately one order of magnitude greater in relation to gain of constituent material, while their bandwidths do not differ much, see Fig. 3(b). What is more, the biasing voltage does not influence g_|| The situation changes dramatically for g_⊥, which behaves in very interesting manner. In this case, the maximum of effective gain g_⊥ exceeds the gain of constituent material a few orders of magnitude, while the effective bandwidth is two orders of magnitude narrower. These effects result from sharp resonance transition of Re(ɛ_⊥) observed for the wavelength $λ_{0}^{(1)}$ at which Type I HMM dispersion changes into elliptic, see Fig. 2, since the resonance transition shape of Re(ɛ_⊥), by Kramers-Kronig relation, mostly determines Im(ɛ_⊥) and both parts of complex permittivity ɛ_⊥ contribute to effective gain g_⊥. It is worth underlining that due to the fact that resonance wavelength $λ_{0}^{(1)}$ can be shifted by applying external voltage, thus we can select, by proper biasing, the wavelength for which the maximum effective gain appears. Naturally, the range of tunability is limited by gain bandwidth of constituent material as well as tunable properties of employed THMM structure [36]. Sufficient value of voltage biasing enables also to shift the resonance wavelength $λ_{0}^{(1)}$ to the spectral range where absorption of constituent material occurs. Figures 3(c) and 3(d) show spectral dependencies of effective absorption α_⊥ and α_||, respectively, for different values of biasing voltage, and absorption of constituent material α illustrated by dashed green curve. Once again, we can observe effects characteristic for the sharp resonant transition, i.e., narrowing of the bandwidth and significant increase of the maximum, appearing for α_⊥ see Fig. 3(c). Similarly, we can select, by proper voltage biasing, the wavelength for which maximum effective absorption α_⊥ appears. Again, tunability is limited by loss bandwidth of constituent material and tunable properties of employed structure. What is more, the spectral behavior of α_|| remains similar to absorption of constituent material and does not depend on applied voltage. Moreover, α_|| achieves much lower values than α_⊥ and the structure reveals loss anisotropy, see Figs. 3(c)-3(d). Analogous characteristics revealing spectral dependencies of effective gain and absorption are obtained for the transition with the resonance wavelength $λ_{0}^{(2)}$ , i.e., ED → Type II HMM, see Figs. 4(a)-4(d).

Fig. 4 Spectral dependencies of effective gain/absorption (g_⊥, g_|| / α_⊥, α_||) corresponding to resonance wavelengths $λ_{0}^{(2)}$ for different values of biasing voltage as well as gain/absorption (g/α) of constituent material: (a) g_⊥ (left axis; blue curves) for V_g = 0.55,0.75,0.9 V and g (right axis; solid green curve); (b) g_|| (left axis; red curves) for V_g = 0.55,0.75,0.9 V and g (right axis; solid green curve); (c) α_⊥ (left axis; blue curves) for V_g = 1.1,1.2,1.4 V and α (right axis; dashed green curve); (d) α_|| (left axis; red curves) for V_g = 1.1,1.2,1.4 V and α (right axis; dashed green curve).

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In contrast to the characteristics for the previously discussed transition, g_|| and α_|| reveal controllable resonant-like and are greater than g_⊥ and α_⊥, respectively. In this case g_⊥ and α_⊥ exhibit similar spectral dependence as gain and absorption of constituent material and they are practically insensitive to the applied voltage. Although, the effective gain g_|| and absorption α_|| peaks are significantly lower and broader than that obtained for g_⊥ and α_⊥ for the Type I HMM → elliptic transition, but still considerable enhancement, with respect to constituent material, is observed. Once more, anisotropy of the THMM structure is confined.

Finally, the spectral dependencies of effective absorption α_|| and α_⊥, observed for the third transition (i.e., ED → Type I HMM) with the resonance wavelength $λ_{0}^{(3)}$ , are shown in Figs. 5(a)–5(b). The α_|| possess similar dependence as in the case of transition with $λ_{0}^{(1)}$ , i.e., the same bandwidth as the constituent material and loss enhancement. However, α_⊥ behaves in different and interesting manner. We observe effective absorption α_⊥ equaled zero for resonance wavelength $λ_{0}^{(3)}$ , which leads to the electromagnetic transparency. Again, by proper biasing we can select the wavelength at which the structure becomes lossless. We can expect that the same transparency effect for this resonance transition will be observed in THMM structure with gain, however achieving this effect requires shifting $λ_{0}^{(3)}$ to the gain band of the constituent material.

Fig. 5 Spectral dependencies of effective absorption (α_⊥, α_||) corresponding to resonance wavelengths $λ_{0}^{(3)}$ for different values of biasing voltage as well as absorption (α) of constituent material: (c) α_⊥ (left axis; blue curves) for V_g = 0,0.5,1 mV and α (right axis; dashed green curve); (d) α_|| (left axis; red curves) for V_g = 0,0.5,1 mV and α (right axis; dashed green curve).

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4. Conclusions

To conclude, we show that THMM structure composed of active material and functional material offers possibility of controllable modification of gain/absorption bandwidth of constituent active medium as well as controllable electromagnetic transparency. Hyperbolic metamaterials, in general, exhibits broadband enhancement effect, confirmed in many scientific studies [12–14,18], which are consistent with our results. Our study also divulges interesting effects appearing in the vicinity of resonance wavelengths connected with transitions between different dispersion regimes. In particular, in case of Type I HMM → ED transition we can observe strong enhancement of effective gain or absorption, encountered by waves propagating along “z” axis, with significantly narrowed bandwidth and weaker broadband enhancement in “x-y” plane. The situation changes for the ED → Type II HMM transition where broadband gain/loss enhancement occurs for wave propagating along “z” axis and stronger enhancement with narrowed bandwidth is obtained for propagation in plane ”x-y”. Other distinct property appears for the third transition, i.e., ED → Type I, HMM which reveals electromagnetic transparency for wave propagating along “z” axis, while for “x-y” we observe moderate broadband enhancement effect. What is noteworthy, by proper biasing of complete structure we can select the wavelength at which the maximum effective gain or absorption appears for waves propagating perpendicularly or parallel to the structure’s layers depending on the transition between certain dispersion regimes. The voltage controlling occurs for resonant transitions for which we observe narrowing of the effective gain/loss bandwidth. Similarly, it is also possible to select wavelength at which electromagnetic transparency appears, while the third transition is considered.

It is worth to underline that, all described effects are a result of dispersion regimes transitions, achievable in HMM structures based on various materials. Thus, replacing graphene by any stimulus-sensitive material providing negative permittivity (e.g., vanadium oxide [37], indium tin oxide [38], nematic liquid crystal [39]) and thulium-doped silica glass by any absorptive/amplifying dielectric material (e.g., dye-doped PMMA) should not influence desired functionality.

In particular, the manufacturing of layered structures consisting of several graphene sheets separated by dielectric layers is technologically achievable by using, e.g., well-developed CVD, MBE methods and cleavage techniques [49,50]. This opens possibility of realization a new class of tunable active/passive media based on THMM structures composed of absorptive/amplifying material paving foundation for platform of novel controllable photonic devices.

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Control of gain/absorption in tunable hyperbolic metamaterials

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusions

References

Cited By

Figures (5)

Equations (12)

Optics Express