## Abstract

The tunability of slow light in graphene-based hyperbolic metamaterial waveguide operating in SCLU telecom bands is investigated. For the first time it has been shown that proper design of a GHMM structure forming waveguide layer and the geometry of the waveguide itself allows stopped light to be obtained in an almost freely selected range of wavelengths within SCLU bands. In particular, the possibility of controlling light propagation in GHMM waveguides by external biasing has been presented. The change of external electric field enables the stop light of the selected wavelength as well as the control of a number of modes, which can be stopped, cut off or supported. Proposed GHMM waveguides could offer great opportunities in the field of integrated photonics that are compatible with CMOS technology, especially since such structures can be utilized as photonic memory cells, tunable optical buffers, delays, optical modulators etc.

© 2017 Optical Society of America

## 1. Introduction

Phenomena of slow and stopped light have, in the last decade, attracted a growing interest due to a wide range of potential applications, including ultra-compact optical modulators and switches, enhanced optical nonlinearities, optical buffering, memory devices etc [1–3]. Initially, slow light was observed in Bose-Einstein condensates [4], Ruby crystals [5], ultra-cold atomic gases [6] and in solid media [7]. More recently, the new mechanisms of achieving slow and stopped light in optical waveguides have been developed [8]. Especially, this effect has been observed in photonic crystal waveguides [8–11]. Another solutions considered for this purpose are plasmonic [12–15] and negative-refractive–index (NRI) metamaterial waveguides [16–22]. In particular, it has been demonstrated both experimentally and theoretically that slow light effect can be achieved in various waveguide configurations, including single-negative [21] as well as double negative materials [18]. Moreover, the general concept of ‘rainbow trapping’ in negative index material (NIM), allowing for slowing or stopping of light over extremely large bandwidths, has been proposed in Ref [18]. What is more, it has been shown, that gain-enhanced nanoplasmonic metamaterials have been investigated in means of improved active imaging, ultrafast nonlinearities as well as cavity-free lasing [23]. Alternatively, the implementation of hyperbolic metamaterial (HMM) in the waveguide structures allows for field enhancing [24], plasmon propagation [25] as well as slowing and stopping light [26–28]. It is also worth to note, that the technology of this kind of materials is available.

In general, hyperbolic metamaterial represents a special class of uniaxially anisotropic metamaterials, realized mostly by sub-wavelength metal-dielectric subsequent layers, which can be described by a diagonal permittivity tensor $\overline{\overline{\epsilon}}=diag[{\epsilon}_{\left|\right|},{\epsilon}_{\left|\right|},{\epsilon}_{\perp}]$, where the principal components have the opposite signs, i.e., *ε _{||}* > 0,

*ε*

_{⊥}< 0 (Type I of dielectric character) or

*ε*< 0,

_{||}*ε*

_{⊥}> 0 (Type II of metallic character) [29–31].

Because of strong anisotropy and hyperbolic dispersion characteristic of HMM, it is shown that a waveguide with core layer formed by Type II HMM provide anomalous modal dispersion leading to superluminal and stopped light [26–28]. This effect is a consequence of TM modal degeneration resulting in the presence of two modes with contrary energy flows. Moreover, there exists certain core layer width for given wavelength for which two propagation constants are identical and energy flows are mutually compensated leading to effective light trapping (critical point of dispersion characteristics).

New functionality of HMM structures can be gained by substituting metal layer by graphene, so called Graphene-based HMM (GHMM) [32–40]. Since the conductivity of single layer graphene is frequency and chemical potential dependent [41,42], the effective tensor components of graphene/dielectric metamaterial structure also reveal similar dependence. Thus, by controlling chemical potential of graphene it is possible to change optical properties of this kind of structure. It is worth noting that, in general 2D materials, such as graphene, provide controllable plasmon propagation over long distances [43,44].

Recently, a novel GHMM-based waveguide has been proposed and numerically analyzed in context of local field enhancement [45], controlled plasmon propagation [46] as well as controlled light slowing in THz region [47].

The study presented in [47] concerned the tunable slow-light waveguide controlled by voltage or temperature and operating in THz range. Especially, the phenomenon of controlling speed of light from slow to fast may find exciting applications in photonic switch, optical buffers and memory devices operating in telecom frequency range.

In this paper we provide detailed study of slow light inside a graphene-based hyperbolic metamaterial (GHMM) waveguide operating in SCLU telecom bands. The analyzed structure is a symmetric planar waveguide composed of Type II GHMM core and air cladding. In our approach, we assume that both of the permittivity tensor components (i.e., *ε _{||}*,

*ε*

_{⊥}) depend on frequency and GHMM structure parameters (i.e., thickness of dielectric and graphene layers) as well as higher order modes propagation is analyzed, in contrast to [47] where the perpendicular permittivity of GHMM,

*ε*was taken to be constant and study was confined to fundamental oscillatory/bulk mode propagation. Moreover, without losing the generality, the considered structure is assumed to be lossless since it does not affect the existence of slow light phenomenon [14].

_{||},Our numerical results revealed possibility of voltage-controlled stopped light of selected wavelength within the SCLU bands range for certain waveguide width. Another proposed approach corresponds to employing waveguide with varying width (i.e., tapered waveguide) which allows for switchable light stopping of all wavelengths within the range of SCLU bands.

## 2. Theoretical model

Figure 1(a) shows an air/GHMM/air planar waveguide with schematically depicted oscillatory mode profile considered in this work. The GHMM consists of subsequent ultrathin graphene-dielectric stacks that single basic cell is composed of a graphene sheet and a dielectric layer (SiO_{2}, *ɛ*_{d} = 2.08). In considered structure we assume “*z*” axis as a propagation direction.

The multilayer metamaterial can be treated as a homogeneous effective medium identified by the permittivity tensor:

**(**

*ǁ**x*,

*y*-axes) and ⊥ (

*z*-axis) refer to the directions parallel and perpendicular to the interfaces in multilayer realization, respectively (see Fig. 1).

As the multilayer structure is subwavelength-scaled (i.e., the thickness of the unit cell is much less than the wavelength of light) the effective medium theory (EMT) can be applied to describe the optical properties of GHMM (so called effective medium homogenization model). By applying EMT method the diagonal components of the relative permittivity tensor are given as follows [25]:

*t*

_{g}and

*ε*

_{g}are the thickness and dielectric permittivity of graphene layer in the unit cell,

*t*

_{d}and

*ɛ*

_{d}are the thickness and dielectric permittivity of dielectric layer, respectively. In contrast to model presented in [47], our approach predicts the influence of structure parameters on tensor components (see Eq. (2), since thicknesses of constituent layers are comparable. It is worth noting that our considerations are limited to real permittivity of dielectric layers, due to the fact that, in general, losses of constituent materials does not prevent slowing light effect from occurring [14].

According to Ref [39]. the graphene’s effective permittivity can be written as:

where*ɛ*

_{0}is the vacuum permittivity and

*σ*is the conductivity of a single-layer graphene. Frequency and chemical potential dependent conductivity of a monolayer graphene can be given by Kubo formula [48]:

*f*

_{D}

*(ξ)*is the Fermi-Dirac function:

*f*(

_{D}*ξ*) = [exp(

*ξ*-

*μ*/

_{C}*k*) + 1]

_{B}T^{−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 electromagnetic wave, and

*τ*is the phenomenological scattering rate, which we set equal 0.1 meV. In case of a few layer graphene its multilayer conductivity can be described by

*σ*

_{ml}=

*σ*∙

*N*, where

_{g}*N*is the number of graphene layers. Considered model can be exploited for

_{g}*N*

_{g}≤ 6 [35].

A change of graphene’s chemical potential and simultaneously the optical properties of the GHMM can be achieved by gate voltage *V*_{g}. Relationship between gate voltage and chemical potential can be given by the following formula [35]:

where *ħ* is Dirac constant, *υ*_{F} is the Fermi velocity of Dirac fermions in graphene (∼10^{6} m/s), *a*_{0} = 9 × 10^{16} m^{−1}V^{−1}, *V*_{dirac} is offset bias which reflects graphene’s doping and/or its impurities.

Considering a symmetric planar GHMM waveguide described by effective parameters (i.e., *ɛ _{||}, ɛ*

_{⊥}) (see Fig. 1.) with an air cladding layer (

*ε*

_{1}= 1) and a graphene-based hyperbolic metamaterial as a core layer, the propagation constant

*β*of the transverse-magnetic (TM) oscillatory modes can be obtained by solving eigenmode equations [26].

*W*is the width of the waveguide.

Using above equations, Eqs. (6) and (7), the dispersion characteristics of symmetric GHMM waveguide, revealing the possibility of slow-light effect, are obtained.

## 3. Results and discussion

We present the dispersion characteristics of the analyzed waveguide with GHMM core layer. The propagation constants of the waveguide modes for different core-layer widths are obtained by solving the eigenmode equations given by Eqs. (6) and (7). Since the slow-light effect is achievable in waveguides with Type II HMM core, the first step of our analysis covers obtaining this kind of GHMM within SCLU bands.

As we presented in [40] the Type II of GHMM structure (i.e., *ε*_{||}<0, *ε*_{⊥} >0) is achievable by proper tailoring of the parameters of basic cell (i.e., thickness and permittivity of dielectric layer, number of graphene layers), as well as applying external electric field. This is shown in Figs. 2(a) and 2(b) where the permittivity tensor components, *ε _{||}* and ε

_{⊥}, are plotted as a function of the gate voltage

*V*

_{g}for the boundary wavelengths of SCLU bands, i.e.,

*λ*= 1.46 µm and

*λ*= 1.675 µm, which are illustrated by blue and red line, respectively. In our case, the basic cell of hyperbolic metamaterial consists of SiO

_{2}with thickness

*t*

_{d}and permittivity

*ε*

_{d}= 2.08 as well as graphene monolayers, which number is denoted by

*N*

_{g}.

In particularly, the Type II is achievable for the structure consisted of dielectric layer with *t*_{d} = 6 nm and six graphene monolayer (*N*_{g} = 6), at the gate voltage *V*_{g} = 0.76 V for the wavelength *λ* = 1.46 µm and at the gate voltage *V*_{g} = 0.44 V for the wavelength *λ* = 1.675 μm, see Figs. 2(a) and 2(b). When we change the geometry of the basic cell (*t*_{d} = 8 nm, *N*_{g} = 4), the gate voltage required to achieve Type II is higher and for *λ* = 1.46 µm takes value *V*_{g} = 1.57 V, while for *λ* = 1.675 µm is equal *V*_{g} = 0.89 V.

The next two figures, Figs. 3(a) and 3(b) show the dispersion characteristics of the waveguide with GHMM core layer for the fundamental mode TM_{0} and different parameters of the basic cell. First of all, we can notice that waveguide of given width layer supports two different propagation constants for each TM mode corresponding to: the forward mode, whose energy flow and wave vector are in the same directions (lower branch), and backward mode, whose energy flow and wave vector are in opposite directions (upper branch). Moreover, for a given core layer width, a critical point appears in which two propagation constants are identical and power flows are mutually compensated leading to effective light stopping (trapping), see Figs. 3(c) and 3(d). In these figures, normalized total power flow *P*, defined by $P=\left({P}_{1}+{P}_{2}\right)/\left({\left|P\right|}_{1}+\left|{P}_{2}\right|\right)$, where ${P}_{1}=2{\displaystyle {\int}_{d/2}^{\infty}{S}_{z}dx}$, ${P}_{2}={\displaystyle {\int}_{-d/2}^{d/2}{S}_{z}dx}$ and *S _{z}* is the z-component of the Poynting vector, is shown.

The negative power flow, *P* < 0, is related to backward modes possessing propagation constants corresponding to the upper branch of dispersion characteristic, i.e., above the critical point, while *P* > 0 is connected with the lower branch, i.e., below critical point, see Figs. 3(a)-3(d), and P = 0 at the critical point. Furthermore, an increase in the number of graphene layers (for given thickness of the dielectric layer) leads to the shift of the critical point towards greater values of waveguide width, see Fig. 3(a). On the contrary, with the increasing thickness of the dielectric layer the light stopping appears for lower values of waveguide width, see Fig. 3(b).

From a technological point of view we should look for such solutions for which light stopping could occur at the possibly largest values of the waveguide width, i.e., possibly large *N _{g}* and simultaneously small

*t*. On the other hand the dielectric layer should not be too thin to avoid the tunneling effect. Thus, for further discussion we will concentrate on the waveguide with GHMM core layer consisting of subsequent bilayers of 6 nm SiO

_{d}_{2}layer (no tunneling effect [49]) and six graphene monolayers.

In Figs. 4(a) and 4(b) the dispersion curves for the wavelengths within the SCLU bands are presented, where blue and green curves correspond to the boundary wavelengths of this range, i.e., *λ* = 1.46 µm and *λ* = 1.675 µm, respectively. It is shown that the proper biasing, in this case *V*_{g} = 1 V in Fig. 4(a) and *V*_{g} = 2 V in Fig. 4(b), as well as fabrication of the tapered waveguide with varying width including the range from 131 nm to 46 nm ([Fig. 4(a)]) and from 198 nm to 127 nm ([Fig. 4(b)]), allow to stop light in the whole SCLU range.

Figure 5 shows the dispersion curves for TM_{0} mode of the considered waveguide structure at λ = 1.55 μm and different biasing. It is worth noting that by changing gate voltage we can select the width of the waveguide for which the light stopping (critical point) can be observed. It means that for given waveguide width, by proper biasing, we can stop (slow down), cut off or provide propagation of the mode. In particularly, for the structure parameters and biasing range presented in Fig. 5, this effect can be obtained for the waveguide width within the range from *W* = 155 nm to *W* = 75 nm. Moreover, by proper tailoring of basic cell geometry and layer parameters this phenomenon can be obtained for different desirable wavelengths.

Furthermore, in the waveguide structure with defined width, it is also possible to control wavelength of stopped light by voltage biasing. This is illustrated in Fig. 6, where the dependence of the stopped light wavelength on the biasing voltage for the waveguide width *W* = 0.12 μm is shown. Particularly, in this structure sweeping voltage bias from 0.9 V to 1.82 V allows to stop light in the whole SCLU range. Moreover, by proper tuning of the biased voltage, it is possible to select wavelength for which the stopping light effect is desirable.

Finally, the dispersion characteristics of the same waveguide structure, for the first four guided modes, the selected wavelength (*λ* = 1.55 μm), and different biasing voltages, are presented in Figs. 7(a)-7(d). As we can see, if the structure is manufactured as a tapered waveguide, with varying width including the range from 620 nm to 8 nm, the biasing provides control of the number of modes which can be stopped, i.e., *m* = 0 for *V*_{g} = 630 mV, *m* = 0-1 for *V*_{g} = 660 mV, *m* = 0-2 for *V*_{g} = 700 mV and finally *m* = 0-3 for *V*_{g} = 800 V. What is worth noting, the higher order mode is stopped for greater values of the waveguide width.

## 4. Conclusions

In this paper we investigate slow-light phenomenon in the GHMM waveguide operating in SCLU telecom bands. The considered structure is a symmetric planar waveguide composed of a Type II GHMM core and air cladding. It has been shown that proper design of GHMM structure forming waveguide layer and geometry of the waveguide itself allows for shaping the waveguide dispersion characteristics by applying external biasing. In particular, for the given width of waveguide it is possible to control the propagation of fundamental mode, i.e., its stopping, supporting or cutting off, as well as to shift critical point (stopping light) for desired wavelength within SCLU bands by employing the proper biasing voltage. Moreover, the realization of a tapered waveguide enables to stop light in the whole SCLU wavelength region and select the number of guided modes which can be stopped. The features described above can lead to potential applications of the proposed waveguides in any application requiring control of light propagation, especially it can be utilize in tunable optical buffers and photonic memory cells as well as optical switches and modulators. It is worth noting, that ultrafast response of complete device, due to graphene relaxation time in order of femtoseconds, is practically limited by driving electrical circuits [50], but still achievable at level of tens of GHz [51]. Moreover, such structures reveals compatibility with CMOS technology, in means of materials (i.e., graphene, SiO_{2}) and technological processes (i.e., PECVD [52], Reactive Magnetron Sputtering [53] or Reactive Ion Etching [54]). We believe that waveguide structures considered within this paper could bring new quality in means of integrated optical systems, all the more being compatible with CMOS technology.

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