## Abstract

We propose in this paper a dielectric-graphene-dielectric tunable infrared waveguide based on multilayer metamaterials with ultrahigh refractive indices. The waveguide modes with different orders are systematically analyzed with numerical simulations based on both multilayer structures and effective medium approach. The waveguide shows hyperbolic dispersion properties from mid-infrared to far-infrared wavelength, which means the modes with ultrahigh mode indices could be supported in the waveguide. Furthermore, the optical properties of the waveguide modes could be tuned by the biased voltages on graphene layers. The waveguide may have various promising applications in the quantum cascade lasers and bio-sensing.

©2013 Optical Society of America

## 1. Introduction

Metamaterials are artificially structured materials that can be composed of dielectric elements or structured metallic with unconventional electromagnetic properties, which have demonstrated potential in various applications, such as left-handed behavior [1] and the cloaking [2]. In recent years, indefinite metamaterial with hyperbolic dispersion has been proposed [3]. The hyperbolic dispersion eliminates the cut off of large wave vectors, modes with ultrahigh refractive indices could propagate along the waveguide. The capability has been applied on the nanoscale waveguides [4, 5] and optical cavities [6].

Graphene, a two-dimensional form of carbon in which the atoms are arranged in a honeycomb lattice [7, 8], which has been emerged as an alternative material in optical devices and optical optoelectronic applications [9]. Recently, the hyperbolic metamaterial with graphene-dielectric multilayers has been studied by I. V. Iorsh *et al*. [10] and M. K. Othman *et al*. [11]. The hyperbolic dispersion characteristics of this stacked structure are investigated at a temperature of 4 K and 300 K, respectively.

In principle, any inductive infinitesimally-thin layer with negative permittivity could be fabricated between dielectric layers to realize hyperbolic metamaterial. However, thin metal layers would suffer from high metallic losses and the spatial and frequency dispersion introduced by periodically patterned conductive layers [11], which may greatly restrict the application of the hyperbolic metamaterials. Recently, heavily doped oxide semiconductors have been shown to exhibit both negative real permittivity and relatively small losses at near- and mid-infrared frequencies. G. V. Naik *et al*. have shown a high-performance hyperbolic metamaterial formed by Al:ZnO and ZnO as the metallic and dielectric components [12]. And the transparent conducting oxides (TCOs) could outperform conventional metals in hyperbolic metamaterials in near-infrared frequencies [13]. C. Rizza *et al*. have proposed a hyperbolic metamaterial made of alternating semiconductor and dielectric layers [14].

The graphene could be treated as a layer of thin metal at THz frequencies. Since the real part of the conductivity (determining the attenuation) of graphene is remarkably small compared with noble metal (e.g. Gold and Silver), which may possess desirable performance on the waveguide modes losses at THz frequencies. What is more, the conductivity of the graphene could be tuned by varying the chemical potential of the graphene sheets via electrostatic biasing, which may provide a robust and flexible method to tune the dispersion characteristics of the waveguide and the optical properties of the waveguide modes. All of these make graphene a good candidate for realizing the hyperbolic metamaterial designs in the THz range.

Recently, the hyperbolic metamaterials have been investigated to achieve high refractive indices at infrared spectroscopy. M. Choi *et al*. have investigated a terahertz metamaterial with unnaturally high refractive index [15], which could achieve highest index of refraction of 33.22 at a frequency of 0.851 THz. J. Shin *et al.* have designed three-dimensional isotropic metamaterials which possess an enhanced refractive index between 5.5~7 in the wavelength range from 3 um~6 um [16].

Since nanosacle waveguides are critical devices in many fundamental studies of optical physics and integrate optics, waveguides based on hyperbolic metamaterials have been recently investigated [4, 5]. Y. He *et al.* have demonstrated that a hyperbolic metamaterial waveguide could achieve ultrahigh effective refractive index up to 62.0 at near-infrared region [4].

In this paper, we propose deep-subwavelength waveguides composed of dielectric-graphene-dielectric multilayer indefinite metamaterials, which support waveguide modes with ultrahigh refractive indices in the mid-infrared and far-infrared frequencies. The optical properties of the modes are investigated by both multilayer structures and the effective medium approach. The dependences of waveguide mode indices and propagation lengths on different mode orders, free-space wavelengths, sizes of waveguide cross and biased voltages on graphene layers are also investigated. Moreover, the mode indices and the propagation lengths could be further adjusted by the biased voltages on graphene layers. The waveguide which supports modes with ultrahigh refractive indices and thus the strong field confinement effect could enhance the light-matter interactions, such as the quantum cascade lasers [17–19], nonlinear optics and bio-sensing [20].

## 2. The hyperbolic dielectric-graphene-dielectric multilayer metamaterial

Graphene's complex conductivity (${\sigma}_{g}={\sigma}_{\mathrm{int}ra}+{\sigma}_{\mathrm{int}er}$) could be determined from the Kubo formalisms, the Eq. (1) and Eq. (2) are due to intraband and interband contributions [21,22], respectively:

*ω*is the angular frequency,

*τ*is the relaxation time which represents the loss mechanism,

*e*is the charge of the electron,

*k*is the reduced Planck's constant,

_{B}*T*is the temperature and

*μ*

_{c}is the chemical potential, which can be defined as [21]:where

*a*≈9 × 10

_{0}^{16}m

^{−1}V

^{−1},

*V*and

_{Dirac}*ν*represent the voltage offset and the Fermi velocity of Dirac fermions in Graphene, respectively. Expression $\tau =5\times {10}^{-13}s$can be considered as the biased voltage

_{F}*V*, which could modify the chemical potential. Moreover, the chemical potential could also be tuned by electric field, magnetic field and chemical doping [23]. It should be noted that, since a factor

_{biased}*ρ*(

*E*) which represents the density of states per spin per unit cell, has been introduced in the deriving process of Eq. (1) and Eq. (2) [24], the effect caused by the finite size of the graphene layer has been neglected in our conductivity model, which has also been applied in the design of graphene optical devices, such as the terahertz gate-controlled metamaterials [25] and the graphene-based modulator [21].

In this work, we assume that the environment temperature is fixed at $\tau =5\times {10}^{-13}s$, the relaxation time is $\tau =5\times {10}^{-13}s$.The conductivity of graphene as functions of free-space wavelength *λ _{0}* and the biased voltage

*V*

_{biased}_{are plotted in Figs. 1(a) and 1(c), respectively. In Fig. 1(a), the voltages are Vbiased = 1.6 volt and 2.5 volt with the corresponding chemical potentials μc = 0.4 eV and 0.5 eV. And the free-space wavelength is fixed at λ0 = 15 um in Fig. 1(c). As shown in Fig. 1(a), with a fixed Vbiased, the real part of graphene conductivity stays near the universal value, σ0 = πe2/2ћ≈6.085x10−5S/m, and the absolute value of the imaginary part increases with the wavelength. The real part of the conductivity is insensitive to the biased voltage, and the imaginary part decreases as the voltage increases. In Fig. 1(c), the real part stays relative high when the biased voltage is below a well-defined threshold, whereas over this threshold the real part recovers the universal value. For the imaginary part, a positive peak value appears at the threshold voltage.}

The multilayer metamaterial waveguide is composed of alternative layers of graphene and dielectric. We choose SiO_{2} as the dielectric material in our subsequent simulation. The permittivity of the dielectric is *ε _{d}* = 2.2 and the refractive index is

*n*= 1.48. The period of the structure is

_{d}*a*= 20 nm. Each period is constructed with a layer of graphene with thickness

*a*1 nm and the layer of dielectric with thickness

_{g}=*a*= 19 nm. The schematic of the waveguide is sketched in Figs. 2(a) and 2(b). The

_{d}*Lx*and

*Ly*represents horizontal and vertical dimensions of the cross section of the waveguide, while the

*Lz*is the length of the waveguide.

Since the period *a* is much less than the incident wavelength, the multilayer metamaterial could be treated as a homogeneous effective medium and the anisotropic permittivity sensor could be defined as [4,26],

*f*0.05, which is the fraction of the multilayer structure period occupied by graphene. The

_{g}=*ε*and

_{x}, ε_{y}*ε*represents the permittivity along

_{z}*x*,

*y*and

*z*directions. The

*ε*is the permittivity of the dielectric. And the

_{d}*ε*and

_{g,t}*ε*represent the tangential and normal components of the permittivity of the graphene layer, respectively. The

_{g,n}*ε*could be derived from the relationship between the relative permittivity and the conductivity [27],where

_{g,t}*d*is the thickness of graphene layer. Since the wavelength

*λ*in the metamaterial is significantly longer than all characteristic dimensions, the general formula thus could be simplified as

*ε*(K = 0,

*ω*) =

*ε*(

*ω*). Moreover, it should be considered that the normal electric field cannot excite any current in the graphene sheet, so the normal component of the permittivity of graphene should be

*ε*= 1. So in our multilayer metamaterial, the normal component of the effective permittivity

_{g,n}*ε*, which could be derived from Eq. (4), is only depending on the filling factor

_{y}*f*. The calculated real parts of permittivity

_{g}*ε*and

_{x}, ε_{y}*ε*as functions of free-space wavelength

_{z}*λ*and the biased voltage $\frac{{k}_{x}^{2}+{k}_{z}^{2}}{{\epsilon}_{y}}+\frac{{k}_{y}^{2}}{{\epsilon}_{x}}={k}_{0}^{2},$ are sketched in Figs. 1(b) and 1(d), respectively. In Fig. 1(b), the voltages are $\frac{{k}_{x}^{2}+{k}_{z}^{2}}{{\epsilon}_{y}}+\frac{{k}_{y}^{2}}{{\epsilon}_{x}}={k}_{0}^{2},$ = 1.6 volt and 2.5 volt and the free-space wavelength is fixed at

_{0}*λ*= 15 um in Fig. 1(d). As shown in Fig. 1(b), when the biased voltage is fixed, the real part of

_{0}*ε*stays positive, while the real parts of

_{y}*ε*and

_{x}*ε*decrease as the wavelength increases. In Fig. 1(d), the curve represents the tangential component of the permittivity shows a similar trend with the imaginary part of conductivity under fixed wavelength, which is shown in Fig. 1(c). It has been shown that the multilayer metamaterial with anisotropic permittivity sensor is equivalent to a uniaxial anisotropic material with dispersion [28]:where

_{z}*k*is the free-space wave vector,

_{0}*k*,

_{x}*k*and

_{y}*k*are the wave factors along

_{z}*x*,

*y*and

*z*direction, respectively. The corresponding effective refractive indices along different directions thus could be achieved through

*k*/

_{i}*k*, where

_{0}*k*is one of the three wave factors. As is shown in Figs. 1(b) and 1(d), with capable biased voltage, the permittivity component

_{i}*ε*could be negative while the normal component

_{x}*ε*is positive, which implies that the dispersion characteristic is hyperbolic type. Furthermore, since the effective permittivity appears as functions of the biased voltages, the dispersion properties of the waveguide could be tuned by adjusting the biased voltages on graphene layers.

_{y}For the hyperbolic metamaterial, the permittivity sensor is anisotropic, which means that one component of the permittivity need to be negative. Therefore, the working wavelength of the proposed metamaterial could span from mid-infrared to far-infrared wavelength region, which could be found in Fig. 1(b). What is more, the working wavelength could be pushed to a shorter wavelength simply by adjusting the biased voltages on graphene layers.

## 3. Dielectric-graphene-dielectric waveguides

The proposed multilayer indefinite waveguide is numerically analyzed with COMSOL MULTIPHYSICS based on finite element method. All materials are nonmagnetic (*μ*_{r} = 1) and the environment is free of external magnetic fields. The biased voltage is *V _{biased}* = 1.6 volt, equivalent to the chemical potential

*μ*

_{c}= 0.4 eV. We calculate the waveguide mode profiles of different mode orders under the dimensions

*Lx*= 200 nm and

*Ly =*200 nm with free-space wavelength

*λ*= 30 μm. The distribution of field components

_{0}*E*and

_{y}*H*for the mode orders (1,

_{x}*my*), which are derived from the effective medium approach and the multilayer structures, are illustrated in Figs. 3(a) and 3(b).

We could note that, for a specific waveguide mode with order (*m _{x}*,

*m*), the

_{y}*m*and

_{x}*m*represent the distributions of the resonant peaks along

_{y}*x*direction and

*y*direction on the cross section. Based on the two methods, we also calculate the effective refractive indices along the propagation direction

*n*and the propagation lengths

_{eff,z}*L*as functions of free-space wavelength

_{m}*λ*, which are plotted in Figs. 3(c) and 3(d). The solid and the dot-dashed lines represent the results calculated from the effective medium approach and the multilayer structure, respectively. The propagation length

_{0}*L*is defined as

_{m}*L*= 1/2Im(

_{m}*k*) =

_{z}*λ*/4πIm(

_{0}*n*). As is shown in Fig. 3(c), the modes with a higher

_{eff,z}*m*may have larger mode indices. And each mode tends to have a smaller indice at longer wavelength. In Fig. 3(d), we could find that the modes with higher indices tend to have less propagation lengths, and the propagation lengths would increase with the wavelength.

_{y}As illustrated in Fig. 3(d), the mode indices derived from the multilayer structure and the effective medium approach agree well for the propagation lengths. While in Fig. 3(c), the differences are obviously concerned with mode orders of the waveguide modes. For higher order modes, there would be more resonant periods of electric (or magnetic) field on the cross section of the waveguide, which are shown in Figs. 3(a) and 3(b). The more periods on the cross section means the less of the graphene-dielectric unit layers in one period, and the effective medium theory would be less precise.

The higher order modes tend to have larger mode indices, while more loss as a consequence of the stronger field confinement effect, we thus only consider the modes with orders (1,*m _{y}*) in the subsequent investigations. Then we calculate the dependences of mode indices and propagation lengths on

*Lx*and

*Ly*. The free-space wavelength is

*λ*= 30 um. The

_{0}*Lx*or

*Ly*spans from 150 nm to 250 nm, while the other is fixed at 200 nm. The result calculated from the multilayer structure and effective medium approach is shown in Fig. 4.

As shown in Figs. 4(a) and 4(c), the mode indices decrease as *Lx* or *Ly* increases. The propagation lengths of all the modes are insensitive to the change of horizontal dimension *Lx*, while they all increase with the vertical dimension *Ly*, which are presented in Figs. 4(b) and 4(d). Finally, we demonstrate the optical properties of the waveguide modes could be tuned by adjusting the biased voltages on graphene layers. The dependences of the mode indices of (1,*m _{y}*) modes on the biased voltage are shown in Fig. 5(a). The propagation lengths as functions of the biased voltage are shown in Fig. 5(b). Here we assume the waveguide cross section is square with

*L = L*= 200 nm and the wavelength is

_{x}= L_{y}*λ*= 30 um. The biased voltage spans from 0.1 volt to 1.6 volt.

_{0}As shown in Fig. 5(a), all the modes tend to have lower mode indices at higher biased voltages. And higher biased voltages lead to less losses, and thus larger propagation lengths, which are shown in Fig. 5(b). Therefore, in our proposed hyperbolic waveguide, the biased voltages on graphene layers could tune not only the dispersion properties of the waveguide, but also the optical properties of the waveguide modes, such as the mode indices and the propagation lengths. The graphene layers could be deposited by chemical vapor deposition (CVD) or by epitaxial growth [29]. The dielectric layers (SiO_{2}) could be fabricated by plasma-enhanced chemical vapor deposition (PECVD) [30, 31].

## 4. Conclusion

In this paper, we have proposed a dielectric-graphene-dielectric waveguide based on multilayer metamaterials. The waveguide mode characteristics have been described through the effective medium approach, which agree with the results calculated from the multilayer structure. Numerical simulation has shown that the multilayer metamaterial waveguide, which appears hyperbolic dispersion characteristics in the mid-infrared and far-infrared wavelength region, could support waveguide modes with ultrahigh refractive indices. The waveguide modes indices and the propagation lengths could be further adjusted by the biased voltages on the graphene layers. The dielectric-graphene-dielectric metamaterial waveguide provides a robust and simple method to achieve strong field confinement and ultrahigh mode indices, which has a promising application in the quantum cascade lasers, nonlinear optics and bio-sensing.

## Acknowledgment

This work is supported in part by the Major State Basic Research Development Program of China (Grant No. 2010CB328206), the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008, 61275092), and the Fundamental Research Funds for the Central Universities, China.

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