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
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  and the cloaking . In recent years, indefinite metamaterial with hyperbolic dispersion has been proposed . 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 .
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 . Recently, the hyperbolic metamaterial with graphene-dielectric multilayers has been studied by I. V. Iorsh et al.  and M. K. Othman et al. . 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 , 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 . And the transparent conducting oxides (TCOs) could outperform conventional metals in hyperbolic metamaterials in near-infrared frequencies . C. Rizza et al. have proposed a hyperbolic metamaterial made of alternating semiconductor and dielectric layers .
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 , 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 .
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 .
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 .
2. The hyperbolic dielectric-graphene-dielectric multilayer metamaterial21]:23]. It should be noted that, since a factor 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) , 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  and the graphene-based modulator .
In this work, we assume that the environment temperature is fixed at , the relaxation time is .The conductivity of graphene as functions of free-space wavelength λ0 and the biased voltage Vbiasedare 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 SiO2 as the dielectric material in our subsequent simulation. The permittivity of the dielectric is εd = 2.2 and the refractive index is nd = 1.48. The period of the structure is a = 20 nm. Each period is constructed with a layer of graphene with thickness ag = 1 nm and the layer of dielectric with thickness ad = 19 nm. The schematic of the waveguide is sketched in Figs. 2(a) and 2(b). The 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],27],Eq. (4), is only depending on the filling factor fg. The calculated real parts of permittivity εx, εy and εz as functions of free-space wavelength λ0 and the biased voltage are sketched in Figs. 1(b) and 1(d), respectively. In Fig. 1(b), the voltages are = 1.6 volt and 2.5 volt and the free-space wavelength is fixed at λ0 = 15 um in Fig. 1(d). As shown in Fig. 1(b), when the biased voltage is fixed, the real part of εy stays positive, while the real parts of εx and εz 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 :Figs. 1(b) and 1(d), with capable biased voltage, the permittivity component εx could be negative while the normal component εy 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.
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 (μr = 1) and the environment is free of external magnetic fields. The biased voltage is Vbiased = 1.6 volt, equivalent to the chemical potential μc = 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 λ0 = 30 μm. The distribution of field components Ey and Hx for the mode orders (1,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 (mx,my), the mx and my represent the distributions of the resonant peaks along 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 neff,z and the propagation lengths Lm as functions of free-space wavelength λ0, 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 Lm is defined as Lm = 1/2Im(kz) = λ0/4πIm(neff,z). As is shown in Fig. 3(c), the modes with a higher my 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.
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,my) in the subsequent investigations. Then we calculate the dependences of mode indices and propagation lengths on Lx and Ly. The free-space wavelength is λ0 = 30 um. The 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,my) 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 = Lx = Ly = 200 nm and the wavelength is λ0 = 30 um. The biased voltage spans from 0.1 volt to 1.6 volt.
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 . The dielectric layers (SiO2) could be fabricated by plasma-enhanced chemical vapor deposition (PECVD) [30, 31].
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.
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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