Engineered surface Bloch waves in graphene- based hyperbolic metamaterials

A kind of tunable hyperbolic metamaterial (HMM) based on the graphene-dielectric layered structure at near-infrared frequencies is presented, and the engineered surface Bloch waves between graphene-based HMM and isotropic medium are investigated. Our calculations demonstrate that the frequency and frequency range of surface Bloch waves existence can be tuned by varying the Fermi energy of graphene sheets via electrostatic biasing. Moreover, we show that the frequency range of surface Bloch waves existence can be broadened by decreasing the thickness of the dielectric in the graphene-dielectric layered structure or by increasing the layer number of graphene sheets. ©2014 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (160.3918) Metamaterials. References and links 1. B. E. Sernelius, Surface Modes in Physics (John Wiley, 2001). 2. J. A. Polo, Jr. and A. Lakhtakia, “Surface electromagnetic waves: a review,” Laser Photonics Rev. 5(2), 234–246 (2011). 3. H. 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Introduction
Surface electromagnetic waves (SEWs) are a special type of waves that are confined at the interface between two media with different properties [1,2]. One of the best known examples of SEWs is surface plasmon polariton (SPP), which is formed at the interfaces between metals and dielectrics [3,4]. SPPs only exist for TM polarization and there is no surface modes exist for TE polarization. SPP has attracted a wide spread attention in the last decade mainly due to their applications in sensing, microscopy, or integrated optics [5,6]. However, when one of the two media at the interface is periodic structures (photonic crystals or superlattices), the Bloch surface waves (BSWs) may appear at the boundary within the photonic bandgaps [7,8]. It has also been shown the lateral conðnement of the BSWs in the waveguide is well described by the 2D Snell's [9]. BSWs have potential applications in optical sensing [10][11][12], enhancement of Goos-Hanchen shift [13], and surface-enhanced Raman spectroscopy [14]. For BSWs, the Bloch wavevector is perpendicular to the interface of the periodic structure and the dielectric layer, and hence the Bloch wavevector is not propagating along the interface. Moreover, most recently, Vukovic et al. proposed another surface wave propagating in metal-dielectric superlattices and isotropic dielectric medium where the Bloch wavevector is parallel to the interface of the two media [15]. This surface wave has been named as surface Bloch waves owing to the Bloch wave propagating along the interface [15].
Recently, graphene has attracted intensive scientific interest owing to its incredible physical properties showing great potential applications in nano-electronic devices and optoelectric devices with ultrahigh electron mobility, ultrafast relaxation time for photoexcited carriers, and gate variable optical conductivity [31]. Graphene plasmonics have generated great interest among scientific community because of the ability of graphene to tune the plasmon dispersion by varying the chemical potential [32]. It seems to be a good candidate for designing tunable optical device that operates in both THz and optical frequency ranges [33]. Most recently, graphene-based HMMs composed of stacked graphene sheets separated by thin dielectric layers have been proposed and investigated [34,35], it has been demonstrated that such graphene-based HMMs can be used for the negative refraction at THz frequencies [36], the spontaneous emission enhancement [34,35], the perfect absorption [37], tunable broadband hyperlens [38], tunable infrared waveguide [39], and so on. However, these HMMs have be realized and investigated mainly at THz frequencies, and mid-or far-infrared frequencies. Graphene-based HMMs at the near-infrared frequencies and visible light are still not demonstrated. In the present paper, we suggest a new class of tunable HMMs for near-infrared frequencies based on the graphene-dielectric layered structure, and reveal that such graphene structures can support a controllable surface Bloch wave between graphene-based HMM and isotropic medium.

The optical properties of graphene sheets
Consider the geometry shown in Fig. 1(a), where the semi-infinite periodic structure created by alternating layers of graphene and conventional dielectric is placed on the left of the homogeneous dielectric. The unit cell of the layered structure is formed by a transparent material of dielectric constant d Experimentally, the multilayer composed of dielectric and graphene can be prepared by the following procedures: the dielectric layer is first coated on a SiO 2 substrate through thermal evaporation, and then the high quality graphene sheets prepared using CVD method are coated on the top of the dielectric layer after a chemical vapor deposition, finally the multilayer is realized by stacking of many graphene/dielectric layers on the SiO 2 substrate. The surface conductivity of graphene is highly dependent on the working wavelength and Fermi energy. Within the random-phase approximation and without external magnetic field, the graphene is isotropic and the surface conductivity σ can be written as a sum of the where ω is the frequency of the incident light, e and  are universal constants related to the electron charge and reduced Planck's constants, respectively. F E and τ are the Fermi energy (or chemical potential) and electron-phonon relaxation time, respectively. B k is the Boltzmann constant, and T is a temperature in K. The Fermi energy F E can be straightforwardly obtained from the carrier density ( 2D n ) in a graphene sheet, 6 10 F ν = m/s is the Fermi velocity of electrons. Here, the carrier density 2D n can be electrically controlled by an applied gate voltage, thereby leading to a voltagecontrolled Fermi energy F E and hence the voltage-controlled surface conductivity σ , this could provide an effective route to achieve electrically controlled surface Bloch wave. We assume that the electronic band structure of a graphene sheet is not affected by the neighboring graphene sheets, hence the graphene's effective permittivity g ε be written as ε is the permittivity in the vacuum.

Bulk Bloch waves of the graphene-dielectric layered structure
Assuming that the electric (magnetic) field is in the form where ( ) for p-polarized mode, and 1 for s-polarized mode.

Dispersion relation of graphene-dielectric layered structure in the subwavelength limit
In the subwavelength limit, i.

Effective permittivity and hyperbolic dispersion of graphene based HMMs
As an example, in Fig. 1(b) we plot the effective permittivities ε ⊥ and || ε . Here, we assume that the Fermi-energy of graphene 0.50

Engineered effective permittivity of graphene based HMMs
The wavelength of the hyperbolic dispersion curve can be electrically controlled by an applied gate voltage on the graphene sheet, which is demonstrated in Fig. 2(a). It is clear that the resonant behavior of || ε can be tuned by varying the Fermi energy F E , and the resonant wavelengths of || ε move to the shorter wavelength with the increases of Fermi energy F E . Actually, the resonant wavelength can be tuned to the visible light if we apply enough voltage to the graphene sheet. Therefore, our graphene-based layered structure has the potential to achieve various HMM-based optical devices.
In addition to the Fermi energy, the resonant behavior of ||