1. Introduction

Graphene has become a subject of intense interest [1, 2] since the graphitic sheet of one-atom thickness has been experimentally realized by A. K. Geim et al. in 2004 [3]. The valence electron dynamics in such a truly two-dimensional (2D) material is governed by a massless Dirac equation. So graphene exhibits many unique electronic properties [1], including half-integer and unconventional quantum Hall effect [4], observation of minimum conductivity [5], and Klein tunneling [6]. The optical-like behaviors of electron waves in graphene [7–12] have also drawn considerable attention recently, such as focusing [7], collimation [8], Bragg reflection [10], and Goos-Hänchen (GH) effect [11, 12]. In this regard, one of the recent work is to investigate the guided modes in monolayer graphene waveguide, by analogy of optical waveguides [13]. The exotic properties of the graphene waveguide are found in two different cases of classical motion and Klein tunneing.

On the other hand, the Dirac point (DP) in photonic crystals (PCs) for the Bloch states [14–17] is found from the similarity of the photonic bands of the 2D PCs with the electronic bands of solids. Several novel optical transport properties near the DP have been shown in [15–17], such as conical diffraction [15], a “pseudodiffusive” scaling [16], and the photon’s Zitterbewegung [17]. Recently, Wang et al. have realized the DP with a double-cone structure for optical field in the negative-zero-positive index metamaterial (NZPIM) [18,19]. Stimulated by these results, we have further found that the tunable transmission gap, Bragg-like reflection, and negative or positive GH shifts in NZPIM slab [20]. All the phenomena such as pseudodiffusive property [18], Zitterbewegung effect [19], GH effect [20] have demonstrated that the DP has great effect on the optical reflection and transmission, which will result in novel optical devices.

In present paper, we will investigate systemically the guided modes in the NZPIM waveguide by using the graphic method. When the wave vector in the waveguide is real in an homogeneous medium, the corresponding modes can be identified as “fast waves”, since their phase velocities in the direction of the guide modes are larger than the phase velocity in such medium. In this case, it is shown that the fundamental mode is absent when the angular frequency is smaller than the DP. Whereas the NZPIM waveguide behaves like conventional dielectric waveguide, when the angular frequency is larger than the DP. The unique properties of the guided modes are very similar as the propagation of electronic wave in graphene waveguide, corresponding to the classical motion and the Klein tunneling [13]. On the other hand, the wave vector becomes purely imaginary for “slow waves”, when the propagation constant exceeds a critical value. We have found that the NZPIM waveguide can only propagate fundamental odd and even surface guided modes-slow waves for ω < ω_D. All these results suggest that one can make the theoretical and even experimental simulations of guide modes in graphene waveguide by the NZPIM waveguide, where such NZPIM can be easily realized in the laboratory nowadays [21].

The paper is organized as follows. In Sec. 2 we introduce the model of NZPIM waveguide and represent the physical properties of the optical DP. Due to the fact that the unique properties of the guided modes with both real and imaginary transverse wave vectors can be supported in negative-refractive-index waveguide [22], we investigate the fast wave and slow wave guided modes in Sec. 3 and 4, respectively. The conclusion is drawn in Sec. 5.

2. Model

 

Fig. 1. Schematic structure of NZPIM waveguide, where the core is the air with the thickness is d and the cladding is the so-called NZPIM.

Download Full Size | PDF

In NZPIM, the medium has a linear dispersion [18–20]

with group velocity ${\nu }_D=\left(\frac{d\omega }{dk}\right){\mid }_{\omega ={\omega }_D}$ , ω_D is the frequency of the DP (corresponding wavelength is λ_D = 2πc/ω_D with the light speed c in vacuum), where two bands touch each other forming a double-cone structure. Near the DP, the light transport obeys the massless Dirac equation as follows:

where $\mathrm{\Psi }=\left(\begin{array}{c}{E}_{z1}(x,y,\omega )\\ E_{z2}(x,y,\omega )\end{array}\right)$ are the eigenfunctions of the electric fields with the same k(ω). It is noted that the condition for realization of the DP in the homogenous optical medium is the index varying from negative to zero and then to positive with frequency [18].

We consider a waveguide structure made of NZPIM, as shown in Fig. 1, where the core is the air with the thickness is d and the cladding is the so-called NZPIM, the incidence angle is θ, and the direction of the guide modes is y axis. There are two types of situations: (a) when the incidence angle is smaller than the total internal reflection (TIR) angle, the modes become radiation modes; (b) when the incidence angle is larger than the critical angle, there will exist oscillating guided modes. What as follows we will focus on the latter case. The TIR angle is defined by sin θ_c = κ₂/κ₁, where κ₁ = ω/c is the wavevector in the air, and κ₂ = (ω − ω_D)/υ_D is the wave vector of the NZPIM. For simplicity, we take the NZPIM cladding in the Drude model with the permittivity and permeability in the following form [18–20]

where ω²_ep and ω²_mp are the electronic and magnetic plasma frequencies, and γ_e and γ_m are the damping rates relating to the absorption of the material. Here we assume γ_e = γ_m = γ ≪ ω²_ep,ω²_em. It is important that when ω_ep = ω_em and γ = 0 (no loss), the DP can be defined as ω_D = ω_ep = ω_em, which can be chosen to be ω_D = 2π × 10GHz for the real negative-zeropositive metamaterial [18, 21]. Then, both ε₂(ω = ω_D) and µ₂(ω = ω_D) may be zero simultaneously. In this case, we obtain κ(ω = ω_D) ≈ 0 and υ_D ≃ c/2.

3. Fast wave guided modes

We consider the transverse electric (TE) guided modes (TM modes can be obtained in the same way), the electric fields in the three regions can be written as

where κ_x = κ₁ cos θ, β = κ₁ sin θ is the propagation constant of the guide modes, and $\alpha =\sqrt{{\beta }^2-{\kappa }_2^2}$ is the decay constant in the cladding region.

 

Fig. 2. (Color online) Graphical determination of κ_xd for fast wave guided modes when ω < ω_D. The solid and dashed curves correspond to tan (κ_xd) and F(κ_xd), respectively. The initial parameters are ω_D = 2π × 10GHz, ω = 0.8ω_D which means the total reflection angle is θ_c = 30°, the thickness of the core are (a) d = 10cm and (b) d = 1cm.

Download Full Size | PDF

Applying the continuity of wave function at the interface x = 0 and x = d, we obtain the corresponding dispersion equation as follow:

We make Eq. (6) in dimensionless form

The dispersion Eq. (7) is a transcendental one and cannot be solved analytically, so we propose a graphical method to determine the solution of κ_xd for the guided modes. We will discuss the properties of the guided modes in two cases ω < ω_D and ω > ω_D, respectively.

Case 1: ω < ω_D. The critical angle is defined as

with the necessary condition $\frac{2}{3}{\omega }_D<\omega <{\omega }_D$ [20].

 

Fig. 3. (Color online) The electric field distribution of guided modes as a function of distance in the NZPIM waveguide corresponding to the intersection in Fig. (2) when ω < ω_D. (a) TE₂: κ_xd = 3.41; (b) TE₃: κ_xd = 6.87; (c) TE₄: κ_xd = 10.49; (d) TE₁: κ_xd = 1.03.

Download Full Size | PDF

As shown in Fig. 2 (a), we plot the dependencies of tan(κ_xd) and F(κ_xd) on κ_xd. The intersections show the existence of the guided modes, as shown in Fig. 3 (a), (b), and (c), corresponding to the TE₂, TE₃, and TE₄ modes, respectively. We find that for some waveguide parameters, the lower-order mode TE₁ can not coexist with higher-order guided modes. So we can not solve TE₁ mode in the same graph Fig. 2 (a). As discussed below, we will see that the absence of TE₁ is due to the fact that the waveguide parameters used in Fig. 2 (a) does not satisfy the dispersion relation of Eq. (9) when m = 1. Actually, we can reduce the thickness of the waveguide to obtain the TE₁ mode in the NZPIM waveguide, as shown in Fig. 3 (d), which corresponds to the dispersion relation graphic of Fig 2 (b) with the waveguide thickness is d = 1cm.

Another interesting property of the guided modes is that the absence of fundamental TE₀ mode for any parameters of the NZPIM waveguide, which is a novel property different from that in conventional waveguide. The unique property is similar to the guide modes of electron waves in graphene waveguide, where the fundamental mode is absent in the Klein tunneling case [13]. For the TE modes, we can write the dispersion relation Eq. (7) as

where

is negative (angular frequency is smaller than the Dirac point, µ₂ < 0, corresponding Klein tunneling in graphene), which represents the phase retardation upon the total internal reflection at the interface between air and the NZPIM. From Eq. (9), we know that for the fundamental mode (m = 0), it does not meet with the required dispersion relation. In fact, the condition for the guided waves to exist in a slab waveguide, has a simple physical meaning: the roundtrip accumulation of phase due to wave propagation across the layer, 2ϕ_prop, including the phase retardation upon the total internal reflection, 2ϕ_refl, should be equal to a multiple of 2π [23]. When the angular frequency is smaller than the Dirac point (the permittivity and the permeability are both negative, NZPIM can be treated as left-handed material), the total phase change does not satisfy the required dispersion relation of Eq. (9), and no fundamental guided modes exist [22–24]. As a matter of fact, this result is also confirmed in the dispersion relation of Fig. 6, where the dispersion of TE₀ mode only exists when ω > ω_D.

 

Fig. 4. (Color online) Graphical determination of κ_xd for fast wave guided modes when ω > ω_D. The solid and dashed curves correspond to tan (κ_xd) and F(κ_xd), respectively. The initial parameters are ω_D = 2π × 10GHz, ω = 4ω_D/3 which means the total reflection angle is θc = 30°, the thickness of the core is d = 10cm.

Download Full Size | PDF

Case 2: ω > ω_D. The critical angle is defined as

with the necessary condition ω_D < ω < 2ω_D [20]. Similarly, we obtain the guided modes of the NZPIM waveguide by using the graphical method, as shown in Fig. 4. It is shown that when ω > ω_D, the properties of the NZPIM waveguide can be treated as a conventional dielectric waveguide. From Fig. 5, we can see that the fundamental odd and even guided modes can coexist with higher-order modes within the same waveguide for general parameters, which is very different from the case when ω < ω_D. Under this condition, it corresponds to the guided modes in graphene waveguide in classical motion [13].

 

Fig. 5. (Color online) The electric field distribution of guided modes as a function of distance in the NZPIM waveguide corresponding to the intersection in Fig. (4) when ω > ω_D. (a) TE₀: κ_xd = 3.03; (b) TE₁: κ_xd = 6.06; (c) TE₂: κ_xd = 9.07; (d) TE₃: κ_xd = 12.07.

Download Full Size | PDF

 

Fig. 6. (Color online) The propagation constant β versus the incident frequency ω near the DP in the NZPIM waveguide.

Download Full Size | PDF

In order to show further the unique properties of guides modes near the DP in the NZPIM waveguide, we plot the dispersion of the guided modes when the incident frequency varies from ω < ω_D to ω > ω_D in Fig. 6. As discussed above, we can see that TE₀ mode only exist when ω > ω_D. In addition, another important and interesting property of the guided modes is that there exists an asymmetric forbidden band for the dispersion. The band will also become wider when the order of the guided modes increases with increasing the incidence angle. The result indicates that the modes are not continuous near the DP. This behavior on the forbidden band discussed here is very similar to the transmission gap in the NZPIM slab [20]. It seems that the guided modes near the DP are quite different from the negative refractive index metamaterial waveguides discussed in Ref. [22, 24], though one can divide NZPIM two parts with positive index and negative index respectively by DP, which corresponds to ω > ω_D and ω < ω_D.

4. Slow wave guided modes

We also find that when ω < ω_D, the NZPIM waveguide can propagate surface guided modes-slow wave. In this case, the function of the modes in core become sinh and cosh with the imaginary transverse κ_x, and the electric fields in three regions can be written as

where κ_x is the transverse decay constant in the core region, and β² = κ²₁ + κ²_x is the propagation constant of the slow wave guided modes, and $\alpha =\sqrt{{\beta }^2-{\kappa }_2^2}$ is the decay constant in the cladding region.

 

Fig. 7. (Color online) The wave function of guided modes as a function of distance of NZPIM waveguide. The initial parameters are ω_D = 2π × 10GHz, ω = 0.69ω_D, and the thickness of the core is d = 10cm.(a) TE₀: κ_xd = 13.8109; (b) TE₁: κ_xd = 13.8112.

Download Full Size | PDF

Similarly, we obtain the corresponding dispersion equation of this system

Write Eq. (13) in dimensionless form as follow

As discussed above, we also propose a graphical method to solve the surface guided modes.

We find that only fundamental odd and even surface guided modes can exist in the waveguide for some parameters. As shown in Fig. 7, higher-order surface modes are forbidden except the TE₀ and TE₁ surface guided modes. These results obtained here also predict the surface mode of electrons and holes in graphene waveguide.

5. Conclusions

In summary, we have investigated the guided modes in the NZPIM waveguide. Due to the linear Dirac dispersion of NZPIM, the unique properties of the guided modes are very similar as the propagation of electronic wave in graphene waveguide. For the fast wave guided modes, it is shown that the fundamental mode is absent when the angular frequency is smaller than the DP. Whereas, the NZPIM waveguide behaves like conventional dielectric waveguide, when the angular frequency is larger than the DP. In addition, we have also found that when ω < ω_D, the NZPIM waveguide can only propagate fundamental odd and even surface guided modes-slow waves. Finally, we will emphasize that these results discussed here do extend the investigations [22–24] and applications [25,26] of the waveguide containing only left-handed material. On one hand, we can control the properties of guides modes for the potential applications by adjusting the angular frequency with respect to the DP. On the other hand, our work will also motivate the further work to simulate many exotic phenomena in graphene with relatively simple optical benchtop experiments, based on the links between Klein paradox and negative refraction [27].