Additional waves in the graphene layered medium

Ruey-Lin Chern; Dezhuan Han; Z. Q. Zhang; C. T. Chan

doi:10.1364/OE.22.031677

1. Introduction

Graphene is a single layer of carbon atoms packed into a honeycomb lattice, which exhibits extraordinary electronic transport properties as strong ambipolar electric field effect [1], massless Dirac Fermions [2], and half-integer quantum Hall effect [3]. It is considered the thinnest material that combines aspects of semiconductors and metals with ultrahigh electron mobility [4] and superior thermal conductivity [5]. If a number of decoupled graphene layers are arranged as a periodic lattice [6–8], with the period much less than the wavelength, the collection of layers can be regarded as a medium with the optical properties characterized by its effective parameters. The effective permittivity strongly depends on the wave vector, showing the spatially dispersive or nonlocal nature of the graphene layered medium [9].

A distinctive character of the nonlocal medium is the emergence of additional waves. Maxwell’s boundary conditions are insufficient to solve the reflection and transmission coefficients for a nonlocal medium [10]. The additional boundary condition is therefore needed to complete the problem [11–14]. This is considered an important feature that characterizes the nonlocal optical properties in insulating crystals with excitons [15]. For the effective medium composed of graphene layers, the additional wave, known as the polariton mode, is manifest on the biquadratic dispersion relation and is attributed to the excitation of plasmon polaritons on the graphene surfaces [9]. In particular, the additional wave exhibits negative refraction with a different physical origin from either the negative index metamaterial [16] or the uniaxially anisotropic medium [17].

In this study, we investigate the features of additional waves that arise in the graphene layered medium, within the framework of effective medium model. Due to the nonlocal nature of the associated medium, multiple waves appear and Maxwell’s boundary conditions are insufficient to solve the reflection and transmission coefficients. For this purpose, an additional boundary condition based on modal expansion is proposed to derive the generalized Fresnel equations, based on which the additional wave is determined. The additional boundary condition is compared with the condition of vanishing currents, which has been employed in the wire medium [18–20], and a good consistency between the two approaches is reached. The feature of additional wave is further illustrated with the Gaussian beams based on the Fourier integral formulation.

2. Effective medium model

2.1. Dispersion relation

Consider a periodic lattice of graphene layers with surface conductivity σ and period a embedded in a background with the dielectric constant ε, as schematically shown in Fig. 1(a). Let the wave vector lie on the xz plane, that is, k = (k_x, 0, k_z), without loss of generality. With the time-harmonic dependence e^−iωt, the TM-polarized dispersion relation of the graphene layers is given by [9]

cos (k_{x} a) = cos (q a) - \frac{i σ q}{2 ω ε_{0} ε} sin (q a),

where

q = \sqrt{ε k_{0}^{2} - k_{z}^{2}}

with k₀ = ω/c. At sufficiently low temperature, kT ≪ μ, where k is the Boltzmann constant, T is the temperature, and μ is the chemical potential of graphene, the surface conductivity σ of graphene is given, within the random phase approximation, by the following relation [21–25]:

\frac{σ}{ε_{0} c} = 4 α \frac{i}{Ω} + π α [θ (Ω - 2) + \frac{i}{π} ln | \frac{Ω - 2}{Ω + 2} |],

where Ω = h̄ω/μ is the dimensionless frequency, h̄ is the reduced Planck constant, α ≈ 1/137 is the fine structure constant, and θ(x) is the Heaviside step function. The first and second terms on the right side of Eq. (2) stem from the intraband and interband contributions, respectively. Note that Eq. (2) is a local approximation of the graphene conductivity, which is valid when the in-plane wave vector components k_x and k_z are well below the Fermi wave vector k_F = μ/(h̄v_F) [21]. Here, v_F ≈ 9 × 10⁵ m s⁻¹ is the Fermi velocity of graphene and k_F is estimated to be on the order of 10⁸ m⁻¹.

Fig. 1 (a) Schematic diagram of the graphene layered medium: a periodic lattice of graphene layers with period a embedded in a background with dielectric constant ε. In this study, a = 2 h̄c/μ (≈ 989 nm) with μ = 0.4 eV and ε = 1.5 are used as the parameters. A small area of the graphene feature is shown on the top layer for illustration. (b) Schematic diagram of the incidence of a Gaussian beam onto the graphene layered medium.

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2.2. Effective permittivity tensor

Assume that the lattice period a is much less than the wavelength λ, so that the graphene layers can be regarded as an effective medium, characterized by the dispersion relation of a homogeneous yet anisotropic medium as

\frac{k_{x}^{2}}{ε_{z}^{eff}} + \frac{k_{z}^{2}}{ε_{x}^{eff}} = k_{0}^{2},

where

ε_{z}^{eff}

and

ε_{x}^{eff}

are the in-plane components of the effective permittivity tensor, given as [9]

ε_{z}^{eff} = \frac{ε_{z}^{0} - \frac{γ}{12} K_{0}^{2}}{1 - \frac{1}{12} K_{x}^{2}}, ε_{x}^{eff} = \frac{ε (1 - \frac{γ}{12 ε_{z}^{0}} K_{0}^{2})}{1 - \frac{γ}{6 ε_{z}^{0}} (K_{0}^{2} - \frac{1}{2 ε} K_{z}^{2})},

with K_x = k_xa, K_z = k_za, K₀ = k₀a,

ε_{z}^{0} = ε (1 + δ)

, γ = ε² (1 + 2δ), δ = iσ̃/(εk₀a), and σ̃ = σ/(ε₀c). The nonlocal nature of the graphene layered medium is manifest on the dependence of the permittivity components

ε_{z}^{eff}

and

ε_{x}^{eff}

on k_x and k_z, respectively. The nonlocal effect pertaining to

ε_{z}^{eff}

, however, is weak as |k_x| is bound by π/a in a periodic lattice. In the present problem, strong nonlocality is associated with

ε_{x}^{eff}

, in which the dependence variable k_z can be large in the layered medium.

In terms of the effective permittivity, the dispersion relation of the effective medium [Eq. (3)] can be characterized by the biquadratic curve (in the xy plane) as:

\frac{x^{2} (1 - ε x^{2})}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,

where a and b are constants, and ε is a positive number. At small x, the biquadratic curve is dominant by the x² and y² terms and is elliptic-like. At large x, the biquadratic curve is dominant by the x⁴ and y² terms and is parabolic-like. For the graphene layered medium, the nonlocal nature is manifest on the hybrid character of the dispersion relation. As will be discussed later, the eigenmodes with the elliptic-like and parabolic-like dispersions are termed as the photon and polariton modes, respectively.

3. Additional wave

3.1. Photon and polariton modes

The feature of additional wave in the graphene layered medium is well illustrated with the polariton mode in the dispersion relation. Figure 2(a) shows the equifrequency surface of the TM-polarized dispersion relation for the graphene layered medium with the parameters in Fig. 1. The dispersion surface consists of two parts: the upper surface is the photon mode, largely conformed to the light cone: $Ω \tilde{a} = \sqrt{(K_{z}^{2} + K_{x}^{2}) / ε}$ ; the lower surface is the polariton mode, arising from the mixing of light wave with the excitations (i.e. plasmons) in the graphene layers. The two modes intersect at a single point on either side of the half space (K_z > 0 or K_z < 0) on the symmetric plane (K_x = 0). Along the plane of constant K_x ≠ 0, the polaritonic band forms an anticrossing (avoided crossing) scheme with the photon band, indicating the existence of couplings between the two modes. Note that the photon and polariton modes determined by the dispersion relation of the graphene layers [Eq. (1)] can also be characterized by that of the effective medium [Eq. (3)]. The latter becomes more accurate when the expansion order (used in the effective medium model) is suitably increased [9]. The effective permittivity components given in Eq. (4) are obtained with the expansion to fourth order, which are sufficient to resolve both the photon and polariton modes.

Fig. 2 (a) Equifrequency surface and (b) equifrequency curves at Ω = 0.06 and Ω = 0.1 of the TM-polarized dispersion relation for the graphene layered medium with the parameters in Fig. 1.

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As the frequency Ω goes above the effective plasma frequency Ω₀ (defined later), there exists two wave numbers K_z for a given frequency. The smaller K_z is associated with the photon mode (P mode) and the larger K_z with the polariton mode (PL mode). In Fig. 2(b), the dispersion curves at Ω = 0.1 (above Ω₀ ≈ 0.0985) are a combination of an ellipse and a parabola (green lines), corresponding to the P mode and PL mode, respectively. This feature can also be explained by the biquadratic curve [cf. Eq. (5)] based on the effective permittivity for the graphene layered medium. The dispersion curves at Ω = 0.06 (below Ω₀), on the other hand, are two open curves (yellow lines) that depict the hyperbolic-like dispersion. In this range, only one K_z exists and the medium behaves like the local hyperbolic material.

3.2. Approximate wave vector components

At small K_x ≈ 0, the wave numbers of the two eigenmodes can be analyzed by the approximate form of Eq. (1):

1 - cosh Q + η Q sinh Q = 0,

where Q = iqa and

η = - \frac{i \tilde{σ}}{2 ε K_{0}}

. One solution of Eq. (6) is determined by sinh(Q/2) = 0 and given as

K_{z} \approx \sqrt{ε} Ω \tilde{a},

where ã = aμ/(h̄c). This solution corresponds to the photon mode that basically conforms to the the light cone. Another solution of Eq. (6) is determined by tanh(Q/2) = ηQ, which exists when η ≤ 1/2 and is approximated as

K_{z} \approx {[ε {(Ω \tilde{a})}^{2} + 12 (1 - 2 η)]}^{1 / 2} .

The condition η = 1/2 gives an approximate frequency:

Ω_{0} \approx 2 {(1 + \frac{ε \tilde{a}}{α})}^{- 1 / 2},

beyond which the second solution to Eq. (6) is valid. This solution corresponds to the polariton mode and is considered an additional wave, for two K_z’s are associated with a given K_x for a single frequency Ω. The frequency Ω₀ serves as the cutoff frequency for the polariton mode and is also termed as the effective plasma frequency for the graphene layers [9]. Using Eq. (9), the additional wave number in Eq. (8) can be expressed as

K_{z} \approx {[ε {(Ω \tilde{a})}^{2} + \frac{48 α}{ε \tilde{a}} (\frac{1}{Ω_{0}^{2}} - \frac{1}{Ω^{2}})]}^{1 / 2} .

In addition to two propagating modes with the real K_z characterized by Eqs. (7) and (10), there exist an infinite number of evanescent modes with the purely imaginary K_z, corresponding to the approximate equation:

1 - cos Q^{'} - η Q^{'} sin Q^{'} = 0,

where Q′ = qa. There are two families of solutions determined by sin(Q′/2) = 0 and tan(Q′/2) = ηQ′, which are approximated as

K_{z} \approx i {[{(2 n π)}^{2} - ε {(Ω \tilde{a})}^{2}]}^{1 / 2},

and

K_{z} \approx i {{[(2 n + 1) π - \frac{2}{(2 n + 1) π η}]}^{2} - ε {(Ω \tilde{a})}^{2}}^{1 / 2},

where n is an integer.

As K_x increases, the photon mode may change from being propagating to evanescent. This feature is similar to the total internal reflection for a wave incident from a dense onto a rare medium. The critical point occurs when K_z = 0 and the dispersion relation becomes

cos K_{x} - cos Q + η Q sin Q = 0,

where

Q = \sqrt{ε} K_{0}

. Using Eq. (9), the critical value of K_x can be approximated as

K_{x} \approx K_{0} {[\frac{4 α}{\tilde{a}} (\frac{1}{Ω_{0}^{2}} - \frac{1}{Ω^{2}})]}^{1 / 2} .

In this regard, there exists a critical angle θ_c for the photon mode as

θ_{c} \approx ArcSin {[\frac{4 α}{\tilde{a}} (\frac{1}{Ω_{0}^{2}} - \frac{1}{Ω^{2}})]}^{1 / 2},

beyond which the totally internal reflection occurs.

Figure 3(a) shows the wave number K_z as the function of Ω at θ = 5° (K_x = K₀ sinθ) for the graphene layered medium with the parameters in Fig. 1. Note that there exist two wave numbers for a given frequency when Ω > Ω₀. The corresponding effective permittivity $ε_{x}^{eff}$ is shown in Fig. 3(b), where $ε_{x}^{eff}$ experiences a drastic variation near Ω₀, with a standard Lorentzian resonance feature. There are also two $ε_{x}^{eff}$ for a given frequency when it is above Ω₀. Note that K_z near Ω₀ can be estimated as $K_{z} \approx \sqrt{ε} Ω_{0} \tilde{a} \approx 2 \sqrt{α \tilde{a}}$ [cf. Eqs. (7)–(9)]. The corresponding wave vector component k_z = K_z/a is on the order of 10⁵ m⁻¹, which is well below the Fermi wave vector k_F of graphene (cf. Sec. 2.1). Another wave vector component k_x is even smaller.

Fig. 3 (a) Wave number K_z and (b) effective permittivity $ε_{x}^{eff}$ as the functions of Ω at θ = 5° (K_x = K₀ sinθ) for the graphene layered medium with the parameters in Fig. 1. P and PL modes correspond to photon and polariton modes, respectively. Dashed line in (b) stands for the permittivity of the background material.

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4. Additional boundary condition

4.1. Modal expansions

Consider a plane wave incident from vacuum onto the graphene layered medium at an angle of incidence θ with respect to the surface normal. Due to the nonlocal nature of the medium, there may exist two transmitted waves. Let r be the reflection coefficient, and t₁ and t₂ be the transmission coefficients associated with the magnetic fields for TM polarization. Maxwell’s boundary conditions are given by the continuity of the tangential electric and magnetic fields at the interface between vacuum and the medium:

1 + r = t_{1} + t_{2},

1 - r = α_{1} t_{1} + α_{2} t_{2},

where

α_{n} = \frac{k_{z, n}}{ε_{x}^{eff} k_{0} cos θ}

for n = 1, 2. Since Eqs. (17) and (18) are insufficient to solve the three unknowns r, t₁, and t₂, the reflection and transmission problem is underdetermined. An additional boundary condition (ABC) is therefore needed to complete the problem. From the viewpoint of the effective medium, the shortage of boundary conditions in this problem comes from the expansion of dispersion relation that amounts to the averaging of fields over the layered structure. The missing information due to the equivalent field averaging shall be recovered from the intrinsic properties of the layered structure.

In this study, we propose a simple strategy to derive the additional boundary condition for the layered medium when it is considered nonlocal and two transmitted waves appear. In the presence of periodic layers with the period at the subwavelength scale, there exists an evanescent wave in vacuum that matches at the interface the high order wave in the layered medium, with the parallel (to interface) wave number k_x + 2mπ/a (m = 0, 1, 2,...). Denote r′ the reflection coefficient of the first order evanescent wave in vacuum and f_n = h_1,n/h_0,n the normalized (to zero order) first order Fourier coefficient of the n-th eigenfield H_n in the layered medium; that is, $H_{n} \approx \sum_{m} h_{m, n} e^{i (k_{x} + 2 m π / a) x}$ and $h_{m, n} = \frac{1}{a} \int_{x_{0}}^{x_{0} + a} H_{n} e^{- i (k_{x} + 2 m π / a) x} d x$ . The continuity of the tangential electric and magnetic fields at the interface for the high order wave is approximated as

r^{'} = f_{1} t_{1} + f_{2} t_{2},

- r^{'} = ρ_{1} f_{1} t_{1} + ρ_{2} f_{1} t_{2},

where

ρ_{n} = \frac{k_{z, n}}{ε_{x}^{eff} \sqrt{k_{0}^{2} - {(k_{x} + 2 π / a)}^{2}}} \approx \frac{k_{z, n} a}{i 2 π ε_{x}^{eff}}

for a ≪ λ. By eliminating r′ in Eqs. (19) and (20), we have

β_{1} t_{1} + β_{2} t_{2} = 0,

where β_n = (1 + ρ_n) f_n (n = 1, 2). Equation (21) is considered the ABC for the layered medium. Note that this condition is derived from the properties of the periodic structure, which is consistent with the common understanding that the ABC comes from the internal structure that composes the medium [11, 13, 15]. Equations (17), (18), and (21) are then solved to give

r = \frac{(1 - α_{1}) β_{2} - (1 - α_{2}) β_{1}}{(1 + α_{1}) β_{2} - (1 + α_{2}) β_{1}},

t_{1} = \frac{2 β_{2}}{(1 + α_{1}) β_{2} - (1 + α_{2}) β_{1}},

t_{2} = - \frac{2 β_{1}}{(1 + α_{1}) β_{2} - (1 + α_{2}) β_{1}},

which are regarded as the generalized Fresnel equations for the nonlocal medium with two transmitted waves. At normal incidence (θ = 0), in particular, t₁ and t₂ represent the eigen-waves with purely symmetric and antisymmetric field patterns, respectively, with respect to the graphene surface. It follows that h_1,1 = 0 and h_0,2 = 0, and therefore f₁ = 0 and f₂ = ∞. The ABC [cf. Eq. (21)] then reduces to t₂ = 0. This feature is reasonable since the purely antisymmetric mode associated with t₂ cannot be excited by the plane wave at normal incidence, which is purely symmetric. In this approach, the number of evanescent waves in vacuum is chosen to be equal to the number of additional waves in the layered medium so that the system equations for the reflection and transmission coefficients are neither overdetermined nor underdetermined.

The ABC presented above is consistent with the modal expansions for the grating structure [26] in case the grating depth is considered infinite. For a TM-polarized wave incident from vacuum, the magnetic wave in vacuum (z < 0) is represented by

H_{y} = e^{i k_{0} (γ_{0} x + \sqrt{1 - γ_{0}^{2}} z)} + \sum_{m = - \infty}^{\infty} R_{m} e^{i k_{0} (γ_{m} x + \sqrt{1 - γ_{m}^{2}} z)},

where γ_m = sinθ + mλ/a. The magnetic field in the layered structure (z > 0) is given by

H_{y} = \sum_{n} T_{n} X_{n} (x) e^{i k_{z, n} z},

where X_n (x) is a separable part (in the x direction) of the n-th eigenmode for the graphene layered structure. The detailed expression of X_n (x) is given in Appendix. In Eqs. (25) and (26), the reflection coefficients R_m and transmission coefficients T_n are determined by the boundary conditions at the interface (z = 0) between vacuum and the layered structure in the integral sense [26] and formulated as

D + R = \underline{χ} T,

\underline{Π} (D - R) = \underline{Ω} T,

where (D)_m = δ_m0, (R)_m = R_m, (T)_n = T_n,

{(\underline{χ})}_{m n} = \frac{1}{a} \int_{x_{0}}^{x_{0} + a} X_{n} (x) e^{- i k_{0} γ_{m} x} d x

,

{(\underline{Ω})}_{m n} = \frac{k_{z, n}}{ε} {(\underline{χ})}_{m n}

, and

{(\underline{Π})}_{m n} = k_{0} \sqrt{1 - γ_{m}^{2}} δ_{m n}

. The detailed expression of

{(\underline{χ})}_{m n}

is given in Appendix. The coefficients R_m and T_n can be uniquely determined when m is equal to n. Note that m ≠ 0 corresponds to the evanescent wave in vacuum (where

\sqrt{1 - γ_{m}^{2}}

is purely imaginary) and n > 1 to the high order wave in the layered structure (which is also considered the additional wave when the layered structure is regarded as a nonlocal medium). Equations (27) and (28) can be solved to give

R = {({\underline{Ω}}^{- 1} \underline{Π} + {\underline{χ}}^{- 1})}^{- 1} ({\underline{Ω}}^{- 1} \underline{Π} - {\underline{χ}}^{- 1}) D,

T = 2 {({\underline{Π}}^{- 1} \underline{Ω} + \underline{χ})}^{- 1} D .

Consider the case where m = 0, 1 and n = 1, 2. Equations (27) and (28) are written as

1 + R_{0} = χ_{01} T_{1} + χ_{02} T_{2},

1 - R_{0} = (Ω_{01} / Π_{00}) T_{1} + (Ω_{02} / Π_{00}) T_{2},

R_{1} = χ_{11} T_{1} + χ_{12} T_{2},

- R_{1} = (Ω_{1} / Π_{11}) T_{1} + (Ω_{2} / Π_{11}) T_{2} .

The last two equations are combined to give

[χ_{11} + (Ω_{11} / Π_{11}) T_{1}] + [χ_{12} + (Ω_{12} / Π_{11})] T_{2} = 0 .

Note that χ_1n = h_1,n and Ω_1n/Π₁₁ = ρ_nh_1,n, so that the above equation can be rewritten as

β_{1} h_{0, 1} T_{1} + β_{2} h_{0, 2} T_{2} = 0,

which is equivalent to the ABC [cf. Eq. (21)] if t₁ = h_0,1T₁ and t₂ = h_0,2T₂. If, further, r = R₀, r′ = R₁, and ε in Ω_mn is replaced by

ε_{x}^{eff}

, the solutions given by Eqs. (29) and (30) will be the same as those by Eqs. (22)–(24).

4.2. Vanishing currents

Another approach to the ABC is imposed by J_z = 0 [18] at the interface between vacuum and the layered medium, which means that the surface currents vanish at the ends of the graphene layers. A similar ABC was proposed to take into account the effect of diffusion in a semiconductor [27]. Based on this condition, the normal (to interface) electric field component E_z satisfies $\frac{\partial^{2} E_{z}}{\partial x^{2}} + \frac{\partial^{2} E_{z}}{\partial z^{2}} + ε k_{0}^{2} E_{z} = 0$ at the interface. Using H_y = t₁e^{i(k₀ sinθx+k_z,1z)} + t₂e^{i(k₀ sinθx+k_z,2z)} and $E_{z} = - \frac{k_{0} sin θ}{ω ε ε_{0}} H_{y}$ , the ABC can be written as

β_{1} t_{1} + β_{2} t_{2} = 0,

where

β_{n} = k_{z, n}^{2} - k_{z, 0}^{2}

(n = 1, 2) and

k_{z, 0} = k_{0} \sqrt{ε - {sin}^{2} θ}

is considered the normal wave vector component in the absence of graphene layers. This ABC has a similar form of Eq. (21), except that the definition of β_n is different. Using this ABC, together with Eqs. (17) and (18), we may have the same expressions of the generalized Fresnel equations for the graphene layered medium [cf. Eqs. (22)–(24)].

Note that in the absence of graphene layers, there is only one normal wave vector component: k_z,1 = k_z,0 and therefore β₁ = 0. The generalized Fresnel equations reduce to

r = \frac{1 - α_{1}}{1 + α_{1}}, t_{1} = \frac{2}{1 + α_{1}}, t_{2} = 0,

which are consistent with the Fresnel equations for ordinary (local) media.

Figure 4(a) shows the Fresnel coefficients as the functions of Ω at θ = 5° based on Eqs. (22)–(24) for the graphene layered medium with the parameters in Fig. 1. The reflection coefficient r and transmission coefficient t₁ exhibit the Lorentzian resonance feature near the effective plasma frequency Ω₀ in a similar manner of $ε_{x}^{eff}$ . The transmission coefficient t₂, which appears as an additional wave, suddenly arises when the frequency goes just beyond Ω₀. The magnitude of the additional wave, however, is gradually reduced as the frequency moves away from Ω₀. In Fig. 4(b), the ratio of t₂/t₁ = −β₁/β₂ based on modal expansions [cf. Eq. (21)] is compared with that based on vanishing currents [cf. Eq. (37)]. Note that although β₁ and β₂ differ somewhat between the two approaches, the ratio t₂/t₁ is very close to each other and the Fresnel coefficients are nearly identical for the two approaches. Note that high order terms in the modal expansions decay rather rapidly so that additional terms will deliver almost the same results as in Fig. 4. The nonlocal character is significant when the frequency approaches Ω₀, around which the additional wave emerges. Away from Ω₀, the nonlocal character is unimportant and the medium is considered local.

Fig. 4 (a) Fresnel coefficients and (b) ratio of t₂/t₁ at θ = 5° as the functions of Ω for the graphene layered medium with the parameters in Fig. 1.

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Note that the half line width of t₂ in Fig. 4(a) is the manifestation of additional wave that emerges in a nonlocal medium. The width will be determined by the effective plasma frequency (which in turn by the layer period, graphene chemical potential, and embedded dielectric constant) and the angle of incidence. The transmission coefficient t₂ is significant near Ω₀ ≈ 0.0985, which is below the interband transition (Ω = 2) and is considered not attributable to damping. Note also that the photon mode will be totally reflected when the angle of incidence θ is above the critical angle θ_c ≈ 7.8° [cf. Eq. (16)]. The generalized Fresnel equations are therefore adequate to characterize the two transmitted waves in the graphene layered medium for θ < θ_c. This feature comes from the rather high aspect ratio of the dispersion curve for the photon mode near the effective plasma frequency Ω₀ (cf. Fig. 2).

At normal incidence (θ = 0), Maxwell’s boundary conditions are simplified to

1 + r = t_{1} + t_{2},

1 - r = \frac{t_{1}}{n_{1}} + \frac{t_{2}}{n_{2}},

where

n_{j} = \sqrt{ε_{x}^{eff} (k_{z, j})}

is considered the refractive index associated with the j-th wave. The corresponding ABC reduces to

(n_{1}^{2} - ε) t_{1} + (n_{2}^{2} - ε) t_{2} = 0,

and the reflection coefficient can be expressed in terms of the effective refractive index n^* as

r = \frac{n^{*} + 1}{n^{*} + 1}, n^{*} = \frac{n_{1} n_{2} (n_{1} + n_{2})}{n_{1}^{2} + n_{1} n_{2} + n_{2}^{2} - ε} .

Note that n^* is different from that in the insulating crystals with excitons [15], where the ABC is given by the zero polarization at the interface [11]. This type of ABC might be applied to the case when the layers are oriented parallel to the interface [28].

Note also that the ABC [Eq. (21) or (37)] gives a ratio between t₁ and t₂, which will not violate the principle of energy conservation. In fact, the continuity of energy flux at the interface is ensured by Maxwell’s boundary conditions alone [29], regardless of the existence of ABC. Multiplying the complex conjugate of Eq. (17) by Eq. (18), we arrive at

1 - {| r |}^{2} - Re [{(t_{1} + t_{2})}^{*} (α_{1} t_{1} + α_{2} t_{2})] = 0,

which states that the energy flux is conserved along the interface normal (the z direction).

4.3. Gaussian beams

The additional wave analyzed in the preceding subsections can be illustrated with the incidence of Gaussian beams, as schematically shown in Fig. 1(b). The Gaussian beam with the center located at x = x₀ and z = −h is well approximated by the Fourier integral on the tangential (to interface) wave vector component k_x as [30]

f (x, z) = \int_{- \infty}^{\infty} ψ (k_{x}) e^{i k_{x} x + i k_{z} z} d k_{x},

ψ (k_{x}) = \frac{w_{0}}{2 cos θ \sqrt{π}} exp [- \frac{w_{0}^{2}}{4 {cos}^{2} θ} {(k_{x} - k_{0} sin θ)}^{2} - i k_{x} x_{0} + i k_{z} h],

where k_z is the normal wave number component, and w₀ is the waist size of the Gaussian beam. Based on this formulation, the TM-polarized incident beam in vacuum is formulated as

H_{i} = \int_{- \infty}^{\infty} h_{i} ψ (k_{x}) e^{i k_{x} x + i k_{0 z} z} d k_{x},

E_{i} = η_{0} \int_{- \infty}^{\infty} e_{i} ψ (k_{x}) e^{i k_{x} x + i k_{0 z} z} d k_{x},

where

η_{0} = \sqrt{μ_{0} / ε_{0}}

,

k_{0 z} = \sqrt{k_{0}^{2} - k_{x}^{2}}

, e_i = (k_0z/k₀, 0, −k_x/k₀), and h_i = (0, 1, 0). The reflected beam is formulated as

H_{r} = \int_{- \infty}^{\infty} h_{r} r (k_{x}) ψ (k_{x}) e^{i k_{x} x - i k_{0 z} z} d k_{x},

E_{r} = η_{0} \int_{- \infty}^{\infty} e_{r} r (k_{x}) ψ (k_{x}) e^{i k_{x} x - i k_{0 z} z} d k_{x},

where e_r = (−k_0z/k₀, 0, −k_x/k₀) and h_r = (0, 1, 0). The transmitted beams in the graphene layered medium are formulated as

H_{t n} = \int_{- \infty}^{\infty} h_{t n} t_{n} (k_{x}) ψ (k_{x}) e^{i k_{x} x + i k_{z n} z} d k_{x},

E_{t n} = η_{0} \int_{- \infty}^{\infty} e_{t n} t_{n} (k_{x}) ψ (k_{x}) e^{i k_{x} x + i k_{z n} z} d k_{x},

where

e_{t n} = (k_{z n} / (ε_{x}^{eff} k_{0}), 0, - k_{x} / (ε_{z}^{eff} k_{0}))

, h_tn = (0, 1, 0), and k_zn are approximately given, based on Eqs. (3) and (4), as

k_{z 1} \approx {[ε k_{0}^{2} - \frac{ε}{γ a^{2}} (6 ε_{z}^{0} - \sqrt{36 {(ε_{z}^{0})}^{2} - 12 γ k_{x}^{2} a^{2} + γ k_{x}^{4} a^{4}})]}^{1 / 2},

k_{z 2} \approx {[ε k_{0}^{2} - \frac{ε}{γ a^{2}} (6 ε_{z}^{0} + \sqrt{36 {(ε_{z}^{0})}^{2} - 12 γ k_{x}^{2} a^{2} + γ k_{x}^{4} a^{4}})]}^{1 / 2} .

The reflection coefficient r (k_x) in Eqs. (48) and (49) and the transmission coefficients t_n (k_x) in Eqs. (50) and (51) are given by Eqs. (22)–(24).

In practice, the infinite integrals in Eqs. (46)–(51) can be approximated by finite integrals over the interval [k₀ sinθ − Δk, k₀ sinθ + Δk], with the error on the order of $erfc (\frac{w_{0} Δ k}{2 cos θ})$ . A moderate value of Δk will be enough to give sufficient accuracy for the related integrals. After obtaining the fields of the incident, reflected, and transmitted beams, the power intensity is given as I = |〈S〉|, where $〈 S 〉 = \frac{1}{2} Re [E \times H^{*}]$ is the time-averaged Poynting vector for the total electric field E and total magnetic field H in vacuum or the layered medium.

Figure 5 is an illustration of the addition wave in the graphene layered medium with the incidence of a Gaussian beam. In Fig. 5(a), the dispersion curve on the wave vector domain is plotted for Ω = 0.0989, where the Poynting vector with the photon mode (S₁) and the polariton mode (S₂) are ordinarily (positively) and negatively refracted, respectively, at θ = 5°. The normalized power intensity of a Gaussian beam incident from vacuum at the same Ω and θ based on the Fourier integral formulation [cf. Eqs. (46)–(51)] is shown in Fig. 5(b). The transmitted beam of the polariton mode, which occurs as the additional wave, is directed toward a negative angle (ϕ₂ ≈ −73°) from the interface normal, compared to the beam of the photon mode with a positive angle (ϕ₁ ≈ 83°).

Fig. 5 (a) Dispersion curve on the wave vector domain at Ω = 0.0989 for the graphene layered medium with the parameters in Fig. 1. Black and gray contours are equifrequency curves for vacuum and the layered medium, respectively. Dashed lines indicate the continuity of K_x across the interface at θ = 5°. (b) Power intensity of a Gaussian beam incident from vacuum onto the graphene layered medium at the same Ω and θ. The intensity is normalized to have a maximum value of unity.

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5. Concluding remarks

In conclusion, we have established a systematic study on the additional waves that arise in the graphene layered medium. The additional wave appears as a distinctive feature of the nonlocal properties in the medium and becomes significant near the effective plasma frequency. An additional boundary condition based on modal expansions is proposed to resolve the underdetermined problem with Maxwell’s boundary conditions and to derive the generalized Fresnel equations. The reflection and transmission coefficients for the graphene layered medium can therefore be determined within the framework of nonlocal effective medium model. The proposed additional boundary condition shows results that are consistent with the condition of vanishing currents that are usually adopted for the wire medium. Finally, the additional wave is illustrated with the incidence of a Gaussian beam based on the Fourier integral formulation.

Appendix

The separable part X_n (x) of the n-th eigenmode for the graphene layered structure is given by the following form [9]:

X_{n} (x) = {\begin{matrix} A_{n} e^{i q x} + B_{n} e^{- i q x}, & 0 \leq x \leq ξ < a \\ C_{n} e^{i q x} + D_{n} e^{- i q x}, & - a < ξ - a \leq x \leq 0 \end{matrix},

where A_n, B_n, C_n, and D_n are the coefficients determined by the boundary conditions on the graphene surface, which satisfy Mx = 0, with x = (A_n, B_n, C_n, D_n)^T and

M = [\begin{matrix} 1 + \frac{σ q}{ω ε_{0} ε} & 1 - \frac{σ q}{ω ε_{0} ε} & - 1 & - 1 \\ 1 & - 1 & - 1 & 1 \\ e^{i q ξ} & e^{- i q ξ} & - e^{i k_{x} a} e^{i q (ξ - a)} & - e^{i k_{x} a} e^{- i q (ξ - a)} \\ e^{i q ξ} & - e^{- i q ξ} & - e^{i k_{x} a} e^{i q (ξ - a)} & e^{i k_{x} a} e^{- i q (ξ - a)} \end{matrix}],

where

q = \sqrt{ε k_{0}^{2} - k_{z}^{2}}

and k_z,n is the n-th root of k_z for a given k_x and ω, determined by the dispersion relation [9]

cos (k_{x} a) = cos (q a) - \frac{i σ q}{2 ω ε_{0} ε} sin (q a) .

If k_z,n satisfies the dispersion relation (56), the coefficients in X_n (x) can be solved by assuming D_n = 1 in Eq. (54). The other coefficients are obtained by solving the first three equations in Mx = 0 and given as

A_{n} = \frac{2 e^{i Q} (e^{i (Q + K_{x})} - 1) ε K_{0} - e^{i K_{x}} (e^{2 i Q} + e^{2 i Q ξ^{'}}) Q σ^{'}}{2 e^{2 i Q ξ^{'}} (e^{i Q} - e^{i K_{x}}) ε K_{0} - e^{i Q} (1 + e^{2 i Q ξ^{'}}) Q σ^{'}},

B_{n} = \frac{e^{i Q} {e^{i K_{x}} [Q σ^{'} + e^{2 i Q (ξ^{'} - 1)} (2 ε K_{0} + Q σ^{'})] - 2 e^{i Q (2 ξ^{'} - 1)} ε K_{0}}}{Q σ^{'} + e^{2 i Q ξ^{'}} [2 (e^{- i (Q - K_{x})} - 1) ε K_{0} + Q σ^{'}]},

C_{n} = \frac{(1 + e^{2 i Q ξ^{'}}) Q σ^{'} - 2 (e^{i (Q + K_{x})} - 1) ε K_{0}}{(1 + e^{2 i Q ξ^{'}}) Q σ^{'} - 2 e^{2 i Q ξ^{'}} (1 - e^{- i (Q - K_{x})}) ε K_{0}},

where Q = qa, K₀ = k₀a, ξ′ = ξ/a, and σ′ = σ/(ε₀c).

Using A_n, B_n, and C_n [Eqs. (57)–(59)] and D_n = 1 in X_n (x) [Eq. (54)], the entry ${(\underline{χ})}_{m n} = \frac{1}{a} \int_{x_{0}}^{x_{0} + a} X_{n} (x) e^{- i (k_{x} + 2 m π / a) x} d x$ is obtained by straightforward integration as

{(\underline{χ})}_{m n} = \frac{4 e^{\frac{1}{2} i (Q + K_{x})} Q {2 (2 m π + K_{x}) cos \frac{Q}{2} sin \frac{K_{x}}{2} σ^{'} - e^{i m π} [2 i ε K_{0} (cos Q - cos K_{x}) + Q sin Q σ^{'}]}}{(2 m π - Q + K_{x}) (2 m π + Q + K_{x}) [(1 + e^{i Q}) Q σ^{'} - 2 (e^{i Q} - e^{i K_{x}}) ε K_{0}]}

where ξ = a/2 and x₀ = −2/a are chosen for convenience.

Acknowledgments

This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 102-2221-E-002-202-MY3 and by National Natural Science Foundation of China under Grant No. 11304038 and Fundamental Research Funds for the Central Universities under Grant No. CQDXWL-2014-Z005. Work in Hong Kong was supported by Research Grant Council grant CUHK1/CRF/12G-1.

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Additional waves in the graphene layered medium

Abstract

1. Introduction

2. Effective medium model

2.1. Dispersion relation

2.2. Effective permittivity tensor

3. Additional wave

3.1. Photon and polariton modes

3.2. Approximate wave vector components

4. Additional boundary condition

4.1. Modal expansions

4.2. Vanishing currents

4.3. Gaussian beams

5. Concluding remarks

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (60)

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