1. Introduction

Graphene, a two-dimensional form of carbon in which the atoms are arranged in a honeycomb lattice, possesses various unique optical, electronic, and mechanical properties, which have important scientific and technological implications [1, 2]. It has already created a significant impact on the fields of photonics and optoelectronics [3], where applications such as light sources [4, 5], photo-detection [6, 7] and light modulation [8] have been identified. Graphene is experimentally observed to support surface plasmonic polaritons (SPPs) [9,10]. Compared with the traditional SPPs in nobel metals, graphene plasmons are highly confined to a much smaller regions, and have much longer propagation distances [11, 12], which render the graphene as a promising material to be used as a deeply-subwavelengthed and lower-lossy waveguide, in the spectrum of infrared and THz regime. More importantly, the surface conductivity of graphene depends on chemical potential and can be tuned by means of chemical doping or gate voltage, locally and ultra-fast. Therefore, graphene provides a unique opportunity in the implementations of tunable ultra-thin waveguides.

A single-atomic layer of graphene confines plasmonic oscillations to the graphene plane but leaving unbounded in graphene plane. Reliable light waveguiding however necessarily deals with systems that are bounded in two or three dimensions, and for that, graphene with finite-extent like graphene ribbons have recently drawn much attentions [13–15]. Graphene waveguide can also be defined by properly patterning the surface conductivity across a flake of doped graphene [16]. Very recently, it was suggested that the lateral confinement of plasmonic oscillations is also possible in extended graphene layer(s), thanks to the arrest of the in-plane diffraction by the intrinsic optical nonlinearity of graphene [17–19]. Graphene is a promising nonlinear optical material, whose third-order nonlinear coefficient has been theoretically predicted [20] and experimentally verified [21, 22] to be very large. The nonlinear optical response of graphene can be further enhanced by the ultra-tight bounding of the optical fields onto the graphene sheet [23, 24]. Thus there have been increasing research interests in the exploitage of graphene to boost various nonlinear optical processes and to decrease the power requirement for novel nonlinear photonic devices [25–27]. In this context, the spatial self-phase modulation(SPM) of the graphene-based nanostructures have been exploited to counteract the beam diffraction leading to the formation of the self-guided beams, namely, the spatial solitons [17–19]. Compared with the usual spatial optical solitons [28] in dielectrics and their counterparts in metallic structures [29, 30], graphene-based spatial solitons are of much tighter confinements. They also have much longer propagation distance than their metallic counterparts. Moreover, due to the strong nonlinear repones of graphene, one may expect that their experimental observations at relatively lower optical powers are possible.

As mentioned above, a very appealing optical property of graphene lies in the fact that the graphene conductivity can be be tuned by means of external ways, such as chemical doping and gate voltage V_bias. A practical way to implement that is to use an uneven ground plane underneath the graphene layer[see an example in Fig. 1(a)], so that a spatially inhomogeneous distance is introduced between the flat graphene and the ground. Being applied a dc biasing voltage, the graphene layer will experience different values of local biasing dc electric fields at the position with different distance from the ground plane. As a result, a spatially inhomogeneous carrier density across the graphene plane will be accumulated and that creates a desirable spatial distributions of chemical potentials. Graphene patterned in this way has been suggested to be a one-atom-thick platform for infrared metamaterials and transformation optics [16]. Relating to the present study, it is therefore understandable that gating the graphene monolayer over a periodically corrugated ground will pattern the graphene with a periodic conductivity lattice. Thus, the doped and patterned graphene provides a unique and practical opportunity to realize photonic lattice at the deep-subwavelength scale.

 

Fig. 1 (a) Surface corrugated ground plane for achieving a conductivity modulation. (b, c) The band structure of the patterned graphene at chemical potential ${\mu }_c^0=0.3\hspace{0.17em}\text{ev}$ (b) and 0.5 ev (c). (d) Evolution of the bandeges versus chemical potential. Solid lines stand for β_r, and dashed for β_i. (e, f) The Bloch mode profiles at the first (e) and second bands (f), corresponding to the points labelled in (b). D = 400 nm, λ = 10μm.

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In this paper, we address the formation of a conductivity lattice across the graphene monolayer by gating a doped graphene over a periodically corrugated ground plane. The aim of the paper is twofold. First, we reveal that the modulation depth of the conductivity lattice can be controlled externally by the gate voltage, basing on which we demonstrate a tunable photonic band structure whose bandgap size and even the closure/opening of the bandgaps can be electrically controlled, dynamically and rapidly. Second, in both the semi-infinite and the finite gaps of such tunable photonic band structures, we predict the existences of graphene lattice solitons locaby a dielectric spacelized in both dimensions, where the in-plane localization is achieved through the balance of coupling between lattices by graphene intrinsic nonlinearity, while the out-of-plane localizations are achieved mainly by the graphene plasmonic effects. We calculate the transverse localization lengths of graphene lattice solitons, and find that they feature deeply-subwavelength scales in both directions, at a relatively lower power level.

2. Theoretical model

We start the analysis by considering the graphene structure as sketched in Fig. 1(a): A monolayer graphene which is infinite in the yz plane is held by a dielectric spacer. The ground plane underneath the spacer is assumed to be periodically corrugated along the y direction, while the uniformity of the ground in the z direction is maintained (z is the beam propagation direction). As described above, if a biased voltage is applied between graphene and ground, the surface conductivity will be periodically modulated across the graphene plane. Following we use a classical simplified capacitor model to understand the dependence of the resulting conductivity lattice on the corrugation geometry and external voltages.

We thus assume the distance between graphene and ground plane is given by, d(y) = d₀ + af(y), where d₀ is their separation before corrugation and the normalized function f(y) (|f(y)|_max = 1) describes the corrugation profile of the ground, with a being the amplitude of the corrugation. For simplicity we assume f(y) is a smooth function with a period D, and as a specific example, we choose f(y) = cos(2πy/D). We also assume that the surface is only weakly corrugated (a ≪ d₀)), therefore the use of a perturbative analysis is justified as detailed below.

As in [1], we treat a doped graphene as a parallel-plate capacitor, with a capacitance given by, $C_g=\frac{{\epsilon }_0{\epsilon }_{\text{r}}}{Ad}$. Here ε_r is the permittivity of the dielectric spacer, A the area of the capacitor and d the distance of plates. For simplicity we assume ε_r = 1.0. With the capacitor model, the structure composed of a graphene and a surface corrugated ground can be looked as a series of infinite narrow parallel capacitors whose capacitance varies continuously with coordinate y. This determines a spatially varying density of surface charges, n(y) = ε₀V_bias/[ed(y)]. Using this expression, one obtains the spatial distribution of chemical potential, ${\mu }_c(y)=\sqrt{n(y)\pi }\overline{h}{V}_F$ [12, 14], as following

Now we consider the linear conductivity of graphene, σ₁. Generally, σ₁ contains both contributions from intra- and interband transitions. However, in the range of THz and far-infrared spectrum, the intraband transition dominates [11] and the conductivity is simplified to ${\sigma }_1=\frac{i{e}^2{\mu }_c}{\pi {\overline{h}}^2(\omega +i{\tau }^{-1})}$ for a highly doped graphene (μ_c ≫ k_BT), where τ is the momentum relax time and K_B Boltzmann’s constant. In the present study, we use the excitation wavelength (in vacuum) λ = 10μm and τ = 0.3ps [14]. Further, as a frequently used technique in numerical simulations, graphene is modelled as an ultra-thin metallic layer with a thickness Δ (we assumed Δ = 0.50nm in the simulation) and endowed with an equivalent dielectric permittivity ${\epsilon }_1=1+\frac{i{\sigma }_1{\eta }_0}{k_0\mathrm{\Delta }}$, which, after the submission of Eq. (1), yields

Equation (2) explicitly shows that a one-dimensional(1D) photonic lattice is defined across the graphene monolayer, and the modulation depth of the lattice can be tuned through the external gate voltage, ${\mu }_c^0(V_{\text{bias}})$, dynamically and quickly. Finally, it should be noted that Eq. (2) only gives the permittivity distribution within the graphene region, and outside that region the permittivity is assumed to be always 1.

3. Results and discussions

3.1. Tunable bandstrutures

As described in the previous section, the periodically patterned graphene defines a tunable in-plane photonic lattice, and thus a tunable photonic band structures can be expected. To demonstration this, we find the eigenmodes (Bloch modes) of the quasi-1D photonic lattice by substituting the electromagnetic field of the form (E, H) = [E₀(x, y), H₀(x, y)]e^i(βz−ωt) into the full sets of Maxwell equations (MEs) and solving the resulting eigenproblem with the help of a mode solver integrated within the COMSOL Multiphysics suit [31]. Here E₀ and H₀ are the electric and magnetic components of the Bloch waves, respectively, and β = β_r + iβ_i is the complex propagation constants. The results are shown in Fig. 1, where both the propagation constants [Fig. 1(b)–1(d)] and some particular Bloch waves [Fig. 1 (e) and 1(f)] are presented.

A notable property observed from Fig. 1 is that, graphene photonic lattice, although by nature supporting plasmonic Bloch waves, their associated band structures exhibit a similar scenario as that of all-dielectric periodic lattice: β_r(k_y = 0) > β_r(k_y = π/D) for the first band, and β_r(k_y = 0) < β_r(k_y = π/D) for the second band, and so on. This is in sharp contrast to the cases for periodic metallic layers or nanowires where the photonic band structure are inverted [29, 30]. The restoration of band structures in graphene photonic lattices should be related to the fact that, despite modulations, the real part of the lattice permittivity defined in Eq. (2) remains negative in the whole graphene region.

As emphasized above, one appealing property of the graphene lattice lies in its highly tunability, and accordingly the tunable band structures. As shown in Fig. 1(b), when ${\mu }_c^0=0.3\hspace{0.17em}\text{ev}$ that corresponds to ε_bg = −214.01 + 4.25i + 10 cos(2πy/D) (Note that the modulation depth is much smaller than the mean value, a condition that was used in the derivation of Eq. (1)), the first and second band exhibits a clear bandgap in-between them. However, by increasing chemical potential to 0.5 ev that now corresponds to ε₁ = −367.79 + 6.73i + 17 cos(2πy/D), the bandgap closes [Fig. 1(c)]. The control of the bandgap-size with chemical potential is further illustrated in Fig. 1(d), where the dependence of propagation constants at the two bandedges on ${\mu }_c^0$ is plotted. Figure 1(d) also shows the lowering of both the first and second bands with the increase of ${\mu }_c^0$, indicating overall the band structure are shifting down along the vertical axis. As a result, after ${\mu }_c^0$ increases to some critical value, the second band starts immersing below the horizontal axis[see an example in Fig. 1(c)], a similar phenomena reported in 1D dielectric nanostructures too [32, 33].

Finally, Figure 1 also plots the imaginary parts of the propagation constants, β_i, that representing the modal loss upon propagation. As the modal loss is given by the spatially weighted average of the imaginary part of the complex permittivity over the modal field profile, β_i is found to be a function of the Bloch wavevectors k_y.

3.2. Graphene lattice solitons

After identifying the band structures and Bloch waves, we proceed with the lattice solitons by taking into account the nonlinear correction to the conductivity of graphene, σ₃, so that the total conductivity now takes as σ_g = σ₁ + σ₃|E_‖|², where E_‖ is the in-plane component of the electric field and the nonlinear conductivity coefficient is [20, 21]

In order to find lattice solitons we adopt a self-consistent method as described in [36]. Both soliton solutions with propagation constants residing in the semi-infinite and the first-finite photonic gaps are found. Figure 2 presents a typical soliton profile from the semi-infinite photonic gap. Such solitons origin from the Bloch modes at the top of the first bands (see mode labelled A in Fig. 1(e)), and thus can be recognized as unstaggered lattice solitons as the phase difference of the longitudinal electric component, E_z, between adjacent lattice is zero [Fig. 2(b)]. Please note that this is in sharp contrast to the previously reported plasmonic lattice solitons where the self-focusing nonlinearity supports only staggered soliton states in the semi-infinite gaps [30, 37], a property being determined by the band structure as descried above. Such 2D soliton states have different localization mechanisms at its two dimensions: in the perpendicular direction(with respect to the graphene plane), the localization is mainly due to the SPPs effect therefore soliton intensity features exponentially decaying away from the both sides of the graphene, see Fig. 2(c). The intensity dip in Fig. 2 (c) indicates the coupling nature of the graphene SPPs between its two interfaces(recall that we treat graphene layer as a ultra-thin metallic layer). In contrast, the wave localization in the in-plane direction is of nonlinearity origin, where the self-focusing nonlinearity arrests the EM tunneling among adjacent lattices [Fig. 2(d)]. As the conventional lattice solitons, the in-plane profile of the soliton features field modulations and on-site peaks.

 

Fig. 2 Profiles of a typical soliton at the semi-infinite gap. (a) and (b) show the transverse distributions of the amplitude (a) and imaginary part of E_z (b) of the electric component of soliton, respectively. (c) and (d) show the 1D profile of soliton along x axis (at y = 0) (c) and y axis (at x = 0), respectively. D = 400 nm, λ = 10μm, δε_g = 4.

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The properties of the unstaggered lattice solitons are summarized in Fig. 3(a)–3(c), where we show the dependence of propagation constants, effective width and soliton power on the nonlinear change of permittivity. As expected, with the increase of δε, the real part of the propagation constant (β_r) increases continuously, indicating the substantially penetrating of soliton solutions into the semi-infinite gap [Fig. 3(a)]. The modal losses (β_i) increase with δε too, but at a rate much slower than that of β_r. The deeply subwavelengthed confinement of solitons are evident from Fig. 3(b), where the localization length of solitons in both dimensions, w_xeff and w_yeff are plotted. The definition of the effective widths are, $w_{\text{xeff}}=\sqrt{\frac{{\int }_{-\infty }^{\infty }{\left|E(x,y)\right|}_{y=0}^2{x}^{2}dx}{{\int }_{-\infty }^{\infty }{\left|E(x,y)\right|}_{y=0}^{2}dx}}$ and $w_{\text{yeff}}=\sqrt{\frac{{\int }_{-\infty }^{\infty }{\left|E(x,y)\right|}_{x=0}^2{y}^{2}dy}{{\int }_{-\infty }^{\infty }{\left|E(x,y)\right|}_{x=0}^{2}dy}}$. The slowly decreasing characteristics of w_xeff(δε) indicate that, although the wave localization in x direction is dominated by graphene plasmonic effect, the nonlinearity further enhances the localization in this direction. The latter means more and more EM energy is pushed towards the graphene layer, thus we see a slowly increase of modal loss in Fig. 3(a). On the other hand, the in-plane localization w_yeff shows initially a strongly dependence on the nonlinearity, and then quickly saturates to a value that is nearly amount to a single lattice period (D = 400 nm). Note that, although deeply subwavelengthed in both dimensions, such lattice solitons require optical power only a few hundred μW to form [Fig. 3(c)]. In terms of the light intensity, a change of δε = 4.0 (corresponds to soliton solution in Fig. 2) requires a peak intensity I_peak = 6.15 ×10¹⁴[V²/m²], or 2.33 ×10⁹[W/cm²], which is well below the damage threshold of graphene for femtosecond pulses [38].

 

Fig. 3 Propagation constants (a,d), effective width (b, e) and power (c, f) for soltions in semi-infinite gap (left column) and the first- finite gap (right column). D = 400 nm, λ = 10μm.

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We also find gap soliton solutons in the first-finite gap (Fig. 4). Under a self-focusing nonlinearity, gap solitons are essentially the nonlinear localized version of the Bloch modes from the edges of the second band(see the Bloch mode labeled as D in Fig. 1(f)). Therefore, gap solitons can be termed as staggered modes since the phase difference of E_z component between adjacent lattices is π [see Fig. 4 (b)]. As conventional gap solitons, graphene gap solitons feature amplitude modulations where the local maximums are achieved at the lattice minimum. Therefore, in contrast to the lattice solitons in the semi-infinite gap that always have a single intensity peak, gap solitons feature pairs of peaks of equal intensity. The properties of gap solitons are summarized in Fig. 3(d)–3(f). Under the same strength of nonlinearity, gap solitons have larger w_yeff than solitons in the semi-infinite gap, while their w_xeff are almost the same. At stronger enough nonlinearity, w_yeff saturates to almost 100 nm which is slight larger than 2D. Due to the same reason, the excitation of the gap solitons requires higher power [Fig. 2 (f)].

 

Fig. 4 Profiles of a typical soliton in the first-finite gap. (a) and (b) show the transverse distributions of the amplitude (a) and imaginary part of E_z (b) of the electric component of soliton, respectively. (c) and (d) show the 1D profile of soliton along x axis(at y = 0) (c) and y axis (at x = 0). D = 400 nm, λ = 10μm, δε_g = 6.

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4. Conclusions

In conclusion, we suggested to use a periodically patterned graphenen monolayer to achieve a quasi-one-dimensional photonic lattice at the deep-subwavelength scales. We demonstrated that the modulation depth of such photonic lattice and their associated band structures can be controlled dynamically and quickly, through an external gate voltage. We also predicted the existence of the graphene lattice solitons whose propagation constants residing in the semi-infinite or the finite photonic gaps. Graphene lattice solitons are deeply-subwavelengthed localized in both dimensions due to the combination of the intrinsic nonlinearity of graphene and its plasmonic characteristics.

It is straightforward to expect that, using the same scheme as in our work but corrugating the surface of the underground plane with other topologies (instead of the periodic pattern), various novel in-plane photonic lattices can be created, such as chirped lattices, defected lattices, binary lattices, and yet their properties can be electrically controlled quickly. Therefore, the underground-mediated patterned-graphene provides an appealing platform to implement the tunable nanophotonics and tunable nonlinear nanophotonics.