All-optical controlling based on nonlinear graphene plasmonic waveguides

Jian Li; Jin Tao; Zan Hui Chen; Xu Guang Huang

doi:10.1364/OE.24.022169

1. Introduction

Benefited from the unique electronic and optical properties of graphene, graphene-plasmon polaritons (GPPs) become an emerging hot research topic in the field of plasmonics. Compared to surface plasmon polaritions supported by noble metals, GPPs have several appealing properties which are the key features for building the next generation photonic and optoelectronic devices [1, 2]. GPPs have the extreme field confinement with the volumes of are of ~10⁶ times smaller than the diffraction limit which has been verified by experiments [3]. Graphene exhibits a relatively large conductivity and high carrier mobility which translate into long optical relaxation times (τ~10⁻¹³ s), and thus could potentially provide a large plasmon wave propagation distances. More attractively, the Fermi level E_f of graphene can be adjusted by means of chemical doping and gate-voltage, making graphene to be a very promising material for active devices [2, 4]. Hence, applications of GPPs, such as waveguides [5–7], photonic modulators [8], filters [9] and metamaterials [10–13] have been widely investigated in the last decade. Actually, in these cases the Fermi level cannot be tuned in real time for chemical doping, and the operation speed of the devices is still limited by the switching time of the gate voltage. Therefore, it is significant to find a more effective way to manipulate the optical properties of GPPs.

Recently, the nonlinear properties of graphene have motivated the increasing attention of research groups. Its remarkable third-order nonlinear optical response and related large nonlinear susceptibility have been put forward both in theories [14, 15] and experiments [14, 16–18]. These works indicate that graphene will be a promising candidate for nonlinear photonic components. A number of applications based on Kerr effect of graphene have been proposed. Nasari et al. studied all-optical tunable notch filter by use of Kerr nonlinearity in the graphene microribbon array [19]. Bao et al and Xiang et al. investigated the optical bistability from a Fabry-Perot filled with graphene at visible [18] and terahertz frequencies [20]. These two studies are based on the self-phase modulation Kerr effect and are not waveguide structures.

In this paper, we address nonlinear cross-phase modulation in GPPs waveguides, and derive the complex effective refractive index with both linear and nonlinear effects. As one of the applications, an all-optical integrated nonlinear graphene plasmonic switch is proposed to achieve high extinction ratio, ultra-compact size and high speed. It may pave the way for all-optical integrated graphene-based devices in infrared applications.

2. Nonlinear properties of graphene plasmonic waveguides

The optical properties of two-dimensional (2D) graphene in the linear case can be described as a function of frequency ω, temperature T, carrier relaxation time τ and the Fermi level E_f, which have been analytically and experimentally studied extensively [21, 22]. They can be adjusted by varying the Fermi level, attributed to that the carrier density of graphene can be changed by chemical doping or gate-voltage. The complex surface conductivity of graphene dominated by intraband and interband transitions is estimated within the random-phase approximation as [23]:

σ (ω) = σ_{int r a} (ω) + σ_{int e r} (ω)

Where

σ_{i n t r a} = \frac{2 i e^{2} k_{B} T}{π ℏ^{2} (ω + i τ^{- 1})} l n [2 \cos h (\frac{E_{f}}{2 k_{B} T})]

σ_{i n t e r} = \frac{e^{2}}{4 ℏ} {\frac{1}{2} + \frac{1}{π} a r c \tan (\frac{ℏ ω - 2 E_{f}}{2 k_{B} T}) - \frac{i}{2 π} l n [\frac{{(ℏ ω + 2 E_{f})}^{2}}{{(ℏ ω - 2 E_{f})}^{2} + {(2 k_{B} T)}^{2}}]}

Here, k_B is the Boltzman constant, ħ is the reduced Planck constant, the carrier relaxation time τ is set to be 0.2 ps, the Fermi level E_f is set to be 0.2 eV in our simulation. Graphene is an anisotropic material with an out of plane permittivity of 2.5 obtained from the dielectric constant of graphite and with an in-plane permittivity characterized by [24, 25]:

ε = 1 + \frac{i σ (ω)}{ε_{0} ω t_{G}}

Where ε₀ is the permittivity of vacuum and t_G is the thickness of graphene which is set to be 1 nm in our simulation.

In the nonlinear case, the induced current of the graphene sheet with the consideration of the third-order nonlinear response can be denote as [20, 26]:

\overset{⇀}{J} = {\overset{⇀}{J}}_{L} + {\overset{⇀}{J}}_{N L} = [σ (ω) + σ_{3} (ω) {| E |}^{2}] \overset{⇀}{E}

σ_{3} (ω) = - i \frac{9}{8} \frac{e^{2}}{π ℏ^{2}} \frac{{(e v_{F})}^{2}}{E_{f} ω^{3}}

Where v_F is the Fermi velocity of 10⁶ m/s, a typical value of graphene and E is the electric field amplitude of the signal light near the graphene sheet. Note that the formula is under the condition of low frequency limit (ħω<2E_f), where the intraband response gives a dominant role and the small contribution of the interband response is not taken into account. It can be seen that the nonlinear response is a self-phase modulation (SPM) of optical Kerr effect, without considering two-photon absorption and the third harmonic generation [20, 26]. However, when the pump light with a different frequency of ω' and the electric field amplitude of E acts on a graphene sheet, the cross-phase modulation (XPM) of optical Kerr effect should be taken into consideration. The nonlinear response of the XPM is approximatively the twice of the one of the SPM under the non-resonant region where the material dispersion can be ignored in small frequency difference between the signal and the pump beams [27], and Eq. (6) turns into:

σ_{3}' (ω') = - i \frac{9}{4} \frac{e^{2}}{π ℏ^{2}} \frac{{(e v_{F})}^{2}}{E_{f} ω'^{3}}

Where ħω'<2E_f according to [20] and [26], thus the total conductivity of graphene with the nonlinear response can be written as:

σ_{t o t a l} (ω, ω') = σ_{i n t r a} (ω) + σ_{3}' (ω') {| E |}^{2}

As it can be seen from the above equations, the nonlinear response of graphene is proportional to the reciprocal of third power of the pump frequency and the square of the electric field amplitude of the pump light on the graphene sheet. It means that a giant nonlinear response of graphene will be triggered under the conditions of a low signal frequency and a high pump power density. On the other hand, when the energy of the pump light ħω'>2E_f (such as near-infrared), the pump light makes the interband excitation of the nonlinear part prominent. Thus the total conductivity including the linear intraband and the nonlinear interband contributions can be written as:

σ_{t o t a l} (ω, ω') = σ_{i n t r a} (ω) + σ'_{3, i n t e r} (ω') {| E |}^{2}

Where $σ'_{3}_{, int e r} (ω') ≅ \frac{2 e^{2}}{ℏ} {(\frac{e v_{F}}{ℏ ω'^{2}})}^{2}$ according to [14] and [28] and the factor 2 for the XPM effect. As an estimate, for the case without the pump light, it gives σ_intra(ω) = 4.41 × 10⁻⁶ + 1.44 × 10⁻⁴i for ω = 26 THz. However, in the present of the pump light with ω' = 281.95 THz (1064 nm) and the maximal E which is 2 × 10⁷ V/m in our paper, it gives the nonlinear contribution σ'_{3, inter}(ω')|E|² = 4.56 × 10⁻⁸, which is two orders of magnitude smaller than that of the intraband excitation (ħω'<2E_f). It only gives very slight contribution to the total conductivity and thus to the XPM effect. Therefore, we will only focus on the intraband contribution in the next sections.

Graphene can support the propagation of GPPs when it sandwiched between two dielectric mediums with permittivities of ε₁ and ε₂ above and below in the terahertz region. Under the condition of the wave vector of the graphene slab waveguide (k_sp) much larger than the free-space wave vector (k₀), the linear effective index of graphene slab waveguide can be calculated by Eq. (4) in [29]. Under the influence of the nonlinear process, the effective index n_eff including both linear and nonlinear parts, can be concluded as:

n_{e f f} = k_{s p} / k_{0} = ε_{0} \frac{ε_{1} + ε_{2}}{2} \frac{2 i ω}{σ_{t o t a l} (ω, ω')}

One can find that n_eff is a function of the signal frequency and the electric field amplitude of the pump light. Thus, a tunable all-optical controlling of GPPs will be formed by mean of adjusting the power density of the pump light.

Figure 1(a) and 1(b) exhibit the real and imaginary part of n_eff as a function of the signal frequency at different electric field amplitudes of the pump light of E = 0 (without pump light), 1 × 10⁷, 1.5 × 10⁷, and 2 × 10⁷ V/m, for a graphene sheet deposited on a silica substrate with the permittivity of 3.9. One can see that both of Real(n_eff) and Imag(n_eff) increase with the increase of E within the frequency range, and which are more obvious at E = 2 × 10⁷ V/m. In addition, the differences of Real(n_eff) and Imag(n_eff) between E = 0 V/m and the other E are enlarged with the increase of the signal frequency. The Imag(n_eff) at E = 0 V/m in Fig. 1(b) is invariable for the reason of the neglect of the interband response. Obviously, the above change of Real(n_eff) is the effect of the photo-induced refractive index change, which can be used to design the interferometric all-optical devices such as cavity-based modulators. Additionally, the change of Imag(n_eff) is the effect of the photo-induced absorption change, which can be used to design the absorption-type all-optical devices. The second effect will be used to construct non-resonant all-optical nonlinear graphene plasmonic switches, the details of which will be described in the next section.

Fig. 1 The Real(n_eff) and the Imag(n_eff) of graphene deposited on a silica substrate as a function of the signal frequency at different field amplitudes of the pump light with ω' = 28.3 THz.

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3. All-optical nonlinear graphene plasmonic switches

The structure of the proposed all-optical nonlinear graphene plasmonic switch is shown in Fig. 2. A graphene nanoribbon is deposited on a silica (SiO₂)/ silicon (Si) substrate, where the permittivities of SiO₂ and Si are set to be 3.9 and 11.9, respectively [30]. The width of the graphene nanoribbon is chosen as 50 nm, with which the nanoribbon can support the quasi-edge mode in the frequency range of interests. The total width of the switch is chosen as W = 500 nm to reduce the crosstalk with other devices. In the configuration, the GPPs as a signal light can be excited by a diffractive grating and propogates along the z direction. A 5 ps high-power CO₂ laser [31] with a spot size of 65 μm and a polarization angle of 45-degree to the x axis vertically incidents onto the switch within the upper limit of the power density that the graphene nanoribbon can withstand [32].

Fig. 2 Schematic of the all-optical nonlinear graphene plasmonic switch. The length and the width of the switch are L = 500 nm and W = 500 nm, respectively. The width of graphene is W_g = 50 nm, and the thickness of each buffer is H₁ = H₂ = 200 nm. The pump light with the electric field direction in the xz-plane is applied on the switch.

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The finite-difference time-domain (FDTD) method with perfectly matched layer (PML) absorbing boundary conditions is used to simulate the optical properties of the switch. A power monitor is set at the output end of the switch to measure the transmittance normalized to the input power. Figure 3 shows the transmittance spectra of the all-optical switch. In the situation of E = 0 V/m, the transmittance decrease monotonously from 0.3 to 0.25 with the increase of the signal frequency from 23 to 28 THz, because the Real(n_eff) and Imag(n_eff) change as the same trend in Figs. 1(a) and 1(b). Under the condition that the pump light with E = 1.5 × 10⁷ V/m irradiates on the switch, the transmittance decrease from 0.025 to almost 0 with the increase of the signal frequency from 23 to 28 THz. Therefore, the all-optical switch can be treated to be “ON” state with E = 0 V/m and can be treated to be “OFF” state with E = 1.5 × 10⁷ V/m. The extinction ratio of the switch (Fig. 3) increases as the increase of the signal frequency, with the minimal value of 11.17 dB at 23 THz and the maximal one of 18.14 dB at 28 THz. Defined that the extinction ratio of 10 dB can be used to distinguish the “OFF” state, the all-optical switch can achieve an over 5 THz bandwidth, showing a great broad-band property.

Fig. 3 Transmittance spectra of the all-optical nonlinear graphene plasmonic switch under the electric field amplitudes of the pump light of E = 0 and E = 1.5 × 10⁷ V/m.

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Figure 4 shows the E_y field profiles of the all-optical nonlinear graphene plasmonic switch at “ON” and “OFF” states, respectively. At the signal frequency of 23 THz, the case of the switch at “OFF” state is presented in Fig. 4(b), one can see that the propagation mode has more tight confinement and more attenuation than it at “ON” state in Fig. 4(a). The two field profiles are explicit proof to demonstrate the efficient operation of the proposed all-optical switch. Besides, at the signal frequency of 28 THz, the E_y field profiles of the “ON” and “OFF” states are also presented in Figs. 4(c) and 4(d). The four field profiles verify the broad-band modulation characteristic of the all-optical switch. Furthermore, the switching speed is determined by the combination of the delay time such as the material response time [33]. Due to the ultrafast response of the carrier relaxation of graphene, the switching speed of the all-optical switch is determined by the pump laser. Therefore, with the improvement of the laser, the switch may realize a potentially very high switching speed.

Fig. 4 The E_y field profiles of the all-optical nonlinear graphene plasmonic switch. (a) and (b) signify respectively the “ON” and “OFF” states at the signal frequency of 23 THz, (c) and (d) indicate the “ON” and “OFF” states at the signal frequency of 28 THz, respectively.

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The extinction ratio of the all-optical nonlinear graphene plasmonic switch can also be tuned by change the electric field amplitude of the pump light. In the case of the signal frequency at 26 THz, the extinction ratio as a function of the electric field amplitude of the pump light E is shown in Fig. 5(a). One can see that the extinction ratio increases from 5.35 dB to 40.97 dB with the increase of E from 0.2 × 10⁷ V/m to 2 × 10⁷ V/m. The modulation depth of 35.62 dB can be achieved at E = 2 × 10⁷ V/m. A larger extinction ratio would be achieved by a further increase of the E that the graphene nanoribbon can withstand. In addition, as it is shown in Fig. 5(b), we also investigate the extinction ratio of the switch according to different plasmonic modes for graphene nanoribbon with the width from 200 nm to 700 nm. It shows that the extinction ratio of the waveguide mode is nearly invariable and is about 15.75 dB, for the signal frequency of 26 THz and the electric field amplitude of the pump light E = 1.5 × 10⁷ V/m. Similarly, the extinction ratio of the edge mode is also nearly invariable and is about 15.5 dB, which is the same as the extinction ratio of the quasi-edge mode for the width of graphene nanaribbon of 50 nm in Fig. 3. Therefore, the width of the graphene nanoribbon has very little influence on the performance of the switch. The little distinction between the two modes is most probably because the optical field of the pump light overlaps with the field of the waveguide mode of the signal light better than that of the edge mode within the graphene nanoribbon.

Fig. 5 (a). The extinction ratio of the all-optical nonlinear plasmonic graphene switch as a function of the electric field amplitude of the pump light with the signal frequency of 26 THz. Figure 5(b). The extinction ratio of the switch as a function of the widths of the graphene nanoribbon for different plasmonic modes.

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4. Conclusion

We have given the complex effective refractive index of graphene plasmonic waveguides with both linear and nonlinear effects based on the nonlinear cross-phase modulation. The non-resonant all-optical nonlinear graphene plasmonic switch as one of the applications has been proposed and investigated. The switch has several features including an ultra-compact size of 0.25 μm², a broad bandwidth over 5 THz, a potentially very high switching speed. Our studies may open a window toward the future all-optical graphene photonic devices.

Funding

Project of Discipline and Specialty Constructions of Colleges and Universities in the Education Department of Guangdong Province (2013CXZDA012); Guangdong Natural Science Foundation (2014A030313446); Program for Changjiang Scholars and Innovative Research Team in University (IRT13064).

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All-optical controlling based on nonlinear graphene plasmonic waveguides

Abstract

1. Introduction

2. Nonlinear properties of graphene plasmonic waveguides

3. All-optical nonlinear graphene plasmonic switches

4. Conclusion

Funding

References and Links

Cited By

Figures (5)

Equations (10)

Optics Express