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We study the propagation characteristics of surface Plasmon polaritons (SPPs) on a patterned graphene sheet incorporating a subwavelength ribbon resonator and a Kerr nonlinear bounding medium (substrate or top cladding) which provides tunable bandpass filtering in the THz regime. We study theoretically and via modeling the tunability of maxima in the transmission spectrum, corresponding to the resonant frequencies of the ribbon resonator, by tuning the graphene Fermi level (via an applied gate voltage) and by altering the intensity of the incident THz wave. We determine the intensity-dependent increase in the refractive index of a Kerr nonlinear medium bounding graphene, via self-phase modulation and via the more efficient process of cross-phase modulation, revealing a noticeable red-shift in the resonant frequencies of the ribbon resonator. These concepts lead to ultrafast switching of SPP transmission through the ribbon (from a high to a low state). Using Kerr nonlinear media to bound graphene increases the tunability of graphene-based devices, enabling nonlinear plasmonic and ultrafast processing in the THz regime.
© 2016 Optical Society of America
1. Introduction
The last decade has produced a vast body of literature on graphene: a single atomic layer of sp2 hybridized carbon atoms which is tightly packed in a symmetric hexagonal honeycomb lattice. Doubt has recently been dispelled about the stability of 2D crystals after the first experimental isolation of graphene in 2004. The interest in graphene is due to its outstanding physical, electrical and optical properties, which originate from the unique smooth-sided conical band structure that converges to a single Dirac point [1–5].
The possibility of dynamic manipulation of electromagnetic waves via the high tunability of the graphene conductivity by means of external electrostatic or magnetostatic fields, chemical doping, or an applied gate voltage [6] has stimulated the use of graphene as a versatile material to realize a large variety of devices embracing a wide spectral range from low THz to optical frequencies, such as filters [7], lenses [8, 9], switches [10], antennas [11], absorbers [12], polarizers [13] and modulators [14].
Graphene can also serve as a good platform for further exploration of plasmonic devices. Surface plasmon polaritons (SPPs) sustained by a single graphene layer display some favorable properties compared to SPPs on noble metal structures. Graphene can support either transverse electric (TE) or transverse magnetic (TM) SPPs depending on the sign of the imaginary part of its conductivity - this is notable as there are no TE SPPs on noble metal structures. Furthermore, in contrast to noble metals which allow the propagation of SPPs primarily in the visible and near-infrared, graphene supports SPPs at low THz frequencies, opening the development of novel plasmonic devices in this part of the electromagnetic spectrum [3, 15, 16]. Actually, the lack of electromagnetic response of most natural materials at low THz frequencies has hindered the development of THz technology, yet this part of the spectrum holds the promise of many applications in, e.g., security, astronomy, communications and biology. Although the advent of metamaterials has partially addressed the materials gap at THz frequencies, the complexity in design and difficulty in fabrication are obstacles [17–19]; graphene as a new material may address this challenge.
Large area graphene sheets with dimensions up to 30 cm can be fabricated by chemical vapor deposition (CVD) [20]. Atomic force microscopy and Raman spectroscopy can be employed to determine the characteristics and number of produced graphene layers [21–23].
The field of nonlinear plasmonics, which involves the nonlinear response of a metal and/or of a bounding medium, is at the hub of significant research effort [24, 25]. The dependency of the nonlinear susceptibility, and the ensuing perturbation of the refractive index, by the intensity of the electric field, which is known as the Kerr effect, is among the most studied nonlinear effects in the literature [26, 27]. While the nonlinear response of bulk dielectric at moderate laser powers is weak, the strong light-matter interaction in the infrared and THz range provided by tunable graphene SPPs, in addition to the high sensitivity of SPPs to the properties of the surrounding dielectric, makes the nonlinear contribution from the Kerr environment a considerable factor [28, 29], this has yet to be employed in the design of functional THz graphene-based devices. In our previously published paper [30] we have employed the Kerr nonlinearity of graphene (modeled by the third order conductivity) in the low terahertz regime, to realize a tunable notch filter; the nonlinear response of bounding media was not taken into account. In this paper, we aim to study the tunability of the resonant frequencies of a subwavelength graphene ribbon resonator, constructed on an infinite graphene sheet, initially through electrical tuning (Fermi level gating), then by the intensity of the incident wave due to the Kerr nonlinear response of the lower or upper cladding via self-phase modulation (SPM) and cross-phase modulation (XPM) processes. (In another recently published work, only the nonlinear contribution of the substrate via the SPM effect was considered in tuning a Bragg reflector [31]).The structure is a novel configuration, providing tunable bandpass filtering, which given the fast response time of plasmonic excitations, the planar nature of graphene, and its potential to be integrated into a silicon-based platform, can find interesting applications in nonlinear plasmonics, and be employed as an efficient element in highly integrated circuits for ultrafast switching and processing in the THz regime.
2. Graphene conductivity
Before proceeding further, we recall some basic formulae for the graphene conductivity. In our analysis, graphene is modeled as a thin layer by a complex surface conductivity which arises from intraband and interband transitions of electrons. In the absence of a magneto-static bias and spatial dispersion, the graphene conductivity can be determined by the well-known Kubo formalism [6]. The intraband contribution, written assuming an exp( + jωt) time harmonic form, is [6]:
where e is the electronic charge, ω is the angular frequency, ℏ is the reduced Planck's constant, T is the temperature, Γ the phenomenological scattering rate, μc is the chemical potential and KB is Boltzmann's constant (note that the opposite sign for the imaginary part is used if an exp(-jωt) time-harmonic form is adopted).The interband term at room temperature and for ℏω < 2μc is very small compared to the intraband one and can be ignored. However for ω ≈2μc/ℏ it is the dominant term and cannot be neglected [29]. The interband term for KBT << |μc| can be approximated as [6]:
In all subsequent modeling, we assume T = 300 K and Γ = 0.43 meV [4]. In the presence of spatial dispersion (which we neglect), the conductivity components become wave number-dependent but still can be derived from the semi-classical Boltzmann’s equation. Spatial dispersion may affect the characteristics of graphene SPPs: in the particular case of magnetically unbiased graphene, it increases SPP losses and slightly reduces SPP confinement. However, in the frequency range of 1-50 THz, these effects are small, and the approximate graphene model described above provides accurate results for SPPs on the structure. For the case of a perpendicular magnetic field applied through the graphene sheet, the conductivity also becomes a tensor by direct and Hall terms [6, 32, 33].
3. Device structure
The structure of interest, depicted in Fig. 1(a), is composed of two semi-infinite graphene sheets, serving as access waveguides for propagating SPPs. A graphene ribbon of length L defines the resonator and is placed at a distance d from the input and output graphene sheets. The Fermi level of graphene can be tuned by doping, or more conveniently by a gate voltage applied to graphene, which renders the device dynamic, providing the ability of electrically tuning the resonant frequencies of the ribbon resonator. For this purpose, the whole structure can be deposited on a conventional SiO2/Si substrate. In this case, the Si substrate, properly doped, can act as a gate electrode, and the voltage can be applied between the substrate and, e.g., a Au electrode on the graphene layer.
In the absence of the resonator and output waveguide, the SPPs propagating on the input waveguide (Fig. 1(b)), are nearly completely reflected at the edge. Although the SPPs are highly confined and very little light is radiated from the edge, placing a graphene ribbon in the vicinity of the edge allows the evanescent field of SPPs on the input waveguide to excite resonating SPPs on the ribbon, and transmitted SPPs on the output waveguide. On resonance, constructive interference should occur for an SPP round trip [34]:
where β is the propagation constant of the SPPs, Re indicates the real part, m is an integer (the first resonance can be determined by placing m = 0) and φr is the phase induced upon reflection at the ends of the ribbon, commonly taken to be -π based on the assumption that the field vanishes at the ribbon edge. In this case, the graphene ribbon is modeled as a Fabry-Perot resonator with ideal mirrors which is oversimplified and fails to fully capture the resonant frequency of graphene ribbons. However by full electromagnetic field computation (using commercial software COMSOL) it has been shown that the introduced phase upon reflection is essentially independent of the intrinsic properties of graphene (Fermi level, temperature….), the wavelength of operation, and the dielectric environment, and is −3π/4. The origin of this phase is the excitation of evanescent modes close to the edge required to satisfy the continuity of electric and magnetic fields [34].The large momentum mismatch between incoming free-space waves and SPPs on graphene poses an excitation challenge. However, several momentum-matching mechanisms can bridge this, such as the Otto prism-coupling configuration, or enhancing the in-plane momentum of incoming light by using the apex of a tip or using plasmonic nano-antennas [9, 22, 23, 35, 36].
4. Simulation method
The two-dimensional subcell finite-difference time-domain (FDTD) numerical technique was used [37], with an equivalent model of the atomic thickness of graphene, to calculate the SPP propagation and power transmission along the x direction (the structure is invariant along y). The outer boundaries of the simulation region are terminated using convolutional perfectly matched layers (CPML) to dissipate outgoing waves [38]. The FDTD grid size is chosen as Δx = Δz = 10 nm which is small compared to the wavelength of propagating SPPs on graphene over the frequency range of interest [39] (the results for a smaller grid converge to the analytical ones). The time step is set to Δt = 0.85/[c(Δx−2 + Δz−2)0.5], following the courant stability condition, where c is the speed of light in free space. We choose our frequency range and the Fermi level of graphene such that the imaginary part of its conductivity is negative, and limit our attention to the propagation of TM SPPs (nonzero Ex, Ez, Hy) [30, 37, 38].
By defining a volume conductivity for a - thick graphene layer as σν ≡ σ/Δ, the volume current density and Ampere’s law can be written as J = σνE and ∇☓H = J + jωε0E = (σν + jωε0)E respectively. In this way, we can ascribe an equivalent complex permittivity εeq≡ε0 + σi/(ωΔ)-jσr/(ωΔ) and thus an equivalent effective linear susceptibility to the Δ - thick graphene layer, where σr and σi denote the real and imaginary parts of the graphene conductivity [4, 31].
By working in the frequency range where only the intraband term of the graphene conductivity contributes, the linear effective susceptibility can be written [31]:
where a is the fraction of the cell occupied by graphene, which in our computations is taken as 0.1. However, this particular value is not essential, as long as the chosen graphene thickness is extremely small compared to the wavelength [4].The key to modeling graphene in the FDTD method is the use of following constitutive relation [38]:
where we have assigned the linear polarization to characterize the linear dispersive nature of graphene and the nonlinear polarization to describe the third order Kerr nonlinearity of abounding medium. These polarization terms can be expressed as [31]: where is the third order susceptibility of the Kerr nonlinear medium. To calculate the electric field E, we use the time stepping expression [31]:where aL = 2G/(ZΔt + G), bL = (ZΔt-G)/(ZΔt + G) and cL = Fε0Δt2/(ZΔt + G), Δt is the time step, and ε∞ denotes the relative permittivity of the bounding dielectric medium and the graphene layer at infinite frequency [31].5. Computational results and discussion
5.1 Linear regime
The FDTD-calculated transmitted and reflected power for the structure of Fig. 1 with L = 1100 nm, μc = 0.1 eV and d = 200 nm are presented in Fig. 2, clearly revealing the formation of resonant modes in the ribbon resonator, and the band-pass filtering performance of the proposed structure. Before investigating the tunability of the structure, we should recall some crucial points relative to the graphene Fermi level and the dispersion relation of propagating SPPs on graphene. The tunability of the graphene Fermi level with applied gate voltage is a significant advantage of graphene-based plasmonic devices compared with their noble metal counterparts. The Fermi level of graphene can be determined via the carrier density, which in turn can be controlled by a gate voltage via [6]:
where ns is the excess charge carrier density, εr and h is the relative permittivity and thickness of the insulating layer (SiO2 in our case - Fig. 1(a)), vF = 9.5☓105 m/s is the Fermi velocity, ε0 is the permittivity of free-space and Vg is the applied gate voltage.
Fig. 2 FDTD-calculated transmittance and reflectance of SPPs propagating along the structure of Fig. 1 with L = 1100 nm, μc = 0.1 eV and d = 200 nm.
The graphene Fermi level for different thicknesses of the SiO2 layer, and distinct gate voltages, is computed and the results are plotted in Fig. 3(a).The dispersion relation for propagating TM SPPs on graphene is given by [15]:
where εr1 and εr2 are the relative permittivity of the bounding media (substrate and cover), β is the SPP wavevector and k0 = ω(μ0ε0)0.5 is the free- space wavenumber. Figure 3(b) plots the effective mode index of SPPs, nspp = Re{β/k0} vs. frequency for different values of relative permittivity of the bounding media and the Fermi level of graphene.Fig. 3 (a) Graphene Fermi level vs. gate voltage for different thicknesses of silicon dioxide. (b) Real part of the effective mode index of propagating SPPs as a function of frequency with the relative permittivity of the surrounding media and Fermi level of graphene as parameters.
To verify the computational accuracy of our numerical method in the linear regime, we compare the guided SPP wavelength computed using the FDTD method with the analytical result computed using (14). The steady-state spatial distribution of Ez of the SPP on suspended (no substrate) graphene is plotted in Fig. 4(a), from which the guided SPP wavelength λspp can be extracted. The results plotted in Figs. 4(b) and 4(c) show very good agreement between the FDTD computations and the analytic ones.
Fig. 4 (a) Spatial distribution of Ez in the x-z plane in the steady-state case for a suspended graphene sheet at the frequency of 5 THz and μc = 0.2 eV. Comparison of λspp extracted from the subcell FDTD results and (14): (b) as a function of the graphene Fermi level at the frequency of 5 THz; and (c) as a function of frequency at μc = 0.1 eV.
Figure 5(a) shows how the proposed band-pass filter can be tuned by altering the graphene Fermi level for the case depicted in Fig. 1(a) with L = 1100 nm and d = 200 nm. Increasing the graphene Fermi level reduces the effective SPP mode index and thus the confinement (Figs. 3(b) and 4(b)), which in turn increases the coupling strength between the input waveguide and the ribbon resonator (due to increased overlap of the evanescent fields). Additionally, as the graphene Fermi level increases, interband transitions decrease leading to lower loss. However to achieve critical coupling, where nearly no reflection on the input graphene waveguide is observed, coupled mode theory requires that the coupling strength and dissipative losses in the resonator be equal, which cannot be satisfied for all resonant modes and all graphene Fermi levels. From Fig. 2 we note that this condition is achieved for the first resonance, but not for the second and third ones (while at the first resonant frequency the reflection reaches to zero this is not observed at other resonant frequencies), due mainly to the losses in the resonator which increase with frequency. In Fig. 5(a), the same Fermi level is assumed for the resonator and waveguide regions. Tuning the Fermi level of only the ribbon resonator leads to the same resonant frequencies, but the frequency where critical coupling is achieved may change [40, 41]. From Fig. 3(b), it is evident that increasing the Fermi level decreases the propagation constant of SPPs, which in turn leads to higher resonant frequency according to (3). Figure 5(b) depicts the spatial distribution of Ex at the frequency of 4.88 THz for two values of graphene Fermi levels. A reduction in graphene Fermi level from 0.16 eV to 0.1 eV leads to a decrease in output power by about 20 dB.
Fig. 5 (a) Tunability of the proposed band-pass filter by altering the graphene Fermi level for the case of L = 1100 nm and d = 200 nm (structure depicted in Fig. 1(a)). (b) Spatial distribution of Ex in the x-z plane in the steady-state case at 4.88 THz for L = 1100 nm and d = 200 nm for two values of graphene Fermi level: μc = 0.1 and 0.16 eV.
Figure 6(a) compares the frequency of the first peak in transmittance vs. the graphene Fermi level computed using the FDTD with that obtained from (3), revealing very good agreement. As shown in Fig. 6(b), the quality factor (Q) of the first resonance, defined as the ratio of the resonant frequency to the peak width, increases with the graphene Fermi level, due mainly to the larger resonant frequency achieved at higher Fermi levels.
Fig. 6 (a) Frequency and (b) quality factor of the first resonant mode vs graphene Fermi level for the results of Fig. 5.
It is noteworthy that the Q’s of the second resonance are larger than those of the first; e.g., for μc = 0.1 eV in Fig. 2 the Q’s of the first and second resonances are 9.42 and 51.2. These values are comparable with the Q of localized surface plasmon resonances on noble metals such as gold Q ≅ 11 and silver Q ≅ 40 [42]. The higher Q of the second resonant mode on one side is attributed to its higher resonant frequency and on the other side to the larger effective mode index and higher confinement of SPPs on graphene at higher resonant frequencies which together overcome to the larger graphene loss at higher frequencies.
Figure 7(a) shows how the transmission spectra red-shift with increasing resonator length, in agreement with (3). The change in the first resonant frequency with resonator length is summarized in Fig. 7(b), showing very good agreement with (3). By increasing the resonator length, Q of the first resonance decreases as shown in Fig. 7(c).
Fig. 7 (a) Transmittance of the band-pass filter for μc = 0.15 eV, d = 200 nm and different resonator lengths (structure depicted in Fig. 1(a)). (b) Resonant frequency and (c) quality factor of the first resonant mode vs. resonator length for the results of Part (a).
The effect of an insulating layer below graphene, on the transmission spectra, is plotted in Fig. 8(a). For this purpose Polymethylpentene (TPX) and SiO2, having εr = 2.1and 3.9 [43], are chosen. Considering the results presented in Fig. 3(b), increasing the permittivity of the insulating layer leads to a red-shift in the first resonant frequency, as depicted in Fig. 8(b), following (3). Figure 8(c) states that sharper resonances are obtained for a larger permittivity of the insulating layer. For the sake of numerical simulation simplicity, the effect of charge accumulation along the edge of the graphene ribbon on its local conductivity was not taken account. However, including this effect may lead to more realistic results [44].
Fig. 8 (a)Transmittance of the band-pass filter with μc = 0.1 eV, L = 1100 nm, d = 200 nm and different values of the relative permittivity of the insulating layer below graphene. (b) Resonant frequency and (c) quality factor of the first resonant mode vs. relative permittivity of the insulating layer below graphene in Part (a).
5.2 Nonlinear regime
We proceed to characterize the nonlinear response of the proposed structure arising from the optical Kerr effect in the substrate. The dielectric constant of a Kerr nonlinear material is given by εd = εl + χ(3)|E|2. For the sake of illustration, in what follows, we take εl = 2.405 and χ(3) ≅ 6.4☓10−12 m2/V2 [29, 45–47], which can be found in semiconductors such as InSb [48–51]. Other nonlinear phenomena such as 2-photon absorption and 3rd harmonic generation are ignored. Including 2-photonabsorption as a loss mechanism leads to reduced efficiency but does not impede its operation under realistic condition. Moreover, to avoid complexity in simulation, we assume that the Kerr nonlinear medium is isotropic, homogenous and dispersionless over our frequency range. However, by incorporating the dispersive nature of InSb more pragmatic results would be expected [52].
In the presence of a Kerr nonlinear substrate, the dispersion relation in (14) is no longer valid by simply replacing the relative permittivity with its intensity dependent one εr1 = εl + χ(3)|E|2 (where |E|2 is taken at the interface of graphene and the nonlinear Kerr medium). It has been shown that in this case that the dispersion relation can be expressed as [28]:
in which it is assumed that the effective mode index of SPPs is much larger than the dielectric constant of the bounding media. Figure 9(a) shows the intensity dependence of λspp for propagating SPPs on the graphene layer for μc = 0.1 eV. The results obtained with the FDTD method show good agreement with those obtained from (15). The spatial distribution of Ez, in the linear and nonlinear regimes, for two different values of nonlinearity (χ(3)|E|2), are depicted in Fig. 9(b). It is noted that the number of guided SPP wavelengths over the domain considered increases from 3 in the linear case to ~4 in the nonlinear case χ(3)|E|2 ≅ 1.8, thus confinement increases with nonlinearity. In these computations εr2 was taken as 1.Fig. 9 (a) Intensity dependence of the guided SPP wavelength along graphene on a Kerr nonlinear substrate. The inset shows a sketch of the structure. (b) Spatial distribution of Ez in the x-z plane in the linear and nonlinear regimes.
Based on the above, the resonant frequency of the bandpass filter is expected to show sensitivity to the intensity. It is important to recall that the change in refractive index of the Kerr nonlinear medium can be achieved via SPM by the intensity of propagating SPPs, or by the more efficient process of XPM where the intensity of a pump light modifies the refractive index of the Kerr medium in the path of propagating SPPs. The refractive index change due to the signal intensity Is and the pump intensity Ip is Δn = n2(Is + 2Ip) where n2Is and 2n2Ip are the index changes associated with SPM and XPM. In what follows we investigate the tunability of the proposed filter due to the Kerr nonlinearity via SPM and XPM. (The device retains its electric tunability even in the nonlinear regime but placing a nonlinear medium in graphene’s environment adds an additional degree of tenability.)
SPM: Firstly, we investigate the tunability of the resonant frequency in the presence of a nonlinear substrate via SPM. For high SPP intensities and a substrate of high dielectric constant, the guided SPP wavelength decreases, and for a fixed resonator length, according to (3) a reduction in resonant frequency is expected. Figure 10(a) shows how the transmission spectrum undergoes a red shift with increasing SPP intensity for L = 625 nm, d = 50 nm and μc = 0.15 eV. For incident intensities up to 0.72 MW/cm2 at 350 GHz reduction in resonant frequency is achieved. The maximum extinction ratio of about 6 dB is computed in this case as illustrated in Fig. 10(b).
Fig. 10 (a) FDTD-calculated transmittance spectra for different values of incident SPP intensity due to a Kerr nonlinearity in the substrate. (b) Extinction ratio vs. frequency for an intensity of 0.72 MW/cm2. The inset shows a schematic of the structure investigated.
Now we consider the case where the substrate is linear (or weakly non-linear) such as SiO2, and the top cladding is a nonlinear medium. The merit of this configuration in comparison with the previous one can be understood by considering the fact that changing the substrate permittivity affects the graphene Fermi level when a fixed gate voltage is applied. For higher SPP intensities a reduction in resonant frequency is expected if the graphene Fermi level remains constant, e.g., when doped to a certain level. However, when a gate voltage is applied between graphene and the substrate, increasing the substrate permittivity through the Kerr nonlinearity increases the graphene Fermi level which in turn increases the resonant frequency. Although in this competition, the red-shift in resonant frequency due to the intensity-dependent increase of the Kerr medium refractive index is stronger than the blue-shift due to the rise in graphene Fermi level [31], the total resonant frequency shift for a certain level of intensity decreases. Thus, if simultaneous tuning via the gate voltage and the incident SPP intensity is required, the second configuration is preferred. The tuning of the transmission spectrum via the SPP intensity for L = 550 nm, d = 50 nm and μc = 0.4 eV is shown in Fig. 11(a). A 520 GHz reduction in resonant frequency is computed in this case, for an incident SPP intensity of up to 0.95 MW/cm2. Also, Fig. 11(b) shows that the maximum attainable extinction ratio is about 5 dB.
Fig. 11 (a) FDTD-calculated transmittance spectra for different values of incident SPP intensity due to a Kerr nonlinearity in the top cladding. (b) Extinction ratio vs. frequency for an intensity of 0.95 MW/cm2. The inset shows a schematic of the structure investigated.
XPM: More interesting results are achieved by employing the more efficient process of XPM. Initially, we study the case where a high-intensity pump is applied to the Kerr nonlinear substrate. The variation of the substrate permittivity in this case strongly affects the transmission characteristics of low intensity SPPs on graphene. We assume a CW (continuous wave) pump source (e.g., a CO2 laser) emitting at a frequency of 28.3 THz applied to the substrate, with L = 625 nm, d = 50 nm and μc = 0.15 eV, and compute transmission spectra. We note a significant red-shift with increasing intensity as shown in Figs. 12(a) and 12(b) for the first and second resonance modes. Figure 12(c) shows that at a pump intensity of 96.8 KW/cm2, maximum extinction ratios of 8.6 and 11.3 dB are obtained around the first and second modes respectively. The red-shifts in resonant frequency of the first and second modes in this case are computed to be 850 and 900 GHz, respectively, as illustrated in Fig. 12(d). These results are compatible with expectations as XPM is a more efficient process, requiring considerably lower levels of intensity compared to SPM, and producing greater tuning of the first and second resonant modes, along with larger extinction ratios for the filter.
Fig. 12 FDTD-calculated transmittance spectra for different values of pump intensity, in the XPM configuration, due to a Kerr nonlinearity in the substrate, near (a) the first and (b) second resonant modes. The insets show a schematic of the structure investigated. (c) Extinction ratio vs. frequency around the first and second resonant modes for a pump intensity of 96.8 KW/cm2. (d) First and second resonant frequencies vs. pump intensity.
The proposed structure in the XPM configuration can be used for ultrafast all-optical on/off switching. Figures 13(a) and 13(b) show a significant reduction of SPP intensity on the output graphene waveguide at 7.4 THz and 11.3 THz, corresponding to the first and second resonant modes, as the pump intensity is increased from 2 W/cm2 to 96.8 KW/cm2 (the pump is omitted from these plots for the sake of clarity because it is more intense than the SPPs).
Fig. 13 FDTD-calculated spatial distribution of Hy in the x-z plane at steady-state for the structure of Fig. 12, near (a) the first and (b) second resonant modes, in the linear and nonlinear regimes, demonstrating switching.
We now turn to the configuration studied in Fig. 11 where the Kerr nonlinearity is present in the top cladding and set L = 650 nm, d = 50 nm and μc = 0.4 eV. For XPM, the pump is assumed applied to the top cladding. Figures 14(a) and 14(b) show the red-shift in the transmission spectrum obtained around the first and second resonant modes with increasing pump intensity. The maximum extinction ratio at a pump intensity of 107.1 KW/cm2 is computed as 5.25 and 8 dB near the first and second modes, respectively, as noted in Fig. 14(c). At this intensity, we note from Fig. 14(d) a 470 and 480 GHz reduction in the first and second resonant frequencies. Optimization of the geometry can likely lead to larger shifts in resonant frequency and larger extinction ratios at lower pump intensities.
Fig. 14 FDTD-calculated transmittance spectra for different values of pump intensity, in the XPM configuration, due to a Kerr nonlinearity in the top cladding, near (a) the first and (b) second resonant modes. The insets show a schematic of the structure investigated. (c) Extinction ratio vs. frequency near the first and second resonant modes for a pump intensity of 107.1 KW/cm2. (d) First and second resonant frequencies vs. pump intensity.
It is expected that the refractive index change via XPM should be twice that achieved via SPM for equal signal and pump intensities. However, the pump employed in the XPM scheme was set to the frequency that maximizes the nonlinear response of the chosen Kerr medium. Furthermore, the SPP frequency is independent and can be selected along with the graphene geometry to decrease the propagation loss introduced by graphene in the structure. Thus, the efficiency of the XPM scheme can be much more than twice that achieved in the SPM scheme at the same signal and pump intensities [26, 53]. Depending on the desired application and power consumption criteria only one of these two processes or both of them simultaneously can be employed to modulate the intensity of propagating SPP wave.
It is worth noting that the mechanisms of laser-induced damage to graphene are different when illuminated by femtosecond (fs) pulsed lasers compared to CW lasers. With fs pulsed lasers, thermal and non-thermal effects contribute, whereas with CW lasers thermal effects play a dominant role. In terms of peak intensity, graphene can tolerate higher intensities for a shorter pulse at lower average intensity. In total, the reported damage threshold for fs pulsed lasers is Ith ~2.7 TW/cm2 which is much higher than observed for CW lasers ~1 MW/cm2.In our study, by working in the low terahertz frequency range, we have chosen the nonlinear dielectric such that the required intensity for an appreciable change in resonant frequency can be kept below this threshold level for CW lasers [30, 54–56]. However, it should be noted that if the graphene dielectric environment is lossy, a reduction in device efficiency (higher intensity for a given shift in resonant frequency) would be expected.
The intensities required for our predicted nonlinear responses can be provided by CO2 lasers or molecular terahertz lasers (up to 10 MW/cm2) [57]. Also, THz lasers with peak electric fields of ~4 MV/m have been constructed recently [58], and single-cycle THz pulses with electric fields greater than 100 MV/m have also been reported [59].
Finally, in comparing the nonlinear conductivity of graphene [30, 60, 61] with the nonlinear permittivity of a dielectric medium bounding graphene reveals that the nonlinear response of former is insignificant compared to the later over our frequency range of interest and for an intensity below 1 MW/cm2. However, it would be worthwhile to reanalyze the nonlinear responses of the system by accounting the nonlinear conductivity of graphene in order to better understand this limitation. Low THz frequencies and low-doped graphene are preferable to exploit the graphene nonlinearity at intensities below the damage threshold.
6. Conclusion
To conclude, we have investigated the tunability of the resonant frequencies of a graphene ribbon resonator constructed from a patterned graphene sheet, through an applied gate voltage modifying the graphene Fermi level, or through modifications of the refractive index of a Kerr nonlinear medium bounding graphene (as a substrate or top cladding), for SPPs propagating along graphene at THz frequencies. We have revealed that by taking advantage of XPM, where the intensity of a pump signal modifies the refractive index of a bounding Kerr nonlinear medium, much more efficient tuning can be achieved in comparison with SPM, where the intensity of the SPPs on graphene changes the refractive index of the Kerr medium. We have shown that noticeable intensity- and voltage-dependent shifts in the resonant frequencies of the ribbon resonator lead to a strong reduction of SPP transmission levels from the input graphene waveguide to the output one. The structure is attractive in the THz regime, for electro-optic and all-optical tunable filtering and ultrafast switching applications.
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