Figure 1a shows a sketch of the graphene/D-shaped fiber (GDF) used in this work. A section of a silica single mode fiber was side-polished, and a monolayer graphene was deposited on top of the planar surface (see Methods for details). Two gold electrodes (channel width ≈ 200 µm) are used for the electrical tuning of the graphene’s Fermi level . The fiber core is 6 µm in diameter and light-graphene interaction occurs via evanescent waves ; the total linear losses of the graphene-fiber heterostructure are < 0.5 dB. Further details on the nanofabrication of the GDF are provided in Supplementary Section S2. In the DFG process used for plasmon generation, graphene acts both as the second-order nonlinear medium and as the nanoscale plasmon waveguide. Driven by the out-of-plane second-order nonlinear susceptibility χ(2) of the GDF, the high frequency pump photon (fpump) splits into a (counter propagating) lower frequency probe photon (fprobe, counter-propagating) and a plasmon (fsp, co-propagating). The DFG requires both energy conservation fsp + fprobe = fpump and momentum conservation kpump = ksp + kprobe, where ħksp, ħkprobe and ħkpump are the momenta of the plasmon, probe and pump respectively. Considering the scalar dispersion relation k = 2πfn/c, the momentum conservation can be re-written as fspnsp = fpumpnpump + fprobenprobe. In order to generate plasmons in graphene via the DFG process, all the optical modes must have transverse magnetic (TM) polarization that probes the non-centrosymmetric out-of-plane direction of the hybrid device, while graphene is centrosymmetric for in-plane excitation [39, 40]. As an example, Fig. 1b simulates the side-view electric field distributions of the pump and the plasmon modes when fpump = 192 THz and fsp = 10 THz.
Figure 1c shows a top-view optical image of the GDF. The fiber core is denoted by a white dashed line and graphene is deposited on the fiber and connected via the source and drain electrodes (Au). Unlike typical back-gate field-effect-transistors , this on-fiber Au-graphene-Au transistor is driven by current rather than gate voltage . Within the Drude’s model, the surface plasmon frequency fsp is defined by the graphene’s dispersion relation which, in turn, depends on the Fermi level: a higher carrier density induces a higher fsp. In our GDF architecture, by changing the source-drain voltage (VD) in the range 0 to 1 V, we can tune the effective graphene Fermi level |EF| from ≈ 0 to 0.4 eV. Fig. 1d (top panel) plots the measured device resistance and the calculated |EF| as a function of VD. When VD = 0 V graphene is intrinsically positively charged (p-doped) with |EF| ≈ 0.1 eV. By increasing VD we drive a current in the graphene channel and shift EF towards the Dirac point and the upper Dirac cone (n-doping). The graphene’s resistance reaches the maximum (Dirac point) when VD = 0.16 V. To confirm the VD dependence of EF, we have also characterized the doping via in-situ Raman spectroscopy (see Supplementary Section S3).
The bottom panel of Fig. 1d plots the two frequency combs that are used for the DFG experiments. Comb 1 (blue curve, pump) is a stabilized Er mode-locked laser with tunable central wavelength ≈ 1560 nm, 3-dB bandwidth ≈ 7.5 THz, and maximum average power of ≈ 30 mW. Meanwhile, Comb 2 (red curve, probe) is spectrally flat in the wavelength region ≈ 1500 nm - 2100 nm and it is obtained by super-continuum generation starting from Comb 1 (see Methods for details). The two combs are locked with the same repetition rate of ≈ 38 MHz and their temporal overlap can be controlled experimentally by a delay line. In the DFG experiments, the two combs are launched into the GDF from opposite directions (the experimental setup is shown in Supplementary Section S3). We further note that in order to achieve co-generation of multiple plasmons and detect them accurately, the pump should be spectrally sharp while the probe must be broadband and spectrally flat. The following scenario can now occur: for those frequencies that satisfy the phase-matching condition, the simultaneous presence (time-overlap) of the pump (Comb 1) and the probe (Comb 2) will lead to the generation of a plasmon and to the enhancement of the counter-propagating probe (Comb 2). Thus, the plasmon generation and the DFG process can be detected as an increase of the probe intensity at a specific phase-matched frequency (ΔIDFG), in analogy with the widely used process of optical parametric amplification .
For example, for a pump wavelength of 1560 nm (192.308 THz) and a nonlinearly excited plasmon at fsp = 10 THz, we will observe a peak on the flat spectrum of Comb 2 (probe enhancement) at ≈ 1645.57 nm (182.308 THz). To further prove that this peak arises from DFG, we modulated the pump at 500 kHz and observed the same modulation in the counter-propagating probe comb (see Supplementary Section S3). In contrast to conventional schemes that use continuous-wave tunable lasers for plasmon generation , our two-combs approach doesn’t require time to scan the laser wavelengths and operates with ≈ 2 orders of magnitude less average optical power. More importantly, due to the large bandwidth (50 THz) of the probe comb, this scheme enables us to find high frequency plasmons beyond the limitation of any near-infrared tunable lasers.
Figure 1e shows the parametric space of the DFG process. To generate the THz graphene plasmons, the phase-matching condition for the counter-pumped DFG and the dispersion of the plasmonic modes must match. For free-standing graphene, the plasmonic dispersion is defined by the Drude’s model (ksp∝ fsp2) : The blue curves in Figure 1e are exemplary cases when |EF| = 0.1 eV and 0.4 eV. On the other hand, the phase-matching condition for DFG can be re-written as (c/2π)ksp=−fspnprobe + fpump(npump + nprobe). When tuning the pump comb from lower to higher frequency (e.g., from 1530 nm to 1610 nm as shown in Figure 1e), the red curve moves from right to left. Since graphene plasmons are generated only at the intersections of the graphene dispersion curves and the DFG phase-matching lines, by tuning the graphene’s Fermi level and the pump frequency, the plasmons’ frequency will shift within the yellow region of Figure 1e.
Figure 2 shows the electrical tunability of our device. In the measurements, we have a central fpump = 192.3 THz (1560 nm) while the probe pulse is broadband. Fig. 2a, 2b and 2c show the graphene’s plasmon dispersion calculated using the random phase approximation (RPA)  at |EF| = 0.1 eV, 0.2 eV and 0.3 eV respectively. Here, we considered two surface-optical phonon resonances [44, 45] of the silica substrate (fiber), located at f1 = 24 THz and f2 = 35 THz (white solid lines). The phase matching condition for the DFG is marked by the white dashed line and we used for the refractive inside the fiber’s core npump ≈ nprobe = 1.45. When |EF| = 0.1 eV (Fig. 2a) the graphene’s plasmon is far below the silicon phonon frequency, hence we can only see one DFG peak at 1623 nm (fsp = 7.5 THz). When |EF| = 0.2 eV (Fig. 2b) the graphene’s plasmon interacts with the 24 THz phonon resonance, dividing the Drude curve into two branches. In this case, we observe two DFG peaks located at 1655 nm and 1804 nm (fsp = 11 THz and 26 THz). Finally, when |EF| = 0.3 eV (Fig. 2c) hybridization of the graphene’s plasmon with the substrate phonons leads to three branches and we observe enhanced DFG peaks at 1672 nm, 1813 nm and 1923 nm (fsp = 13 THz, 27 THz and 36 THz). In addition, based on the dispersion relation nsp = (fpumpnpump+ fprobenprobe)/(fpump – fprobe), we can estimate the effective refractive index of the plasmonic modes. For instance, when fsp ≈ 7.5, 27 THz and 36 THz we obtain nsp ≈ 68, 19 and 14, suggesting strong confinement of the plasmons. The simulated fiber mode indices are shown in Supplementary Section S1.
In order to further confirm the nature of the observed signal and distinguish it from other possible side effects (e.g., saturable absorption), in Fig. 2c and 2d we analyze the conversion efficiency of ΔIDFG. In particular, the generation of plasmons and the consequent intensity enhancement at fprobe relies on the consumption of the pump. We thus fix the probe power to 10 mW while increasing the pump power, and we measure both the pump transmission and the total ΔIDFG at different values of |EF|. For low pump power values (< 8 mW), the curves at |EF| = 0.1 eV and 0.2 eV are almost identical, while the curve at |EF| = 0.3 eV shows higher transmission, likely due to reduced absorption in the thermally broadened Fermi Dirac distribution (considering the pump photon energy of ≈ 0.8 eV). Initially, the transmission curves for all |EF| values increase with pump power and reach a plateau at ≈ 12 mW due to saturable absorption . Subsequently, for pump powers > 12 mW, graphene is fully saturated and a further increase of the power leads to a lower transmission due to pump consumption via the DFG process. For the same value of the pump power (> 12 mW) we observe that the ΔIDFG raises from the noise floor and subsequently increases linearly (Fig. 2d), as expected for DFG considering Isp = ΔIDFG(fsp/fprobe) = [χ(2)]2IpumpIprobe/Lsp2, where Lsp is the transmission loss of the plasmon. From a linear fit of the DFG conversion efficiency in Fig. 2d, we obtain a [χ(2)]2/Lsp2 at |EF| = 0.3 eV of ≈ 10−4 W−1, in agreement with theoretical values .
In Fig. 3 we show the ultrafast nature of our method by scanning the delay between the pump and probe combs while measuring the DFG enhanced signal (ΔIDFG), in analogy with an auto-correlation measurement (Fig. 3a, see also the implementation in Supplementary Video S1). Fig. 3b, 3c and 3d show the time-dependent DFG for VD = 0 V (|EF| = 0.1 eV), VD = 0.4 V (|EF| = 0.2 eV), and VD = 0.6 V (|EF| = 0.3 eV) respectively. At zero delay time we observe the largest nonlinear enhancement ΔIDFG for all EF values. In this measurement, the central pump wavelength is 1560 nm and the time delay is tunable from -5 ps to +5 ps. The soliton pulse width for the pump and probe combs are ≈ 430 fs and ≈ 110 fs respectively (Supplementary Section S3). The enhanced probe intensities are plotted in the bottom panels of Fig. 3b, 3c and 3d: all the ΔIDFG peaks have a time-width of ≈ 500 fs in sech2 fitting, as expected from a parametric process given the pulse duration of the pump and probe pulses used in our experiments. Such fast response of the GDF device is highly promising for high-speed devices and logic operations.
Finally, the co-existence of multiple electrically tunable plasmons allows us to perform logic operations, as schematically shown in Fig. 4a. Two parallel electrical signal generators (VA and VB, the other contact is grounded) are used as input and each of them can provide either 0 V (OFF state, digital signal 0) or 0.5 V (ON state, digital signal 1). By combining the different states of VA and VB (i.e., both ON, only one ON or both OFF), the graphene’s EF can be tuned to 0 eV, 0.24 eV and 0.4 eV. The pump and probe beams are launched into the GDF in opposite direction and we detect the gate-tunable plasmon generation via the DFG as an increase in the probe intensity (ΔIDFG) at specific frequencies. For logic operations, three ΔIDFG peaks (defined by the three plasmonic dispersion branches) with fsp in the 0 ~ 50 THz band are filtered using three band-pass filters (BPF) based on fiber Bragg gratings (the spectral characterization of the filters is shown in Supplementary Section S3).
Figure 4b explains the filtering scheme more in details in the retrieved ‘EF-fsp’ map. When increasing the |EF| from 0.1 eV to 0.4 eV, the fsp of the three branches shifts in different ways. For instance, the low frequency branch shifts from 7.5 THz to 12.2 THz, while the middle and high frequency branches experience almost no shift (< 1.5 THz). On the other hand, as previously discussed (see Fig. 2), the middle frequency branch at 26 ~ 27 THz appears only for |EF| > 0.15 eV, while the high frequency branch at ≈ 37 THz appears for |EF| > 0.25 eV due to plasmon-phonon interaction. By selecting the filtering frequency to 7.5 THz, 27 THz and 37 THz with respect to the pump frequency, we can thus obtain the three different optical logic outputs. In particular, the ΔIDFG from BPF3 is detectable only when both VA and VB are ‘ON’ (AND gate); the ΔIDFG from BPF2 is detectable when either VA or VB is ‘ON’ (OR gate) while the ΔIDFG from BPF1 is detectable only when both VA and VB are ‘OFF’ (NOR gate). Fig. 4c shows a measured example of the optical logic operations. We design two square-wave modulated signal traces for both VA and VB (0.5 V RZ code, sampling rate 1 MHz, data stream 100 kbps). Accordingly, the different BPFs provide at the output the logic operation expected from the AND, OR and NOR gates. Specifically, the signal-to-noise ratios of the AND, OR, and NOR gate outputs are higher than 89%, 82%, and 77% respectively, thus allowing a precise discrimination of the logic operation.