Electronically tunable extraordinary optical transmission in graphene plasmonic ribbons coupled to subwavelength metallic slit arrays

Subwavelength metallic slit arrays have been shown to exhibit extraordinary optical transmission, whereby tunnelling surface plasmonic waves constructively interfere to create large forward light propagation. The intricate balancing needed for this interference to occur allows for resonant transmission to be highly sensitive to changes in the environment. Here we demonstrate that extraordinary optical transmission resonance can be coupled to electrostatically tunable graphene plasmonic ribbons to create electrostatic modulation of mid-infrared light. Absorption in graphene plasmonic ribbons situated inside metallic slits can efficiently block the coupling channel for resonant transmission, leading to a suppression of transmission. Full-wave simulations predict a transmission modulation of 95.7% via this mechanism. Experimental measurements reveal a modulation efficiency of 28.6% in transmission at 1,397 cm−1, corresponding to a 2.67-fold improvement over transmission without a metallic slit array. This work paves the way for enhancing light modulation in graphene plasmonics by employing noble metal plasmonic structures.

and (d), the "EOT only" indicates the subwavelength metallic slit array without GPRs.
Weighted sum

Supplementary Note 1. Anti-crossing behavior in the coupled structure
As shown in Supplementary Fig. 3a, the coupled structure shows anti-crossing behavior at a crossing between the graphene plasmonic resonant mode and extraordinary optical transmission (EOT) resonant mode. We also observed that the frequency splitting depends on the number of graphene plasmonic ribbons (GPRs) inside the subwavelength metallic slit (N GPRs ) 1,2 , as shown in Supplementary   Fig. 3b. In this calculation, the pitch of the ribbons was determined by 800nm divided by N GPRs , where the 800nm corresponds to the subwavelength metallic slit width, and the graphene Fermi level for each N GPRs was tuned to minimize the transmission. To evaluate this relationship, we used the classical oscillator ΔΩ=frequency splitting, α=coupling coefficient, res −1 =resonance frequency

NGPRs=number of GPRs, γG & γE =absorption linewidth of bare GPRs and GPRs-EOT, respectively
In this model, we fitted our data of frequency splitting as a function of number of graphene ribbons N GPRs to extract the coupling coefficient α. The best fit was obtained with α=2.04×10 4 cm -2 , and the root mean square error was 2.39cm -1 , as shown in Supplementary Fig. 3c. We believe that the small deviation between the model and the calculated frequency splitting comes from assumptions in the classical oscillator model. The model assumes that a single GPR does not interact with adjacent GPRs, and the coupling coefficient is identical for all GPRs. In a real system, the graphene plasmons are a collective oscillation, which would affect the linewidth γ G . In addition, the coupling coefficient α would be altered depending on the position of each GPR. Regardless of discrepancies between the assumptions in the model and the real system, this model shows very good agreement with the calculated frequency splitting.
In addition, the coupled system exhibits a strong coupling when it contains six or more GPRs, as we can see by considering the frequency splitting and the average linewidth of the two resonant modes, as shown in Supplementary Fig. 3c. As a result of this coupling between two resonant modes, the splitting is also exhibited in transmission spectra, as shown Supplementary Fig. 3d.
To create strong coupling, the energy exchange rate should be faster than the decay rate of each resonant mode 3 . Therefore, the anti-crossing behavior disappears if the Q-factor of one resonant mode becomes too low, which happens with a low graphene carrier mobility. As shown in Supplementary Figs.   4a and b, the Q-factor of the GPRs becomes lower as the graphene carrier mobility is decreased. As a 8 result, the anti-crossing behavior in the coupled structure is nearly indiscernible at μ h =1,000cm 2 V -1 sec -1 , and completely disappears at μ h =450cm 2 V -1 sec -1 , as shown in Supplementary Figs. 4c and d. At a low graphene carrier mobility, there is no dip in the absorption spectra. A clear dip in the absorption spectra begins to emerge at μ h =1,500cm 2 V -1 sec -1 , and the frequency splitting is nearly saturated above μ h =5,000cm 2 V -1 sec -1 . This tendency is also observed in the transmission spectra, as shown in Supplementary Figs. 4e and f.

Fermi level
Silicon nitride (SiN x ) exhibits photoluminescence (PL) emission over the visible range 4 . Since the PL signal is much stronger than the Raman signal from graphene, the Raman peaks are almost indiscernible when the graphene is transferred onto the SiN x membrane. Therefore, we measured the Raman spectrum after transferring the CVD-grown graphene onto SiO 2 substrate, as shown in Supplementary Fig. 5a. The G-peak and the 2D-peak are located at 1595cm -1 and 2694cm -1 , respectively, and their ratio of I 2D /I G =2.04. The Raman spectrum shows that the D-peak (1348cm -1 ), which corresponds to defects in graphene, is very small, and the ratio of I G /I D =17.9.
To calculate the graphene Fermi level of graphene on SiN x membrane from the gate voltage (V g ) between the graphene and the back contact, we used a capacitor model 5 based on the charge neutral point (CNP) measured by a gate dependent resistance measurement of graphene 6,7 , as shown in Supplementary   Fig. 5b. In the calculation, we assumed the dielectric constant of SiN x as 10 (Supplementary Ref. 6). As shown in the Supplementary Fig. 5c, the graphene plasmon resonance frequency depending on graphene Fermi level between simulations and mid-infrared transmission measurement shows good agreement with this dielectric constant. The slight discrepancy between the simulation and the measurement results could come from atmospheric and substrate impurities 8,9 . Non-uniform DC electric field along the graphene ribbons, such as the lightning rod effect at the edges, could also affect the doping level.

Supplementary Note 3. Numerical fitting of the simulation with measured transmission spectra
A real experiment differs from simulations in several ways. First, the finite numerical aperture (NA) of the objective lens used in Fourier transform infrared (FTIR) microscope induces a broad angular 9 distribution in incoming light in contrast to purely normal incident light used in simulations. Second, some imperfections in fabrication could lower the graphene quality. Such factors could cause broad linewidth in the modulation spectrum and low modulation efficiency. To take into account these factors, we employed a low graphene carrier mobility to fit the linewidth, and a scaling factor to compensate the modulation efficiency 6 .
When it comes to the finite NA (0.58) of the objective lens, we simulated the structure varying the incident angle from -35° to 35° with 1° step. In the case of bare GPRs, the simulation results show that the NA effect is not so large, as shown in Supplementary Fig. 6a. Although the modulation efficiency decreases slightly as the incident angle increases, the line shape or the peak position do not change significantly. In this simulation, we used a graphene carrier mobility of 450cm 2 V -1 sec -1 , which results in good agreement between the simulation and measurement results in terms of linewidth of the modulation spectrum.
Supplementary Figure 6b shows the simulation and experimental data with E F =-0.542eV, which corresponds to the graphene Fermi level showing the maximum modulation efficiency in the coupled structure (GPRs-EOT) device. With aforementioned broad angular distribution of incoming light, a graphene carrier mobility of 450cm 2 V -1 sec -1 and a scaling factor of 0.633 to account for degradation, the simulation result matches the measurement result very well.
In contrast to the bare GPRs, the broad angular distribution of incoming light significantly affects the GPRs-EOT device because the EOT resonance itself strongly depends on the incident angle, as shown in Fig. 5. Such a strong dependence of modulation is shown in Supplementary Fig. 6c. In this simulation, we used the same graphene carrier mobility of bare GPRs. Similar to the EOT spectrum in Fig. 5, the modulation peak also blue-shifts with oblique incident light. As a result, the maximum modulation efficiency of the weighted sum spectrum is reduced by 20.3% compared with the modulation spectrum using purely normal incoming light.
In Supplementary Fig. 6d, we compared the measurement data with simulation results with a scaling factor of 0.734 to account for degradation. This value is slightly higher compared to the scaling factor for bare GPRs. We expect that there are less dead resonators in the GPRs inside the subwavelength metal slits than in the bare GPRs device because the dimension in transverse direction is much shorter compared with the bare GPRs structure, and therefore could reduce the chance of disconnection.