Fig. 1. (a) Schematic diagram of the graphene-supported MDM plasmonic waveguide. An external gate voltage Vg is exerted on the graphene. The inset shows the section-cross profile of the waveguide with graphene. Here, d1 (d3) represents the distance between the graphene and metal film below (above) the dielectric layer (silica, ϵd=2.25). d2 and L are the thickness and length of graphene, respectively. (b) Real and imaginary parts of the relative permittivity ϵg of graphene with the different bias voltages |Vb| at the wavelength of 1.49 μm in the waveguide with d1=45  nm and d3=5  nm.

Fig. 2. (a) Real and (b) imaginary parts of ERI (neff) of the plasmonic mode at λ=1.49  μm as a function of the bias voltage of graphene in the MDM plasmonic waveguide with d1=45  nm and d3=5  nm. The curves and circles denote the theoretical and simulation results obtained by the equations and FEM, respectively.

Fig. 3. (a)–(d) Normalized electric field and magnetic field profiles in the MDM waveguide at λ=1.49  μm when |Vb|=6 and 7.76 V.

Fig. 4. Transmission spectra in the graphene-supported MDM plasmonic waveguide [shown in Fig. 1(a)] when |Vb|=6 and 7.76 V. The inset shows the transmission as a function of the bias voltage in the waveguide with d1=45  nm, d3=5  nm, and L=670  nm.

Fig. 5. (a) Schematic diagram of the graphene-supported MDM plasmonic waveguide with the double side-coupled stubs. The inset shows the cross-section diagram of the plasmonic waveguide. l1 and w1 (l2 and w2) stand for the height and width of the stub 1 (stub 2), respectively. (b) Transmission spectra in the graphene-supported MDM plasmonic waveguide with the stub 1, stub 2, and double stubs when |Vb|=6  V. (c) Transmission spectra in the waveguide with the double stubs when |Vb|=7.76  V. The inset shows the ER of the modulation at different wavelengths. Here, d1=45  nm, d3=5  nm, l1=130  nm, w1=50  nm, l2=190  nm, w2=50  nm, and L=670  nm.

Fig. 6. (a)–(d) Normalized electric field and magnetic field distributions at λ=1.49  μm in the MDM plasmonic waveguide when |Vb|=6 and 7.76 V.

Fig. 7. ER of the EOM as a function of the carrier mobility μ of graphene in the MDM plasmonic waveguide without and with the double side-coupled stubs.