## 3 Design Optimization
### \(w\)  
For the waveguide structure in Figure 1, the width of the Ag strips \(w\) will definitely affect the mode field distribution. Therefore, Re(\(N_{\rm eff}\)), Re(\(\Delta N_{\rm eff}\)), MD and PL of the TM mode and complex mode in the DBOSW modulator versus the Ag strips \(w\) are investigated, when \(h_{\rm b}\) = 180 nm, \(h\) = 100 nm, and \(h_{\rm t}\) = 200 nm. As shown in Figure 9a, Re(\(N_{\rm eff}\)) of the TM mode gradually increases from 1.631 to 1.675 at \(\mu_{\rm c}\) = 0 eV, when \(w\) varies from 50 to 150 nm. Re(\(N_{\rm eff}\)) of the TM mode remains at 1.675, without changing with the increment in \(w\). Inversely, Re(\(N_{\rm eff}\)) of complex mode decreases from 1.712 to 1.673 at \(\mu_{\rm c}\) = 0 eV, when  
Figure 7: Re(\(N_{\rm eff}\)), Re(\(\Delta N_{\rm eff}\)), MD and PL of the TM mode and complex mode in the DBOSW modulator versus \(h\) are shown in (**a**), (**b**) and (**c**), respectively. The illustration in (**c**) shows the variation of electric field distribution for both modes versus \(h\). varies from 50 to 150 nm. A negligible change of Re(\(N_{\text{eff}}\)) of the complex mode is observed when \(w\) is over 150 nm. When \(\mu_{\text{c}}\) = 1 eV, Re(\(N_{\text{eff}}\)) of the TM mode and complex mode exhibit similar inverse variation with the increment in w, which leads to a turning point of Re(\(\Delta N_{\text{eff}}\)) at \(w\) = 150 nm, as shown in Figure 9b. As \(w\) changes from 50 nm to 200 nm, the MDs of the TM mode and complex mode decrease by 55.1 and 30.5%, respectively. Both modes have the same MD at \(w\) = 160 nm. When \(w\) ranges from 50 to 150 nm, the PL of the TM mode decreases from 0.16 to 0.14 dB/\(\upmu\)m, while PL of complex mode from 0.21 to 0.15 dB/\(\upmu\)m. Both PLs become stable when \(w\) is larger than 150 nm, as shown in Figure 9c. The decrement of the MD and PL can be attributed to the extension of the mode field region with the increment in w, which weakens the graphene-light interaction strength, as shown in the illustration in Figure 9c. Since the TM mode and complex mode are distributed differently in the middle slot and the top  
Figure 8: Re(\(N_{\text{eff}}\)), Re(\(\Delta N_{\text{eff}}\)), MD as well as PL of the TM mode and complex mode in the DBOSW modulator versus \(h_{\text{t}}\) are shown in (**a**), (**b**) and (**c**), respectively. The illustration in (**c**) shows the variation of electric field distribution for the TM mode and complex mode versus \(h_{\text{t}}\). Figure 9: Re(\(N_{\text{eff}}\)), Re(\(\Delta N_{\text{eff}}\)), modulation depth as well as propagation loss of the TM mode and complex mode in the DBOSW modulator versus \(w\) are shown in (**a**), (**b**) and (**c**), respectively. The illustration in (**c**) shows the variation of electric field distribution for both modes versus \(w\). metal slot, the increment in \(w\) has a greater impact on the MD of the TM mode. Moreover, the enlargement of the mode field area will weaken the field intensity, and then the metal absorption, leading to a lower PL. From the above, \(\text{Re}(\Delta N_{\text{eff}})\) shows a minimum, while the MDs and PLs of the two modes are very close at \(w\) = 150 nm. Therefore, we choose \(w\) to be 150 nm in following work. Another parameter to be confirmed in the proposed design is the gap width \(g\) between two upper metal parts. When \(h_{\text{b}}\) = 180 nm, \(h\) = 100 nm, \(w\) = 150 nm and \(h_{\text{t}}\) = 200 nm, \(\text{Re}(N_{\text{eff}})\), \(\text{Re}(\Delta N_{\text{eff}})\), MD and PL of the two modes in the DBOSW modulator versus metal slot width \(g\) are investigated. As shown in Figure 10, similar variations of \(\text{Re}(N_{\text{eff}})\) and \(\text{Re}(\Delta N_{\text{eff}})\) to that in Figure 9 can be observed. The difference is that the change of \(g\) has a greater impact on \(\text{Re}(\Delta N_{\text{eff}})\) than that on \(\text{Re}(N_{\text{eff}})\). As \(g\) increases, \(\text{Re}(\Delta N_{\text{eff}})\) decreases firstly and then gradually increases.