## 3 Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(b\)/\(a\) & \(n_{\text{eff}}\) & \(L_{\text{P}}\) (\(\upmu\)m) & \(FWHM\) (\(nm\)) \\ \hline
2 & 3.0709 + 0.19168\(i\) & 83 & 140 \\
3 & 2.7599 + 0.17064\(i\) & 93 & 100 \\
5 & 2.4004 + 0.14512\(i\) & 110 & 75 \\
10 & 2.0191 + 0.11677\(i\) & 136 & 40 \\
15 & 1.8561 + 0.10439\(i\) & 152 & 36 \\
20 & 1.7646 + 0.09751\(i\) & 163 & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The mode properties versus \(b\)/\(a\), and \(a\) = 0.5 \(\upmu\)m, \(u_{\text{c}}\) = 0.5 eV, \(\varepsilon_{1}\) = 2.25, and \(\varepsilon_{2}\) = 1. Figure 2: The time-averaged Poynting vector \(S_{\text{z}}\) along \(y\) axis for (**a**) \(b\)/\(a\) = 2, \(u_{\text{c}}\) = 0.5 eV, and (**b**) \(b\)/\(a\) = 10, \(u_{\text{c}}\) = 0.5 eV, 1 eV at 3 THz. The center coordinates of the ellipses are (0, 2 \(\upmu\)m) and (0, 10 \(\upmu\)m). Here, \(a\) = 0.5 \(\upmu\)m, \(\varepsilon_{1}\) = 2.25, and \(\varepsilon_{2}\) = 1. The insets show the enlarged detail of \(S_{\text{z}}\). the graphene-coated circular nanowire (GCCNW) is not the optimal choice. The GP mode in GCENW had superior performances (\(b/a\) ranging from 1 to 3) both in propagation length and mode area when compared with the GCCNW (\(b/a=1\)). We will discuss this in Section 4. In addition, the \(FoM\) had a minimum value of \(b/a=1.5\). Therefore, in order to achieve a longer propagation length and smaller mode area, larger \(b/a\) values are preferred. The results are in consistent with that of Figure 2.  
Figure 3: (a) Real part of \(n_{\text{eff}}\) and \(L_{\text{P}}\), and (b) \(A_{\text{eff}}/A_{0}\), and figure of merit (\(FoM\)) of the GP mode as a function of frequency. Here \(u_{\text{c}}=0.5\) eV, and \(a=0.5\) μm. The thick and thin lines stand for \(b/a=2\) and \(b/a=3\), respectively. The permittivity of nanowire provided another degree of freedom to be adjusted, which also had a great influence on GP mode in GCENW. Figure 5a shows the relationship between Re(\(n_{\text{eff}}\)), \(L_{\text{P}}\) and the permittivity at 3 THz. With increasing \(\varepsilon_{1}\), Re(\(n_{\text{eff}}\)) almost linearly increased as well as the loss (i.e., decrease of \(L_{\text{P}}\)). Figure 5b presents \(A_{\text{eff}}\)/\(A_{0}\) and \(FoM\) with respect to nanowire permittivity. The change of \(A_{\text{eff}}\)/\(A_{0}\) with increasing permittivity was very small. For \(\varepsilon_{1}\) = 10 and \(\varepsilon_{1}\) = 2, the relative error of the normalized mode area was less than 10%. Thus, the permittivity seemed to have a very slight impact on \(A_{\text{eff}}\)/\(A_{0}\). Therefore, smaller dielectric permittivity results in better performance of the GP mode. The surface conductivity \(\sigma_{\text{g}}\) of graphene could be adjusted by changing chemical potential \(u_{\text{c}}\). Figure 6 shows the dependences of the GP mode on the chemical potential at \(f_{0}\) = 3 THz. When the chemical potential increased from 0.2 eV to 1 eV, the effective mode index gradually decreased, while \(L_{\text{P}}\) linearly increased, as depicted in Figure 6a. The normalized mode area was enlarged by less than 2-fold when \(u_{\text{c}}\) ranging from 0.2 eV to 1 eV, shown in Figure 6b. For \(b\)/\(a\) = 2 and \(u_{\text{c}}\) = 1 eV, \(L_{\text{P}}\) and \(A_{\text{eff}}\)/\(A_{0}\) are 147 \(\upmu\)m and 1.6 \(\times\) 10\({}^{-3}A_{0}\), respectively. When further increasing \(b\)/\(a\) to 10, the effective mode index of the GP mode was \(n_{\text{eff}}\) = 1.4792 + 0.055868i. The corresponding \(L_{\text{P}}\) and \(A_{\text{eff}}\)/\(A_{0}\) were about 285 \(\upmu\)m and 1.4 \(\times\) 10\({}^{-3}\). As shown in Figure 2b, it appeared that when increasing \(u_{\text{c}}\) from 0.5 eV to 1 eV, the normalized \(S_{\text{z}}\) was overlapped with that of \(u_{\text{c}}\) = 0.5 eV. Thus, the FWHM of the focal spots was still 40 nm. Finally, the increase of \(u_{\text{c}}\) resulted in the enlargement of \(FoM\).