## 3 Results  
We considered the GCENW waveguide to be surrounded by a dielectric medium (\(\varepsilon_{2}\)). The elliptical dielectric nanowire (\(\varepsilon_{1}\)) was covered by monolayer graphene. As shown in Figure 1a, the short axis and long axis were \(a\) and \(b\). Here we placed emphasis on the lowest order mode for its long propagation distance. Figure 1b shows the energy distribution of the lowest order GP mode for \(a\) = 0.5 \(\upmu\)m and \(b\) = 1 \(\upmu\)m at 3 THz, the majority of the optical energy was restricted in the tips, which is more evident for a larger \(b\)/\(a\). The Poynting vector in \(z\) direction (\(S_{z}\)) had the largest value at the surface and decayed rapidly away from the interface, seen in Figure 2. Since the elliptical nanowire has two tips, two focal spots with same size could be obtained. Figure 2a, and 2b present the FWHM [51] of \(S_{z}\) at the tips for \(a\) = 0.5 \(\upmu\)m, \(b\) = 1 \(\upmu\)m and \(a\) = 0.5 \(\upmu\)m, \(b\) = 5 \(\upmu\)m, which were about 140 nm and 40 nm, respectively. Surprisingly, when increasing \(b\)/\(a\) from 2 to 10, the FWHM of the focal spots reduced from 140 nm to 40 nm, and the propagation length increased from 83 \(\upmu\)m to 136 \(\upmu\)m. Hence, the ratio of \(b\)/\(a\) had significant influence on the spot size and propagation length. As shown in Table 1, increasing \(b\)/\(a\), the FWHM could be further reduced. However, when \(b\)/\(a\) increased to a certain value, the elliptical nanowire gradually becomes an approximate parallel plate. Therefore, the ratio of \(b\)/\(a\) should not be much larger. Figure 1: (**a**) The cross section of the graphene-coated elliptical nanowire (GCENW) waveguide. (**b**) Normalized electromagnetic energy distribution of the graphene plasmon (GP) mode at 3 THz. Red arrows indicate the polarization directions. The short axis and long axis are \(a\) = 0.5 \(\upmu\)m and \(b\) = 1 \(\upmu\)m, respectively. The other parameters are \(u_{\text{c}}\) = 0.5 eV, \(\varepsilon_{1}\) = 2.25, and \(\varepsilon_{2}\) = 1. Figure 3a demonstrates the dependencies of the effective mode index and propagation length on frequency for the GP mode. As the frequency ranges from 1 THz to 5 THz, Re(\(n_{\text{eff}}\)) = Re(\(\beta\))/\(k_{0}\) gradually increased, while the propagation length \(L_{\text{P}}\) decreased with frequency increasing. At higher frequencies, the much larger absorption of graphene led to the enhancement of the propagation loss. Figure 3b shows the normalized mode area (\(A_{\text{eff}}\)/\(A_{0}\)), and \(FoM\) of the GP mode as a function frequency. When \(f_{0}\) = 1 THz and \(b\mu\) = 2, \(L_{\text{P}}\) is approximately 125 \(\upmu\)m, and \(A_{\text{eff}}\)/\(A_{0}\) is only \(2.44\times 10^{-4}\). When \(b\mu\) = 3, \(L_{\text{P}}\) is approximately 146 \(\upmu\)m, i.e., increased by 16.8%, while the normalized mode area became slightly smaller, seen in Figure 3b. As a result, the GCENW with a larger \(b\mu\) showed better performance both in propagation length and mode area. We further discuss the impact of the ratio of \(b\)/\(a\) on the modal behavior in GCENW. Here, it is necessary to clarify that one can keep \(a\) or \(b\) constant while changing another. Thus, we set \(a\) = 1 \(\upmu\)m and swept \(b\) from 0.5 \(\upmu\)m to 3 \(\upmu\)m (i. e., \(b\)/\(a\) ranging from 0.5 to 3) at 3 THz. Figure 4a depicts Re(\(n_{\text{eff}}\)) and \(L_{\text{P}}\) with respect to \(b\), and one can see that Re(\(n_{\text{eff}}\)) decreased with the increase of \(b\). For the lowest order mode, propagation length increased with \(b\) increasing, and \(L_{\text{P}}\) is about 83 \(\upmu\)m and 107.7 \(\upmu\)m for \(b\)/\(a\) = 0.5 and \(b\)/\(a\) = 3, respectively. As for the normalized mode area shown in Figure 4b, there was a maximum value at \(b\)/\(a\) = 1, which is depicted by the vertical black dotted line. This result implies that