The waveguide structure based on graphene materials has been a research hotspot in recent years. By employing the finite element method (FEM), the characteristics of the five lowest-order modes supported by the waveguide based on graphene-coated double elliptical and cylindrical parallel nanowires were reported. Since a purely numerical method is adopted in this study, it is impossible to give a clear physical image of the mode formation mechanism. To this end, we intend to employ the multipole method (MPM) to reanalyze the fundamental mode of the waveguide structure discussed before, and give a clear physical image of the mode formation mechanism. Meanwhile, the MPM correctness is verified by comparing the relative error between the results of the two calculation methods with the maximum value of the term number expanded by the MPM, the working wavelength, the Fermi energy, the semi-major and semi-minor axes of the elliptical cylindrical nanowires, the lateral spacing between the surfaces of the nanowires, and the relative height of the cylindrical nanowires.

We leverage the MPM to calculate the characteristics of modes supported by the waveguide based on graphene-coated double elliptical and cylindrical parallel nanowires. First, we assume that the double elliptical cylindrical nanowires and the cylindrical nanowire exist alone and that the longitudinal components of the field are expanded into series form in their coordinate systems respectively. Then, according to the field superposition principle, the longitudinal components of the field in each region of the combined waveguide are obtained. Then, the radial and angular components of the field are obtained by the relationship between the lateral component and the longitudinal component of the field. The involved derivatives can be obtained via the gradient of the scalar field and the point product of the unit vector. Then, graphene is regarded as a conductor boundary without thickness, and a linear algebraic equation system is established by the boundary relationship and point-by-point matching method. Finally, the effective refractive index and field distribution of modes supported by the waveguide can be obtained by solving this system of linear algebraic equations.

Any change in the number of series expansion terms, the operating wavelength, the Fermi energy, and the structure parameters of the waveguide will affect the MPM accuracy. The relative errors of the real and imaginary parts of the effective refractive index calculated by the MPM and the FEM decrease as the Mmax values increase (Fig. 3). As the working wavelength increases from 8.0 to 10.0μm and Fermi energy increases from 0.35 to 0.60eV, the relative error rises (Figs. 4 and 5). When the radius of the cylindrical dielectric nanowire increases from 42 to 58nm, the semi-major axis of the elliptic cylindrical nanowire grows from 96 to 104nm, with the increased relative error of the real part of the effective refractive index and decreased relative error of the imaginary part of the effective refractive index (Figs. 6 and 7). When the short half axis of the elliptic cylindrical nanowire increases from 91 to 99nm, the relative error of the real part of the effective refractive index reduces and the imaginary part of the effective refractive index rises (Fig. 8). As the transverse spacing between the nanowire surfaces increases from 12 to 28nm, and the relative height of the cylindrical nanowire rises from 50 to 66nm, the relative error of the effective refractive index decreases (Figs. 9 and 10). These phenomena can be explained by the field distribution in space. Since the MPM ignores the nonlinear superposition effect of the field, the relative error increases under stronger coupling between the fields on the nanowire surface.

The results show that the larger number of series expansion terms leads to closer results of the MPM to those of the FEM, and the increasing working wavelength and Fermi energy bring about rising relative errors of the real and imaginary parts of the effective refractive index. As the radius of cylindrical dielectric nanowires and the major and semi-axial axes of elliptical cylindrical nanowires increase, the relative error of the real part of the effective refractive index rises, and that of the imaginary part of the effective refractive index decreases. Under the increasing short semi-axis of elliptical cylindrical nanowires, the relative error of the real part of the effective refractive index decreases, and that of the imaginary part of the effective refractive index rises. When the lateral spacing between the nanowire surfaces and the relative height of the cylindrical nanowires increases, the relative errors of the real and imaginary parts of the effective refractive index decrease. These phenomena can be explained by the field distribution. Within our calculation range, the relative errors are maintained on the order of 10^-3.