In realizing integrated nanophotonic circuits, a promising key is the use of surface plasmons, collective electron oscillations that allow the localization, confinement, and transmission of electromagnetic energy [1, 4]. Various geometries for plasmonic waveguides and integrated components have been suggested; these include grooves and slots in metal films [2-5], metal strips [6-8], dielectric strips deposited on a metal film [9], metal caps on a silicon-on-insulator rib waveguide [10-12], and others [13-15].

An alternative approach to achieve integrated nanophotonic circuits involves utilizing the nanowire (NW)-based plasmonic waveguides, which are typical one-dimensional waveguide structures with relatively low losses in visible and near-infrared spectral ranges [16-20]. To understand the properties of NW-based waveguides, it is essential to investigate their modal dispersion behavior. Schmidt and Russell established a connection between the dispersion of propagating modes on NW geometry and the dispersion of those on planar metal-insulator geometry with a single interface [17]. For an NW radius less than 100 nm, however, there is wide discrepancy owing to interaction with the electromagnetic field.

In this study, we propose a simple model to elucidate the dispersion behavior of spiraling modes on silver nanowire by building a relationship with the planar insulator-metal- insulator geometry. This approximated analysis is compared with a more exact solution obtained by solving Maxwell’s equation in cylindrical coordinates. Schmidt and Russel previously established a connection between the dispersion of propagating modes on an NW geometry and those on a planar metal-insulator geometry when the radius of the NW is much larger than the skin depth [17]. For a Ag NW with a radius of 100 nm, the average percentage error for the fundamental mode of the real component of the effective refractive index was reduced from 11.9% (previous result [17]) to 5% in the 400-800 nm wavelength range. The average percentage error for the 1^st order mode was reduced from 24.8% (previous result [17]) to 1.5% in the 400-800 nm wavelength range. This approach provides important insights into the behavior of NW waveguides. Further, it has much broader applicability than previous methods and it can be extended to provide insights into more complicated NW-based waveguides.

An infinitely long and straight Ag NW (with radius a) was oriented along the z-axis and surrounded by silica as illustrated in Fig. 1(a). We found the modal dispersion relation of the Ag NW by building a relationship with the planar insulator-metal-insulator (IMI) geometry as shown in Fig. 1(b). It is worth noting that there has been previous work to investigate the surface plasmon propagation and confinement in IMI geometry [22]. Solving Maxwell’s equations using the boundary conditions for the IMI geometry, we obtained the dispersion relation (details are given in the Appendix.),

where , k₀ = ω/c is the free-space wavevector, ^ε_m , ^ε_d are the relative permittivity in the Ag and SiO₂, respectively. The dielectric constant for Ag is taken from experimental data [21], the real-valued dielectric constant for SiO₂ is fixed to (1.47)². The effective thickness of the metal layer ^Dν is expressed by

where ν is the mode order. The propagating mode in the Ag NW is influenced by the finite cross-sectional area of NW, however, the propagating mode in the IMI geometry is infinitely extended in the x-direction. Therefore we needed to define the effective width along to the x-axis to build an equivalent relationship with Ag NW, and we assumed that the cross-sectional area of the NW was equal to the effective metal area of the IMI geometry, described by

for the fundamental mode. Here, ^W_eff is defined as a perimeter of an effective mode area for fundamental mode, that is to say ^W_eff = 2π(a+δ), where δ represents the effective field radius and defined as the distance required for the electric field intensity to decay by a factor of 1/e along the radial direction of the Ag NW, as shown in the inset image of Fig. 2. We assumed that the propagating mode had its total wavevector determined by the IMI dispersion relation,

where β is the component of the wavevector along the propagating axis of the Ag NW [15, 17]. Combining Eqs. (1)-(4), we obtain the equivalent dispersion relation for the Ag NW, described by

To obtain the effective field radius δ , we used a direct a direct solution of the dispersion relation of the Ag NW obtained by solving Maxwell’s equation in cylindrical coordinates, as follows [23]:

where J_ν is a Bessel function of the first kind, is a derivative of J_ν, K _ν is a modified Bessel function of the second kind, is a derivative of K _ν, , .

The propagation constant β was calculated by solving Eq. (6). Using these results, we obtained the radial depen-dence of the z-components of the electric field and the δ as shown in inset image of Fig. 2. Figure 2 shows the effective field radius δ as a function of the radius a and the wavelength λ. The δ obtained from Eq. (6) is indicated by the black dots. The analytical expression indicated by the black dashed lines was obtained by a fitting of the δ obtained from Eq. (6) and is described by

where A=0.01349, B=33.55, and τ_a=611.6. Here, f (a) = f_∞(1-γ₁e^-a/a₀₁ - γ₂e^-a/a₀₂ where f_∞=1.58587, γ₁=0.4437, a₀₁=1499.3, γ₂=0.1271, and a₀₂=66.2.

The complex values of the effective refractive index n_e = n_eR + in_eI are yielded from Eq. (5) and (6). Figure 3 shows the effective refractive index of the first three azimuthal mode orders as a function of the wavelength. The top panel shows the real and imaginary parts of the effective refractive index for a radius of 100 nm, and the bottom panel shows the same for a radius of 500 nm. For a radius of 100 nm, the real part of the effective refractive index for the exact solution was in good agreement with that for our approximate value in the 400-1400 nm wave-length range. Small discrepancies were noticed between the exact and approximate solutions for the imaginary part of the effective refractive index (shown in the inset of the top panel of Fig. 3. For a radius of 500 nm, the real and imaginary parts of the effective refractive index for the exact solution were in good agreement with those for our approximate value in the 400-1400 nm wavelength range.

We now examine in detail the percentage error of the real part of the effective refractive index and the propagation length in the 400-800 nm wavelength range as shown in Fig. 4. For the real part of the effective refractive index, the percentage error (ⁿ_eR) is defined by 100×(| ⁿ_eR,exact - ⁿ_{eR, approx}. | / n_eR,exact), and the approximated model proposed is low-value for the whole wavelength range compared with the previous planar metal-insulator geometry as shown in Fig. 4(a) [17]. The average percentage error for the fundamental mode was reduced to 5% compared with the previous result of 11.9%, and that for the 1^st order mode was reduced to 1.5% compared with the previous result of 24.8%.

Figure 4(b) shows the percentage error (L_P) of the propagation length. The propagation length is given by L_P = 1 / (2n_elk₀) and the percentage error (L_P) is defined by 100×(| ^L_P,exact - ^L_{P, approx}. | / L_P,exact). The average percentage error for the fundamental mode was reduced to 48% compared with the previous result of 121.5%, and that for the 1^st order mode was slightly increased 39.4% compared with the previous result of 36.2%.

In summary, we proposed a simple model to elucidate the dispersion behavior of spiraling modes on Ag NW by building a relationship with the planar IMI geometry. The previous model with planar metal-insulator geometry has large discrepancies compared with the exact solution when the radius of NW is compatible with the skin depth, due to interactions with the electric field around the NW. Our equivalent model solved this problem by introducing plasmon modes that exist on thin metal films sandwiched between two regions of identical dielectrics, and the percentage error was remarkably reduced. This approach provides important insights into the behavior of NW wave-guides. In addition, it has much broader applicability than the previous model and it can be extended to provide insights into more complicated NW-based waveguides.