Single-mode plasmonic waveguiding properties of metal nanowires with dielectric substrates are investigated using a finite-element method. Au and Ag are selected as plasmonic materials for nanowire waveguides with diameters down to 5-nm-level. Typical dielectric materials with relatively low to high refractive indices, including magnesium fluoride (MgF2), silica (SiO2), indium tin oxide (ITO) and titanium dioxide (TiO2), are used as supporting substrates. Basic waveguiding properties, including propagation constants, power distributions, effective mode areas, propagation distances and losses are obtained at the typical plasmonic resonance wavelength of 660 nm. Compared to that of a freestanding nanowire, the mode area of a substrate-supported nanowire could be much smaller while maintaining an acceptable propagation length. For example, the mode area and propagation length of a 100-nm-diameter Ag nanowire with a MgF2 substrate are about 0.004 μm2 and 3.4 μm, respectively. The dependences of waveguiding properties on geometric and material parameters of the nanowire-substrate system are also provided. Our results may provide valuable references for waveguiding dielectric-supported metal nanowires for practical applications.
©2012 Optical Society of America
Owing to their capability of manipulating electromagnetic fields on the deep-subwavelength scale by converting light into surface plasmon polaritons (SPPs), plasmonic metal nanostructures have inspired a variety of potentials ranging from ultra-compact optoelectronic circuits [1,2], optical sensors [3,4] to quantum electrodynamics research [5–8]. So far, various types of metal nanostructures have been proposed for guiding SPPs [9–15], among which Au and Ag nanowires are typical one-dimension waveguide structures with relatively low losses at visible and near-infrared spectral ranges. In the past years, waveguiding properties of metal nanowires have been extensively investigated theoretically [9,16–19], but mostly on free-standing nanowires with symmetric dielectric surroundings. In experimental cases [20–22], a supporting substrate is usually indispensable for nanowire manipulation and characterization, and therefore waveguiding properties of nanowires with proper substrates are of great importance for practical applications.
Recently, several groups reported the plasmon modes of Ag nanowires coupled with a silica or a silicon substrate [23,24], but mostly focused on nanowire-substrate system with a certain gap, and the propagation lengths obtained in Ref.  showed relatively large discrepancy with both theoretical and experimental results [25–27]. More recent works compared the propagation lengths of Au and Ag nanowires with glass or indium tin oxide (ITO) substrates [28,29], but the power distribution, effective mode area have not been studied. In contrast to the well-studied free-standing nanowires, waveguiding properties of substrate-supported SPP nanowires have not been adequately investigated.
In this paper, we investigate waveguiding properties of metal nanowires with dielectric substrates using a Comsol Multiphysics finite element method. Aiming for operating SPP nanowires with tight confinement, here we focus on nanowires with subwavelength diameters down to 5-nm-level, and study only the lowest order mode in proposed nanowires under the following considerations: (1) single-mode operation is favorable or required in most practical applications; (2) the fundamental mode plays the central role and dominates the propagation properties ; (3) experimentally, although both of the m = 0 and 1 modes have no cutoff diameter, the m = 1 mode is difficult to excite . Also, single-mode SPP waveguiding nanowires can be realized by controlling the incident polarization .
Using waveguiding systems with Au and Ag nanowires and dielectric substrates including magnesium fluoride (MgF2), silica (SiO2), indium tin oxide (ITO) and titanium dioxide (TiO2), we obtained single-mode waveguiding properties of these nanowires including propagation constants, power distributions, effective mode areas, propagation distances and losses at the typical plasmonic resonance wavelength of 660 nm. The propagation distances and losses reported here agree well with experimental results [22,27,29].
2. Basic model
The mathematical model for our numerical simulation is illustrated in Fig. 1 , in which an infinite long and straight nanowire with a diameter of D is placed on a dielectric substrate.
The propagation constant (β) and propagation length (Lm) of the nanowire are defined as 
The propagation loss (α) is inversely proportional to Lm as33–37]38] and εAg = −17.6986-1.1786i , and d(ε(r)ω) /dω is obtained by analytical models from Refs  and , respectively. For simplicity, we assume that the nanowire has a perfectly smooth sidewall without surface-roughness-induced scattering. The refractive index of air is assumed to be 1.0, and the indices of four substrates chosen in this work at 660 nm wavelength are listed in Table 1 .
3. Modal profiles and power distributions
Based on the model set up in Section 2, we investigate waveguiding properties of the nanowire-substrate system using a Comsol Multiphysics finite element method. The computational domain is discretized into a triangular mesh with an element size of one tenth of the nanowire diameter (e.g., 10 nm for a 100 nm diameter nanowire), terminated by perfectly matched layer (PML) boundaries.
Calculated modal profiles in terms of energy density distributions for Au nanowires guiding 660-nm wavelength light are shown in Fig. 2 , in which Au nanowires with different diameters (50, 100, 200 nm) and substrates (air, MgF2, SiO2, ITO and TiO2) are considered. While the free-standing 50-nm-diameter Au nanowire shows a symmetric mode profile (Fig. 2(a)), the introduction of the substrates breaks the symmetry and confines the energy to the interface between the nanowire and the substrate (Figs. 2(b)-2(d)), as has been reported previously . Moreover, the fractional energy confined around the interface increases with the increasing nanowire diameter (e.g., modal profiles in Fig. 2(d) vs. Fig. 2(c)). The energy density enhancement can be explained by polarized substrate (insets of Fig. 2(a) and Fig. 2(b)), which is similar to nanoparticle-substrate system [43,44]. Meanwhile, for Au nanowires with the same 100 nm diameter (Figs. 2(e)-2(h)), the energy confinement around the interface increases with the increasing substrate indices (e.g., modal profiles in Fig. 2(h) vs. Fig. 2(g)), which can be attributed to higher polarizability in the substrate with a higher refractive index.
The propagation constants (β) of Au and Ag nanowires are obtained from the eigenvalue equations by the COMSOL software. Figure 3 and Fig. 4 give the calculated Re(β) and Im(β) of Au and Ag nanowires, respectively. It shows that, at 660-nm wavelength, Re(β) is approximately a constant when D is larger than about 100 nm; however, when D goes below 100 nm, Re(β) starts to increase with the decreasing D. It is noticed that, when D is very small (e.g., kD << 1, k represents the wavevector), Re(β) exhibits a 1/D behavior, agrees well with previous works [30,45]. The small D case can be regarded as the quasi-static configuration of electric field and associated charge density wave [30,46], in which the substrate has little impact on the plasmon mode. For comparison, Fig. 5 gives calculated modal profiles of SiO2-substrate-support Ag nanowires with diameters of 10, 20, 50 and 100 nm, clearly showing that the substrate-induced impact on the modal profiles decreases with the nanowire diameters: the modal profiles are obviously asymmetric in 50 and 100-nm nanowires, while show better symmetries in 10 and 20-nm nanowire. Also, Re(β) is affected by refractive index of the substrate (nsub): for the same nanowire, the higher the nsub, the larger the Re(β), which can be attributed to the lower light propagation velocity in higher-nsub medium.
To quantify the power distribution of the fundamental modes inside the nanowire core and the substrates, we define the fractional power inside the core (ƞc) and the substrate (ƞs) asFigure 6 and Fig. 7 are the D-dependent ƞc and ƞs of Au and Ag nanowires, respectively. It shows that, ƞc increases monotonically with decreasing D. When D is very small (e.g., <5 nm for Au nanowire and < 3 nm for Ag nanowire), ƞc approach a constant value of about 72% for Au nanowire and 52% for Ag nanowire. In contrast, with increasing D, ƞs increases monotonically resulting in higher fractional energy confined inside the substrate.
4. Optical confinement and effective mode areas
Owing to their possibility of confining propagation light below the diffraction limits, optical confinement (or equivalently the effective mode area) is one of the most important merits of the plasmonic waveguides. To quantify the confinement, here we calculate the effective mode area (Am) of nanowire waveguides with Eq. (4). Figures 8(a and b) show calculated mode areas of Au and Ag nanowires with dielectric substrates, respectively. For reference, mode areas of free-standing Au and Ag nanowires (without substrate) are also given (black line). It shows that, unlike that of a tightly confined dielectric waveguides (e.g., a silicon nanowire), in which Am increases with decreasing D, Am of the plasmonic nanowire decreases monotonically with D making it possible to realize ultra-tightly confined waveguiding modes. For example, Am of a 50-nm-diameter Au nanowire is about 0.0026 μm2, which is about 0.6% of λ2; for comparison, the minimum Am of a silicon nanowire is about 3% of λ2 . Moreover, compared to that of the free-standing nanowire, Am of the substrate-supported nanowire is significantly reduced. For example, for a 100-nm-diameter free-standing Ag nanowire, Am is about 0.021μm2. When it is supported by a SiO2 substrate, Am reduces to 0.003 μm2, much smaller than that of the free-standing one. It is also noticed that, for the same nanowire, the higher the index of the substrate, the smaller the Am. Besides, with the same diameter and substrate, Am of a Au nanowire is always smaller than a Ag nanowire. In addition, for reference, Fig. 9 gives the normalized amplitude of the electric filed along y direction (see inset) of a 200-nm-diameter Au nanowires. The electric filed is much stronger at the substrate side than that at the opposite interface. Also, the higher the index of the substrate, the higher peak intensity of the electric field.
5. Propagation lengths and waveguiding losses
Around the surface plasmonic resonance frequency, a waveguiding metal nanowire with subwavelength confinement usually suffers from high ohmic losses. By substituting the calculated Im(β) in Fig. 4 into the definition in Eq. (2), we obtained the propagation length (Lm) shown in Figs. 10(a) and 10(b), in which the propagation loss (α) is obtained by Eq. (3). It shows that, Lm increases monotonically with the increasing diameters (D), reflecting the trade-off relations between the confinement and the loss of the plasmonic waveguide. For the Au nanowire, Lm of the freestanding nanowire is always larger than those with the substrate; while for the Ag nanowire, this is only true when D is smaller than 120 nm. Moreover, Fig. 11 gives nsub-dependent Lm of Au and Ag nanowires with diameters of 100 and 200 nm, respectively. It shows that, for the Au nanowire, Lm decreases monotonically with nsub increasing from MgF2, SiO2, ITO to TiO2 for both the 100- and 200-nm diameter nanowires. For the Ag nanowire, when D = 100 nm, Lm decreases monotonically with nsub increasing from MgF2, SiO2, ITO to TiO2; however, when D = 200 nm, Lm increases with increasing nsub (from MgF2 to SiO2) until 1.7, and then drops down with nsub increasing from ITO to TiO2, which agrees well with those shown in Fig. 10(b).
Figure 10(b) also shows that, for Ag nanowires with diameters larger than 120 nm, the substrate-supported nanowire exhibits a larger Lm (or equivalently lower loss) than the free-standing one, suggesting a possible explanation for the larger experimentally measured Lm of Ag nanowires (with substrates)  than that (without substrate) obtained by numerical calculation . To verify the validity of the simulation results, we have also compared the calculated Lm of both Au and Ag nanowires with experimental results reported previously [22,27,29]. For example, Ma et al. reported a propagation length of 10.56 μm in a 260 nm diameter Ag nanowire supported by SiO2 substrates at 633 nm wavelength , agrees well with our calculation result of about 10.86 μm at 660 nm wavelength.
In summary, we have investigated the single-mode waveguiding properties of Au and Ag nanowires with typical dielectric substrates. Using a FEM method, propagation constants, power distributions, effective mode areas, propagation lengths and losses are obtained numerically. It shows that, for a substrate-supported nanowire, the larger the nanowire diameter, the more asymmetric the plasmon mode profile. Also, with increasing substrate index, the fractional energy confined around the interface increases, and the effective mode area decreases. Meanwhile, compared to a free-standing nanowire, the nanowire-substrate structure can simultaneously offer a much tighter confinement and a relatively large propagation length, when the parameters of the nanowire and the substrate are properly chosen. For example, for a 100-nm-diameter free-standing Ag nanowire, Am and Lm are 0.021 μm2 and 3.6 μm, respectively. When it is supported by a MgF2 substrate, Am and Lm are 0.004 μm2 and 3.4 μm, respectively. It should also be mentioned that, in our calculation, the nanowires are assumed to have smooth surfaces and uniform diameters, while real nanowires are usually not ideally uniform, and the dielectric constants may be slightly different from those used in this work. Compared with previous experimental results, waveguiding properties we obtained here are reasonable and meaningful. Particularly, for experimental study or practical applications, plasmonic nanowires are usually supported by a certain substrate (usually dielectric) and operated in single mode, therefore, the single-mode plasmonic waveguiding properties of a metal nanowire shown here could be very helpful for design, selection and utilization of plasmonic waveguides and building blocks for a variety of applications from nanophotonic/plasmonic passive waveguiding circuits to active devices.
This work was supported by the National Natural Science Foundation of China (Nos. 61036012, 61108048) and Fundamental Research Funds for the Central Universities.
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