Single-mode plasmonic waveguiding properties of metal nanowires ...

Single-mode plasmonic waveguiding properties of metal nanowires with dielectric substrates Yipei Wang, Yaoguang Ma, Xin Guo, and Limin Tong* State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Zhejiang University, Hangzhou 310027, China *[email protected]

Abstract: 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 OCIS codes: (240.6680) Surface plasmons; (230.7370) Waveguides; (999.9999) Nanowires.

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Received 15 Jun 2012; revised 15 Jul 2012; accepted 15 Jul 2012; published 2 Aug 2012

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1. Introduction 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 freestanding 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. [23] 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 [9]; (3) experimentally, although both of the m = 0 and 1 modes have no cutoff diameter, the m = 1 mode is difficult to excite [30]. Also, single-mode SPP waveguiding nanowires can be realized by controlling the incident polarization [31]. 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 [32]

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Fig. 1. Mathematic model for simulation of a nanowire-substrate waveguiding system.

= β Re ( β ) + i Im ( β ) , Lm =

1 . 2 Im ( β )

(1) (2)

The propagation loss (α) is inversely proportional to Lm as

α =

−10 log (1 e ) Lm

≈

4.343 . Lm

(3)

The effective mode area (Am) is defined as [33–37]

Am =

Wm

{

max W ( r )}

,

(4)

where Wm is the total mode energy and W(r) is the energy density (per unit length flowed along the direction of propagation). For dispersive and lossy materials, the W(r) inside can be calculated as [33–37]

W (r ) =

1  d ( ε (r )ω ) 2 2 E (r ) + µ0 H ( r )  ,  2 dω 

(5)

where ε(r) is the complex dielectric function. At the wavelength of 660 nm used in this work, the dielectric constants of the Au (εAu) and Ag (εAg) were assumed to be εAu = −13.68151.0356i [38] and εAg = −17.6986-1.1786i [39], and d(ε(r)ω) /dω is obtained by analytical models from Refs [40] and [41], 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. Table 1. Refractive indices of substrates at 660 nm wavelength Substrate material MgF2 SiO2 ITO TiO2

Refractive Index 1.3767 1.45 1.842 2.7

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.

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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 [42]. 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.

Fig. 2. Energy density distribution on the cross section of Au nanowires. (a) D = 50 nm, no substrate; inset, schematic illustration of the polarized fields. (b) D = 50 nm, MgF2 substrate; inset, schematic illustration of the polarized fields. (c) D = 100 nm, MgF2 substrate; (d) D = 200 nm, MgF2 substrate; (e) D = 100 nm, no substrate; (f) D = 100 nm, SiO2 substrate; (g) D = 100 nm, ITO substrate; (h) D = 100 nm, TiO2 substrate. The wavelength of light used here is 660 nm.

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