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the opposite behaviors. This tradeoff between confine- (a) 1 ment and absorption loss is well known in the surface plasmon mode. For the nanowire on silica substrate, comparing with the nanowire in air, neff is larger and Aeff is smaller (b) with all d. These obvious differences come from the pres- ence of large refractive index material near the metal nanowire. In addition, the mode area of the nanowire on silica substrate is only about 1/5 of that in air. To characterize the field pulling by the silica substrate, we (c) present the confinement factor for g = 0 in Fig.2(d). The η increases with d to nearly 80% when d = 200nm. In Fig.(2), Aeff simply increases with the increase of d. However, the other three parameters approach satu- ration values. In order to get a physic interpretation, we (d) can approximately treat this hybrid system as the metal plane-substrate for large d. The analytical solution to Eq. p (1) in the plane condition gives n eff = ǫmǫs/ (ǫm + ǫs), where ǫm and ǫs is the electric permittivityp of metal and p p substrate. So we have n eff = 1.537, L = 53µm, and ηp = 90% in the plane limit. FIG. 2: (color online) Effective index (a), propagation loss Now, we turn to discuss the experimental condition, (b), mode areas (c), and energy confinement in the substrate where the nanowire is near the substrate with a finite (d) of surface plasmon modes for the nanowire on the sub- gap g > 0. This is a realistic case for many experiments ∞ strate (red dashed, g = 0) and in air (blue solid, g = ) because the substrate is not perfect smooth, or in spe- versus the diameter of silver nanowire. cific experimental config, the nanowire is suspended in air[17]. By fixing the diameter d = 100 nm, we study the 2 2 Here, |E(r)| and |H(r)| are the intensity of electric and propagation loss and effective index (Fig.3) for different gaps between nanowire and substrate. magnetic fields, respectively. ε(r) and µ0 are the electric and vacuum magnetic permittivities. With the increase of gap g, neff monotonously de- To describe the impact from the substrate, we also creases and gradually approaches the in-air case. This is define the confinement factor η as the portion of energy because the influence of substrate becomes smaller when in substrate, the nanowire departs from the substrate. When g = 200 nm, the substrate has a very minor effect on the plas- mon mode of the silver. However, the dynamics of L is η = W (r)ds/ W (r)ds. (5) Zsub Zall much more complicated than that of neff . When the silver nanowire moves away from the silica substrate, L The confinement factor η can also quantify the interac- increases first, then quickly drops to 2 µm. After that, tion of SPP and substrate, it slowly increases, and finally approaches to 25 µm [19]. A finite element method is used to investigate the plas- We can divide the gap into three regimes: the near field, mon modes with a fixed wavelength at 637 nm. The en- dissipation, and far field regimes, which correspond to ergy distribution profiles of the plasmon modes in the different dynamics of L and different loss mechanisms. nanowire are shown in Fig. 1(b) and (c), with g = ∞, 0, (i) Near field regime. In this regime, we have neff > respectively. Here, nair = 1 and nsub =1.45. It is not dif- nsub. Due to the phase mismatching, the dominant loss ficult to find a rotation symmetric field distribution when of the plasmon mode comes from the metal absorption. the nanowire is in air (g = ∞), while a strongly asym- By increasing the gap, neff gradually decreases as the metric field distribution is presented in the presence of nanowire moves away from the substrate. A smaller neff the silica substrate (g = 0). Moreover, the maximum of indicates a smaller energy portion in the metal, so that the electric field is distributed in the substrate-nanowire the absorption loss is reduced with increasing g. gap. In other words, the substrate can significantly pull (ii) Dissipation Regime. When the gap further in- the mode field. creases, neff may be smaller than nsub. Meanwhile, the We first investigate the impact of the substrate on the plasmon mode still has much energy in the substrate, as plasmon modes for various nanowire sizes. Fig.2 shows can be seen in the inset of Fig.3 (g = 50 nm). From the effective index (neff ), propagation length (L), effec- Eq. (1), when neff < nsub, the solutions are traveling tive mode area (A ) and portion of energy (η) versus eff ∝ 2 2 d. As a comparison, the ideal case without substrate waves in substrate as ψ exp(i nsub − neff x). The en- q (g = ∞) is also depicted. It is found that neff de- ergy in traveling wave will dissipate into infinite space. creases with the increase of d, while L and Aeff shows Therefore, in this regime, the mode energy quickly dis- 3

is confined near nanowire-substrate interface and neff is large. It should be more efficient to coupling light to the surface plasmon mode through the substrate, especially 2nm 50nm 200nm for large d. The efficiently excitation of surface plas- mon mode through substrate have been demonstrated recently[10, 11]. It is worth noticing that the propa- gation loss strongly depends on the gap. In a recent experiment, near field surface plasmon mode detection is demonstrated[17], where the silver nanowire is put on the Ge nanowire, with g = 40nm, d = 100nm and λ = 600nm. From Fig. 3(b), it works in the dissipation regime with L ≈ 2µm. For more efficiently detection, g should be larger than 200nm or smaller than 10nm . In conclusion, we have numerically studied the mode profile, propagation loss and mode area of plasmon modes of a silver nanowire near the silica substrate. To reduce the propagation loss, our result suggests that the gap between silver nanowire and the substrate should be as FIG. 3: (color online) Effective index (a) and propagation small as possible. When the silver nanowire is attached length (b) of the silver nanowire with d=100 nm versus the to the substrate (g = 0), the propagation length is lim- gap. Insets: mode field profiles of the surface plasmon modes with g=2 nm, 50 nm, 200 nm, respectively. ited by metal absorption and the mode area is reduced by a factor of 5. In addition, for nanowire with d = 200 nm, the field energy portion of the plasmon mode in the sipates by coupling to the dissipative traveling waves in substrate is as large as 80%, which is suitable for coupling the substrate. As a result, L has a sudden decrease. to QDs or molecules embed in substrate. Accompanying (iii) The far field regime. When the nanowire further with the scanning near field optical microscope, the silver moves away from the substrate, for example, g = 200 nanowire can be precisely moved on the substrate, and nm, the coupling between the plasmon mode and the dis- selectively coupled to single QDs in substrate. This is sipative traveling modes in substrate becomes very weak potential for experimental realization of the single pho- as it is proportional to their field overlap. Thus, even ton transistors, or coupling two distant QDs to generate entanglement for quantum information science. neff

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