1. Introduction

Semiconductor nanowires offer tremendous applications in nanophotonics, such as waveguides, sensors, photodetectors, and lasers [1–4]. The use of current nanofabrication techniques enables us to easily control the sizes and dimensions of nanowires. However, the optical modes are very weakly confined due to the limited index-contrast when the diameters of nanowires are much smaller than the operation wavelength. This makes it difficult to achieve deep subwavelength optical scale, which would limit their practical applications in nanophotonics, such as nanolasers, where small mode volumes and wire radius are highly required. Furthermore, strong optical confinement is highly required for enhanced optical field strength and gradient of light field, which will highly enhance the optical force in the nanoscale region.

Surface plasmon polaritons (SPPs), coherent electron oscillations that exist at the metal-dielectric interface, are promising candidates for deep subwavelength confinement [5, 6]. However, SPP based waveguides suffer huge propagation loss because of the intrinsic ohmic loss. In order to address this critical challenge, a kind of hybrid plasmonic waveguide that is comprised of a dielectric waveguide on one side of a metal surface has recently been proposed, which is able to support long range propagation length, while maintaining an ultra-small mode size due to the coupling between the dielectric cylindrical waveguide mode and the SPP mode at the metal-dielectric interface [7–24]. Recent experimental report on a plasmonic nanolaser using the hybrid plasmonic waveguide consisting of a CdS semiconductor nanowire on top of a silver substrate shows that the generated optical modes are a hundred times smaller than the diffraction limit, which offers the possibility of exploring extreme interactions between light and matter [16]. For the purpose of further increasing the propagation length of hybrid plasmonic modes, a symmetric hybrid LRSPP waveguide based on the coupling between the traditional LRSPP mode [25] and rectangle-shaped dielectric waveguide mode has been proposed by several groups [26–28]. It is shown to be capable of strongly confining light energy in the low index gaps between the dielectric slabs and thin metal strips, while the propagation length is still comparable to that of traditional LRSPP modes supported by dielectric-metal-dielectric waveguides. However, unlike dielectric cylinder based hybrid plasmonic waveguides, rectangle-shaped waveguides above metal tend to yield singular fields at the corner of the rectangle-shaped waveguides, which results in a much larger propagation loss [11]. Additionally, for dielectric nanowire based hybrid plasmonic waveguides, the full width at half maximum (FWHM) along the metal surface direction can be made much smaller than the diameter of the nanowire, hence the total modal area is much smaller [7]. Quite recently, it has been reported that the gradient optical forces on a hybrid plasmonic waveguide using a dielectric nanowire can be enhanced over ten times as opposed to that in a conventional dielectric slot waveguide [18].

In this article, we propose a novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film. With the strong interaction between the dielectric cylindrical waveguide mode and LRSPP mode of a thin metal film, deep-subwavelength mode as well as long-range propagation length can be simultaneously achieved. It is shown that it is capable of achieving the same degree mode confinement, while its propagation length is one order of magnitude larger than that for the previous dielectric nanowire based hybrid plasmonic waveguide. On one hand, an ultra-long propagation length is very important to design high-quality plasmonic nanolaser. On the other hand, the combination of ultra-low loss and strong coupling strength between the cylinder waveguide mode and LRSPP mode would lead to a much larger optical force with unit propagation length, which maybe very useful for constructing practical nanoscale optical tweezers to manipulate a single nanoparticle.

2. Geometry and modal properties of the proposed hybrid plasmonic waveguide

Figure 1 schematically shows the proposed hybrid plasmonic waveguide, where two identical cylindrical Si nanowires are placed on each side of a thin metallic film with a small gap distance h. The surrounding dielectric layer is SiO₂ and the metallic film is silver with thickness of t = 20nm and permittivity of ${\epsilon }_{\text{m}}$ = −129 + 3.3i at λ = 1.55μm [29]. Here we consider an intrinsic Si nanowire which has a negligible material loss at the window around 1550nm. For a thin metal film surrounded by uniform dielectric medium, there exists a symmetric mode and an anti-symmetric mode. The symmetric one is the so-called LRSPP mode, which is able to support long-rang propagation length but with a weak mode confinement. On the contrary, the propagation length of the anti-symmetric mode can become several orders of magnitude shorter than that of LRSPP mode. The dielectric cylindrical mode couples with both of the two modes. However, in this work only the symmetric mode is of our interest because it supports much longer propagation length.

 

Fig. 1 Schematic of a hybrid long-range plasmonic waveguide, where two identical cylindrical Si nanowires of permittivity ${\epsilon }_{\text{n}}$ and diameter d are placed on each side of a thin metallic film with a gap distance of h. The surrounding dielectric layer is SiO₂ of permittivity ${\epsilon }_{\text{d}}$. ${\epsilon }_{\text{n}}$ and ${\epsilon }_{\text{d}}$ are 12.25 and 2.25 at λ = 1.55μm. The metallic film is silver with a permittivity of ${\epsilon }_{\text{m}}$ = −129 + 3.3i and thickness of t = 20nm.

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In this paper, the modal properties are investigated by means of the finite-element method (FEM) using COMSOL. A perfect electric conductor boundary condition (PEC, ) was set on the horizontal boundary and a perfect magnetic boundary condition (PMC, ) on the vertical boundary. In the convergence analysis, the calculation region in the x-direction (4λ) and y-direction (8λ) is assumed to be sufficiently large to ensure an accurate eigenvalue.

In the following, we modulate the diameter, d, and the dielectric gap width, h, to adjust the mode field distribution, and propagation length, ${\text{L}}_{\text{m}}$, of a single hybrid mode at λ = 1.55μm. In order to have a meaningful comparison with the previous hybrid plasmonic waveguide consisting of a dielectric nanowire on a metal substrate, the propagation length and modal area are defined as the same as that in [7]. The propagation length is given by ${\text{L}}_{\text{m}}=\lambda /[4\pi \mathrm{Im}(n_{eff})]$, where $n_{eff}$is the effective index of the hybrid mode. In order to have a meaningful comparison of the modal area with the previous hybrid plasmonic waveguide consisting of a dielectric nanowire on a metal substrate, the modal area, A_m, is defined as the ratio of the total mode energy and the peak energy density:

Figure 2(a) shows the dependence of the normalized modal area, A_m/A₀, on $d$ and h. For a small gap distance h, high electromagnetic energy is localized within the gaps between the metal and dielectric nanowire [Figs. 2(b) and 2(c)]. It is interesting to note that, despite the strong confinement, the propagation length of the hybrid mode, ${\text{L}}_{\text{m}}$, is much larger than that of the LRSPP mode in a pure Si-Ag-Si model [Fig. 3 ]. When $d$ is larger than a certain value, the propagation length even exceeds that of the LRSPP mode in a pure SiO₂-Ag-SiO₂ model. As the gap distance increases, the hybrid mode tends to confine light energy both in the gaps and dielectric nanowires [Fig. 2(d)]. For a large cylinder diameter and gap distance, the hybrid waveguide supports a cylinder-like dielectric guided mode that the electromagnetic energy is confined in the two dielectric nanowires [Fig. 2(e)]. In this case, the propagation length is much bigger than that of the LRSPP mode in a pure SiO₂-Ag- SiO₂, but at the cost of much larger modal area as opposed to that in Figs. 2(b) and 2(c), where a small gap distance is employed. To balance the modal area and propagation length, an optimum combination of the gap distance and cylinder diameter is required. We can also see from Fig. 2(a) and Fig. 3 that there exists a point with$d$ = 240nm, that both of the modal area and propagation length is minimum.

 

Fig. 2 (a) Normalized modal area (A_m/A₀) versus the cylindrical diameter $\text{d}$ for different gap distance h. (b-e) Electromagnetic energy distributions for [h, d] = [5, 300] nm (b), [h, d] = [10, 240] nm (c), [h, d] = [50, 240] nm (d), and [h, d] = [100, 400] nm (e).

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Fig. 3 Propagation length (${\text{L}}_{\text{m}}$) versus the cylindrical diameter d for different gap distance h. The propagation lengths of LRSPP modes in Si-Ag-Si and SiO₂-Ag-SiO₂ are denoted as black dashed line and black solid line, respectively.

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Figure 4 shows the normalized energy density along x = 0 and y = t/2 + h at $d$ = 240nm for different gap distance h. For a small gap distance h, a large portion of light energy resides within the gaps between the metal film and dielectric nanowire. With the increase of h, the field confinement along the y direction reduces, hence more light energy will be localized in the two dielectric nanowires [Figs. 4(a), 4(c), 4(e), 4(g), 4(i)]. It is interesting to note that, the energy density profile along the x direction also shows subwavelength localization [Figs. 4(b), 4(d), 4(f), 4(h), 4(j))]. The FWHM along the x direction is 56nm at h = 5nm, which is approximately 23% of the nanowire diameter [Fig. 4(b)]. For previous rectangle-shaped hybrid plasmonic waveguides, the confinement along the same direction is highly limited to the width of high-index dielectric layer, and then the FWHM along the metal surface direction can be much bigger [26–28]. As h is enhanced monotonically, the corresponding FWHM along the x direction for the hybrid LRSPP waveguide increases. The FWHM along the x direction is 172nm at h = 100nm, which is still smaller than nanowire diameter [Fig. 4(j)].

 

Fig. 4 Normalized energy density along x = 0 [vertical dashed line in the inset of (a)] at h = 5nm (a), 10nm (c), 30nm (e), 50nm (g), and 100nm (i). Normalized energy density along y = t/2 + h [horizontal dashed line in the inset of (a)] at h = 5nm (b), 10nm (d), 30nm (f), 50nm (h), and 100nm (j). The cylindrical diameter d is set at 240nm.

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3. Mode character and coupling strength

Figure 5(a) shows the effective refractive index of the hybrid mode on $d$ for different gap distance h. The guided hybrid mode is beyond the dielectric cylinder mode or pure LRSPP mode in a pure SiO₂-Ag- SiO₂ model. This is because that the LRSPP mode is coupled with the dielectric cylinder waveguide mode, which induces a much higher effective index [7, 28]. The mode’s effective index can be increased by reducing the gap distance for a fixed $d$, or enhancing the diameter of the cylinder nanowire, $d$, for a fixed gap distance, h. This can be explained that the LRSPP mode couples with the dielectric cylinder mode more effectively as $d$ increases or h reduces.

 

Fig. 5 (a) The dependence of the mode effective index of the hybrid LRSPP mode, ${\text{n}}_{\text{hyb}}$, on d for different gap distance h. As a comparison, the mode effective index of a pure cylindrical dielectric waveguides, ${\text{n}}_{\text{d}}$, versus d is depicted in the solid black line. The dependence of the effective index of the pure LRSPP mode at SiO₂-silver-SiO₂ waveguides is shown in the dashed black line. (b) The mode character derived from Eq. (2). (c) The dependence of coupling strength $\kappa$ on d and h.

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In order to describe the mode characteristics of the hybrid mode, a mode character, ${\left|{\text{a}}_{+}(\text{d},h)\right|}^2$, is employed to represent the superposition of the cylinder waveguide mode and the LRSPP mode based on the coupled-mode theory [7]

The mode character can be used to evaluate the degree to which the hybrid mode is a dielectric cylinder-like waveguide mode or a SPP-like mode [7]. If $n_c(d)$ is larger than $n_L$, ${\left|{\text{a}}_{+}(\text{d},h)\right|}^2$exceeds 0.5 that the hybrid mode more likes guided cylinder waveguide modes. In this case, a large portion of light energy is located in the cylinder nanowire. On the contrary, if $n_c(d)$ is smaller than $n_L$, ${\left|{\text{a}}_{+}(\text{d},h)\right|}^2$is less than 0.5 that the hybrid mode more likes a SPP mode. A relatively larger portion of light energy tends to be confined in the gap region between the dielectric nanowire and metal film. Figure 5(b) shows the dependence of the mode character derived from Eq. (2) on the cylinder diameter $d$ for different gap distance h. From the figure we can clearly see how to modulate the geometrical parameters to achieve the hybrid mode with a small modal area. For a fixed value of h, the mode character increases with$d$. A large h and $d$ leads to a large mode character that the hybrid mode is a dielectric cylinder-like waveguide mode [Fig. 2(e)]. For a fixed value of h, one can decrease the cylinder diameter to achieve a smaller mode character that the hybrid mode is more SPP-like mode [Figs. 2(b), 2(c)]. The hybrid mode resembles both dielectric cylinder-like waveguide mode and SPP-like mode for a moderate cylinder diameter [Fig. 2(d)]. The hybrid mode has equal cylinder and SPP characteristics corresponding to the condition of $n_c(d)$ = $n_L$.

According to the coupled-mode theory for two optical waveguides system, the coupling strength between the LRSPP mode and dielectric cylinder waveguide mode can be written as [7, 18]

4. Figure of merit

The figure of merit (FoM) can be applied to quantitatively measure plasmonic waveguides, and help trade-off mode confinement against attenuation. Here the FoM is defined as the ratio of the propagation length to the effective mode size defined as the diameter of A_m [30]

For simplicity, here we only consider the case of a small gap distance in the hybrid LRSPP waveguide that a large portion of light energy is located in the gap region between the dielectric nanowire and metal film. The definition of mode size by Eq. (1) has only been used to evaluate the mode confinement for hybrid plasmonic waveguides. In a previous literature, the mode size, A_e, defined as the area bounded by the closed 1/e field magnitude contour relative to the global field maximum, has been widely used to measure the mode area of plasmonic waveguide [30]. Here we also use this method to calculate the mode size of the hybrid LRSPP waveguide and previous hybrid plasmonic waveguide. Figure 6 shows the dependence of mode size, A_e, for the two waveguides on the cylindrical diameter $d$ for h = 5nm, and 10nm. We can see that, for the hybrid LRSPP waveguide, A_e is always larger than that for the previous hybrid plasmonic mode. This is because the electromagnetic field is located on both sides of metal film for the hybrid LRSPP waveguide. When the cylindrical diameter is very small, the confinement for both structures is relatively weak as a large portion of light field expands into the cladding. We can get an optimum mode confinement at around $d$ = 150nm for both structures. Further increasing the cylindrical diameter will slightly enhance the mode size. It should be noted that, A_e can be smaller than A_m as $d$ is approaching 600nm for the hybrid LRSPP waveguide. For example, for the hybrid LRSPP waveguide with h = 5nm, A_e is larger than A_m when $d$ is smaller than 540nm. As $d$ further increases, A_e becomes smaller than A_m.

 

Fig. 6 The dependence of the mode size,${\text{A}}_{\text{e}}$, of the hybrid LRSPP waveguide and previous hybrid plasmonic waveguide on the cylindrical diameter d for h = 5nm, and 10nm. The mode size,$\delta \text{w}$, for the LRSPP mode in Si-Ag-Si and SiO₂-Ag-SiO₂ are denoted as black dashed line and black solid line, respectively.

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Figure 7 shows the dependence of FoM of the hybrid LRSPP waveguide and previous hybrid plasmonic waveguide on the cylindrical diameter d for h = 5nm, and 10nm. The mode size used to evaluate FoM in Fig. 7(a) is A_m, while that for Fig. 7(b) is A_e. It can be seen that, the FoM for the hybrid LRSPP mode is always larger than that for the previous hybrid plasmonic mode in the whole range of nanowire diameter d. For both of hybrid waveguides, the minimum FoM occurs at approximately d = 250nm, at which the FoM for the hybrid LRSPP mode is 7 times as large as that for the previous hybrid plasmonic mode in Fig. 7(a), while this value is approximately 7.8 times in Fig. 7(b). We can see that, the present hybrid LRSPP waveguide is superior to the previous plasmonic waveguides since it shows a much longer propagation distance for similar degrees of confinement.

 

Fig. 7 The dependence of the figure of merit (FoM) of the present hybrid LRSPP waveguide and previous hybrid plasmonic waveguide on the cylindrical diameter d for h = 5nm, and 10nm. (a) A_m is used to evaluate FoM. (b) A_e is used to evaluate FoM. (1): the present hybrid LRSPP waveguide. (2): the previous hybrid plasmonic waveguide.

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For future experimental fabrication of the present hybrid LRSPP waveguide, we can use ‘vapour-liquid-solid’ method to produce Si nanowires with precise sizes and dimensions [31]. We can then position the Si nanowire on a SiO₂ substrate and cover the nanowire with a SiO₂ cladding. The silver film will later be deposited using sputtering and/or E-beam evaporation with a high precision. Using the same method, we can form the upper SiO₂ layer, Si nanowire, and SiO₂ cladding in succession. We believe that, at least in principle, the present structure can be fabricated by current nano-fabrication technology. However, compared with the rectangular-shaped waveguides, the fabrication process for the present hybrid plasmonic waveguide is by no means easy since placing nanowires precisely to implement an integrated nanophotonic circuit is very challenging by current nanofabrication technology.

5. Conclusion

We have proposed a novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically positioned on each side of a thin metal film. The strong coupling between the dielectric cylinder waveguide mode and long-range SPP mode induces deep-subwavelength optical field within the gaps between the dielectric nanowire and metal film. Compared with the previous hybrid plasmonic waveguide consisting of only one dielectric nanowire above a metal film, a much larger propagation length as well as a comparable modal area can be achieved, which greatly improves the figure of merit. The combination of ultra-low loss and strong coupling strength between the dielectric waveguide mode and plasmonic mode is expected to give rise to enhanced optical forces with unit propagation length. The hybrid long-range surface plasmon waveguides are very useful to construct various functional devices and find potential applications such as nanophotonic waveguides, high-quality nanolasers, and optical trapping and transportation of nanoparticles and biomolecules.