1. Introduction

In order to have high-integration density, it is essential to develop a nano-scale optical waveguide which is the basic element for photonic integration circuits. Currently there are three kinds of popular nano-scale waveguides, namely, silicon-on-insulator (SOI) nanowires with an ultra-high index-contrast [1,2], photonic crystals [3], and surface plasmon (SP) waveguides [4–16]. For the former two nano-scale waveguides, the optical field confinement is limited to the order of a wavelength in each direction. In contrast, surface plasmon (SP) waveguides could provide a true nano-scale waveguiding and confinement of light.

In the past years people have presented several three-dimensional structures which can support highly localized fields, e.g. narrow gaps between two metal interfaces [8–10,15,16] and V-grooves in metals [11,12]. However, it is well known that such a nano-scale optical waveguide has a large loss and the propagation distance is usually at the scale of several micrometers. Recently, a hybrid plasmonic waveguide with a dielectric cylinder above a metal surface has been presented for subwavelength confinement and long propagation distance [17]. However, it is not easy to fabricate such a waveguide structure due to the cylindrical structure. A rectangular plasmonic waveguide should be more attractive because it is possible to fabricate by using the standard planar lightwave circuit technology. In Ref [18], the authors has given an analyses for the dispersion relation and loss of subwavelength confined mode of several metal-GaAs-gap waveguides, e.g., a GaAs-cylinder with a gap of SiO₂ on metal, a rectangular GaAs strip above Ag-substrate, and a Ag-gap-GaAs strip on a SiO₂ substrate.

It is well known that recently silicon photonics has become very attractive because its fabrication compatibility to the standard CMOS microelectronics technology. It will be therefore interesting to develop a silicon-based hybrid plasmonic waveguide with simplified fabrication processes. In our previous paper, we have presented a SOI nanowire with a metal cap which is for a submicron-heater [19]. In that case, the SiO₂ layer between silicon core and the metal cap is thick enough to prevent the absorption due to the metal. Here we consider the case when the SiO₂ layer is very thin (e.g., several tens of nanometers). For such a SOI rib with a metal cap, when we consider the quasi-TM polarization (whose electrical field is vertical), there will be a field enhancement at the thin-SiO₂ region due to the boundary condition of the electrical field, which is some similar to the horizontal slot waveguides [20]. Particularly, when the Si rib height is zero, one obtains a hybrid plasmonic waveguide with a Si slab and the etching depth is much shallow, which makes the fabrication much easier. The present hybrid plasmonic waveguide is also good to realize a low-voltage compact optical modulator when the nano-layer material between the Si layer and the metal layer has a high electro-optical coefficient. In this paper, we give a theoretical investigation on the modal characteristics of such a Si-based hybrid plasmonic waveguide.

2. Waveguide Structure and Analysis

Figure 1 shows the cross section of the present structure, which consists of a SOI rib with a metal cap. For such a structure, the fabrication is simple and CMOS compatible. One could use a standard SOI wafer. An alternative way is using the method of depositing SiO₂ and alpha-Si thin films on a Si substrate with the PECVD (plasma enhanced chemical vapor deposition) technology [21]. The next step is to form a SiO₂ thin film with a thickness of several tens of nanometers by using the PECVD technology or the process of thermal oxidation. By slowing down the speed of deposition or oxidation, it is possible to control the thickness precisely. Then a metal film (e.g., Ag or gold) is deposited on the SiO₂ layer. The photoresist thin film is then formed on the metal layer and the waveguide patterns are then defined by using E-beam lithography to have a high resolution. An RIE (Reactive Ion Etching) process is then used to etch through the layers of metal, SiO₂ (as shown in Fig. 1). The Si layer could be etched through or partially, which will be discussed below. For the fabrication, the metal cap is used as the mask for etching. More importantly, for the present structure, the metal cap contributes to the nano-scale waveguiding and confinement.

 

Fig. 1 The cross section of the present hybrid plasmonic waveguide with a metal cap on a silicon-on-insulator rib. When the height h _{Si_rib} = 0, the Si part becomes a slab waveguide and consequently the fabrication is very easy because the etching becomes shallow.

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When the thickness of the SiO₂ layer between Si and metal is large (e.g., 0.5 μm), the fundamental mode field is confined well in the Si region and the metal layer almost does not influence the mode field distribution. In this case, the present structure is like a regular SOI nanowire. However, when the SiO₂ thickness becomes smaller (e.g., <50 nm), the metal layer will introduce a significant influence on the field distribution of the guided mode. As an example, we choose the geometrical dimensions as follows: w _co = 200 nm, h _m = 100 nm, h _SiO2 = 50 nm, and h _{Si_rib} = h _Si = 300 nm. We choose the wavelength λ = 1550 nm and the corresponding refractive indices for all the involved materials as n _metal = 0.1453 + 11.3587i (Ag) [17], n _SiO2 = 1.445, and n _Si = 3.455. Here we consider an intrinsic Si layer which has a negligible material loss at the window around 1550nm. When a doped Si layer is necessary for some special situations (e.g., when P-/N- contact is introduced), the loss due to the doping could be estimated by using the formula given in Ref [22]. For example, when the doping density N = 10¹⁹/cm³, the imaginary part of the refractive index n _im = 0.2097, which should be included when estimating the loss of the waveguide. For the case with an intrinsic Si layer (which is usually used for passive optical components), Fig. 2 shows the field distribution of the major-component E_y(x, y) for the quasi-TM fundamental mode calculated by using an FEM (finite element mothod)-based mode solver. In order to see the profile more clearly, we also plot the field profles E_y(x, 0) and E_y(0, y).

 

Fig. 2 The calculated field distribution for the major component E_y(x,y) of the quasi-TM fundamental mode of the present hybrid plasmonic waveguide with w _co = 200nm and h _SiO2 = 50nm. In this figure, the field distributions E_y(0, y) and E_y(x, 0) are also shown. One sees that the field at 50nm-SiO₂ nano-layer is enhanced greatly.

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From the curve of E_y(0, y), one sees that the field at 50nm-SiO₂ nano-layer is enhanced greatly. It is well known that there is a similar field enhancement in a low-index region in a pure-dielectric horizontal slot waveguide because of the strong discontinuity of the normal component of the electric field at the high-index-contrast interface [2,20]. For the present hybrid plasmonic waveguide, the principle is different partially. At the Si-SiO₂ interface, there is a strong discontinuity of the normal component of the electric field, which is the same as that in a pure-dielectric horizontal slot waveguide. On the other hand, at the SiO₂-metal interface, surface plasmon (SP) wave is excited. The electrical field of the excited SP wave decays exponentially at both sides of the interface and has a peak at the interface. In the thin SiO₂ layer, the field distribution could be regarded as the sum of two exponential functions. When the SiO₂ layer is very thin (smaller than the evanescent penetration depth), the field at SiO₂ layer is enhanced greatly, as shown in Fig. 2. In the following parts, we consider the designs with different thicknesses of SiO₂ (h _SiO2 = 50 nm, 20 nm, or 5 nm) and the other parameters are chosen as h _m = 100 nm and h _Si = 300 nm.

First we consider the case of h _{Si_rib} = H _Si = 300 nm (i.e., the Si layer is etched through). Figure 3 (a) shows the real part of the effective refractive index of the present hybrid optical waveguide with different SiO₂ nano-layer thicknesses as the core width w _co decreases. For the case with a thinner SiO₂ layer, the effective index n _eff becomes larger. When the core width decreases, the effective index decreases. Even when the core width decreases to 50nm, there is still a guided mode supported in the hybrid optical waveguide, which is very interesting to have nano-scale light confinement (similar to the other metal waveguides). On the other hand, for the realization of plasmonic waveguide devices, it is very important to allow a long propagation distance, L _prop, which is defined as the distance that the amplitude of the field attenuates to 1/e, i.e., L _prop = 1/(n _im k ₀) where n _im is imaginary part of the effective refractive index n _eff, k ₀ is the wave number in vacuum (k ₀ = 2π/λ). The effective refractive index n _eff is obtained by using an FEM-based mode solver in this paper. The previous pure plasmonic metal waveguide usually has a propagation distance of several micrometers (e.g., 3~5 μm [23]). For the present hybrid structure, the calculated propagation distance is shown in Fig. 3 (b) as the core width varies. From this figure, one sees that the propagation distance is on the order of 10² μm (similar to that reported in Ref [17].), which is several tens of times of that for the nano-scale pure plasmonic metal waveguide [23].

 

Fig. 3 For the cases of h _SiO2 = 50 nm, 20 nm, and 5 nm, the real part of the effective refractive index n _eff (a), and the propagation distance L _prop (b) as core width w _co. The insets in Fig. 3(b) show the field distribution of the major component E_y of the quasi-TM fundamental mode for the cases of w _co = 100nm, 300nm, and 500nm. One sees that there is more optical field confined in the Si layer when the core width increases. This is why the effective index real(n _eff) and the propagation distance L _prop increases as the core width increases.

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The insets in Fig. 3 (b) show the field distribution of the major-component E_y of the electrical fields. From these figures, one sees that there is more optical confinement in the Si layer when the core width increases. This is why the effective index real(n _eff) and the propagation distance L _prop increases as the core width increases, as shown in Fig. 3 (a) and 3(b), respectively. We also see that the thickness h _SiO2 of the SiO₂ nano-layer plays an important role for the propagation distance. When choosing a thinner SiO₂ layer, the propagation distance becomes smaller. For the case with a relatively large thickness h _SiO2 (e.g., 50 nm), more power confined in silicon region. Therefore, when the core width decreases, the power confined in the silicon region will change greatly. This is why the influence of the core width on the propagation distances is significant when the thickness h _SiO2 is relatively large. For example, the propagation distance decreases from 432 μm to 76 μm when the core width decreases from 0.5 μm to 50 nm. In contrast, for the case with a very thin SiO₂ layer (e.g., 5 nm), the propagation distance is around 50 μm and does not change greatly as the core width decreases. In summary, the calculation results in Fig. 3 (b) show that the present hybrid plasmonic waveguide supports a propagation distance on the order of several tens of wavelength λ, which is useful to develop plasmonic waveguide devices.

One should note that there is a trade-off between the dimension of the plasmon waveguide and its propagation distance. Since it is easy to realize a propagation distance over 10⁴ μm by using a singlemode SOI nanowire when the core width w _co > 300 nm, in this paper, we focus on the potential of the present hybrid plasmonic waveguides for a relatively long propagation as well as a nano-scale (<100 nm) optical confinement (which is beyond the ability of conventional pure dielectric optical waveguides, e.g., SOI nanowires).

In the analysis above, the Si layer is etched through. We note that the aspect ratio of such a waveguide will be high when the core width becomes very small (e.g., ~100 nm). This will make the fabrication difficult in some degree. A solution to avoid this problem is using a shallowly-etched Si layer (i.e., h _{Si_rib}<H _Si, as shown in Fig. 1). Figure 4 shows the propagation distance L _prop and the real part of the effective index as the Si rib height h _{Si_rib} decreases from 0.3 μm to 0. Here we consider the case with h _SiO2 = 5 nm (about λ/300) and the core wdith w _co = 100 nm (about λ/15) in this example. From this figure, one sees that the propagation distance increases as the rib height h _{Si_rib} decreases. According to the effective index method, a shallow silicon rib makes an equivalent layer with a larger index. This will make more power confined in silicon layer. Therefore, when the silicon rib decreases until zero, both the propagation distance and the real part of the effective index increase, as shown in Fig. 4.

 

Fig. 4 For the cases of h _SiO2 = 5 nm and w _co = 100 nm, the real part of the effective refractive index n _eff and the propagation distance L _prop as the rib height h _{Si_rib} decreases. When the height h _{Si_rib} = 0, the Si part becomes a slab waveguide, in which case the propagation distance is close to 100μm (~60λ) and the fabrication is very easy because the etching becomes shallow.

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From Fig. 4, one sees that for this case (h _SiO2 = 5 nm, which is about λ/300) the propagation distance increases monotonously as the rib height h _{Si_rib} decreases. Particularly, when h _{Si_rib} = 0, one obtains a hybrid plasmonic waveguide with a Si slab and the etching depth is much shallow. This makes the fabrication much simpler and easier. Meanwhile, the propagation distance is close to 100 μm, which is good for ultra-dense photonic integrations. In order to show the optical confinement for such a design with a Si slab, we calculate the field distribution of major-component E_y(x, y) for the quasi-TM polarization in the case with a ultra-small core width w _co = 50 nm (which is about λ/30), as shown in Fig. 5 . In the insets, we also show the field distributions E_y(x, 0) and E_y (0, y) for a clear view. From this figure, one sees the field is confined tightly in the low-index nano-layer. For the present case, the optical confinement is about 50nm × 5nm (~λ/300 × λ/30), which are related with the thickness of the low-index nano-layer and the core width.

 

Fig. 5 The field distribution of the quasi-TM fundamental mode with a nano-scale light confinement when h _{Si_rib} = 0, h _SiO2 = 5 nm, and w _co = 50 nm. The insets show the field distributions E_y(x, 0) and E_y (0, y). One sees that the optical field confinement at the vertical direction is on the order of 5 nm (~λ/300), which is related with the thickness of the low-index nano-layer. At the lateral direction, the optical field confinement is about 50nm (~λ/30).

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Figure 6 shows the calculated coupling length of two parallel hybrid plasmonic waveguides with the same parameters as those used for Fig. 5 (h _{Si_rib} = 0, h _SiO2 = 5 nm, and w _co = 50 nm). The coupling length is given by L _c = π/(β _o–β _e), where β _e and β _o are the propagation constants of the even and odd super-modes of the system of the parallel waveguides (as shown by the inset). The coupling length is almost increases exponentially as the separation D increases, which is similar to the conventional dielectric optical waveguides. When the separation is decreased to 100nm, the coupling length is as small as 2.8μm. This makes it possible to realize a compact directional coupler (which is a basic element for photonic integration circuits).

 

Fig. 6 The calculated coupling length as the separation between two parallel hybrid plasmonic waveguides. The waveguide parameters are: h _{Si_rib} = 0, h _SiO2 = 5nm (~λ/300), and w _co = 50nm (~λ/30).

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3. Conclusion

We have studied a Si-based hybrid plasmonic waveguide with a metal cap for nano-scale light confinement. The present theoretical investigation has shown that a nano-scale (e.g., 50nm × 5nm) optical confinement is obtained with this hybrid plasmonic waveguide when it operates at 1550nm. At the same time, the low-loss enables the present hybrid plasmonic waveguide to have a relatively long propagation distance (on the order of 100 wavelengths). The fabrication the present hybrid plasmonic waveguide is simple and compatible with the standard processes for SOI wafers. Furthermore, our calculation has also shown that one could use a Si slab (instead of Si rib) under the metal cap (see Fig. 5), in which way the fabrication becomes much simpler and easier. With the present hybrid plasmonic waveguide, it is also possible to realize a low-voltage compact optical modulator when the nano-layer material between the Si layer and the metal layer has a high electro-optical coefficient. In order to connect with pure SOI nanowire when necessary, it is possible to introduce mode transformers in the similar way shown in Ref [24]. by consisting of several adiabatic tapers.