1. Introduction

Optical waveguides at the order of nano-scale are desired to achieve photonic integrated circuits (PICs) with a high-integration density. Among various nanophotonic waveguides, the surface plasmon (SP) waveguide has become one of the most attractive candidates because it can break the diffraction limit and thus enables a true nano-scale waveguiding and confinement of light. In contrast, small optical waveguides based on pure dielectric materials are still limited to the order of a wavelength in each direction, due to the diffraction limit [1, 2]. Furthermore, plasmonics offers a way to transfer and process both photonic and electronic signals along the same plasmonic circuit, which is desirable in order to combine the advantage of both photonics and electronics for high signal processing speed and an easy realization of active components. Many types of three-dimensional plasmonic waveguides have been proposed to support highly localized fields, e.g., narrow gaps between two metal interfaces [3–9] and V-grooves in metals [10, 11]. However, such nano-scale optical waveguides are subjected to large losses and the propagation distance is usually at the scale of only several micrometers. This drawback severely limits the application of nano-scale plasmonic waveguides.

In order to achieve a nano-scale light confinement as well as relatively long propagation distance, hybrid plasmonic waveguides have been proposed in recent years and attracted a lot of attention [12–16]. Hybrid plasmonic waveguides can also provide the ability for submicron bending [17], which is very important to achieve ultra-dense photonic integration circuits. The hybrid plasmonic waveguide usually has a high-index region, a metal region, and a low-index nano-slot between them. In Ref [12], a hybrid plasmonic waveguide is realized by putting a dielectric cylinder above a metal surface. For the application of photonic integrated circuits, a planar plasmonic waveguide is desired. For example, one can realize a hybrid plasmonic waveguide for TM polarization by putting a metal cap [16] or metal plate [18] on the top of a SOI (silicon-on-insulator) rib waveguide with SiO2 on the surface. Another type of hybrid plasmonic waveguide was presented for TE polarization by e.g., introducing double low-index vertical nano-slots [21–23]. Considering the fabrication, the structure with metal plate gives good adherence between metal and the low-index material. Therefore, in this paper, we consider the hybrid plasmonic waveguide with a metal plate on the top. Due to the field enhancement in the nano-slot region, hybrid plasmonic waveguides have been used to achieve an enhanced nonlinear effect of optics (e.g., optical parametric amplifier [24]), and highly-efficient optical modulation [25]. Recently some experimental results have also been demonstrated for low-loss Si-based hybrid plasmonic waveguides [18, 26-27].

Even though a hybrid plasmonic waveguide has a relatively low loss and enables a relatively long propagation distance, it is still desirable to reduce the loss further and extend the applications of hybrid plasmonic waveguides in large-scale PICs. It is well known that combining a gain medium is a potential way to overcome the intrinsic loss of plasmonic waveguides. Table 1 gives a brief summary for the gain and operation wavelength of the gain medium reported previously [28–42]. From this table, it can be seen that quantum-dots and III-V semiconductors are very good options to achieve a high gain at long wavelengths. For short wavelengths, dye is a good option. People have proposed gain-assisted plasmonic waveguides with various gain medium [28]-[42]. In Ref [28], a net gain of 85.5dB/cm for long-range surface plasmonic waveguides has been demonstrated by using a dipolar gain medium with a gain of about 420 cm−1. In Ref [35], a deep subwavelenth plasmonic laser has been experimentally demonstrated, by using a hybrid plasmonic waveguide, which has a CdS cylinder above a silver plate. To the best of our knowledge, not much work has been done in the area of silicon-based hybrid plasmonic waveguides with a gain medium. For silicon photonics, the gain medium could be realized by using Er-doping [30]-[31], quantum dots [32, 39, 41], silicon nano-crystals [34], etc.

Table 1. The gain reported in the literatures.

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In this paper, we present a theoretical investigation of a hybrid plasmonic waveguide with a high-index or low-index gain medium. Due to the field enhancement in the nano-slot region, a gain enhancement is observed. Furthermore, since the intrinsic loss of the present hybrid plasmonic waveguide is relatively low, one can compensate the loss or even achieve a pure gain by combining a medium with a moderate gain.

2. Structure and analysis of hybrid plasmonic waveguides

In order to obtain a good adherence between the metal and the low-index material, in this paper we consider the hybrid plasmonic waveguide with a metal plate on the top [18, 24]. Figure 1(a) shows the cross section of the hybrid plasmonic waveguide, which consists of a Si substrate, buffer layer, a high-index dielectric rib, a low-index cladding, a low-index nano-slot and an inverted metal rib. There is a low-index nano-slot region between the high-index region and the metal region. When the size of the slot between the dielectric and metal ribs is of nano-scale, there is a significant field enhancement. As an example, Fig. 1(b) shows the power profile in a hybrid plasmonic waveguide with the present structure. The parameters in this example are: the high index n _H=3.455, n _L=1.445, n _metal=0.1453 +11.3587i, the index of the buffer layer n _buf=1.445, the heights H=300nm, h _rib=250nm, h _slot=10nm, h _m=100nm, and the width w _co=200nm.It can be seen that there is a significant field enhancement, which is similar to that shown in Ref [16].

 

Fig. 1 (a) The cross section of the present hybrid plasmonic waveguide with an inverted metal rib. The materials in the slot region and the cladding region have the same (low) refractive index; (b) the power distribution in a hybrid plasmonic waveguide with the present structure. The parameters in this example are: the high index n _H=3.455, n _L=1.445, nmetal=0.1453+11.3587i, the heights H=300nm, h _rib=250nm, h _slot=10nm, h _m=100nm, and the width w _co=200nm.

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Usually the low-index material could be SiO₂, Al₂O₃, SiN, or polymer while the high-index material could be silicon, or III-V semiconductor. Since silicon photonics has become very attractive because of its fabrication compatibility to the standard CMOS microelectronics technology, here we choose silicon-on-insulator (SOI) wafers for the present hybrid plasmonic structure as an example. Accordingly, the fabrication process is proposed as shown in Fig. 2(a) -2(f). The first step is forming a low-index material (e.g., SiO₂ or polymer) thin film with a thickness of several tens of nanometers, as shown in Fig. 2(a). Figure 2(b) shows the following step of making a patterned hard-mask layer with an appropriate material (e.g., SiN). Then form a rib with a dry-etching process (see Fig. 2(c)). The next step is to deposit a low-index thin film to cover the whole wafer and make a flat top surface with a CMP (chemical-mechanical polishing) process if needed, as shown in Fig. 2(d). The mask layer is then removed with a wet-etching process and thus one has a notch as shown in Fig. 2(e). In the final step a metal thin film is deposited on the top (see Fig. 2(f)).

 

Fig. 2 The proposed fabrication processes for hybrid plasmonic waveguide with a metal plate on the top, (a) form a low-index thin film; (b) make a patterned hard-mask layer with an appropriate material (e.g., SiN); (c) form a rib with a dry-etching process; (d) deposit a low-index thin film to cover the whole wafer and make a flat top surface with a CMP (chemical-mechanical polishing) process; (e) remove the hard-mask with a wet-etching process; (f) deposit a thin metal film on the top.

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In order to give a characteristics analysis for the present hybrid plasmonic waveguide, we use an FEM (finite-element method)-based mode solver to calculate the eigenmodes. In our calculation, we choose the parameters as follows: the total silicon thickness H=300nm, the Si rib height h _rib=250nm. The refractive indices @1550nm for the materials of Si, SiO₂, and silver are n _Si=3.455, n _SiO2=1.445, n _Ag=0.1453 +11.3587i. Here we consider the cases with different metal rib height (h _m=5, 50, and 100nm) and different slot thickness (h _slot=10nm, and 50nm).

Figure 3(a) shows the real part of the effective refractive index of the present hybrid plasmonic waveguide as the core width w _co varies. We also show the power confinement ratios in the cladding (Г _cl), the nano-slot (Г _slot), the Si region (Г _Si), and the buffer layer (Г _bf) for the cases of h _slot=10nm, and 50nm, respectively, in Fig. 3(b) and 3(c). In these figures, we do not include the power confinement ratio in the metal region because it is too small to show clearly. Instead it will be given in Fig. 3(h) below to explain the loss. The power confinement ratio is defined as the ratio of the power confined in a certain region to the total power.

 

Fig. 3 The effective index (a), the power confinement factors in different regions (b) and (c), the power density in the slot region (d), the effective area (e), the loss (f), and the power confinement ratio Γ _metal in the metal region (h), as the waveguide width varies when h _slot=10, and 50nm. The metal rib height h _m is assumed to be 5, 50, and 100nm.

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From Fig. 3(a), one sees that the effective index becomes smaller as the core width decreases. In a hybrid plasmonic waveguide with a smaller core width w _co, the power confinement ratios Г _slot and Г _Si decrease while the power confinement factor Г _cl in the cladding region increases. For example, when the core width decreases to sub-100nm, the power confinement factor Г _cl in the cladding region becomes about 60% or more, as shown in Fig. 3(b) and 3(c). From Fig. 3(a), it can also be seen that the effective index n _eff for a given waveguide width w _co is larger in the case with a thinner slot layer. This could also be explained by comparing the power confinement ratios shown in Fig. 3(b) and 3(c). When the slot thickness is smaller, the field enhancement in the nano-slot region is stronger. Consequently less power is confined in the cladding region and the buffer region, while more power is confined in the Si rib region (see Fig. 3(b), and 3(c)). Therefore, one has a smaller effective index when choosing a thinner slot.

From Fig. 3(b) and 3(c), it can be seen that the power ratio Г _slot changes only slightly for the cases with different metal rib heights when the rib width is larger (e.g., w _co>300nm). In contrast, when the rib width is small (<300nm), one can obtain a higher power ratio Г _slot by choosing a larger metal rib height than for the smaller rib height. As the rib width decreases, the power confinement ratio Г _slot in the low-index slot region decreases while the area of the slot region also becomes smaller. In this case, there is an optimal rib width to achieve a maximal power density in the slot region, as shown in Fig. 3(d). Here the power density is normalized by the total waveguide optical power [21]. For nonlinear applications, it is desired to achieve a high power density, which can be obtained by choosing a smaller slot height, as shown in Fig. 3(d). We also note that a larger metal rib height is helpful to achieve a higher power density. For example, for the case of h _slot=10nm, the maximal power density is up to about 88μm^–2, and 151μm^–2, when the metal rib height h _m=5nm, and 100nm, respectively. Their corresponding optimal core widths are w _co=70, and 140nm, respectively. According to Fig. 3(d), one can tune the optimal core width for maximal power density by appropriately choosing the metal rib height.

Figure 3(e) shows the calculated effective area of the hybrid plasmonic mode. The effective area A _eff is defined as [12]

From the analysis above, one sees that the present hybrid plasmonic waveguide enables a nano-scale light confinement. On the other hand, for the realization of plasmonic waveguide devices, it is very important to have a low loss due to metal absorption. The loss of an optical waveguide is given by L=8.86n _im k ₀ where n _im is the imaginary part of the effective refractive index n _eff, and k ₀ is the wave number in vacuum (k ₀=2π/λ). In Fig. 3(f), we show the calculated loss for the cases of h _slot=10 and 50nm. In order to explain the loss behavior, we also calculate the power confinement ratio Γ _metal in the metal region, as shown in Fig. 3(h). From Fig. 3(h), one sees power confinement ratio Γ _metal is negative, which indicates a Joule loss [44-45].

When the slot becomes thicker, the field enhancement in the slot region becomes weaker (see the power density shown in Fig. 3(d)) and less field is confined in the metal region (see Fig. 3(h)). Consequently one has lower loss when choosing a larger slot thickness, as shown in Fig. 3(f). For example, for the case of h _m=100nm and w _co=100nm (where the effective area is minimal), the loss is reduced from 0.12dB/μm to 0.0454dB/μm when increasing the slot height h _slot from 10nm to 50nm. Figure 3(f) also shows that the metal rib height play an important role for the loss. A smaller metal rib height helps to reduce the loss because less power is inside the metal region (as shown in Fig. 3(h)). For example, for the case of h _slot=50nm and w _co=100nm (where the effective area is minimal), the loss is reduced from 0.0454dB/μm to 0.0287dB/μm, when decreasing the metal rib height h _m from 100nm to 5nm. On the other hand, when the loss is reduced by increasing the slot height or decreasing the metal rib height, the effective area becomes larger. Therefore, one should make a tradeoff when choosing the slot height and the metal rib height for acceptable loss as well as effective area.

3. Hybrid plasmonic waveguides with a gain medium

Even though the hybrid plasmonic waveguide provides a way to achieve a relatively low loss propagation as well as a nano-scale optical confinement, its intrinsic loss limits its applications in a long-distance optical interconnect. In order to overcome this limitation, an intuitive way is to introduce a medium with gain. In the following parts, we give analyses for a hybrid plasmonic waveguide with a low-index or high-index gain region. Usually the pump is fed in the electrical or optical ways. For plasmonic waveguides and devices, optical pump is often used because of the convenience. For the present plasmonic waveguide, the light pump could be applied by the back illumination since there is a metal on the top. On the other hand, it is also possible to introduce an electrical pump for the present plasmonic waveguide. For the case with a high-index gain medium, the carrier injection can be done by putting two electrodes on the high-index slab at two sides of the ridge. For the case with a low-index gain medium, the carrier injection can be done by putting an electrode on the slab region while the metal on the top could be used as the other electrode. The electrical pump way is more complex and one should design it carefully according to the structure and material of the gain medium. More work is needed to discuss the optical as well as electrical pump ways in the future.

The low-index gain medium could be silicon nanocrystal, polymer with quantum dots (or Er doping), etc. The silicon nanocrystal could be formed by some thin-film deposition technologies. For polymer with quantum dots, one can use a spin-coating technology. According to the reported gain values [32]-[38], here we assume moderate gain g for the low-index gain medium, e.g., 35.2, 176, and 352dB/cm. Figure 4(a) and 4(b) show the calculated loss (or gain) for hybrid plasmonic waveguides with h _slot=10nm, and 50nm, respectively. The metal rib height is chosen as h _m=5nm to have a low loss. In our calculation, we assume that the low-index gain medium fills the cladding region and the slot region.

 

Fig. 4 The gain or loss of a hybrid plasmonic waveguide with low-index gain medium for the cases of h _slot=10nm (a), and h _slot=50nm (b); (c) the ratio of ∂G/∂g; (d) the gain or loss as the gain g of the low-index gain medium increases; h _m is chosen to be 5nm.

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From Fig. 4(a) and 4(b), one sees that the loss of the plasmonic waveguide becomes smaller as expected when the gain of the low-index medium increases. When the core width becomes small, it is possible to compensate the intrinsic loss and even achieve a pure gain by a moderate gain medium. For example, for the case of h _slot=10nm, one obtains a pure gain G of up to 67dB/cm when the core width w _co=30nm and the low-index medium has a gain of 352dB/cm. In contrast, when choosing a larger slot height, the intrinsic loss becomes lower and consequently it is easier to compensate the intrinsic loss as well as obtain a pure gain. For example, when h _slot=50nm, the intrinsic loss of a 30nm-wide hybrid plasmonic waveguide could almost be compensated by using a medium with a gain of only 176dB/cm (see Fig. 4(b)).

Figure 4(c) shows the ratio ∂G/∂g as the core width varies, where g and G are the gains of the gain medium, and the TM fundamental mode of the plasmonic waveguide, respectively. One sees that a larger ratio ∂G/∂g is obtained by choosing a larger slot height or a smaller core width. This is because more power is confined in the gain region (i.e., the cladding and the nano-slot regions) when increasing the slot height or decreasing the core width. Since only the optical field in the cladding and the slot region “sees” the gain medium, the ratio ∂G/∂g of a guided mode of the hybrid plasmonic waveguide is intuitively assumed to be smaller than 1.0. However, for a hybrid plasmonic waveguide with a small core width (e.g., <60nm), a gain enhancement is observed, i.e., the ratio ∂G/∂g becomes larger than 1.0. Nevertheless, this does not violate the fundamental physics according to the analysis for a high index-contrast optical waveguide in Refs [46]-[47]. The reason for ∂G/∂g>1 in a high-index contrast optical waveguide has been given in Ref [47]. For high-index contrast waveguides, the waveguide gain is not proportional to the power confinement ratio in the gain region (i.e., the percentage of the guided mode power which overlaps with the gain medium) due to the large electric field discontinuities at dielectric interfaces. According to Ref [47], the ratio ∂G/∂g is determined by two parts, i.e., ∂G/∂g≈n _g/n _Aγ, where n _g is the group index, n _A is the real part of the refractive index of the gain medium, and γ is the spatial confinement of the energy density to the active region of the waveguide. It has been proved that since n _g/n _A is usually larger than 1.0 one could have ∂G/∂g>1, e.g., when γ is close to 1.0 [47]. Since the present hybrid plasmonic waveguide has a high index-contrast and consequently the TM polarization mode has a large electric field discontinuities at dielectric interfaces, it is reasonable to have ∂G/∂g>1 in a certain case as shown in Fig. 4(c).

Such a gain enhancement is helpful to reduce the threshold of gain required for compensating the intrinsic loss and even achieving a pure gain in a hybrid plasmonic waveguide. As an example, in Fig. 4(d) we show the gain (or loss) G of a hybrid plasmonic waveguide with w _co=100nm as the gain g of the low-index medium increases. One sees that the gain (or loss) G increases linearly when the gain g increases. The case of h _slot=50nm has a larger slope ∂G/∂g and a higher gain (or lower loss). In this case, the intrinsic loss could be compensated when the gain medium has a gain of about 300dB/cm, which is achievable [32]-[38].

The high-index gain medium could be silicon with Er doping, a III-V semiconductor, etc. Here the gain g of the high-index gain medium is assumed to be 35.2, 176, and 352dB/cm. Figure 5(a) and 5(b) show the calculated loss or gain for hybrid plasmonic waveguides with h _slot=10nm, and 50nm, respectively. The metal rib height is chosen as h _m=5nm to have a low loss.

 

Fig. 5 The gain or loss of a hybrid plasmonic waveguide with a high-index gain medium for the cases of h _slot=10nm (a), and h _slo=50nm (b); (c) the ratio ∂G/∂g; and (d) the gain or loss as the gain g of the high-index gain medium increases; h _m is chosen to be 5nm.

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As shown in Fig. 5(a) and 5(b), the loss of the plasmonic waveguide becomes smaller, as expected, when the gain of the high-index medium increases. When the core width becomes larger, the signal amplification is more effective because more power is confined in the high-index gain region. When choosing a relatively large slot height (to have a low intrinsic loss), one could compensate the intrinsic loss and even achieve a pure gain by a moderate gain medium. For example, for the case of h _slot=50nm, one obtains a pure gain G of up to 60dB/cm when the core width w _co=300nm and the high-index medium has a gain of 352dB/cm. In contrast, when choosing a smaller slot height, the intrinsic loss becomes larger and consequently one has to introduce more gain to compensate the intrinsic loss.

Figure 5(c) shows the ratio of ∂G/∂g as the core width w _co varies, where g and G are the gains of the high-index gain medium and the TM fundamental mode of the plasmonic waveguide, respectively. A larger ratio ∂G/∂g can be obtained when choosing a larger core width because more power is confined in the high-index gain region. In the present example, for the case of h _slot=50nm, the maximal ratio ∂G/∂g is about 0.86 when w _co=500nm. This maximal ratio is much larger than the power confinement factor in the gain region (Γ _Si≈0.72), which is also due to the enhancement analyzed in Ref [46].

As an example, in Fig. 5(d) we show the gain (or loss) G of a hybrid plasmonic waveguide with w _co=500nm when introducing different gain g for the high-index medium. One sees that the gain (or loss) G increases linearly when the gain g increases. And the design of h _slot=50nm has a larger slope ∂G/∂g and a higher gain than the design of h _slot=10nm. For the case of h _slot=50nm and w _co=500nm, the intrinsic loss could be compensated when the gain medium has a gain of about 200dB/cm, which is achievable [32]-[38].

3. Conclusion

We have presented a theoretical investigation of a hybrid plasmonic waveguide with a low-index as well as high-index gain medium. The present hybrid plasmonic waveguide has been shown to have a relatively low intrinsic loss and a nano-scale optical confinement. One can make a trade-off between the optical confinement and the propagation distance by adjusting the waveguide width, slot height, and the metal rib height. For the hybrid plasmonic waveguide with a low-index gain medium, a gain enhancement has been observed (i.e., the ratio of ∂G/∂g >1), where g and G are the gains of the gain medium and the TM fundamental mode of the hybrid plasmonic waveguide, respectively. The intrinsic loss could be compensated when using a low-index gain medium with a moderate gain. For example, when the core width of w _co=30nm and the slot height h _slot=50nm, one can compensate the intrinsic loss of the hybrid plasmonic waveguide by using a low-index medium with a gain of only 176dB/cm. In contrast, for the hybrid plasmonic waveguide with a high-index gain medium, one obtains a higher gain when choosing a wider core width. For the case of w _co=500nm and h _slot=50nm, the intrinsic loss could be compensated when the high-index medium has a gain of about 200dB/cm.