## Abstract

Characteristic analyses are given for a bent silicon hybrid plasmonic waveguide, which has the ability of submicron bending (e.g., *R* = 500nm) even when operating at the infrared wavelength range (1.2μm~2μm). A silicon hybrid plasmonic submicron-donut resonator is then presented by utilizing the sharp-bending ability of the hybrid plasmonic waveguide. In order to enable long-distance optical interconnects, a pure dielectric access waveguide is introduced for the present hybrid plasmonic submicron-donut resonator by utilizing the evanescent coupling between this pure dielectric waveguide and the submicron hybrid plasmonic resonator. Since the hybrid plasmonic waveguide has a relatively low intrinsic loss, the theoretical intrinsic Q-value is up to 2000 even when the bending radius is reduced to 800nm. By using a three-dimensional finite-difference time-domain (FDTD) method, the spectral response of hybrid plasmonic submicron-donut resonators with a bending radius of 800nm is simulated. The critical coupling of the resonance at around 1423nm is achieved by choosing a 80nm-wide gap between the access waveguide and the resonator. The corresponding loaded Q-value of the submicron-donut resonator is about 220.

©2011 Optical Society of America

## 1. Introduction

It is well known that a surface plasmon (SP) waveguides can break the diffraction limit and thus enable nano-scale optical waveguiding and light confinement. Therefore, surface plasmon (SP) waveguides have attracted a lot of attention to achieve photonic integrated circuits (PICs) with an ultrahigh integration density. Furthermore, plasmonics offers a way in order to transfer and process both photonic and electronic signals along the same plasmonic circuit, which is desired to combine photonics and electronics for high-speed signal processing and to easily realize some easy realization of active components. In the past decades, many types of three-dimensional plasmonic waveguides have been proposed to support highly localized fields, e.g., narrow gaps between two metal interfaces [1–7] and V-grooves in metals [8, 9]. The problem is that the propagation distance is usually limited to the order of several micrometers due to the huge intrinsic loss. Recently hybrid plasmonic waveguides have been proposed and attracted a lot of attention as a good option to realize nano-scale light confinement as well as relatively long propagation distance [10–26]. This makes it interesting to realize elements of various functionality by using hybrid plasmonic waveguides.

As a versatile element for photonic integrated circuits, optical cavities have been used in many applications, e.g., light sources [27], optical filters [28], optical modulators/switches [29], optical sensing [30], nonlinear optics [31], etc. People have developed various optical cavities based on different materials/structures (especially pure dielectric platforms), e.g., photonic crystal cavities, silicon microring/microdisk, etc. However, there is not much work on optical cavities based on hybrid plasmonic waveguides. It is well known that for many applications one usually desires to have an optical cavity with a high-Q factor and low volume. Therefore, it is very important to use an optical waveguide, which allows an ultrasharp bending. In Ref [24], an analysis is given for a semiconductor-insulator-metal strip waveguide (which is a kind of hybrid plasmonic waveguide). In this paper, we focus on the bent hybrid plasmonic waveguide with a metal cap, which has not been characterized well. Our results show that the present hybrid plasmonic waveguide enables a submicron bending radius. Consequently this makes it available for realization of submicron resonators.

In Ref [26], a subwavelength hybrid plasmonic nanodisk without access waveguides is investigated theoretically and shows that it is possible to simultaneously achieve a relatively high Q-factor and Purcell factor. Even though a hybrid plasmonic waveguide has a relatively low loss and enables a relatively long propagation distance (~10^{2} μm), it is still not a good option for long-distance (e.g., 10^{3}~10^{4}μm) optical interconnects if there is no assistance from gain mediums [32].Therefore, in this paper, we propose a silicon hybrid plasmonic submicron-donut resonator with pure dielectric access waveguides by utilizing the evanescent coupling between the pure dielectric waveguide and the submicron hybrid plasmonic resonator. This way, the pure dielectric waveguide enables a long-distance optical interconnect because of its low intrinsic low loss. Meanwhile, no additional mode converter is needed to combine the hybrid plasmonic circuits and the pure dielectric waveguides. A microdonut resonator is essentially a microdisk with an inner hole perforated at the center, and a single mode operation can be made by adjusting the radius of the inner hole. In comparison with a microring resonator, a microdonut resonator is expected to have a higher quality factor because the fundamental mode only interacts with the outer sidewall of the resonator by adjusting the radius of the inner hole [33]. In this paper, the submicron-donut resonator operating at the near-infrared wavelength range (1200nm~2000nm) is realized by using Si hybrid plasmonic waveguides with the ability of sharp-bending. Furthermore, particularly the hybrid plasmonic waveguide used for the submicron-donut resonator has a large contact area between the metal layer and the low-index layer, which helps to maintain a good adherence of the metal layer and makes the fabrication easy and robust.

## 2. Structure and Theory

Figure 1(a)
shows the schematic configuration of the present hybrid plasmonic submicron-donut resonator. The hybrid plasmonic waveguide consists of a Si substrate, buffer layer, a high-index dielectric rib, a low-index cladding, a low-index nano-slot and a metal cap. A low-index nano-slot region is inserted between the high-index region and the metal region. For the TM fundamental mode, there is a significant field enhancement in the low-index nano-slot region due to the boundary condition of the perpendicular electrical field. Usually the low-index material could be SiO_{2}, Al_{2}O_{3}, SiN, or polymer while the high-index material could be silicon [14], or III-V semiconductor [25]. In present hybrid plasmonic submicron-donut resonator, one can achieve a single-mode operation since the fundamental mode of the resonator is mostly confined around the outer perimeter of the microdonut while the higher-order modes are pushed into the leaky zone. This is the difference between microdonut resonators and microdisk resonators. Furthermore, in a microdonut resonator, the fundamental mode only interacts with the outer sidewall of the resonator (which will be seen in Fig. 6 below) by adjusting the radius of the inner hole. In contrast, in a microring resonator, the mode interacts with two sidewalls in microring resonators. Consequently, a higher quality factor for the fundamental mode is expected when using microdonut resonator. Since the hybrid plasmonic waveguide used for submicron-donut resonators is wide, there is a large contact area between the metal layer and the low-index layer, which helps to achieve a good adherence of the metal layer and consequently makes an easy and robust fabrication.

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. The wavelength-dependent refractive indices of the materials involved are given by the following formulas.

#### 2.1. Metal* [*34*]*

The frequency-dependent complex refractive index *ε*(*ω*) of any metal can be appropriately given by the Drude formula as follows .

*ω*c is the collision frequency and

*ω*p is the plasma frequency. The temperature dependence of the plasma frequency owing to its volumetric effects is written aswhere γ

_{e}is the expansion coefficient of the metal and

*T*

_{0}is the reference temperature (e.g., room temperature).

The collision frequency *ω*_{c} is given by

*ω*

_{cp}and

*ω*

_{ce}are corresponding to contributions from the phonon–electron scattering and electron–electron scattering, respectively.

The phonon-electron scattering frequency *ω*_{cp} is given by [35]

*T*

_{D}is the Debye temperature.

The electron-electron scattering frequency *ω*_{ce} is given by [36]

*E*

_{F}is the Fermi energy,

*k*

_{B}is Boltzmann constant,

*h*is Plank constant. Γ is a constant giving the average over the Fermi surface of the scattering probability and ∆ is the fractional umklapp scattering. All the parameters involved for Ag are given in Table 1 .

#### 2.2. SiO_{2} and Si

The refractive index of a dielectric is usually given by the Sellmeier formula. For SiO_{2}, one has [34]

*λ*is the wavelength given in micron (μm),

*B*

_{1}=0.6961663,

*B*

_{2}=0.4079426,

*B*

_{3}=0.8974994,

*λ*

_{1}

^{2}=0.004679148μm

^{2},

*λ*

_{2}

^{2}=0.013512068 μm

^{2},

*λ*

_{3}

^{2}=97.934002 μm

^{2}.

For Si, one has [37]

*n*

_{0}=3.41696,

*A*

_{1}=0.138497,

*A*

_{2}=0.013924,

*A*

_{3}=22.09×10

^{25},

*A*

_{4}=1.48×10

^{27}, Λ

_{1}=0.028 (for undoped silicon),

*T*

_{0}is the ambient temperature (

*T*

_{0}= 300K), d

*n*/d

*T*is the temperature coefficient, and

*T*

_{0}=293K, d

*n/*d

*T*= 1.5×10

^{−4}1/K.

## 3. Results and discussions

By using Eqs. (1), (6) and (7), we obtain the dependences of the refractive indices of SiO_{2} Si, and Ag on the wavelength, as shown in Fig. 2(a)
and 2(b). The experimental data from Ref [38]. for the silver’s refractive index *n*_{Ag} is also shown here. From this figure, it can be seen that the Drude formula gives a good estimation for the refractive index of silver in the wavelength range from 1.2μm to 2.0μm.

#### 3.1. Ultrasharply bent hybrid plasmonic waveguides

In order to give a characteristic analysis for the bent hybrid plasmonic waveguide, first we consider the case of λ=1550nm, which is one of the most concerned window. The corresponding parameters in this example are: the high index *n*_{H}=3.4777 (Si), *n*_{L}=1.444 (SiO_{2}), *n*_{metal}=0.159+11.245i (Ag), the index of the buffer layer n_{buf}=1.444 (SiO_{2}). The silicon height *H*=300nm, and the metal height *h*_{m}=200nm. In order to have a sharp bending, we choose a deep etching, i.e., *h*_{rib}=*H*=300nm.

For the calculation of the complex propagation constants *β* of the eigenmodes of bent hybrid plasmonic waveguides, we use a full-vectorial finite-difference method (FV-FDM) mode-solver in a cylindrical coordinate system. The complex effective index *n*_{eff} is then given by *n*_{eff} = *β*/*k*_{0}, where *k*_{0} is the wavenumber in vacuum. Figure 3(a)
–3(c) show the real part of the calculated effective index *n*_{eff} for the TM fundamental mode of the bent hybrid plasmonic waveguides as the bending radius *R* decreases from 2μm to 0.5μm. Here we consider the cases with *h*_{slot} = 10, 20, and 50nm. The waveguide widths are set at *w* = *R*, 400, 300, and 200nm, respectively. The radius *R* is for the outer sidewall of the bent waveguide, as shown in Fig. 1. Note that the submicron-donut becomes a submicro-disk when choosing *w* = *R*. From Fig. 3(a)–3(c), it can be seen that one has a larger effective index when choosing a larger waveguide width, especially when the bending radius is larger (e.g., *R* = 1~2μm). This is because more power is confined in the core region. As the bending radius decreases, the peak of the mode profile shifts outward and consequently the inner surface influence the mode profile less. Therefore, the effective index of the bent waveguide becomes less sensitive to the waveguide width.

Figure 4(a)
–4(c) show the imaginary part *n*_{eff_im} of the effective index for bent hybrid plasmonic waveguides as the bending radius decreases. There are two sources contributing to the imaginary part (i.e, the loss). One is from the intrinsic loss due to the metal absorption. This intrinsic loss per unit length is not sensitive to the bending radius. The other one is from the leakage due to the bending, which increases exponentially as the bending radius decreases. Therefore, from Fig. 4(a)–4(c), one sees that the imaginary part *n*_{eff_im} is not sensitive to the bending radius when the bending radius is relatively large. When one reduces the bending radius further to a certain value (*R*_{0}), a significant increase is observed. The low-index slot height *h*_{slot} also plays an important role for the loss in a bent hybrid plasmonic waveguide. According to our previous analysis, a straight hybrid plasmonic waveguide with a smaller slot height *h*_{slot} has a higher loss because more power sees the metal [14, 32]. Since a bent waveguide with a relatively large bending radius is similar to a straight waveguide, one has a larger imaginary part (loss) when choosing a smaller *h*_{slot}. However, on the other hand, a hybrid plasmonic waveguide with a smaller slot height has a stronger confinement (see the effective index shown in Fig. 3(a)–3(c)), and consequently lower bending loss can be achieved and a smaller bending radius is allowed. For example, when *h*_{slot} = 50nm, a significant imaginary part is observed when *R* <*R*_{0} (where *R*_{0} = 1.2μm). In contrast, when *h*_{slot} = 10nm, the radius *R*_{0} is as small as 0.75μm. Figure 4(a)–4(c) also show that the imaginary part (loss) becomes lower by choosing a wider waveguide. This is because there is less power confined in the metal region for a hybrid plasmonic waveguide with a larger core width.

We also calculate the loss for a 90°-bending hybrid plasmonic waveguide with the imaginary part *n*_{eff_im} of the effective index, i.e., *L* = 20log_{10}[exp(–*n*_{eff_im}*k*_{0}*R*π/2)], wehre *k*_{0} is the wavenumber in vacuum. The calculated results for the cases of *h*_{SiO2} = 10, 20, and 50nm are shown in Fig. 5(a)
–5(c), respectively. From these figures, it can be seen that there is an optimal bending radius, which gives a minimal total loss for a 90°-bending. This is due to the joint contributions from the bending leakage (which increases exponentially as the bending radius decreases) and the intrinsic loss due to the metal absorption. This intrinsic loss per 90°-bending is proportional to the bending radius. Therefore, one has a minimal total loss at an optimal bending radius *R*_{opt}. When *R*>*R*_{opt}, the intrinsic loss due to the metal absorption is dominant and the loss decreases linearly almost as the bending radius decreases. Otherwise, the bending loss becomes dominant and the total loss increases exponentially as the bending radius decreases. The low-index slot height *h*_{slot} also plays an important role for the loss in a bent hybrid plasmonic waveguide. From Fig. 5(a)–5(c), one sees that the optimal bending radius *R*_{opt} for a minimal loss is sensitive to the slot height *h*_{slot}. When choosing a smaller slot height *h*_{slot}, one has a smaller optimal bending radius *R*_{opt}. For example, one has *R*_{opt} = 600nm, 800nm, and 1.1μm when choosing *h*_{slot} = 10, 20, and 50nm, respectively. The reason is that the smaller slot height provides a stronger ability of light confinement, as discussed in Ref [14,32]. For the case with a small slot height (e.g., *h*_{slot} = 10nm), the bending loss is observed until the bending radius becomes very small. In contrast, when there is a large slot height (e.g., *h*_{slot} = 50nm), the hybrid plasmonic effect becomes weaker and one has a smaller intrinsic loss [32]. In this case, however, the behavior of the hybrid plasmonic waveguide is like a dielectric waveguide and the bending loss becomes huge when the bending radius is smaller than 1μm. Figure 5(a)–5(c) also show that the loss becomes lower by choosing a wider waveguide. This is because there is a stronger confinement for a hybrid plasmonic waveguide with a larger core width.

#### 3.2. Submicron-donut resonator

For the design of submicron-donut resonators, we consider the case of *h*_{slot} = 20nm and *w*_{co} = 400nm as an example. Figure 6(a)
–6(d) show the electrical field distribution *E _{y}*(

*x*,

*y*) of the TM fundamental mode for the cases of

*R*= 2μm, 1μm, 800nm, and 500nm, respectively. From these figures, it can be seen that there is a field enhancement in the low-index slot region. We note that the light is still well confined in the slot region even when the bending radius is as small as 500nm (about 1/3 of the operation wavelength). When the radius decreases, one sees that the peak of the electrical field shifts outward as expected. Consequently the TM fundamental mode less interacts with the inner sidewall, which helps to achieve low scattering loss. For example, when the bending radius is reduced to 800nm, the field hardly interacts with the inner sidewall at tall.

Figure 7(a)
shows the calculated real part of the effective index and the loss of a bent hybrid plasmonic waveguide when the wavelength varies from 1.2μm to 2μm. From this figure, it can be seen that the effective index decreases as the wavelength increases, which is similar to a pure dielectric waveguide. Such a negative dispersion comes from the material dispersion and the waveguide dispersion. The dispersion coefficient *D* = (∂*n*_{eff}/∂λ) is about –0.8~–0.4μm^{–1}. As the wavelength increases, the loss of the bent hybrid plasmonic waveguide becomes higher. There are two reasons. The first reason is related with the metal absorption loss. For the hybrid plasmonic waveguide, the power confined in the silicon region at the short wavelength is more than that at long wavelength. It indicates that less light see the metal and consequently less metal absorption at the shorter wavelength in the range from 1.2μm to 2μm. The other reason is that the confinement ability becomes weaker at longer wavelength and consequently the loss due to the bending leakage increases. When the wavelength varies from 1200nm to 2000nm, the loss is roughly doubled.

Figure 7(b) shows the group index *N*_{g} of the bent hybrid plasmonic waveguide. The group index *N*_{g} is given by *N*_{g} = *n*_{eff}–*Dλ*, where *D* is the dispersion coefficient. The intrinsic Q-value of the submicron-donut resonators based on the bent hybrid plasmonic waveguide is also shown in Fig. 7(b). The intrinsic Q-value is calculated by the following formula: Q = *N*_{g}2π/(αλ_{0}) [39], where α is the attenuation coefficient, λ_{0} is the wavelength in vacuum. Since α = 2*n*_{eff_im}*k*_{0}, one has Q = *N*_{g}/(2*n*_{eff_im}), where *n*_{eff_im} is the imaginary part of the effective index. From this figure, it can be seen that the intrinsic Q-value decreases as the wavelength increases, which is due to the higher loss at longer wavelength. Because of the relatively low loss of the hybrid plasmonic waveguide, the intrinsic Q-value is expected to be as high as 10^{2}~10^{3} even when choosing a very small bending radius, *R* = 800nm. This Q-value for the present submicron-donut resonator is comparable to that for a plasmonic microring resonator with *R* = 5μm calculated in Ref [40]. Meanwhile, the small volume of the submicron-donut resonator helps to achieve a high Purcell factor *F*_{P} given by *F*_{P} = 3(4π^{2})^{–1}(λ_{0}/*n*)^{3}(*Q*/*V*_{eff}) [26], where *V*_{eff} is the effective mode volume, *V*_{eff} = ∫*P*(*r*)d^{3}*r/P*_{max} [41], where *P*(*r*) is the electromagnetic energy density of the mode, and *P*_{max} is the peak value of *P*(*r*). With this definition, one has an effective modal volume *V*_{eff}≈0.01μm^{3} for the present case (*R* = 800nm and *h*_{slot} = 20nm) and the corresponding Purcell factor *F*_{P} is about 1450.

In order to achieve the spectral responses of the hybrid plasmonic submicron-donut resonator, we use a three-dimensional finite-difference time-domain (3D-FDTD) method to simulate the light propagation in the present submicron-donut resonator. The grid sizes are chosen as ∆*x* = ∆*z* = 20nm, and ∆*y* = 5nm, respectively. According to the analysis for the loss of a bent hybrid plasmonic waveguide, we choose *h*_{slot} = 20nm, *w*_{co} = 400nm, and *R* = 800nm as an example. Figure 8(a)
shows the calculated wavelength responses for a submicron-donut resonator when the gap width *w*_{g} is chosen as 60, 80, 100, and 120nm. In this example, the width of the pure dielectric access waveguide is chosen as 350nm to be singlemode. From this figure, it can be seen that there are four resonances in the wavelength range from 1200nm to 1800nm, and that the free spectral range (FSR) is not uniform due to the dispersion. The separation between the two adjacent resonances around 1550nm is about 148nm. Such a large FSR is due to the ultrasmall radius.

For a resonator with coupled waveguides, the Q value is given by 1/Q=1/Q_{c} + 1/Q_{0}, where Q_{c} is the Q-value due to the coupling, and Q_{0} is the intrinsic Q-value. When critical coupling occurs, one has Q_{c} = Q_{0}, and Q = Q_{0}/2. For the present case, the intrinsic Q-value estimated by using the formula of Q = *N*_{g}/(2*n*_{im}) is about 800 (see the curve for the case of *R* = 800nm shown in Fig. 7(b)). The Q-value for the submicron-donut with a coupled access waveguide is expected to be as high as 400. From the spectral response given in Fig. 8(a), the Q-value can be calculated by using Q = λ/∆λ_{3dB}, where ∆λ_{3dB} is the 3dB bandwidth. The Q-value at the resonance around 1423nm is about 220, which is lower than the expected value (Q_{0}/2 = 400). This might be due to some excess scattering loss at the coupling region between the straight access waveguide and the resonator (due to the very sharp bending).

For a resonator with a single access waveguide, it is well known that there is a critical coupling that gives a maximal extinction ratio at resonances. In Fig. 8(b), the extinction ratio at the resonance wavelength 1423.67nm is shown as the gap *w*_{g} varies from 60nm to 120nm. The extinction ratio is defined as the ratio between the powers at the statuses of off-resonance and on-resonance. From this figure, one can see that a critical coupling is achieved when choosing the gap width about 80nm and the corresponding extinction ratio is close to 20dB.

Figure 9(a)
and 9(b) show the simulated electrical field distribution *E _{y}*(

*x*,

*z*) in the hybrid plasmonic submicron-donut resonator when the input wavelength is chosen as 1423.67nm (on-resonance), and 1437nm (off-resonance), respectively. The

*y*-position of this

*xz*plane is at the middle of the low-index slot region. From Fig. 9(a) and 9(b), one can see that there is a field enhancement in the submicron-donut due to the resonance, which is similar to the regular pure dielectric micro-resonator. Almost no bending leakage is observed from this FDTD simulation. With the present submicron-donut resonator, one can also realize some ultrasmall active hybrid plasmonic optoelectronics devices. For example, hybrid plasmonic optical modulators/switches could be realized by introducing some electro-optical material (e.g., EO polymer with a high EO efficiency [42]). It is also possible to realize submicron-donut lasers when a gain medium is introduced to achieve a net gain [32].

## 3. Conclusion

We have given a characteristic analysis for ultra-sharp bent silicon hybrid plasmonic waveguides. It has shown that the hybrid plasmonic waveguide has the ability for submicron bending (e.g., 500nm) with very low leakage loss due to bending. This provides a way to realize submicron resonators. In this paper, we have proposed a silicon hybrid plasmonic submicron-donut resonator with pure dielectric access waveguides by utilizing the evanescent coupling between the pure dielectric waveguide and the submicron hybrid plasmonic resonator. The pure dielectric waveguide enables a long-distance optical interconnect due to its intrinsic loss. Furthermore, no additional mode converter is needed to combine the hybrid plasmonic circuits and the pure dielectric waveguides. Because of the relatively low loss of a hybrid plasmonic waveguide, the intrinsic Q-value of the present submicron-donut resonator is up to 2000 even when the bending radius is reduced to 800nm. Finally a 3D-FDTD simulation method has been used to calculate the spectral response of hybrid plasmonic submicron-donut resonators, which has a bending radius of 800nm and the corresponding loaded Q-value is about 220. The relatively high Q-value makes the present submicron-donut resonator very promising for realizing ultrasmall active hybrid plasmonic devices by introducing some active mediums, e.g., EO polymer with a high EO efficiency [42] for hybrid plasmonic optical modulators/switches, gain medium for submicron-donut laser [32].

## Acknowledgment

This project was partially supported by Zhejiang Provincial Natural Science Foundation (No. R1080193), the National Nature Science Foundation of China (No. 61077040), the “111” Project (No. B07031), a 863 project (Ministry of Science and Technology of China, No. 2011AA010301) and also supported by the Fundamental Research Funds for the Central Universities.

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