## Abstract

We propose a new strip/slot hybrid waveguide with double slots, which exhibits a flat and low dispersion over a 1098-nm bandwidth with four zero-dispersion wavelengths. Dispersion of dual-slot silicon waveguide is mainly determined by mode transition from a strip mode to a slot mode rather than by material dispersion. Dispersion tailoring is investigated by tuning different structural parameters of waveguides. Moreover, nonlinear coefficient of dual-slot silicon waveguide and phase-matching condition in FWM are both explored in detail. The dual-slot waveguide can be used to generate supercontinuum with bandwidth extending up to 1630 nm pumped by femtosecond pulses. This waveguide will have a great potential for ultrabroadband signal processing applications from near-infrared region to mid-infrared region.

©2012 Optical Society of America

## 1. Introduction

Silicon-on-insulator (SOI) waveguides have attracted much research interest due to its varied potential applications in a wide spectral region. The SOI waveguide was proposed to be used for telecom wavelengths, which was first reported as early as 1985 by Soref [1]. In particular, nonlinearity of SOI waveguides has been extensively investigated in recent years. Nonlinear optical mechanisms such as self-phase modulation (SPM) [2–4], cross-phase modulation (XPM) [5–7], four-wave mixing (FWM) [8–12], stimulated Raman scattering (SRS) [13–15], and two-photon absorption (TPA) [16, 17] have all been studied widely for all-optical signal processing including wavelength conversion, parametric amplification, soliton propagation and pulse compression. Chromatic dispersion as an essential property of SOI waveguide plays a critical role in the nonlinear processes as mentioned above. Low and flat dispersion over a broad wavelength range can be used to optimize phase matching, which is beneficial for many nonlinear processes. However, it is difficult to obtain flat dispersion because of large light confinement and high waveguide dispersion which may change rapidly over wavelength.

Careful control of width, height and other geometry parameters of strip or rib silicon waveguides allows for anomalous-GVD, which can enhance numerous nonlinear optical processes [18–24]. In general, dispersion tailoring through the methods above can get an anomalous dispersion region with one or two zero-dispersion wavelengths (ZDWLs), but the low-dispersion range is narrow. To reduce phase mismatch, fourth-order or higher-order dispersion should be strictly controlled by accurately designing geometry parameters, which can suppress the contribution from second-order dispersion. However, a large wavelength range with small phase mismatch can be achieved more easily when a flat and low dispersion region with a broad bandwidth is realized. To get an ultrabroadband flat dispersion region, one can take other trials, such as designing different waveguide configuration or replacing the substrate and cladding of silicon waveguides by special materials [25–27]. In particular, much research work on slot waveguides has been done in recent years. Light enhancement and confinement in a nanometer-wide low-index material has been demonstrated by Almeida et al. as a significant step toward investigation of slot waveguides [28]. Komatsu et al. obtained low and flat dispersion region over a 266-nm bandwidth by using highly-nonlinear horizontal slot waveguides [29]. Dispersion-engineered silicon slot waveguides have also been presented to realize broadband wavelength conversion [30]. Yang et al. and Yoo et al. reported optical power confinement in multi-slot silicon waveguides, which are very different from the dual-slot silicon waveguide in this paper [31, 32]. Zhang et al. described a strip/slot hybrid waveguide with a single slot [25, 26]. The hybrid waveguide in Ref. 25 has flat dispersion properties with four ZDWLs over a 670-nm bandwidth. However, the proposed dual-slot silicon waveguide in our work further improve the structure of this hybrid waveguide and shows more excellent dispersion and nonlinear properties.

The dual-slot silicon waveguide exhibits flat and low dispersion properties with four ZDWLs. Dispersion varies between −24 and + 22 ps/(nm·km) over a 1098-nm bandwidth, which is potential for both telecom and mid-infrared applications. A mode transition from a strip mode to a slot mode introduces additional negative and broadband waveguide dispersion, which can flatten the dispersion profile. The dispersion can be flexibly tailored by tuning structural parameters of this waveguide. Furthermore, we calculate the phase-matching condition and simulate supercontinuum generation by use of the dual-slot silicon waveguide.

## 2. Structure and nonlinear coefficient of silicon waveguide

Figure 1(a)
shows a single-slot waveguide. This structure can ensure that TE mode is well-guided in a wide wavelength range [33]. The vertical slot is full of silicon nanocrystals (Si-nc) with silicon excess of 8%, two silicon strips surround the slot with air cladding, and the substrate is 3-µm buried oxide. Three slot waveguides with different slot offset variations are shown in Fig. 2(a)
. The slot width, height and total width of three waveguides are set to be 74 nm, 400 nm and 653 nm, respectively. We change the slot offset to one side to observe the evolution of mode patterns. Compared to asymmetric slot waveguide, the modal pattern of 1.55 µm wavelength is much more sensitive to slot offset changes in symmetric slot waveguide (the top one) as illustrated in Fig. 2(a). When the slot offset is 189.5 nm, its dispersion profile is flat with two ZDWLs between 1.27 µm and 2.53 µm as depicted in Fig. 2(c). Dispersion bandwidth over 1200 nm is obtained, while the dispersion is high and close to 900 ps/(nm·km). Material dispersions of Si, SiO_{2} and Si-nc are all taken into account [34–36]. Between 1.2 µm to 2.4 µm over a wide wavelength range, material refractive index of Si-nc is about 1.72. To get an ultrabroadband flat and low dispersion profile, the structure of asymmetric slot waveguides still needs to be improved.

The dual-slot silicon waveguide is designed, which is based on the asymmetric slot waveguide with 189.5 nm slot offset as shown in Fig. 1(b). The optimized waveguide parameters are: the width of left and right silicon parts W_{LR} = 137 nm, the width of central silicon part W_{CS} = 516 nm, the width of left slot W_{LS} = 74 nm, the width of right slot W_{RS} = 73 nm, and the height H = 400 nm. Impacts of various structure parameters on dispersion and the origin of optimized parameters are carefully explained in section 3. We use finite element method to calculate the dispersion of quasi-TE and quasi-TM mode for different waveguides. Dispersion of three different waveguides shows a obvious contrast in Fig. 2(c). The ridge waveguide with dimensions of 516 × 400 nm^{2} (see the cyan solid curve) has one ZDWL with high dispersion, and TE mode is cutoff at much shorter wavelengths compared to the dual-slot waveguide. TM mode of the dual-slot waveguide cannot be confined in the longer wavelength and the dispersion variation is large. The inset of Fig. 2(c) corresponding to the red curve shows a unique dispersion profile of TE mode of the dual-slot waveguide with optimized waveguide parameters. Four ZDWLs are located at 1572 nm, 1867 nm, 2314 nm and 2576 nm, respectively. Flat dispersion varies between −24 and + 22 ps/(nm·km) from 1527 nm to 2625 nm over a 1098 nm bandwidth. Figure 2(b) shows modal distributions at 1200 nm, 1572 nm, 1867 nm, 2314 nm, 2576 nm and 2800 nm, respectively. Mode patterns obviously fluctuate as wavelength changes. At short wavelengths, the mode remains mainly in central silicon part and seems like a quasi-strip mode, while it gradually transits to a quasi-slot mode at longer wavelengths. In fact, the dual-slot silicon waveguide is multi-mode waveguide. The higher-order TE mode has relatively complicated field profiles and largely different dispersion properties, which can be excited in proper conditions. Compared to ridge waveguide and single-slot waveguide as depicted in Fig. 2(c), losses of long wavelengths by substrate leakage can be reduced in dual-slot waveguide, so mid-infrared wavelengths can also propagate.

The guiding mechanism of high-index-contrast silicon waveguides such as channel or nanowire waveguides is based on total internal reflection [18–24]. The mechanism of slot waveguides is also relying on total internal reflection. Due to the large discontinuity of electric field on the high index contrast interface, light can be well guided in low-index materials surrounded by high-index materials [28–32]. The proposed strip/slot hybrid waveguides (i.e., single-slot and dual-slot waveguides) combine these two guiding characteristics as mentioned above. Clearly, the structure of dual-slot waveguide is designed based on single-slot waveguide. Dispersion of single-slot and dual-slot waveguides is dramatically tailored by the mode transition from a strip mode to a slot mode, which can be only occurred in these strip/slot hybrid waveguides. However, dispersion of the dual-slot waveguide will be tuned more greatly after adding an additional slot, because the mode transition can be generated more easily compared to single-slot waveguide in our simulation. The special structure with double slots introduces unique waveguide dispersion in dual-slot waveguide. Due to the double slots, we have more design freedom to tailor dispersion. We choose vertical slots to investigate dispersion properties of waveguides. Then TE mode of the dual-slot waveguide can be extensively investigated, which exhibits an ultrabroadband flat and low dispersion profile rather than TM mode. In this case, Raman effect can be taken into account and contributes to some nonlinear processes. Moreover, large negative waveguide dispersion can be introduced by using vertical slots. Compared to single-slot waveguide, more wide and flat dispersion region is obtained in this way.

Impacts of material and waveguide dispersion on overall dispersion of waveguides are obviously different as wavelength changes. Material dispersion of silicon plays a dominant role, so the overall dispersion of waveguides is negative at shorter wavelengths [25, 34]. While waveguide dispersion gradually controls the overall dispersion in longer wavelength range, which introduces positive dispersion. This can be clearly seen from the forming of dispersion profile of ridge waveguide as depicted in Fig. 2(c). In order to confine longer wavelength, the single-slot waveguide is used as shown in Fig. 1(a). Compared to ridge waveguide, dispersion decreasing of the single-slot waveguide in middle wavelength range is caused by mode transition from quasi-strip mode to quasi-slot mode. The transition can bring certain negative waveguide dispersion, which introduces two ZDWLs in dispersion profile of single-slot waveguide. By using the waveguide with double slots in Fig. 1(b), dispersion profile with four ZDWLs can be generated in the middle wavelength range. Mode transition introduced by double slots greatly affects the dispersion of dual-slot waveguide. Thus, a flat and low dispersion with an ultrabroad bandwidth is realized under the control of double slots.

We now investigate the nonlinear coefficient *γ _{e}* and effective area

*A*, which are important parameters of silicon waveguide. The lowest-order optical nonlinearity is third order in silicon waveguides because crystal lattice of silicon is centrosymmetric. Nonlinear Kerr index

_{eff}*n*

_{2}is 5 × 10

^{−14}cm

^{2}/W for Si-nc with 8% silicon excess, annealed at 1250°C [37], along with

*n*

_{2}= 5.52 × 10

^{−14}cm

^{2}/W for silicon at 1.5 μm wavelength [38]. The nonlinear coefficient of dual-slot silicon waveguide is given by [39],

*F*(

*x,y*) is the profile of the field, and

*λ*is the wavelength. Effective area is given by [39],

We calculate the nonlinear coefficient and effective area of dual-slot silicon waveguide with optimized waveguide parameters as mentioned above. Figure 3 shows spectral dependence of these two parameters. The effective area becomes larger as wavelength increases. When most of light is confined in central silicon at short wavelengths, the nonlinear coefficient is mainly determined by silicon. Nonlinear Kerr index of Si-nc is less than silicon and more leakage of guided mode are produced at much longer wavelengths. As wavelength increases, more power of guided mode enter the slots and substrate, so the nonlinear coefficient gradually decreases.

## 3. Principle of dispersion tailoring

Dispersion can be tailored by tuning the structural parameters of dual-slot waveguide. Numerous numerical calculations are performed to study how the dispersion depends on the structural parameters. All of the parameters are gradually tuned, which are based on the optimized waveguide parameters as mentioned in section 2.

We first verify how the width of left and right silicon parts W_{LR} affects dispersion properties of dual-slot waveguides. W_{LR} is increased from 127 nm to 147 nm individually as described in Fig. 4(a)
, and other structural parameters are kept the same. The overall dispersion in the middle wavelength range is moved from anomalous to normal dispersion regime, and dispersion variations of two peak parts are 153 ps/(nm·km) and 248 ps/(nm·km), respectively. Dispersion values in long wavelength range are more sensitive to fluctuations of W_{LR}. The shape and slope of dispersion profile are slightly changed. Figure 4(b) shows that dispersion curves are changed as the width of central silicon part W_{CS} decreases. The curves are almost entirely moved down without little change of shape and slope. Decreasing of W_{CS} promotes mode transition from quasi-strip mode to quasi-slot mode. Clearly, negative waveguide dispersion by this transition affects dispersion profile too much and moves down the curves.

When the height varies from 430 nm to 370 nm, the dispersion shape and slope are dramatically tailored as described in Fig. 5(a)
. Dispersion in long wavelength is changed more rapidly and dispersion flatness is destroyed. The guided mode will be coupled with the substrate mode as the height decreases, which introduces negative dispersion. At longer wavelengths, dispersion changes more largely. Dispersion is also calculated for different widths of left and right slots (i.e., W_{LS} and W_{RS}), which synchronously increase with the same value as depicted in Fig. 5(b). The dispersion shape and slope are also greatly changed. When the slot widths vary, the dispersion curves nearly rotate at 1820 nm, and the long wavelength are affected more greatly. Therefore, the slope or third-order dispersion can be flexibly modified by tailoring the height and slot widths of waveguides.

As shown in Fig. 4 and Fig. 5, the dispersion profile is sensitive to dimension deviations from desired values in dual-slot waveguides. Dimension deviations caused by fabrication errors can greatly affect the dispersion of dual-slot waveguides. Uncertainties of mask dimensions, undercutting deviations of etching or errors in other processes will all result in dimension deviations during the fabrication of waveguides. However, the dual-slot waveguide has high flexibility to tailor dispersion properties.

A new phenomenon is found about dispersion tailoring as illustrated in Fig. 6
. New introduced parameters are the width of left silicon part W_{L} and the width of right silicon part W_{R}, which are different from W_{LR} as mentioned before. Figure 6 indicates four situations with different tailoring methods. When we set W_{LS} = 84 nm and W_{RS} = 73 nm, the dispersion curve is nearly identical with the curve corresponding to W_{LS} = 79 nm and W_{RS} = 78 nm, and the maximal dispersion variation is less than 7 ps/(nm·km). The same trend occurs in cases of W_{LS} = 64 nm, W_{RS} = 73 nm and W_{LS} = 69 nm, W_{RS} = 68 nm, respectively. By tailoring width of left silicon part without tuning two sides, approximate curves can be achieved. It is predictable that one will find similar results when tailoring the width of right slot. Figure 6(b) reflects a high coincidence and the maximal dispersion variations are 0.5 ps/(nm·km) and 1.5 ps/(nm·km) corresponding to upper and lower curves.

## 4. Phase matching condition in FWM

Parametric amplification through FWM in silicon waveguide is a nonlinear optical process arising from third-order optical nonlinearity. In degenerate four-wave mixing, two pump photons at frequency *ω _{pump}* are passing their energy to signal and idler photons at respective frequency

*ω*and

_{signal}*ω*(

_{idler}*2ω*), which amplify the signal wave and generate the idler wave. Moreover, the pump experiences the effects of self-phase modulation (SPM) and cross-phase modulation (XPM). The wavelength conversion is phase-sensitive and its bandwidth is dominated by the phase-matching condition. The linear contribution related to dispersion profile and the nonlinear contribution due to Kerr effect mainly determine the phase mismatch in dual-slot silicon waveguide. Phase mismatch

_{pump}= ω_{signal}+ ω_{idler}*Δβ*is given by [10],

*P*is pump power,

_{pump}*Δβ*is linear phase-mismatch and

_{linear}= β_{signal}+ β_{idler}-2β_{pump}*β*,

_{pump}*β*and

_{signal}*β*are pump, signal and idler wavenumbers, respectively. Phase-matching is achieved under the condition of

_{idler}*Δβ = 0*.

We now focus on phase matching by using the dual-slot waveguide with optimized parameters as mentioned in section 2. Figure 7(a) shows two phase mismatch curves for different pump wavelengths (i.e., 1.58 μm and 1.86 μm) close to two different ZDWLs, while the pump power are both set to be 0.1W. When the pump is 1.86μm, phase matching bandwidth is much wider which is determined by the unique dispersion. It is beneficial for broadband wavelength conversion because of small phase mismatch from near-infrared region to mid-infrared region. Figure 7(b) shows phase mismatch by using the same pump wavelength (1.86 μm) with different power. Even with a low pump power, phase matching can be realized. Therefore, power consumption can be decreased using this dual-slot silicon waveguide.

## 5. Supercontinuum generation in dual-slot silicon waveguide

When optical pulses propagate through a silicon waveguide, pulse spectral broadening is affected not only by the nonlinear effects such as SPM, FWM, SRS and siliton fission, but also by the dispersion properties. Dispersion is an important factor for supercontinuum generation. A flat dispersion profile with multiple ZDWLs over an ultrabroad wavelength range can be used to generate supercontinuum. The optimized dispersion with four ZDWLs is shown in Fig. 3 and other dispersion properties are clearly discussed above. Both ends of the bandwidth are in the normal dispersion region, which is helpful for dispersive wave generation, so pulse spectrum can be further broadened [25]. Here a femtosecond pulse is launched in the TE direction to achieve supercontinuum generation. The following nonlinear envelope equation is used in our model [40]:

*A = A*(

*z, t*) describes complex amplitude as a function of longitudinal coordinate

*z*along the waveguide and retarded time

*t*,

*β*is

_{m}*m*th-order dispersion coefficient at central frequency

*ω*

_{0}_{,}

*α*is linear loss, and

_{l}*γ*is complex nonlinear coefficient. The response function

*R*(

*τ*)

*=*(1

*-f*)

_{R}*δ*(

*τ*)

*+ f*(

_{R}h_{R}*τ*) includes instantaneous electronic response and the Raman response function. Here,

*f*is 10.32%, and the function

_{R}*h*is given by ${h}_{\text{R}}(t)=({\tau}_{1}^{2}+{\tau}_{2}^{2})/({\tau}_{1}^{2}{\tau}_{2}^{2})\mathrm{exp}(-t/{\tau}_{2})\mathrm{sin}(t/{\tau}_{1})$, where

_{R}*τ*

_{1}= 10 fs,

*τ*

_{2}= 3.03 ps [40].

A hyperbolic secant pulse is launched into the dual-slot silicon waveguide. The center wavelength and full width at half maximum (FWHM) are 1.58 μm and 50 fs, respectively. The nonlinear coefficient *γ _{e}* is 93.6W

^{−1}m

^{−1}as mentioned in section 4, and find TPA coefficient

*β*= 5 × 10

_{TPA}^{−10}cm/W for silicon at 1.58 μm wavelength [38]. The length of the waveguide and the linear losses are set to be 8 mm and 10 dB/cm, respectively. Figure 8 shows supercontinuum generation with different pump power. The inset shows the temporal shape of output and input pulses. Within such a short propagation distance, dramatic spectral broadening takes place as a result of higher-order soliton fission and dispersive wave generation. The spectrum bandwidth at −30 dB level is 1630 nm from 1150 nm to 2780 nm when the peak power is 120W, while the bandwidth is 1520 nm under the peak power of 60W. Clearly, an ultra-broad supercontinuum is generated, which is useful for practical applications.

## 6. Conclusion

In this paper, a unique dual-slot structure was determined to achieve low and flat dispersion with four ZDWLs in silicon waveguide. A flexible and effective way was found to tailor dispersion. The bandwidth of flat dispersion varying between −24 and + 22 ps/(nm·km) reached as high as 1098 nm. The dependence of dispersion on varied waveguide geometry parameters has been comprehensively researched. In addition, we calculated the nonlinearity coefficient of dual-slot waveguide and phase matching condition in FWM. Broadband phase matching can be achieved by using the optimized configuration. Supercontinuum with 1630-nm bandwidth was generated by using the dual-slot waveguide. The ability to tailor waveguide properties could enables applications such as wavelength conversion, pulse compression and soliton communications.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029 and 61178023.

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