1. Introduction

Low-chromatic-dispersion light-guiding can be very important for a variety of applications, including: (a) phase matching of data signals for maximizing nonlinear interactions in signal processing [1], (b) low distortion of analog signals for accurate beam steering [2], and (c) transmission of high-bandwidth data signals [3]. A key challenge for optical fibers and integrated waveguides is to realize a very flat and low dispersion over a wide wavelength range. Flattened near-zero dispersion has been achieved in optical fibers [4–6]. Although one may argue that the on-chip silicon waveguides are typically so short that the chromatic dispersion does not have a significant effect on picosecond pulse evolution, it does matter in many other important applications such as femtosecond pulse propagation [7,8], ultra wideband wavelength conversion [9,10] and frequency comb generation [11]. Moreover, an optimized dispersion profile in silicon waveguides could be beneficial to reduce phase mismatching and release the requirement for high pump power in nonlinear processes, which has potential to mitigate two-photon absorption (TPA) and achieve ‘green’ signal processing.

However, for silicon waveguides or nanophotonic wires, dispersion flattening has been a difficult task due to the tight light confinement and strong waveguide dispersion in the highly nonlinear integrated waveguides. Silicon strip and rib waveguides would have to have a quite large transverse size to produce one zero-dispersion wavelength (ZDW) in a wavelength range of interest, typically around 1550 nm [12,13]. If the waveguide size is shrunk to generate a smaller mode area, the ZDW is blue-shifted to even 1200 nm wavelength, where TPA is significantly higher, limiting the efficiency of nonlinear interactions. In addition, strong waveguide dispersion makes overall dispersion highly wavelength-dependent and degrades dispersion flatness (defined as max/min dispersion variation divided by bandwidth, in unit of ps/(nm²∙km)). In Table 1 , we list some simulation results on recent progresses to release the trade-off between tight mode confinement and flat dispersion bandwidth [14–17]. Improved flatness of dispersion curves is obtained by adding a conformal silicon nitride overlayer [16] or a slot structure [17]. A laudable goal would be to continue the progress for low dispersion waveguides that can be operated over wide wavelength ranges, potentially enabling high nonlinear efficiency, low power requirements, a broad operation bandwidth and controllable propagation of femtosecond optical pulses.

Table 1. Dispersion and nonlinearity comparison between the published and proposed waveguides

View Table

We propose a silicon strip/slot hybrid waveguide that generates low dispersion of 0 ± 16 ps/(nm∙km) over a 553-nm wavelength range, which is more than 20 times better than previous results in terms of dispersion flatness, as shown in Table 1. Different from previously reported slot waveguides, the proposed waveguide has a critically important feature, that is, a mode transition from a strip mode to a slot mode, which achieves flattened dispersion by introducing additional slightly negative and broadband waveguide dispersion. A dispersion profile with three ZDWs is, for the first time to the best of our knowledge, obtained in silicon waveguides. The waveguide exhibits flattened dispersion from 1562-nm to 2115-nm wavelength, which is potentially useful for both telecom and mid-infrared applications.

2. Principle and modeling of dispersion flattening

A dielectric slot waveguide typically consists of at least two parts of high-index materials that are closely placed with a layer of low-index material between them. The electric field of light polarized normal to the interfaces of the high-index and low-index materials is enhanced in the low-index layer due to index discontinuity [18]. Figure 1(a) shows a horizontally slotted waveguide, in which upper and lower silicon layers have high refractive index and a low-index slot is made of Si nano-crystals (Si-nc, i.e., silicon-rich silicon dioxide) [19,20]. Cladding is air, and substrate is 2-μm-thick silicon dioxide.

 

Fig. 1 (a) Previously reported silicon slot waveguide that exhibits convex dispersion profile. (b) A strip waveguide is added to produce additional negative waveguide dispersion. (c) The strip/slot waveguide coupler is flipped to improve sequent fabrication procedure. (d) A strip/slot hybrid waveguide with spacing layer removed for achieving flattened dispersion.

Download Full Size | PDF

Slot waveguides may provide additional design freedom to tailor chromatic dispersion [21,22]. Although previous work has shown that relatively low dispersion can be obtained with a chalcogenide or Si-nc slot [17], there is always a convex dispersion profile with a dispersion variation of >300 ps/(nm∙km) over about 250-nm bandwidth. Further flattening the dispersion requires introducing additional waveguide dispersion, which is broad and slightly negative in order to compensate for the existent convex profile. As reported in Ref. 23, strong negative dispersion can be produced by an anti-crossing effect due to mode coupling from a strip mode to a slot mode in a strip/slot waveguide coupler. Figure 1(b) shows such a coupler, in which a strip waveguide is integrated on the top of a slot waveguide [23]. The peak value and bandwidth of the negative dispersion are controllable by modifying the spacing between the two waveguides [23]. The SiO₂ spacing layer here has to be reduced or even removed to broaden the newly induced waveguide dispersion and make it slightly negative. However, a problem in doing this is that one end up having a thick top silicon layer, which is likely to be lossy amorphous silicon deposited in fabrication process. We thus flip the strip and slot waveguides and then remove the SiO₂ spacing layer as shown in Fig. 1(c) and 1(d). Finally, in the proposed structure, the lower silicon layer of the slot waveguide and the silicon strip merge. Although it looks like a slot waveguide, we would emphasize that this is essentially a strip/slot hybrid waveguide according to the light-guiding physics. At short wavelengths, the low-index slot layer serves as a barrier to confine most of light within the lower silicon part, which forms a strip mode. As the wavelength becomes longer, light extends more to the upper silicon part, and the electric field in the slot is enhanced to form a mode that is more like a slot mode. Such mode transition, although subtle, is in principle similar to the mode coupling described above and produces additional shallow and concave waveguide dispersion to compensate for the existent convex dispersion. This light-guiding mechanism differentiates the proposed waveguide from previously reported slot waveguides.

We choose the following structural parameters: upper silicon height H_u = 265 nm, slot height H_s = 50 nm, lower silicon height H_l = 510 nm, and waveguide width W = 500 nm. The silica substrate is 2-μm-thick. A quasi-TM mode (vertically polarized) forms a slot mode [18]. The slot material is silicon nano-crystal with silicon excess of 8%, annealed at 800 °C. It exhibits relatively high nonlinearity compared to samples with higher silicon excess [19]. Material dispersions in Si-nc [20], silicon [24], and silica are taken into account. Over a wavelength range from 1200 to 2400 nm, material refractive index of Si-nc is between 1.72 and 1.725. Figure 2(a) shows the material dispersions for Si-nc and silicon. We obtain the effective index of the guided mode using full-vector finite-element-method software (COMSOL), with an element size of 5, 40, and 100 nm for slot, silicon and other regions, respectively. The overall dispersion is calculated by taking the 2nd-order derivative of the effective index with respect to wavelength, i.e., D = -(c/λ)∙(d²n_eff/dλ²). As an example of convergence of the calculated dispersion, we show the dispersion values obtained at 1550 nm using increasingly reduced element size from 30 to 3 nm in the slot layer in Fig. 2(b). It is noted that an element size of 5 nm is good enough to produce accurate dispersion values.

 

Fig. 2 (a) Material dispersion for silicon and Si-nc, calculated using Sellmeier equations [20,24]. (b) Calculated dispersion value converges as element size is reduced.

Download Full Size | PDF

3. Flattened dispersion profiles

Figure 3 shows the calculated dispersion for the proposed slot waveguide with the parameters mentioned above, and the corresponding dispersion coefficient β₂ is also presented. Flat dispersion of 0 ± 16 ps/(nm∙km) is obtained over a 553-nm wavelength range, from 1562 to 2115 nm, and β₂ is within 0 ± 0.04 ps²/m. Three ZDWs are located at 1595, 1828, and 2062 nm. We show the modal distributions at 1550, 2150, and 2750 nm in Fig. 3. At a short wavelength, the mode looks like a quasi-strip mode and remains mainly in the lower silicon part, while it starts to transition to a quasi-slot mode at long wavelengths. This confirms our explanation on the principle of dispersion flattening. The contributions of silicon and Si-nc to the overall dispersion vary with wavelength due to the mode transition. The optical power within the slot increases from 2.2 to only 10.3 percent of the total power as the wavelength is increased from 1400 to 2200 nm. This means that one could replace the Si-nc slot with other materials with similar refractive index such as SiO₂, keeping the low and flat dispersion.

 

Fig. 3 Flattened dispersion profile within 0 ± 16 ps/(nm∙km) is obtained from 1562 to 2115 nm. The corresponding dispersion coefficient β₂ and modal field distributions are shown.

Download Full Size | PDF

We now examine the dependence of the dispersion properties on the slot height, H_s, with H_u = 265 nm, H_l = 510 nm, and W = 500 nm. Figure 4(a) shows that, as H_s increases from 44 to 56 nm, the dispersion curve rotates, with a nearly zero dispersion value at around 1866 nm. In this way, one can modify the third order dispersion (i.e., dispersion slope), without changing the average dispersion much. Another important parameter is the waveguide width. With H_u = 265 nm, H_s = 50 nm, and H_l = 510 nm, we calculate the dispersion curves for different widths ranging from 420 to 540 nm. As seen in Fig. 4(b), the dispersion at a long wavelength is more sensitive to the width and becomes more negative, with the local minimum value shifting from a wavelength of 1902 nm to 2025 nm as the width decreases. Dispersion flatness is thus destroyed. This is because the decreased width causes a reduced effective index of the guided mode and makes it more likely to be coupled with the substrate mode, and such a mode transition also results in negative dispersion at longer wavelengths.

 

Fig. 4 In silicon strip/slot hybrid waveguides, dispersion profiles are tailored by changing (a) slot height and (b) waveguide width, respectively.

Download Full Size | PDF

The influence of the lower silicon height, H_l, on the dispersion is also examined, with H_s = 50 nm, H_u = 265 nm, and W = 500 nm, as shown in Fig. 5(a) . The dispersion profile is almost entirely moved up without a dramatic change in its shape and slope, as the lower silicon height H_l is increased from 490 to 530 nm. At 1850-nm wavelength, the dispersion value increases from –104 to 80 ps/(nm∙km), with a rate of 45 ps/(nm∙km) per 10 nm. As mentioned above, the dispersion is flattened by the mode transition. When the lower silicon height is decreased, the effective index of the strip mode drops with wavelength more rapidly, which causes a sharper mode transition from the strip mode to the slot mode and over-balanced dispersion, and moves the whole dispersion curve down. The averaged dispersion value can be changed more effectively by modifying the upper silicon height H_u. When keeping H_s = 50 nm, H_l = 510 nm, and W = 500 nm, we change H_u from 235 to 295 nm. As shown in Fig. 5(b), the dispersion curve is moved down, and the dispersion value at 1850-nm wavelength is reduced from 162 to −153 ps/(nm∙km), with a rate of 52 ps/(nm∙km) per 10 nm. In this case, the dispersion slope is slightly changed.

 

Fig. 5 In silicon strip/slot hybrid waveguides, dispersion profiles are tailored by changing (a) lower silicon height and (b) upper silicon height, respectively.

Download Full Size | PDF

Although the goal of this paper is to show a flattened near-zero dispersion profile, we would emphasize that this just serves as an example of dispersion tailorability enabled by the mode transition in the proposed strip/slot hybrid waveguide. For a specific application, a perfectly flat dispersion may not be most desirable, since many other important parameters such as the nonlinear coefficient and loss are not as flat as the dispersion over wavelength. We find that an appropriate combination of the slot height and lower silicon height provides a very effective way to tailor the dispersion profile, which is critically important for manipulating ultrafast pulse dynamics and nonlinear signal processing in silicon photonics. More sophisticated dispersion tailoring (e.g., designing 4th- and 5th-order dispersion terms) may be needed for a specific application, in which case advanced optimization algorithm such as generic algorithm [5,6] could be used.

4. Discussion

To calculate the nonlinear coefficient, we choose a nonlinear Kerr index n₂ = 4.8 × 10⁻¹⁷ m²/W and TPA coefficient β_TPA = 7 × 10⁻¹¹ m/W for Si-nc with 8% silicon excess, annealed at 800 °C [19], along with n₂ = 4.06 × 10⁻¹⁸ m²/W and β_TPA = 7.45 × 10⁻¹² m/W for silicon at 1550 nm [25,26]. To obtain the parameters in silicon, we fit the measurement results in Refs. 25 and 26 using six-order polynomials and average them. The value of n₂ is close to the commonly used one [27]. Using a full-vector model [28], the nonlinear coefficient γ in high-index-contrast waveguides is computed with contribution of both transverse and longitudinal field components. We obtain γ = 134 + j28 /(W∙m) at 1550-nm wavelength, and the imaginary part of γ characterizes the TPA property of the waveguide.

The mode transition to a quasi-slot mode at long wavelengths helps lift the guided mode up, which is beneficial to reduce the loss by substrate leakage. The proposed waveguide is multi-mode, and the higher-order mode has greatly different dispersion profiles, which requires a careful mode excitement. A single-mode nano-taper [29] may be helpful in doing beam coupling and mode screening simultaneously. The dispersion is highly polarization dependent, and the quasi-TE mode has a strong anomalous dispersion >1000 ps/(nm∙km) when the wavelength is beyond 1500 nm.

5. Conclusion

We have proposed the use of slot structures to achieve flat and near-zero chromatic dispersion in silicon waveguides. Flattened dispersion of 0 ± 16 ps/(nm∙km) was obtained over a 553-nm wavelength range, with three ZDWs. The dispersion profile was found to be highly tailorable in terms of the average dispersion value and dispersion slope.