We propose highly nonlinear slot waveguides with flat and low dispersion over a wide wavelength range. Si nano-crystal and chalcogenide glass are considered as slot materials. Over a 244-nm bandwidth, dispersion of 0±0.16 ps/(nm∙m) is achieved in a silicon nano-crystal slot waveguide, with a nonlinear coefficient of 2874 /(W∙m). The As2S3 slot waveguide exhibits dispersion of 0±0.17 ps/(nm∙m) over a bandwidth of 249 nm, with a nonlinear coefficient 16 times larger than that of As2S3 rib waveguides and a nonlinear figure of merit three times larger than that of Si strip waveguides.
©2010 Optical Society of America
Integrated highly nonlinear waveguides and photonic nano-wires can form the backbone of on-chip optical signal processing [1–3], especially for high bit-rate signals . Such waveguides can be composed of silicon [1–3, 5–8], silicon nitride [9, 10], Si nano-crystals (Si-nc or Si-rich Si dioxide) [11, 12], III-V compound semiconductors [13, 14], and chalcogenide glasses [4, 15, 16]. Flat and low chromatic dispersion over a wide wavelength range is crucial for enhancing nonlinear interactions of optical waves. Recently, there has been much interest in tailoring dispersion in the waveguides [17–21]. A tight confinement of light with high index contrast may introduce strong and fast-changing waveguide dispersion, making it difficult to obtain a wideband and flat dispersion profile. A slotted waveguide structure  could provide extra design freedom to tailor the waveguide dispersion [23–27], while keeping a large fraction of the guided mode in a thin slot layer. However, there has been little reported on the wideband low dispersion in these waveguides.
We propose highly nonlinear slot waveguides for achieving a flat and near-zero dispersion profile. Two types of nonlinear materials are considered: (i) chalcogenide glass (As2S3), which reduces the overall two-photon-absorption (TPA) coefficient and (ii) Si-nc, which greatly increases the nonlinear Kerr coefficient compared to silicon strip waveguides. The As2S3 slot waveguide exhibits a flat dispersion of 0±0.17 ps/(nm∙m) over a bandwidth of 249 nm, with a nonlinear coefficient of 160 /(W∙m) and a significantly improved nonlinear figure of merit (FOM). The silicon-nc slot waveguide features a flat dispersion of 0±0.16 ps/(nm∙m) over a bandwidth of 244 nm, with a high nonlinear coefficient of 2874 /(W∙m) at 1550 nm. The proposed waveguides are compared to previously published results in Table 1 , in terms of dispersion and nonlinearity properties. Accumulated dispersion in the waveguides is low, since, for most on-chip applications, the waveguides are typically a few centimeters in length.
2. Waveguide structure and numerical model
As shown in Fig. 1(a) , a horizontal slot is surrounded by two silicon layers with air cladding, and waveguide substrate is 2-µm buried oxide. For the quasi-TM mode (vertically polarized), due to the discontinuity of its electric field at the interfaces of the slot and the silicon layers, a large fraction of the guided mode is confined in the slot layer . The effective index of quasi-TM mode as a function of wavelength is obtained by using a finite-element mode solver (COMSOL Multiphysics 3.4), with an element size of 5, 40, and 100 nm for slot, silicon and other regions, respectively. Material dispersion is taken into account for the slot (As2S3  and Si nano-crystal ), silicon  and silica substrate. Group velocity dispersion, D = -(c/λ)∙(d2n/dλ2), is calculated, where n is the effective index of refraction, and c and λ are the speed of light and wavelength in vacuum, respectively.
We note that the Kerr nonlinear index of refraction n2 and TPA coefficient βTPA, corresponding to the real and imaginary parts of nonlinear coefficient γ respectively, vary with wavelength [30, 31]. For silicon, the measurement results given in Refs. 30 and 31 are fitted using six-order polynomials and averaged to take the dispersion of nonlinearity into account, when we calculate γ as a function of wavelength. For As2S3, different n2 and βTPA values have been reported [32–36] at individual wavelengths, and the measured n2 is 2.5×10−18 , (averaged) 3×10−18 [33,34], 2.93×10−18 , and 3.05×10−18 m2/W  at 1.064, 1.31, 1.54 and 1.57 μm, respectively, which is roughly unchanged. We thus choose constant n2 =3×10−18 m2/W and βTPA = 6.2×10−15 m/W for our simulations at all wavelengths. For Si nano-crystal with 8% silicon excess, annealed at 800 °C, we choose n2 = 4.8×10−17 m2/W and βTPA = 7×10−11 m/W at 1550 nm . For silica, n2 = 2.6×10−20 m2/W is used, and βTPA is neglected. Nonlinear coefficient γ is computed with a space step of 1 nm using a full-vector model , in which the contributions of different materials to nonlinearity are weighted by optical mode distribution, and the longitudinal electric field of the mode is also considered. We use a FOM defined as γ's real part divided by γ's imaginary part times 4π, i.e., γre/4πγim. In a scalar model with a single nonlinear material, γre = 2πn2/λAeff and γim = βTPA/2Aeff, where Aeff is effective mode area. The FOM that we use here is equivalent to the widely used FOM n2/λβTPA .
3. Dispersion and nonlinearity in chalcogenide and Si nano-crystal slot waveguides
For As2S3 slot waveguides, we set W = 280 nm, Hu = Hl = 180 nm, and Hs = 115 nm. Figure 1(b), shows a flat dispersion profile within 0±0.017 ps/nm obtained from 1460 to 1709 nm wavelength (bandwidth of 249 nm) for a 10-cm-long waveguide on chip. There are two zero dispersion wavelengths (ZDWs), located at 1500 and 1677 nm, respectively. The maximum dispersion of 0.1695 ps/(nm∙m) occurs at 1595-nm wavelength. The flat and low dispersion is obtained by employing slot structures. As shown in Ref. 27, the effective index of a slot waveguide can change rapidly with wavelength, and this makes the effective index of the slot mode closer to that of a substrate mode at long wavelengths (e.g., around 2100 nm in this case), causing mode coupling and negative dispersion . Thus, the total dispersion curve is bent and becomes small and flat at the wavelength of interest near 1550 nm.
Varying slot height can significantly increase the dispersion, and Fig. 1(b) shows an increase in the value of the dispersion peak from 0.0971 to 0.3301 ps/(nm∙m) as the slot height decreases from 120 to 105 nm. Accordingly, dispersion peak blue-shifts from 1600 to 1580 nm. Dispersion can also be tailored by changing the waveguide width, as shown in Fig. 2(a) . Widening the waveguide produces a dispersion curve that is shifted to longer wavelength. The right ZDW red-shifts by 111 nm, from 1677 to 1788 nm as the waveguide width is changed from 280 to 310 nm, while the left ZDW blue-shifts by only 32 nm. Changing the slot height and waveguide width together enables tailoring the center wavelength and ZDWs while keeping flat and low dispersion. For example, one can start with the structure parameters given above and then increase both the waveguide width and the slot height. These modifications produce a red-shift of dispersion curves while maintaining the low peak value of dispersion. Figure 2(b) shows the tailored dispersion as the upper Si height Hu is reduced from 190 to 160 nm. It is noted that the right ZDW has a shift towards long wavelength by 65 nm, larger than the left ZDW. Moreover, the peak dispersion value is relatively tolerant to the change of Hu, and the dispersion flatness is improved by increasing Hu.
We examine the nonlinear coefficient γ and FOM as a function of wavelength with varied slot height in Fig. 3(a) . For a slot height Hs = 115 nm, γ decreases from 178.3 to 135.6 /(W∙m) as wavelength increases from 1400 to 1800 nm. A similar trend is found for other slot heights. In contrast, the FOM increases with wavelength from 0.87 to 1.52 for Hs = 115 nm. This is explained as follows. First, the dispersion of nonlinearity in silicon is considered. From 1400 to 1800 nm, silicon's material FOM defined as n2/λβTPA increases from 0.252 to 0.739. Second, we assume that n2 and βTPA in As2S3 do not change with wavelength, so As2S3 material FOM decreases with wavelength. Third, the material index and mode distribution change with wavelength, and contributions of different materials to FOM depend on wavelength. With Hs of around 120 nm and a smaller index contrast between Si and As2S3 (compared to Si and Si-nc), the field enhancement in the As2S3 slot is less than that shown in Fig. 5 insets for an 120-nm Si-nc slot, and thus contribution of silicon layers to FOM is more than that of the As2S3 slot. This is why the overall FOM increases with wavelength. The third factor has limited effect. For a larger slot height, more power is confined in the As2S3 slot, and the FOM value becomes higher. There are similar changes in γ and FOM versus wavelength in Fig. 3(b), with the waveguide width changed, but the FOM is insensitive to the width change.
For silicon nano-crystal slot waveguides, we choose W = 500 nm, Hu = Hl = 180 nm, and Hs = 47 nm. Figure 4(a) shows a flat dispersion profile within 0±0.16 ps/(nm∙m) obtained over a 244-nm wavelength range, from 1539 to 1783 nm. There are two ZDWs at 1580 and 1751 nm, respectively. The peak dispersion of 0.156 ps/(nm∙m) is found at 1670 nm. Figure 4(a) shows that the dispersion peak value is decreased from 0.2101 to 0.0508 ps/(nm∙m) as slot height Hs varies from 46 to 49 nm, at a rate of 0.053 ps/(nm∙m) per nm. We note that, as compared to As2S3 slot waveguides, the Si nano-crystal slot waveguides have a larger index contrast between the slot and silicon layers and a smaller slot height, and this causes stronger field enhancement in the slot. The overall dispersion is dominated by waveguide dispersion, which results in the high sensitivity of dispersion to the slot height. We examine the dispersion change caused by increasing the lower silicon height Hl, as shown in Fig. 4(b). The thicker the lower silicon layer, the flatter the dispersion profile near its peak value. The dispersion curve is red-shifted for a larger Hl. The right ZDW shifts by 106 nm, from 1696 to 1802 nm as Hl is changed from 170 to 190 nm. Generally, similar trends of dispersion tailoring are found for the As2S3 and Si nano-crystal slot waveguides as a structural parameter is changed, so we do not show dispersion profiles with varied W and Hu repeatedly. However, for a smaller slot height, dispersion is more sensitive to a change in the slot height. As an example, Fig. 5 shows dispersion profiles modified by a 10-nm change of Hs in a 10-cm-long Si nano-crystal slot waveguide when Hs = 40, 80 and 120 nm, (W = 500 nm, Hu = Hl = 180 nm), and accordingly dispersion value is changed by 0.698, 0.339, and 0.195 ps/(nm∙m) at 1650-nm wavelength. A small Hs induces strong field enhancement shown in Fig. 5 insets, and light is tightly confined in a small area, which causes an increased dispersion sensitivity to the slot height.
Calculated nonlinear coefficient γ and FOM with the slot height Hs of 47 nm are 2874 /(W∙m) and 0.447, respectively, at 1550-nm wavelength. A small change in the slot height Hs, from 46 to 49 nm, does not change γ and FOM much. Due to the strong field enhancement in the slot, FOM is dominated by the material properties of the Si-nc slot. This is confirmed by noting that silicon's material FOM is 0.352 at 1550-nm wavelength, but the Si-nc’s material FOM n2/λβTPA is 0.4424, which is close to the computed FOM. Due to the lack of measurements on the dispersion of nonlinearity in Si-nc, γ is not calculated as a function of wavelength.
4. Discussion and conclusion
Chalcogenide glasses are good photonic platforms for on-chip nonlinear signal processing [15, 39]. Typically, the nonlinear coefficient γ of As2S3 rib waveguides is found to be ~10 /(W∙m) [39, 40], which is much lower than that of silicon waveguides [1, 3, 20]. This is partially because the As2S3 waveguides have a smaller index contrast and larger mode area, compared to silicon waveguides. However, the relatively small refractive index in As2S3 allows the formation of a chalcogenide slot waveguide, greatly enhancing light confinement and increasing γ from ~10 to 160 /(W∙m). Moreover, a benefit of introducing the As2S3 slot is that it increases the waveguide FOM from ~0.35 (for Si) to 1.15 at 1.55 μm, which approaches saturation of nonlinear performance . By properly combining different materials, one can have a slot waveguide with a designable (instead of geometry independent) FOM. Though the Si nano-crystal slot waveguides have relatively small FOMs, the high nonlinear coefficient γ allows a great reduction of pump power for nonlinear optical signal processing on a chip. Nonlinear length LNL = 1/γP may still be comparable to dispersion length for high-speed signals , which requires the optimization of the dispersion properties of the waveguides.
We have proposed highly nonlinear slot waveguides with flat and low dispersion for on-chip nonlinear signal processing. Flat dispersion within 0±0.16 ps/(nm∙m) is obtained over a 244-nm wavelength range for Si nano-crystal slot waveguides, and nonlinear coefficient is 2874 /(m∙W) at 1550 nm. We have shown that As2S3 slot waveguides can produce a dispersion profile within 0±0.17 ps/(nm∙m) over a 249-nm bandwidth, with greatly increased nonlinear coefficient as compared with previously reported chalcogenide rib waveguides.
The authors would thank Prof. Jurgen Michel, Prof. Lorenzo Pavesi, Dr. Rita Spano, Dr. Shahraam Afshar V. and Dr. Qiang Lin for helpful discussions. This work is sponsored by DARPA (under contract number HR0011-09-C-0124) and HP Laboratories.
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