Tailoring the dispersion behavior of silicon nanophotonic slot waveguides

We investigate the chromatic dispersion properties of silicon channel slot waveguides in a broad spectral region centered at ~1.5 μm. The variation of the dispersion profile as a function of the slot fill factor, i.e., the ratio between the slot and waveguide widths, is analyzed. Symmetric as well as asymmetric geometries are considered. In general, two different dispersion regimes are identified. Furthermore, our analysis shows that the zero and/or the peak dispersion wavelengths can be tailored by a careful control of the geometrical waveguide parameters including the crosssectional area, the slot fill factor, and the slot asymmetry degree. ©2010 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (230.7370) Waveguides; (260.2030) Dispersion References and links 1. M. Lipson, “Guiding, modulating, and emitting light on silicon – Challenges and opportunities,” J. Lightwave Technol. 23(12), 4222–4238 (2005). 2. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678– 1687 (2006). 3. B. Jalali, and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24(12), 4600–4615 (2006). 4. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002). 5. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31(9), 1295–1297 (2006). 6. E. Dulkeith, F. Xia, L. Schares, W. M. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14(9), 3853–3863 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3853. 7. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14(10), 4357–4362 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4357. 8. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). 9. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29(14), 1626–1628 (2004). 10. P. Sanchis, J. Blasco, A. Martínez, and J. Martí, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol. 25(5), 1298–1305 (2007). 11. Z. Zheng, M. Iqbal, and J. Liu, “Dispersion characteristics of SOI-based slot optical waveguides,” Opt. Commun. 281(20), 5151–5155 (2008). 12. R. Spano, J. V. Galán, P. Sanchis, A. Martínez, J. Martí, and L. Pavesi, “Group velocity dispersion in horizontal slot waveguides filled by Si nanocrystals,” International Conf. Group IV Photon., 314–316 (2008). 13. L. Zhang, Y. Yue, Y. Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express 17(9), 7095–7101 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7095. 14. P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14(20), 9197–9202 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9197. 15. C. Ma, Q. Zhang, and E. Van Keuren, “Analysis of symmetric and asymmetric nanoscale slab slot waveguides,” Opt. Commun. 282(2), 324–328 (2009). 16. K. Okamoto, “Beam propagation method,” in Fundamentals of Optical Waveguides (2000), Chap. 7, 273–281. 17. B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23(24), 4477–4485 (1984). 18. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-1300. #124635 $15.00 USD Received 23 Feb 2010; revised 31 Mar 2010; accepted 8 Apr 2010; published 12 Oct 2010 (C) 2010 OSA 27 September 2010 / Vol. 18, No. 20 / OPTICS EXPRESS 20839


Introduction
In the last few years, silicon photonics has emerged as an attractive and promising technology in the field of integrated optoelectronics. The recent progress in nanofabrication techniques has enabled the development of basic photonic building blocks on a silicon-on-insulator platform including light sources, modulators, and photodetectors [1][2][3]. The silicon photonics approach provides some advantages, including lower-cost and higher-integration, compared with more traditional solutions based on other materials, e. g., III-V semiconductor compounds or LiNbO 3 .
In general, any silicon-based photonic component is affected by chromatic dispersion. Then, the design and optimization of silicon photonic devices requires a very precise knowledge of the dispersion properties. In this context, the chromatic dispersion of a simple silicon waveguide with ~6 μm 2 cross-sectional area was first measured by Tsang et at [4]. In this dimension regime, the light confinement is weak and the dispersion profile is primarily determined by the intrinsic silicon dispersion. In contrast, when the cross-sectional area is reduced, the optical confinement is stronger and, then, the effective dispersion is the result of the interplay between the material and the waveguide or geometrical dispersion [5][6][7]. In fact, a careful control of the waveguide shape and size allows for the tailoring of the group velocity dispersion (GVD) so that normal, anomalous, or even zero GVD can be achieved in the spectral region centered at ~1.5 μm [5][6][7].
On the other hand, the so-called silicon nanophotonic slot waveguides have been proposed and fabricated for different applications [8][9][10]. In these waveguide structures, the optical field is strongly confined in a very thin region of low refractive index material, which is sandwiched between two silicon layers. As a result, a high optical intensity is produced in a small area so that the nonlinear optical performance is highly enhanced [10]. An early analysis of the dispersion properties of symmetric slot waveguides was reported by Zheng et al. [11]. In [11], numerical simulations demonstrate that the GVD of slot waveguides is, in general, significantly quite different compared with the dispersion of traditional channel waveguides. However, the analysis by Zheng et al. was limited to a small spectral range of only 0.15 μm centered at ~1.55 μm and the dispersion control capabilities were unexplored. Simulated results of GVD in a horizontal slot waveguide filled with silicon nanocrystals have also been reported [12]. In addition, more complex slotted waveguides have been proposed for dispersion compensation purposes [13].
In this paper, we perform a detailed analysis of chromatic dispersion in silicon channel slot waveguides. First, a simple channel waveguide is assumed [7] and two different dispersion regimes are identified. Next, we analyze the influence of a slotted region on the GVD of the channel waveguide by considering different slot fill factors. Again, the same previous two different dispersion regimes are distinguished. In this case, the slot fill factor determines the dispersion regime in which the guiding structure operates. We also analyze the GVD in asymmetric slot-based structures, i.e., when the slot is placed in a region different than the geometrical center of the waveguide [14,15].

Dispersion in nanophotonic slot waveguides
Let us first consider a conventional silicon channel waveguide consisting on a silicon channel embedded in a silica cladding, as shown in Fig. 1(a). Throughout this paper, three different cross-sectional areas, A = hw, will be considered, 1 μm 2 , 0.5 μm 2 , and 0.1 μm 2 , for both, conventional and slot waveguides, with h and w being the height and width of the waveguide, respectively, as shown in Fig. 1. These transversal areas are similar to those considered in other related works in silicon [7] so fabrication should not be a problem by using conventional nanofabrication techniques [6,7,9]. In addition, for simplicity, a fixed aspect ratio of 1-to-1.5 (height-to-width) will be assumed in all cases. However, it is important to mention that the GVD profile also depends on this parameter, as demonstrated in [7]. Our numerical simulations are performed by using a full-vector mode solver based on the beam propagation method [16]. In Fig. 1(a) a typical electric field profile of the fundamental quasi-TE mode in the x axis is plotted. The field distribution is confined in the silicon core although evanescent tails are found in the silica region. By using the mode solver, we compute the effective index, n eff (λ), in a broad spectral range and by numerical differentiation the GVD parameter as a function of wavelength, D λ = -(λ/c o )d 2 n eff /dλ 2 , is obtained. It is worth mentioning that our analysis includes the contribution of material dispersion to the GVD, by considering the Sellmeier equations for both silicon and silica [17]. In Fig. 2, we show the resultant GVD profiles for the three crosssectional areas under analysis. For comparison, the normal dispersion of pure crystalline silicon is also plotted. On the one hand, note that for larger cross-sectional areas, the GVD profile is similar to that corresponding to the silicon dispersion in such a way that the GVD gradually increases for longer wavelengths. We name this GVD behavior as material dispersion regime. A vertical up shifting in the dispersion profile is observed so that, eventually, we find a zero-GVD wavelength and, then, a region with anomalous GVD. The more the cross-sectional area is reduced, the more the zero-GVD wavelength is decreased. On the other hand, for smaller areas, the GVD profile is quite different having a maximum GVD along the spectral region and two zero-GVD wavelengths when the maximum dispersion value is positive, as shown in the figure. These GVD characteristics describe the geometrical dispersion regime. For other aspect ratios in this dispersion regime, similar dispersion profiles are obtained but we find normal GVD in the whole spectral range [7]. Two different qualitative dispersion characteristics are then observed depending on the cross-sectional area of the channel waveguide. Although beyond the scope of this paper, we mention that a more rigorous description of these two dispersion regimes can be performed by analyzing the sign of the next higher order dispersion parameter [5]. In the material dispersion regime, this term will be strictly positive in the whole spectral region but in the geometric dispersion regime we could find spectral regions with different sign and thus specific wavelengths where the term is cancelled.

Symmetric slot waveguides
We now turn our attention to the case of silicon waveguides with a vertical slot. In Fig. 1(b), a typical geometry of a slot waveguide is shown. Note that the modal electric-field distribution has a strong discontinuity at the high-index-contrast interfaces and the optical field is significantly increased in the slot region. We have computed the dispersion properties of three different slot waveguides with the above introduced cross-sectional areas, i.e., 1 μm 2 , 0.5 μm 2 , and 0.1 μm 2 , which include the slot region. The same aspect ratio is assumed. The resultant GVD curves are shown in Fig. 3(a-c), respectively. For each cross-sectional area, different slot fill factors have been considered, namely, 1:5, 1:10, 1:25, and 1:50. The fill factor is defined as the normalized ratio between the slot and the waveguide widths, i.e., t/w, see Fig. 1(b). Generally, in Figs. 3(a)-3(c) the dispersion profiles can be grouped into the two dispersion regimes previously defined for a conventional waveguide. As expected, when the slot fill factor is decreased, the GVD profiles asymptotically converge to the dispersion of conventional channel waveguides.  The effect of the slot on the waveguide dispersion is different for each particular crosssectional area. For a cross-sectional area equal to 1 μm 2 , Fig. 3(a), a change in the slot fill factor translates into a relatively small variation in the GVD curve. In fact, all the dispersion profiles lie in the so-called material dispersion regime. Note that for larger fill factors, the dispersion profile exhibits a zero-GVD wavelength and, as a result, a spectral region with anomalous dispersion is found. For intermediate cross-sectional areas, ~0.5 μm 2 , Fig. 3(b), we find that the slot dimension strongly determines the dispersion regime in which the waveguide operates. More particularly, for the fill factors 1:5 and 1:10 we have GVD profiles in the geometrical dispersion regime while the fill factors 1:25 and 1:50 present GVD curves quite similar to the silicon material dispersion profile. For small cross-sectional areas, 0.1 μm 2 , Fig. 3(c), we find that the slot waveguide mostly works in the geometrical dispersion regime. Note that the dispersion curve is significantly down shifted when the slot fill factor is increased while the wavelength with maximum-GVD is nearly constant at ~1.4 μm (dashed vertical line). Transversal dimensions h = 258 nm and w = 387 nm are considered in this latter case, according to the assumed aspect ratio.

Asymmetric slot waveguides
In asymmetric silicon slot waveguides, the slot location is different than the geometrical center of the waveguide [14,15], as shown in Fig. 1(c). We define the asymmetry degree as k = 2s/w, where s is the distance from the center of the waveguide to the center of the slot, in absolute value, and w/2 is half of the total width of the waveguide, as shown in Fig. 1. With this definition, symmetric slot waveguides have an asymmetry degree equal to zero. We have analyzed the GVD for different asymmetry degrees, namely, k = 0, 0.25, 0.5, and 0.75 while keeping the same cross-sectional area. Our aim is to investigate the influence of the waveguide asymmetry on the chromatic dispersion properties for the same cross-sectional areas assumed in previous simulations. Figure 4, shows the results from numerical simulations for the three different above introduced cross-sectional areas (columns) and three different slot fill factors, 1:5, 1:10, and 1:25 (rows). Note that, according to our asymmetry degree definition, a specific slot shift to the right or to the left from the center yields the same asymmetry degree. As expected, the same GVD is obtained for both cases, as verified by numerical simulations. In general, we note from obtained results that the GVD is more sensitive to asymmetry changes when smaller cross-sectional areas are considered. For large areas, 1 μm 2 , and for all the fill factors, Figs. 4(a)-4(c), the waveguide always operates in the material dispersion regime and a small change in the GVD is observed when the asymmetry degree is increased. For intermediate areas, 0.5 μm 2 , more significant changes are found in the GVD. On the one hand, in Figs. 4(d-4e), by starting in the geometrical dispersion regime, a change in the asymmetry degree modifies the dispersion profile in such a way that the maximum-dispersion wavelength shifts to longer values. For larger asymmetry degrees, k = 0.75, the GVD is switched from the geometrical to the material dispersion regime. On the other hand, for a smaller fill factor, 1:25, Fig. 4(f), the GVD first changes from the material to the geometrical dispersion regime as the asymmetry degree is increased returning to the original behavior when the asymmetry is further increased. Finally, for small cross-sectional areas, 0.1 μm 2 , Figs. 4(g)-4(i), we find a significant larger GVD variation compared with previous examples. Note the different scales in the dispersion axis. For a fill factor equal to 1:5, Fig. 4(g), when the asymmetry degree is increased the GVD enters into a new dispersion regime in which the dispersion profile exhibits both a maximum and a minimum dispersion value along the spectral region. This dispersive behavior is consistent with the GVD profiles obtained for conventional channel waveguides with cross-sectional areas smaller than ~0.1 μm 2 [18]. For larger fill factors, Figs. 4(h),4(i) the waveguide exclusively operates in the geometrical dispersion regime for all the asymmetry degrees but exhibits a variation in the maximum-dispersion wavelength for a 1:10 fill factor, Fig. 4(h), whereas a nearly constant maximum-dispersion wavelength is observed for a 1:25 fill factor, Fig. 4(i). Interestingly, the cases k = 0.5 in Fig. 4(e) and k = 0 in Fig. 4(g) exhibit quite flat dispersion profiles over a certain spectral range. We attribute this behavior to a transition between different dispersion regimes. For all the cross-sectional areas, we have performed similar simulations by considering a fill factor equal to 1:50 obtaining small changes in the dispersion curves when the asymmetry in the waveguide is varied.  Fig. 4. Dispersion profiles of a slot waveguide for different asymmetry degrees. Three different cross-sectional areas are considered (a-c) 1 μm 2 , (d-f) 0.5 μm 2 , and (g-i) 0.1 μm 2 . Each row corresponds to a constant slot fill factor, 1:5, 1:10, and 1:25, respectively. The aspect ratio in all cases is 1:1.5.

Conclusions
A detailed analysis of the dispersion properties of silicon-on-insulator vertical slot waveguides has been performed. Our study shows that, in general, the dispersion behavior of slot waveguides strongly depends on the slot dimension and location. Two different dispersion regimes have been qualitative distinguished by analyzing and comparing several waveguide examples with different cross-sectional areas and slot fill factors. Our results show that a careful control of the slot geometrical parameters, i.e., width and position, enables the tuning of the GVD characteristics, including the maximum and/or the zero-GVD wavelengths. Furthermore, constant dispersion in a broad wavelength range can be achieved by properly designing slot waveguides. Dispersion tailoring of slot waveguides may be interesting for controlling several relevant phenomena including ultrashort pulse propagation and nonlinear optical effects.